Physical Constraints on the Size and Shape of Microalgae

PHYSICAL CONSTRAINTS ON THE SIZE AND SHAPE OF MICROALGAE

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF BIOLOGY AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Kevin Alan Miklasz March 2012

This work is licensed under a Creative Commons Attribution- Noncommercial-Share Alike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/

This dissertation is online at: http://purl.stanford.edu/mz210dd1320

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Mark Denny, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Stephen Monismith

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

George Somero

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Judith Connor

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract

Scaling laws provide a link between biology and physics. They provide a means by which patterns in biology can be quantified and an explanation from physics can be correlated with such patterns. In this thesis, I use scaling laws to look for connections between biology and physics in two systems, diatom frustules and coralline algal reproduction.

The first part of this thesis addresses several size and shape related constraints in the unicellular group of phytoplankton known as diatoms. In Chapter 1, an empirically- derived scaling law is related back to theory by modeling how cellular density changes with size. In Chapter 2, several further expressions for sinking speed are derived for various shapes and configurations of diatoms. In Chapter 3, empirical relationships between size and frustule thickness are developed and several hypotheses related to the theoretical basis behind these scaling laws are tested. In Chapter 4, the relationship between carbon flux and diatom size is tested in a theoretical model of diatom bloom dynamics.

In the second part of this thesis, I explore patterns with macro-algal spores, an entirely new system for which few scaling laws have been derived. In Chapter 5, I conduct a meta-analysis on one group of macro-algae, coralline algae, to identify empirical scaling laws in the reproductive parameters of this group. In Chapter 6, I report field measurements of reproductive rates of four local species of coralline algae to unravel some of the difficulties encountered in Chapter 5 and noted in the literature.

iv Finally in Chapter 7, I compare one component of the fitness between two local species of coralline algae that have different life history strategies.

I had various success correlating physics to biology in these two systems through the use of scaling laws. The incomplete success is likely attributed to the fact that the physical explanations were not providing strong constraints on the biological systems and thus were not driving the observed patterns, contrary to expectation. I can thus generally conclude that the geometry of the diatom frustule does not seem to be constrained by sinking speed constraints, and that attachment speed and strength may not be as great a limiting factor on the survival of coralline algal spores as expected.

v Acknowledgements

It is quite difficult to accomplish any enterprise that takes over five years without the help and support of many people. I am indebted to many people for their support and advice throughout my PhD, and can only barely scratch the surface of the needed thanks here.

First and foremost, I want to thank my advisor, Mark Denny. Mark has a natural ability to advise. He has always given his students as much freedom or direction as needed, and knows how to adjust this balance over the course of a thesis. His advice on issues of science or otherwise is always exceptional and greatly appreciated. Over the course of my PhD, I have become convinced that one learns the process of science much as one learns any trade with an apprenticeship, through the example of your mentor.

Mark sets an excellent example as a scientist and as an advisor, and I am sure that my growth as a scientist in the past five years is mostly attributed to following his lead and guidance. I can only hope to live up to the example he has set as I continue in my career.

I have also been lucky to have an exceptional dissertation committee. George

Somero has always been available to offer his abilities as a superb editor, or patiently discuss the nuances of biological terminology that my physics sense has botched. One of my favorite classes at Stanford has been George’s “Philosophy of Science” course, which

I still think about today when doing research. Judith Connor has always been both an encouraging positive influence, and harshest critic when needed. Stephen Monismith has offered a helpful hand from my first year on Stanford’s campus when I spent a quarter

vi researching in his lab. And Manu Prakash has offered a novel perspective and insight into my thesis.

My passion for algae originated in the “Marine Botany” course taught at Friday

Harbor Labs by Paul Gabrielson and Charles O’Kelley. I have both of my instructors to thank for such an inspiring course, as well as all of the other students in that course. I’d like to particularly thank Paul, who has graciously put up with my nagging about the

California crustose corallines during and after the course. Paul has helped me to figure out how to identify the species used in this thesis, as well as sparking my interest in the nuances of taxonomy. I also want to thank all of the phycologists that have particularly excited my passion for algae, including Kathy Ann Miller, Gayle Hansen, and Tom

Mumford.

The Denny lab has comprised an exceptional group of people during my time here, and I have them to thank in addition to Mark for my understanding of what it means to be a scientist. Patrick Martone was the first to put the buzz in my ear concerning coralline algae, and is in a large part responsible for my mid-thesis abandonment of diatoms for “pink rocks.” His enthusiasm for corallines is infectious and made my research all that much more exciting. Luke Miller embodies to me a scientist with a balanced life perspective. Luke was never too busy, even when consumed with the end of his thesis, to take the time to build a ridiculously unnecessary device, or to help another labmate with whatever problem they are having. I have tried as best I can to adopt Luke’s example as I’ve gone through my own thesis. I have thoroughly enjoyed the generous lab environment created by everyone I have overlapped with: Mike Boller as my first officemate, Katie Mach as my immediate senior student and an impossible act to follow,

vii Anton Staaf’s infinite knowledge in all things engineering, Sarah Tepler’s wit and spite,

Megan Jensen as the lab’s other token physicist/engineer, and Tom Hata as both a collaborator on research projects and a co-conspirator on non-research projects.

My field work would not be possible without the help of many people who not only helped record my data but also ensured my survival at 3:00 AM on a stormy winter night: James Bohnhoff, Christine O’neill, Kimberly Vincent, Sarah Tepler, Carolyn

Tepolt, Alyssa Gehman, Katie Mach, and Hannah Jaris. I want to particularly acknowledge James Bohnhoff, a Stanford undergrad, majoring in chemistry but spending two quarters at Hopkins to broaden his horizons, help out with my research, and undertake a spectacular research project of his own.

My knowledge and appreciation for statistics has grown over the course of my

PhD, in particular through discussion with Jim Watanabe, our local stats expert. I was always surprised by the hours Jim would put aside when I’d walk into his office, unannounced, to pester him with a complex statistics issue. In addition to Jim, there’s several other people I need to thank for their time and knowledge: Alyssa Gehman,

Kristy Kroeker, Alison Haupt, and Chelsea Wood.

The staff at Hopkins has made this thesis process much smoother than it would have been otherwise. Chris Patton offered help with a bit of everything from safety issues to acidity sensors to SEM prep work. Judy Thompson generally made everything work out even when you were convinced you did it wrong, and Doreen Zelles would patiently help with any issues that come up (even if you were asking for the fifth time how to send a FedEx package). The main campus staff has been equally vital to the ease of the PhD

viii process, in particular Valeria Kizka’s tireless work, but also the help from Jennifer

Mason, Dan King, and Matt Pinheiro.

I have to acknowledge everyone at Hopkins who has made my thesis a thoroughly fun and enjoyable experience, especially my cohort, Nishad Jayasundara, Jason Ladner,

Julie Stewart, Malin Pinsky, and Aaron Carlisle. My experience at Hopkins and Stanford would not be the same without Kristin Hunter-Thompson, Alison Haupt, Michelle Ow,

Ernest Daghir, Judit Pungor, Malithi Jayasundara, Beth Pringle, Posy Busby, Danna

Staaf, Carolyn Tepolt, Kris Ingram, Kelly Barr, Cheryl Logan, Salvador Jorgensen, Liz

Scheimer, Chelsea Wood, Rebecca Martone, Ashley Booth, and others I’ve undoubtedly forgotten but will also be sorely missed. And lastly, the encouragement and love from

Hannah Jaris that have made the last few years especially wonderful.

Last, I’d like to acknowledge my family, for their support and guidance throughout the years. Before my time here, they had never visited the west coast, but at least some members of my family have visited me pretty much every year of my PhD.

The continual support and encouragement from my parents has contributed my success, both here and throughout my life.

ix Table of Contents

Abstract...... iv Acknowledgements ...... vi Table of Contents ...... x List of Figures ...... xv List of Tables ...... xvi General Introduction...... 1

Part I: Diatom sinking and related issue of size and shape Summary...... 8

Introduction to diatoms...... 8 Basic biology ...... 8 Frustules...... 10 Buoyancy control ...... 13 The evolutionary importance of sinking...... 14

Chapter 1 Diatom sinking speeds: from observations to theory

1.1 Introduction ...... 20 1.2 Methods...... 21 1.2.1 Species ...... 21 1.2.2 Culturing techniques...... 21 1.2.3 Sinking speed measurements ...... 22 1.2.4 Frustule geometry measurements...... 24 1.2.5 Statistical analyses...... 24 1.3 Model ...... 26 1.3.1 Stokes’ law...... 26 1.3.2 Cellular densities ...... 27 1.3.3 Spherical model...... 28 1.4 Results ...... 29 1.4.1 Sinking speed and the frustule geometry...... 29 1.4.2 Theoretical model of sinking speed...... 30

x 1.5 Discussion...... 31 1.5.1 Comparison with previous studies...... 32 1.5.2 The sinking of large diatoms...... 33 1.5.3 Conclusion...... 34

Chapter 2 Theoretical modeling on the effect of shape 2.1 Introduction ...... 42 2.2 Alternate models for the sinking of simple centric diatoms...... 43 2.2.1 Cylindrical model ...... 44 2.2.2 Elliptical model ...... 45 2.2.3 Inclusion of vacuole...... 47 2.2.4 Discussion ...... 50 2.3 Additional models for the sinking of more complex diatom shapes ...... 51 2.3.1 Cylindrical model for elongate diatoms...... 52 2.3.2 Elliptical model for pennate diatoms...... 55 2.3.3 Model for triangular-shaped diatoms...... 55 2.3.4 Models for spines ...... 56 2.3.5 Models for mucilage...... 58 2.3.6 Models for linear chain-forming diatoms ...... 59 2.3.7 Discussion ...... 64

Chapter 3 Defining empirical scaling laws for frustule thickness 3.1 Introduction ...... 78 3.2 Methods...... 80 3.3 Results ...... 81 3.3.1 Symmetric and asymmetric diatoms...... 81 3.3.2 Solitary and chain-forming diatoms ...... 82 3.3.3 Benthic and planktonic ...... 83 3.4 Discussion...... 83 3.4.1 Correlation with previous scaling laws...... 83 3.4.2 Correlation between thickness and silica content...... 85 3.4.3 Evolutionary implications of frustule geometry...... 87 3.4.4 Implications for sinking speed predictions ...... 90

xi Chapter 4 Effect of diatom sinking speed on bloom formation 4.1 Introduction ...... 99 4.2 Methods...... 101 4.2.1 The model ...... 101 4.2.2 Parameter selection...... 103 4.2.3 Defining metrics for sensitivity analysis...... 105 4.3 Results ...... 106 4.3.1 Effect of size...... 106 4.3.2 Effect of sinking speed model...... 106 4.3.3 Effect of growth rate...... 107 4.3.4 Effect of shear rate...... 107 4.3.5 Effect of coagulation kernel...... 108 4.3.6 Other parameters ...... 108 4.4 Discussion...... 109 4.4.1 Diatom density and developing predictive models ...... 109 4.4.2 Implication on bloom species selection ...... 110 4.4.3 Mixed species blooms...... 111 4.4.4 Out of the mixed layer ...... 112

Part II: Size scaling in coralline algal reproduction

Summary...... 122

Introduction to coralline algae ...... 122 What are calcified algae?...... 122 Ecological importance...... 126 Reproductive features...... 127 Reproduction energetics ...... 130

xii Chapter 5 A meta-analysis of spore size and number 5.1 Introduction ...... 137 5.2 Methods...... 138 5.3 Results ...... 141 5.4 Discussion...... 143 5.4.1 Patterns in algal reproductive energetics ...... 143 5.4.2 Remarks about conceptacle...... 145 5.4.3 Are crustose and articulated corallines comparable?...... 147

Chapter 6 Coralline algal reproductive rates and size 6.1 Introduction ...... 164 6.2 Methods...... 167 6.2.1 Species ...... 167 6.2.2 Seasonality ...... 169 6.2.3 Spatial variability...... 174 6.2.4 Statistical analyses and conversion parameters...... 174 6.3 Results ...... 177 6.3.1 Seasonality ...... 177 6.3.2 Spatial variability...... 178 6.3.3 Reproductive rates ...... 179 6.3.4 Reproduction and size...... 180 6.4 Discussion...... 181 6.4.1 The r vs. K selection paradigm...... 181 6.4.2 Seasonality ...... 185 6.4.3 Individuality in algae ...... 187 6.4.4 Are crustose and articulated corallines comparable?...... 189

Chapter 7 Temporal characteristics of the attachment strength of coralline algal spores 7.1 Introduction ...... 212 7.2 Methods...... 215 7.2.1 Shear flume design ...... 215 7.2.2 Calibration of shear flume...... 217 7.2.3 Species collection and testing ...... 219 7.2.4 Shear measurements ...... 222 7.2.5 Variability in spore release...... 223

xiii 7.3 Results ...... 225 7.3.1 Attachment profile...... 225 7.3.2 Variability in spore release...... 226 7.4 Discussion...... 227 7.4.1 Observations on the stages of the attachment process...... 227 7.4.2 Comparisons to previous data ...... 231 7.4.3 Benefit of attachment profile shape...... 233 7.4.4 Variability in attachment and individuality in algae ...... 236 7.4.4 Evolution of spore size ...... 238

References...... 254

xiv List of Tables

Table 1-1. Geometric and sinking data for six diatom species ...... 35 Table 1-2. Regression results for four frustule parameters against frustule diameter...... 36 Table 2-1. Ratio of chain sinking speed to an equivalent volume sphere ...... 66 Table 3-1. Diatom data used in the meta-analysis ...... 92 Table 3-2. Regression results for frustule thickness against diameter ...... 94 Table 4-1. Parameters used in the model...... 114 Table 4-2. The effect of density model on the four metrics ...... 115 Table 4-3. The effect of various parameters on the four metrics...... 116 Table 5-1. Crustose coralline algae reproductive data used in the meta-analysis...... 150 Table 5-2. Articulated coralline algae reproductive data used in the meta-analysis...... 156 Table 6-1. Reproductive parameters ...... 190 Table 6-2. ANOVA results for reproduction...... 191 Table 6-3. ANOVA results for frond length...... 192 Table 6-4. ANOVA results for frond area...... 193 Table 6-5. Regression results for conceptacle density against plant length for the seasonality data...... 194 Table 6-6. Regression results for conceptacle density against plant area for the seasonality data...... 195 Table 6-7. Regression results between reproduction and size for the spatial variability data, shown by frond...... 196 Table 6-7. Regression results between reproduction and size for the spatial variability data, shown by crust...... 197 Table 7-1. ANOVA results of spore release with date, frond and branch as factors...... 244 Table 7-2. ANOVA results of spore release with frond, branch and hour as factors .....245 Table 7-3. ANOVA results of attachment with date, frond and branch as factors ...... 246 Table 7-4. ANOVA results of attachment with frond, branch and hour as factors...... 247

xv List of Figures

Figure 0-1. Picture of a diatom frustule under SEM...... 18 Figure 0-2. Illustration of a diatom frustule with relevant planes and axes ...... 19 Figure 1-1. Diatom frustule with geometric parameters labeled ...... 37 Figure 1-2. Diatom sinking speed against cell diameter for live and dead diatoms, with model shown...... 38 Figure 1-3. Four geometric parameters against cell radius...... 39 Figure 1-4. Diatom sinking speed against cell diameter with Stokes’ law, constant t and variable t...... 40 Figure 1-5. Diatom sinking speed against diameter with previous data, models shown with confidence intervals ...... 41 Figure 2-1. Spherical and cylindrical models at constant and variable parameters...... 67 Figure 2-2. Spherical and elliptical models at constant and variable parameters...... 68 Figure 2-3. Model with vacuole, for a constant cytoplasm volume...... 69 Figure 2-4. Model with vacuole, for a constant cytoplasm area...... 70 Figure 2-5. Illustration of elongate cuboid diatom...... 71 Figure 2-6. Illustration of elongate cyllindrical diatom...... 72 Figure 2-7. Diatom sinking speed against cell diameter with elongate diatoms and a elongate diatom model...... 73 Figure 2-8. Illustration of elliptical diatoms ...... 74 Figure 2-9. Illustration of triangular diatoms...... 75 Figure 2-10. Illustration of diatom chain...... 76 Figure 2-11. Diatom sinking speed modeled against cell volume for various shapes ...... 77 Figure 3-1. Valve thickness against diameter broken by symmetric vs. asymmetric diatoms ...... 95 Figure 3-2. Valve thickness against volume broken by symmetric vs. asymmetric diatoms ...... 96 Figure 3-3. Valve thickness against volume broken by solitary vs. chain diatoms ...... 97 Figure 3-4. Valve thickness against volume broken by benthic vs. planktonic diatoms...98

xvi Figure 4-1. The sinking speed models...... 117 Figure 4-2. The four metrics shown across sinking speed model and size ...... 118 Figure 4-3. The four metrics shown across growth rate...... 119 Figure 4-4. The four metrics shown across shear rate...... 120 Figure 4-5. The four metrics shown across coagulation kernel...... 121 Figure 5-1. Conceptacle diameter against tetrasporangial diameter ...... 159 Figure 5-2. Algal thickness against tetrasporangial diameter...... 160 Figure 5-3. Algal extent against tetrasporangial diameter...... 161 Figure 5-4. Tetrasporangial number against diameter...... 162 Figure 5-5. Conceptacle volume against total tetrasporangial volume ...... 163 Figure 6-1. Reproductive rates shown over a year...... 198 Figure 6-2. Reproductive rates for P. neofarlowii shown over two years...... 199 Figure 6-3. An autocorrelation of the reproductive rates data...... 200 Figure 6-4. Conceptacle densities averaged over the year ...... 201 Figure 6-5. Tetraspore densities averaged over the year...... 202 Figure 6-6. Reproductive volume densities averaged over the year ...... 203 Figure 6-7. Tetraspore release densities averaged over the year ...... 204 Figure 6-8. Reproductive volume release densities averaged over the year...... 205 Figure 6-9. Various reproductive rates against tetraspore diameter...... 206 Figure 6-10. Conceptacle density against plant length for the seasonality data ...... 207 Figure 6-11. Conceptacle density against plant length for the seasonality data ...... 208 Figure 6-12. Conceptacle density against plant length and area for the spatial variability data, shown by frond...... 209 Figure 6-13. Conceptacle density against plant length and area for the spatial variability data, shown by crust...... 210 Figure 6-14. Tetrasporangial number against diameter with data from Chapter 5...... 211 Figure 7-1. Diagram of shear flume and slide mount...... 248 Figure 7-2. Average attachment profile for C. vancouveriensis...... 250 Figure 7-3. Individual attachment profiles for C. vancouveriensis...... 251 Figure 7-4. Average attachment profile for C. tuberculosum...... 252 Figure 7-5. Individual attachment profiles for C. tuberculosum ...... 253

xvii General Introduction

Physics has the potential to provide sweeping insights into biological systems.

However there is a classic disparity between the fields of biology and physics: physicists prefer simplified, idealized situations that can be exactly predicted (think perfects spheres on infinite, frictionless planes), whereas biologists revel in the complexities and non- uniformities of their systems. Any attempt to work in both fields calls for careful tight- rope walking, in order to simplify the complexities of biology to the point where physics can be applied without losing sight of the interesting biology that gave rise to such complexities. For example, pretending that a limpet is a sphere means predictions can be easily made with physical formulae, but they are unlikely to be accurate or biologically useful. Additionally, an exact simulation of the precise geometry of a limpet shell is likely to be much more accurate, but incorporating all of the complexity means the results are only applicable to the single exact limpet shell used in the simulation. A different individual shell with slightly different nooks and crannies would need a different simulation, making generalization across a population problematic and across species out of the question. The trick is to develop a formulation that is simple enough to be generalizable but detailed enough to be biologically accurate. Finding and walking the line between physical simplicity and biological complexity is more of a heuristic art than a precise formula, but a concept known as scaling laws can provide the link.

A scaling law is a simplified equation. Scaling laws ignore constants, and simply relate how two variables scale with each other. An example is y∝x, which simply states that y scale directly with x. Another example is y∝x2, which says y scales with the square

1 of x, so if x doubles, y quadruples. Scaling laws are seldom exact, and therefore do not typically make the most precise predictive tools. But scaling laws excel in defining and explaining underlying trends across similar populations or species. When used appropriately, they take in enough biological complexity to be meaningful, but still possess the generality of a physical formula.

Scaling laws are flexible: one can start a scaling analysis from either physics or biology. To derive a scaling law from physics, one rearranges an equation so that the two variables of interest align on opposite sides of the equation, drops out all other variables as constants, and switches the “equals” sign to a “scales like” sign. To describe a scaling law from biology, one identifies and measures the two variables of interest on an organism and fits a linear regression on logarithmic axes to the data to find the scaling exponent. Of course one loses accuracy in both perspectives. By dropping the other variables in the physical formula one loses predictive power, and by focusing on only two variables of an organism one loses sight of the rest of the organism’s complexity. But when the starting equation and biological measurements are chosen carefully, one can achieve agreement between physics and biology and ultimately gain an understanding of how physical processes constrain biological shape or function.

Although a scaling analysis can begin from either perspective, it is really most complete once both biological and physical perspectives have been established. If the biology of a system is relatively well-known, one may start from biological data to empirically derive a scaling law. In this case, the physical explanation for the scaling law is offered post-hoc. An example of this can be found in the metabolic ecology literature, where a ¾ scaling law between metabolic rate and body mass has been suggested and

2 hotly debated for some time (Kleiber 1932), but a good physical explanation for the scaling law is yet to be offered (for one attempt at a physical explanation, see West,

Brown, and Enquist 1997).

To start from a physical perspective, a theoretical understanding of the situation is postulated from basic physics. In this case, a scaling law and accompanying physical mechanism are postulated to explain the scaling relationship between two variables.

Biological data are then collected post-hoc, to test whether the data fit the hypothetical physical scaling law. An example of this is the theory of accelerational force predicting an upper hydrodynamic limit to the size of intertidal algae (Gaylord et al. 1994).

A scaling law can be used to relate any two variables, but often one or both of those variables is a size-related parameter. Size is one of the most important constraints on organisms, as much of biological diversity is simply due to organisms living at different sizes (Brown, West and Enquist, 2000, Schmidt-Nielson 1984). Many factors differ across size, from physical concerns like diffusion efficiency and the hydrodynamic effect of viscosity versus inertia, to biological concerns like metabolic rates and predation risk. Because of the abundance of size-related constraints, and therefore the potential of size to explain trends, size is often employed in scaling laws. Accordingly, nearly every section of this thesis will take size into account in some manner. In some senses this thesis is truly about the constraints of size for micro-algae, as expressed through scaling laws.

At this point, it is useful to introduce the concepts of allometry and isometry.

These terms have had various definitions throughout time and between different authors, so some clarification is necessary. Since many scaling laws relate two size parameters to

3 each other, there are certain expectations for how those size parameters should scale. If shape is constant with size (what is called geometric similarity), then one expects the length of one part of an organism to be directly related to other lengths (L∝L1), to the square root of area (A∝L2), and to the cube root of volume (V∝L3). Traditionally, the sorts of relationships which satisfy geometric similarity are called isometric relationships, but in modern times researchers have more loosely referred to any scaling law with an exponent of 1, or direct proportionality, to be an isometric relationship. This can be confusing, since if one is relating a length to a volume and the shape stays constant with size, one would expect a scaling exponent of 1/3, which is isometric by the definition of geometric similarity but not by direct proportionality. Allometric scaling laws describe shapes that change with size or do not exhibit geometric similarity, which occurs if the size parameters are not related as given above (L∝L1, A∝L2, V∝L3). Again, modern usage has more loosely referred to any scaling exponent between any sorts of variables which is significantly different from 1 as allometric (i.e. in the metabolic ecology literature given above). In this thesis, I will use these terms in their traditional sense of geometric similarity only, and so will restrict their use to relationships between two size parameters.

Scaling laws can have a particular use for functional evolution and body size evolution, and this is the primary context in which I will employ the scaling laws in this thesis. Often when an allometric relationship is found, it is because there is some constraint acting on the biological system that causes shape to change with size (Bonner and Horn, 2000). Relating a physically-derived scaling law to a biological one is a one way to find the mechanical or functional basis for the constraint. The implicit assumption

4 behind this reasoning is that isometric relationships constitute a sort of “null hypothesis,”

or that geometric similarity indicates that no constraint affects shape across size.

This use of allometry for determining evolutionary constraints is best illustrated

by an example. It has been found that tree height (h) is empirically related to trunk radius

(r) by a scaling exponent of 2/3 (McMahon and Bonner 1983). Thinking about this

situation physically, we expect that a tree can only get as high as it can hold its own

weight. For this reason, let us examine the situation where the force of a tree’s weight

(Fg) is equal to the force sufficient to cause local tree failure by buckling (FE) (Eqs. are

derived from Vogel, 2003).

Fg = FE

n" 2EI mg = h 2 !

3 4 2 n# Er "#r hg = 2 ! 4h where m is mass, g gravitational acceleration, E material stiffness, I second moment of

! inertia, ρ the density of the tree, and the tree is treated as a cylinder. For the purposes of

the scaling analysis, all of these terms are constants that will be dropped. Turning this

into a scaling law, we get:

r 4 r 2h " h 2

2 h " r 3 ! The result is that if the trees were constrained by their potential for mechanical failure by

! bending under their own body weight, then one would expect h∝ r2/3. The

correspondence of this physical scaling law with the empirically measured biological

5 scaling law is some evidence that tree diameter and height are limited by this physical constraint (McMahon and Bonner 1983).

One further note must be addressed before moving on to the bulk of the thesis.

Scaling laws differ from other mathematical expressions in that they imply correlation, not causality. This is especially true for biologically derived scaling laws, as the variables of interest are simply related to, but not necessarily caused by, each other. The two variables can be switched, the scaling exponent can be written on the left hand variable rather than the right, with no loss or distortion in meaning. This is in contrast to something like a response function, for instance, in which the independent variable in the function clearly causes the output of the function (the dependant variable) to change, thus implying a cause and effect type of relationship.

In this thesis, I explore several issues in scaling laws and attempt to bridge a gap between biology and physics. As this is an overly general topic, I have decided to limit myself. Physically, I will constrain my exploration to small objects in water, or what might be called a low Reynolds number world. Biologically, I have decided to constrain myself to non-motile algae. I have chosen non-motile particles - what might be considered passive particles – because they are more likely to be constrained by direct interaction with physical constraints. As will be seen, though, it is nearly impossible to consider any living biological organism as a passive particle and this will be a continual source of interesting complications. My chapters will work with all aspects of a scaling analysis- from situations where we have a biologically-derived scaling law but no physically derived ones, vice versa, and from situations in which there are interesting questions to be asked but scaling laws have yet to be derived.

6 In Part I, I investigate a system for which a biologically derived scaling law has been known for some time, but is yet to be explained by a physically derived correlate

(Chapter 1). Once I solve this discrepancy between the physical and biological scaling laws, I then spend the rest of the chapter deriving several other related scaling laws, from physical (Chapters 2 and 4) and biological (Chapter 3) perspectives.

Part II works with an entirely new system for which few scaling laws have been derived, the spores of red algae. The second part of my thesis works primarily from a biological perspective, as I empirically derive scaling laws and try to make sense of a new system.

Part I

DIATOM SINKING AND RELATED ISSUES OF

SIZE AND SHAPE

Summary

Introduction to diatoms

Basic biology

Diatoms are an abundant, diverse group of organisms. They are the third most diverse group of organisms on the planet, containing an estimated 100,000 species

(Raven et al. 2005). In this respect, they are bested by only insects and flowering plants.

Along with this diversity is a plethora of ecological and morphological diversity (Round

1990). Diatoms have invaded every aquatic habitat on the planet, from freshwater lakes

8 to open ocean. They inhabit the polar regions to the tropics, and shallow, mixed coastal zones to the open ocean. Morphologically, diatoms exhibit a wide variety of shapes for a single celled organism, from short fat cylinders to extremely elongate cylinders, with lengths more than 100 times their width (Waite and Nodder 2001).

Diatoms occupy a relatively large size range for a single taxonomic group built on essentially one design. Diatoms range in diameter from about 2 µm to 2 mm, a difference of three orders of magnitude in length accounting for a difference of nine orders of magnitude in volume (Round, 1990). Very few organisms can occupy such a large size range without undergoing major structural differences. For comparison, the difference between the smallest and largest mammal (shrews to blue whales) is about 8 orders of magnitudes by volume, less than that of diatoms. The largest mammals (whales), though, can only live in aquatic environments, since their bone strength is not sufficient to support their own weight on land. So, the largest and smallest mammals live in very different environments in part because of their difference in size, yet the smallest and largest diatoms can be found living side by side in an essentially similar ocean environment.

The diatom life-cycle is especially interesting from the perspective of size. After sexual reproduction, diatoms start off their life cycle at their largest size. Diatoms then asexually reproduce about once a day, each cycle producing two diatoms, one the same size as the original and one which is a little bit smaller (Round 1990). The result is that over time the average size of the cells in a diatom population decreases. This continues until the cells receive a trigger, which causes them to undergo sexual reproduction and release free-swimming gametes. These gametes fuse and grow to produce the enlarged

9 diatoms. The cause of this trigger is still unknown (Round 1990), but probably related to the size of the diatoms and some sort of light or temperature signal, since shipment of diatom cultures through the mail often induces sexual reproduction (Duffield, pers. com.). The result of this process is that a single diatom species can experience up to a four-fold difference in size throughout its lifecycle (pers. obs.), which is no extraordinary feat for animals with juvenile and adult phases but unique for a primarily asexually reproducing organism.

Diatoms are the basis of many aquatic food webs. By numbers, they are bested in abundance in the ocean only by the much smaller bacteria. By biomass, diatoms are the most abundant eukaryote in the ocean (Worden et al. 2004, Field et al. 1998). For this reason, understanding their population dynamics is important to modeling ocean ecosystems. To complicate this, some diatom population dynamics can be quite unpredictable, appearing in large blooms that arise in a matter of days and disappear just as quickly. These blooms can provide a ready source of food for zooplankton, but are sometimes toxic when dominated by certain diatom species and can lead to hypoxia, anoxia, or shellfish poisoning (Trainer et al. 2010).

Frustules

The characteristic feature of a diatom is its frustule. The frustule is a siliceous covering on the outside of the cell wall (Round 1990). The frustule is made primarily of amorphous silica, but is embedded in an organic matrix (Swift 1992). Features of the frustule are the primary means of identifying different diatoms, and are also the primary source of their morphological diversity.

10 The frustule is made up of two components, the valve and girdle. If one thinks of a diatom as a Petri dish shape, then the valves are the endcaps of the Petri dish and the girdles are bands that encircle the lip of the valve (Fig. 0-1). Each diatom has at least two girdle bands, one for each valve. Sometimes more than two girdle bands can be found between the valves, essentially elongating the diatom from valve tip to valve tip. The valves and girdles are held together by some sort of organic glue, which is often dissolved by the bleaching preparation used for Scanning Electron Microscopy analysis

(Round 1990).

Some terminology must be introduced at this point. If one views the frustule by looking straight onto the face of the valve, this is called valve view. Similarly, if one views the frustule from 90° to the valve by looking straight onto the face of the girdle, this is termed the girdle view (Round 1990). The two main groups of diatoms are distinguished by their valve symmetry. The centric diatoms are distinguished by having radial symmetry in valve view, whereas pennate diatoms have bilateral symmetry in valve view (Round 1990). Although diatoms can have valve shapes that do not always visually fall into these categories, if one follows the growth of such valves they either proceed from a radial or bisymmetric core, even if the resulting valve shape is not radially or bilaterally symmetric. Most diatoms feature bilateral symmetry in girdle view, though sometime the diatoms are “curved,” such that one side of the girdle band is shorter than the other, leading to a slight asymmetry. This asymmetry can sometimes be drastic, causing a 180° rotation of the frustule such that the two valves lie on almost parallel planes (Round 1990).

11 Some additional terminology must be offered for discussion of the frustules’ morphology. This terminology will follow Round (1990, Fig. 19 in particular), except where I offer my own terminology for geometric simplicity. Although here I will refer to the various planes of the frustule for consistency with Round (1990), I will refer to the axes rather than the planes in the rest of this thesis.

Centric diatoms, due to their radial symmetry, have two primary axes or planes of interest (Fig. 0-2A). The valvar planes lie parallel to the valve face, cutting through the girdle bands. The pervalvar axis lies perpendicular to the valvar planes, projecting out through the center of the valve face. This axis is also known as the azimuthal axis of the cylinder. Thus in valve view, one is looking down the pervalvar or azimuthal axis. The radial planes cuts through the valve and girdle, perpendicular to the valvar planes. The radial axes projects perpendicular to the radial planes, as in typical geometric terminology in cylindrical coordinates.

For pennate diatoms, there are three planes and axes of interest (Fig. 0-2B). The valvar planes and pervalvar axis are defined similarly as that for centric diatoms, (the valvar planes are parallel to the valve face). In valve view, the bilateral symmetry often leads to an elliptical valve face. For geometric simplicity, I will simply refer to these two axes as the short valve diameter and long valve diameter, as I could not find a name for these axes in the diatom literature. What have been defined in the literature are the apical and transapical planes. The apical planes lie perpendicular to the valvar axis and along the long diameter of the valve. Thus the axis of the short valve diameter projects perpendicular to the apical plane. Similarly, the transapical planes lie along the axis of the

12 short valve diameter, such that the axis of the long valve diameter projects perpendicular from this plane.

In this thesis, I use valve diameter when discussing and plotting simple, cylindrical centric diatoms. But when discussing pennate diatoms or centric diatoms with a functionally bilateral shape, I always specify long or short valve diameter. Valve or girdle height always refers to the height of one valve or girdle, measured in girdle view along the pervalvar axis.

Buoyancy control

Considering that frustules are denser than water, they cause diatoms to sink in the water due to an overall negative cellular buoyancy. Diatoms can mediate their sinking by modifying their cellular density. This seems to occur primarily in the cell vacuole, which diatoms regulate by exchanging high for low molecular weight ions, a metabolically costly process (Anderson 1978, Woods and Villareal 2008). This physiological manipulation of density explains much of the observed variability in sinking speeds of metabolically active diatoms (Waite et al. 1992, Waite et al. 1997). This physiological process is sensitive to a variety of environmental variables, such as light (Fisher et al.

1996, Patel et al 2005), temperature (Takabayashi et al. 2006), nutrients (Bienfang et al.

1982, Takabayashi et al. 2006), salinity (Ignatiades and Smayda 1970) and even turbulence levels of the surrounding water (Patel et al 2005).

In nutrient-deprived or physiologically inhibited treatments, in which diatoms are metabolically constrained, sinking speed approaches a size-dependant maximum value

(Waite et al. 1997). I will contend in this thesis that this size-dependant maximum sinking speed should be governed solely by physical processes, and should be described by

13 Stokes’ law. Conversely, the amount to which diatoms can regulate their buoyancy is also size dependant, with larger diatoms containing a greater percentage of vacuole in their cell’s cytoplasm (Woods and Villareal 2008) and therefore have a larger volume with which to counterbalance the relatively heavy frustule. Thus larger diatoms are able to achieve a slower minimal sinking speed than smaller diatoms. This size-dependant minimum sinking speed leads naturally to two consequences; first that there should exist a size at which the minimum and maximum sinking speeds meet and no buoyancy regulation is possible, presumably due to the absence of a significantly large vacuole in such small diatoms. Waite et al. (1997) finds that this point is reached for diatoms of 7.8

µm in volumetric equivalent diameter, what will be termed Waite’s critical diameter.

Second, there should exist a size at which diatoms become positively buoyant. This second point is also true, and Villareal (1988) finds this point should be reached for diatoms greater than about 30 – 140 µm in diameter.

The evolutionary importance of sinking

As stated, diatoms are typically negatively buoyant due to their frustules, only achieving some positive buoyancy when in a physiologically active state. Additionally, diatoms have no cilia or flagella outside of the sexual phase of their life cycle, meaning that they spend the majority of their lives without active locomotion. This means that the fundamental state of a diatom is one of negative buoyancy, i.e., sinking through the water. This fact raises the question of the evolutionary value of sinking. The common paradigm in the literature is that sinking is generally bad for diatoms since it moves them away from sunlight (Smayda 1970). The detrimental value of sinking is the hypothesized explanation for why diatoms have spines or form chains, structures which increase their

14 drag and reduce their sinking speed (Round 1990, Reynolds 2006). The idea seems to be that although high-drag diatoms are still sinking, they are not sinking quite as fast as they would be without drag-inducing structures, and so are not as unfit as they might be.

S. Vogel (pers. com.) offered another perspective. If diatoms were to be positively buoyant by nature, they would quite commonly be near the surface of the water. In still water, they could actually become permanently entrained in the surface due to surface tension effects at the water’s edge. A diatom stuck at the water’s surface or even near the surface would be exposed to ultraviolet radiation, leading to over-exposed, damaged pigments and imminent death. On the other hand, it can be a long way to the bottom of the mixed layer (about 10 – 100 m), and diatoms sink slowly, so although sinking can lead to death there is always a chance that ocean mixing can resuspend diatoms and bring them back into the photic zone. Clearly the optimal state is neutral buoyancy slightly below the water’s surface in still water. But given that neutral buoyancy is quite difficult to achieve and turbulence is ubiquitous, Vogel suggests that it is better to be slightly negatively buoyant and chance ocean mixing than to be slightly positively buoyant and face imminent death.

Raven (2004) offers another potential benefit to diatom sinking. Since diatoms can achieve positive buoyancy only through a physiologically expensive process, a diatom infected with a parasite is unlikely to be able to devote as much energy to buoyancy control as an unparasitized diatom. This means that infected diatoms are more likely to sink out of the mixed layer (and the corresponding diatom population) than unparasitized diatoms. Thus sinking may be a way to remove infected hosts from a population and prevent infectious parasites from spreading, conferring greater fitness to

15 diatom populations as a whole. To back up this hypothesis, the rate of parasitism in diatoms is exceptionally low compared to other groups of organisms (Raven 2004).

When thinking about the evolutionary value of a phenomenon like sinking, it is important to consider how much of a change in sinking speed is meaningful. As noted, diatoms are already capable of at least an order of magnitude variation in sinking speed through ion exchange. As will be seen, this order of magnitude difference is actually the largest source of variation in sinking speed. Any features that cause a change much smaller than this are comparatively negligible adjustments to sinking speed. But where exactly should the line be drawn? What benchmark for a “negligibly small change” should one choose? The choice of such benchmarks is by nature arbitrary, but is still useful. Benchmarks provide a single measuring stick whereby different kinds of analyses of different types of features can be easily compared and put into a holistic perspective.

I will consider a sinking speed change of a factor of 10 or more to be large, as this is the scale of changes caused by ion exchange, the largest modifier of sinking speed. But it is not entirely clear what is a small change to sinking speed. Some theoretical adjustments can change sinking speed by less than 1%, or a factor of 1.01, which makes the lower limit to sinking speed variability so close to 0 as to essentially be considered unchanged. In contrast, thickness varies naturally within a species by up to 50% of the mean, which can cause up to a 20% variability in sinking speed (see Chapter 1 for the measurements and analysis). This natural variability can provide a ready benchmark for characterizing small changes to sinking speed. Therefore, I will refer to changes of less than 20% as relatively small changes to sinking speed, both because they are much less

16 than the order of magnitude variability caused by ion exchange and because such changes are encompassed by the natural variation in thickness within a species.

There is still quite a bit of ground between a 20% (1.2 fold) change, and a 1000%

(10 fold, or order of magnitude) change. Thus I will define an intermediate benchmark between these values, for changes that are not so small as to be easily ignored, but not so large as the extreme variability caused by ion exchange. For simplicity, I will use a logarithmically defined intermediate between 20% and 1000%. The geometric mean of these two values is 141%, which for convenience will be simplified to a 100% (2 fold) change.

Thus I will use two arbitrary benchmarks to characterize changes to sinking speed: 20% (1.2 fold change) and 100% (2 fold change). Changes to sinking speed of less than 20% will be referred to as relatively small scale changes, those between 20% and

100% as relatively moderate scale changes, and those greater than 100% as relatively large scale changes.

17 Figure 0-1

Picture of a diatom, with valve and girdle labeled. Notice this is a partically interior view, with one valve missing.

18 Figure 0-2

A simplistic illustration of a diatom, with the relevant planes and axes show. The arrows are coded the same color as the plane to which they project perpendicularly. A) A short, fat, symmetric centric diatom. B) A typical elongate pennate diatom. Green is the valvar plane, purple the radial plane, blue the apical plane, and orange the transapical plane.

19 Chapter 1 Diatom sinking speeds: from observations to theory

1.1 Introduction

Despite extensive research on the fluid dynamics of small particles moving at slow speeds, the sinking speed of individual diatoms has yet to be accurately predicted by theory. Stokes’ law is often invoked to explain diatom sinking, predicting that velocity scales with radius squared, that is, velocity as a function of radius has a scaling exponent of 2 (Happel and Brenner 1991). Diatoms fit the assumptions of Stokes’ law: they are small, roughly spherical, and slow moving. Nonetheless over the last fifty years, empirical measurements of diatom sinking speeds have shown that the power law relationship between radius and velocity has a scaling exponent between 1.2 and 4

(Smayda 1970, Waite et al. 1997). This mismatch between empirical observation and theory has often been noted, (Reynolds 2006, Walsby and Reynolds 1980), but the physical basis for the mismatch is yet to be resolved.

Here, we offer a new model for the relationship between diatom size and maximum sinking speed (a modified Stokes’ law) that accounts for the discrepancy between the traditional Stokes’-law prediction and measured sinking speeds. We parameterize a key component of the model, frustule thickness, and broadly confirm the model with empirical measurements.

20 1.2 Methods

1.2.1 Species

We used diatom cultures from the Culture Center for Marine Phytoplankton

(CCMP) at the Bigelow Laboratory for Ocean Sciences, listed here from largest to smallest: Coscinodiscus wailesii (CCMP 2513), C. sp. (CCMP 1583), C. radiatus

(CCMP312), C. radiatus (CCMP 310), Porosira glacialis (CCMP 652), and Minidiscus variabilis (CCMP 495). These cells range in diameter from 225 µm to 3 µm. To minimize complications due to shape and ecology, we selected cultures of centric diatoms that were planktonic, solitary, marine and close to spherical in shape. C. wailesii is the only one of these species previously studied with respect to sinking (Smayda,

1970).

Each diatom in this study is shaped roughly like a short, fat cylinder. The nomenclature used in this section is as described for the frustule in section 1.2.2., but is also depicted pictorially in Fig. 1-1. The two smallest species, P. glacialis and M. variabilis, have processes or chitinous threads that project out from the valve face.

1.2.2 Culturing techniques

Diatoms were batch cultured at Hopkins Marine Station. Cultures were grown at their preferred growth temperatures (20°C for C. wailesii and C. sp., 12°C for both C. radiatus cultures and M. variabilis, and 2°C for P. glacialis). Each culture was given f/2 media (Guillard 1975) every three weeks, at which point all cultures had reached a stationary phase of growth. Sinking measurements were performed between two and four weeks after a transfer to fresh media, so that all cultures tested were in a similar stationary phase of growth, likely near nutrient deprived conditions. In such nutrient

21 deprived conditions, cell division is slowed or halted (Brzezinski et al. 1990) and sinking rate is increased (Bienfang et al. 1982), possibly due to accumulation of thicker frustules in slow growth conditions (Z. Finkel pers. com.). This choice of culturing method may bias our cultures towards thicker frustules and therefore faster sinking speeds compared to other methods, but this should not affect the accuracy of our models since the frustule thicknesses used in our models were directly measured from the cultured diatoms (see below).

1.2.3 Sinking speed measurements

Many previous studies used the SETCOL (Bienfang 1981) procedure to measure diatom sinking speeds in which the temporal change in concentration of diatoms at the top and bottom of a water column allows for a calculation of average sinking speed. This procedure is not ideal because wall effects and effects due to high particle concentrations are neither controlled for nor accounted for in terms of fluid dynamics. O’Brien et al.

(2006) found tracking individual diatoms (as done here, see below) to be more accurate and appropriate for size-sinking speed relationships than the time and mass averaged

SETCOL procedure. In particular, SETCOL measures average sinking speed, underestimating maximum sinking speed by a factor of 2-3.

Sinking speed and valve diameter of individual diatoms were measured simultaneously in a density stratified water column (Δ4 g L-1 salinity over a column height of 30 cm). The water column was covered and surrounded by a temperature bath at

20°C, eliminating any convection due to temperature variation. The water column was 21 cm by 21 cm in cross section. Diatoms were inserted using a pipette centered in the top of the chamber and tracked as they fell through the column. The diameter of the largest

22 diatom was approximately 1000 times smaller than the distance from the diatom to the nearest wall, rendering wall effects on sinking negligible (Happel and Brenner 1991).

A telemicroscope (Questar DR1, Questar Inc.) was used to visualize individual diatoms in the column. Diatoms were backlit, with the light positioned at a slight angle to the viewing plane of the telemicroscope so that only light refracted from each diatom’s frustule reached the imaging system. Time lapse photography (Canon EOS 40D, Canon

Inc.) was used to record sinking, and ImageJ (NIH) was used to calculate sinking speed and size of individual diatoms.

For the four largest species, image quality was sufficient to distinguish pigmented intact diatoms from unpigmented empty frustrules. For these species, only intact diatoms were included in the analysis. For P. glacialis, pigmentation was not discernable and all observed diatoms were included in the analysis. Due to its small size, the diameter and pigmentation of M. variabilis could not be reliably determined, but these cells could nonetheless be accurately located by their frustule’s diffraction pattern and their sinking speed thereby measured. M. variabilis’s diameters ranged from 3 µm to 5 µm, and an average size of 4 µm was assumed for all recorded diatoms of this species.

The accuracy of the method was tested with polystyrene beads of known size (20

µm, 45 µm, 90 µm) and density (1050 kg m-3). The scaling exponent between the beads’ sinking speeds and size was 1.94 ± 0.13 (slope ± 95% CI, r2 = 0.97), statistically indistinguishable from the Stokes’ law scaling exponent of 2.

All species were tested both alive and dead. In the ‘live’ treatment, a sample was taken directly from culture and inserted into the water column. In the ‘dead’ treatment, a sample was taken from a culture, heated to boiling in a microwave oven, and then rapidly

23 cooled to room temperature before insertion into the water column (Fisher and Harrison

1996). All sinking measurements were performed within an hour of heat shock, after

which diatoms were observed to lose their cytoplasmic components.

The sole drawback of tracking individual diatoms rather than using the SETCOL

procedure is that the size of diatoms smaller than 15 µm could not be accurately

measured with the optical arrangement used in the present study.

1.2.4 Frustule geometry measurements

A scanning electron microscope (Hitachi S-3400N VP-SEM) was used to

characterize the geometry of the frustule of each species. During the course of our study,

the SEM was calibrated on a quarterly basis using “SIRA TEST” stubs (Electron

Microscopy Science). For our working distances of less than 10 mm and our maximum

accelerating voltage of 5 kV, the maximum calibration error was 3%. The relevant

measurements were valve diameter, valve and girdle thickness (Fig. 1A,B), and valve and

girdle height (Fig. 1B,C). The SEM stage could be tilted, allowing height to be measured

on frustules that lay flat on the stage (Fig. 1B). If hʹ is the girdle height measured in Fig.

1B, actual girdle height (hg) is:

h " h = (1-1) g sin(#)

where α is the tilt angle of the stage. To avoid measurement error at small angles, the

! stage was tilted as much as possible, typically between 15° and 25°.

1.2.5 Statistical analyses

For simplicity, average density values were used in our models. To account for

uncertainties in these density values and in our thickness measurements, we calculated

95% confidence intervals of the models using a bootstrap method. We assumed a uniform

24 distribution of cytoplasm and frustule densities across the density ranges given below.

We used a normal distribution for thickness with a standard deviation of 25% of the mean

value, based upon our SEM measurements (Table 1). We randomly selected parameter

values from these specified distributions and calculated sinking speed over 10,000 trials,

taking the upper and lower 2.5% of sinking speeds at each size to define the 95%

confidence interval. We used these 95% confidence intervals to compare how much our

models differed given the known uncertainties in our parameters. Since we could

estimate the value and uncertainty of our parameters independently of our sinking speed

data, this bootstrapping technique is more appropriate than a model selection criterion

such as the Akaike information criterion, which estimates the parameters from the data

itself. Considering the relative scarcity and variability in our data and the complexity of

our model, the Akaike information criterion would provide relatively poor estimates of

our parameters. Thus more statistical power can be gained by independently estimating

the parameters and using a bootstrapping technique to compare the models.

To quantify the degree to which each model explains the upper limit of the data,

we binned the data according to logarithmic size units and took the top five sinking

speeds in each size bin. We then calculated the sum of squared deviations (SSD) of the

binned, log-transformed maximal sinking speeds from each model. This model-specific

SSDmodel was used, along with the SSDtotal of the data from the overall mean, to calculate

R2:

SSD " SSD R 2 = total model (1-2) SSDtotal

For the traditional Stokes’ law, we left ρD as a free parameter to be optimized, which

! 2 generated the largest possible R (or smallest possible SSDmodel) given the form of

25 Stokes’ law. Even with this parameter optimization, the SSDmodel of Stokes’ law was

greater than the SSDtotal, indicating that Stokes’ law did not describe the trend in the data.

For the geometric measurements of the frustule, we performed a least-squares

power law regression between frustule parameters and frustule diameter, both within and

between species (Schmidt-Nielsen 1984). Since there was an unequal number of data

points collected among species, for the between-species regression one average value was

used per species to give each species equal statistical weight.

1.3 Model

1.3.1 Stokes’ law

Stokes’ law predicts that a spherical diatom’s sinking speed (U) is (Happel and

Brenner 1991):

2(" # " )gr 2 U = tot w (1-3) 9µ

-3 where ρtot is overall diatom density, r is radius, ρw = 1023 kg m is the density of

! seawater at 20°C and 33 g L-1 salinity, g = 9.8 m s-2 is gravitational acceleration, and

-3 -1 µ = 1.07 x 10 Pa s is the dynamic viscosity of seawater at 20°C and 33 g L salinity.

Stokes’ Law holds true for Reynolds numbers less than 1, where Reynolds number (Re)

is defined as:

U (2r)" Re = w (1-4) µ

All diatoms in this study sink at Re < 0.1.

! Stokes’ law predicts how quickly a spherical diatom should sink given a certain

overall cell density, ρtot. As previously described, diatoms reduce their density through

ion exchange, a physiologically costly process (Anderson et al. 1978, Waite et al. 1992).

26 In the absence of physiological density reduction, diatoms sink at their maximal speed, which should be governed solely by fluid mechanics (Waite et al. 1997). This study is primarily concerned with modeling this maximal speed, which can occur either in dead diatoms or in live but physiologically inactive diatoms. Therefore the model is fit to the maximal sinking speed of both live and dead diatoms.

As noted, Stokes’ law is derived for a spherical object. Some implications of a non-spherical shape are explored in section 2.

1.3.2 Cellular densities

A diatom is composed of two distinct components: the frustule and the cytoplasm.

The thin siliceous frustule is relatively dense and covers the outer surface of the diatom.

Diatom frustules are 10% to 70% amorphous silica (density ≈ 2600 kg m-3), the rest of the frustule being composed of proteins and sugars (density ≈ 1300 kg m-3) (Csoegoer et

-3 al. 1999, Schmid et al. 1981). This gives a range of frustule density, ρfr, from 1400 kg m to 2200 kg m-3. If we assume that the frustule has a constant, small thickness over a large range of diatom sizes, this relatively dense mass of the frustule scales as a surface area.

To our knowledge, there have been no direct measurements of marine diatom cytoplasm density, but based on scant data from animal and plant cells living in fresh and salt water, cytoplasm likely has a density only slightly greater than seawater (ρcyt = 1030

- 1100 kg m-3, Smayda 1970). The range of 1030 to 1100 kg m-3 will be used in this study, but this range may not account for the presence of a vacuole, which can be less dense than seawater (Woods and Villareal 2008). Inclusion of a vacuole will be discussed in section 2.2.3.

27 -3 -3 For simplicity, median values of ρfr = 1800 kg m and ρcyt = 1065 kg m were

used in our models.

In light of differences in density between the frustule and cytoplasm, the overall

diatom density term in Stokes’ law can be split into two components:

Vcyt Vfr "tot = "cyt + "fr (1-5) Vtot Vtot

where V is volume and the subscripts fr, cyt, and tot, refer to the frustule, cytoplasm, and

! total for the entire diatom, respectively.

1.3.3 Spherical model

To solve Eq. 1-5 analytically, we need to derive expressions for the volumes. As a

first approximation, we model the diatom as a sphere. In this case:

4 3 Vtot = 3 "r (1-6)

4 3 Vcyt = 3 " (r # t) (1-7)

! 4 2 2 3 Vfr = Vtot "Vcyt = 3 #(3r t " 3rt + t ) (1-8) ! where t is the thickness of the frustule. We can then express Eq. 1.5 as:

! (r # t)3 3r 2t # 3rt 2 + t 3 " = " + " (1-9) tot cyt r 3 fr r 3

Inserting Eq. 1-9 into Eq. 1-3 predicts the sinking speed of a spherical diatom:

! 3 2 2 3 2g $ (r # t) (3r t # 3rt + t ) 2 ' U = & "cyt + "fr # "w r ) (1-10) 9µ % r r (

2 If r >> t, Vfr scales approximately with r , and Eq. 1-9 and 1-10 can be simplified:

! 3t " # " + " $ " (1-11) tot cyt ( fr cyt ) r

2 2(#cyt $ #w )gr 2(#fr $ #cyt )grt U " + (1-12) ! 9µ 3µ

28 ! If (ρcyt - ρw)r >> 3(ρfr – ρcyt)t, velocity scales as radius squared, in agreement with the

traditional Stokes’ law. Using typical densities and frustule thicknesses (t = 1 µm), a

scaling exponent of 2 is predicted to hold for r > 75 µm. As size decreases below 75 µm,

the exponent should approach 1, except that at small radii (r < 10 µm), the condition that

r >> t is no longer is valid. At these small radii (where r and t are of the same

magnitude), Eq. 1-10 must be used instead of Eq. 1-12, and velocity again scales as

radius squared.

The simple model described above assumes a constant frustule thickness over all

diatom sizes. If instead thickness is a function of frustule radius, the appropriate scaling

law for thickness (of the form t = bra ) can be inserted into Eq. 1-10 to derive a variable

thickness model:

a 3 2 a 2a 3a 2g $ (r # br ) (3r br # 3rbr + br ) 2 ' U = & "cyt + "fr # "w r ) (1-13) 9µ % r r (

Note that if thickness scales directly with radius, i.e., a = 1, then the traditional Stokes’-

! law scaling exponent of 2 between sinking speed and radius is recovered.

1.4 Results

1.4.1 Sinking speed and the frustule geometry

Sinking speed data are shown in Table 1-1 and Fig. 1-2. For all cultures except P.

glacialis, dead diatoms sank faster than live diatoms, although differences in mean

sinking speed between live and dead cells were statistically significant for only three of

the larger cultures (C. wailesii, C. radiatus CCMP310, C. sp.). The sinking speed of live

cells had a slightly significant trend with size (0.34 ± 0.28, scaling exponent ± 95% CI),

whereas the dead cells exhibited a strongly significantly positive trend between sinking

29 speed and size (0.90 ± 0.21 , scaling exponent ± 95% CI) which was also significantly different from that of live cells. Note that these scaling exponents are for all of our data, not just the maximal sinking rates predicted by our models.

The maximal sinking speeds had a significantly positive scaling exponent with size (0.96 ± 0.15, scaling exponent ± 95% CI). If C. wailesii is removed from the analysis

(see Discussion for why this may be reasonable), the scaling exponent is even greater

(1.18 ± 0.14, scaling exponent ± 95% CI).

No significant intraspecific trend with size was found for any frustule dimension in any of the species (p > 0.05), despite a 50% - 120% range in frustule thickness

(relative to the mean frustule thickness) within species, and occasionally within a single frustule. Between species, all frustule dimensions varied significantly with size (Table 1-

2, Fig. 1-3). Valve height does not scale significantly differently from isometry, whereas the scaling of valve and girdle thicknesses and girdle height differ significantly from isometry (Table 1-2). The valve tends to be twice as thick as the girdle, on average.

1.4.2 Theoretical model of sinking speed

Predicted sinking speeds are shown in Fig. 1-2. Our new model (based on Eq. 1-

10) provides a better fit to the upper edge of the data than does the traditional Stokes’-law prediction. The new model is curvilinear with an exponent that varies from 1.2 to 1.9 for the size range of diatoms in this study.

Given significant size-dependant variation of frustule parameters among species

(Table 1-2), we show the model that accounts for this variable thickness (Eq. 1-13) in

Fig. 1-4. This figure also highlights the maximal sinking speed data points that were used to calculate the R2 values. The curvilinear nature of the constant thickness model is not

30 evident in the variable thickness model. Instead, the scaling exponent (the slope in this logarithmic plot) is relatively constant at a value of approximately 1.6. Constant and variable thickness models converge at large diameters, but the variable thickness model predicts slower sinking speeds for small diatoms than does the constant thickness model.

The 95% confidence intervals of the spherical model with constant and variable thicknesses are shown in Fig. 1-5, across the entire range of diatom sizes. Although the spherical model with constant thickness performs better (R2=0.57), it is statistically indistinguishable from the spherical model with variable thickness (R2=0.38) at all sizes, based on the considerable overlap of their 95% confidence intervals. Both models diverge from the traditional Stoke’s law at small and large sizes, indicating that both models are significantly different from and, on the basis of their R2 values, significantly better fits to the data than the traditional Stokes’ law.

1.5 Discussion

Smayda (1970) suggested that the discrepancy between the traditional Stokes’-law prediction and empirically measured sinking speeds for diatoms is due to variations in shape or physiology. This study minimized the effect of shape, yet the mismatch between measurement and theory remains (Fig. 1-2). Physiology can account for much of the variation in sinking speed at any given size, but does not explain why the maximal sinking speed does not follow Stokes’ law across sizes (Waite et al. 1997).

Our modification to the density term in Stokes’ law is capable of explaining the wide variety of scaling exponents previously found for the maximal sinking speed of diatoms (Smayda 1970, Waite et al. 1997). With this new model, any scaling exponent between 1 and 2 can be observed for the maximal sinking speed and size relationship,

31 depending upon size, density, and frustule thickness of the species in question. Frustule thickness is an especially important parameter that can greatly affect the scaling relationship between velocity and size (Fig. 1-4). Use of a constant versus an allometrically-scaled thickness can cause a difference in sinking speed of up to 80% in sinking speed, a relatively moderate effect on sinking speed. Yet the use of these different models does not create a statistically significant difference in the prediction of maximal sinking speed. Thus the constant thickness model is recommended for the sake of simplicity. Thus, with a simple mathematical adjustment, we have brought sinking speed theory and empirical data an order of magnitude closer together.

1.5.1 Comparison with previous studies

Fig. 1-5 compares previously measured sinking speeds to the data obtained in this study. As described in the section 1.2.3, the previous SETCOL procedure employed by some studies is less accurate than this study’s measurement techniques in measuring maximal sinking speed (O'Brien et al. 2006, Bienfang 1981). To account for some of

SETCOL’s inaccuracies, SETCOL sinking speeds are multiplied by a factor of 2.5 in Fig.

1-5.

For many species with diameters in the range of 20 µm to 100 µm, the present study found higher maximal sinking speeds than previous studies, even with the

SETCOL correction (Fig. 1-5). This could be due to the use of irregularly shaped diatoms in previous studies, or to the inaccuracies of previous measurement techniques, both

SETCOL and otherwise, or to the faster sinking rates produced in our culturing methods compared to those of other studies (see section 1.2.2). Our measurements of maximal sinking speeds are larger than those made by O’Brien et al. (2006) using a similar

32 technique to ours, but O’Brien et al. tested only live diatoms, which likely had physiologically reduced densities causing slower maximal sinking speeds. In both this and previous studies, there is a 1.5 to 2 order of magnitude range of sinking speeds for a given size, a relatively large variation in sinking speed (Smayda 1970, Waite et al. 1997).

Aside from higher maximal sinking speeds, the data from this study and the variable thickness spherical model both show good agreement with previous data (Fig. 1-

5). Velocity scales with radius to the 1.4 – 1.6 power depending on the model employed, which is in the range of previous values of 1.2 – 1.6 (Smayda 1970, Waite et al. 1997).

Additionally, our thicknesses correspond well with previous studies. In particular, the diatom frustule images (Semina 2003, Ferrario et al. 2008) exhibit a range of thicknesses from 0.2 to 5.0 µm, which encompasses our measured thickness range (Table

1-1).

1.5.2 The sinking of large diatoms

None of our models can explain the relatively slow maximal sinking speed of C. wailesii, the largest species studied (Fig. 1-4). It is unlikely that this discrepancy is due to errors in measured sinking rates; our maximal sinking speed for C. wailesii (35 m day-1) is similar to previous measurements (29 m day-1) (Smayda 1970, Waite et al. 1997).

One potential explanation for C. wailesii’s slow sinking is porosity, or percentage of void space, of the frustule, which has a notable allometric scaling pattern of increasing with size (pers. obs.). Yet the change in areolae cover was slight across cell size (10% in

M. variabilis to 30% in C. wailesii) and when incorporated in the models had a relatively small effect on sinking rates.

33 The presence of copious amounts of mucilage could explain the slow sinking of

C. wailesii through adding positively buoyant material to the outside of the diatom. Yet as will be shown in section 2.3.4., mucilage seems to have little overall effect on the sinking of diatoms, and an especially small effect on the sinking of large diatoms.

As a third alternative, the presence of a low density vacuole is the most likely explanation for C. wailesii’s slow sinking. I will discuss potential models that incorporate a cell vacuole in detail in section 2.2.3.

1.5.3 Conclusion

In summary, a modification to Stokes’ law can accurately predict maximal diatom sinking speeds across a wide range of cell size. Although it has been empirically demonstrated that the scaling law relating diatom sinking speed to size has an exponent between 1.2 and 1.6, this analysis is the first to offer theoretical support and a mechanistic explanation for this empirical result. This study shows that modifying the density term in Stokes’ law is an important aspect of mechanistically predicting maximal diatom sinking speeds.

34 Table 1-1

Measured sizes, valve thicknesses, and sinking speeds for various diatom cultures. The p-value is based on a student’s t-test between the mean sinking speeds of live and dead cells, with significant values in bold. CW = Coscinodiscus wailesii, CS = Coscinodiscus sp., CR = Coscinodiscus radiatus, PG = Porosira glacialis, MV = Minidiscus variabilis.

NA indicates that there were not enough data points for this species to perform the t-test.

-1 r (µm) tv (µm) U (m d ) Mean for Mean for Culture mean (range) mean (range) live cells dead cells Range p CW 108 (84 - 132) 1.66 (1.02 - 2.32) 4.1 17.5 (0.9 - 35.2) <0.01 CS 38 (25 - 54) 1.33 (0.8 - 2.06) 7.9 13.6 (0.4 - 30.0) 0.01 CR 312 28 (21 - 36) 1.27 (0.85 - 1.65) 4.2 5.6 (1.5 - 18.6) 0.54 CR 310 17 (10 - 25) 0.75 (0.58 - 0.89) 3.9 5.9 (1.9 - 8.7) 0.02 PG 10 (8 - 13) 0.5 (0.24 - 0.84) 2.4 1.7 (0.2 - 6.8) 0.48 MV 2 0.05 0.3 0.7 (0.3 - 0.8) NA All 60 (2 - 132) 1.03 ( 0.05 - 2.32) 5.2 8.5 (0.4 - 35.2) <0.01

35 Table 1-2

The results of the regression for the four frustule parameters against frustule diameter.

The regression model was of the form parameter = b ra. The thickness measurements were both significantly different from no trend (a=0) and isometry (a=1). The height measurements both had significant trends, but the valve height trend did not differ from isometry.

Parameter a ( ± 95% CI) b (± 95% CI) df r2 Valve thickness 0.45 (0.27) 0.23 (0.25) 4 0.9 Girdle thickness 0.40 (0.36) 0.14 (0.20) 4 0.79 Valve height 0.97 (0.31) 0.24 (0.33) 4 0.97 Girdle height 0.69 (0.24) 1.09 (1.14) 4 0.96

36 Figure 1-1

(A) A diatom valve and (B,C) a valve and girdle shown in three different views. (A) Top- down view of valve interior. (B) Valve and girdle interior in an oblique top-down view.

(C) Valve and girdle in girdle view. In the left column, the valve is shown in light grey and the girdle in dark grey. Refer to the text for the meaning of labels.

37 Figure 1-2

Data for all six species shown on double logarithmic axes, separated by live and dead treatment. The traditional Stokes’ Law prediction (Eq. 1-3) is compared to the density- modified Stokes’ law prediction, with tfr=1 µm (Eq. 1-10). The traditional Stokes’ law

-3 prediction uses ρtot=1800 kg m , such that both models converge at small diameters.

38 Figure 1-3

The geometric parameters of the frustule from Fig. 1-1 plotted against frustule radius for the six diatom species in this paper. The statistical parameters of the power law regression are shown in Table 1-2.

39 Figure 1-4

A comparison of the two models in the study and the traditional Stokes’ law. Data from

Fig. 1-2 are shown with the maximum sinking speeds in each size bin indicated by black circles. The SSD of each model is calculate with respect to these maximal sinking speed values. In the constant parameter models, tfr = 1 µm. The traditional Stokes’ law is shown

-3 with ρtot=1124 kg m , a value which minimizes the SSD of this model.

40 Figure 1-5

A comparison of the sinking data from this study, the spherical models (Eqs. 1-10 and 1-

13) from this study, the traditional Stokes’ law, and data from several previous studies.

The grey area around the model lines show 95% confidence intervals in sinking speed prediction due to uncertainties in the values for tv, ρfr, and ρcyt. The overlap in these models is shaded a darker grey- notice that the confidence limits overlap at all sizes shown. The data from O’Brien et al. (2006) are individual diatom sizes and sinking, unlike the other studies which are averaged sinking rates. Rather than showing all of the numerous individual data points, the range of the data from O’Brien et al. (2006) is traced out by the given lines (CS = Coscinodiscus sp., SC = Skeletonema costatum, SD=small diatoms).

41 Chapter 2 Theoretical modeling on the effect of shape

2.1 Introduction

Diatoms have a plethora of shapes (Round 1990), and any treatment of diatom diversity would be incomplete without a discussion of shape. The most obvious inaccuracy in the treatment of diatoms in Chapter 1 is the assumption of spherical shape.

Additionally, non-spherical diatom shapes have previously been invoked as adaptations to lower sinking (Smayda 1970). For these reasons, I would be particularly remiss if I did not address how the issue of shape specifically affects the sinking speed of diatoms.

Although this chapter will mostly concern the shape of the frustule, it is not entirely limited to the frustule. In this sense, this chapter is more a discussion of the general complications that can arise when predicting sinking speed, whether these arise from frustule morphology, cell vacuole morphology, or whole diatom colony morphology.

This discussion will be mostly theoretical and proceed in two sections. In section

2.2 I address the how the non-spherical shape of the centric diatoms used in Chapter 1 affects predicted sinking speeds. In 2.3, I discuss how the shapes of more geometrically complicated diatoms affect their sinking characteristics.

The discussion will center around two issues. First, in what contexts does it become important to consider the effect of shape when modeling diatom sinking speed?

Second, what are the evolutionary implications of the effects of different shapes on sinking speed?

42 2.2 Alternate models for the sinking of simple centric diatoms

Even though I chose diatoms with as simple a geometry as possible in Chapter 1,

they are not exactly spherical. Rather, they appear more like short, fat cylinders with

rounded edges (Fig. 1-1), which affects the predictions of the model in two ways.

First, to account for drag incurred by non-spherical shapes, the Stokes’-law

prediction (Eq. 2) should include a form resistance term, Φ (Smayda 1970):

2(" # " )gr 2 U = tot water eq (2-1) 9µ$

where req is the equivalent radius, or radius of a sphere with the same volume as the shape

! under investigation. Here, Φ is the ratio between the sinking speed of a non-spherical

object to that of a sphere of equivalent volume. By definition, spherical objects have a

form resistance of 1. Form resistance has been calculated for a variety of non-spherical

shapes (Happel and Brenner 1991). For the simple short fat cylindrical shape of the

diatoms in Chapter 1, the form resistance term is approximately 1.05 resulting in a 5%

reduction in sinking speed, a relatively small adjustment. Additionally, this factor would

affect only the magnitude coefficient, not the scaling exponent, of the relationship

between sinking speed and size, and therefore cannot explain the primary discrepancy

between the Stokes’-law prediction and empirical data.

Additionally, in accounting for form resistance, one should use the equivalent

radius of a diatom, which for diatoms of the shapes considered in Chapter 1 is about 14%

greater than that of the more traditionally used valve radius. Radius is potentially squared

in Stokes’ law, which with the form resistance taken into account can increase the sinking

speed prediction by as much as 25%, though probably more around 15%. This is a

relatively small correction to sinking speed, so I will ignore the form resistance term and

43 equivalent radius for the nearly spherical shapes considered in section 2.2, but both

corrections will become important in section 2.3.

Second, a cylindrical shape changes the expression for the density scaling term

(Eq. 1-5). The main driver of the density change with size occurs because frustule mass

scales with surface area whereas the cytoplasm mass scales with cell volume, and these

scaling patterns hold regardless of the object’s shape. However, it is not known how

different the prediction of alternate geometries is across size for diatoms, and so this issue

will be explored in the next section.

2.2.1 Cylindrical model

A squat cylinder is a better approximation for a centric diatom’s true shape than is

a sphere. In addition, a cylindrical model can allow for differences in frustule thickness

between valve and girdle, which is not possible in the radially symmetric spherical

model.

To construct the cylindrical model, consider a Petri-dish shaped diatom composed

of two valves and two girdles. Eqs. 1-6 to 1-8 can be modified in the following manner:

2 Vtot = 2"r (hv + hg ) (2-2)

2 2 Vcyt = 2" (r # tv ) (hv # tv ) + 2" (r # tg ) hg (2-3)

! 2 Vfr = Vtot "Vcyt = 2#tv (hv (2r " tv ) + (r " tv ) ) + 2#hg tg (2r " tg ) (2-4) ! where h is height (see Fig. 1) and the subscripts v and g refer to the valve and girdle,

! respectively. Inserting Eqs. 2-2 to 2-4 into Eq. 1-5 yields,

2 2 2 (r # tv ) (hv # tv ) + (r # tg ) hg tv (hv (2r # tv ) + (r # tv ) ) + hg tg (2r # tg ) "tot = "cyt 2 + "fr 2 r (hv + hg ) r (hv + hg )

(2-5)

44 Inserting Eq. 2-5 into Eq. 1-3 predicts sinking speeds for a cylindrical diatom:

$ 2 2 2 ' 2g (r # tv ) (hv # tv ) + (r # tg ) hg tv (hv (2r # tv ) + (r # tv ) ) + hgtg (2r # tg ) 2 U = & "cyt + " fr #" w r ) 9µ % (hv + hg ) (hv + hg ) (

(2-6) ! Analogous to the spherical model (Eq. 1-10), this cylindrical model assumes

constant tv, tg, hg, and hv. A variable parameter model can be formulated by substituting

ay the appropriate scaling laws (of the form y = byr , where y is a frustule parameter) for tv,

tg, hg, and hv in Eq. 2-6:

a a a a a $ (r # b r tv )2 (b r hv # b r tv ) + (r # b r tg )2 b r hg ' & tv hv tv tg hg ) "cyt a a + & (b r hv + b r hg ) ) 2g hv hg U = & ) a a a a a +a a 9µ & b r tv (b r hv (2r # b r tv ) + (r # b r tv )2 ) + b b r hg tg (2r # b r tg ) ) tv hv tv tv hg tg tg 2 & " fr a a # "w r ) (b r hv + b r hg ) %& hv hg ()

(2-7)

! A cylindrical model was fit to the data in Fig. 2-1, for both constant thickness

(R2=0.36) and variable thickness (R2=0.28) models, using the scaling relationships in

Table 1-2 and R2 calculated as by Eq. 1-2. Note that this was a slightly worse R2 value

than the spherical model, but this difference is not statistically significant given the

uncertainty in our parameter values. In any case, the spherical model is clearly the

recommended model for the sake of simplicity.

2.2.2 Elliptical model

Diatoms are not perfect cylinders with sharp edges, but have rounded valves. The

radius of curvature of the valve edge is often quite small, so that large diatoms are best

approximated as having the sharp edges of a cylinder. For smaller diatoms, the radius of

45 curvature of the valve edge is on the same order of magnitude as the radius of the valve,

such that an elliptical model for the valve may be a better approximation. An elliptical

model can be derived using a half-spheroid for the valves and a cylindrical model for the

girdles:

4 2 2 Vtot = 3 "r hv + 2"r hg (2-8)

4 2 2 Vcyt = 3 "(r # tv ) (hv # tv ) + 2"(r # tg ) hg (2-9)

! 4 2 V fr = Vtot "Vcyt = 3 #tv (hv (2r " tv ) + (r " tv ) ) + 2#tghg (2r " tg ) (2-10) ! Inserting these expressions into Eqs. 1-5 and 1-3 gives:

! 4 2 2 4 2 3 (r # tv ) (hv # tv ) + 2(r # tg ) hg 3 tv (hv (2r # tv ) + (r # tv ) ) + 2tghg (2r # tg ) "tot = "cyt 2 4 + "fr 2 4 r ( 3 hv + 2hg ) r ( 3 hv + 2hg ) (2-11)

$ 4 2 2 4 2 ' 2g 3 (r # t ) (h # t ) + 2(r # t ) h 3 t (h (2r # t ) + (r # t ) ) + 2t h (2r # t ) ! v v v g g v v v v g g g 2 U = & "cyt 4 + " fr 4 #" w r ) 9µ % ( 3 hv + 2hg ) ( 3 hv + 2hg ) ( (2-12)

A variable parameter model can be formulated by substituting the appropriate !

ay scaling laws (of the form y = byr , where y is a frustule parameter) for tv, tg, hg, and hv in

Eq. 2-12:

a a a at ah $ 4 tv 2 hv tv g 2 g ' 3 (r # bt r ) (bh r # bt r ) + 2(r # bt r ) bh r & " v v v g g + ) cyt a ah & ( 4 b r hv + 2b r g ) ) 2g 3 hv hg U = & ) a a a a a +a a 9µ & 4 tv hv tv tv 2 hg tg tg ) 3 bt r (bh r (2r # bt r ) + (r # bt r ) ) + bh bt r (2r # bt r ) & " v v v v g g g # " r 2 ) fr a ah w & ( 4 b r hv + 2b r g ) ) % 3 hv hg ( (2-13)

An elliptical model was fit to the data in Fig. 2-2, for both a constant thickness ! (R2=0.41) and variable thickness (R2=0.30) model, using the scaling relationships in

Table 1-2. Note that this was a slightly worse R2 value than the spherical model, but this

46 difference is not statistically significant given the uncertainty in our parameter values.

Again, we recommend the spherical model over the elliptical model for the sake of simplicity.

2.2.3 Inclusion of vacuole

The maximal sinking speed of the largest diatoms (>100 mm) was not well predicted by the models in Chapter 1, potentially because the large central vacuole was not included in the density model. Flotation (positive overall cell buoyancy) seems to be a phenomenon of large diatoms, occurring when the cell’s vacuole exchanges high- density ions for low density ions (Moore and Villareal 1996, Woods and Villareal 2008).

As previously stated, Waite et al. (1997) have defined a critical cell diameter of 7.8 µm; in cells smaller than this the vacuole appears to be a negligible component of cell density.

However, the vacuole forms an increasing fraction of cell volume as overall cell size increases, filling up to 99% of the cell in the largest diatoms and significantly affecting sinking rates (Woods and Villareal 2008, Villareal 1988). Our models diverge from the data at a diameter of about 100 µm (Fig. 1-4), indicating that a vacuole may be a necessary component of sinking models for diatoms larger than 100 µm.

It’s unclear exactly how to incorporate a vacuole into the model. Clearly the vacuole increases with cell size faster than the rest of the cellular components, though without more measurements it is difficult to determine how exactly it increases with size.

At the very least, we know that the vacuole volume increases faster than, or scales allometrically with, the rest of the cytoplasmic contents. The vacuole density is also lighter than that of the rest of the cytoplasm, indicating a further mechanism for density

47 to decrease with size in diatoms, and most likely explaining why the models in Chapter 1

overpredicted the sinking speed of diatoms >100 µm diameter.

It is unclear what would be a reasonable estimate for the density of the cell

-3 vacuole, ρvac. When physiologically active, ρvac can be as low as 1000 kg m (Woods and

Villareal 2008), but the relevant values for the predictions in Chapter 1 concern

physiologically inactive diatoms. The vacuole density for dead cells is estimated to be

just slightly less than that of seawater (i.e. ~3 kg m-3, Smayda 1970), but Villareal (pers.

com.). This suggests that vacuole density may be somewhere very close to that of

seawater, but possibly slightly more dense than seawater in some species. For our

-3 purposes, we will assume that ρvac is in the range of 1015-1025 kg m , encompassing

values below and slightly above the density of seawater in my models (1023 kg m-3).

I formulate two potential possibilities for how to model the cell vacuole. The key

to me seems to be what happens to the cytoplasm. If the point of the vacuole is to

maximize buoyancy, then the vacuole should be as large as possible. The only constraint

on the vacuole would be how much room to leave for the rest of the cytoplasm, or how

much is needed for the cell to operate. In the first formulation, I assume that the cell

needs a certain volume of cytoplasm to stay alive. In this model, the volume of cytoplasm

(Vcyt) stays constant as size increases, with the remaining volume filled in by the vacuole

(subscript vac). This can be achieved by modifying Eq. 1-5:

Vcyt Vvac V fr "tot = "cyt + "vac + " fr (2-14) Vtot Vtot Vtot

3 Note that the size of the cytoplasm is set to Waite’s critical cell volume (Vcyt=200 µm ),

! and that the volume can be rewritten:

48 Vcyt Vtot #Vcyt #V fr V fr "tot = "cyt + "vac + " fr (2-15) Vtot Vtot Vtot

Since Vcyt is constant, expressions are only needed for Vtot and Vfr. I will use the simple

! spherical model of Eq 1-6 and 1-8 for the Vtot and Vfr, respectively, with the allometric

models used for thickness (not written out), which gives:

3 3 3 2 2 3 4# Vcyt (r $ t) $ 4# Vcyt 3r t $ 3rt + t " = " + " + " (2-16) tot cyt r 3 vac r 3 fr r 3

% 3 3 3 2 2 3 ( 2g 4# Vcyt (r $ t) $ 4# Vcyt 3r t $ 3rt + t 2 U = ' "cyt + "vac + " fr $" w r * (2-17) ! 9µ & r r r )

The results of this model are shown in Fig. 2-3, encompassing a range of ρvac. ! In the second model, I assume that the cytoplasm lies just inside the cell

membrane with a constant thickness tcyt. Woods and Villareal (2008) and Round (1990)

both describe the vacuole as a large centrally located object, with the cytoplasm forming

a thin layer on the outside of the vacuole. My reasoning is that the cytoplasm is where the

chloroplasts are located. A bigger cell has more surface area to capture light, and

probably a proportionate increase in chloroplasts for capturing this light. Each chloroplast

needs some volume of cytoplasmic materials to maintain its function, which give the

cytoplasm some set thickness tcyt. This thickness tcyt stays constant with size, and is set to

a thickness that gives essentially no vacuole at Waite’s critical cell diameter of 7.8 µm3

(i.e. tcyt =3.9 µm). As before I use Eq 1-6 and 1-8 for Vtot and Vfr, respectively with the

allometric models used for thickness (not written out), and also:

4 3 4 3 Vcyt = 3 "(r # t) # 3 "(r # t # tcyt ) (2-18)

4 3 Vvac = 3 "(r # t # tcyt ) (2-19) ! Putting this all into 2-14 and 1-3:

! 49 3 3 3 3 3 (r # t) # (r # t # tcyt ) (r # t # tcyt ) r # (r # t) " = " + " + " tot cyt r 3 vac r 3 fr r 3

(2-20)

! $ 3 3 3 3 3 ' 2g (r # t) # (r # t # tcyt ) (r # t # tcyt ) r # (r # t) 2 U = & "cyt + "vac + " fr #" w r ) 9µ % r r r (

(2-21) ! The results for this second model are shown in Fig. 2-4.

Note that both models cause a similar reduction in the sinking speed of large, but

not small, diatoms. The vacuole models substantially diverge from the spherical models

at a cell diameter of 100 µm, though the constant volume model drops off slightly faster

than the constant surface area model. For comparison, at 200 µm or the diameter of C.

wailesii in Fig. 2-3 and 2-4, the vacuole drops the prediction by about 30%-40%, a

relatively moderate adjustment to sinking speed, whereas the maximal sinking speed of

C. wailesii is 80% (a factor of 4) lower than the prediction of the variable thickness

spherical model. A vacuole correction falls closer to, but is still a fair bit faster than, the

sinking of large (>100 µm) diatoms.

2.2.4 Discussion

The results for both the cylindrical and elliptical model show poorer fits by their

R2 value than the simple spherical model. This indicates that these more realistic shapes

do not necessarily lead to better predictions, and that approximating the diatom as the

most geometrically simple shape is satisfactory for mapping out the sinking speed trend.

Therefore, for the sake of simplicity the spherical model (eq. 1-13) is recommended for

studies that desire a general size-dependant model of diatom sinking speed (for example,

Chapter 4). The evolutionary implications of this lack of difference are quite interesting,

50 indicating that slight deviations from a spherical shape are not enough to drive changes in sinking speeds. Use of a cylindrical or elliptical model differed from the spherical model by anywhere between <1% to 45% (a relatively small to moderate adjustment to sinking speed), depending on the cell size and type of frustule thickness model used. What this suggests is that from a physical perspective, the diatoms from Chapter 1 can be considered functionally spherical.

The vacuole model does seem to be a necessary component of predicting sinking speeds for large (>100 µm in diameter) diatoms. Uncertainties in vacuole density as well as the scaling of vacuole volume with size prevent a recommendation of which vacuole model to use. Note that both models presented here provide a non-negligible impact on the sinking speed of large diatoms even when not physiologically active, though not enough to account for the discrepancy between C. wailesii and the models in Chapter 1.

2.3 Additional models for the sinking of more complex diatom shapes

Although we have just seen that the cylindrical shape of some diatoms has a relatively small to moderate effect on sinking speed, diatoms with more extreme morphologies can differ substantially from the predictions of Stokes’ law, both through the form resistance term and through the scaling of density. This section presents the scaling laws specific to different diatom geometries. These models are purely theoretical and (except for one of the models in section 2.3.1) have not been tested against empirical data. They are based on the same principles that led to the predictions given in Chapter 1 and section 2.2: only the geometry is different.

The expressions in Chapter 1 are general expressions that, with some error, hold true for many diatom shapes. If one wants to generate general predictions across all

51 diatom species, then Eq. 1-13 is recommended. If instead one wants expressions relevant

for a particular group of diatoms, or for a particular species of diatoms, then I would

recommend selecting from the most appropriate geometries given below. Unless

otherwise stated, I derive alternate expressions for the volume of the frustule, cytoplasm,

and total cell (Eqs. 1-6 through 1-8).

2.3.1 Cylindrical model for elongate diatoms

A diatom with a rectangular-shaped valve (e.g. pennate and some centric diatoms)

can be modeled using an adjusted cylindrical model:

Vtot = lw(2hv + 2hg ) (2-22)

Vcyt = (l " 2tv )(w " 2tv )(2hv " tv ) + (l " 2tv )(w " 2tv )2hg (2-23) ! V fr = Vtot "Vcyt (2-24) ! where l and w are the length and width of the valve, respectively. This essentially models

! the diatom as a cylinder with a rectangular cross-section, or a cuboid (Fig. 2.5). If the

edges of the valve are sufficiently rounded, then this is probably an inaccurate

formulation. In these cases, one should model the diatom as a cylinder with the long

diameter orientated on the azimuthal axis of the cylinder (Fig. 2.6). For this case, we can

slightly modify the equations given in section 2.2.1:

2 Vtot = 2"rs rl (2-25)

2 Vcyt = 2"(rs # t) (rl # t) (2-26)

! 2 Vfr = Vtot "Vcyt = 2#t(rl (2rs " t) + (rs + t) ) (2-27) ! where rs and rl are the short and long radius of the valve, respectively. Not that in this

! model, the thickness of the valve and girdle are confounded in the calculations, and

cannot be treated separately, thus only one thickness is used. This model is a bit of an

52 approximation in several respects, as it treats the transapical cross-section of the diatom

as elliptical, even though the valves are rounded but the girdles are not (Fig. 0-2B, Fig. 2-

6B).

For these cylindrical models, Φ can be approximated as (Berg 1983):

4 " par = (2-28) 9(ln( 2rl ) 1 ) rs # 2

8 " perp = (2-29) 9(ln( 2rl ) + 1 ) ! rs 2 where the subscripts par and perp refer to a cylinder with its long axis oriented parallel or

! perpendicular to the direction of sinking, respectively. This is an analytic formulation,

and it should be noted that it is derived assuming req in Eq. 2-1 is equal to rs, and that

rs<

We demonstrate the usefulness of the above analysis (despite its approximate

nature) using the sinking speed from Waite and Nodder (2001). The two largest species

examined by Waite and Nodder (Thalassiothrix antartica and Trichotoxon reimboldtii)

did not follow the sinking speed trend formed by the smaller diatoms of that and other

studies (Waite et al 1997). These two large species are exceptional in that they are

extremely elongate cylinders approximately 20 µm in diameter and a few millimeters in

length (along the long valve diameter). These two species of diatoms sink faster than a

diatom with a diameter and length of 20 µm, but not as fast as a diatom with a diameter

and height of a few millimeters. It was unclear which dimension, diameter or height, is

the more relevant measurement of size for these elongate diatoms, when comparing them

to other, non-elongate diatoms. Waite and Nodder (2001) decided to plot sinking speed

against major axis length (MAL), defined as the longest dimension of the diatom in any

53 direction. MAL is valve diameter or cylinder diameter for the smaller species (because

they are short fat cylinders), but long valve diameter or cylindrical height for the two

largest species (because they are elongate cylinders, Fig. 2.6). Whereas the models in the

present study all use valve diameter as the variable size parameter, MAL treats the

cylinder height as the variable size parameter since it is cylinder height, not diameter, that

increases in the elongate diatoms. A model using cylinder height rather than diameter as

the increasing size parameter can be used to describe the size-sinking speed trend formed

by the two largest species in Waite and Nodder (2001).

Let’s assume the short valve diameter and total pervalvar height are both fixed at

2.5 µm, whereas the long valve diameter increases in size. We can then construct a model

that shows the change in sinking speed as a function of increasing long valve diameter (or

MAL). For this purpose, we can use Eq. 2-29 for the form resistance term in Eq. 2-1 (Eq.

2-28 could be used for form resistance as well, giving a 2x lesser form resistance):

(" # " )gr 2 U = tot water s ln( rl ) + 1 (2-30) 4µ ( rs 2)

where rl is the increasing size parameter. For ρtot, we use a cylindrical model with t=

! 0.3 µm (the reported valve thickness for these diatoms, Semina 2003) and Eq. 2-25

through 2-27 inserted for the relevant volumes in Eq. 1-5:

$ 2 t r (2r # t) + (r + t)2 ' g (rs # t) (rl # t) ( l s s ) 2 rl U = & " + " # " r ) ln + 1 & cyt fr w s )( ( rs ) 2) 4µ % rl rl (

(2-31)

! Note that ρtot, stays relatively constant at the larger sizes and that form resistance is the

major driver of changes to sinking speed at these sizes.

54 The result of this model is shown in Fig. 2-7. This new model follows the trend of

the data from Waite and Nodder (2001) quite well, offering a relatively large orders-of-

magnitude lower prediction for sinking speed at the largest sizes than the variable

thickness spherical model (Eqn. 1-13). Thus, a model that accounts for the exceptional

shape of these two species can explain their exceptional relationship between size and

sinking speed.

2.3.2 Elliptical model for pennate diatoms

This section is in part a more general form of the expressions given in 2.2.2. We

will model the valve as a half-ellipsoid, and the girdle as a cylinder with an elliptical

cross-section (Fig. 2-8):

4 Vtot = 3 "rl rs hv + 2"rl rs hg (2-32)

4 Vcyt = 3 " (rl # tv )(rs # tv )(hv # tv ) + 2" (rl # tg )(rs # tg )hg (2-33) ! V fr = Vtot "Vcyt (2-34) ! Notice that if one sets rl=rs, one derives the equations given in section 2.2.2. for a

! symmetric centric diatom. These expressions are not specific to pennate diatoms, but can

also be used for centric diatoms with non-symmetric shapes, if the valve-view shape of

their valve is elliptical in nature.

If rs << rl, for the purposes of form resistance the diatoms can be treated as

cylinders, and Eqs. 2-28 and 2-29 can be used for the form resistance.

2.3.3 Model for triangular-shaped valve diatoms

Some diatoms are triangular in valve view, including Ditylum brightwellii (a

species extensively studied with respect to sinking characteristics). An equilateral

triangular valve can be modeled as:

55 3 2 Vtot = 4 b (2hv + 2hg ) (2-35)

6 3 Vcyt = (b " 3 t)( 2 b " 3t)(hv + hg " t) (2-36) ! V fr = Vtot "Vcyt (2-37) ! where b is the length of one of the sides of the triangle (Fig. 2-9). If the total pervalvar

! height ( 2(hv+hg) ) is on the same order of magnitude as b, then the form resistance term

for such diatoms is unlikely to be significantly different from 1, meaning use of a form

resistance term will have a relatively small effect on sinking speed.

2.3.4 Models for spines and processes

To modify the density term, protuberances from the diatom’s shell can be

incorporated in the frustule volume term. Siliceous cylindrical protuberances can be

included as an additional volume:

2 V fr = VD "Vcyt +V prot = VD "Vcyt + n#rprot h prot (2-38)

where n is the number of protuberances, rprot is the radius of the protuberance, and hprot is

! the length of the protuberance. A conical protuberance can be added as:

1 2 V fr = VD "Vcyt +V prot = VD "Vcyt + 3 n#rbase h prot (2-39)

where rbase is the radius of the base of the cone.

! Incorporating the form resistance of spines and processes is a more complicated

matter. The processes, and occasionally the spines, are located in a haphazard manner on

the surface of the diatom. This means that the form resistance term is difficult to calculate

analytically, though it can be solved numerically or estimated empirically. I do not know

of any numerical estimates of the form resistance of processes, but Walsby and Xypolyta

(1977) empirically estimated the form resistance of chitinous processes on Thalassiosira

fluviatilis. They found that for a diatom 6 µm in diameter that had 76 processes per valve

56 that were 60-80 µm long, the form resistance was 1.9. This estimate seems reasonable for most diatoms that have similar processes. Notable though, the processes also increased the overall density of the cell, which countered the decrease in sinking due to form resistance. The result was that sinking speed was reduced by only a factor of 1.7, a relatively moderate adjustment.

Note that the two smallest species in Chapter 1, P. glacialis and M. variabilis, likely had processes, though probably fewer than T. fluviatilis, and so a prediction for their sinking speed might do better to take this minor correction into account. Yet a correction on this scale is not that significant compared to the hundred-fold range in variability in the sinking speed of P. glacialis.

This case study by Walsby and Xypolyta (1977) proves an important point: spines and processes will always increase the density of the cell, which counters the decrease in sinking caused by form resistance. Although there is still a net decrease in sinking speed, this decrease is small compared to the changes cause by physiological regulation of density. The result is that one finds it hard to conclude that the function of such spines is to reduce sinking when more effective sinking regulation mechanisms exist. At the very least, one must show that the relatively moderate adjustment to sinking speed of 70% caused by the addition of spines is enough to affect the fitness of the diatom.

The conclusion from this analysis is that spines and processes cause a relatively moderate change to the sinking speed of diatoms, but can often be ignored in diatom models with little loss to accuracy. The evolutionary implication is that spines and processes should not be thought of as adaptations to reduce sinking speed, unless it is shown that the 70% reduction in sinking speed conferred by spines is enough to increase

57 their fitness. In all likelihood, spines may serve some function for the organism other than

sinking speed reduction, such as predator defense or as a coagulation mechanism.

2.3.5 Models for mucilage

Mucilage can be modeled as a layer on the exterior of the frustule following

Hutchinson (1967). The resulting expression for cell density is:

*3 $ tm ' # tot" = #m +& 1+ ) (#tot * #m) (2-40) % r (

where tm is the thickness of the mucilaginous coating around the cell, and ρm is the ! density of the mucilage. Although ρm is not well characterized, it is probably slightly

denser than seawater (pers. com. Villareal). Although such mucilage will without a doubt

reduce the overall density of the organism thereby reducing sinking speed, it also

increases the size of the cell, which increases sinking speed. The resulting expressing for

sinking, given that the cell is now slightly bigger, is (Hutchinson 1967):

# )3 & # tm & 2 2% "m +% 1+ ( ("tot ) "m) ) "w ( g(r + tm ) $ $ r ' ' U = (2-41) 9µ

The result is that even for mucilaginous layers up to 20% of the cell diameter, there is a

! relatively small effect (<15%) on predicted sinking speeds of the diatoms. This is

particularly true for large diatoms due to their relatively low surface area to volume ratio.

Anything which thinly coats the outside of a large cell area is by geometry a small

proportion of the total cell volume. Additionally, mucilage does not significantly affect

the shape of the cell, and so the form resistance term should remain unchanged. The

conclusion for mucilage is similar to that for spines and processes: there is little ultimate

effect of mucilage on sinking speed, meaning that it’s not worth incorporating in

58 predictive models of diatom sinking. Additionally, mucilage probably has some

evolutionary importance other than modifying sinking speed, such as promoting

coagulation.

2.3.6 Models for linear chain-forming diatoms

The density of a chain of linear diatoms is essentially the same as the density of

any individual cell in the chain, with the minor caveat that spacing between the cells is

filled with seawater and connecting filaments (Fig. 2-10). At low Reynolds numbers, the

connecting filaments can entrain seawater, which could decrease the overall density of

the chain somewhat. If ρtot is the density of the cells in the chain, sv is the spacing

between the valves, and l is the length of an individual cell in the direction of the chain,

then the density of the chain is:

m c (m tot + m w )n "totVtot "wVg "ch = = = + (2-42) Vc (Vtot +Vg )n Vtot +Vg Vtot +Vg

" " " = tot + w (2-43) ch s l ! 1+ v 1+ l sv

where mw is the mass of water in the gaps between cells, and Vg is the volume of water in

! the gaps between cells. To get a 20% reduction in sinking speed, the excess density of the

chains has to be 20% less than the excess density of a single cell. Thus, we would need

the following equation to hold:

" " tot + w # " s l w 1+ v 1+ " # " l s 0.8 < ch w = v (2-44) "tot # "w "tot # "w

For typical densities used in Chapter 1, this equation holds true only when sv/l >

! 0.25, or when the spacing is at least 25% of the length of the cell, which is true only for

59 some diatom chains (Round 1990). When considering such chain-forming diatoms, the

density reduction due to cell spacing should be taken into account. For the purposes of

this section, we will consider the more common chain forming species with sv/l > 0.25

such that the density of the chain is functionally the same as the density of a single cell in

the chain.

If we treat the chain of n cells as a long, slim cylinder, then Φ can be

approximated as (Berg 1983):

4 " = (2-45) par nl 1 9(ln( r )# 2 )

8 " = (2-46) perp 9(ln( nl ) + 1 ) ! r 2 where r is the chain cylinder radius, and the subscripts par and perp refer to a cylinder

! with its long axis oriented parallel or perpendicular to the direction of sinking,

respectively, and n is the number of cells in the chain (Fig. 2-10). Alternatively, nl can be

replace by the total length of the chain, if cells overlap or other constraints make such a

substitution more convenient. Note that this expression is valid only where the length of

the cylinder, nl, is much greater than the radius of the cylinder.

Inserting Eqn. 2-46 into Eqn. 2-1 derives the sinking speed of a perpendicularly

oriented diatom chain:

2 ("tot # "water )gr nl 1 U = (ln( r ) + 2) (2-47) 4µ

For a given chain radius r, making a chain longer, that is, increasing l, will always make

! the chain sink faster. This theoretical prediction has been empirically demonstrated

(Smayda 1970), but one must be careful in thinking about the functional significance of

60 chains. If one compares ten diatoms sinking individually and the same ten diatoms

arranged into a chain, the chain sinking speed will always be greater than the individual

sinking speed. On the other hand, if one compares ten diatoms arranged in a chain to a

single spherical diatom that has a volume equal to that of ten small chain diatoms, it is

possible that the spherical diatom may sink faster than the chain. Thus, diatom chains can

be considered as a way to increase “individual” biomass without significantly increasing

sinking speed, a point first noted by Walsby and Reynolds (1980). To additionally

complicate this issue, I also note that a chain of diatoms has a greatly increased surface

area to volume ratio relative to the equivalent volume spherical diatom, which would

increase total cell density due to a larger contribution of frustule to overall density.

Therefore, the approach offered in this thesis can add a new element to this debate.

Let’s arrange n spherical diatoms in a linear chain (Fig. 2-10). The diatoms have a

radius r such that the cylinder also has a radius r, and has a length of 2nr. The sinking

speed expression can then be described by Eq. 2-47, with 2r substituted for l:

2 ("tot # "water )gr 1 U = (ln(2n) + 2) (2-48) 4µ

Let’s also create a spherical diatom of equivalent volume to the diatom chain, with radius

! rʹ, density ρʹtot and sinking speed Uʹ. The sinking speed of this diatom follows Eqn. 1-3:

(# " $ # )gr " 2 U " = tot water (2-49) 4µ

Since the volume of the large diatom, Vʹ, is equal to the volume of n chain diatoms each

! with volume V, we can solve for rʹ in terms of r:

V " = nV (2-50)

! 61 4 4 "r # 3 = n "r 3 (2-51) 3 3

1 r " = n 3 r (2-52)

! We desire to know if the diatom chain sinks slower than the single large diatom,

! so we will take the ratio of their sinking speeds:

1 U 9(ln(2n) + 2) ($tot % $water ) = 2 # (2-53) 3 U " 8n ($ tot" % $water )

For ρtot and ρʹtot, we will use the approximation for a large spherical diatom with a

! constant frustule thickness (Eq. 1-11) and Eq. 2-52:

3t 1 ($cyt % $water + $ fr ) U 9(ln(2n) + 2) r = 2 # (2-54) U " 8n 3 3t ($cyt % $water + $ fr 1 ) n 3 r

If this ratio is less than 1, the chain sinks slower than the large spherical diatom. The ratio

! depends on two variables, n, the number of cells in the chain, and t/r, the thickness of the

frustule relative to the radius. Table 2-1 shows this ratio against these two variables. A

minimal cell length of n=10 is used since this guarantees that the length of cylinder is an

order of magnitude longer than the cylinder radius, thus making the use of Eqn. 2-46

appropriate. Also t/r was set to a maximum of 0.01, the maximal value for which Eqn. 1-

11 is appropriate. Also note that because thickness scales allometrically with size, t/r is

also related to cell size, meaning that the relatively thin cells are big cells and the

relatively thick cells are small cells.

In the first column, t/r=0 to show what would happen if the frustule was absent, or

if the density of the chain and single large cell were equal (Table 2-1). This is the case

that Walsby and Reynolds (1980) argued for, and is also applicable to non-diatomaceous

62 chain forming species. When t/r=0, the chain forming cells do have a relatively moderate to large reduction in sinking speed compared to their spherically equivalent counterparts, but the story is not so simple for diatoms with a frustule. Chains of diatoms sink slower than large equivalent volume cells only when the chain is long and the frustule is relatively thin (i.e. t/r is small, Table 2-1). Therefore the only situation in which chain formation can be considered to have a relatively moderate to large effect in reducing sinking speed is in the case of large diatoms linked in long chains, a rare situation in diatoms (Round 1990). For most diatom chains, chain-formation produces a relatively small to moderate effect on sinking speed. For chain formation to be considered an adaptation for sinking speed reduction, one would need to show that changes to sinking speed of less than 50% can increase the fitness of diatoms. But it should be noted that for many diatom species, chain formation would actually have no effect on or increase their sinking speed relative to their spherical equivalent counterpart. For this reason, it is difficult to consider chain formation as an adaptation for sinking speed reduction.

One caveat should be added to the preceding point- the analysis above assumed that frustule thickness of the chain diatoms and the large diatoms was equal, whereas

Chapter 1 found an allometric relationship between thickness and radius. Smaller diatoms have slightly thinner frustules, which would give the smaller chain forming diatom a slightly reduced sinking speed. However, at this point it is unclear if constant thickness or an allometrically defined thickness gives a more appropriate comparison between a single large cell and a chain of smaller cells. A better understanding of the functional importance of the allometric relationship could explain why the allometric relationship exists and whether it would be appropriate to use it in this comparison. In any case, this

63 effect would be most prominent for the longest chains of diatoms, where the size difference is greatest between a single large cell and a chain. Considering that most diatoms do not form chains of this length, this is likely to be a small adjustment for most chain-forming species.

2.3.7 Discussion

At this point, one may ask how the various shapes modeled in this section affect the sinking speed across size. In other words, what does it mean for a diatom to increase in size according to different shapes? To answer this question, I will go through a couple of the more informative cases presented in section 2.3. Note that for all models, frustule thickness is set to a constant value of 1 µm for consistency.

The elongate diatom model was shown in Fig. 2.7, and essentially models diatoms as a cylinder that increases in length. The elliptical model for pennate diatoms can model the same situation, except that the diatom is now an ellipse with a long radius that increases and a constant short radius. This model converges with the elongate diatom model at large sizes, but exhibits a relatively small increase in sinking speeds at smaller sizes (<15%) due to the fact that cylinders have slightly greater surface area and therefore density for a given rl and rs than ellipsoids (Fig. 2-11).

We can also model a moderately elongate diatom which does not change in shape across size (say, by setting 10rs = rl ) using the elliptical model. In this case, at large sizes

(>104 µm3) the moderately elongate diatom shows a relatively large increase in sinking speed, compared to either an elliptical model with a fixed rs or even a spherical model

(Fig. 2-11). The reasoning is simple: such a shape is not elongate enough to have a large form resistance, but is elongate enough to have increased surface area to volume ratios

64 relative to a sphere, and thus increased density. It is interesting to note that for a given volume, a sphere is the shape that minimizes surface area to volume ratios, so the lightest densities and slowest sinking speeds are best achieved by assuming as close to a spherical shape as possible. It is interesting to note that enlarged diatoms do tend to have either subspherical shapes which minimize density or extremely elongate shapes which maximize form resistance. Ethmodiscus rex, the largest diatom by volume, is a short, fat cylinder, and the diatom genus with the longest valves (Trichotaxon) has a cylindrical shape with a relatively small radius (Round 1990). In contrast, many diatoms in the small

6 3 to medium size range (<10 µm ) do have a shape approximating rl = 10rs (see Chapter

3).

The three final models in Fig. 2.11 show little difference from the simple spherical model. These models include triangular valves (with dimensions close to

Ditylum brightwellii, Round 1990), spines, and mucilage. Whatever benefits these shapes confer to the diatom, if any, must be unrelated to sinking speed, since all of these shapes confer a relatively small to moderate adjustment to sinking speed compared to a spherical shape. Spines and mucilage extend from the exterior of a diatom, and thus may have a function in allowing a diatom to interact with its immediate physical environment. But the triangular shape is very similar to a sphere from a physical standpoint (as was a short, fat cylinder). Why a diatom should develop into a short, fat triangular cylinder rather than a short, fat round cylinder remains to be established.

65 Table 2-1

The ratio of chain sinking speed to an equivalent volume spherical diatom is shown against two variables, n and t/r (see Eq. 2-54). Values less than 1, for which the chain diatoms sink slower, are underlined. The last column (t/r=0) represents a cell with no frustule, or a non-diatomaceous plankton. The other columns apply to diatoms with some frustule thickness.

t/r 1/10 1/15 1/20 1/30 1/50 1/100 0 10 1.54 1.45 1.38 1.29 1.17 1.04 0.85 12 1.50 1.40 1.33 1.23 1.11 0.98 0.79 15 1.44 1.34 1.27 1.16 1.04 0.91 0.72 20 1.36 1.26 1.18 1.07 0.95 0.82 0.64 n 25 1.30 1.19 1.11 1.00 0.88 0.75 0.58 30 1.24 1.13 1.05 0.94 0.82 0.70 0.54 40 1.16 1.04 0.96 0.85 0.74 0.62 0.47 50 1.09 0.98 0.90 0.79 0.68 0.56 0.42 100 0.89 0.78 0.70 0.60 0.51 0.41 0.30

66 Figure 2-1

The spherical and cylindrical models at constant and variable parameters.

67 Figure 2-2

The spherical and elliptical models at constant and variable parameters.

68 Figure 2-3

This figure shows a model that incorporates a vacuole into the variable thickness spherical model. Each line is a shift in vacuole density of 1 kg m-3, with the range of density from 1015-1025 kg m-3 shown by all the solid lines. For comparison, the variable thickness model without a vacuole is shown by a dashed line for comparison. These predictions are for the constant cytoplasm volume model.

69 Figure 2-4

Similar to Fig. 2-3, but these predictions are for the constant cytoplasm area model.

70 Figure 2.5

Valve (A) and girdle (B) view of the elongate cuboid diatom shape described in section

2.3.1.

71 Figure 2.6

Valve (A) and girdle (B) view of the elongate cylindrical diatom shape described in section 2.3.1. Also note that here, rl is MAL (the major axis length).

72 Figure 2-7

This figure shows the data from chapter 1 along with the anomalous data from Waite and

Nodder (2001). Notice that the predictions derived for the anomalous shape of these diatoms (dashed grey line, Eq. 2-31), also accurately describes the sinking speed of these diatoms, especially compared to a spherical model (dotted line, Eq. 1-10).

73 Figure 2.8

Valve (A) and girdle (B) view of the elliptical diatom shape described in section 2.3.2.

74 Figure 2.9

Valve (A) and girdle (B) view of the triangular diatom shape described in section 2.3.3.

75 Figure 2.10

Valve (A) and girdle (B) view of the diatom chains described in section 2.3.6. In this figure, the diatoms are modeled as spheres, but this does not necessarily need to be true, meaning l does not have to be equal to 2r.

76 Figure 2-11

Shows the various diatom shapes across size on one plot. Lines were extended to the cell volume that is reasonable for a diatom of that shape, where appropriate. Where expressions for sinking speed were given in the text, equations were listed. Otherwise, just the section was listed. See text for more details on the parameters used in the equations.

77 Chapter 3

Defining empirical scaling laws for frustule thickness

3.1 Introduction

Besides shape, the most important potential influence on the sinking speed predictions in Chapter 1 is the scaling of frustule thickness. Whereas shape often either speeds or slows the sinking speed of diatoms by a set fraction across size (through form resistance), thickness actually changes the sinking speed relationship across size.

Therefore, frustule thickness could play an important evolutionary role when considering the functional importance of size and sinking speed.

The intent of this section is to further empirically define the scaling relationship between thickness and size, and to test evolutionary hypotheses about the functional importance of this relationship. This will be done with a meta-analysis, by collecting size and thickness information from the extensive literature on diatoms.

The first question I wish to address concerns diatom size. Based on Chapter 1, it seems sensible to assume that thickness increases allometrically with size, but size is a general term. With what measure of size does thickness seem to increase? To answer this question I look at the relationship between thickness and several measures of size.

The second question concerns potential evolutionary constraints on scaling of thickness across size. I present two hypotheses related to this question.

First, I hypothesize that the allometric scaling of thickness across size serves to beneficially adjust the sinking of diatoms across size. Since thickness increases with size at less than isometry, this means that sinking speeds are more uniform across size than

78 might be the case given isometry. If there is an optimal sinking speed for diatoms, then this sort of scaling might best allow all diatoms, regardless of their size, to approach that sinking rate. This hypotheses can be easily tested- if the function of the allometric scaling of thickness is to keep sinking speed constant, then we should expect to see this scaling law in planktonic diatoms that sink, but not in benthic diatoms. In contrast, benthic diatoms should either show no scaling, or exhibit isometry (depending on what might be produced by a lack of constraints).

Second, I hypothesize that the allometric scaling of thickness across size might serve to structurally reinforce the diatoms as they get bigger. As diatoms get bigger, they could be crushed by predators or turbulence or other physical forces, if they did not also get thicker. Pondaven (2007) finds that a species of diatoms thickens its frustule in the presence of herbivores, so it may be sensible to consider that scaling across size is related to defense. If, for example, it was beneficial for diatoms to maintain a constant structural safety factor across size, one would expect diatoms to get thicker as they got bigger, but probably not isometrically. The exact scaling of thickness would depend on both how resistance to breaking and breaking force scale with size. Determining such forces is an empirical and modeling project unto itself, and is beyond the scope of the present analysis. Here, we can begin to test this hypothesis by comparing the thickness scaling of diatoms of different shapes (symmetric vs. asymmetric) and configurations (chains vs. solitary). In both these comparisons, an increase in size will cause a different shape in the two groups. And different shapes have different structural integrity, which would require different scaling laws with size.

79 3.2 Methods

Data were compiled from anthology collections of SEM images of diatom frustules, which are usually taxonomically based. A list of the sources is given in Table

3-1.

Measurements were taken from plates on which valve thickness and valve diameter (long and short, if appropriate) were clearly visible. Species were only included if there was a good picture of the frustule that allowed for its thickness to be calculated, and a scale bar was present. The valve diameter often could not be calculated on the same plate as frustule thickness, and so in these cases valve diameter was measured on a different plate. When possible, I tried to use plates that featured the same frustule in high and low magnification allowing both valve diameter and frustule thickness measurements on the same individual. When this was not possible, I used the diameter of another individual in the same species from the same source material. For Trichotaxon reinboldtii, the long diameter was too large to be measured by SEM, so I used a value typical for this species of 1500 µm (Semina 2003).

Information was also collected about the diatom species, habitat, and presence in chains. The diatom anthologies used for the frustule measurements tend to lack species- specific ecological information. Information about habitat and chain formation, if not given in the anthology, was characterized from Round (1990) using information about the genus. If the genus was said to live in multiple habitats or found in chains and solitarily, those species were excluded from the analysis. Habitat information was either taken from the anthology, or the data were not used in the analysis.

80 When transforming from cell diameter to cell volume, I assumed simple shapes

for the diatoms. For the symmetric centric diatoms (termed symmetric due to their radial

symmetry), the diatom’s height was set equal to its diameter (d), such that the diatom was

modeled as a short, fat cylinder.

"d 3 V = (3-1) tot 4

where Vtot is the total volume of the cell, and d is valve diameter. For pennate diatoms

! and especially asymmetric centric diatoms (together termed asymmetric diatoms), I

assumed an ellipsoid with the radius of the pervalvar axis being equal to the short valve

diameter or the diatom (thus producing an elongate spheroid):

"d 2d V = s l (3-2) tot 6

where ds is the short diameter, and dl is the long diameter.

! All regressions were linear regressions performed on the log-transformed data.

Confidence intervals were calculated from the standard errors of the parameters given by

the lm function in the programming language R.

3.3 Results

3.3.1 Symmetric and asymmetric diatoms

Frustule thickness is plotted against valve diameter in Fig. 3-1, for both

symmetric and asymmetric diatoms. Notice that the symmetric diatoms tend on the whole

to be larger than the asymmetric diatoms. Also note that symmetric and asymmetric

diatoms by short diameter seem to follow a very similar scaling law or slope (Table 3-2),

but the curves are shifted relative to each other on the axis of valve thickness. The

asymmetric diatoms by long diameter show a different but statistically indistinguishable

81 scaling law from the other two comparisons (Table 3-2). The scaling law of asymmetric diatoms by long diameter is actually statistically indistinguishable from no trend. Notice that the farthest right point (largest diameter diatom species) seems to form an outlier to this trend. This point is Trichotaxon reinboldtii, and if this point is removed, the slope is

0.162 +/- 0.32 for asymmetric diatoms by long diameter, still quite different from the scaling laws by short diameter and for symmetric diatoms (Table 3-2), but still indistinguishable from no trend.

If long diameter of asymmetric diatoms is compared to the diameter of symmetric diatoms, we note that for a given diameter, the asymmetric diatoms seem to have thinner frustules than the symmetric diatoms. Yet we know that an asymmetric diatom with a long diameter equivalent to the diameter of a symmetric diatom is much smaller by volume than the symmetric diatom.

For this reason, it seems sensible to compare thickness against cell volume. When the data were transformed to cell volume in Fig. 3-2, the points now all fall along a similar line. The exponents for symmetric and asymmetric diatoms are no longer significantly different from each other, such that it is reasonable to fit all of the points to one line. The scaling exponent between thickness and cell volume for all diatoms is 0.18

+/- 0.05 (Value +/- 95% CI).

3.3.2 Solitary and chain-forming diatoms

Thickness is plotted against cell and chain volume in Fig. 3-3. Notice that only smaller species tend to be chain forming species, and that no statistically significant difference exists between the scaling laws for these two groups (Table 3-2), even though

82 the chains have a lower scaling exponent based on these limited data. The scaling law for the chain-forming species is also not statistically different from no trend.

3.3.3 Benthic and Planktonic

Thickness is plotted against cell volume in Fig. 3-4 for both benthic and planktonic diatoms. Notice that large diatoms are only planktonic, although small diatoms can be either benthic or planktonic. Also notice that no significant difference is found in the scaling laws between benthic and planktonic diatoms, although both are significantly different from no trend (Table 3-2). The scaling exponents are quite similar, although the benthic diatoms are shifted towards thinner frustules.

3.4 Discussion

3.4.1 Correlation with previous scaling laws

The scaling exponents derived in this section for symmetric diatoms as a function of diameter (0.41) show good agreement with the exponent used in Chapter 1 (0.45), indicating that Chapter 1 was using an accurate formulation for frustule thickness. Note that slightly different scaling exponents were found for asymmetric diatoms (notable 0.52 by ds), but this difference disappeared when transforming to cell volume. Notice that if we transform the volumetric scaling exponent derived for all diatoms (0.180) to an exponent for equivalent diameter, we get 0.54 +/- 0.15, which is again similar to our previous measurements. Thus, we can generally conclude that thickness seems to scale with radius with an exponent somewhere between 0.41 and 0.54.

Several studies have examined the scaling of silica content (mass of silica per cell) with cell size, and have found varying results. If frustule thickness is constant, silica content should increase as V2/3 (the scaling exponent, a, is 2/3). Brzezinski (1985) found a

83 lack of pattern: silicon to carbon ratios were uncorrelated to surface area to volume ratios,

as expected if silicon follows a frustule-like scaling and carbon follows a total cell

volume scaling. Conley et al. (1989) did find a trend: silica content scaled directly with

cell volume for freshwater diatoms (a scaling exponent of a=1), but was slightly

allometric for marine diatoms (a=0.91). In contrast, Sicko-Goad et al. (1984) found a

scaling exponent of a=0.75 between frustule volume and cell volume, whereas Reynolds

(1986) finds a scaling exponent of a=0.71 between silica content and cell volume.

Assuming that the frustule covers the diatom as a surface area with thickness t, and Si has

a scaling exponent a with volume:

Si "V a (3-3)

tr 2 " r 3a (3-4) ! t " r 3a#2 (3-5)

! 1 In this case, Conley et al.’s results indicate that for freshwater diatoms, t∝r , and for

! 0.73 0.25 marine diatoms t∝r . The results of Sicko-Goad et al. yield t∝r , and those of

Reynolds, t∝r0.13. Notice first that small deviations in the silica relationship lead to large

variation in the thickness scaling relationship, due to the nature of Eq. 3-5. The silica

scaling relationships found in the literature include almost the entire range of scaling

possibilities, from isometry (a=1) to constant (a=0). Conley et al.’s analysis not only

includes the most data, but it includes the data from Sicko-Goad et al. and Reynolds, and

therefore is the most complete and potentially trustworthy data set. But there are several

complications that can arise when fitting curves to scaling relationships (Schmidt-Nielson

1984), and Conley et al. do not specify the type of statistics they used to get their scaling

parameters. Thus even their data must be taken with a grain of salt. At the very least, the

84 silica content literature encompasses the range of thickness scaling exponents found in

this section and Chapter 1, offering some further support for the relationship given here.

More data and better analysis are needed to compare the scaling exponents derived for

silica content and thickness.

3.4.2 Correlation between thickness and silica content

Although to my knowledge there have been no attempts to measure or

characterize frustule thickness as I have done, there are many studies that have

characterized silica content. To compare my frustule thickness measurements to silica

content, I can estimate frustule thickness given some measurement of silica surface

density (σSi), where σSi is the mass of silica per surface area:

" Si Afr = #SiVfr (3-6)

where Afr and Vfr are the surface area and volume of the frustule, respectively, and ρSi is

! the density of amorphous silica. Assuming that the frustule covers the outside of the cell

as a thin layer:

" Si Afr # $Si (AfrtSi ) (3-7)

# t " Si (3-8) Si $ ! Si

where tSi is the thickness of a layer of pure silica. Using the density of amorphous silica

! -3 -5 -2 (SiO2), ρfr = 2600 kg m , and a range of surface densities, σ = 0.24 – 50.0 x 10 kg m

of silica (Paasche 1980), the approximate thickness of the frustule’s silica layer is

between 0.0009 and 0.19 µm. The range in surface density is mostly due to cell size

(Conley et al. 1989), such that the range of calculated thickness can be considered the

thickness variation across size, comparable to that measured in Chapter 1 (Table 1-1).

The calculated range is clearly smaller than the thickness range measured in Chapter 1

85 (0.05 – 2.32 µm), but I measured the thickness of the entire frustule, not a layer of just

pure silica. The solid part of the frustule is between 10% and 70% silica; the remainder

consists of organic components (Schmid et al. 1981, Swift 1992). Thus I need to apply

several correction factors to get to the thickness measured in this study, as follows:

# 1 t " Si (3-9) $fr n nonporen inorg

where nnonpore is the portion of nonporous surface (0.80 on average, pers. obs.) and ninorg

! is the portion of inorganic volume. The amount of hydration in the silica is included as

organic material in determining the value for ninorg. The portion of inorganic mass in the

frustule is 0.10 – 0.70 (Schmid et al. 1981); we will use an average value of 0.40. To

calculate the portion of inorganic material by volume (rather than by mass), we assume

that the organic matter has a density of 1300 kg m-3, about half that of the inorganic

matter (silica, 2600 kg m-3). In this case, the 0.40 inorganic matter by mass is about 0.25

inorganic matter by volume.

The thickness in Eq. 3-9 should correspond to the thickness of a frustule with

hydrated silica, organic matter and pores, i.e. the thickness measured in this thesis. The

calculated thickness range is now 0.0045 – 0.95 µm, whereas my measured range is 0.05

– 2.32 µm (Table 1-1). Although some discrepancy still exists, the diatoms analyzed in

Brzezinski (1985) were mostly small ( r < 10 µm), with the majority of the species

producing a calculated thickness range of 0.0045 – 0.23 µm. The one species in my study

for which r < 10 µm, M. variabilis, has a thickness in this range (t = 0.05 µm). The upper

edge of the calculated thickness range in Brzezinski (1985) was primarily determined by

one species, Coscinodiscus granii (r = 30 µm). Based on my measured thicknesses (Table

86 1-1), the calculated thickness of 0.95 µm for C. granii is reasonable for a diatom with r =

30 µm (for C. radiatus 312, t = 0.85 – 1.65 µm).

In summary, there is good correlation between measured frustule thickness and silica content, once all relevant adjustments are taken into account, justifying the use of frustule thickness rather than the more commonly measured silica content.

3.4.3 Evolutionary implications of frustule geometry

I presented two hypotheses for the allometric scaling of frustule thickness: first that it is an adaptation to adjust sinking speed across size, and second, that it is related to maintaining a constant structural integrity across size. This first hypothesis has no support, whereas there is some support for the second hypothesis.

The first hypothesis is invalidated by comparing the sinking of benthic to planktonic diatoms. If the allometric scaling of thickness was an adaptation to adjust sinking speed across size, then we should expect to see the allometric scaling only in planktonic diatoms. Instead, benthic and planktonic diatoms share a statistically indistinguishable scaling law between frustule thickness and cell volume (Fig. 3-4). Most confusingly, the benthic diatoms have thinner frustules, which would result in slower sinking as compared to the planktonic diatoms. The implication is that planktonic diatoms benefit from thicker frustules, and therefore greater sinking rates. The only sensible conclusion seems to be that sinking is not related to the scaling of thickness across size.

The second hypothesis is somewhat supported in this study. We have noted that the scaling law for asymmetric diatoms by long diameter is quite different from the scaling law for asymmetric diatoms by short diameter or that for symmetric diatoms. If

87 we envision, for a moment, that frustule thickness is for structural reinforcement, then the frustule ellipsoid is most prone to mechanical failure by forces imposed along its transapical plane (Fig. 0-2, radial forces imposed partway down the long diameter axis of the valve), which is also the plane along which predators might crush a diatom. If we treat these elongate diatoms as cylinders, then as the small diameter or radius of the cylinder increases, the frustule will become structurally weaker unless the thickness of the frustule is accordingly increased. To maintain a constant safety factor across size, we would expect frustule thickness to increase with short diameter. By this reasoning, we should also expect the long diameter to be unrelated to the thickness of the diatom. Both of these predictions are held up by the data (Table 3-2).

An argument against this hypothesis is that once the transformation to volume occurs, the asymmetric and symmetric diatoms fall along a similar scaling law, regardless of their shape. This suggests that volume is a more important consideration than shape, and that thickness is related to something to do with balancing cell volume and frustule content, like buoyancy control (although buoyancy control was ruled out by the first hypothesis).

Through a similar argument, we might expect chains and solitary diatoms to have different functional constraints, such that diatoms in chains might be more structurally reinforced for their size than solitary diatoms. Although a difference is seen in their scaling laws, it is not statistically significant and there is a clear size preference for each group; the solitary diatoms are larger than the chain-forming diatoms (Fig. 3-3). It is unclear if this difference in scaling laws is due to the size difference of the diatoms or to

88 their solitary vs. communal nature, or if it is simply due to a small sample size. In any case, the evidence for the structural hypothesis can only be treated as tentative here.

Most interestingly for the structural hypothesis is the benthic vs. planktonic comparison. If predators cause structural failure, one might expect to see a different subset of predators in benthic vs. planktonic communities. In this case, we might expect to see similar scaling laws across size in both groups, so that a constant safety factor is maintained, but the scaling laws to be shifted with respect to each other to account for generally weaker predators in one of the communities (as in Fig. 3-4). This is of course highly speculative and in need of predator strength measurements, both in different communities and across size.

In any case, the tentative evidence for the mechanical constraint hypothesis, coupled with the literature on frustule thickness responding to herbivores (Pondaven

2007) is impetus for further analysis on how structural constraints and forces involved in herbivory may scale with size.

A third possibility exists, that thickness is simply a by-product of the differential size of diatoms, and has no functional significance. I have two arguments against this.

First, the scaling pattern is distinctly allometric, which suggests it is not simply a by- product of size as an isometric pattern might be. Second, note a curiosity- through my direct measurements of frustule thickness in Chapter 1, I noticed that within a species frustule thickness remained constant across size, even though many species spanned a significant range of sizes. The constancy of frustule thickness was particularly striking in one species analyzed by SEM but not used in sinking speed trials: Coscinodiscus asteromphalus (CCMP 1814). This species underwent sexual reproduction during

89 transport from the CCMP, causing almost a five fold increase in valve diameter (15 µm to 70 µm) with no apparent change in frustule thickness (~ 1 µm). Based on the allometric scaling law between species (Table 1-2), a five-fold increase in valve diameter should correspond to a doubling of valve thickness. Also of note, the valve and girdle height was constant across this size range.

The constancy of frustule parameters within a species seems to indicate that some features of the frustule design are hardwired in some fashion, and do not simply change because the diameter of the diatom changes. A different diatom species, though, that occupies a different size range, will have a different thickness. For this reason, I think thickness is actually an adaptation of some sort to living in a different size range, and not simply a consequence of building a frustule of a different size.

3.4.4 Implications for sinking speed predictions

Although sinking speed does not seem to be the functional cause behind the allometric scaling of frustule thickness, thickness still does have a significant effect on sinking speed. As noted, the scaling laws in this section are similar to the scaling laws found using only six species in Chapter 1 (shown in grey circles in Fig. 3-1). This confirms that the allometric scaling law for frustule thickness is a general feature of all diatoms, and should be taken into account when predicting sinking speed.

Most interestingly, a single scaling law can be derived across symmetric and asymmetric diatoms when plotting thickness against cell volume. This indicates that it might be more appropriate to use cell volume rather than diameter in sinking speed models. Most models already use equivalent diameter in their models, which is the diameter of the shape if the volume were arranged into a sphere, which is essentially the

90 same as using cell volume in the model. This analysis indicates that this transformation is not only convenient but potentially more appropriate when considering how thickness scales across size.

91 Table 3-1

Data used in the meta-analysis with sources. All units are in µm. Legend: S- solitary, C- chain forming, SC- mostly solitary, also chain forming, CS- mostly chain forming, also solitary, B- benthic, P-planktonic, BP-benthic and planktonic.

Species Chains Type Habitat Shape dl ds tv Source Fragilaria sp. C Araphid B Cylindrical 22.0 8.5 0.38 Choi et al. 2008 Fragilaria striatula C Araphid B Cylindrical 32.0 8.0 0.40 Choi et al. 2008 Stellarima microtrias S Centric P Simple 18.2 0.40 Choi et al. 2008 Coscinodiscus alboranii S Centric P Simple 89.0 0.78 Ferrario et al. 2008 Coscinodiscus asteromphalus S Centric P Simple 100.0 1.50 Ferrario et al. 2008 Coscinodiscus bouvet S Centric P Simple 210.0 2.40 Ferrario et al. 2008 Coscinodiscus granii S Centric P Simple 68.0 0.67 Ferrario et al. 2008 Coscinodiscus janischii S Centric P Simple 180.0 1.75 Ferrario et al. 2008 Coscinodiscus jonesianus S Centric P Simple 116.0 0.20 Ferrario et al. 2008 Coscinodiscus oculoides S Centric P Simple 164.0 1.75 Ferrario et al. 2008 Coscinodiscus radiatus S Centric P Simple 50.0 1.25 Ferrario et al. 2008 Coscinodiscus wailesii S Centric P Simple 230.0 1.75 Ferrario et al. 2008 Parlibellus phoebeae C Raphid B Cylindrical 13.9 2.9 0.18 Hein et al. 2008 Amphora paraveneta S Raphid B Elliptical 59.2 8.5 0.25 Lange-Bertolot et al. 2003 Hantzschia anguis S Raphid B Cylindrical 201.7 13.3 0.25 Lange-Bertolot et al. 2003 Hantzschia obtusa S Raphid B Cylindrical 101.2 15.1 0.22 Lange-Bertolot et al. 2003 Nitzschia andicola SC Raphid BP Cylindrical 82.0 3.0 0.15 Lange-Bertolot et al. 2003 Stauroneis acuta C Raphid B Elliptical 83.3 14.7 0.88 Lange-Bertolot et al. 2003 Stauroneis amphibia S Raphid B Elliptical 30.7 7.8 0.32 Lange-Bertolot et al. 2003 Stauroneis hyperborea S Raphid B Elliptical 49.5 10.0 0.69 Lange-Bertolot et al. 2003 Cymbella cucumis SC Raphid B Elliptical 84.7 27.3 0.34 Metzeltin 2002 Encyonopsis mantasoana C Raphid B Elliptical 50.5 6.1 0.27 Metzeltin 2002 Eunotia bilunaris var. mucophila SC Raphid B Elliptical 20.7 2.2 0.12 Metzeltin 2002 Eunotia desmogonioides SC Raphid B Elliptical 124.0 5.6 0.31 Metzeltin 2002 Eunotia rhomboidea SC Raphid B Elliptical 15.5 3.8 0.19 Metzeltin 2002 Melosira varians C Centric B Simple 23.3 0.42 Metzeltin 2002 Navicula horstii S Raphid B Elliptical 57.6 9.1 0.42 Metzeltin 2002 Neidium iridis var. amphigomphus S Raphid B Elliptical 67.5 21.8 0.28 Metzeltin 2002 Peronia fibula S Raphid B Cylindrical 26.7 5.1 0.14 Metzeltin 2002 Stauroneis madagascariensis S Raphid B Elliptical 80.7 15.3 0.60 Metzeltin 2002 Surirella costei S Raphid B Elliptical 71.3 14.0 0.59 Metzeltin 2002 Surirella muscicola S Raphid B Elliptical 115.3 14.0 0.91 Metzeltin 2002 Actinocyclus exiguus S Centric P Simple 13.3 0.25 Semina 2003 Asteromphalus hoockeri S Centric P Simple 20.3 0.25 Semina 2003 Asteromphalus hyalinus S Centric P Simple 23.7 0.50 Semina 2003 Asteromphalus roperianus S Centric P Simple 96.3 1.25 Semina 2003

92 Azpeitia neocrenulata S Centric P Simple 19.3 0.75 Semina 2003 Azpeitia neocrenulata S Centric P Simple 27.3 0.80 Semina 2003 Azpeitia tabularis S Centric P Simple 42.5 1.10 Semina 2003 Coscinodiscus oculus-iridis S Centric P Simple 197.5 1.20 Semina 2003 Eucampia antartica C Centric P Elliptical 42.0 13.5 2.00 Semina 2003 Fragilariopsis curta C Raphid P Cylindrical 21.6 6.6 0.40 Semina 2003 Fragilariopsis cylindrus C Raphid P Cylindrical 25.0 3.0 0.30 Semina 2003 Fragilariopsis doliolus C Raphid P Cylindrical 60.0 6.8 0.50 Semina 2003 Fragilariopsis kerguelensis C Raphid P Cylindrical 21.8 10.2 0.67 Semina 2003 Fragilariopsis rhombica C Raphid P Cylindrical 20.7 13.5 0.50 Semina 2003 Fragilariopsis rhombica C Raphid P Cylindrical 35.3 16.3 0.50 Semina 2003 Fragilariopsis ritscheri C Raphid P Cylindrical 46.9 11.5 0.58 Semina 2003 Fragilariopsis sublinearis C Raphid P Cylindrical 52.5 7.5 0.65 Semina 2003 Fragilariopsis vancheurickii C Raphid P Cylindrical 29.3 6.0 0.67 Semina 2003 Nitzschia sicula var. bicuneata SC Raphid P Cylindrical 31.3 6.8 0.40 Semina 2003 Odontella aurita C Centric P Elliptical 47.5 19.5 1.33 Semina 2003 Odontella weissflogii C Centric P Elliptical 41.3 34.6 0.58 Semina 2003 Planktoniella sol S Centric P Simple 28.0 0.80 Semina 2003 Porosira glacialis C Centric P Simple 25.7 0.50 Semina 2003 Porosira pseudodenticulata C Centric P Simple 49.0 0.25 Semina 2003 Roperia tesselata S Centric P Simple 46.0 0.70 Semina 2003 Thalassionema nitxchioides C Araphid P Cylindrical 15.3 2.8 0.30 Semina 2003 Thalassiosira anguste- lineata CS Centric P Simple 61.4 1.10 Semina 2003 Thalassiosira antartica var. borealis CS Centric P Simple 33.1 0.70 Semina 2003 Thalassiosira cf. frenguelli CS Centric P Simple 20.5 2.00 Semina 2003 Thalassiosira echinata CS Centric P Simple 20.3 0.63 Semina 2003 Thalassiosira gracilis CS Centric P Simple 12.3 1.14 Semina 2003 Thalassiosira gravida CS Centric P Simple 28.5 0.92 Semina 2003 Thalassiosira ignota CS Centric P Simple 30.8 1.73 Semina 2003 Thalassiosira latimarginata CS Centric P Simple 51.9 2.19 Semina 2003 Thalassiosira lentiginosa CS Centric P Simple 47.8 1.00 Semina 2003 Thalassiosira lineata CS Centric P Simple 21.3 0.90 Semina 2003 Thalassiosira maculata CS Centric P Simple 78.0 5.00 Semina 2003 Thalassiosira oestrupii var. venrikae CS Centric P Simple 15.0 0.95 Semina 2003 Thalassiosira porroiregulata CS Centric P Simple 14.2 0.32 Semina 2003 Thalassiosira tumida CS Centric P Simple 96.5 2.67 Semina 2003 Trichotaxon reinboldii S Araphid P Cylindrical 1500.0 4.2 0.40 Semina 2003

93 Table 3-2

The results of the linear regressions for frustule thickness against frustule diameter in log space. The “Figure” column refers to the figure that contain the data and fit that the parameters shown here correspond to.

Group Figure Scaling Exponent df r2 Value +/- 95% CI All, by Vtot All 0.180 0.050 73 0.41

Symmetric, by d 1.5-1 0.411 0.259 37 0.20 Asymmetric, by ds 1.5-1 0.515 0.260 34 0.30 Asymmetric, by dl 1.5-1 0.087 0.244 34 0.01

Symmetric, by Vtot 1.5-2 0.137 0.086 37 0.20 Asymmetric, by Vtot 1.5-2 0.170 0.105 34 0.22

Chains 1.5-3 0.114 0.130 19 0.11 Solitary 1.5-3 0.188 0.067 30 0.51

Benthic 1.5-4 0.178 0.113 19 0.33 Planktonic 1.5-4 0.132 0.058 49 0.28

Figure 3-1

Frustule thickness against cell diameter, with the data separated by symmetric and asymmetric diatoms. Data is plotted twice for asymmetric diatoms, against their long and short valve diameters. The six species from chapter 1 are shown in grey for comparison, though their data is not included in the regressions.

95 Figure 3-2

Frustule thickness against approximate cell volume, with the data separated by symmetric and asymmetric diatoms.

96 Figure 3-3

Frustule thickness against approximate cell volume, with the data separated by solitary and chain-forming diatoms.

97 Figure 3-4

Frustule thickness against approximate cell volume, with the data separated by benthic and planktonic diatoms.

98 Chapter 4

Effect of diatom sinking speed on bloom formation

4.1 Introduction

Diatoms are the most productive group of eukaryotic phytoplankton, generating half of the ocean’s fixed carbon every year (Field et al. 1998, Nelson et al. 1995). For this reason, they are potentially one of the greatest contributors to the carbon cycle

(Ragueneau et al., 2000). Diatoms transport significant amounts of carbon and silicon to the deep ocean (Beaulieu and Smith 1998, Armbrust 2009). The rate of mineral transport is governed primarily by the maximal sinking speed of dead diatoms, either as individual cells or aggregates (Jackson 1990, Jackson and Kiorboe 2008, Ragueneau et al. 2000).

The process of aggregation is particularly important- due to the strongly size dependant nature of sinking speeds at low Reynolds numbers, aggregates can sequester carbon as much as 100 times faster than the same biomass of individual diatoms. When dissolution and consumption by herbivores are also considered, it seems even less likely that individual diatoms would reach the ocean bottom before their carbon was absorbed and recycled into the upper layers of the ocean. Even the fastest-sinking individual large diatoms sinking at their maximal recorded rate would take over 130 days to reach the bottom of a 4,000 m ocean basin. For smaller diatoms, this process could take several years. For this reason, aggregates have been suggested to be the main way that carbon from organisms like diatoms might ever get sequestered to the ocean floor (Smetacek

1985, Alldredge and Silver 1988).

99 Predicting these fluxes in current and future ocean conditions is of great importance. Such fluxes can be predicted in two ways, through indirect or direct models.

Mineral fluxes are most often calculated using indirect, inverse models in which sinking is parameterized to fit measured concentrations of diatoms in the ocean, rather than calculated directly from hydrodynamic principles (for a review, see Ragueneau et al.

2000). Although these measurements lead to accurate current predictions, they have no mechanistic underpinnings and so extrapolations to future, different ocean conditions are next to impossible.

Some studies have used direct models of diatom sinking speeds to estimate carbon fluxes. These models start with a mechanistic understanding of hydrodynamic principles, and often (as in Jackson 1990 or Jokulsdottir and Archer in prep) incorporate aggregation. A drawback of such models is that they are often for idealized, one or two dimensional oceans. Simulating a three-dimensional ocean with topography and currents is currently computationally unrealistic. This is not to say such models are unhelpful-

Jackson and Kiorboe (2008) has correlated his models with diatom bloom dynamics,

Jokulsdottir (pers. com.) has been able to recreate realistic depth-integrated carbon fluxes with her simple model, and Pondaven (1998) initiates her model with measurement in surface waters. In all these cases though, the sinking speed used in the models is based on a best guess from empirical data, and differences in sinking speed caused by diatom size are often ignored.

What we desire is a model somewhere in between direct and indirect modeling- one would prefer the mechanistic underpinning of a direct model with the specificity and accuracy of indirect models. Creating such a model is beyond the scope of this thesis.

100 What I wish to do in this section is help move direct models from a highly idealized treatment of diatoms to one that is more biologically accurate. To achieve this goal, I will incorporate the size-dependant sinking speeds developed in Chapter 1 into direct models of diatoms. I will then evaluate the extent to which size-dependant sinking speeds can improve the accuracy of direct models.

In this section I show the effect of incorporating the models derived in Chapter 1 into a simple carbon flux model. I take a simple aggregation model and determine the importance of size and size-dependant diatom density relative to other parameters in the model, thus determining how important it is to consider the corrections offered in Chapter

1 in models of carbon flux.

4.2 Methods

4.2.1 The model

I use a modification of the coagulation model from Jackson (1990). This model essentially simulates a monodisperse (single species) bloom. The model only uses base particles of one size. The model divides particles into logarithmically spaced size bins, the larger size bins corresponding to aggregations of smaller particles. This model is not lagrangian, meaning it does not track individual particles. It assumes particles are uniformly mixed throughout the mixed layer (the depth of which can be specified in the model). It then calculates the probability with which particles of various sizes are likely to coagulate with each other, and then moves that fraction of the particles into a larger size class. This model also ignores nutrient status and predator abundance, two factors than serve to “end” a diatom bloom; thus the blooms in this model do not end. The only factors at play in the model are growth, coagulation, and sinking; thus this model creates

101 an upper physical limit to the size of bloom events, which may be further limited by other nutrient-driven or predator-driven processes. Jackson and Kiorboe (2008) successfully used this model to predict the maximum possible concentration of diatoms in bloom events, and he has also attempted to incorporate nutrient (Jackson and Lockmann 1993) or herbivore effects (Jackson 2001) into his models, but such complications are beyond the scope of the present study. For more information on this model, refer to Jackson

(1990).

The model proceeds in essentially two stages, which will be referred to as the two states of the model. The first state involves exponential growth of the smallest size class of particles. Once concentrations get high enough, small particles coagulate into larger particles. Eventually, coagulation becomes dominate enough to produce large, fast sinking particles which leave the mixed layer at a rate that balances growth. This produces an equilibrium state with a stable concentration-size distribution, and it is the nature of this equilibrium state that is of most interest.

The model uses several different kernels to define its coagulation rates. The rectilinear kernel essentially considers particles to interact in a vacuum. In this case, particles intersect if the volume mapped by their sinking trajectory intersects another particle’s trajectory. The curvilinear kernel, on the other hand, assumes particles follow

Stokes’ flow type streamlines. In this kernel, if the particles intersect by following streamlines, they coagulate. The fractal kernel is a correction to the curvilinear kernel.

Since particle aggregates are fractal-like in nature, streamlines do not fully divert around aggregates. Instead aggregates are “leaky”, allowing some streamlines to flow through gaps in them. In the fractal kernel, particles still follow streamlines, but the streamlines

102 are adjusted for the leaky nature of aggregates when predicting coagulation rates. For more information on coagulation kernels, refer to Jackson (1990).

I added two intermediate-Reynolds-number corrections to the model: an intermediate Reynolds number term to the sinking speed expression (Brown and Lawler

2003) and an inertial-type kernel, as described in Humphries (2006). The inertial kernel is a correction for inertial effects to the basic curvilinear kernel. These effects can be present for Re<1, but are especially relevant for Re>1. In the inertial kernel, particles do not exactly follow Stokes’ flow streamlines, but instead exhibit some inertia, diverting the particles from their streamlines in some cases. A potentially more accurate kernel is one in which the inertial and fractal components are taken into account together, but since both derivations depend on empirically derived data, it is at this point unclear how to create such a kernel.

4.2.2 Parameter selection

As stated, this model produces a prediction, based primarily on physics, for what the maximal concentration of particles in a bloom should be. My interest was to determining the sensitivity of the model’s prediction to variations in parameters. In particular, I am interested in how various sinking speed models affect the prediction of the coagulation model in comparison to other factors that have been manipulated previously (Jackson 1990, Jackson 1995).

The main new parameter we varied in the model was sinking speed model across size. I also vary several physical and biological parameters that have been varied before: initial particle size, shear rate, growth rate, coagulation kernel, and number of size classes

(Jackson 1990, Jackson and Lochmann 1992), and intermediate Re effects both to the

103 coagulation kernel and the expression for sinking speed (Humphries 2009). Parameter values are given in Table 4-1. I created a default set of parameters, such that when looking for the effect of a parameter on the results of the model, only that parameter was varied while all other parameters (except for particle size and sinking speed model) were set to their default value. Particle size and density model vary through their entire range for each parameter- thus interactions between certain parameters and size/density model could be observed.

I vary particle size logarithmically. I start with a cell diameter of 20 µm, which is the typical default size used by Jackson (1990) in his models. I then increase and decrease this size by a factor of 2 and 4 to span a large range of cell sizes. I constrain the size range of individual particles to 80 µm maximally, since in Chapter 1 it was determined that the density models should not be applied to diatoms larger than 100 µm in diameter.

To vary the sinking speed model, I use 5 different size-dependant sinking speed models. Although called “sinking speed” models, I actually modify the size-dependant density term in Stokes’ law to vary sinking speed model. For comparison purposes, I kept the original model used by Jackson which is derived from data in Smayda (1970), called the Jackson-Smayda model (Fig. 4-1). For comparison purposes, I use three models from

Chapter 1 (Miklasz and Denny 2010) and label them as such: I use a density model assuming a constant frustule thickness (Eq. 1-10), one assuming an empirically derived allometric scaling to frustule thickness (Eq. 1-13), and one which assumes that density is constant across size, using a density value derived from the maximum sinking speed data of Chapter 1 and shown in Fig. 4-1. Since the Miklasz-Denny models are meant to capture maximal sinking speeds and the Jackson-Smayda model is fit to average sinking

104 speed, the models are potentially not comparable. The Jackson-Smayda model does not only have a different exponent between velocity and size, but it is shifted to slower sinking speeds. I want to separately test the effects of a different scaling exponent and the effect of using average rather than maximal sinking speed. To do this, I create a fifth model, the adjusted Smayda-Jackson model. This model is multiplied by a factor of 9, which makes it cross with all of the Miklasz-Denny models at a diameter of 80 µm. The adjusted model can therefore be thought to model maximal diatom sinking speeds, albeit with a different exponent than the other models. The sinking predictions of all five density models are shown in Fig. 4-1.

The number of size classes used depends on the initial particle size. Since the model’s assumptions are only valid for Re<1, I set the number of size classes such that the largest possible aggregates were sinking at Re<1. To test the effect of adding larger size classes, I use a model with intermediate Re effects incorporated into the model (not the default parameter set). I then either constrain the number of size classes to Re<1, or I include more size classes such that Re was <40 (the maximal allowed Re using the intermediate Re corrections). Thus I test the effect of size classes only in the situation where it was appropriate to increase the number of size classes without violating the assumptions of the model.

4.2.3 Defining metrics for sensitivity analysis

I define four metrics of the equilibrium distribution to compare across models: equilibrium volumetric flux (φeq), equilibrium volumetric concentration (Ceq), time to reach maximum flux (tflux), and time to reach maximum concentration (tC). The timing variables are defined by maximal values because the distributions almost always

105 overshoot their equilibrium flux and concentration, before settling back to an equilibrium value. I use this “overshoot” to define the start of the equilibrium stage and the timing metrics. For a few of the larger sizes in the slower growth models, this overshoot does not occur as the values simply asymptotically approached equilibrium. In the cases where this occurs, the timing metrics blow up to infinity and are not necessarily comparable, and were excluded from the calculations.

In comparing different runs of the model, I average over size class or sinking speed model to report the results. In this manner, I first calculate the ratio of the metric of interest for each size class or sinking speed model, and then average the ratio across size or sinking speed model.

4.3 Results

4.3.1 Effect of size

Size has an important effect on the results of the model, mostly through the effect of sinking speed. Size affects the sinking speed and density of the particle, which in turn affect the rate at which biomass leaves the mixed layer (Fig. 4-2). Additionally, the initial starting size determines how much coagulation must occur for particles to get large enough to sink quickly out of the mixed layer, which affects the timing of equilibrium as well as the final concentrations and fluxes. For this reason, size has a strong effect on timing and equilibrium volume concentration (Fig. 4-2).

4.3.2 Effect of sinking speed model

The effect of sinking speed model on the four metrics under consideration is shown in Table 4-2 and Fig 4-2. Equilibrium volume concentration is the only parameter that shows significant variation across sinking speed models. This variation also interacts

106 with particle size. In figure 4-2D, this effect is strongest at the smallest sizes, causing over a 10-fold variation in concentration. The Miklasz-Denny models are on average a factor of 3 smaller in equilibrium volume concentration from the original Smayda-

Jackson formulation. Oddly enough, equilibrium flux is nearly constant across sinking speed model and size even though equilibrium volume concentration varies greatly.

Both the magnitude of the sinking speed model (top 2 rows, Table 4-2), and the exponent of the sinking speed model across size (bottom 3 rows, Table 4-2), have significant effects on equilibrium volume concentration. The magnitude of the sinking compares average diatom sinking or maximal diatom sinking, whereas the slope relates to our current understanding of the physics of sinking. Use of more accurate physics affects the predictions by up to a factor of 2.5 across size, whereas the use of average vs. maximal sinking can affect the prediction by up to a factor of 4 across size (Table 4-2).

4.3.3 Effect of growth rate

Changing the growth rate of the particles had a predictably important effect on all four metrics (Fig. 4-3). Decreasing growth rate by a factor of two increased the time it took to reach equilibrium by a factor of two (Table 4-3). Additionally, a slower growth rate meant that lower values for flux and volume concentration were needed to balance the slower growth. These metrics also decreased by a factor of two. Growth rate did not interact with sinking speed model or size.

4.3.4 Effect of shear rate

Including shear in the model has an almost two-fold effect on the equilibrium flux and equilibrium volume concentrations (Table 4-3). The presence of shear increases coagulation rates, which causes particle loss through coagulation and sinking to balance

107 growth at lower final concentrations. Notice that shear does interact with sinking speed model (Fig. 4-4). A sinking speed model like the original Jackson-Smayda that predicts slower sinking speeds will have lower differential sedimentation coagulation rates, which means that increasing shear coagulation will have a relatively greater effect on such models.

4.3.5 Effect of coagulation kernel

The type of coagulation kernel has a very large effect on all four parameters of the model (Fig. 4-5). Kernels which give minimal (curvilinear) and maximal (rectilinear) metric values are unrealistic theoretical extremes, though they put absolute bounds on the range of possibilities. The two potential “realistic” intermediate kernels (fractal and intermediate Re) differ by a factor of 3 in their parameter estimates, with the intermediate

Re kernel offering higher equilibrium fluxes and concentrations (Table 4-3). This can be considered our current uncertainty in the model predictions due to kernel choice. Put simply, kernels which allow for greater coagulation rates result in lower equilibrium flux and equilibrium volume concentrations.

4.3.6 Other parameters

Adding an intermediate Reynolds number correction into the sinking speed expression has no noticeable effect on the model’s predictions (Table 4-3). Although the largest aggregates in the model do have Reynolds numbers greater than 1, these particles are already sinking so quickly out of the mixed layer than any minor modification to their sinking characteristics does not change the fact that they are still lost in a single timestep of the model.

108 Extending the model size classes to include larger aggregates increases the flux, but not any other metrics (Table 4-3). This allows larger particles to form which sink faster, but does not effect the basic coagulation process so has a limited effect on a metric like volume concentration. It is unclear how much more accurate these additional size classes make the model: the largest particles are likely to dis-aggregate under even modest shear, which is a process not considered here.

4.4 Discussion

4.4.1 Diatom density and developing predictive models

My results indicate that taking sinking speed model into account has at least as important an effect on the predictions of a bloom model as several other parameters already considered, such as shear rate and growth rate. Additionally, size has an especially large effect on the model, and can only be properly accounted for with an accurate sinking speed model. For this reason, choice of sinking speed model should be a more active concern for those involved in mechanistic modeling of diatom carbon flux or bloom events.

The different sinking speed models look at different components of the sinking speed trend, in particular the scaling exponent and magnitude of sinking speed across size. The fact that such a large difference in bloom metrics resulted between the Smayda-

Jackson models and the Miklasz-Denny models raises an important question: which sinking speed, average or maximal, is more appropriate in these models? A more complex model, like that in Jackson and Lochmann (1993), which takes nutrient status and physiological state of diatoms into account, is the best solution. Even in this case, however, the incorporation of physiological status was overly simplistic. When

109 physiologically active, Jackson and Lachmann (1993) reduced the particles sinking by a factor of 2, yet sinking speed can be reduced by over an order of magnitude, and the amount of reduction is a size dependant process, as discussed in the Introduction to

Diatoms. Additionally, even if the sinking of “live” cells is more relevant for a bloom than of “dead” cells, recent studies (Waite 1997), including the data I have collected in

Chapter 1, point out that the sinking of live cells is often constant across size, rather than following the slope given by Smayda (1970) and used by Jackson (1990). In any case, the results here certainly provide justification for the use of a more sophisticated treatment of diatom density and sinking speed in coagulation models.

4.4.2 Implication on bloom species selection

Size has an important effect on several parameters in the coagulation model, which in turn affect all metrics of the model. One interesting possibility of our models is that it appears that high particle concentrations are limited to the smaller species of diatoms. The equilibrium volume concentration always increases as particle size decreases. If we transformed from volume concentration to particle concentration, this trend would be even more drastic. Our results then indicate that only small species can be bloom forming.

If one wants to identify potential high-concentration, bloom-forming species, the results here indicate that faster growing species can reach higher concentration densities.

Additionally, if nutrients or predators are taken into account, the speed at which species can bloom may also become important in helping them achieve a high concentration faster before other factors limit their growth. This would make growth rate a potentially even more important factor for species that can achieve high concentrations.

110 4.4.3 Mixed species blooms

The model described above assumed monodisperse (i.e. single species) blooms.

To consider more ecologically relevant situations we clearly need to consider mixed species blooms. Although some studies have attempted analytical models for mixed species bloom (Jackson 2005, Kiorboe et al. 1994), numeric models are difficult as they require a lot more computing power (eg., Jokulsdottir and Archer in prep). Additionally, due the to mass-averaged techniques used in the model described here, models that have particles of more than one size or density are difficult to incorporate, without invoking a multidimensional particle-mass spectrum. The number of dimensions to the particle-mass spectrum is equal to the number of different particles in the distribution, making ecologically relevant models that incorporate tens or hundreds of species computationally impractical.

How might mixed species assemblages affect the physics of bloom dynamics?

The coagulation rates of large-to-small particles is higher than the rates of medium-to- medium particles, indicating that mixed species assemblages can increase coagulation- type processes for the same starting biomass, potentially decreasing final bloom abundance. Bigger particles, though, have lower cell densities than smaller particles, and therefore for a similar sized aggregate, a mixed particle assemblage could have lower densities and sink slower than a monodisperse assemblage of only small particles.

One may argue that although blooms are often dominated by one species, and although other species are still present, they are in much lower numbers and can be ignored. Such reasoning, though, ignores the important point that similarly sized particles are highly unlikely to coagulate due to differential sedimentation, since their

111 sedimentation rates are so similar. Coagulation with the “background” distribution of particles, even if the background is present in low numbers, can have important effects on triggering coagulation processes, and therefore on the timing of blooms (Jackson, pers. com.). Essentially the same equilibrium state will be reached, but not in the same manner.

I did not vary the background distribution of particles in this study but used the default distribution (Jackson 1990). Varying such a parameter is one potential way to test mixed species dynamics.

The take-away point here is that dynamics derived for monodisperse distributions may not hold for mixed-particle assemblages. The potential interactions are complex, and the only real way to know what happens to mixed particle communities is to employ numeric models.

4.4.4 Out of the mixed layer

Once particles are out of the mixed layer, shear coagulation becomes irrelevant

(except for the occasional wake left by large animals), and particles further coagulate only through differential sedimentation. In this region, sinking predictions for dead diatoms, such as those presented in Chapter 1, become even more important in predicting coagulation and sinking rates. This was essentially simulated when comparing models with and without shear, and it was noted that shear interacts with sinking speed model.

Thus, going from a model of bloom formation in the mixed layer to carbon flux at depth requires especially accurate models of sinking speeds.

Some authors have speculated that the only way that particles reach depth is through coagulation to extremely large particles which sink very quickly (Smetacek

1985). These particles are not accurately modeled in Jackson’s current model, as they

112 exist in an intermediate Re regime, and his original model has no intermediate Re corrections. Yet some of the factors we added to account for intermediate Re effect (such as the sinking modification or inclusion of intermediate Re size classes) had little to no effect on the predictions of the model. I believe this is primarily due to limiting this model to the mixed layer. The largest particles were already sinking so fast that they were leaving the mixed layer in one time step of the model. An intermediate Re correction would only cause them to sink even faster, but in this sort of discrete time-step model, they essentially could not sink any faster than they were already sinking. Out of the mixed layer, of course, they would sink much faster, and be more likely to reach the bottom before something either ate them or they dissolved. Thus, intermediate Re corrections are probably most relevant when looking at processes below the mixed layer.

113 Table 4-1

The parameter values used in the model. Where multiple values are given, the bold value indicates the default case. *This value varied as described in the text.

Description Variable name Values used Units

Fractal dimension fr 2.2 Number of size classes nsect * Mixed layer depth Lz 65 m Time steps delttim 0.5 day Stickiness alpha 1 Seawater density rhofl 1.0275 g cm3 Seawater kinematic viscosity visc 0.01 cm2 s-1 Initial particle concentration num_1 1 part cm-3 Radius ratio ratrig 1.36 Solver choice sol 3 Aggregate size section at which growth stops nsecgro 4

Particle diameter dia0 5, 10, 20, 40, 80 µm Density models delrhochoice 1, 2, 3, 4, 5

Shear rate Lgam, gamma 0, 1 s-1 Growth rate Lmu 1.5, 1 day-1 Kernel choice nkernal 1, 2, 3, 4

114 Table 4-2

Shows the effect of sinking speed model on the four metrics, averaged across size. The values are the ratio of the first parameter case: second parameter case. Values are dimensionless, bolded when the values differ by at least a factor of two.

tflux φeq tC Ceq New models vs. old SJ 0.99 0.77 0.98 0.28 New models vs. SJ slope 1.04 1.22 1.05 2.67 MD constant vs. Stokes' 0.95 0.90 0.95 0.51 MD variable vs. Stokes' 1.00 0.93 1.02 0.58 MD variable vs. MD constant 1.08 1.07 1.08 1.43

115 Table 4-3

Shows the effect of various parameters on the four metrics, averaged across density model and size. The values are the ratio of the first parameter case: second parameter case. Values are dimensionless, bolded when the values differ by at least a factor of two.

tflux feq tC Ceq Effect of shear (0 s-1 vs. 1 s-1) 0.95 0.60 0.96 0.55

Int RE v curvilinear 0.88 0.32 0.88 0.33 fractal v curvilinear 0.73 0.09 0.71 0.09 rectilinear v curvilinear 0.60 0.02 0.58 0.03

Re<1 size classes vs. Re<40 size classes 1.00 2.38 1.01 1.03

Effect of growth (1.5 d-1 vs. 1 d-1) 2.09 0.56 2.15 0.61

Re<1 for sinking vs. Re<40 for sinking 1.00 1.00 1.00 1.00

116 Figure 4-1

The sinking predictions of all five sinking speed models.

117 Figure 4-2

This shows the four metrics, with sinking speed model and size varied as parameters.

118 Figure 4-3

The four metrics, with growth rate varied as a parameter. Only two sinking speed models are shown. The bold line refers to the default case (Table 4-1).

119 Figure 4-4

The four metrics, with shear rate varied as a parameter. Only two sinking speed models are shown. The bold line refers to the default case (Table 4-1).

120 Figure 4-5

Shows the four metrics, varying the type of coagulation kernel as a parameter. Only two sinking speed models are shown. The bold line refers to the default case (Table 4-1).

121

Part II

SIZE SCALING IN CORALLINE ALGAL

REPRODUCTION

Summary

In this section, I explore patterns with macro-algal spores, an entirely new system for which few scaling laws have been derived. Although scaling in reproductive biology has been well studied (Smith and Fretwell 1974) and scaling laws have been described for other taxonomic groups such as land plants (Fenner 2005), the first attempts to apply the scaling laws developed for those groups to algae have been unsuccessful (Santelices

1990). In Chapter 5, I conduct a meta-analysis on one group of macro-algae, coralline algae, to identify empirical scaling laws in the reproductive parameters of this group. In

Chapter 6, I report field measurements of reproductive rates of four local species of coralline algae to unravel some of the difficulties encountered in Chapter 5 and noted in the literature. Finally in Chapter 7, I compare one component of the fitness between two local species of coralline algae that have different life history strategies.

Introduction to coralline algal

What are calcified algae?

Corallines are a family of red algae that incorporate calcium carbonate outside their cell walls. Corallines have been commonly referred to as “living rocks,” as their hard appearance gives them a resemblance to pink rocks. These algae have essentially two forms, a crustose form in which the corallines adhere strongly to rocks, and an

122 upright or “articulated” form, so labeled because of articulations or uncalcified joints throughout the algae. These joints confer flexibility to what would otherwise be a rigid upright branch. Articulation has risen at least three distinct times in coralline algae

(Johansen 1981, Broom et al. 2008). Such flexibility allows them to “bend with the flow” and survive in high wave environments (Martone and Denny 2008).

Corallines live in nearly every marine habitat of the world. In Monterey, they are one of the highest intertidal organisms (Pseudolithophyllum neofarlowii, pers. obs.), whereas they are also the deepest recorded living macroalgae (Littler 1972). Their distribution ranges from the tropics to the polar regions (Johansen 1981). They are often found in greatest relative abundance in high wave environments, but are also found in areas with intense herbivory, as their calcified nature is likely an adaptation to suit them to both types of environment (Steneck 1986). The crustose corallines can be found with or without protuberances, or bumps, on their surface.

Some coralline algae are extremely long-lived. Clathromorphum nereostratum can grow to over 5 cm in thickness, which corresponds to an age of over 100 years

(Halfar et al. 2007). On the other end of the spectrum are thin epiphytic crustose corallines, such as those in Melobesia or Hydrolithon. Melobesia can become reproductive in as little as a few weeks after settlement, causing Morcom et al. (1997) to compare such corallines to “ephemeral weeds.”

The taxonomy of corallines has changed considerably over the years (see

Woelkerling 1988 for a review). In this thesis, I will adopt the system of taxonomy used by Woelkerling (1988) for crustose corallines. He recognizes four subfamilies of crustose corallines: Lithophylloideae, Mastophoroideae, Melobesioideae, and Choreonematoideae,

123 although the Choreonematoideae, comprising a single genus of uncalcified, parasitic corallines, will not be considered in this thesis. To these crustose corallines I add two subfamilies of articulated corallines, following Johansen (1981): Corallinoideae and

Metagoniolithoideae. The genera Amphoroa and Lithothrix are articulated genera, but are generally considered to be part of the subfamily Lithophyllioideae due to similarities with their sister crustose coralline in this subfamily (Broom et al. 2008). Recent molecular taxonomy (Broom et al. 2008, Miklasz and Gabrielson unpublished data) indicates that

Lithophylloiodeae, Melobesioideae, Corallinoideae, and Metagoniolithoideae are well- defined, monophyletic lineages, whereas Mastophoroideae is a poorly defined, polyphyletic lineage that should be broken into many smaller subfamilies (Kato et al

2011).

Crustose coralline algae have three types of tissues in their thallus: the hypothallus, perithallus, and epithallus (Woelkerling 1988). The hypothallus is the most basal tissue, and usually grows parallel to the substratum. The hypothallus is the crustose coralline’s main means of lateral growth. Further from the substrate, the direction of growth curves away from the substrate and becomes perpendicular to it. This region of perpendicular growth is termed the perithallus. At the peripheral edge of the perithallus is the meristem, a layer of cells which actively divide, adding to the thickness of the crust.

The epithallus extends from the meristem to the surface of the crust, and can be composed of one or more cells. The epithallus is commonly grazed or sloughed off in many species, causing it to be constantly regrown. The outermost cell wall of the apical epithelial cells is uncalcified, and is hypothesized to be one of the main means by which corallines exchange nutrients with the surrounding seawater (Woelkerling 1988).

124 Typically, only the epithallus or upper cells of the perithallus are pigmented and thus actively photosynthetic.

Articulated corallines have a crustose basal region, and one or more articulated or jointed fronds which arise from the base. The structure of the basal region is similar to that for crustose species. In contrast, the frond is composed of alternating calcified intergenicula and uncalcified genicula. The intergenicula have an outer epithallus, with a growing meristem forming its innermost layer. Inside the meristem is the cortex

(Johansen 1981). In the articulated subfamily Corallinoideae studied in this thesis, the genicula are not formed directly by the meristem, but rather by the subsequent decalcification of already formed intergenicular tissue.

The result of the above morphologies is that corallines can often be thought of as a set of parallel filaments of cells, growing side by side. These parallel filaments do not necessarily act independently, as they are linked by either secondary pit connections or cell fusions. Like most red algae, corallines have pit connections linking cells. The connections are formed when cells divide, and allow for the transport of nutrients from cell to cell. Therefore, each cell can share resources with all of the other cells in its

“growth filament.” But corallines also form secondary connections between adjacent filaments, in the form of secondary pit connections or cell fusions. In subfamilies that produce secondary pit connections, multiple pit connections can be formed with neighboring cells, allowing nutrients to be exchanged with multiple neighboring growth filaments (Johansen 1981). In subfamilies that form cell fusions, multiple cells can fuse together, forming one enlarged cell with a single nucleus. Because of the abundance of connections between neighboring filaments, coralline algae are extremely effective at

125 transporting nutrients compared to other crustose algae. Dethier and Steneck (2001) found that when a part of a crustose alga was shaded, most species showed either no growth or a loss of tissue in the shaded region, whereas corallines were the only group to show positive growth in shaded regions, presumably from nutrient transport from non- shaded regions of the crust.

Many crustose corallines engage in fierce space competition with one another.

Thicker corallines are slower growing, but nearly always overgrow thinner neighboring corallines allowing them to win out in space competitions (Paine 1984). Additionally, the presence of grazers causes as a “competitive reversal,” by damaging some of the thicker but weakly protected species enough to slow their growth rate and allow the thinner species to overgrow what is typically the dominant species (Steneck et al. 1991). Some species are so good at overgrowing neighbors that the crust’s tissue overgrows itself, forming a “swirly” surface to the crust and giving rise to a unique internal morphology

(Steneck and Paine 1986).

Ecological importance

Within the coralline group, corallines have different morphologies which appear to fulfill several different ecological niches. Padilla (1984) found that the finely branched articulated Corallina vancouveriensis excels at resisting desiccation stress, while the stouter Calliarthron tuberculosum can prevent limpets and other grazers from crawling on its frond. In contrast, the crustose corallines were best adapted for space competition

(Padilla 1984). Guenther and Martone (unpublished data) found the differences between

C. vancouveriensis and C. tuberculosum to be carried over in their physiology- C. vancouveriensis can maintain photosynthesis even when partially dried, and C.

126 vancouveriensis stays hydrated longer than C. tuberculosum. Steneck and Adey (1976) found that Lithophyllum congestum would change its external morphology depending on local water motion and light. Steneck (1986) also found that the corallines differ widely in their resistance to herbivory, and have several different techniques for protecting themselves from herbivory. Some species grow especially thick epithalli, preventing herbivores such as grazing molluscs from reaching the important meristematic tissue.

Other species grow regularly spaces protuberances to prevent herbivores from reaching the majority of their surface. Not only has this difference in herbivory resistance through morphology been found for extant corallines, but in the geological record the morphologies that currently are associated with herbivory resistance appeared only after the rise of molluscan herbivores (Steneck 1983). Clearly algae are influenced by and respond to their ecological habitat.

In addition, coralline algae can influence the ecology of their communities.

Corallines themselves are home to many animals, in particular burrowing worms (pers. obs. for Lithophyllum grumosum). Corallines foster the recruitment of many kelps and animals such as corals (Nelson 2009). Coralline algae are currently relatively dominant, abundant members of many communities (Sapper and Murray 2003). With future climatic changes, corallines have been suggested to both become more abundant as they replace species more vulnerable to environmental changes (Wootton 2001), or to become less abundant as they themselves perish due to climate induced changes (Nelson 2009).

Reproductive features

Coralline algae reproduce using structures called conceptacles, which are essentially hollow cavities near the surface of the plant’s calcified skeleton in which

127 spores are produced (Johansen 1981). The use of conceptacles as reproductive structures is unique to coralline algae. Conceptacles are just visible to the naked eye, and can be readily visualized with a low-power microscope. The ease with which conceptacles can be identified and counted in the field makes coralline algae ideal for studies on reproduction.

Like most red algae, coralline algae alternate generations between sexual and asexual life history stages. The asexual, diploid life history phase is called the tetrasporophyte, which produces tetrasporangia, or packets of four tetraspores. These tetrasporangia are formed inside the conceptacle, from perithallial cells at the base of the conceptacle (Johansen 1981), and grow into either male or female haploid gametophytes.

The male gametophytes produce spermatangia inside their conceptacles, which often have a pointed apex (Johansen 1981). After fertilization, female plants produce the third life history phase, the diploid carposporophyte, insides their conceptacles. The carposporophyte conceptacles on the female gametophyte can be similar in appearance to those on the tetrasporophyte, but are often a little larger on average (Abbott and

Hollenberg 1976). The carposporophyte releases carpospores that grow into tetrasporophytes. It has been noted that that tetrasporophytes often greatly outnumber gametophytes in coralline algae, and that some tetrasporophytes are found to produce diploid bispores, which bypass the gametophyte phase and grow directly into tetrasporophytes (Johansen 1981). Due to their abundance, this thesis primarily focuses on the conceptacles of tetrasporophytic coralline algae, unless otherwise noted.

Conceptacles can be found anywhere on the surface of crustose species. In articulated corallines, conceptacles are located solely on the upright fronds. The location

128 of the conceptacles on the intergenicula is quite variable between different articulated corallines, and is used for taxonomic distinction (Johansen 1981). Some genera, like

Calliarthron, can have several conceptacles per intergeniculum, and some like Corallina, only have one (Abbott and Hollenberg 1976).

There are 4 types of tetrasporangial conceptacles, and type can help to distinguish between subfamilies of coralline algae (Johansen 1981). Recent molecular phylogenies show that three of these conceptacle types define monophyletic clades of corallines, but the fourth type (Type 2 conceptacles) is quite polyphyletic (Broom et al. 2008). There are also several types of gametangial conceptacles which also have taxonomic relevance, but since those are not the concern of this thesis I will refrain from discussing them (see

Woelkerling 1988 for more info).

Type 1 conceptacle development occurs when a ring of meristematic cells overgrow themselves to create a hollow cavity containing a single pore opening.

Tetrasporangia are formed at the base of the conceptacle. This conceptacle type is most common to the most specious group of articulated corallines, the Corallinoideae, and is the conceptacle type used by both articulated coralline species studied in Chapter 6 and 7.

The formation of Type 2 and 3 conceptacles is similar. They form by the degradation of calcified, perithallial tissue. Tetrasporangia are formed from perithallial cells are the bottom of the conceptacle. The main difference between these two types is whether the conceptacles have one pore (uniporate) or several pores (multiporate). In

Type 3 conceptacles (Melobesoideae), each tetrasporangium forms an apical plug in the roof of the conceptacle, which leads to a pore opening for each spore. Thus, these conceptacles are multiporate. In Type 2 conceptacles (Lithophylloideae and

129 Mastophoroideae), only one central pore opening is created in the roof of the conceptacle, but it is otherwise similar to Type 3 conceptacles. The Lithophylloideae is additionally defined by having secondary pit connections, and is monophyletic, whereas the Mastophoroideae are additionally defined by having cell fusions and are polyphyletic

(Broom et al. 2008). The two crustose species used in Chapter 6 have Type 2 conceptacle development, although one belongs in the Lithophylloideae and one in the

Mastophoroideae. Although conventional taxonomy held these as sister groups and thus led me to their initial selection, recent work on the species on the West Coast (Miklasz and Gabrielson, unpublished data) determined that the two species used in this study are not actually all that closely related.

The final tetrasporangial conceptacle type relates to the most basal Sporolithon genus. Although placed in the subfamily Melobesoideae by the taxonomy used here

(Woelkerling 1988), molecular phylogenies commonly place this genus as a sister group to all other coralline algae (Broom et al. 2008). The Sporolithon or Type 4 conceptacles are comprised of sori, thin-walled cavities that each contain one tetrasporangium. In a sense, these are not true conceptacles since each cavity only contains one tetrasporangium. Type 4 conceptacles will not be considered further in this thesis.

Reproduction energetics

Species exhibit very different sorts of strategies when proportioning energy to reproduction. From an evolutionary perspective, the more surviving offspring an individual can produce, the greater its evolutionary fitness. Yet offspring numbers are limited by how much energy can be devoted to reproduction, which is in turn limited by how much energy is spent in growth or repair and how much energy can be acquired.

130 Size also plays into this issue: bigger organisms acquire more total energy than smaller organisms and have lower metabolic rates per unit mass than smaller organisms (at least for mammals), potentially giving bigger organisms more energy for reproduction (Brown et al. 1993).

Another evolutionary choice concerns how to use the energy devoted to reproduction. Should energy be spent increasing the fitness of the offspring, or producing more offspring? Often producing a bigger offspring increases its fitness, so the classic choice is one between a few, highly fit large offspring, or many less fit, smaller offspring

(Smith and Fretwell 1974). Two general strategies, termed r-selected and K-selected, are used to take advantage of these different reproductive styles (Pianka 1970). r-selected species are characterized as short-lived, fast growing species. These species benefit from an opportunistic type of lifestyle. They often live in disturbance-prone habitats and produce many low-quality offspring, since in such habitats some of the offspring are guaranteed to die no matter what sort of parental investment occurs. As a consequence, it is best not to invest too much in any one offspring. K-selected species tend to be long- lived, slow growing and have competitive life-styles, meaning that they survive when others may not. Such species typically produce a few high quality offspring that are very likely to survive.

Smith and Fretwell (1974) suggest that there is an optimal size/number of offspring that will maximize an organism’s fitness. Thus, one can theoretically determine the evolutionarily optimal combination of offspring size/number for an organism.

Hypothetically, this sort of model can also be used to derive offspring size/number relationships between different organisms. This sort of argument relies on several

131 assumptions. First, it assumes that a reproductive adult has a set supply of energy for reproduction. However, different species, or even a single individual at different periods of its life, can have very different supplies of energy for reproduction. Second the argument assumes that offspring fitness always increases with offspring size, but with diminishing returns. Neither of these assumptions is necessarily true for organisms like plants or algae.

For plants, Grime (1977) has suggested that there are essentially three types of reproductive strategies, and that the binary r-K gradient created for animals is not sufficient for plants. In Grime’s scheme, there are two main forces that drive reproductive strategies: stress and disturbance. Stress “consists of conditions that restrict production,” in other words recurring, difficult environments. Disturbance consists of “partial or total destruction of plant biomass,” and can be caused by sudden physical, ecological, pathological, or human-induced changes. In this framework, there are three viable strategies, the competitors (low stress and low disturbance), the stress tolerant species

(high stress, low disturbance), and the disturbance tolerant species (low stress, high disturbance). In regions of high stress and high disturbance, no viable strategy predominates. In low stress environments, we recover our typical r vs. K selection paradigm- the competitors do well in low stress and low disturbance, and the opportunists

(or disturbance-tolerant species) due well in low stress, high disturbance environments. In the environments of low disturbance but high stress, where it is continually difficult but not impossible to survive, a third strategy is developed of the stress-tolerant species.

Most interesting are some of the predictions offered by Grime for these three strategies. Grime (1977) predicts disturbance-tolerant plants to direct a large component

132 of their energy to reproduction compared to the other two strategies. Competitive plants are expected to show strong seasonality in reproduction, whereas stress-tolerant plant show no seasonality, and r-selected plants show bursts of reproduction after temporarily favorable conditions. The presence of multiple reproductive strategies might explain why patterns between offspring size and number can be difficult to determine for plants

(Fenner 2005).

It is currently unclear which and how many reproductive strategies might be viable for algae. In a marine environment, storms and waves certainly cause a disturbance, but only in exposed environments. In an intertidal environment, low tides leave organisms exposed to thermal and desiccation stress for a portion of the day. Thus algae can live in both high stress and high disturbance environments, which makes it possible that all three strategies described by Grime might be viable for intertidal algae.

As such, looking for simple binary patterns between offspring size and number may not be valid for algae.

Attempts to find patterns in reproduction for algae have been mostly unsuccessful

(Santelices 1990). In the indepth study of red algal spore size by Ngan and Price (1979), only three patterns were found: carpospores tend to be larger than the tetraspores for a given species, spore size maps out along taxonomy, and larger-spored species tend to occur only at lower tide levels, while smaller-spored species are found throughout the entire tidal range. Relationships between number of offspring and offspring size have not been found in algae (Santelices 1990). Interestingly Fenner (2005) notes that such broadscale patterns are generally found for plants, with an inverse correlation between offspring size and number. This raises the question of why plants should exhibit this

133 pattern but not algae. Santelices (1990) notes that although in plant seeds greater size relates to greater investment, algae spores carry little nutritional store from their parent and can begin photosynthesizing immediately. This means larger algal spores do not necessarily have the greater nutritional stores that would translate into greater fitness as plant seeds do, potentially complicating the relationship between spore size and number.

Instead, variation in algal spore size has been correlated to physical concerns like buoyancy and dispersal abilities (Santelices 1990), without a connection to concerns that directly affect fitness such as survivability and germination. Additionally, generalizing across all algae is difficult because often different methods are used to characterize reproduction for different groups (Hoffmann 1987). Even so it seems difficult to make an argument that such patterns may exist. One of the largest kelps, Macrocystis pyrifera, produces some of the smaller spores amongst algae and produces them in relatively great number (Gaylord et al. 2006). Thus the largest algae do not produce the largest offspring, as is generally true for plants and animals. It is useful to note that algae do not form a monophyletic lineage; brown algae are actually quite evolutionarily distinct from red and green algae, despite general similarities in their form and habitat (Keeling et al. 2005).

Thus, it may be more useful to look for patterns solely within taxonomically similar algae.

When considering reproductive patterns in algae, it should be noted that algae have a relatively small range of offspring size. Algae span roughly 1.5 orders of magnitude in linear spore size, compared to the nearly 4 orders of magnitude exhibited by seeds in plants (Santelices 1990). Additionally, algal spores are comparatively smaller than the offspring of other macro-organisms. The range of algal spores varies from 15 µm

134 to 120 µm (according to Santelices 1990, though I find a slightly larger size range in

Chapter 5), which is in the range of the smallest of plant seeds (100 µm). The small size of algal spores is partly due to the fact that algal offspring are always released from their parents in a single-celled form (Abbot and Hollenberg 1976), whereas plants and animals are often, but not always, multicellular at “birth.” A multicellular offspring is able to occupy a much larger range of sizes than a single-celled form. This fact still raises the question of why algal offspring are only unicellular whereas other groups have developed multicellular offspring, but it can explain why the range of algal offspring size is relatively small.

In summary, reproductive theory suggests that there should be a tradeoff between spore size and number, with the optimal position determined by how offspring fitness scales with size. Tradeoffs between size and number only occur if individuals are constrained to a set amount of energy, otherwise a species with more energy stores might simply produce more and bigger offspring than another species. Despite the uncertainty in this assumption, tradeoffs between spore size and number have been found in plants but not in algae. Although algal spores size has been related to physical factors like spore buoyancy and tidal height, it has yet to be related to spore number. This may be because

1) algal reproduction is measured differently in different taxonomic groups, preventing broadscale comparisons, 2) algae violate the assumption of a set amount of energy to reproduction, and so spore size and number are not constrained to interact, 3) large spore size is limited by physical or other constraints, both limiting algae to relatively small sizes and a small range of sizes, and violating the assumption that larger offspring have

135 greater fitness. This thesis evaluates the validity of these three points in attempting to discover patterns between spore size and number in algae.

136 Chapter 5

A meta-analysis of spore size and number

5.1 Introduction

One of the potential issues involved in determining relationships between spore size and number in algae is that reproductive rates in algae are often measured in non- comparable ways in different groups of algae (Hoffman 1987). Therefore algae may actually have significant scaling patterns with size, but we may be unable to detect them until we use more consistent measurement techniques. Controlling for the differences in measurement technique and taxonomy is the necessary first step in determining what sort of scaling patterns, if any, exist in algal reproduction.

Coralline algae are an ideal group in which to start identifying patterns in reproductive energetics. The size range of coralline algal spores (5µm to 200 µm, Table

5-1) spans the previously reported spore size range for red algae (Santelice 1990). Yet the corallines have a consistent reproductive structure across their taxonomic diversity, meaning that reproduction can be characterized the same way across the entire spore size range (Woelkerling 1988). If there are any patterns in reproductive energetics in red algae, we might expect to see them across coralline algae.

This chapter addresses the following question: if reproduction is measured using a single scale across a range of spore sizes, do clear reproductive patterns emerge for algae? I address this question by looking for patterns between spore size, conceptacle size, adult thallus size, and spore number in coralline algae. These parameters are often recorded in the taxonomic literature for corallines, and so this question can be addressed

137 using a meta-analysis. In particular, I will answer the following specific questions: In coralline algae, are offspring size and number inversely correlated, as they are in other groups of organisms? In coralline algae, is offspring size directly related to adult size or the size of the reproductive organ?

5.2 Methods

I used a variety of taxonomic sources in this literature survey. Only reproductive features of the tetrasporophyte life history stage were used in this analysis, and I used only taxonomic entries that reported the diameter of the tetrasporangia. When noted, I also recorded tetrasporangial length, conceptacle height and diameter, the thickness of the crust or intergenicula, the maximum extent of the entire alga, and the number of spores inside a conceptacle. For crustose corallines, the extent referred to the maximal diameter of the crust, if noted. For articulated corallines, extent refers to the maximal distance the fronds extend from the substrate. For multiporate conceptacles, the number of pore openings is equal to the number of tetrasporangia in the conceptacle, so this was used as a measure of spore number in these cases. Table 5-1 and 5-2 show the data, with sources. I recorded both the species names that were given in the original taxonomic sources for easy reference and the currently accepted taxonomic name (Guiry and Guiry 2012).

The taxonomic literature often reports size parameters as a range, though occasionally only one value is given, or only an upper or lower limit to size. For consistency, all results are analyzed using the maximal reported size, which was the measurement that provided the most data. The data were also analyzed across minimal size and median size (halfway between maximum and minimum), and the trends were in the same direction as those found using only maximal sizes.

138 In some cases, the same species was reported in two different sources. As these sources covered different geographic areas, these records often refer to different populations of the same species, or sometimes to cosmopolitan species that were given a common name but are likely not as widespread as previously thought. In many cases, the values reported for these species are different despite being reported under the same species name, indicating that populations that live in different areas are probably quite distinct from each other. For an example, compare the records for Corallina officinalis reported from Great Britain (Irvine and Chamberlain 1994), the Southeastern USA

(Schneider and Searles 1991) and China (Tseng 1983). In these sorts of cases, the records can still be considered, from a statistical perspective, as independent data points representing distinct measurements of the parameters of interest. In some cases, where sources cover neighboring geographic records, the species record from one source is used in the source for the neighboring geographic area, with little to no change to the parameters measured for the species. In these cases, the records hardly represent two

“independent” measures of the parameters investigated in this study. When these cases could be clearly identified, the duplicate records were removed so that only one data point was used for such species. In these cases, data in the most recent reference are used and other records removed.

For some records, two different entries from the same source that were originally considered separate species are now known to be the same species. In this case, one of two options was used. If the records have sufficiently overlapping ranges, then the duplicate records were combined into one entry with overlapping ranges. This was done for Lithophyllum pustulatum in both Taylor (1957) and Irvine and Chamberlain (1994)

139 and for Mesophyllum crassiusculum in Mason (1953). If the ranges were not considerably

overlapping, then the records were kept as separate entries, as it is possible that these

might still represent separate, but wrongly classified, species.

In total, I identified 17 (14%) duplicate records to be removed or combined with

other entries for the crustose species and 10 duplicate records (33%) for the articulated

species. Thus, there are 101 crustose entries and 20 articulated independent records used

for this analysis. Of these, based on our current understanding of the taxonomic names of

these entities, there are 75 unique crustose species names and 12 unique articulated

species names. Note that the data were also analyzed with the duplicate records included

and the same trends were found as when such data were excluded, indicating that such

duplicate records have a minimal impact on the patterns identified in this study. There

was one analysis (extent vs. tetrasporangial diameter for crustose corallines) which

showed a significant effect with the duplicates but an insignificant effect without the

duplicates. The implications of this difference will be addressed more fully in section 5.3.

All size parameters were tested against tetrasporangial diameter for correlations

using Pearson’s correlation test, using the cor.test() function in the Programming

language R. The basic Pearson’s correlation test was chosen because it can best indicate

what simple patterns, if any, are present among the data, without having to specify a

dependent and independent variable.

The volume of the conceptacle was estimated assuming that the conceptacle

consisted of a short, fat, circular cylinder. In this case the volume of the conceptacle, Vcon,

is:

2 dcon (5-1) Vcon = "( 2 ) hcon

! 140 where dcon is the diameter of the conceptacle and hcon is the height of the conceptacle.

Note that depending on the shape of the conceptacle and the presence or absence of a

centrally located column of calcified tissues (called a collumella), this estimate can

slightly overestimate or underestimate the true volume of the conceptacle.

The volume of a tetrasporangium (Vtet) was calculated as an elongate circular

cylinder:

2 dtet (5-2) Vtet = "( 2 ) ltet

where dtet and ltet are the diameter and length of the tetrasporangium, respectively. This

! probably slightly over-estimates the volume of the spores, since the ends of the cylinder

taper to an apex rather extend into an idealized cylinder as assumed by this calculation.

To get the total volume of all the spores in a conceptacle, Vtet is multiplied by the total

number of spores in a conceptacle.

When comparing the volume of a conceptacle to the total tetrasporangial volume

inside that conceptacle, I used a Type II linear regression on the log-transformed data

using the sma function in the smatr package for R. The regression allowed the calculation

of a scaling exponent, which I tested against isometry (an exponent of 1). The regression

was used in this case rather than Pearson’s correlation test because there is a clear

hypothesis (of isometry) that can be tested.

5.3 Results

The range of maximal tetrasporangial diameter in this study varied from 20 µm –

200 µm, one order of magnitude. The middle 90% of spores exhibited a slightly smaller

range, from 25 µm – 104 µm, only a factor of 4 difference.

141 There was a significant relationship between the diameter of the conceptacle and the tetrasporangia diameter (Table 5-3, Fig. 5-1). The patterns held for both the combined data for all corallines (p<0.001) and crustose corallines (p<0.001), but no significant pattern was found for articulated corallines (p=0.83). The general pattern suggests that larger conceptacles contain larger spores.

Spore size is shown against two measures of adult size, thickness (Fig. 5-2) and extent (Fig. 5-3). For crustose corallines, a significant direct relationship was found for thickness (p=0.045) though extent did not show a significant trend (p=0.064). In articulated corallines neither measure showed a significant relationship (p=0.21 and p=0.09 respectively, Table 5-3). This indicates that in crustose corallines, spore size somewhat increases with adult size.

Spores size is shown against spore number in Fig. 5-4. There is a significant positive correlation between these two parameters for crustose species (p=0.012, Table 5-

3). There were too few articulated species for which there were data (n=3) to perform a similar analysis. This indicates that in the groups for which there are enough data, there is a direct correlation between spore size and spore number.

Since for some of the data there appears to be heterogeneous variance across the independent variable, the correlations were also performed on log-transformed data

(Table 5-4). Trends that were significant on the untransformed data were even more significant on the transformed data. Likewise trends that were not significant on the untransformed data were also not significant on the transformed data, excepting crust extent. On the untransformed data, crust extent showed a barely insignificant (p=0.064,

Table 5-3) trend with tetrasporangia diameter, whereas on the transformed data it was

142 highly statistically significant (p=0.007, Table 5-4). Also of note, this comparison was the only one to show a different trend after the removal of duplicate records. With the duplicate records included, this comparison was highly statistically significant on either the untransformed (p=0.002) or transformed (p<0.001) data, whereas without the duplicate records, this trend was only statistically significant on the transformed data

(p=0.007). To summarize, there is a slightly insignificant to very significant positive correlation between crust extent and tetrasporangial diameter for crustose corallines.

Total conceptacle volume is shown against total volume of all of the tetrasporangia inside of a conceptacle (Fig. 5-5). The scaling exponent of the data is significantly different from isometry (exponent is 0.64, Table 5-5). The points do generally fall around a line of equivalence (Fig. 5-5), indicating that the volume of spores is roughly equal to the total volume of the conceptacle, i.e. that the conceptacle is completely filled with spores. Considering the crudeness of the volume estimates, it is interesting that the volume estimates are this close together.

5.4 Discussion

5.4.1 Patterns in algal reproductive energetics

In agreement with previous results, coralline algae span a relatively small range of spore sizes (Santelices 1990), only one order of magnitude. Over this range of sizes, a pattern emerged for crustose coralline algae: thicker crusts contain bigger conceptacles filled with many large spores, and vice versa. These patterns mostly align with our expectations- bigger organisms have bigger reproductive structure and bigger offspring.

Note that although these patterns were found to be generally true, there are definite outliers to the general trend. For example, Neogoniolithon brassica-florida has

143 some of the largest conceptacles in the data, but still has relatively small tetrasporangia

(Fig. 5-1). Additionally, there is only one species, Leptophytum laeve, that has spores larger than 150 µm, and it also has conceptacles as large as Neogoniolithon brassica- florida, yet it follows the general trend of the other data. What might be causing these species to deviate from the general pattern may be as interesting as what is causing the pattern itself, and deserves further study. As a check on the data, removing all of these large-conceptacled outliers produced a highly statistically significant trend between conceptacle diameter and tetrasporangial diameter (p<0.001) with a similar correlation coefficient (R=0.39) to the one found using all of the data (R=0.47). Removing only the large-spored outlier, Leptophytum laeve, still produced a highly statistically significant trend (p<0.001) with a similar correlation coefficient (R=0.40).

Another interesting outlier is Amphiroa hancockia. This species has the smallest tetrasporangia of any species in the data set, though it also has by far the thickest intergenicula of any species in the analysis (Fig. 5-2). This species more than any other drives the negative correlation coefficient for articulated species (even though this correlation was not statistically significant), yet it is clearly an outlier. This species also deserves further study.

The most shocking result is that within a conceptacle, one either finds a few small spores or many big spores. This is in contrast to patterns found in other organisms, in which spore size is typically inversely related to spore number. There are several possible explanations for this result.

First, this result was found when looking at patterns inside a conceptacle, but different species can have widely different numbers of conceptacles. The proposed

144 patterns of spore size and spore number are based on the idea that there is a constant energy available to reproduction, which is proportioned into either many small spores or a few large spores. In either case, there is the same total biomass or energy being put into offspring. Energy input to algae comes via sunlight, so one might expect algae to have a similar energetic input per surface area rather than per conceptacle. To add to this point, the small conceptacles that contain few small spores can be geometrically packed tighter together than the larger conceptacles that contain a few large spores. Thus, it is possible to get larger densities of smaller conceptacles, and by looking on a per surface area basis, we might expect to find that spore size and spore number to be inversely correlated. This possibility will be explored in Chapter 6.

Second, it is possible that algae simply do not follow the patterns of other organisms, for one of the reasons given in the Introduction to Part II. Perhaps larger species have more energy to devote to reproduction, allowing them to produce more reproductive material than species with smaller conceptacles. Or perhaps larger spores do not necessarily have greater fitness than smaller spores, indicating that large spores must still be produced in great numbers to produce as many viable offspring as smaller individuals.

Chapter 6 will allow for a further distinction between these possibilities.

5.4.2 Remarks about conceptacles

The conceptacle constitutes a unique structure in the coralline algae, as a hollowed out cavity for reproductive tissue. As stated in Introduction to Part II, conceptacles come in all manner of sizes and shapes. These sizes and shapes have been

145 related to ecological factors, such as herbivory resistance (Steneck 1986), but no one has yet related the conceptacle morphology to issues in reproduction itself.

Conceptacle diameter increases with tetrasporangial diameter, indicating that larger conceptacles hold larger spores (Fig. 5-1), a result that is not exactly surprising but still interesting. More interesting is the result between conceptacle volume and total tetrasporangial volume (Fig. 5-5). The close relationship between these two parameters indicates that the conceptacle is nearly completely filled with tetrasporangia.

This opens some interesting questions as to what drives the relationships between conceptacle size, spore size, and spore number. For example, one possibility is that the conceptacle size is set first, for ecological or other reasons, and then tetrasporangial spore size and number are set in such a way as to fill up the conceptacle. This possibility wouldn’t account for the correlation between spore diameter and conceptacle diameter, as those parameters have no reason to be correlated.

A second possibility is that the spore size drives the relationship. If the alga has a certain spore size/number constraints that must be maintained to maximum reproductive fitness, then the conceptacles are made to be the right size to hold that many spores of that size. Most befuddling here, though, is the inverse correlation between spore size and number. Why do we not find large conceptacles filled with many small spores, or a small conceptacle that contains just a handful of giant spores?

In any case, it is probably overly simplistic to think of the process as solely driven by one constraint acting on either spore or conceptacle size. It is also possible that the sizes of spores and conceptacles are developmentally linked, or that there are a variety of constraints acting on multiple elements of this system. Therefore, it is difficult at this

146 point to determine what might be the functional cause of a conceptacle’s size or shape based on the patterns found here, though these patterns can provide basis for further investigation.

5.4.3 Are crustose and articulated corallines comparable?

Crustose and articulated corallines have very different morphologies, and grouping them together in one comparison might be inappropriate. If some of the correlations found in this chapter have a functional basis, then algae that have different shapes could have very different functional mechanisms driving the patterns we see. For example, articulated corallines live erect off the surface, but attached to a basal crust. In such a position, the articulated corallines can face dislodgement due to wave or abrasion forces (Martone and Denny 2008). Since the conceptacles are located on the fronds, they are also subject to being removed by such physical, disturbance-type forces. Removal of a frond though, does not mean death for the individual, since the surviving basal crust can always give rise to more fronds. Thus, articulated corallines might be more affected by disturbance type phenomena, and adopt disturbance-tolerant (r-selected) life history strategies (Grime 1977). For this strategy, Grime predicts rapid growth, annual lifestyles, and large reproductive investment after temporarily favorable conditions.

Crustose corallines, on the other hand, live close to surfaces. Although they can get torn off by winter storm events, their more common concern is herbivore grazers that can literally scrape the tops off of conceptacles (Steneck 1986). This can provide a stress- type phenomenon to crustose corallines, causing them to adopt a stress-tolerant life history strategy (Grime 1977). For this strategy, Grime predicts slow growth, perennial lifestyles, and a small investment into relatively constant reproduction.

147 Articulated corallines in particular show few significant correlations across the different parameters examined in this chapter. Articulated corallines, though, are a very heterogeneous group. The location of the conceptacle on the intergenicula varies from genus to genus, and can affect both the size of the conceptacle and the potential density of conceptacles on the frond. Thus comparing across all articulated corallines may not be reasonable. Part of the difference among articulated corallines is due to their polyphyletic nature: articulation in corallines has arisen on at least 3 separate instances (Johansen

1981). The genus Amphiroa, for instance, has conceptacles and spores that greatly resemble their closest relative, Lithophyllum, in size, shape, and sunken placement on the surface. Amphiroa essentially treats its intergenicula as a crustose surface, and scatters many small conceptacles over that surface. This limits the conceptacles to small sizes, since a sunken conceptacle that was close in diameter to the diameter of the articulation, would cause major structure problems. The genus Corallina, on the other hand, occupies another extreme. Corallina features conceptacles on the apical intergenicula of branches.

The intergenicula that contain a conceptacle are almost entirely composed of the conceptacle itself, and little else. The conceptacles also can have diameters slightly exceeding the normal diameter of intergenicula, allowing them to be quite large.

The result of this taxonomic difference in conceptacle type between representatives of the sub-families Lithophylloiodeae and Corallinoideae means that any correlation across conceptacle or tetrasporangial size will actually just be testing the difference between the small conceptacled Lithophylloiodeae and the large conceptacled

Corallinoideae. In this sense, it may not even be useful to draw comparisons across articulated corallines (or at least without accounting for phylogenetic or morphological

148 relatedness), much less drawing comparisons across crustose and articulated corallines as a whole.

149 Table 5-1

Crustose coralline reproductive data with sources. The values given are a range, from minimum to maximum. If one of these values is missing, then the taxonomic source only reported a maximum or minimum measure for that species. Thallus diameter is a maximal value, as this refers to the average, “adult” size of the algae.

150

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152 4) 4) 4) 4)

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153

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154

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155 Table 5-2

Articulated coralline reproductive data with sources for the meta-analysis. The values given are a range, from minimum to maximum. If one of these values is missing, then the taxonomic source only reported a maximum or minimum measure for that species.

Thallus diameter is a maximal value, as this refers to the average “adult” size of the algae.

156 ler tt e i ain ain ain ain L (2008) (2008) (2008) (2008) (2008) (2008) (2008) (2008) and and and and and and and and and and and and

(1975)

erenc and mberl mberl mberl mberl (1994) (1994) (1994) (1994) (2000) ieson ieson ieson ieson ieson ieson ieson ieson tins Ref Irvine Irvine Irvine Irvine ler Dawes Dawes Dawes Dawes Dawes Dawes Dawes Dawes Cha Cha Cha Cha tt ath ath ath ath ath ath ath ath Git Li M M M M M M M M

-10 -10 -10 Spore Number lus

5 5 4 3 4 1 8 8 10 12 20 2.5 2.5 (cm) Thal Expansion ar cul (um)

80-400 90-200 45-100 90-240 400-650 150-600 125-280 120-440 150-600 300-1500 200-1800 500-1000 600-1000 width Intergeni a (um)

-50 -100 38-65 35-65 40-50 70-120 80-120 80-100 80-105 100-150 110-225 200-260 140-225 180-220 trasporangi length Te a m) (u ter -40 17-25 25-35 20-25 30-50 20-70 60-70 30-40 40-60 40-60 35-55 50-60 20-25 me 100-130 trasporangi Dia Te e cl ter ta

me -300 (um) 270-420 300-400 240-300 240-300 250-300 450-600 250-420 375-500 260-375 300-340 Dia Concep ta (if a a Current name different) Jani cubensis Jani squama a a m ta s ii is is im im tum var. llu ies al cea al ana iss iss li la Spec beauvois brasi fragil officin cubense adhaeren capil rubens aspergi officin squama rubens rubens cornicula fragil ina ilon ina ilon ll ll a a a a a ipt ipt thothryx Genus Amphiroa Amphiroa Amphiroa Cora Hal Jani Jani Jani Li Cora Hal Jani Jani Amphiroa

157 ler ler ler tt tt tt e i i i and and L L L (1991) (1991) er er (1983)

erenc and and and

les les (2000) (2000) (2000) Ref ler ler ler tt tt tt Tseng Schneid Schneid Sear Sear Li Li Li

Spore Number lus 1 8 6 8 5 15 (cm) Thal Expansion ar cul (um)

-10000 45-100 65-280 500-1300 200-1700 width Intergeni a (um)

30-50 38-63 60-80 80-105 72-125 100-150 trasporangi length Te a m) (u ter 15-20 25-38 45-60 17-25 32-48 30-40 me trasporangi Dia Te e cl ter ta

me (um) 210-270 249-300 270-850 200-320 Dia Concep ii (if Current name different) Amphiroa beauvois i ii is ies cea al a la Spec hancocki capil rubens beauvois officin zonat ina ll a a Genus Amphiroa Jani Jani Amphiroa Cora Amphiroa

158 Figure 5-1

Conceptacle diameter versus tetrasporangial diameter shown for A) crustose and B) articulated species.

159 Figure 5-2

Algal thickness, either of intergenicula or of the crust, versus tetrasporangial diameter shown for A) crustose and B) articulated species.

160 Figure 5-3

The extent of the algae plotted against tetrasporangia diameter shown for A) crustose and

B) articulated species.

161 Figure 5-4

Tetrasporangial number vs. tetrasporangia diameter shown for A) crustose and B) articulated species.

162 Figure 5-5

Estimated conceptacle volume versus estimated total tetrasporangial volume for the crustose corallines, since conceptacle height is not often recorded for articulated corallines. The line indicates a linear relationship, with the two volumes exactly equal to each other. The regression is shown by the dashed grey line, and is significantly different from an isometric relationship (Table 5-5).

163 Chapter 6

Coralline algal reproductive rates and size

6.1 Introduction

In the last chapter, I noted a paradox, in which small conceptacles contain a few small spores and large conceptacles contain many large spores. Although this pattern is in contrast to patterns expected by reproductive energetics (Smith and Fretwell 1974), it can be solved if smaller conceptacles are found in greater density than larger conceptacles. In such a case, one might then find that per surface area, species contain the expected pattern of either many small spores or a few large spores. And as noted in section 5.4.1, looking for a pattern in reproduction energetics on a per surface area basis may be more appropriate than looking for patterns on a per conceptacle basis.

The paradox noted above provides the justification for the main question addressed in this chapter: what is the pattern between spore number and spore size when looking on a per surface area basis? Answering this question requires dealing with two issues, which will form the two other main foci of this chapter.

First, algal species can show very different sorts of seasonal patterns in reproduction, based on their taxonomic group and location (Hoffmann 1987). Many different seasonal patterns have been found for coralline algae, from a lack of pattern or constant reproduction (Corallina elongata, Neto 2000; Corallina sps., Baba et al. 1988,

Serraticardia maxima Chihara 1973), to reproduction during only a few months of the year (Lithophyllum sps. Chihara 1974; Amphiroa sps. Chihara 1973; Lithophyllum yessoense and Neogoniolithon sp. Noro et al. 1981), to reproduction for most of the year

164 except for a few months (Lithophyllum incrustans, Edyvean and Ford 1986; Mesophyllum erubescens Chihara 1974; several corallinoid sps. Chihara 1973; Corallina officinalis,

Baba et al. 1988). Some of the patterns are correlated with taxonomy, with members of the mostly articulated subfamily Corallinoideae tending to show constant or near constant reproduction, and members of the mostly crustose subfamily Lithophylloideae tending to show some sort of seasonal pattern. I have found data on only one Melobesioid crust and one Mastophoroid crust, so it is not possible to draw conclusions for these subfamilies.

The effect of seasonality is that any attempt to characterize reproduction using measurements at a single point in time is potentially biased. Depending on the time chosen, one species could be at a high point of reproduction while another was at a low point. Thus a single measurement at a single point in time can be biased. To avoid this dilemma, one must measure the reproduction of all species of interest for at least a whole year, and then compare yearly averaged reproductive rates.

Thus, the second major focus of this chapter will be on the reproductive patterns formed by the species under question. Although the intent of looking at year-long reproduction is to better answer the first question presented in this section, an added benefit is that I will also be able to address a second question: do local coralline algae display seasonal patterns in reproduction, and does the presence or absence of such patterns agree with previous results for related species?

The third focus of this chapter concerns issues revolving around individuality in algae. Reproductive energetics are typically defined as reproductive output per individual. The difficulties in Chapter 5 and this chapter are derived in part because individuality is so difficult to define for algae. Without clear individuals, reproduction is

165 difficult to characterize on a per individual basis. And with indeterminate growth, one might expect reproductive output to be related to an individual’s size, rather than being constant per individual. Because of these issues, reproduction must be characterized by some other means than per individual, such as per conceptacle or per surface area. As an example of the difficulties encountered, articulated corallines consist of individual fronds arising from a single basal crust. The fronds are genetically identical, but are unlikely to exchange nutrients or be otherwise linked with each other (P. Martone, pers. com.). Thus, it is unclear whether the individual fronds or the crust as a whole should be considered as a functional individual.

As a second example, consider one of the crustose species examined in this study,

Pseudolithophyllum neofarlowii. This crustose coralline grows indeterminately over the surfaces of rocks, sometimes covering an unbroken extent of a meter or more (per. obs.).

This large extent has probably arisen from the fusion of conspecifics when they come into contact, which produces no noticeable connecting tissue once the fusion has occurred

(personal observation). Although physically linked, portions of a crust that are meters apart are probably not physiologically linked, and at the very least experience very different micro-environments, which can trigger different responses in coralline morphology (Steneck 1986). Thus, the spatial extent to which such a crust acts as an individual is likely much smaller than a meter.

I do not expect to definitively answer the question of what is individuality in algae, if such a question is even answerable, but I answer two related questions in this study. First, on what level do articulated corallines exhibit a similar response to reproduction: on the level of frond, crust, or even greater spatial scales? Second, do

166 corallines show any intra-specific patterns between reproduction and size? If corallines do not act as coordinated individuals, but rather as an aggregation of uncoordinated parts

(defined as physically linked groups of functionally independent organisms), then one would expect that the reproduction measured per surface area should not show a pattern with measures of algal size. Lack of such a pattern would indicate that portions of the algal crust are not dependent on the size of the algae as a whole in determining their reproductive output. This would also provide justification for only measuring arbitrarily small sections of crusts like P. neofarlowii that are composed of immeasurably large segments of a chimera of fused individuals.

In this section, I will address four questions surrounding these three foci. First, I search for patterns in the seasonality of four local coralline algae species. I will then characterize on which spatial scale articulated coralline algae behave in a coordinated fashion, or display individuality. Next, I will use the data from both the seasonality analysis and the spatial variability analysis to compare the reproduction output of the four species. Finally, I will look for correlations between algal size and reproduction, to see if quantifying reproduction per surface area is a good, size-independent measure of reproduction.

6.2 Methods

6.2.1 Species

I worked with four species of coralline algae: Corallina vancouveriensis,

Calliarthron tuberculosum, Lithophyllum grumosum, and Pseudolithophyllum neofarlowii. The species were chosen due to their local prevalence at Hopkins Marine

Station and the fact that they span a relatively large range in spore size (33 µm - 70µm in

167 diameter). C. vancouveriensis and C. tuberculosum are both in the subfamily

Corallinoideae, while L. grumosum is in the subfamily Lithophylloideae and P. neofarlowii is of uncertain taxonomic status, but in a sister clade to the subfamily

Lithophylloideae (Miklasz and Gabrielson, unpublished data). Using genetics as a guide,

I identified the range of the crustose species’ morphological diversity so that I could accurately identify them in the field (Miklasz and Gabrielson, in prep).

Because it did not seem equitable to compare articulated corallines with crustose corallines, I primarily treated these four species as two separate comparisons, P. neofarlowii vs. L. grumosum and C. vancouveriensis vs. C. tuberculosum for the crustose and articulated species, respectively. For the articulated species, a related Corallina species (Corallina officinalis) has faster growth rates than C. tuberculosum (Colthart and

Johansen 1973). Considering that C. vancouveriensis is even thinner and therefore probably faster growing than C. officinalis, I assume that C. vancouveriensis has a faster growth rate than C. tuberculosum. C. vancouveriensis also has smaller spores as compared to C. tuberculosum (Table 6-1). Although the breaking stress of the joints in both species is quite similar (P. Martone, unpublished data), C. tuberculosum can withstand greater forces than C. vancouveriensis due to the greater cross-sectional area of its intergenicula. Thus in that it has faster growth rate, smaller offspring, and greater vulnerability to disturbances, C. vancouveriensis appears to have a “fast” or r-type life history compared to C. tuberculosum.

For the crustose species, it is accepted that thicker crusts have slower growth rates

(Steneck 1986). Thus, we can expect the thicker species (L. grumosum) to grow more slowly than its thinner counterpart (P. neofarlowii). P. neofarlowii also has smaller

168 spores than L. grumosum (Table 6-1). Thus because of its faster growth rate and smaller offspring, P. neofarlowii appears to have a “fast” or r-type life history relative to L. grumosum.

I should note that whether a species has slow or fast life history does depend in some part on its reproductive output, which is measured in this study, and several other parameters not yet measured for these species. The life history characteristics assigned to the species (fast or slow) are based on a few traits, and on assumptions of how those traits correlate to traditional paradigms (Smith and Fretwell, 1974). These should not be taken as firm categories, but rather as an hypothesis to be tested in this study about how the traditional paradigm might explain patterns formed by these four species. There is of course the possibility that the traditional paradigm simply does not apply to the algal species studied here, in which case the hypothesized classification of life history categories are irrelevant.

As an additional note, both of the crustose species have significantly smaller spores than the articulated species (Table 6-1). To compare reproduction rates across spore size, I also looked beyond the two pairwise comparison defined here, evaluating trends across all four species.

For simplicity, I measured reproduction of only the asexual life history phase of these species, the tetrasporophytes.

6.2.2 Seasonality

I measured the reproductive output of the same individuals in the field for a year.

Twelve individuals were tagged at each of two sites (a “Protected” gradually sloped shoreline that received dampened waves, and an “Exposed” sharply sloped channel that

169 received relatively strong waves), for 24 total individuals of each species except for L. grumosum. A horizontal transect was laid across each site, and every meter the closest individual to the tape was tagged. Although prevalent, L. grumosum was mostly found in tidepools, but unlike C. tuberculosum its conceptacles were quite small and difficult to quantify without use of a field microscope. The microscope could not be submerged in deep tidepools. Thus only individuals of L. grumosum which were out of tidepools in the low intertidal zone could be used. As a consequence, only 3 individuals of L. grumosum were tagged in each site (6 total) in the low intertidal zone or edge of tidepools, where they could be easily sampled.

For the crustose species, a Z-spar marking was put next to the crust so that the same individual could be found month after month. As previously noted for P. neofarlowii, crusts fuse when encountering conspecifics, so distinguishing individuals was difficult. Arbitrary cutoff points were defined for ambiguous individuals of this species. To ensure that the same portion of the individual was counted each time, pictures were taken of each individual with the counted area indicated and used as a reference.

P. neofarlowii and L. grumosum were measured in a similar fashion. A transparency with 0.5 cm grid was overlaid on the crust, and the number of squares which were at least half-filled with crust and which contained fertile crust were counted. Areas with dense conceptacles could be identified by eye on these crusts and counted. Where ambiguous, a field microscope was used to confirm whether the areas identified by eye did in fact contain healthy conceptacles. This technique identified the percentage of the crust that is reproductive. A monocular field microscope (Steindorff Shop Portable

Microscope 20x) was then used to count the number of conceptacles in a fertile region of

170 the plant. 2-3 regions were counted, and the region with the most conceptacles was used.

In this way, the maximal reproductive output, rather than the average reproductive output was estimated. To get the total conceptacle density over an individual, I multiplied the conceptacle density in a fertile region measured under the microscope by the percentage of the crust that was fertile, to get the number of conceptacles per surface area over the whole crust. To measure the linear size of the crust, the extension of the crust in the longest dimension was measured.

Since a strong pattern in reproduction was found in P. neofarlowii (section 6.3.1),

12 individuals of this species were studied for an additional year to see if the timing and magnitude of this species’ reproduction was consistent across years. In choosing which

12 individuals to use, I intentionally chose individuals that had not been damaged in the first year, and that showed especially clear seasonal patterns in the first year.

For C. tuberculosum, I initially attempted to tag individual fronds with zipties, while also tagging the crust that the frond came from with Z-spar. The majority of the zipties were torn off the plants in the first few months after being tagged. Either the frond was ripped off with the ziptie, or the ziptie was ripped off the frond on its own. Some individuals kept their zipties and those fronds were measured repeatedly until they eventually lost the ziptie or were removed by wave action. For individuals that lost the ziptie in the first two months, a random reproductive frond (if one could be found) was chosen from the crust marked with Z-spar. Two tagged individuals were upon later and closer investigation found to be gametophytes, and so were excluded from the analysis, leaving 22 tetrasporophytic individuals of C. tuberculosum.

171 For C. tuberculosum, reproductive output was measured in the following manner: a branching point on the frond was chosen such that 100-200 intergenicula were distal to that branching point. The numbers of total intergenicula and fertile intergenicula were then counted to calculate the percentage of the frond that was reproductive. Next, 7 fertile intergenicula were chosen haphazardly, and the number of conceptacles on those intergenicula were counted. Those seven numbers were averaged and divided by the surface area of an average intergeniculum to get the total conceptacles per surface area on a fertile intergeniculum. This was multiplied by the percentage of fertile intergenicula on a frond to calculate the conceptacles per surface area on the entire frond. The length and width of the fronds, measured in the axial and perpendicular to axial direction, respectively, were also measured. Length was measured from the branching point previously identified to the distal tip. Since the portion of the frond that extended from the first branch point to the crust was never reproductive and never counted, this method could overestimate reproductive output. Additionally, because I specifically looked for reproductive individuals when resampling individuals that lost their tag, I could also have biased the measurements towards high levels of reproduction.

For C. vancouveriensis, intergenicula were too small to be counted directly as in

C. tuberculosum. Additionally, many fronds were in short, tangled clumps, so it was too difficult to measure conceptacle density with a field microscope. The conceptacles were big enough to be identified by the naked eye, though. A subjective scale of conceptacle density was created from 0-3, with 0 being the least reproductive, 1 being only slightly reproductive and 2 and 3 being increasing levels of highly reproductive. This subjective scale was calibrated in the lab every three month by collecting several random fronds of

172 C. vancouveriensis, assigning them a reproductive category, and then exactly measuring the density of conceptacles under a microscope. The categorization system was found to vary slightly in magnitude with time, but was reasonably consistent in creating non- overlapping categories of conceptacle density. Therefore, a time-varying calibration was used to transform this subjective scale to approximate conceptacles per surface area. To measure frond area, either in the lab or field, care was taken to make sure that the frond was fully extended and untangled, but only a minimal effort was given to spreading out the numerous side branches. Although this procedure might underestimate the area of this species, it corresponds to how the sub-branches may overlap during a low tide, and thus corresponds to effective, or photosynthetic, surface area. The number of squares that were more than half-filled with C. vancouveriensis was counted to get an area measurement.

Five haphazardly chosen individuals from each marked crust of C. vancouveriensis were chosen each month to be surveyed. Using the aforementioned grid, their length and area were measured, and a reproductive scale was assigned. After transforming the reproductive scale to conceptacle per surface area, the five frond’s reproductive densities were averaged to give one score of conceptacle per surface area score for each crust.

Note that for the articulated species, both sides of the frond were used when estimating conceptacle density per surface area.

The first two months were not used in the analysis for any of the species due to initial measurement difficulties like the one described above for C. tuberculosum. Thus

173 the measurements were actually performed for 14 months, but only the last 12 months were used.

6.2.3 Spatial variability

Considering that there was no significant seasonal variation in C. vancouveriensis and C. tuberculosum (see section 6.3.1), a more accurate estimation of their reproduction was developed. This was done both to provide a potentially less biased data set to compare reproductive rates between the two species, and to address the question of individuality in articulated corallines. For these data, I measured the reproduction of these species at a single point in time, but more accurately and intensely characterized their reproduction across different spatial scales. I collected five fronds from each of twelve individual crusts from each of two sites in the field, using a haphazardly located horizontal transect and sampling the nearest crust at every meter. The fronds were collected and brought back to the lab. All conceptacles on each frond were counted under a microscope, length was measured from base to tip using a ruler, and area was quantified by taking a picture of the crust with a camera against a white background. The image was transformed into a binary black and white image in ImageJ, and the area of the crust was then calculated. The same method was used for both C. vancouveriensis and C. tuberculosum, eliminating biases caused by different measurement techniques between the two species.

6.2.4 Statistical analyses and conversion parameters

To analyze seasonal patterns of reproduction, an autocorrelation analysis was performed on the yearlong seasonality data. For the individuals for which it was available, the two years of data were used for P. neofarlowii. Autocorrelation coefficients

174 were calculated on the data for each individual. Thus, although I am looking for a period of 12 months using only 12 months’ worth of data, I have 24 individuals, or 24 separate

2 time series, to use. Since the uncertainty at any given lag is given by N , where N is the number of data points at that time lag, using 24 records gave more power than I would otherwise have to detect significant auto-correlations. !

To analyze the results of the spatial variability study, a nested ANOVA design was used with Site, Crust, and Frond as factors. The levels were tested for significance, as well as counting the total percentage of the variance contained in each level. Because the parameters measured represented count data, the data were transformed for normality before analysis using a square root transformation. The data were also measured without the transformation and the same statistically significant patterns were found as with the transformation.

To compare between species the results of the yearlong seasonality data, I averaged reproduction over the year to calculate the conceptacles found on plants per surface area for each individual. I then averaged the values of each individual to calculate a mean and standard deviation for each species. To compare the results of the spatial variability data between species, I calculated an average and standard deviation of the data for all 120 fronds. A student’s t-test was used to test for significant differences in each of the paired comparisons, P. neofarlowii vs. L. grumosum and C. vancouveriensis vs. C. tuberculosum.

To look for patterns in reproduction and size for the seasonality data, I averaged size-related parameters and reproductive rates over the entire year for each individual.

Major axis length was used as a measure of length for the two crustose species. Frond

175 length was used as a measure of length for the two articulated species. Surface area was directly measured for P. neofarlowii, L. grumosum, and C. vancouveriensis. For C. tuberculosum, I used the number of intergenicula as a measure of area.

To look for patterns in reproduction and size for the spatial variability data, I simply used the length, area, and reproduction data measured directly on the two articulated species. In the spatial variability data, a significant effect of crust was seen on reproduction, frond length, and frond area for both articulated species (see section 6.3.2).

For this reason, it can be argued that it is not appropriate to compare reproductive rates to frond length or area for each frond, since it is known that fronds arising from the same crust are more similar to each other than fronds arising from a different crust. However, I always used five fronds from each crust, thus weighting the effect of crust equally across the data, ensuring that although the data are not independent, they are not biased. In any case, I decided to take the regression between reproduction and size both over frond, and over crust (by averaging the values for the five fronds arising from that crust).

In both sets of data, the size parameter and reproductive rate were tested for significant trends by fitting the data to a linear regression and testing for a significant slope between the two variables of interest.

A series of conversion factors needed to be measured to allow me to transform the data from conceptacles per surface area to other measurements of interest; spores per surface area, reproductive biomass per surface area, spores released per surface area per year and biomass released per surface area per year. The three needed conversion factors

(shown in Table 6-1) are spores per conceptacle, spore diameter, and spore release per conceptacle per time. I calculated the number of spores per conceptacle for each species

176 by cracking open conceptacles and counting the number of tetrasporangia, and multiplying by 4. I measured spore size as the diameter of tetraspores after the tetraspores had split apart from the tetrasporangia and settled on the surface of a glass slide. The volume of a spore was calculated using the diameter of the tetraspore and assuming that the tetraspores are spheres. I calculated spore release rate per conceptacle by letting pieces of each species sit in a container overnight and counting the number of spore released and the number of mature conceptacle found on the segment of algal tissue.

6.3 Results

6.3.1 Seasonality

The results of all four species across the year of sampling are shown in Fig. 6-1.

Both P. neofarlowii and L. grumosum have a period of increased reproduction in the fall and decreased reproduction in late spring. C. vancouveriensis and C. tuberculosum show a steady level of reproduction throughout the year.

P. neofarlowii showed the most dramatic trend in reproduction, so it was studied for an additional year (Fig. 6-2). In both years, P. neofarlowii shows a strong increase in reproduction between September and October. Peak reproduction occurs in November and December, followed by a steady decline in reproduction through March. A sharp decline occurred in both years near March and April, until conceptacles became nearly absent on the surface of the crust between April and June. Reproduction then once again begins to increase between June and July.

A statistically significant cyclic pattern was found in the two year data for P. neofarlowii (Fig. 6-3A), with a period of 1 year. L. grumosum shows what might be the semblance of a significant cycle with an annual period (notice the negative correlation at

177 7 months and the positive correlation at 11 months, Fig. 6-3B). But the pattern is not statistically significant, preventing a definite conclusion. Both C. vancouveriensis and C. tuberculosum show a general lack of any seasonal pattern in their autocorrelations (Fig.

6-3C,D). Although both species do exhibit a barely statistically significant negative correlation at a lag of 5-9 months, there is little indication of a positive correlation at 11 months and little semblance of a seasonal pattern, in either the auto-correlation (Fig. 6-3) or the data over time (Fig. 6-1).

6.3.2 Spatial variability

Reproductive rate shows a similar trend for both C. vancouveriensis and C. tuberculosum (Table 6-2). For both species, a significant effect was found at the levels of crust, with the majority of the variability being explained by this level. Still, a sizable

(36%) of the variability can be attributed to differences between fronds. Notice that over

80% of the variation can be explained by these two levels of the analysis.

Frond length showed different patterns between C. vancouveriensis and C. tuberculosum (Table 6-3). In C. vancouveriensis, a significant effect of frond length was seen at both site and crust, with the variance spread across all three levels. The most variance was on the level of site, with longer fronds found in the exposed site. In C. tuberculosum, only crust showed a significant effect, with 65% of the variation explained by this factor. Notice that for both species, over 24% of the variation is between different fronds of the same crust.

Frond area also showed different patterns between the two species under investigation (Table 6-4). For C. vancouveriensis, a significant effect was seen for both site and crust, but the majority of the variability (56%) was on the level of frond,

178 indicating that fronds growing from the same crust can have quite different areas. For C. tuberculosum, the results are similar, but the variance is more evenly spread across the three levels.

6.3.3 Reproductive rates

Reproductive rates are calculated from both the seasonality data and the spatial variability data, and shown together in Fig. 6-4 through Fig. 6-8.

In Fig. 6-4, the data are shown for the four species as average number of conceptacles per surface area, or conceptacle density. Noting that species are arranged from left to right in increasing spore and conceptacle size, one generally sees fewer conceptacles per surface area in the species with larger conceptacles and bigger spores

(Fig. 6-9A). Significant differences are found between the two crustose species (p<0.001) and between the two articulated species using either the seasonality data (p<0.001) or the spatial variability data (p<0.001).

For the purposes of correlating these results to reproductive energetics, one cares more about the number of spores per surface area, namely spore density, than the number of conceptacles per surface area. After making this transformation, all four species have strikingly similar numbers of spores per surface area, centered around 2000 spores cm-2

(Fig. 6-5). The only significant difference was for the articulated species in the spatial variability data (p<0.001).

Reproductive energetics often assumes that there is a constant energy source applied to reproduction. The next three graphs are concerned with testing this assumption. First, I compared the amount of reproductive biomass per surface area, or reproductive volume density, among species (Fig. 6-6). No significant differences were

179 found in any of the paired comparisons, though there was a general trend of greater reproductive biomass per surface area in the species with larger spores (Fig. 6-9C), with the articulated species generally containing more reproductive mass per surface area than the crustose species.

Next I calculated the number of spore released per surface area per time to develop a measure of spore production rates (Fig. 6-7). In each paired test, a significant effect was found with the species with faster life history traits releasing more spores.

Thus P. neofarlowii produced more spores than L. grumosum (p<0.001), and C. vancouveriensis produced more spores than C. tuberculosum in both the year-long seasonality data (p<0.001) and the spatial variability data (p<0.001). Note that there was no general trend with spore size (Fig. 6-9D), as the values for the articulated and crustose species are quite similar. Note that the number of spores released from a single cm2 of surface area numbers in the millions.

Finally, the data were converted to reproductive mass released per surface area per time (Fig. 6-8). Significant differences in release of reproductive biomass were found between the crustose species (p<0.001) and between the articulated species with the spatial variability data (p<0.001). In both pairwise comparisons that were significant, the species that had faster life history traits released a greater amount of reproductive biomass per surface area. In general though, the articulated corallines showed greater release rates than the crustose corallines.

6.3.4 Reproduction and size

I looked for patterns between reproductive rate and size for both the seasonality data in section 6.3.1 (shown in Fig. 6-10,11 and Table 6-5,6) and the spatial variability

180 data collected for the two articulated species in section 6.3.2 (shown in Fig. 6-12,13 and

Tables 6-7,8).

For the seasonality data, average conceptacle density is shown against a measure of plant length (Fig. 6-10) and plant area (Fig. 6-11) for each of the four species. The two crustose species, P. neofarlowii and L. grumosum, showed no significant trend against either length or area (Table 6-5,6). In contrast, C. vancouveriensis showed a significant positive trend with reproduction against both length and area (Table 6-5,6), indicating that larger fronds are more reproductive. C. tuberculosum showed a negative correlation between reproduction and size, though only a significant trend against frond area (Table

6-6), indicating that larger fronds are less reproductive.

For the spatial variability data, the results are plotted across frond in Fig. 6-12.

Notice that the only significant trend is for C. vancouveriensis when plotted against frond area (Table 6-7, Fig. 6-12B). The results are plotted across crust in Fig. 6-13. In this case, no significant trends are found for either species, or either measure of size (Table 6-8).

6.4 Discussion

6.4.1 The r vs. K selection paradigm

In regards to the main focus of this chapter, namely to determine whether coralline algae follow the traditional patterns typically found in reproductive energetics, the data do not provide a conclusive answer. The main test for patterns between spore number and spore size was Fig. 6-5, in which only one of the paired comparisons showed a significant difference, but in all the tests, spores numbers were quite similar. For example, from P. neofarlowii to C. tuberculosum there is 2.1-fold increase in spore diameter, or a 9.8-fold increase in spore volume, with virtually no change in spore

181 numbers per surface area. In the one test in which there was a significant difference, between C. vancouveriensis and C. tuberculosum, there was a 1.8-fold difference in spore volume with a 2.6-fold difference in spore numbers per surface area. This result does fall in line with the traditional patterns found in reproductive energetics: the species that has smaller spores by a factor of 1.8 produces 2.6 times as many spores (Fig. 6-6). For the other species, since spore numbers are the same, we see that the species with larger spores accordingly have a slightly larger reproductive volume density, indicating that the species with larger spores are putting more energy per surface area into reproduction.

Now, this analysis assumes that the relevant measurement for comparison to traditional patterns in reproductive energetics was the average number of spores on the surface at a given point in time. Although C. vancouveriensus and C. tuberculosum have similar spore densities, C. vancouveriensis produces and releases spores at almost twice the rate as C. tuberculosum (Table 6-1). Thus, when integrating across time, one finds twice as many spores produced by C.vancouveriensis as C. tuberculosum. Therefore, looking at spore production rates may be a more relevant comparison in correlating spore size and numbers to expectations from reproductive energetics. I do not have a way to measure production rates, but I can measure a close proxy, spore release rates. Using such a measurement, one recovers the pattern in Fig. 6-7: the species with the smaller spores and faster life history traits (with fast and slow life history defined in section

2.3.1) releases more spores per surface area in each pair-wise comparison. This is the sort of result I would expect based on theory (Smith and Fretwell 1974), though if I try to extend this result over both crustose and articulated corallines, I find that the there is no

182 clear pattern from P. neofarlowii to C. tuberculosum. This can just mean, of course, that articulated and crustose corallines are not comparable.

Finally, I checked the assumptions of traditional reproductive energetics model

(Smith and Fretwell 1974) by plotting reproductive volume released per surface area. If one takes this as a measure of the amount of energy each species puts into reproduction, one can then test one of the assumptions behind Smith and Fretwell’s model (1974): energy invested into reproduction is constant between species. On the other hand,

Grime’s (1977) three-part model for plant reproductive strategies based on disturbance and stress suggests that plants living at different ends of the three-part spectrum actually invest different amounts of energy into growth and reproduction. In any case, the data in

Fig. 6-8 offer a glimpse at how the species invest energy into reproduction. Although some significant differences are found, there is a generally similar investment in reproduction in each pairwise comparison, at least compared to the large difference between articulated and crustose corallines. If we take these results at face value, this indicates that the relatively low energy investment, seasonal reproduction, and perennial nature put crustose corallines in the competitive or stress- and disturbance-free regime

(Grime 1977). Articulated corallines, with their higher reproductive output, constant reproduction, and generally perennial nature (though they can be torn off in winter storms) lie somewhere between stress-tolerant and disturbance-tolerant regimes (Grime

1977). Neither description provides a perfect fit for articulated corallines, but they share traits of organisms that employ strategies based on both disturbance and stress.

As a last note, the numbers derived here are interesting when compared to other species. The tetrasporangial number per conceptacle was compared to the other species

183 analyzed in Chapter 5 (Fig. 6-14). These four species fall inside the trend formed by other species, indicating that they are not outliers or extreme in their reproductive characteristics. Edyvean and Ford (1986) found conceptacle density for Lithophyllum incrustans to be in the range of 100-500 conceptacles cm-2, a fair bit higher than the average values found here (Fig. 6-4), though individuals of P. neofarlowii were able to reach comparable rates of 400 conceptacles cm-2 in peak reproductive months (Fig. 6-2).

In a slightly different measure of reproduction, Schoschina (1996) estimated that a single individual of the fleshy red alga Phycodrys rubens can contain up to 150,000 (550,000 maximally) tetraspores in its autumn peak of reproduction. Based on the size estimates for this species in that same publication, plant area is probably on the order of 100 cm2, such that individuals probably contain on the order of 1,500 (5,500 maximally) tetraspores per cm2 at the peak of their reproduction. This is quite similar to the average tetraspore densities of the four coralline species in this study (Fig. 6-5), which had on average ~2,000 tetraspores per cm2.

In Fig. 6-7 spore production rates are estimated between 10-60 thousand tetraspores for each cm2 of surface per year. This is a bit lower than the numbers estimated for other species. Boney (1978) estimates that the fleshy red alga Rhodymenia pertusa releases about 15000 carpospores per carposporangial mass (the gametophyte reproductive structure) per day. Using other numbers in this publication, and assuming that this species is reproductive year-round (thus making this estimate biased towards high numbers if this assumption is wrong), this equates to 35 million carpospores per cm2 per yr, three orders of magnitude higher than the estimates in the present study. Gaylord et al. (2006) estimate that the giant kelp Macrocystis pyrifera releases about 108 spores

184 per plant per day. If we estimate that the total surface area of a M. pyrifera plant is on the order of 10 m2, or 105 cm2, then M. pyrifera produces about 370 thousand spores for each cm2 per year, about an order or magnitude higher than the estimates for the coralline algal spores here (Fig. 6-7). Scagel (1961) report higher rates for Nereocystis leutkeana, which produces 37x 1011 spores per individual per yr. Using a similar estimate for surface area of 105 cm2, we estimate the reproduction of this species at 37 million spores per cm2 per yr.

In conclusion, when looking at spore release rates per surface area, I was able to recover a familiar pattern in coralline algae: the species with faster life history traits were found to release significantly larger numbers of smaller spores, and vice versa. This pattern held only within taxonomic comparisons, however; generalizations between articulated and crustose corallines did not show significant differences.

6.4.2 Seasonality

A second focus of this chapter has been on seasonality in algal reproduction. The articulated species showed little seasonality in their reproduction (Fig. 6-3) in agreement with previous data for the subfamily Corallinioideae, but exhibited a large degree of intra-specific and even intra-individual variability (Table 6-2). P. neofarlowii showed a significant seasonal pattern, while L. grumosum showed the semblance of a seasonal pattern (Fig. 6-3). The pattern for P. neofarlowii was hard to fit into the previously defined categories for corallines, of either constant reproduction with a decline for a few months, or no reproduction with a peak for a few months (Chihara 1974). P. neafarlowii has both a strong peak for a few months and decays to no reproduction for a few months, with intermediate reproduction rates for the rest of the year. P. neofarlowii belongs to the

185 Mastophoroideae subfamily, which is neither monophyletic (Broom et al. 2008) nor has a well-defined pattern with respect to seasonality. Therefore, there is no expected pattern for this species, but it generally follows the trend apparent for crustose corallines of having a strong seasonality.

Is there any environmental reason for why P. neofarlowii might choose to be unreproductive in late spring or early summer? Helmuth et al. (2010) found for other intertidal organisms that the late spring and early fall period is most likely to cause extreme thermal events for intertidal organisms at Hopkins Marine Station. P. neafarlowii responds to temperature stress by bleaching its surface (pers. obs.). In the yearly seasonality data, I noted that sections of some but not all crusts appeared bleached in the late spring and summer months of 2010 and 2011, indicating that they were likely experiencing temperature stress during this period. This generally overlapped with the period in which P. neofarlowii was not reproductive. Bleached tissue was first seen in early spring, and disappeared by late summer when the crusts had repaired themselves from the damage (pers. obs.). If the surface of the crust bleached while it was reproductive, it would no longer have energy to devote to reproduction and its spores would likely die. Thus, the seasonal pattern may be a way for the alga to prevent its reproductive structures from experiencing the most stressful portion of the year. Although

P. neofarlowii avoids the late spring extreme temperatures, its reproductive peak occurs during October and November, a period in which thermal extreme events are still relatively common. Therefore, avoiding temperature extremes is not by itself a satisfying explanation for P. neofarlowii’s reproductive pattern.

186 6.4.3 Individuality in algae

By looking at reproduction per surface area rather than per individual, this chapter attempted to look at reproductive energetics from an unconventional angle. Individuality can be hard to define in algae, but considering the importance of the concept of an

“individual” in considering evolutionary fitness and reproductive output, it would seem fair to at least try to understand what an “individual” alga would be if such a thing even exists.

First looking at the crustose species, I found no pattern for either species between conceptacle density and plant size (Fig. 6-10 and 6-11). This suggests that this measure of reproduction is independent of size, a common confounding issue for photosynthetic organisms. When a photosynthetic organism has indeterminate growth, larger individuals have more energy available to them and tend to be more reproductive per individual, as shown by Ford et al. (1983) for a crustose coralline. Thus one cannot look at reproduction per individual without controlling for size. By looking at reproduction per surface area, I am looking at a measure of reproduction that is proportional to energy input through sunlight. In this sense, I am looking at the proportion of energy input converted to reproductive material between different organisms, or how similarly sized portions of algae that have a different total size use the same amount of energy in different ways.

This is different than comparing the reproduction rate per individual of two animals of different sizes, since those animals have access to very different energetic resource pools for reproduction. For crustose corallines, I found no pattern between reproduction per surface area and size, which indicates that the total size of the organism does not matter from the perspective of reproductive energy investment per area. In a sense, this fact

187 allows one to treat the alga as an aggregation of uncoordinated parts, in which each local group of cells acquires and uses its own energy independent of the rest of the colony.

For articulated corallines, the issue of individuality is especially apparent; should one consider an individual to be the fronds or the crust from which the fronds arise?

Since the fronds are all genetically identical, the genetic individual certainly exists on the level of crust. But for the purposes of reproductive energy investment, what an individual is may be very different. The spatial variability data indicate that although a decent percentage (36%) of variability can be explained on the level of frond (indicating that fronds are effectively independent of each other), the majority (52%-58%) of the variability can be explained on the level of crust, indicating that crusts act differently from other crusts, or that fronds within a crust show a degree of similarity. This is evidence that for articulated corallines, the level of crust may be the more appropriate definition for an individual.

Also interesting is the issue of size in articulated corallines. The most persistent pattern between reproduction and size was in C. vancouveriensis. These results indicate that bigger C. vancouveriensis fronds tend to be more reproductive (Fig. 6-10,11,12).

Although the fronds have indeterminate growth, Corallina fronds tend to get more reproductive as they get bigger, suggesting either that bigger fronds have more total energy available to them, or big fronds dedicate less energy to growth and more to reproduction. In either case, the frond appears to be acting more as a cohesive whole than as an aggregation of uncoordinated parts. The results were slightly different for C. tuberculosum, in which no pattern between size and reproduction was found for the majority of cases, although even in this case it should be noted that conceptacles are

188 rarely found on the most basal intergenicula, indicating that a certain size must be reached for reproduction to occur. After this size is reached, the results indicate no further trend between size and reproduction. In summary, for C. vancouveriensis at least, it may be somewhat appropriate to consider fronds, or even whole crusts, as individuals.

6.4.4 Are crustose and articulated corallines comparable?

Although this question was touched on in Chapter 5, it is worth revisiting here.

There were several major differences found in this chapter between articulated corallines and crustose corallines. The crustose species showed seasonality in their reproduction, whereas the articulated corallines did not (Fig. 6-3). The crustose species produced a considerably lower amount of reproductive biomass per unit area than articulated corallines (Fig. 6-8). Lastly, because of differences in how their conceptacle density scales with size, I reached different conclusions about the individuality of the crustose and articulated corallines (section 6.4.3).

All of these differences lead one to conclude that in terms of reproductive energetics, articulated and crustose corallines may not be comparable. Thus, when attempting to look at patterns across a range of species that span a multitude of spore sizes, it is recommended that one either focus on articulated corallines or crustose corallines, but not try to generalize across both groups.

189 Table 6-1

Various reproductive parameters for the four species under consideration, used for the transformations in Fig. 6-4 through 6-8.

Estimated Tetraspore Tetrasporangia Tetraspore tetraspore volume release rate Species per conceptacle diameter (µm) (µm3 x103) (# yr-1) Pseudolithophyllum neofarlowii 8 32.5 18 876 Lithophyllum grumosum 35 37.5 28 493

Corallina vancouveriensis 18 57.5 100 1606 Calliathron tuberculosum 45 69.5 176 2373

190 Table 6-2

Results of the nested ANOVA analysis on reproduction for both Corallina vancouveriensis and Calliarthron tuberculosum based on the spatial study. Conceptacle density was analyzed across 3 factors: site, crust, and frond.

Sum of F % of total Species Factor DF Squares value p Variance variance Corallina vancouveriensis Site 1 55 2.8 0.11 0.92 12.3% Crust 22 430 7.2 <0.001 3.9 51.8% Frond (residuals) 96 260 2.7 36.0%

Calliarthron tuberculosum Site 1 7 1.3 0.26 0.11 6.5% Crust 22 108 8.0 <0.001 1.0 57.5% Frond (residuals) 96 59 0.61 36.0%

191 Table 6-3

Results of the nested ANOVA analysis on frond length for both Corallina vancouveriensis and Calliarthron tuberculosum based on the spatial study. Frond length was analyzed across 3 factors: site, crust, and frond.

Sum of F % of total Species Factor DF Squares value p Variance variance Corallina vancouveriensis Site 1 5.8 32.9 <0.001 0.096 55.6% Crust 22 3.9 4.2 <0.001 0.035 20.3% Frond (residuals) 96 4.0 0.042 24.1%

Calliarthron tuberculosum Site 1 0.6 0.7 0.40 0.010 3.9% Crust 22 18 10.5 <0.001 0.17 65.0% Frond (residuals) 96 7.6 0.080 31.1%

192 Table 6-4

Results of the nested ANOVA analysis on frond area for both Corallina vancouveriensis and Calliarthron tuberculosum based on the spatial study. Frond area was analyzed across 3 factors: site, crust, and frond.

Sum of F % of total Species Factor DF Squares value p Variance variance Corallina vancouveriensis Site 1 1.5 6.5 0.018 0.024 15.3% Crust 22 5.0 2.5 0.001 0.045 28.3% Frond (residuals) 96 8.7 0.090 56.4%

Calliarthron tuberculosum Site 1 44 12.3 0.002 0.74 33.9% Crust 22 79 5.0 <0.001 0.72 32.9% Frond (residuals) 96 70 0.73 33.2%

193 Table 6-5

The results of the regressions between conceptacle density and a measure of plant length for the seasonality data. Significant effects are highlighted in bold.

Species Slope 95% CI R2 p DF Pseudolithophyllum neofarlowii 3.2 4.9 0.08 0.19 22 Lithophyllum grumosum -1.2 15.3 0.01 0.84 4

Corallina vancouveriensis 3.6 3.0 0.22 0.021 22 Calliathron tuberculosum -3.8 4.8 0.12 0.11 20

194 Table 6-6

The results of the regressions between conceptacle density and a measure of plant area for the seasonality data. Significant effects are highlighted in bold.

Species Slope 95% CI R2 p DF Pseudolithophyllum neofarlowii 0.50 0.83 0.07 0.22 22 Lithophyllum grumosum 0.77 5.8 0.03 0.73 4

Corallina vancouveriensis 7.2 4.6 0.32 0.0038 22 Calliathron tuberculosum -0.14 0.10 0.28 0.011 20

195 Table 6-7

Results of the regressions for Corallina vancouveriensis and Calliathron tuberculosum, using the spatial variability data. Results are analyzed by frond. Significant effects are highlighted in bold.

Species test Slope 95% CI R2 p DF Corallina vancouveriensis Length -0.58 3.2 0.001 0.72 118 Area 4.6 3.7 0.05 0.015 118

Calliathron tuberculosum Length 0.40 0.50 0.02 0.12 118 Area 0.0016 0.13 <0.001 0.98 118

196 Table 6-8

Results of the regressions for Corallina vancouveriensis and Calliathron tuberculosum, using the spatial variability data. Results are analyzed by crust. Significant effects are highlighted in bold.

Species test Slope 95% CI R2 p DF Corallina vancouveriensis Length -2.7 7.3 0.03 0.45 22 Area 2.8 11 0.01 0.62 22

Calliathron tuberculosum Length 0.49 1.2 0.03 0.41 22 Area -0.046 0.34 0.004 0.78 22

197 Figure 6-1

The mean reproductive rates of the four experimental species, shown across a year. Error bars are 95% confidence intervals based upon all of the individuals measured in that month.

198 Figure 6-2

Reproductive rates for the 12 individuals of Pseudolithophyllum neofarlowii that were followed for two years. Not that a spike in reproductive rate of similar timing and magnitude was present in both years.

199 Figure 6-3

An autocorrelation of the data from Fig. 6-1 and 6-2. Significance lines are shown in grey, colors correspond to the colors in Fig. 6-1.

200 Figure 6-4

The number of conceptacles per surface area for each of the four species, averaged over a year to account for seasonal patterns in reproductive rates. Note that this measurement was performed twice for the articulated coralline species- once over 24 individuals over the course of a year, and once at a single point in time over 60 fronds across 24 individuals to derive a less biased estimate. Note that the species are arranged from left to right in order of increasing spore and conceptacle size. Error bars show the 95% confidence intervals. PN= Pseudolithophyllum neofarlowii, LG=Lithophyllum grumosum,

CV=Corallina vancouveriensis, CT=Calliarthron tuberculosum.

201 Figure 6-5

Bar graph of spores per surface area. Note that the species are arranged from left to right in order of increasing spore and conceptacle size. Error bars show the 95% confidence intervals. PN= Pseudolithophyllum neofarlowii, LG=Lithophyllum grumosum,

CV=Corallina vancouveriensis, CT=Calliarthron tuberculosum.

202 Figure 6-6

Bar graph of reproductive volume per surface area. Note that the species are arranged from left to right in order of increasing spore and conceptacle size. Error bars show the

95% confidence intervals. PN= Pseudolithophyllum neofarlowii, LG=Lithophyllum grumosum, CV=Corallina vancouveriensis, CT=Calliarthron tuberculosum.

203 Figure 6-7

Bar graph of number of spores released per surface area per unit time, which is a proxy for spore production rates. Note that the species are arranged from left to right in order of increasing spore and conceptacle size. Error bars show the 95% confidence intervals.

PN= Pseudolithophyllum neofarlowii, LG=Lithophyllum grumosum, CV=Corallina vancouveriensis, CT=Calliarthron tuberculosum.

204 Figure 6-8

Bar graph of reproductive volume released per unit time. Note that the species are arranged from left to right in order of increasing spore and conceptacle size. Error bars show the 95% confidence intervals. PN= Pseudolithophyllum neofarlowii,

LG=Lithophyllum grumosum, CV=Corallina vancouveriensis, CT=Calliarthron tuberculosum.

205 Figure 6-9

The averaged reproductive rates from the seasonality data shown across spore size for the various measures of reproduction. These are the same data shown in Fig. 6-4 through 6-8: refer to those figures for units on the reproductive rates (not shown here due to space).

206 Figure 6-10

The four panels show number of conceptacles per surface area against a measure of plant length for the four species in the study, based on the seasonality data. The best fit regression line is shown. The parameters of the fit are given in Table 6-5. PN=

Pseudolithophyllum neofarlowii, LG=Lithophyllum grumosum, CV=Corallina vancouveriensis, CT=Calliarthron tuberculosum.

207 Figure 6-11

The four panels show number of conceptacles per surface area against a measure of plant area for the four species in the study, based on the seasonality data. The best fit regression line is shown. The parameters of the fit are given in Table 6-6. PN=

Pseudolithophyllum neofarlowii, LG=Lithophyllum grumosum, CV=Corallina vancouveriensis, CT=Calliarthron tuberculosum.

208 Figure 6-12

Conceptacle density is shown against both length and area for the two species used in the spatial variability data. The best fit regression line is shown. Results are plotted by frond.

The parameters of the fit are given in Table 6-7. CV=Corallina vancouveriensis,

CT=Calliarthron tuberculosum.

209 Figure 6-13

Conceptacle density is shown against both length and area for the two species used in the spatial variability data. The best fit regression line is shown. Results are plotted by crust.

The parameters of the fit are given in Table 6-8. CV=Corallina vancouveriensis,

CT=Calliarthron tuberculosum.

210 Figure 6-14

The spore size against spore number as shown in Chapter 5, but with the four species from this study included and highlighted in bold. To be consistent, maximal spore diameter was used for the species in this study.

211 Chapter 7

Temporal characteristics of the attachment strength of

coralline algal spores

7.1 Introduction

Offspring fitness can play a part in determining the type of reproductive strategy employed by an organism. In the traditional r-K paradigm set out by Smith and Fretwell

(1974), K-selected species are expected to produce offspring with greater fitness, or probability of surviving. Yet little is known about the relative fitness of algal spores from different species. For instance, we do not know if spore fitness even varies significantly between species or correlates with traits like spore size (Santelices 1990). For algal spores, there are many chances for spores to die between being released and germinating on a surface. To greatly simplify the process: spores must first disperse through the water column, then they must settle on the surface, and finally they must germinate into adults

(Santelices 1990). Loss of spores at any of these stages can reduce their fitness, though it is currently unknown how factors like spore size might influence the potential for spores to be lost at the various stages of this process (Steneck 1986). Of particular interest is the first stage of the settlement process, in which spores initially attach to surfaces. The size of a spore influences both its attachment force to the surface, and the extent to which the spores sticks up into the boundary layer and thereby experiences strong detachment forces. Therefore in looking for a determinate of spore fitness, the attachment process seems like a good place to begin.

212 Attachment in algal spores is known mostly from the work done on biofouling.

Biofouling studies have primarily investigated the effect of attachment on different types of surface, mostly for the purpose of developing a surface to which algae cannot attach

(Santelices 1999, Callow and Fletcher 1994). Most of this work has been performed on the spores of Ulva, a genus of green algae, and it is unclear how relevant such results may be to other groups of algae, such as the corallines studied in this thesis. Studies typically quantify attachment on a binary scale of attached or unattached at single time for a single force of detachment. Yet attachment is a time-dependent process, in which spores become more attached through time, through developing filaments that attach to micro- structures in surfaces (Steneck 1986), or through mucilage and epoxy-like resins that can harden over time (Fletcher and Callow 1992). When only looking at one detachment force at one point in time, it is unclear whether differences in attachment can be explained by the spores of one species being weaker, taking longer to attach, or by some combination of the two. Thus, to fully understand the differences in the attachment process between species of corallines, one must look at what I will term the attachment profile, or the patterns of attachment of spores over a range of both detachment forces and settlement times (for an example of this, see Christie 1970).

In this chapter I compare the attachment profile of two of the species used in

Chapter 6: Corallina vancouveriensis and Calliarthron tuberculosum. These two species are chosen because they are relatively closely related (Miklasz and Gabrielson, unpublished data) and many aspects of their biology have been well studied in the past.

For instance, we know that growth rate seems inversely related to thickness in corallines

(Colthart and Johansen 1973), and that C. vancouveriensis is thinner than C.

213 tuberculosum (Abbott and Hollenberg 1976). Thus, I will assume that C. vancouveriensis grows faster than C. tuberculosum. Padilla (1984) found that C. vancouveriensis was adapted to the desiccation stress of exposed intertidal conditions, whereas C. tuberculosum was adapted to the intense herbivory of tidepool conditions. Guenther and

Martone (in prep) have shown that C. vancouveriensis retains more water in its branches than C. tuberculosum, but also that the photosynthesis of C. vancouveriensis continues even when partially desiccated, and that it recovers from desiccation more consistently than C. tuberculosum. Although the breaking stress of the joints in both species is quite similar (P. Martone, unpublished data), C. tuberculosum can withstand greater forces than C. vancouveriensis due to the greater cross-sectional area of its intergenicula. In addition to this wealth of comparative studies, I have shown in Chapter 6 that by several different measures, C. vancouveriensis produces more smaller spores than C. tuberculosum, although both seem to be investing a similar amount of energy into reproduction.

Based on these traits, it appears that C. vancouveriensis has fast, or r-selected, life history traits as compared to C. tuberculosum. As C. vancouveriensis produces more, smaller spores that those of C. tuberculosum and both put the same total energy into reproduction, one would (according to this scenario) expect that the spores of C. vancouveriensis would be less fit than those of C. tuberculosum (Smith and Fretwell

1974). Although measuring the evolutionary fitness of these spores is beyond the scope of this study, we can measure one factor that contributes to spore fitness, namely spore attachment ability.

214 In this chapter, I will compare the attachment profile of the spores of C. vancouveriensis and C. tuberculosum. This will determine both if there is much variability in attachment ability between closely related species, and if that attachment ability follows the expectations based on reproductive theory (Smith and Fretwell 1974) and Chapter 6. In collecting data on attachment profiles, several interesting question arose concerning the individuality of algal fronds and the potential for spores arising from the same parent plant to differ from each other. Thus, data were also collected on the release and attachment of spores to determine the extent to which algal fronds acted like an coordinated individual or an aggregation of uncoordinated parts

7.2 Methods

7.2.1 Shear flume design

To measure the attachment strength of spores, I constructed a shear flume. Shear is the force produced on a surface due to the viscosity of a moving fluid and its interaction with the no slip boundary condition. Shear acts in the direction of flow and parallel to the surface. Shear is typically the greatest force on small objects that are located near to surfaces, and thus is the most likely force to dislodge small algal spores

(Vogel 1994). The shear flume design was modeled after Schultz et al (2000), except that

I used an open rather than closed flume type design and gravity rather than a pump to generate a pressure head to drive the flow. This design was chosen because a gravity powered design has less loss of energy due to turbulent mixing compared to a pump, and has a theoretically infinite achievable shear stress. To increase stress, one simply increases the height of a gravity powered pressure head.

215 A schematic diagram of the shear flume is shown in Fig. 7-1A. The main device consists of a stand pipe which feeds into the working section. The stand pipe has several overflow ports and valves which allow the height of the water, and therefore shear stress, to be maintained at a constant level throughout a trial. A reservoir sits above the stand pipe and fills the stand pipe in trials. A valve links the stand pipe to the reservoir, and allows the inflow of water in the stand pipe to match the outflow of water from the working section and the overflow valves. The reservoir is kept constantly full by a sump pump contained in a large container, which receives seawater both from the outflow of the working section and from a fresh seawater tap. A cap was made to fit on the end of the working section, such that the flow could be halted allowing measurements to be made while keeping the stand pipe filled with seawater.

The working section consists of an acrylic pipe with a 9 cm square cross-section, which at one end narrows to a 2.1 mm tall by 9 cm wide acrylic channel. A trip (sudden contraction of the pipe which “trips” the flow and induces turbulence) which blocks

>15% of the area of the opening was introduced as the upstream end of the channel

(Durst et al. 1998). This ensures that turbulence is fully developed in the downstream end of the working section, making the conditions similar to what spores are likely to experience in natural conditions.

In order to create an area where spores could be mounted in the shear flume, a section was cut out of the acrylic channel at the downstream end of the working section,

29.5 cm (140 channel heights) from the upstream end of the channel. This distance was used because it was well in excess of the 60 channel heights Durst et al. (1998) found were needed for turbulence to fully develop. A glass plate was mounted on the top side of

216 the channel, and an opening for three glass slides on which spores could be settled was

cut out of the bottom portion of the channel.

To allow the glass slides to be attached and removed from the mount, three holes

were drilled partway into the top face of the mount. A second hole was drilled into the

side of the mount, overlapping with the holes drilled through the top face and connected

by tubing to a vacuum pump. Slides were then placed over the three holes on the top face

of the mount and held to that face through suction (Fig. 7-1B). The recessed edge of the

mount was lined with silicone vacuum grease, and it was held in place during trials with

clamps. Aluminum bars were added to the bottom face of the mount and the top face of

the shear flume, so that the clamp did not stress the acrylic directly, and to offer further

structural support to prevent the narrow channel from bowing due to the force of the

clamp.

7.2.2 Calibration of the shear flume

The shear stress values imposed on spores by the shear flume can be derived as

given in Schultz et al. (2000), using the following formula and based on the assumptions

that the shear stress is measured in a short, wide and uniform channel:

h dp " # $ 7-1 w 2 dx

where τw is the shear stress, h is the height of the channel, and dp is the pressure gradient

! between two points in the channel separated by a distance dx. The negative sign simply

indicates that the shear stress acts in the direction of flow, or in the opposite direction of

the pressure gradient. Basically, one finds the pressure loss over a section of the channel,

and uses that to define the average shear stress over that section of the channel. I

measured the pressure distribution in two ways. First, I drilled holes in the working

217 section upstream and downstream of the slide mount, measuring the pressure drop across the entire mount. Second, I manufactured a second mount whose top surface extended into the shear flume as much as the slides on the mount would extend into the flume. I drilled two holes in this artificial mount to measure the shear stress over the section of the bottom surface of the shear flume where slides would be located

My two measurements of shear stress did not match, indicating that the assumptions behind the formula given in Schultz et al. (2000) were not correct. Namely, the pressure distribution was not uniform along the length of the shear flume.

Specifically, the shear stress measured on the local surface of the mount was 2-3x lower than the shear stress measured over the entire mount. A large pressure drop was found to occur around the downstream end of the mount, indicating that there was a sudden contraction of the channel at this end. Using the dial micrometer of a light microscope, I measured the interior gap between the top acrylic plate of the channel and the top surface of the mount, and found it to be significantly greater (>40%) than the height of the rest of the channel. Presumably, this local expansion of the channel height caused a local reduction in the flow speed, meaning that the shear stresses over the mount were reduced compared to the rest of the channel. Thus, the shear stresses found using the artificial mount were deemed more accurate and used to calibrate the shear flume.

As an additional complication, some of the grease used to seal the mount to the working section entered the channel, and caused an even greater constriction and pressure drop at the downstream end of the mount. This increased the pressure drop of the downstream end and caused lower shear stresses in the mount when this grease was present. The grease was cleaned out when this issue was discovered, but was present for

218 the measurements on C. vancouveriensis, but not C. tuberculosum. As a result, slightly

different shear stress values were used for each species.

The pressure difference between two points in the flume was measured by

attaching tubes to the holes in the working section of the shear flume. The tubes were

connected to glass tubes filled with water, such that the height difference (hy) between the

water in the two tubes corresponded to the pressure drop over that section of the shear

flume by thus formula:

dp = "ghy 7-2

where ρ is water density and g is gravitational acceleration. Note that differences in water

! height of less than 0.5 cm could not be measured accurately, which translates into a less

than 1 Pa uncertainty in the estimate for the shear stress. The pressure drop was measured

for each of the 5 experimental pressure heads set by the overflow valves in the shear

flume, thus allowing spores to be tested against 5 different shear stresses.

7.2.3 Species collection and testing

As noted in the introduction, experiments were limited to C. vancouveriensis and

C. tuberculosum. Fronds were collected and immediately stored in a sheltered, outdoor

seawater table in running seawater for at least 24 hr before testing. After sitting in the

seawater table for a week, algal fronds were observed to develop unnatural growth

patterns, and conceptacles were seen to degrade. Therefore, all tests were done within 5

days of collection.

For a shear flume test, a 1.5 x1.5 cm square was marked on a glass slide with a

permanent marker to indicate where to subsequently look for spores on the slide. The

marked side of the slide was placed face down, so that spores were not in contact with the

219 marker ink. Glass slides were used straight from the box (Caroline Biological supply slides, 25x75 mm), and were not reused. I placed three glass slides in a sealed 300 ml container filled with seawater. The slides were inserted into the water an hour before spores were released, thus allowing no time for a biofilm to develop. It is possible that the timing of the attachment process may be affected if a biofilm was allowed to develop on my glass slides.

For each test, seawater was collected from a seawater tap. This water is sand filtered but otherwise untreated. I have observed that the water often contains copepods and small worms. The water was collected in two glass jars and bubbled with atmospheric air to equilibrate it with atmospheric conditions. This was necessary because the water coming from the tap was collected at 17 m depth and can often be slightly acidic and/or hypoxic; I did not want daily variation in such conditions to affect my measurements. It was found that on a mildly acidic day, 20 min of bubbling in this manner was sufficient to bring the water into equilibrium with the atmospheric conditions

(i.e. the pH measured by a glass electrode was unchanged by more bubbling). To be safe, and to control for potentially extremely acidic conditions, seawater was always bubbled for at least 30 min. After bubbling, the water was poured into two containers, and the glass slides were inserted into the containers. The containers were then placed in a constant temperature bath at 13°C. Once the water in each container was under 14°C, fronds were inserted into the containers.

Each test involved six glass slides spread out over the two containers. Six fronds were haphazardly selected from the seawater tables for each test and brought back to the lab. Branches were trimmed from each frond and placed inside the square markings on

220 the slides. The size and number of branches was set by the number of conceptacles I estimated would be necessary to consistently release 50 tetraspores per slide per hour on average. Lids were placed on the containers, and the fronds were left to release spores naturally for 1 hour. After 1 hr, the lids were removed and the fronds were taken out.

Spores were then left to settle for a specified settlement time, measured from the middle of the release period. Accordingly, settlement times have an uncertainty of 30 min, meaning spores could have been released ± 30 min from the desired settlement time. For example, a nominal 2 hr settlement period includes spores that have between 1.5 and 2.5 hr to settle. Shear stress testing began as soon as the settlement period ended and proceeded to completion as quickly as possible, typically 20 min, adding an additional level of variability to settlement time.

A conservative approach was used in assessing spore release and determining which individuals to use. Among the six slides set up to release spores, shear tests would only be run if at least three of the slides produced good release events, where good release was defined as at least 16 spores, or 4 tetrasporangia, released. This was necessary because spores release is a temporally stochastic process for these species

(Bohnhoff et al., in prep). Not only did this approach tend to limit the analysis to days in which a sufficiently large number of spores were released to allow for the calculation of attachment percentage, but it also limited the analysis to “healthy spores days.” There were some days in which generally poor release was observed, and of the release that did occur, especially poor motility and attachment was also observed. In other words, on some days, all fronds investigated would release especially small numbers of what seemed “unhealthy”, poor quality spores. It is unclear if these days are caused by

221 previous stress to the frond, or from some uncontrolled conditions in the water. Some further testing indicated that both of these possibilities are playing a role. In any case, since poor numbers of release were correlated with poor spore quality, my conservative selection technique eliminated these poor quality spores from the analysis. In this manner, the attachment profiles can be thought of as the profile of healthy spores. When more than three slides produced good spores, I used the three slides with the highest release rates.

7.2.4 Shear measurements

Once the settlement period ended, several measurements were taken in the lab before subjecting spores to the shear flume. First, the number of spores released was counted. Then of the six slides, the three with the greatest release rates were rinsed, by lightly removing the slides from the water and plunging them back in three times. After rinsing, spore numbers were counted again. On the attachment profiles, this “rinse” was treated as a shear stress of 0 Pa.

Slides were then inserted into the shear flume, and a slight amount of water was run over them to keep them submerged. This amount of water created a low level of shear stress, and so was done in a controlled manner. The top 5.5 cm of the working section would be filled, producing 5.5 cm of pressure head and creating shear stresses of less than 0.5 Pa on the spores. The exact amount of shear stress created here was less than the resolution of the calibration technique described in section 7.2. For visual clarity, these points were assigned a value of 0.5 Pa. The end of the shear flume was then capped, cutting off the flow and allowing the spore number to be counted with a field microscope while the flume was filled to the next pressure level. Once the water level reached the

222 next higher overflow marker, the cap was removed and the water flowed at the specified pressure head and shear stress for at least 15 s, at which point the cap was replaced. This timing was chosen because Christie et al. (1970) found that, for Ulva intestinalis spores, most of the detachment occurred in the first 10 s of applied shear stress; longer times produced little additional removal of spores. Spore numbers were again counted and the process was repeated until all shear stresses were imposed. The slides were then counted one last time in the lab under a microscope after the final shear stress, which allowed for a duplicate measurement of spore numbers remaining after the final shear stress.

Sometimes it was found that spores counted in the lab under a higher quality microscope were missed with the field microscope in the shear flume. When possible, these spore numbers were added back into the totals.

This process was repeated on five different days for each desired settlement time.

With three slides used on five different days, this led to 15 different fronds being used for each settlement time.

7.2.5 Variability in spore release

I characterized the variability in spore attachment found for C. vancouveriensis, to understand how much fronds acted like “individuals” in that they released spores with similar attachment characteristics.

The preparation technique was similar to that used in the shear flume testing.

Fronds were collected 1-5 days before the experiment, and stored in an outdoor seawater table. On the day of the experiment, seawater was collected from the tap, bubbled to equilibrate the water to atmospheric pH, and cooled in a temperature bath. Fronds were then haphazardly selected from the seawater table, and branches were trimmed from the

223 frond and placed in the cooled seawater. The branches were then left in the seawater for one hour to allow time for spores to release. After an hour, branches were removed, and spores were given a settlement time of 4 hr (7.5 hr after the removal of the branches) to attach to the glass slides. A settlement time of four hours was chosen because it was the settlement time that showed the largest variability in spore attachment (see section 7.3.1).

To assess attachment strength, the tip of a Pasteur pipette was placed near the spores, and the bulb of the pipette was quickly pressed to induce a flow near the spore and thus subject them to a strong shear stress. The shear stress imposed by pipettes in the manner exceeded that of the highest shear stress level in the shear flume. Evidence of the magnitude of shear stress is provided by the fact that spores that were able to remain attached in the shear flume could not always remain attached after the pipette test. The exact amount of imposed shear could not be calculated, and likely varied a bit from slide to slide depending on the position of the pipette relative to the spores and the speed at which the bulb was compressed. To control for this variability, three to four blasts of water from the pipette were used on each slide. This technique was used rather than the shear flume because it allowed for a large number of tests to be conducted in a short time.

Two experiments were performed to assess variability, both using a three-factor nested ANOVA approach. For the first analysis, I used date, frond, and branch as my factors. Once a week for 5 successive weeks I collected a sample of seawater from the tap in a large flask, enough to fill 5 containers of slides. The water was bubbled for an hour and spread among the containers. On each day, five branches from each of nine fronds were used. Each branch from one frond was placed into a different container, such that each frond had one branch in each container. “Branch” is used as a loose term; the

224 branches were relatively large in that they contained several subbranches and between

10-30 conceptacles on average.

For the second analysis, I used frond, branch, and hour as my factors. This test was performed on one day, in which I used 3 branches from each of 3 fronds. Each branch was used in 5 successive hours to generate 5 samples of spores that originated from the same frond but differed in the hour in which they were released. Each of the nine branches was transferred to a new container after an hour of release.

In both studies, conceptacle numbers were counted and calculated to estimate release rates per conceptacle. Spore numbers were counted before and after a blast with a

Pasteur pipette to determine the percentage of attached spores.

7.3 Results

7.3.1 Attachment profile

The average attachment profile for C. vancouveriensis is shown in Fig. 7-2. As seen in this profile, a greater percentage of the spores are able to attach at any given shear stress when given more time to settle. C. vancouveriensis spores were “fully attached” or able to survive the strongest shear of the shear flume, after 24 hr of settlement, with 50% of the spores achieving full attachment in as little as 4 hr. Note that some spores (~10% on average) were able to achieve attachment within 1 hr ( ±30 min) of release.

The average attachment profile tells only part of the story. All fifteen individuals tested at each settlement time are plotted together in Fig. 7-3. The total range of attachment at each settlement time is shown in the shaded region. Notice that although the average profile indicates that attachment increases with time, there is substantial variability between spores from different individual fronds in their attachment ability.

225 These ranges are often overlapping, meaning the individual from which the spores come may have a greater effect on spore release than does settlement time. Most striking is the four hour settlement time tests, which show the greatest variability. Anywhere between 0-

90% of spores show full attachment in this time period.

The average settlement profiles for C. tuberculosum are shown in Fig. 7-4. The general shape of the profile is similar to that of C. vancouveriensis, but the attachment process itself occurs on a much slower timescale. After 24 hr of settlement, only 50% of

C. tuberculosum spores have reached full attachment on average, while 100% attachment seems to take more than 48 hr. Similar to C. vancouveriensis, C. tuberculosum shows high variability in its attachment profile between individuals, with the highest variability occurring at the intermediate attachment time of 24 hr (Fig. 7-5).

7.3.2 Variability in spore release

Variability in spore release rates is shown in Table 7-1 and 7-2. Notice that in both tests, the greatest percentage of the variability (around 60%) was always found at the lowest level of the analysis, for branches within fronds (Table 7-1), and for hours within branches (Table 7-2). A significant effect of frond on release rate was found in the first test (Table 7-1), but not the second (Table 7-2), indicating that release rates may significantly vary between fronds. However, there is about 1.6 times as much variation between different branches in a single frond as between different fronds in both tests. The great variability at the lowest level indicates that release rates are extremely variable, even within a single branch tested at subsequent hours. Note that an almost statistically significant effect is seen for date, an indication that there are some days that show strong

226 or poor release rates. This was mostly due to one day that showed particularly bad release rates: if that day is removed from the analysis, date has a highly insignificant effect.

Variability in spore attachment ability is shown in Table 7-3 and 7-4. Note that again a majority of the variability (~60%) is always present at the lowest level of the analysis. In this case, though, significant effects were found at the level of date and frond in both analyses, indicating that spores from different fronds, or collected on different days, show different levels of attachment. Notice that there was no effect of branch on spore attachment. Once again, most of the variability occurs between spores that originate from the same conceptacles but are released at different times, indicating that the majority of the variability in spore attachment is related more to individual spores themselves than anything about the parent frond from which they originate.

7.4 Discussion

7.4.1 Observations on the stages of the attachment process

As is common for spores of many algal species, I observed the coralline algal spores in this study to undergo a multi-stage attachment process. First, let’s examine the attachment process know for other algae. In Ulva, researchers have found a multistage attachment process (see Fletcher and Callow 1992 for a review). In the first stage, a spore finds and attaches to a surface through the use of its flagella (Callow et al. 1997). After this initial attachment, spores release a mucilaginous substance which attaches the spore to the surface. Later, around the time of spore germination, a second, more permanent, adhesive material is released from the Golgi bodies (Fletcher and Callow 1992). Red algal spores do not have flagella, but there is some evidence that they too may have a multi-stage attachment process. Although red algae spores are released in a

227 polysaccharide-based mucilaginous package which plays a part in initial attachment, they also release a secondary glycoprotein adhesive from their Golgi bodies (Fletcher and

Callow 1992). Red algal spores also locomote on surfaces (Pickett-Heaps et al. 2001), a process which must occur after a primary attachment to a surface has occurred, but before a permanent adhesive has been released. Johansen (1981) noted that coralline algal spores in particular will at some point “flatten” to the substrate, transforming from a nearly spherical to a discoid shape. Jones and Moorjani (1974) also report that coralline spores undergo a flattening in the attachment process, which causes a 20% increase in spore diameter. Additionally, Jones and Moorjani (1974) report a “halo” of refractive material surrounding the spores, and in one of their species the halo had several concentric rings, indicating different types of material being released at different times.

I was able to observe the distinct flattening process in spores of C. vancouveriensis, and looked for the presence of stages in the attachment profile data.

Specifically, was the flattening process related to observed increases in attachment strength? After spores were observed to flatten, they always survived even the highest test of the shear flume, and survived the shear produced by the pipette tests. However, some unflattened spores also had the potential to survive the strongest shear stresses of the shear flume and the pipette test, so flattening always was correlated with, but was not necessary for achieving, high attachment strength.

Spores of C. tuberculosum were never observed to flatten, even in the 48 hr tests.

At one point, I left C. tuberculsoum spores in a container for a week in the temperature bath, and at the end of the week still failed to observe spore flattening. Additionally, those spores did not withstand the shear stress induced by the pipette test after any length

228 of settlement time. In this sense, although nearly 100% of the spores of C. tuberculosum were able to achieve sufficient attachment in 48 hr to withstand the shear forces created by the shear flume, 0% of spores were ever able to survive the pipette test. The implications of this are discussed in section 7.4.2

Flattening did seem to be a prerequisite for germination in either species.

Germination in C. vancouveriensis occurred in some spores in as little as 8 hours, but only for spores that were flattened. C. tuberculosum’s spores never flattened and also did not germinate.

Based on the shape of the attachment profile (Fig. 7-2 and 7-4), I can additionally describe the attachment of C. vancouveriensis and C. tuberculosum as an asynchronous but extremely rapid process. The process is asynchronous because some, but not all, of the spores reach strong attachment at an intermediate settlement time, indicating that different spores initiate the attachment process at different times. This is in contrast to a synchronous process, in which all spores would initiate the stages of the attachment process at the same time. Extreme synchrony can be define as all spores being either attached or unattached at a given settlement time. If this were to happen, I would observe values of only 0 or 100% in the attachment profile. Extreme asynchrony, on the other hand, is in part a subjective measure that depends on the timescale of asynchrony. That is to say, spores may be asynchronous on the time scale of seconds, meaning the timing of the attachment process is misaligned between spores on the order of seconds. Given the measurement techniques in this study, this short term asynchrony could not be observed, and asynchrony over timescales of less than 20 min would appear synchronous. The timescale of measurement thus in some sense informs the scale of asynchrony. Since my

229 settlement time occurs on the scale of hours and we still see asynchrony in the data, the settlement process of spores is misaligned by hours. In short, settlement in these species is asynchronous at the scale of hours.

The attachment process is rapid because spores go from low to high attachment very quickly, at least quickly relative to the time scales of settlement in these tests

(hours). The rapidity of the process can be estimated from the fact that very few spores become unattached between the initial exposure to 1-3 Pa and their subsequent exposure to 12-15 Pa twenty minutes later. To illustrate this point, let’s assume that a spore’s attachment is extremely rapid, to the point that attachment is essentially instantaneous. In this scenario, the spore may or may not wash off the slide after the initial application of

1-3 Pa of shear stress, depending on whether the spore has begun to attach by this point in time. But once the spore has begun to attach, it becomes fully attached nearly instantaneously, and applications of higher shear stress will fail to remove the spore.

Thus only these spores that have attached will survive the imposition of 3 Pa., and because the survivors are fully attached, those that survive 3 Pa. will also survive 15 Pa.

Therefore a flat, horizontal line between attachment percentage and shear stress at a given settlement time is evidence of rapid attachment. Conversely, let’s assume that a spore has a slow attachment process that occurs on the timescale of hours. In this case, a spore that has begun to “attach” and does not wash off with a low shear stress of 1-3 Pa may still be only partway through its attachment process. There is the possibility, then, that this spore will wash away at some intermediate shear stress. Slow attachment of spores would thus appear as a significant negative slope between attachment percentage and shear stress at a given settlement time. Therefore for comparative purposes, the rapidity of the attachment

230 process can be qualitatively evaluated by the slope of the attachment percentage across shear stress. In the present data, this slope is relatively shallow (compared to other data, see section 7.4.2), indicating that spores attach rapidly.

To summarize the multistage attachment process of the spores in this study, coralline spores undergo an initial attachment which is often quite weak (cannot survive shear stresses in excess of 3 Pa). Spores can then proceed to a secondary attachment phase. The transition to this phase can proceed quite rapidly, potentially on the order of minutes or less. Once in this phase, spores are quite adherent, and can withstand shear forces in excess of 12 Pa. The transition to this phase, though, can be delayed for as much as tens of hours. In this phase, spores are still capable of locomotion (see section 7.4.4).

Spores then can proceed to a flattened state. Once flattened, spores are rarely if ever dislodged by shear forces, though they are no longer capable of locomotion. The transition to this flattened phase is likely rapid. Once the spore is flattened, it can proceed to the final phase, germination.

7.4.2 Comparisons to previous data

The results here compare well to the conclusions drawn from other algal species based on incomplete attachment profiles (either across time or shear stress, but not both).

Charters et al. (1973) present sparse data for the attachment profile of several red algal species. To summarize, they found an asynchronous attachment process for the three species examined, with the one species for which they had enough data also showing a relatively rapid attachment process. In two of the species, attachment increased with time on a similar timescale to the species in the present study, with 50% attachment occurring in the 10-15 hr range and with full attachment occurring in 24 hr or more. Christie et al.

231 (1970) show a full attachment profile for Ulva intestinalis. Their data show an asynchronous, but slower process for these spores as compared to coralline spores. Pang and Shan (2008) looked at the attachment of brown algal zoospores across shear stress, and, unlike the current study, found a maximal shear stress at which the brown algal spores could stay attached. They used a single settlement time of only 1 hr in their study, so it is possible that with longer settlement times brown algal spores could achieve much stronger attachment. Note that Pang and Shan studied a subtidal species, and the authors noted that within an hour, the spores were attached with enough strength to withstand most subtidal currents.

Most useful for comparative purposes, Jones and Moorjani (1974) looked at attachment in two articulated coralline species across settlement time. Jania rubens had spores similar in size to C. tuberculosum, 75 µm. J. rubens began to attach in as little as 2 hr, though all spores were not attached even after 4 hr. The first cell division was reported at around 9 hr, which is a quite similar timescale to that of C. vancouveriensis in both attachment and germination. The other species examined by Jones and Moorjani (1974) was Corallina officinalis, which had an average spore size similar to C. vancouveriensis,

58 µm. C. officinalis first began attachment in as little as 5 hr, but not all spores were attached after 32 hr (a timescale for the first cell division was not given). There are two interesting conclusions from this work. First, Jones and Moorjani (1974) were led to the same conclusion as myself when considering their incomplete attachment profiles: that the attachment process itself was likely quite rapid, but the onset of the attachment process both takes several hours and is asynchronous, or “delayed between individuals” to use their language. Second, the species which showed faster attachment (J. rubens)

232 also was the thinner, faster growing species of the two, but also had the larger spores.

Thus in both Jones and Moorjani (1974) and the present study, the species with the faster growth shows faster attachment, though attachment speed did not correlate with spore size.

As a further comparison to the crustose species used in Chapter 6, I noted some interesting observations when measuring the release rates of those algal species. For this measurement, I allowed each of the four species to release spores continually for 24 hr, at which point I counted the number of spores released for use in the calculations employed in Chapter 6. The spores left after these tests had settled on the slides for anywhere between 1-24 hr. For C. vancouveriensis, the results were as expected, some spores showed flattening and some even germination, but many were hardly attached. For C. tuberculosum, no spores were flattened or germinating. For the two crustose species, though, flattening and subsequent germination were observed, and in a greater percentage of the spores than in C. vancouveriensis. For P. neofarlowii in particular, many spores had already reached an 8-celled stage, farther along in the germination process than C. vancouveriensis. Thus the crustose species seem to show faster attachment and germination than the articulated species. Considering that the crustose species also had smaller spores than the articulated species, and that P. neofarlowii had the smallest spores, we can note a general pattern, in these four species at least, that the attachment and subsequent germination process increased in speed as spore size decreased.

7.4.3 Benefit of attachment profile shape

As noted, the attachment profiles for both C. vancouveriensis and C. tuberculosum show an asynchronous, but extremely rapid attachment process, likely

233 occurring on the timescale of minutes or less. Although it is somewhat surprising that spores can achieve attachment so quickly, waves and the corresponding forces that might dislodge corallines occur on the timescale of tens of second. For algal spores to be able settle in between wave cycles, they must have an attachment process that works on a similar timescale. Thus, to correlate the results derived here to field measurements, a better analysis on the timescale of the attachment process itself should be performed. To do so is extremely difficult, however. As noted, spores seem to delay the onset of their attachment process by hours to days, before quickly attaching to a surface in minutes or less. How one might study a process which may stochastically occur anywhere between 0 and 8 hours, but when it does occur happens in a minute or less, is a challenge in itself.

At the very least, a much different method of measuring attachment must be found, since the one used here takes 20 minutes.

C. tuberculosum displayed an extremely slow attachment profile. This apparent slowness can be explained in one of two ways. First, it is possible that the attachment process is still as rapid as that of C. vancouveriensis, but just delayed for a longer period of time. In a wave swept environment, this may give C. tuberculosum spores more time to disperse farther from their parent plant. In this scenario, one runs into a complication.

If one assumes that the attachment for C. tuberculsoum is equivalent to but simply slower than the process in C. vancouveriensis, then one must also conclude that C. tuberculsoum is a fair bit weaker than that of C. vancouveriensis based on the results with the pipette test, and does not involve the typical flattening process described for other coralline species. If C. tuberculosum is actually weaker than C. vancouveriensis, then combined

234 with the results from Chapter 6, we can conclude that C. tuberculosum is producing fewer and weaker spores than C. vancouveriensis, opposite of our expectation.

The second possibility is that the C. tuberculosum spores are doing something quite different from C. vancouveriensis spores, and did not actually complete the attachment process over the course of the experiment. In particular, C. tuberculosum spores might differ from C. vancouveriensis spores by requiring some sort of cue to complete their attachment process, perhaps because they are more selective than C. vancouveriensis spores. Consider that the spores experienced a relatively artificial environment; a smooth, glass slide in constant temperature and light conditions, in still water. For a more r-selected species like C. vancouveriensis, it may just be capable of settling in whatever environment it can find, as soon as it can find one. But as a more K- selected species, C. tuberculosum may need a specific set of cues to induce settlement in a particularly favorable environment, and those cues may not be present in a laboratory setting.

Another interesting component to this story is the asynchrony evident in the shape of the attachment profile. There are several possible causes for this asynchrony. First, the spores may not attach in synchrony because they experience different microenvironments. In particular, spores were often seen to settle in clumps. Clumps of spores also were often better attached than individual spores (pers. obs.), indicating that presence of conspecifics (Pang and Shan 2008, Callow et al. 1997) or of a edge (Callow et al. 2002) was inducing settlement and attachment, as had been shown in other algae.

Under this view, spores that had happened to end up in clumps attached sooner than those that had not, causing an asynchrony in attachment due to differential local stimuli.

235 Therefore spores would have attached in more or less synchrony, or their process is governed by the same set of triggers, but variability in external conditions causes asynchrony. Thus asynchrony might be caused by the variable rate at which spores find suitable habitats.

The second possibility is that asynchrony is an intrinsic property of the spores.

Under this view, two spores experiencing the exact same external conditions would still settle at different times, since the cellular machinery that governs the attachment process in inherently stochastic. Since there was still differential attachment ability between spores not in clumps, there is some evidence that this may be the case.

Differentiating between these two causes of asynchrony is important in extrapolating results to natural conditions. If spore asynchrony is caused by external stimuli, then it seems possible that when spores find a suitable environmental cue, the spores will settle in that location. In this case, the asynchrony could represent a sorting of spores to favorable microenvironments. On the other hand, if asynchrony is an inherent part of the cellular machinery, then some spores would settle quite soon after release, close to their parent plant, and some might disperse quite far away from their parent.

Thus, in this case, asynchrony would be a mechanism to ensure a wide dispersal kernel.

7.4.4 Variability in attachment and individuality in algae

There was a great amount of variability in the attachment process of the spores of both species (Fig. 7-3 and 7-5). Although the variability in part could be a cause of the asynchrony in attachment, there is still an enormous variability in the percentage of attachment occurring at any given settlement time. In other words, although the fact that at some settlement time only 50% of the spores are attached indicates asynchrony, the

236 percentage of spores that are attached at a given settlement time is also extremely variable from frond to frond. And as was shown in the variability analysis, this variability occurs not only between different fronds, but between different branches on the same frond (Table 7-3), or even between spores released at different times from the same branch, or the same set of conceptacles (Table 7-4).

There are several possibilities for why spores show such variation in attachment ability. First, it is possible that spores simply have differential fitness. Some spores may simply have been given more resources by the parent plant than others, allowing them to attach faster or stronger. In support of this idea, there is some variation in spore size around the mean (Table 6-1), on the order of ±10 µm. Second, as noted before, spore attachment may be induced by proximity to neighbors. Spore release from conceptacles is a temporally stochastic process in C. vancouveriensis (Bohnhoff et al., in prep), such that the proximity of spores to neighbors is randomly determined by the number of conceptacle that release spores around a particular spore’s current position. Thus this stochastic nature of spore release can lead to a random distribution of spores whereby some have close neighbors and some do not. This can lead to spores in clumps attaching better than those that are not, creating variability in spore attachment that exists on the level of individual spores. Third, it has been noticed that red algal spores tend to locomote on surfaces (Pickett-heaps et al. 2001). Spores that locomote may not be as fully attached as those that do not, potentially explaining the variability in attachment. Of course this still leaves open the question of why is there exists variability in when, and by how much, spores move. Some preliminary experiments indicate that locomoting spores

237 can still survive the shear stress created by the pipette test and thus do not appear weaker than spores that are not locomoting, but this issue certainly deserves further study.

Lastly, I’d like to return to the issue brought up in Chapter 6, of algal individuality. In the last section, it was suggested that fronds of C. vancouveriensis may act in a coordinated manner on the level of frond or even crust. The results of this chapter on spore release and attachment rates offer a slightly different picture. The majority of the variability in spore release or attachment rate was always explained by the lowest level of the analysis, on the level of branches, or of spores released from branches. This indicates that individual spores themselves act more differently from each other than spores coming from the same branches or fronds. This indicates that from the perspective of the spores, different components of the same frond act functionally different from other components of the same fronds, supporting a view of the frond as an aggregation of uncoordinated parts. Counter to this point though, a significant effect on attachment rate in both analyses was found on the level of frond (Table 7-3 and 7-4), indicating that there was a sense in which spores coming from the same frond did act similar to each other.

Although statistically significant, frond could account for only about 24% of the variance in each test, compared to the approximately 62% of variance explained by the lowest level of each nested ANOVA. Thus even here, there is a sense in which frond seems to be acting as a coordinated individual.

7.4.5 Evolution of spore size

In determining patterns between spores size and number, it is crucial to understand how spore fitness scales with spore size (Smith and Fretweel, 1974). In the introduction to this chapter, I suggested that attachment ability may be one of the main

238 factors that constrains spore fitness across size. The data presented here (Fig. 7-2 and 7-4)

found no upper limit to the forces that a spore could withstand, indicating that they could

attach well in excess of the forces generated by the shear flume. Although shear stress

was not a constraining factor on the attachment of spores of the size used in this study, it

is possible that such forces could limit the maximal size of a spore, once spores achieved

sizes in excess of the sizes examined here. If large spores experience forces in excess of

what they can withstand, such spores would be unviable in many wave-swept conditions.

In support of this, Ngan and Price (1979) find larger-spored species only in low levels of

the shore where water velocities and forces are reduced.

From a fluid mechanics perspective, there is good reason to suspect that

deattachment forces on spores would increase with spore size. Let’s model a flattened,

attached spores as a cylinder projecting from the surface with radius r and height of 0.9r.

This shape correspond to an equivalent volume sphere with a radius 22% larger than the

cylinder, close to the 20% increase that occurs when spores flatten (Jones and Moorjani

1974). Forces acting to detach this “spore” include, shearing force (Fsh) and form drag

(FD), both of which act in the direction of fluid flow. Fsh is given by the following

expression:

Fsh = µkAsh (3-3)

where µ is viscosity of seaweater, 10-3 Pa s at 20°C, k is the rate at which water velocity ! increases with distance away from a solid surface, and Ash is the wetted surface area of

the spore. In the laminar sublayer of the boundary layer, velocity increases linearly with

distance. Thus, the near boundary velocity gradient is linear such that I assume k =U/y,

where U is the water velocity at some distance y from the surface. For a wave-swept

239 intertidal environment, k can be conservatively estimated as 104 s-1, though it may

potentially be 2-3 times as large (T. Hata, unpublished data). Substituting in the surface

area of the cylindrical spore:

2 Fsh = 2.8µk"r (3-4)

2 Thus, Fsh scales with r .

! On the other hand, the typical expression for form drag, FD, is given by (Vogel

1994):

1 2 FD = 2 "A prC DU (y) (3-5)

3 -3 where ρ is the density of seawater (approximately 10 kg m ), Apr is the projected surface ! area in the direction of flow, CD is the drag coefficient, and U(y) is the water velocity at a

distance y above the surface. If I consider the cylinder as a series of infinitesimally thin

circular sections, then the projected area of each section is 2rdy. To calculate forces on

the whole cylinder, I integrate over height, such that:

0.9r 2 FD = "r # C D (ky) dy (3-6) 0

If for simplicity I set CD to a value of 1:

! 2 4 FD = 0.24"#k r (3-7)

In this case, the form drag increases with r4, due to the fact that as size increases, the

! spore both has greater surface area and sticks up farther into boundary layer, both causing

the spore to experience greater form drag.

The spore sticks to a surface with an adhesive force (Fad). The force of adhesion

can be expressed simply as:

Fad = TAad (3-8)

! 240 -2 where T is the tenacity of the adhesive expressed in N m , and Aad is the area of the

adhesive surface. I will assume T stays constant with size. For our hypothetical spores:

2 Fad = T"r (3-9)

Thus attachment strength of spores scales with r2.

! For a spore to stay attached to a surface, the adhesive force must exceed the

combined shear force and form drag:

Fad > Fsh + FD (3-10)

which can be expressed more exactly as:

! T"r 2 > 2.8µk"r 2 + 0.24#"k 2r 4 (3-11)

T > 2.8µk + 0.24"k 2r 2 (3-12)

! 2 Since both Fad and Fsh scale with r , the adhesive strength can keep up with the shearing

4 ! force through size. FD, though, scales with r and so in conjunction with Fsh results in an

overall force that exceeds attachment force at some large size, creating an upper limit to

spore size. Of course, this also means that at small enough sizes, form drag decreases in

magnitude much faster than shear stress, and so should become a negligible component

of forces on small spores. Determining the point at which FD exerts a significant force on

spores be determined by taking the ratio of FD to Fsh, which is in essence a calculation of

Reynolds number of the spores (Resp) averaged over the boundary layer (Vogel 1994).

When I do such a calculation, I arrive at the following result:

2 FD 2.4"kr 2.4"U (y)r = = = Resp (3-13) Fsh 28µ 28µ

It is interesting to note that a ratio of 1, meaning a balance between shear and drag forces,

! occurs at a spore radius of about 30 µm. By a spore size of 100 µm, FD has becomes 10

241 times as great a force as that of Fsh. Also noteworthy, 100 µm is in the same order of

magnitude as the maximal size of spores (roughly 200 µm, Chapter 5). Thus, the maximal

size of spores occurs around the size at which form drag becomes the significant force on

the spores.

This exercise suggests that form drag increases fast enough with size to

potentially constrain the size range of spores. Under this hypothesis, we should expect

large algal spores to experience forces that may potentially exceed their attachment

strength, detaching at values for the velocity gradient, k, which smaller spores can survive

just fine. In particular, we can write out the expression for the critical value for the

velocity gradient (kcrit) at which spores become detached. When Resp<<1, Fsh is the

primary detachment force on the spores acting against Fad. Setting these forces (Eq. 3-4

and 3-9) equal to each other:

T = 2.8µkcrit (3-14)

kcrit "T (3-15) ! As can be seen in Eq. 3-15, kcrit has no dependence on size. Thus when Resp<<1, there is ! one value for kcrit and the corresponding shear stress at which spores become detached,

regardless of their size. On the other hand, when Resp>>1, FD is the primary detachment

force on the spores. In this case, using Eq. 3-7 and 3-9:

2 2 T = 0.24"kcritr (3-16)

#1 kcrit " r (3-17) ! Thus when Resp>>1, kcrit should decrease linearly with increasing radius. Thus kcrit should

! be constant as size increases until Resp=1, after which kcrit begins to decrease with

increasing spore size.

242 This analysis offers a ready hypothesis for the maximum size of spores, and how spore attachment should scale with size. In particular, if one can measure both T and typical values for k on a wave-swept shoreline, one can use Eq. 3-16 to find a critical size at which spores cannot attach strongly enough to survive days with even moderate fluid dynamic forces. Unfortunately, measurements of T for spores and kcrit for a turbulent, wave-swept intertidal environment may prove difficult. kcrit may be primarily determined by turbulent sweeps that penetrate the boundary layer and which vary in frequency and magnitude. The nature of turbulent sweeps is currently too ill-defined to be incorporated in the present analysis, but could require a more sophisticated model than the current one which ignores time dependant, extreme processes. In addition, T may be difficult to measure on a spore with a flexible cell membrane, even with appropriate micro-scale force measurement devices. Short of being able to measure T directly, one can measure the maximal shear stress (or kcrit) that spores can withstand across spore size, allowing one to test the scaling laws derived in Eq 3-15 and 3-17. Shear stresses of the magnitude required for this investigation were not achieved in the current study, but may be a useful focus for future studies.

243 Table 7-1

The results of the nested ANOVA to quantify variability in spore release rates in

Corallina vancouveriensis, with date, frond, and branch as factors. A square root transformation was used to normalize the data.

Sum of Percentage of Factor df Squares F value P Variance total variance Date 4 1.19 2.5 0.055 0.01 9.60% Frond 40 4.70 3.0 <0.001 0.02 34.06% Branch (residuals) 180 7.00 0.04 56.34%

244 Table 7-2

The results of the nested ANOVA to quantify variability in spore release rates in

Corallina vancouveriensis with frond, branch, and hour as factors. A square root transformation was used to normalize the data. This analysis goes to a level deeper than the results in Table 7-1, by breaking the variability in “branch” into variability between

“branch” and subsequent “hours”.

Sum of Percentage of Factor df Squares F value P Variance total variance Frond 2 0.24 1.9 0.234 0.01 14.71% Branch 6 0.39 1.9 0.105 0.01 23.64% Hour (residuals) 36 1.22 0.03 61.65%

245 Table 7-3

The results of the nested ANOVA to quantify variability in spore attachment rates in

Corallina vancouveriensis, with date, frond, and branch as factors. An arcsine transformation was used to normalize the data.

Sum of Percentage of Factor df Squares F value P Variance total variance Date 4 9.83 5.0 0.003 0.05 13.53% Frond 37 18.23 2.0 0.004 0.10 24.42% Branch (residuals) 113 28.29 0.25 62.05%

246 Table 7-4

The results of the nested ANOVA to quantify variability in spore attachment rates in

Corallina vancouveriensis with frond, branch, and hour as factors. An arcsine transformation was used to normalize the data. This analysis goes a level deeper than the results in Table 7-3, by breaking the variability in “branch” into variability between

“branch” and subsequent “hours”. Since the effect of branch was highly non-significant

(p>0.25), the Sum of Squares and degrees of freedom were pooled when calculating the F and p values for the effect of Frond.

Sum of Percentage of Factor df Squares F value P Variance total variance Frond 2 0.99 5.6 0.007 0.03 23.69% Branch 6 0.61 1.2 0.330 0.02 14.75% Hour (residuals) 36 3.07 0.09 61.56%

247 Figure 7-1

A) Diagram of the shear flume in side view. The reservoir is filled with water through a submersible pump in a cooler. The water flows from the reservoir to the stand pipe through a control valve, until it reaches one of 5 overflow ports. The water travels through the stand pipe into the working section, where the spores are mounted on the glass slides on the slide mount, shown in purple. A cap fits on the end of the working section as shown to stop the flow between shear tests. Water that leaves the working section enters a cooler that contains the submersible pump, thus recycling the seawater back into the reservoir. The set-up is illustrated with water held at the fourth overflow port. B) Close-up diagram of the slide mount, in both top and side view. Slides are placed over the O-rings shown in red, and held to the mount through suction. Internal tubing

(shown in blue dotted lines) connects to a vacuum pump.

248 249 Figure 7-2

The attachment profile for Corallina vancouveriensis, shown across shear stress and attachment time.

250 Figure 7-3

The attachment profile for Corallina vancouveriensis, shown across shear stress and attachment time. The plots are broken up by settlement time, and the data from all 15 individual fronds are shown by the lines. The shaded area maps out the range of attachment profiles of each frond, with the average profile for each settlement time shown by a bold line. These averages are the same lines shown in Fig. 7-2.

251 Figure 7-4

The attachment profile for Calliarthron tuberculosum, shown across shear stress and attachment time.

252 Figure 7-5

The attachment profile for Calliarthron tuberculosum, shown across shear stress and attachment time. The plots are broken up by settlement time, and the data from all 15 individual fronds are shown by the lines. The shaded area maps out the range of attachment profiles of each frond, with the average profile for each settlement time shown by a bold line. These averages are the same lines shown in Fig. 7-4.

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