Investigations of Swimming Microorganisms under

Variable Apparent Weight

by

Iiyong Jung

BS : Kyungpook National University, South Korea 2002

MS : Kyungpook National University, South Korea 2005

A dissertation submitted in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

in the Department of Physics at Brown University

Providence, Rhode Island

May, 2015 c Copyright 2015 by Ilyong Jung This dissertation by Ilyong Jung is accepted in its present form by

the Department of Physics as satisfying the dissertation requirement

for the degree of Doctor of Philosophy.

Date

James M Valles, Jr., Advisor

Recommended to the Graduate Council

Date

Thomas R Powers, Reader

Date

Jay X Tang, Reader

Approved by the Graduate Council

Date

Peter M Weber, Dean of the Graduate School

iii Curriculum Vitae

 Author

• Ilyong Jung

• Date of Birth : August 20, 1976

• Nationality : South Korea

 Education

• BS : Kyungpook National University, Physics (2002)

• MS : Kyungpook National University, Nano-Science and Technology (2005)

• PhD : Brown University, Physics (2014)

 Honors and Awards

• Shirley Chan Award from the American Physical Society (2014)

• Brain Korea 21 Scholarship (2003-2006)

 Membership in Professional and Honor Societies

• Sigma Xi (2014-Present)

• The American Physical Society (APS) (2011-Present)

• The Division of Biological Physics of The APS (2011-Present)

• The Division of Fluid Dynamics of The APS (2011-Present)

• Korean Physical Society (2005-Present)

iv  Publications

7) Ilyong Jung, Karine Guevorkian, and James M. Valles, Trapping of Swimming

Microorganisms at Lower Surfaces with Increasing Buoyancy, Physical Review Letters

113, 218101 (2014)

6) Ilyong Jung, Thomas R. Powers, and James M. Valles, Evidence for Two Ex- tremes of Ciliary Motor Response in a Single Swimming Microorganism, Biophysical

Journal 106, 106 (2014)

5) Jung, I. Y., Lee, S. M., and Sohn, S. H, Optical Properties of (Y, Gd)BO3:

3+ 2+ Eu and BaMgAl10O17: Eu Phosphors Coated with SiO2 Nano Particles for a

Plasma Display Panel, SID Symposium Digest of Technical Papers 38, 1325 (2012)

4) Lee, S. M., Chung, K. Y., Jung, I. Y., Sohn, S. H., Jeong, B. H., Jeong, Y.

C., Kwak, M. G. and Park, L. S, Optimization of Discharge Gases for a Mercury-free

Flat Fluorescent Lamp, SID Symposium Digest of Technical Papers 38, 503 (2012)

3) James M. Valles, Jr., and Ilyong Jung, Probing the Force Sensitivity of Swim-

ming Micro-organisms with Intense Magnetic Fields, Mag Lab Reports 18, 5 (2011)

2) Il Yong Jung, Sang Mock Lee, Jung Hyun Lee, and Sang Ho Sohn, Optical

2+ Characteristics of the BaMgAl10O17: Eu Phosphor Coated with SiO2 Nano Film by

Using a Radio Frequency Sputtering Method, Molecular Crystals and Liquid Crystals

470, 129 (2007)

1) I. Y. Jung, Y. Cho, S. G. Lee, S. H. Sohn, D. K. Kim, D. K. Lee, and Y. M.

2+ Kweon, Optical Properties of the BaMgAl10O17: Eu Phosphor Coated with SiO2

for a Plasma Display Panel, Applied Physics Letters 87, 191908 (2005)

v Acknowledgments

I would first like to thank my supervisor, James M Valles for all of his help and guidance. Words cannot express my deep appreciation and gratitude to him. Without his support, I could not complete my dissertation at Brown. Jim taught me how to be a good physicist, but I learned much more than physics itself. His good deeds, personality, patience, generosity, consideration, and curiosity became exemplary to my life in all ways. Thank you very much Jim. You are the best luck in my life.

I also want to thank my thesis committee members, Thomas Powers and Jay

Tang. They happily became my committee members and gave me valuable advices.

Thank you Tom and Jay for your time and effort in reading my thesis and papers and guiding me.

The Valles lab is a wonderful community where I have made many lifelong friends.

Thanks especially to Jimmy Joy, Xue Zhang, Shawna Hollen, Hung Nguyen, Chenwei

Zhao, Christine Sunu, Michael Wagman, Ryan Provencher, Cole Van Krieken, Hojin

Park, Harry Mickalide, Eva Lyubich, Chenming Jiang, and Kevin Argueta for their friendship and support. I wish all of you the best of luck.

Finally and most importantly, Sunkyung, you have my deepest thanks. Thank you for your love, support, patience, and prayer. Seojoon and Seoha, you are the best gifts in my life. This thesis is dedicated to my family.

vi Contents

1 Introduction 1

2 Paramecium 6

2.1 Introduction ...... 7

2.2 Ciliary motion ...... 8

2.2.1 Microtubule motors ...... 10

2.2.2 Ciliary beating cycle ...... 10

2.2.3 Metachronal wave ...... 12

2.3 Membrane potentials and ion channels ...... 15

2.3.1 Mechanosensitive ciliary regulation ...... 15

2.3.2 Physiological background ...... 17

3 Paramecium in fields 21

3.1 Introduction ...... 22

3.2 Paramecium in gravity ...... 25

3.2.1 Gravitaxis ...... 25

3.2.2 Gravikinesis ...... 32

vii 3.3 Paramecium in magnetic fields ...... 45

3.3.1 Paramecium under a static magnetic field ...... 45

3.3.2 Diamagnetic levitation ...... 51

4 Apparatus and methods 57

4.1 Introduction ...... 58

4.2 Apparatus ...... 59

4.2.1 NHMFL magnet ...... 59

4.2.2 Illumination ...... 61

4.2.3 Temperature control ...... 62

4.2.4 Sample chambers ...... 64

4.2.5 Borescope and video recording ...... 66

4.3 Image analysis ...... 67

4.3.1 Tracking swimming trajectories ...... 67

4.3.2 Fundamental swimming parameters ...... 68

4.4 Experimental solutions ...... 69

4.4.1 Test solution ...... 69

4.4.2 Methyl Cellulose solution ...... 71

4.4.3 Viscosity measurement ...... 72

4.5 Paramecia culturing ...... 75

4.5.1 General procedure ...... 75

4.5.2 Bacteria culturing ...... 76

4.5.3 Culturing medium ...... 77

viii 4.5.4 Autoclaving ...... 78

4.5.5 Inoculation ...... 79

4.5.6 Maintaining Cultures ...... 80

5 Two Ciliary Motors in a Single Microorganism 81

5.1 Introduction ...... 82

5.2 Materials and methods ...... 86

5.2.1 Culturing Paramecium caudatum ...... 86

5.2.2 Viscous solution ...... 86

5.2.3 Swimming trajectory experimental setup ...... 87

5.2.4 Magnetic force buoyancy variation ...... 89

5.2.5 Beating frequency measurement ...... 90

5.3 Results ...... 91

5.3.1 High speed imaging of ciliary motion ...... 91

5.3.2 Swimming trajectory investigations ...... 95

5.3.3 Phenomenological model of ciliary propulsion ...... 103

5.4 Discussion ...... 109

6 Trapping Microorganisms under Varying Buoyancy 111

6.1 Introduction ...... 112

6.2 Materials and methods ...... 115

6.2.1 Paramecium tetraurelia and caudatum ...... 115

6.2.2 Magnetic force buoyancy variation ...... 116

6.2.3 Magnetic torques on swimmers ...... 118

ix 6.3 Results ...... 120

6.3.1 Accumulation of Paramecium tetraurelia ...... 120

6.3.2 Accumulation of Paramecium caudatum ...... 121

6.3.3 Summary of observations ...... 124

6.3.4 Force and torque balance model ...... 124

6.4 Discussion ...... 129

7 Conclusion and Applications 131

x List of Figures

2.1 A Paramecium diagram (lucidhysteria.deviantart.com) along with its

various components...... 7

2.2 An image of a Paramecium caudatum (www.dlr.de)...... 9

2.3 (a) A general structure of a cilium. (b) A cross-section of a cilium

showing an axoneme, dynein arms, and other various components. (c)

Beating mechanism in a cilium (www.rpi.edu)...... 11

2.4 Sequence of an individual cilium motion showing a power and recovey

stroke. This figure is adopted from Paramecium: A current survey,

(1974)...... 12

2.5 (a) The metachronal waves on a Paramecium. (b) A more detailed

picture of (a) showing antiplectic metachronal waves. This figure is

adopted from Paramecium: A current survey, (1974)...... 13

2.6 Side view of a symplectic (a) and antiplectic (b) metachronal wave

coordination. The red cilia indicate the positions of the current wave

crests while the blue cilia show the future cilia which will form next

wave crests...... 14

xi 2.7 Avoiding reaction of Paramecium toward a physical obstacle. This

figure is adopted from Behavior of the lower organisms, (1906). . . . 16

3.1 The orientation of swimming trajectories of Paramecia in a magnetic

field at 0 T (a) and 9 T (b). The corresponding orientational dis-

tributions of each trajectory are shown in the circular histograms (c)

and (d). The bin size is 15◦. This figure is adopted from Biophysical

Journal, 90, 3004 (2006)...... 23

3.2 (a) A photo of Paramecia population in a tube. At first, Paramecia

were distributed randomly. (b) About 10 mins later, the Paramecia

gathered at the top of the tube due to their gravitaxis. (c) Time ex-

posure trajectories of swimming Paramecia in the vertical observation

chamber for 5 secs. (d) Trajectories of initially horizontally swimming

Paramecia in the vertical chamber for 13 secs. At the beginning, most

cells were moving from left to right. Gravity is directed down the

page. (c) and (d) are adopted from Journal of Experimental Biolody,

24, 4158 (2010)...... 26

3.3 (a) Histogram showing experimentally measured β from curvature tra-

jectories (n=42). (b) Histogram showing the predicted values of β

from the shape analysis (n=25). This figure is adopted from Journal

of Experimental Biolody, 24, 4158 (2010)...... 30

xii 3.4 The effects of sedimentation and gravikinesis on the trajectory of a

Paramecium swimming against gravity, g. The Paramecium swims

along the inclination angle, θ. Arrows indicate direction of the vec-

tors. P , S, VU , ∆, α, θ, and a indicate the propulsion, sedimentation,

upward swimming velocity, gravikinesis factor, an angle between the

propulsion and veloticy, resulting velocity angle, and anterior of the

cell, respectively. This figure is adopted from Journal of Theoretical

Biology, 185, 201 (1997)...... 34

3.5 The deformation of the lower cell membrane. (a) The electrophys-

iological model of gravikinesis in a Paramecium presumes that the

gravity-induced deformation of the anterior (A) membrane during the

downward swimming leads to depolarization (+). The depolarization

results in the depression of ciliary frequency and thus reduces propul-

sive speed. (b) In the upward swimming, the deformation induced by

the gravity hyperpolarizes (−) the posterior end (P) and increases in

ciliary frequency and propulsion. (c) In the horizontal swimming, both

depolarization (+) and hyperpolarization (−) lead to the neutraliza-

tion of the gravity responses. Therefore, the horizontal swimmer moves

at the constant rate. The big and small arrows indicate the directions

of swimming and deformaion, respectively. This figure is adopted from

Acta protozoologica, 31, 185 (1991)...... 36

xiii 3.6 Asymmetrical distribution of Ca2+ and K+ channels over a Parame-

cium body. Ca2+ channels are more abundant at the anterior end (A)

while the Paramecium has more K+ channels at the posterior end (P).

This figure is adopted from Journal of Comparative Physiology A, 185,

517 (1999)...... 37

3.7 Orientation-dependent changes in active propulsion of Paramecium

caudatum under normal gravity. 0◦ indicates that the Paramecium

swims upwards. Each 15◦ sector contains at least 408 Paramecia. This

figure is adopted from Naturwissenschaften, 86, 352 (1999)...... 39

3.8 Swimming speed of Paramecia in simulated gravity for upward (N)

and downward (H) swimmers. The bars indicate the widths of the

speed distributions. The solid lines denote the predicted change in

the swimming speed in the absence of the gravikinesis. Their widths

indicate the uncertainty in the mean sedimentation rate. This figure is

adopted from Proceedings of the National Academy of Sciences, 103,

13051 (2006)...... 42

3.9 The averaged gravikinetic factor measured from three trials. The lines

are least-square fits to the data. The negative slope implies that

Paramecia have negative gravikinesis. The uncertainty bars are ±

SD. This figure is adopted from Proceedings of the National Academy

of Sciences, 103, 13051 (2006)...... 43

xiv 3.10 The coordinate system for the calculation of the orientation energy of

an anisotropic rod in a homogeneous magnetic field B which lies in the

x-z plane. θ indicates the angle between the magnetic field and the z

axis. This figure is adopted from Biosystems, 36, 187 (1995)...... 46

3.11 (a) Image sequence of a immobilized and neutrally buoyant Parame-

cium in a homogeneous magnetic field in Ficoll solution. (b) Orienta-

tion rate of immobilized and neutrally buoyant Paramecia as a function

of time where t(s) = 0 indicates the time when θ = 45◦. (c) ln(tan θ)

−1 2 vs. ts = t(BB0 ) in various magnetic field B, where B0 = 6.4 T . (d)

The averaged ∆χ (the dashed line) obtained from swimming Parame-

cia. This figure is adopted from Biophysical journal, 90, 3004 (2006). 50

3.12 The magnetic field profiles for B (black) and B(dB/dz) (gray) as a

function of the distance from the center of the magnet. B is max when

z = 0 while the magnitude of B(dB/dz) is max when z = ±83. The

data are normalized to the center magnetic field value B0...... 52

4.1 (a) A photo of the magnet system at the NHMFL. (b) A cross sectional

drawing of the resistive magnet (www.magnet.fsu.edu). Dimensions

are in millimeters. The sketch is not scale drawn to scale...... 60

4.2 The pictures of the thin LED green backlight from the front (a), the

back (b), and the side (c). A piece of black tape was used to provide a

uniform black background. The LED was attached to a cupper plate

to keep room temperature...... 62

xv 4.3 (a) The picture of the apparatus. Various components such as the 6

mm diameter borescope, CCD camera, LED, and support structure

are shown. (b) Close up view of the bottom of the apparatus. The

copper tubing connected to a water circular bath runs through the

copper plate. The position of the sample chamber was adjusted using

the sample distance controller. (c) Close up view of the apparatus near

the sample chamber. The chamber was fixed at the sample holder using

screws. The height of the borescope was adjusted using the borescope

height controller...... 63

4.4 Experimental chambers. (a) 2×16×16 mm square chamber for observ-

ing Paramecia population. (b) 5 mm diameter circular chamber for

tracking single Paramecium...... 64

4.5 Sample fit of a swimming track (black dots) to a sine wave (gray line)

along with the parameters employed...... 70

4.6 Viscometer for transparent solutions...... 73

xvi 5.1 Experimental chamber. The small dots indicate Paramecia. The solid

line in the middle of the chamber is a glass strip that separates the 2

mm×16 mm×16 mm chamber into two chambers. The fact that there

are two chambers is not important for this work. The gray circle and

black box regions inside the circle indicate field of view of a camera and

areas of interest, respectively, where swimming analysis was done. Note

that two holes were needed to inject Paramecia to avoid air bubbles.

This figure is adopted from Biophysical Journal, 106, 106 (2014). . . 88

5.2 Viscosity dependence of ciliary beat frequencies. (a) Phase contrast

image of a Paramecium. The brackets define 4 observation regions:

O–oral-groove, A–anterior, B–mid-body, and P–posterior. Examples

of 2 metachronal wave crests can be seen within the dotted ellipse.

The scale bar is 50 microns. (b) Ciliary beat frequencies fw measured

in the four regions (solid symbols) normalized to their average value

in standard test solution plotted versus ηw. Results obtained for the

oral-groove by Machemer (open circles) are shown for comparison. The

lines are guided to the eye and not fits. This figure is adopted from

Biophysical Journal, 106, 106 (2014)...... 92

xvii 5.3 (a) Sketch of a Paramecium indicating the translational velocity vector

~v and the rotational velocity vector ~ω that produce its motion along

a left handed helical trajectory as shown in (b). A and P denote the

anterior and posterior regions, respectively. The lightly shaded patch

corresponds to the oral-groove. This figure is adopted from Biophysical

Journal, 106, 106 (2014)...... 97

5.4 (a)-(d) The swimming speed distributions for different η. The bin size

is the SD of the swimming distributions. (e)-(f) The mean swimming

speeds and δv/v as a function of ηw. (g)-(h) Swimming trajectory

parameters, radius and pitch. Each bar represents the SD...... 99

5.5 The averages of ωk and ω⊥ derived from analysis of more than 50

trajectories at each viscosity. The bars give the standard deviation of

the populations. This figure is adopted from Biophysical Journal, 106,

106 (2014)...... 100

5.6 Comparison of Paramecia swimming speeds as a function of viscos-

ity in Methyl Cellulose (square) solution and Ficoll solution (circle).

Paramecia show the nearly identical swimming speeds in both solu-

tions. The bars give the standard deviation of the populations. The

lines are guides to the eye. This figure is adopted from Biophysical

Journal, 106, 106 (2014)...... 101

xviii 5.7 Viscosity dependence of propulsive forces and torques. (a)-(c) Products

of the viscosity with each of the rates shown in Figure 5.4(e) and Figure

5.5. The bars give the standard deviation of the populations. This

figure is adopted from Biophysical Journal, 106, 106 (2014)...... 102

5.8 Ciliary beaing of a Paramecium. The body cilia (blue) of Paramecium

beat with an oblique angle while the oral groove cilia (green) of the

Paramecium beat perpendicular to the direction of its propulsion. The

numbers indicate the positions of the cilia as a function of time. . . . 105

5.9 Insights from the phenomenological model. (a) Plot of ωk/v vs. ηw to

compare to Equation (5.10). This ratio is proportional to tan θ, where

θ gives the direction of the force exerted by the average patch of cilia.

(b) ω⊥ vs. v to compare to Equation (5.11) at different viscosities. The

intercept gives the oral-groove cilia contribution to the torque produc-

ing ω⊥. The line is a fit that tests the expected linear dependence of

ω⊥ on v. The bars give the standard deviation of the populations. This

figure is adopted from Biophysical Journal, 106, 106 (2014)...... 107

xix 5.10 Comparisons of motor characteristics. Propulsive force from these ex-

periments pw and force predictions based on the measured beat fre-

quencies, mid-body, B and oral-groove, O, shown in Figure 5.2(b) and

numerical model calculations. The forces are the product of the vis-

cosity and the rate (i.e. speed or frequency). (a) Forces as a function

of viscosity. (b) Forces as a function of rates. All forces and rates are

normalized to their values in water. The lines are guided to the eye

and not fits. This figure is adopted from Biophysical Journal, 106, 106

(2014)...... 109

6.1 Schematic of the magnet. For Paramecia placed above the center of the

magnet, the levitation position, magnetic forces are upward (negative)

while swimmers below the center of the magnet, the sedimentation

position, are exposed to downward (positive) magnetic forces. At the

center of the magnet where dB/dz is zero, there is no magnetic force.

This figure is adopted from Physical Review Letters, 113, 218101 (2014).117

xx 6.2 (a) The surface trapping probability (STP) of P. tetraurelia at the

bottom. Frame showing 3 P. tetraurelia at the bottom surface when

w/w1g = −2. The long axes of their bodies were canted away from

horizontal as they swam along the surface. (b) STP of P. tetraurelia

near the bottom surface as w/w1g changes from 1 to −2. The negative

force indicates that the force direction is from the bottom to the top.

(c) STP of P. tetraurelia and w/w1g (solid line) as a function of time.

w was changed in steps. (d) Trapping of P. caudatum at the bottom

surface. Upward apparent weight force (w/w1g = −6) is applied. (e)

STP of P. caudatum at the top (N) and bottom (O) surface. (f) STP

of P. caudatum at the top surface minus the bottom surface and w/w1g

as a function of time. The solid line marks the changes in w/w1g. The

dotted line denotes where both the probability and w/w1g are zero.

The bars on the points give the uncertainty estimated presuming the

measurements followed a binomial distribution. This figure is adopted

from Physical Review Letters, 113, 218101 (2014)...... 119

6.3 STP at the top (tringle) and bottom (downward triangle) surface as

a function of homogeneous magnetic field at constant apparent weight

of w/w1g = 1. The bars on the points give the uncertainty estimated

presuming the measurements followed a binomial distribution. This

figure is adopted from Physical Review Letters, 113, 218101 (2014). . 123

xxi 6.4 (a) A sketch of a canted Paramecium in contact with and swimming

along the bottom surface. In this figure, the Paramecium swims against

~w. (b) Phase diagram of P. caudatum behavior at the bottom surface.

The upper four frames give schematics of the swimming behavior in

the magnetic field regions I–IV. The gray region specifies the range

of θmax expected for a typical population of swimmers. This figure is

adopted from Physical Review Letters, 113, 218101 (2014)...... 125

7.1 The gravikinesis under different viscosity of (a) 1.0 cP , (b) 2.3 cP ,

and (c) 4.1 cP . The trendlines correspond to the gravikinesis of (a)

−54 µm/s, (b) −27 µm/s, and (c) −18 µm/s, respectively. (Bars)

Standard deviation of the population...... 135

xxii List of Tables

4.1 Recommended viscosity range for each model and modified viscometer

constant at 20◦C...... 72

5.1 Averaged ciliary beat frequencies in the four different regions from five

measurements. This table is adopted from Biophysical Journal, 106,

106 (2014)...... 94

5.2 Averaged distances between crests in the mid-body region from three

measurements...... 94

xxiii Chapter 1

Introduction

1 Paramecium is a unicellular ciliated protozoan covered by thousands of cilia. It is commonly studied in biology as representative of the ciliates due to its being widespread in nature and its relatively large size. Moreover it shows clear quan- tifiable responses to environmental stimuli such as gravity [1, 2], magnetic field [3, 4], electric field [5], temperature [6], light [7], or chemical gradients [8, 9]. The organism can move forward and backward or even pivot to quickly and properly respond to ex- ternal stimulations that change from moment to moment [10]. Of particular interest has been its response to perturbations that play important roles in cell life. These ex- ternal force responses including their graviresponses are clear signs that Paramecium senses forces (either actively or passively). For example, Paramecium under gravity tends to collect at the top of a container with the anterior end directed upwards.

This behavior is called negative gravitaxis and has been investigated since it was

first reported [10, 11]. It has also been proposed that membrane potential changes through the activation of mechanically sensitive ion channels induce a regulation of

Paramecium swimming propulsion, called gravikinesis, when it is exposed to changes in gravity [1, 12, 13, 14, 15, 16]. Both graviresponses are surprising because the cell’s apparent weight in water is only about 80 pN. In chapter 2 and 3, we will introduce

Paramecium and describe their graviresponses. In spite of its importance and many studies of responses to physical stimulation in microorganisms, some crucial proper- ties such as ciliary motor characteristics [17, 18, 19] and cell interactions with surfaces

[20, 21] have not been clearly elucidated. Investigating physical responses requires methods to apply forces and torques to living cells. Guevorkian [1] developed the use of intense static magnetic fields for investigations of gravikinesis in populations

2 of Paramecia.

There is much evidence that magnetic fields play an important role in living

organisms. Magnetic fields can affect long term cell development [22], human cancer

cell growth [23], and bacteria mutations [24]. Most often, these responses are active.

However, there also exist passive responses such as static magnetic field effects on

cell orientation without a physiological change [25, 26]. For example, immobilized

and motile Paramecia orient in a static magnetic field. When Paramecia are exposed to strong homogeneous magnetic fields (∼ 4 T ), the cells swim either parallel or antiparallel to the field direction [4]. In this study, we employed this passive method to investigate cell trajectories and thus to see ciliary responses in Paramecia to viscous drag forces. The trajectory alignment provides a lot of benefits for observing and analyzing cells’ swimming trajectories. In addition, a strong inhomogeneous magnetic

field can also physically exert a magnetic force on a diamagnetic object [27, 28]. Since

Beaugnon [29] first levitated water using a inhomogeneous magnetic field, Valles and cowokers [1, 30] have developed the method by applying it to living organisms such as Xenopus laevis and Paramecium. We used the technique, called Magnetic Force

Buoyancy Variation (MFBV), to observe Paramecia responses to variable surface contact forces. The MFBV could provide an alternative to centrifugation experiments for investigating force sensitivity and responses in living cells. In chapter 3, we will introduce details about a method to apply magnetic torques and forces on swimming cells as well as describe the experimental setup for this method in chapter 4.

The experiments in chapter 5 were motivated by the desire to understand how motile cilia respond to mechanical drag forces such as medium viscosity. Motile cilia

3 in microorganisms have been under scrutiny due to their multi-functional roles such

as sensing extracellular signals, exerting propulsive force for locomotion, and nutrient

uptake [10, 31]. We employed strong magnetic fields to align the cells to observe

their swimming trajectories and to provide a neutrally buoyant environment. We

measured their swimming parameters and ciliary beating frequencies to determine

how the speed and direction of the cilia beating changes with viscosity. Interestingly,

we found that the beat frequency in the oral groove changes little while the beat

frequency in the mid-body region was inversely proportional to viscosity changes.

The results suggested that there are ciliary motors with two extremely different motor

characteristics. More details will be discussed in chapter 5.

The ability of microorganisms to interact with other cells or surfaces is a very

important characteristic [20, 21, 32, 33]. During these interactions, they send and

receive various physical and chemical signals to coordinate their behavior. Remark-

ably, experiments on bacteria suggest that their collection near surfaces is not due

to an active response to touching the surfaces. They do not change either the mag-

nitude or the direction of their propulsive force. However, Paramecium has been

known as a cell which actively responds to physical perturbations. In chapter 6, we

test how Paramecia transduce mechanical forces such as their collisions caused by planar surfaces. We used the MFBV method to control the mechanical force from

flat surfaces. Surprisingly, making them buoyant led them to become trapped at the bottom surface and vice versa. These counterintuitive behaviors occur because they failed to turn around after swimming into a surface. We introduced a purely passive mechanical model indicating that making them buoyant decreases the turning torque

4 on Paramecia and concluded that the Paramecia interact passively with flat surfaces.

This result provides the new and interesting insight into cell responses to mechanical force. More details will be discussed in chapter 6.

5 Chapter 2

Paramecium

6 Figure 2.1: A Paramecium diagram (lucidhysteria.deviantart.com) along with its various components.

2.1 Introduction

Paramecium is an ellipsoidally shaped eukaryote. Its anterior part is slender and blunt while its posterior is thicker and sharp as shown in Figure 2.1. The body of a

Paramecium is not symmetrical, but slightly spiral shaped. Its body length roughly varies from 100 µm to 250 µm depending on species, and its cross sectional radius is

about 40 µm. Its body density (1040 Kg/m3) is slightly higher than that of water.

A schematic figure of a Paramecium in Figure 2.1 shows its various components

7 which can be divided into three categories, cortex, endoplasm, and nuclei. The cortex

is an outer layer approximately 4 µm thick. It contains several complex and impor-

tant biological structures of Paramecia such as cilia, cell membrane, pellicle, and ectoplasm. The endoplasm is the great inner mass of cytoplasm which is constantly moving. It includes certain components, called organelles and inclusions. A Parame- cium has dimorphic nuclei. It possesses one large macronucleus and one or more small micronuclei depending on the species. In this chapter, we will present more details in ciliary motion, one of the most important components for cell’s locomotion, as well as a general physiological background in Paramecium which are necessary to investigate its swimming characteristics.

2.2 Ciliary motion

The entire body of a Paramecium is covered by thousands of cilia. The cell uses them for various purposes such as locomotion, food uptake, and reproduction. In particular, a Paramecium has two types of cilia, body cilia and oral groove cilia shown in Figure

2.2. The body cilia control the cell’s locomotion such as turning and forward and backward swimming. These cilia beat toward the posteriors with an oblique angle at a frequency of ∼ 30 Hz in water [34]. The oral groove cilia provide nutritive currents toward its gullet [31]. The right side of Figure 2.2 shows the broad and oblique oral groove. The oral groove extends from the cell mouth to the anterior. The oral groove cilia beat at a frequency ∼ 35 Hz in water. Contrary to the body cilia, the oral groove cilia beat parallel to the long body axis [34]. The length and diameter of cilia

8 Body Cilia

Oral-groove Cilia

Figure 2.2: An image of a Paramecium caudatum (www.dlr.de).

are about 10 µm and 0.2 µm, respectively. Cilia lengths are almost uniform over the body but slightly increases at the posterior end (∼ 16 µm).

The locomotion of the cell is driven by ciliary beating as shown in Figure 2.2.

The ciliary beating consists of two phases, the power stroke and the recovery stroke.

During the power stroke, stiff and straight cilia swing fast through the perpendicular to the surface of the cell. This stroke is responsible for forward swimming. During the recovery stroke, softly bent cilia rotate counter-clockwise to return to the starting position of the power stroke. Sometimes, this recovery stroke is responsible for back- ward swimming. More information about the beating, the metachronal wave what reflects coordination among the individual cilia, and its mechanism will be described in the next section.

9 2.2.1 Microtubule motors

Figure 2.3(a) shows a cilium bounded by the plasma membrane. The cilium grown

from the basal body has the axoneme, a complex of microtubules, and associated

proteins. In particular, the axoneme consists of a 9+2 pattern where 9 and 2 indicate

nine doublet microtubules around the periphery and two singlet microtubules in the

center surrounded by the sheath as shown in Figure 2.3(b) [35, 36, 37]. Each doublet

microtubule can be divided into two components, the A and B tubules. In particular,

the A tubule of each doublet is connected to the dynein arms. The central pair of microtubules determines the length of the cilium. The nexin links and radial spokes provide elastic connections between microtubule doublets and between the A tubule of each doublet and the central sheath. Bending of the cilium is produced by the A

tubule dynein arms. When ATP is provided, the A tubule dynein arms walk along

the adjacent B tubule toward minus (−) end (Figure 2.3(c)). The minus ends are

fixed in the basal body, and the flexible links such as the nexin links and radial spokes

limit the bending.

2.2.2 Ciliary beating cycle

The movement of cilia consists of two steps, a power or effective stroke and a recovery

stroke as shown in Figure 2.4. During the effective stroke (1-3) in Figure 2.4, the tip

of the cilium is obliquely beating from the left anterior to the right posterior in a

vertical plane of a cell’s surface. Paramecium generally swims along a left handed helical trajectory due to this ciliary beating motion. During the recovery stroke (4-

10 (a) (b)

+ + (c) Movement of dynein arms

_ _

Figure 2.3: (a) A general structure of a cilium. (b) A cross-section of a cilium showing an axoneme, dynein arms, and other various components. (c) Beating mechanism in a cilium (www.rpi.edu).

11 Figure 2.4: Sequence of an individual cilium motion showing a power and recovey

stroke. This figure is adopted from Paramecium: A current survey, (1974).

8) in Figure 2.4, the cilium starts a counter-clockwise rotation parallel to the cell

surface. As a result of the whole cycle, the cell moves in the opposite direction of the

power stroke. The effective stroke operates 6 times faster than the recovery stroke.

A Reynolds number of a cilium during the strokes is ∼ 0.01.

2.2.3 Metachronal wave

A Paramecium is covered by about 3000-4000 cilia in a square array spaced by about

2 to 3 µm [37]. The cilia beat simultaneously based on the stroke mechanism de-

scribed in the previous section. This simultaneous beating appears as a metachroal

wave pattern shown in Figure 2.5. This longitudinal and oblique row pattern occurs

because cilia do not beat in the same phase but at a constant phase difference be-

tween neighboring cilia. In general, about 12 metachronal wave peaks are observable

in a Paramecium implying that the metachronal wavelength is about 10 µm with its

12 (a) (b)

Figure 2.5: (a) The metachronal waves on a Paramecium. (b) A more detailed picture of (a) showing antiplectic metachronal waves. This figure is adopted from

Paramecium: A current survey, (1974).

13 (a) Power stroke Wave crest Power stroke

(b) Power stroke Wave crest Power stroke

Figure 2.6: Side view of a symplectic (a) and antiplectic (b) metachronal wave co- ordination. The red cilia indicate the positions of the current wave crests while the blue cilia show the future cilia which will form next wave crests. body length 200 µm.

There are several forms of metachronal coordination. Figure 2.6(a) shows a sym- plectic metachronal coordination. In this metachrony, the power strokes are in the same directions as the wave crest motion where the red cilia indicate the crests of the waves. Because the direction of the power stroke is from left to right, the blue cilia next to the right side of the red cilia will form the next crest implying that the wave crests will move from the left to the right. On the other hand, an antiplectic metachronal wave occurs when the direction of power strokes is opposed to the motion of the wave crests (Figure 2.6(b)). In this case, because the direction of the power strokes switches from right to left, the blue cilia next to the right side of the red cilia will form wave crests from the left to the right.

14 In particular, Paramecia show a series of metachronal waves under normal condi- tion as shown in Figure 2.5(a). These metachronal waves move from the left anterior to the right posterior at a slightly oblique angle from the long axis. The speed of the wave is ∼ 200 µm/s [37]. This pattern corresponds to an antiplectic metachronal coordination when viewed from the side. On the other hand, Machmer [38] reported symplectic metachronal waves when viscosity of solution was increased to 5-6 cP .

2.3 Membrane potentials and ion channels

It has been proposed that gravity forces can alter membrane potentials in cells and stimulate them. In particular, Machemer and Ooya [13, 39] argued that gravity can affect the swimming behavior of Paramecium by expanding its lower membrane.

Paramecium has 11 types of distinct membrane channels. Calcium- and potassium- dependent ion channels have been investigated extensively due to Paramecium’s cil- iary beating dependence on these ion channels [14]. The asymmetrical distributions of these ion channels make the response of the electrochemical potential in the cell dependent on where forces are applied to the cell surface since the beating frequency depends on the potential. Paramecium changes its swimming when they experience forces.

2.3.1 Mechanosensitive ciliary regulation

When a Paramecium encounters a stimulus, the cell may respond to it either posi- tively or negatively. If they meet a favorable stimulus such as food, they may move

15 Figure 2.7: Avoiding reaction of Paramecium toward a physical obstacle. This figure is adopted from Behavior of the lower organisms, (1906).

toward the source. On the other hand, when they undergo a sudden interruption or unfavorable stimulus such as a physical obstacle and strong temperature or chemical gradient, they may swim backward to react negatively. In particular, when a cell meets a mechanical obstacle, it shows an avoiding reaction [10]. During the avoid- ing reaction, the cell shows three steps, 1) a short backward swimming (1-3), 2) a pivoting to the aboral side of its body (3-5), and 3) a forward swimming to a new direction (5-6) (Figure 2.7). Both of these positive (forward) and negative (backward) motile reactions may depend on how cilia beat such as beating angle, frequency, and amplitude.

The avoiding reaction can be explained from a physiological point of view. A forward swimming Paramecium has a membrane potential at a resting level, ∼ 25

mV . During forward swimming, the potential is maintained with a very low Ca2+

concentration, and normal metachronal beats (symplectic) propel the cell forward.

16 When the cell bumps into an obstacle, the stimulus on its anterior depolarizes ciliary

membrane. Then the cell shows backward swimming away from the obstacle. The

backward swimming results from ciliary reversal controlled by electrical potentials

across the cell membrane. The physical stimulation initiates action potentials in the

cell leading to Ca2+ influx [40]. That induces depolarization and reverses ciliary beating direction for the backward swimming. More details such as the physiological background and gravity induced ciliary beating regulation will be presented in the next section and chapter 3.

2.3.2 Physiological background

A cell membrane can be considered as an electrical capacitor due to its lipid bilayer

(insulator) and intracellular and extracellular solutions (two conductors). Therefore, the membrane can store charges and maintain an electrical potential across its thick- ness. Most cells have a negative membrane potential, but its sign and magnitude can be altered when ions flow through the membrane. The mechanisms of the cell mem- brane potential shift and the ion flow depend on three factors: the ion’s concentration gradient, the existing electrical gradient, and the permeability of membrane.

Nernst equilibrium

Let’s assume that a membrane separates the inside and outside of a cell and allows a specific type of ions, K+, for example, to flow through it. If we assume that K+ concentration inside the membrane is higher than outside, then the permeant K+ ions

inside the cell will move to outside due to concentration gradient. However, as the

17 outside K+ ions relocated from the inside build up near the outside membrane, they may repel additional K+ ions which are forced to move from the inside to the outside due to the concentration gradient. Finally, when the diffusion due the concentration gradient is balanced by the repulsive force, there is no net flux of the ions even though some permeant ions can still flow in both directions. This stable state is called Nernst equilibrium, and the transmembrane voltage gradient, EN , is given by

  RT Cin EN = Vin − Vout = − ln (2.1) ZF Cout

where V , R, T , Z, F , and C indicate the voltage, the gas constant, the absolute

temperature, the valence of a permeant ion, the Faraday constant, and the equilibrium

concentration of ions, respectively [41]. The subscripts in and out denote the inside

and outside of a membrane. The Nernst equilibrium potential can be used for only

one type of permeant ion. It can measure the membrane potential produced by the

ions and predict the movement direction of the ions through the cell’s membrane.

Goldman–Hodgkin–Katz equation

More generally, Goldman–Hodgkin–Katz voltage equation can be used to predict cell

membrane potential for the more complex situation where more than one type of ion

is involved. Assuming that ions are maintained in a dynamic steady state and using

ionic permeability, the Goldman–Hodgkin–Katz voltage [42] is given by

" + + − # RT PNa[Na ]in + PK [K ]in + PCl[Cl ]out EG = ln + + − (2.2) F PNa[Na ]out + PK [K ]out + PCl[Cl ]in

+ + where PNa, PK , and PCl are the relative membrane permeabilities of Na , K , and

Cl−, respectively. The square bracket indicates the corresponding ion’s concentra-

18 tion. This equation implies that the membrane potential is weighted by the perme-

ability of relevant monovalent ions. If considering only one type of ion, the Gold-

man–Hodgkin–Katz can be reduced to the Nernst equation except that the Nernst

equation requires the valance of the ion, Z.

Membrane potentials in Paramecium

Based on the theories above, Baba and coworkers [42] estimated the gravity-induced shift of membrane potential to be ∼ 1 mV . From Equation (2.1), they calculated

membrane potentials for K+ and Ca2+ which are necessary to explain graviresponses

◦ + 2+ in Paramecium. At 20 , the equilibrium potentials EK and ECa for K and Ca ,

respectively, were −81 mV and 116 mV . The signs indicated that K+ ions are

abundant inside a cell while Ca2+ ion concentration inside the cell is lower than

outside. The concentration ratio outside Paramecium menbrane to inside for K+

and Ca2+ are 1/25 and 104, respectively. Randall [43] reported the total membrane

potential dependence in a Paramecium on ionic conductance, given by

gK gCa EM = EK + ECa (2.3) gK + gCa gK + gCa

where EM and g are the total membrane potential in the cell and the ionic conductance

for corresponding ion, respectively. Using the membrane potential of Paramecium,

−29 mV , and Equation (2.3), Baba and coworkers [42] estimated gCa/gK to be 0.36.

From Equation (2.3), they reported that 1% change in either gK or gCa will produce

a membrane potential shift by 0.6 mV or 0.3 mV , respectively. The order of these

membrane potential shift is comparable to the shift induced by gravity which is es-

19 timated to be ∼ 1 mV . Therefore, the gravity-induced membrane potential shift to this extent may result from 1% mechanoreceptive changes in K+ and Ca2+ conduc-

tance. In particular, Machemer [44] reported that Paramecium body cilia beat at

∼ 17 Hz at resting potential corresponding to a swimming speed of ∼ 1000 µm/s.

This result indicates that active propulsion regulation under normal gravity, ∼ 60

µm/s, corresponds to ciliary beating frequency of ∼ 1 Hz or membrane potential

shift of ∼ 1 mV .

20 Chapter 3

Paramecium in fields

21 3.1 Introduction

Microorganisms as well as plants and animals respond to various types of stimuli such as chemical, temperature, light, electric field, magnetic field, and gravity which are ubiquitous in their environment [1, 2, 5, 6, 8, 9, 14, 45, 46]. They utilize these responses to select suitable mechanisms to survive, eat, move, and reproduce.

In particular, all microorganisms live and swim under the influence of gravity while exhibiting various responses to gravity. For example, Paramecia, ciliates, are known for sensing gravity. Their gravisensitivities are very surprising because their body density, 1.04 g/mL, is only slightly higher than their surrounding medium.

Accordingly, their apparent weight is only about 80 pN. However, they can orient their swimming direction and regulate their propulsion with respect to the gravity vector without gravisensing organelles, such as amyloplasts, as in a plant cell [47].

These graviresponses can be divided into two behaviors, gravitaxis and graviki- nesis. Under normal gravitational conditions, Paramecia show a negative gravitaxis.

It is called negative because the direction of their movement is away from the source of the stimulus, the center of the earth. In other words, Paramecia prefer to swim upward or toward the top of a container. Its degree of orientation depends on various factors such as feeding status, culture age, temperature, and oxygen concentration of the medium [14]. In addition, Paramecia use regulation of their swimming propulsion to compensate for their sedimentation. When Paramecia swim horizontally, they ex- ert a constant propulsive force to swim at a constant speed. However, when swiming against gravity, they exert a larger constant propulsive force while they simultane-

22 Figure 3.1: The orientation of swimming trajectories of Paramecia in a magnetic

field at 0 T (a) and 9 T (b). The corresponding orientational distributions of each trajectory are shown in the circular histograms (c) and (d). The bin size is 15◦. This

figure is adopted from Biophysical Journal, 90, 3004 (2006).

ously swim and sediment. In this chapter, we will investigate the effects of gravity on

swimming Paramecia.

Many organisms are known to respond to Earth’s magnetic field. Accordingly,

they regulate their movement and direction. For example, large organisms such as

birds and dolphins use it to navigate [48, 49] while most small organisms except some

bacteria [50] respond little to the magnetic field due to their weak magnetic properties.

23 However, under strong static magnetic field (∼ 1.7 T ), most microorganisms such as bull sperm become orient [51]. In particular, the trajectories of Paramecium caudatum become oriented under intense static magnetic fields (∼ 3 T ) [4]. The Paramecia align with their long body axis either parallel or antiparallel to the direction of the magnetic

field as shown in Figure 3.1. Initial swimming trajectories in 0 T are randomly distributed (Figure 3.1(a)). However, when the Paramecia are placed under the static magnetic field of 9 T , their upward or downward trajectories are aligned along the magnetic vector (Figure 3.1(b)). The circular histograms present further evidence that the tracks orient at all directions at 0 T (Figure 3.1(c)) while most tracks at 9

T orient within 10◦ of the vertical axis (Figure 3.1(d)). This passive response is due to a magnetic torque exerted on the diamagnetically anisotropic components of the

Paramecia [4].

Diamagnetic levitation is another interesting response to magnetic fields. Mag- netic fields can repel diamagnetic materials including all living organisms. If the field is strong enough, the force can balance gravity and even move the diamagnetic object upward against gravity. Valles and coworkers first levitated a living biological organ- ism, embryos of the frog Xenopus laevis, using a large inhomogeneous magnetic field

[30]. Guevorkian developed the magnetic levitation technique to investigate gravita- tional sensitivity in swimming Paramecia [1]. These studies demonstrated a method for controlling continuously varying gravity forces acting on diamagnetic materials or biological cells. In the previous experiments, the technique, Magnetic Force Buoy- ancy Variation (MFBV), was employed to provide the variable forces. The MFBV required current carrying solenoids to produce intense inhomogeneous magnetic fields.

24 These solenoids must provide a large field and gradient because the magnetic force is proportional to (B~ · ∇~ )B~ . Particularly, the magnetic field could alter the buoyancy of immobilized or swimming cells in solution by adjusting the magnetic field and its gradient. At the National High Magnetic Field Laboratory, it was possible to gen- erate magnetic field and gradient sufficient to adjust the apparent weight of cells by as much as a factor of 10. In the last section of this chapter, we will introduce the

MFBV method in more detail.

3.2 Paramecium in gravity

3.2.1 Gravitaxis

Figure 3.2 shows the negative gravitaxis in Paramecia. This negative gravitaxis is the tendency to be collected near the top area of a container. As a result of gravitaxis, initially uniformly distributed Paramecia (Figure 3.2(a)) are gathered near the top

(Figure 3.2(b)). Because their body density is heavier then water, the mechanism underlying this phenomenon has been in dispute since it was first observed [11]. It has been widely assumed that both physiological and physical mechanisms are needed to account for the characteristics of the gravitaxis. The physiological hypothesis as- sumes a biological mechanism suggesting that cells can sense the gravity directly and control their ciliary beating and upward swimming speed. On the other hand, phys- ical hypotheses assume purely passive responses to gravity to explain the principal phenomena of the gravitaxis.

25 10 mins

(a) (b)

(c) (d)

Figure 3.2: (a) A photo of Paramecia population in a tube. At first, Paramecia were distributed randomly. (b) About 10 mins later, the Paramecia gathered at the top of the tube due to their gravitaxis. (c) Time exposure trajectories of swimming

Paramecia in the vertical observation chamber for 5 secs. (d) Trajectories of initially horizontally swimming Paramecia in the vertical chamber for 13 secs. At the begin- ning, most cells were moving from left to right. Gravity is directed down the page.

(c) and (d) are adopted from Journal of Experimental Biolody, 24, 4158 (2010).

26 In particular, Roberts [2] recently analyzed swimming patterns in swimming Parame-

cium to investigate the physical hypotheses such as the density variation model (also

known as gravity-buoyancy or bottom heaviness model) and the asymmetry body

shape model (or drag-gravity model) by immersing cells in a medium of the same

density as Paramecia. The time exposure trajectories in Figure 3.2(c) shows that the

Paramecia initially swim all directions. However, the trajectories of initially horizon-

tally swimming Paramecia in the vertical chamber show systematic smooth upward

curvatures as shown in Figure 3.2(d). Roberts suggested that if the upward curvature

is a purely passive response (i.e. physical), then the reorientation can be described

by the equation [52], dθ = −β sin θ (3.1) dt

where θ, t, and β indicate the orientation of the cells relative to the vertical, time, and maximum rate of orientation, respectively. In the experiment, he measured β from the trajectory curvatures and compared it with a predicted value of β obtained

from cell shape measurements. He concluded that the primary reason for gravitaxis

is the front-rear shape asymmetry of Paramecia. More details of this model will be

shown at the end of this section.

Physiological hypothesis

It has been believed that Paramecia can physiologically and directly sense gravity

[14]. The physiological hypothesis assumed that a hydrostatic pressure difference

across a cell can adjust the ciliary beating and thus the upward swimming speed

to be collected near the top surface. The assumption has been validated in many

27 ways. Gebauer and cowokers [12] measured changes in the cell membrane potential of Paramecia induced by rotating impaled Paramecia by 180◦ indicating a sensitivity to the directional change of the gravity. Machemer and coworkers [15] suggested that stretch sensitive ion channels in the cell membrane may be activated by the hydrostatic pressure variation across the cell. The closing and opening of Ca2+ and

K+ ion channels regulate the beating frequency and thus swimming propulsion. This physiological behavior called gravikinesis, a tendency to swim faster when cells swim upwards opposite to the gravity vector compared to downwards, has been considered as one mechanism of the gravitaxis from a physiological point of view.

However, the gravikinesis contribution on the gravitaxis obtained from the phys- iological speed change depending on the gravity vector could not necessarily explain mechanisms of gravitaxis. Machemer [16] estimated the pressure using Stoke’s law and the buoyancy force. Using a volume of a Paramecium, 2.5 × 105 µm3, and den- sity difference between the Paramecium and water, 0.04 g/mL, he approximated the pressure, 0.053 P a, over a circular surface of the Paramecium with 50 µm diameter.

In order for gravekinesis to contribute to gravitaxis, the upward speed increment due to the gravikinesis must be, at least, 2 times bigger than cell sedimentation. How- ever, the estimated pressure differences are much smaller than the pressure required to open stretch activated K+ ion channel to account for the gravitaxis [53]. The speed increment due to gravikinesis under normal gravity, ∼ 50 µm, corresponded to only about 50% of the cell sedimentaion.

Although the pressure is a potential candidate for the mechnosensation relating to both the gravikinesis and gravitaxis, some mechanisms for amplifying such a small

28 effect due to gravity may be necessary for this physiological hypothesis [54]. Recent

studies suggested two independent gravity response systems for the gravikinesis and

gravitaxis. Gravitaxis measurements using Paramecia populations concluded that the activation and relaxation times of the gravitaxis and gravikinesis in Paramecium caudatum and biaurelia are different, implying separate gravisensor systems [55, 56].

Roberts [2, 57] also reported that the gravikinesis may result from the interaction of the ciliary propulsion system with the surrounding sedimentary flow and be unlikely to make any significant contribution to the gravitaxis.

Physical hypothesis

The density variation model [11, 58] is a purely physical mechanism produced by the buoy effect resulting from the back heaviness of a Paramecium. This hypothesis assumed that there is a constant density gradient within a Paramecium. If the body density of the Paramecium is not homogeneous, the center of mass does not necessarily coincide with the center of buoyancy. If the center of mass is shifted posteriorly, the heavier posterior end tends to direct the Paramecium upward. Another physical hypothesis, the asymmetry body shape model with the constant body density, was suggested by Roberts on the basis of the low Reynolds number hydrodynamics of swimming microorganisms that have an asymmetrical geometry [52]. This model assumes that the reaction center of hydrodynamic stress is located more anteriorly than the center of mass and buoyancy. According to Stokes’ law, the larger posterior of the cell will sink faster than the smaller anterior, at a rate that grows with the square of the ratio of the diameters.

29 (a)

(b)

Figure 3.3: (a) Histogram showing experimentally measured β from curvature tra- jectories (n=42). (b) Histogram showing the predicted values of β from the shape analysis (n=25). This figure is adopted from Journal of Experimental Biolody, 24,

4158 (2010).

30 The density variation model was tested by experiments on cells immersed in solu- tions with densities different from the cells [59]. If the gravitactic reorientations was due to an inhomogeneous density within the cell, then the reorientation rate would be independent of solution density. Taneda showed that the reorientation rate of the swimming Paramecium in a higher density solution decreased and reached zero when the density of the solution is about 1.1 g/mL. These results implied that the density variation model failed and that gravitaxis is independent of the density of the solution.

Robert and Mogami [2, 52, 60] conducted experiments on motile and immobilized

Paramecium and concluded that the shape asymmetry model can explain the prin- cipal mechanism of gravitaxis while the density variation model plays a minor role in the gravitaxis. In particular, Robert showed that negative gravitaxis in Parame- cia primarily resulted from their upwardly curving trajectories. In the experiment, cells initially swimming horizontally exhibited a systematic upward curvature. This initially horizontal swimming was obtained by 1) collecting cells near the top surface using gravitaxis, 2) rotating the chamber by 180◦ in the vertical plane to have the cells at the bottom, 3) waiting until the cells start swim upward due to gravitaxis, and 4) rotating the chamber by additional 90◦ when the cell swim near the middle of the chamber. Immediately after the stage 4), the cells swim horizontally as shown in

Figure 3.2(d). The experimentally measured maximum rate of the orientation, β, in

Equation (3.1), was ∼ 7◦/s as shown in Figure 3.3(a).

31 They also measured cell shape given by

ab r(ϕ) = p + c × cos ϕ (3.2) a2 sin2 ϕ + b2 cos2 ϕ where the first term on the right indicates an ellipse with semi-major and semi-minor axes of a and b and the second term a degree of fore-rear asymmetry specified by c. r(ϕ) is the distance from the center of the coordinate system to the cell surface at angle ϕ. When ϕ = 0, r(ϕ) points toward the posterior end of the cell. Roberts obtained the shape parameters using photo images of cells and Equation (3.2). The averaged shape parameter values (n=25), a, b, and c, were 123 µm, 29 µm, and 6.9

µm, respectively. These parameters were used to estimate β given by [61]

ρ − ρ β ∼ (0.056 ± 0.004) 0 gc (3.3) η where ρ, ρ0, and η are the density of cell and solution and viscosity of the solution, respectively. Note that at low Reynolds number, β depends only on c and β = 0 when the density of a cell and solution is identical as this model hypothesized. The averaged value of β from the shape measurements was ∼ 9◦/s (Figure 3.3(b)) slightly higher than the experimental measurements. This correspondence suggested that a purely physical mechanism is responsible for gravitaxis. The front-rear asymmetry body shape observed in populations of the Paramecia seemed to be sufficient to account for the observed orientation rate distribution in gravitaxis.

3.2.2 Gravikinesis

Freely swimming microorganisms modulate their swimming locomotion rate to obtain food, to escape from predators, or to respond to other stimuli. These types of motile

32 responses in swimming microorganisms are called kinesis. In general, these responses are strictly independent of the direction of stimulus. For example, photokinesis is a tendency for microorganisms to regulate their swimming speed depending on the light intensity [14]. If a cell swims faster in a bright place than a dark environment, the response is called positive photokinesis. If the cell swims slower in brightness than darkness, the behavior is described as negative photokinesis. In some special cases, swimming cells become motionless in darkness or immotile in bright light. The kinesis responses have also been observed from various microorganisms induced by other stimuli such as chemical [9, 62, 63].

Certain ciliates such as Paramecium, Didinium, Tetrahymena, and Loxodes show a kinetic behavior induced by gravity called gravikinesis [13, 14, 39, 54, 64]. However, contrary to a classical definition of the kinesis, this response is not independent of the direction of the stimulus, i.e. the gravitational field direction. Their propulsion depends on the orientation of the vector from the posterior to anterior relative to the gravity vector. Figure 3.4 shows the schematic of gravikinesis of an upward swimming Paramecium with an arbitrary inclined angle from the vertical, the gravity vector in this case [65]. The Paramecium initially exerting its propulsion, P , at an angle, θ −α, tends to increase the propulsion by ∆, gravikinetic factor. Subsequently, it sinks downward due to the sedimentation, S, and moves at a speed, VU , along an angle, θ. What we observe during an experiment from a video recording is the resulting trajectory from VU . Machemer [16] suggested a modifed definition of the gravikinetic factors for cells swimming with an arbitrary angle. From the geometry

33 g Δ 0

α 0 VU

Figure 3.4: The effects of sedimentation and gravikinesis on the trajectory of a

Paramecium swimming against gravity, g. The Paramecium swims along the in-

clination angle, θ. Arrows indicate direction of the vectors. P , S, VU , ∆, α, θ, and a indicate the propulsion, sedimentation, upward swimming velocity, gravikine- sis factor, an angle between the propulsion and veloticy, resulting velocity angle, and anterior of the cell, respectively. This figure is adopted from Journal of Theoretical

Biology, 185, 201 (1997).

34 in Figure 3.4, he introduced

S sin θ α = arc tan (3.4) S cos θ + VU q 2 2 ∆ = VD + S + 2VDS cos θ − P. (3.5)

However, the measurement of the gravikinesis for cells swimming with an inclined angle may be challenging. Detailed studies such as instant swimming cell orienta- tion measurement, drag coefficient changes during the rotation, membrane potential measurement for Parameica swimming with an arbitrary angle, and helical trajectory analysis may be required to investigate the cell orientation effects on the gravikinesis.

The gravikinesis in Paramecia differs from the gravitaxis described in the previous section. Even though gravikinesis depends on cell orientation as gravitaxis does, it does not generate cell orientation or cell accumulation. In this section, we will introduce an electrophysiological model of the gravikinesis which has been supported by experiments on several different species of Paramecia using analysis of motile and immotile Paramecia.

Electrophysiological model of gravikinesis

A central assumption of the electrophysiological model is that only the lower mem- brane of Parameda is deformed due to the gravity-induced pressure gradient across the membranes to cause an orientation dependent swimming propulsion modulation

[12, 15]. This assumption is based on another assumption that the cytoplasm of

Paramecia consists of a viscoelastic fluid with a density that is higher than that of the surrounding medium. Figure 3.5 shows the detailed description for the deforma-

35 P A

P A A P

(a) (b) (c)

Figure 3.5: The deformation of the lower cell membrane. (a) The electrophysiological model of gravikinesis in a Paramecium presumes that the gravity-induced deformation of the anterior (A) membrane during the downward swimming leads to depolariza- tion (+). The depolarization results in the depression of ciliary frequency and thus reduces propulsive speed. (b) In the upward swimming, the deformation induced by the gravity hyperpolarizes (−) the posterior end (P) and increases in ciliary frequency and propulsion. (c) In the horizontal swimming, both depolarization (+) and hyper- polarization (−) lead to the neutralization of the gravity responses. Therefore, the horizontal swimmer moves at the constant rate. The big and small arrows indicate the directions of swimming and deformaion, respectively. This figure is adopted from

Acta protozoologica, 31, 185 (1991).

36 P K-Channel Ca-Channel A

Figure 3.6: Asymmetrical distribution of Ca2+ and K+ channels over a Paramecium body. Ca2+ channels are more abundant at the anterior end (A) while the Paramecium has more K+ channels at the posterior end (P). This figure is adopted from Journal of Comparative Physiology A, 185, 517 (1999). tions of the lower membrane shape as Paramecia swim downwards (a), upwards (b), and horizontally (c). Outward deformation of the sensitive anterior (a) and posterior

(b) membrane or both (c) can cause gravitransduction. This model predicts that the anterior membrane deformation during the downward swimming results in depolar- ization as shown in Figure 3.5(a). Subsequently, the depolarization in the anterior end leads to the decreased ciliary beating frequency. Finally, this low beating frequency induces a smaller propulsive speed. On the other hand, the deformation in the poste- rior end during the upward swimming hyperpolarizes the posterior membrane (Figure

3.5(b)). The hyperpolarization increases the ciliary beating frequency and the propul- sive speed. Cells swimming horizontally do not respond to the gravitational vector because they simultaneously experience both depolarization and hyperpolarization during the constant horizontal swimming as shown in Figure 3.5(c).

The model also assumes a bipolar mechanosensitivity of Paramecia. Paramecia

37 have two antiparallel gradients of mechanosensitivity channels, K+ and Ca2+, in their

membranes extending from the anterior end to the posterior end of the cell as shown

in Figure 3.6 [15, 66]. Ca2+ channels are most abundant at the anterior end and

gradually diminished at the posterior end. On the other hand, the cell contains most

K+ channels at the posterior end while it has fewer Ca2+ channels at its anterior

end. The asymmetrical distribution of Ca2+ and K+ channels allows the Paramecia

to respond to the gravitational vector. Activation of anterior Ca2+ mechanoreceptor

channels generates a depolarizing potential while stimulation of K+ mechanoreceptor

channels hyperpolarizes their membranes. The polarity and amplitude of the mem-

brane potential determine how the cilia beat. For example, the electrophysiological

theory of the gravikinesis predicts that Paramecia swimming upward orientation have

their deformed posterior membrane. This stimulus opens most of K+ mechanorecep-

tor channels which are most abundant near the posterior area. These opened K+ channels result in hyperpolarization and thus increased ciliary beating frequency and the propulsive speed to compensate their sedimentation [67]. On the other hand, for

Paramecia swimming downwards, the gravity-induced outward deformation in the anterior membrane opens mostly Ca2+ channels. The Ca2+ channels induce weak depolarization and reduces the ciliary beating frequency and the propulsive rate re- sulting in a slowed downward swimming speed. This regulation in propulsive speed is called negative gravikinesis. The negative sign is due to its direction opposite to that of the gravity vector and sedimentation.

The electrophysiological model was developed by Gebauer and coworkers [12]. In the study, they measured the change in the propulsion rate as a function of swimming

38 Figure 3.7: Orientation-dependent changes in active propulsion of Paramecium cau- datum under normal gravity. 0◦ indicates that the Paramecium swims upwards. Each

15◦ sector contains at least 408 Paramecia. This figure is adopted from Naturwis- senschaften, 86, 352 (1999).

39 cell orientation (Figure 3.7) and compared the results with orientation-dependent membrane potential obtained from impaled Paramecia by rotating the Paramecia

180◦. The symmetrical propulsive rate distribution as a function of swimming orien- tations in Figure 3.7 agreed with the electrophysiological model. When the Paramecia swam horizontally, the gravikinetic response was minimum or absent. On the other hand, the gravikinetic response was maximized at the vertical orientations. The maximal propulsion changes, 46 µm/s, occurred when cells were oriented between

80◦ and 100◦. They suggested that an upward reorientation of a Paramecium from the horizontal position shifted the effective area affected by the gravity vector more posteriorly resulting in increased cell propulsion. Similarly, a downward reorientation of the horizontally directed cell resulted in the effective area more anteriorly and thus caused decreased propulsion of the cell.

Measurement of gravikinesis

The electrophysiological model predicted that the swimming velocity of cells is the resulting vector sum of propulsion for a free swimmer, P~ , passive sedimentation, S~, and propulsion changes due to gravikinesis factor, ∆~ [15]. For upward swimming

Paramecium, it is evident that the gravity and sedimentation tends to reduce the speed of the Paramecium while they increase the speed of the cell swimming down- ward. To investigate the gravikinesis responses in a Paramecium, it is necessary to observe its passive tendency to sediment. If we assume that the sedimentation is linearly proportional to gravity and that there is no gravikinesis effect, then we may

40 have

VU = P − S, (3.6)

VD = P + S (3.7)

where VU and VD indicate the upward and downward swimming speeds, respectively.

Therefore, this tendency to sediment without the gravikinesis may lead to

VD − VU = 2S (3.8) where the sedimentation speed (∼ 100µm/s under normal gravity) is a linearly pro- portional to varying gravity force, fgm(g). To test the active regulation in propulsion,

Guevorkian [1] measured the swimming speed of Paramecia under varying fgm(g) for both upward and downward swimmers as shown in Figure 3.8 where the solid line corresponded to the passive response without gravikinesis as described in Equation

(3.8). If gravikinesis does not exist in Paramecia, then Equation (3.8) should hold.

The speed distribution in Figure 3.8, however, showed the existence of gravikinesis in Paramecia. The measurements of the upward (N) and downward (H) swimming speed failed to match the passive response. This result indicated that Paramecia exhibit a negative gravikinesis in response to the variable gravity force.

To explain the compressed speed distribution inside the passive line in Figure 3.8,

Guevorkian used the definition of gravikinesis, introduced by Machemer [15] using two gravikinetic factors, ∆U and ∆D, for an upward and downward swimmers. The corresponding relationships were given by

VU = P − S − ∆U (3.9)

41 Figure 3.8: Swimming speed of Paramecia in simulated gravity for upward (N) and downward (H) swimmers. The bars indicate the widths of the speed distributions.

The solid lines denote the predicted change in the swimming speed in the absence of the gravikinesis. Their widths indicate the uncertainty in the mean sedimentation rate. This figure is adopted from Proceedings of the National Academy of Sciences,

103, 13051 (2006).

42 Figure 3.9: The averaged gravikinetic factor measured from three trials. The lines are least-square fits to the data. The negative slope implies that Paramecia have negative gravikinesis. The uncertainty bars are ± SD. This figure is adopted from

Proceedings of the National Academy of Sciences, 103, 13051 (2006).

43 VD = P + S + ∆D (3.10)

∆ = (∆D + ∆U )/2 = (VD − VU − 2S)/2 (3.11) where ∆ is the averaged gravikinetic factor. The measurements for the averaged gravikinetic factor from three different Paramecia under varying forces were shown in Figure 3.9. The averaged slopes, −50µm/s, from the three trials showed the gravikinetic factor dependence on the varying gravity forces. The measurement agreed with other studies [13, 15, 16, 39].

Other effects on gravikinesis

Some factors such as nutrient conditions and swimming direction of cells can affect the values of the gravikinetic factors. For example, gravikinetic factors can substantially vary depending on nutritional status [68]. In this experiment, variable gravity between

1.5 and 5.4 g was applied to investigate the negative gravikinetic in Paramecium. The gravikinetic factors of well-fed cells (3 day cultures) was −48 µm/s. On the other hand, the gravikinetic factors of starved cells (7 day cultures) substantially decreased to −66 µm/s. This study reported that the primary reason for this huge difference

(∼ 40%) was due to the gravikinetic response of the upward swimmers. However, the reason for the propulsion changes only in upper swimmers is still unknown.

44 3.3 Paramecium in magnetic fields

3.3.1 Paramecium under a static magnetic field

Experiments on highly ordered biomaterials such as isolated retinal rods [69] or

Paramecium [4] showed that their orientations can be rotated in a homogeneous

magnetic field due to the magneto-orientation effect. This effect arises from a dia-

magnetic anisotropy which most living organisms have. Assuming a cylindrically

symmetric body shape, they found that the cylindrical rods tended to be rotated and

reoriented with their long body axes parallel or antiparallel to the homogeneous mag-

netic field. They also observed that the reorientation depended on the field strength.

In this section, we will discuss the magneto-orientation based on the diamagnetic

anisotropy in Paramecia as well as another possible magnetic torque effect caused by

shape anisotropy.

Magneto-orientation based on diamagnetic anisotropy

In this model, we assume that a Parameicium has a cylindrical symmetry body shape

as shown in Figure 3.10 where z axis is located along the long axis of the rod. The directions of x and y axes were arbitrarily chosen, and the center of the cylindrical body located at the origin, o, of the reference coordinate system. We assume that the magnetic field, B, lies on the x − z plane. The volume susceptibility, a second order

45 B = Bcosθ B z

By = 0

Bx = Bsinθ

Figure 3.10: The coordinate system for the calculation of the orientation energy of an anisotropic rod in a homogeneous magnetic field B which lies in the x-z plane. θ indicates the angle between the magnetic field and the z axis. This figure is adopted from Biosystems, 36, 187 (1995).

46 tensor ←→χ , in the reference coordinate system is given by a constant diagonal matrix,   χ 0 0  ⊥    ←→   χ =  0 χ 0  (3.12)  ⊥      0 0 χk

where χ⊥ and χ⊥ are the radial and axial principal volume susceptibilities of the

cylindrical body, respectively [70]. We assume that the variable magnetic field in the

x − z plane is given by a column vector or a row vector,   B sin θ       B~ =  0  (3.13)       B cos θ   B~ = B sin θ 0 B cos θ (3.14)

where θ is the angle between the magnetic field and the z axis as shown in Figure 3.10.

To calculate the magnetic potential energy, UB, we consider the induced magnetic moment, ~µ,       χ 0 0 B sin θ χ sin θ  ⊥     ⊥        ←→       ~µ = V χ · B~ = V  0 χ 0   0  = VB  0  (3.15)  ⊥                  0 0 χk B cos θ χk cos θ where V is the volume of the rod. If we assume that the magnetic field is constant and homogeneous, then the magnetic potential energy is given by,

Z B 2 ~ B 2 UB = − dB · ~µ = − (∆χ cos θ + χ⊥) (3.16) 0 2µ0

where ∆χ and µ0 indicate ∆χ = (χk−χ⊥)V and the magnetic permeability of vacuum,

respectively. The magnetic potential predicts two possible cases for a diamagnetic

47 object, χk < χ⊥ < 0 or χ⊥ < χk < 0. Guevorkian [4] found that the second case

holds for Paramecium. This result suggests that there are two stable equilibrium orientations of Paramecium when θ = 0◦ or θ = 180◦. From Equation (3.16), the

magnetic torque due to the diamagnetic anisotropy is given by

dU 1 2 τB = − = − ∆χB sin 2θ. (3.17) dθ 2µ0

Magneto-orientation based on shape anisotropy

Magneto-orientation based on the diamagnetic anisotropy is not the only possible

mechanism of the reorientation. Even an object having a perfect diamagnetic isotropic

possibly experiences a torque exerted by a homogeneous magnetic field. If the shape of

the object is anisotropy, then the interaction between the externally applied magnetic

field and the locally induced field causes a torque on the isotropic diamagnetic object

[70]. The resultant magnetic field is not homogeneous. In other words, the initially

homogeneous magnetic field is not perfectly homogeneous anymore. Therefore, the

object tends to rotate in the homogeneous magnetic field. Based on theories [71, 72],

Guevorkian [4] estimated the shape anisotropy torque, τS, acting on a Paramecium,

−27 1 2 τS = 1.8 × 10 B sin 2θ. (3.18) 2µ0

Considering that ∆χ in Equation (3.17) is of the order of 10−23, comparing the shape

anisotropy torque with the diamagnetic anisotropy torque suggests that the orienta-

tion energy due to the shape anisotropy torque is much smaller than the thermal noise

5 [70] and that the magnitude of τS is approximately 10 times smaller than τB. In this study, therefore, we ignore the orientational effects due to the shape anisotropy.

48 Aligning Paramecium along static magnetic field

Since Paramecia swim in low Reynolds world, the magnetic torque is linear to orien- tation rate, θ˙. Therefore

˙ τB = βθ (3.19)

where β is the drag coefficient for a radial rotation. For objects that are ellipsoids of

revolution, the rotational drag coefficient can be approximated [73]

8πηa3 β = (3.20) 3 ln(2a/b − 1/2)

where a and b indicate the semimajor and semiminor axis, respectively. If we integrate

Equation (3.17) and Equation (3.19), we can obtain the time dependence on the rotation ∆χB2t ln(tan θ) = ln(tan θ0) − . (3.21) µ0β

Figure 3.11(a) shows image sequence of an immobilized Paramecium in 4 mins.

The Paramecium is neutrally buoyant using a magnetic field levitation method which

will be introduced in the next section. In a Ficoll solution of 6.5 times the viscosity

of water and at 4 T magnetic field, the initially horizontally oriented Paramecium

rotated parallel to the magnetic field. Figure 3.11(b) shows the time and magnetic

field magnitude dependence of the orientation. The rotation rate is proportional to

the strength of the magnetic field and maximized at 45◦ suggesting that the response

is passive. The plotting and resultant slope in Figure 3.11(c) from Equation (3.21)

can yield the average of ∆χ. The result from three different trials on immobilized

Paramecia estimated the mean value of ∆χ = 6.7 × 10−23 m3. Guevorkian also

49 (d)

Figure 3.11: (a) Image sequence of a immobilized and neutrally buoyant Paramecium

in a homogeneous magnetic field in Ficoll solution. (b) Orientation rate of immobi-

lized and neutrally buoyant Paramecia as a function of time where t(s) = 0 indicates

◦ −1 2 the time when θ = 45 . (c) ln(tan θ) vs. ts = t(BB0 ) in various magnetic field B, where B0 = 6.4 T . (d) The averaged ∆χ (the dashed line) obtained from swimming

Paramecia. This figure is adopted from Biophysical journal, 90, 3004 (2006).

50 measured ∆χ for swimming Paramecia. She reported that the averaged ∆χ, 8.3 ×

10−23 m3, varied little in magnetic field change up to ∼10 T (Figure 3.11(d)).

3.3.2 Diamagnetic levitation

Levitation of objects without material support has been investigated and developed in many ways [27]. In particular, Beaugnon [29, 28] reported that magnetic field can levitate diamagnetic organic materials. Although a very strong magnetic field and gradient are required to exert forces sufficient to levitate the materials, the diamag- netic levitation opened a new branch of biophysics such as levitation of biological organisms which allowed investigation of the gravisensitivity of biological systems

[30]. For example, Paramecium has been known as one of the most sensitive and tini- est gravity sensors. As discussed previously, they reorient their swimming direction antiparallel to the gravity vector (negative gravitaxis) and incerase their propulsion as they swim against the gravity (negative gravikinesis). In this section, we will introduce a general introduction to the magnetic field induced force on a diamag- netic object and its application to living microorganisms. The technique, Magnetic

Force Buoyancy Variation (MFBV), can vary the apparent weight of Paramecia to investigate their swimming properties under variable gravity forces.

Magnetic field gradient levitation

It is basically always possible to levitate every living creature using a magnet since all organisms are, at least, weakly diamagnetic. Diamagnetic materials such as water and microorganisms develop persistent atomic or molecular currents which oppose

51 1 1.0

) 1

-

(m

0 0

2 0

0 0.5

)/B B/B

dz

B(dB/ -1 0.0

-200 -100 0 100 200 z (mm)

Figure 3.12: The magnetic field profiles for B (black) and B(dB/dz) (gray) as a function of the distance from the center of the magnet. B is max when z = 0 while the magnitude of B(dB/dz) is max when z = ±83. The data are normalized to the center magnetic field value B0.

52 externally applied magnetic fields [74]. However, the molecular magnetism is very weak and thus not noticeable in everyday life.

If a biological object of magnetic susceptibility, χO, is located in a very strong magnetic field, its potential energy per unit volume is given by [74]

χ U(z) = − O B2(z). (3.22) 2µ0

Then we can obtain the magnetic force per unit volume, FM (z), exerted on the diamagnetic substance in the z direction

χ dB(z) FM (z) = −∇U(z) = B(z) . (3.23) µ0 dz

Figure 3.12 shows a schematic of the normalized magnetic field and gradient profile in a solenoid magnet used in this study. The magnetic field (black), B, is maximum at the middle of the magnet while the magnetic field gradient (gray), B(dB/dz), has two peaks at two equidistant points from the center. As Equation (3.23) indicates, the levitation force at the center of the magnet is zero since the magnetic gradient is zero. The magnetic force is upward when the object is placed above the magnet

(z > 0). The object can be levitated at this upper position since the diamagnetic force opposes the gravity vector. At the upper position, the magnetic levitation under the normal gravity is obtained by adjusting B(z)(dB(z)/dz) so that

dB(z) µ ρg B(z) = 0 . (3.24) dz χ

On the other hand, if the object is located at the lower position, the magnetic force is downward (z < 0). At this position, the diamagnetic force combines with earth’s gravity to exert a stronger downward force. With the 50 mm bore magnetic system

53 at National High Magnetic Field Laboratory (NHMFL) in Tallahassee, Florida, the

maximum magnetic field, 31 T , and maximum B(dB/dz), 4700 T 2/m were available.

For a diamagnetic material such as a cell in a solution, we should consider the

buoyancy and drag of the cell. Then the total force acting on the cell is given by

~ ~ ~ ~ FT otal = FM + FD + FBuoy (3.25)

~ ~ where FD and FBuoy denote the linear drag force and buoyancy force, respectively.

From Equation (3.23), the diamagnetic force acting on both the cell and solution at

the same time is given by

~ (χO − χS)V dB FM = B zˆ (3.26) µ0 dz

where χO and χS are the magnetic susceptibilities of the object and solution, respec-

tively. At low Reynold number, the drag force is linearly proportional to the moving

speed of the object,

~ FD = −ξvzz.ˆ (3.27)

where ξ and vz are linear drag coefficient of the object and speed of the object in z

direction. Here we assumed that the object moves along the z direction. The total

buoyancy force is given by

~ FBuoy = −(ρO − ρS)V gzˆ (3.28)

where ρO and ρS indicate the densities of the object and solution, respectively. The zero total force at low Reynolds number yields the apparent weight per unit volume, w,

(χO − χS) dB ξ w = (ρO − ρS)g − B = − vz. (3.29) µ0 dz V

54 The apparent weight per unit volume under normal gravity and without the magnetic

field, w1g, is defined as

w1g = (ρO − ρS)g. (3.30)

Then we can obtain the normalized apparent weight on the cell

w (χ − χ ) B dB = 1 ± O S (3.31) w1g (ρO − ρS) gµ0 dz where − and + signs are for the levitation (z > 0) and sedimentation (z < 0) positions (Figure 3.12), respectively. With the magnet at the NHMFL, we could apply the range of the apparent weight from −8g to 10g on swimming Paramecia.

Equation 3.31 can be rewritten as

w  B 2 = 1 ± (3.32) w1g Bneut

where Bneut is the magnetic field at neutral buoyancy. Bneut was measured as the

field at which immobilized Paramecia stopped to sink or rise in solution. We derive this form by writing the field as a function of z as

B(z) = Bf(z) (3.33) where B is the field at the center of the bore. f(z) depends only on the geometry of the magnet windings. Consequently,

dB(z) df(z) B(z) = B2f(z) . (3.34) dz dz

For neutral buoyancy, w = 0, from Equation (3.31), we have

dB(z) (ρP − ρs) Bneut |B=Bneut = gµ0. (3.35) dz (χP − χs)

55 Thus, we can derive

dB(z) !2  2 w B(z) dz B = 1 ± dB(z) = 1 ± . (3.36) w1g Bneut Bneut dz |B=Bneut

56 Chapter 4

Apparatus and methods

57 4.1 Introduction

The majority of this study used a method for continuously varying forces acting on

swimming Paramecia, an optical system to record their locomotion for a long period,

and image analysis software. To obtain the varying forces, we employ a technique,

Magnetic Force Buoyancy Variation (MFBV), at the National High Magnetic Field

Laboratory (NHMFL) in Tallahassee, Florida. This method employs intense and in-

homogeneous static magnetic fields generated by current carrying solenoids to apply

magnetic forces to cells and solutions. At the NHMFL, the forces generated were

sufficient to even stall swimming Paramecia. This magnet also afforded the space

necessary for an optical system to investigate the effects of the buoyancy changes on

the swimmers. The optical system was designed to be nonmagnetic not to compro-

mise the strong magnetic field of the magnet. The experimental setup also required

an image analysis technique to trace their trajectories and investigate them. The

images from a 90◦ side view borescope and a charge coupled device camera were ana- lyzed using Image Pro Analyzer 7.0 (Media Cybernetics, Rockville, MD). In addition, other detailed experimental methods such as Paramecia culturing and making viscous

solution will be described in this chapter.

58 4.2 Apparatus

4.2.1 NHMFL magnet

Paramecia were exposed to magnetic fields generated by a DC resistive magnet at the NHMFL shown in Figure 4.1(a)-(b). This 50 mm bore resistive magnet offered magnetic fields up to 31 T and large B(dB/dz) up to 4700 T 2m−1. Figure 3.12 shows the normalized field profiles for B (black) and B(dB/dz) (gray) produced by this magnet. The magnetic field B is maximum at the center of the magnet where z = 0.

The magnetic field gradient profile is also shown in Figure 3.12. The magnitude and direction of the varying magnetic force depended on the magnetic field gradient. The magnetic force is zero at the center of the magnet because the field gradient is zero at that position. The magnitude of the force is maximum at two equidistant positions z = ±83 mm from the center of the magnet. For a diamagnetic object above the center of the magnet, the total applied magnetic force is upwards. This upward magnetic force allowed us to levitate the diamagnetic object. On the other hand, the total force is downwards for a sample below the center of the magnet. At this position, we could obtain the maximized apparent weight for the sample. We adjusted the height of the sample position from the bottom of the magnet and inserted our sample from the top of the magnet. See the chapter 4 for the details of the principle of the varying magnetic force and its application.

59 (a) (b)

Figure 4.1: (a) A photo of the magnet system at the NHMFL. (b) A cross sectional drawing of the resistive magnet (www.magnet.fsu.edu). Dimensions are in millime- ters. The sketch is not scale drawn to scale.

60 4.2.2 Illumination

Guevorkian [75] had designed an optical system using green LEDs. Since these LEDs

were magnetic, she used long optical fibers as light guides to illuminate samples deep

inside the bore of the magnet. However, controlling this optical system was difficult

because the imaging qualities of tiny samples were very sensitive to the position of

the light guides. In this study, we designed a green LED backlight system (TBL 1×1,

Metaphase Technologies, Bensalem, PA) shown in Figure 4.2. The new LED light

array was tightly held to a copper plate for heat sinking. A piece of black tape was

placed over the center of the array to produce uniform dark field illumination with a

clean background as shown in Figure 4.2(a). To minimize the magnetic property of

the LED light, we replaced all magnetic components with nonmagnetic substitutes.

This LED illumination system was also chosen to minimize phototactic effects on

Paramecia swimming motion. Paramecia show the least changes in their swimming

with green monochromatic light (565 nm) [55]. This monochromatic illumination also

reduced heating which affects on their motion [76, 77]. The recommended driving

voltage for this LED light source was 24 V . To minimize the heat from the light source, however, we used 15 V which required a current less than 0.01 A. This reduced voltage provided sufficient intensity to observe their swimming. The heat from the backlight LED increased the temperature of the sample by about 2 degrees, but the temperature could be easily reduced using the temperature control system.

61 (a) (b) (c)

Figure 4.2: The pictures of the thin LED green backlight from the front (a), the back

(b), and the side (c). A piece of black tape was used to provide a uniform black background. The LED was attached to a cupper plate to keep room temperature.

4.2.3 Temperature control

The temperature of a sample chamber in the magnet bore changed due to magnetic

field ramping and the heat from the LED light source. As we indicated above, changes in temperature alter Paramecia swimming. To keep a constant chamber temperature, we designed a water circulation system. As seen in Figure 4.2 and Figure 4.3(a)-(c), we developed a support structure to mount both the LED backlight and the experimental chambers on a copper plate and a copper sample holder using screws. The copper plate and the sample holder was directly connected to two legs of a copper u-tube

(Figure 4.3(b)-(c)). Water from a temperature regulated bath was circulated through the u tube while we kept supplying ice cubes to the water bath. Five meter long

62 Camera

Water tube (b) (Tygon) Borescope

Thermal couple Sample distance controller Water tube (cupper) (c) Borescope height controller

Sample LED holder backlight

(a) Sample height controller

Figure 4.3: (a) The picture of the apparatus. Various components such as the 6 mm diameter borescope, CCD camera, LED, and support structure are shown. (b) Close up view of the bottom of the apparatus. The copper tubing connected to a water circular bath runs through the copper plate. The position of the sample chamber was adjusted using the sample distance controller. (c) Close up view of the apparatus near the sample chamber. The chamber was fixed at the sample holder using screws.

The height of the borescope was adjusted using the borescope height controller.

63 (a) (b) Holes for Groove for injection a glass strip

Holes for air bubble

Holes for thermal couple

Figure 4.4: Experimental chambers. (a) 2×16×16 mm square chamber for observ- ing Paramecia population. (b) 5 mm diameter circular chamber for tracking single

Paramecium.

Tygon tubes connected the two copper tubes to the water bath. The sample chamber was directly connected to a thermal couple to monitor the temperature of the sample

(Figure 4.3(a)). The temperature of the sample was constant to ± 1◦.

4.2.4 Sample chambers

We employed two different types of experimental chambers, a 2×16×16 mm square chamber and a circular chamber with 5 mm diameter. Frames of both chambers were made of Plexiglas (Rohm and Haas, Philadelphia, PA) sandwiched by two coverslips. These coverslips were sealed to the frames with VALAP (1:1:1, vase- line/lanolin/paraffin) which is known to be nontoxic for microorganisms. Both cham-

64 bers had a depth of 2 mm which was deep enough to avoid frequent collisions with the coverslips and small enough to provide a 2 dimensional environment. We designed both chambers to have symmetrical shape to eliminate asymmetrical swimming dis- tributions caused by the geometry of a chamber [15]. Each chamber type had two holes, one for the injection of solution containing Paramecia with a syringe and the other to remove air bubbles. These two holes also were sealed with the VALAP after the injection. Both chambers had additional two holes in their frames to insert ther- mal couples to monitor temperature changes in the sample. The chambers were fixed in the sample holder. A sample height controller was first used for the macro control of the chamber height and a second sample distance controller for the micro control as shown in Figure 4.3(a)-(c).

The first type of the chamber shown in Figure 4.4(a) was designed for investigating populations of swimming Paramecia. This chamber also allowed us to observe the swimming behavior of Paramecia near surfaces by inserting a glass strip into the grooves at the middle of the chamber. The second type of the chamber had a circular geometry (Figure 4.4(b)). This chamber was specially designed to trace a single

Paramecium trajectory. Its circular geometry allowed most of Paramecia in the chamber to swim at the middle of the chamber without collisions on the side walls.

This geometry of the chamber maximized the frequency at which the Parameica pass through the field of view and the number of the trajectories from a single Paramecium at a time.

65 4.2.5 Borescope and video recording

We used two different types of borescopes, a 6 mm diameter and a 8 mm diameter

90◦ side view borescopes (123006 and 123008, ITI, Westfield, MA) to view swimming

Paramecia in the 50 mm bore of the magnet. Their small diameter (6 mm and 8 mm, respectively) and long working distance (36 cm and 70 cm, respectively) were suitable to fit deep into the small diameter bore. The proper vertical positions of the borescopes were adjusted using the borescope height controller in Figure 4.3(c).

Both borescopes had a 40◦ viewing angle and weak magnetization. Paramecia that were very close to the top and the bottom cover glasses were filtered by focusing the borescope on the middle of the chamber. The depth of focus of the borescope was

∼ 1 mm. The end of the borescopes connected to a charge coupled device camera

(XCD-SX 90, Sony, Tokyo, Japan) through a C-mount adapter. Recording were made at 7.5 frames/s.

To record ciliary motion of swimming Paramecia, we emplyed a high speed camera

(Fastcam PCI R2, Photron USA, San Diego, CA), its associated software (Fastcam

Viewer, Photron USA, San Diego, CA), and an inverted light microscope (Nikon

TE2000, Tokyo, Japan). However, the measurements of ciliary frequency of swimming cells using the high speed camera (500 frames per second) and the microscope required to solve two challenging problems, 1) the small field of the view obtained from the high magnification of the microscope and 2) the short depth of the field of the high magnification microscope. These problems could be solved by 1) an extremely tiny container to provide highest appearance rate for an observation and 2) a low height

66 container but tall enough not to affect their swimming to continuously trace the cells.

We used a tiny drop of solution (∼ 10 µL) containing manually taken 10 cells and sandwiched the solutions between coverslips separated by parafilm (Sigma Aldrich, St.

Louse, MO) sheets of approximately 125 µm thickness. This constrained geometry allowed the cells to be shown frequently and us to observe them continuously. Their swimming trajectories seemed to deviate from normal, but their swimming speeds changed little.

4.3 Image analysis

4.3.1 Tracking swimming trajectories

The images of the swimming trajectories obtained from the borescope and the CCD camera were analyzed using Image Pro Analyzer 7.0 (Media Cybernetics, Rockville,

MD). The following steps were repeated to trace the trajectories of swimming Parame- cia:

1. Merge 150 uncompressed images into one video file from the Sequence-Merge

files menu. Errors often occurred when using compressed images and more than 300

images.

2. Set proper Area of interest (AOI) from the menu at least 10 body lengths away from the plexiglass frame or the cover glass strip to avoid wall effects.

3. Open the Tracking data table from the Measure-Track objects menu.

4. Open the Tracking options. From the Measurements tab, select suitable

67 variables to measure. In this study, mean velocity, X and Y coordinates, body an-

gle, and time were usually measured. From the View/Output tab, choose desired options. Especially, set the Frame interval to 0.133 to correspond to the 7.5 frame rate. Carefully set Track parameters from the Auto tracking tab. In general, set- ting the Velocity limit to 30 pixels/frame and Minimum total track length to 50 pixels without calibration resulted in the maximum number of tracking trajectories.

5. Click Find all tracks automatically, and select the appropriate threshold for the objects from the Intensity range selection. Most of the cases, Manual-Select ranges were used. Since our objects are bright, choose the intensity range from 50 to 255.

6. Trajectories 3 times faster or slower than average speed were manually dis- carded. These trajectories were either from noise or wrong tracking. In the next section, the specific methods used to eliminate inaccurate trajectories and to calcu- late various swimming parameters will be described in more detail.

4.3.2 Fundamental swimming parameters

As we described in the previous chapter, Paramecia swimming trajectories aligned with intense static magnetic fields. However, these alignments were not perfect enough to obtain their fundamental swimming parameters. In general, most of their helical trajectories were rotated several degrees from the vertical axis resulting in inaccurate

fitting of the projection of their helical trajectories with a sine function. In order to obtain the approximate rotation angle, we first fitted the trajectories with a linear

68 function, z = ax + b, where z, x, a, and b are the vertical and horizontal coordi- nates and constants, respectively. Most of tilted trajectories were well fitted with x = R sin(2πz/λ + φ) after the rotation of the original trajectory with an angle α where φ, R, λ, and α = tan−1(a) are a phase constant, amplitude, wavelength, and rotation angle, respectively, as shown in Figure 4.5. Note that the z-axis points along

the applied magnetic field, which is the direction along which the trajectories tended

◦ 2 to align [4]. Additional trajectories were discarded if α > 10 , Rreduced < 0.9, or the

total length of the track was shorter than one wavelength. Finally, from the parame-

ters describing the 2D swimming trajectories, we could obtain two additional funda-

mental parameters, the angular frequency of the motion, ω = (2π/λ)(∆z/∆t), and q the averaged swimming speed along the 3D helical trajectories, v = ω R2 + (λ/2π)2

[78].

4.4 Experimental solutions

4.4.1 Test solution

All Paramecium experiments in this study were done in test solution to provide constant experimental environment. The final test solution contained 1 mM CaCl2,

1 mM KCL, 0.1 mM MgSO4, and 1.5 mM MOPS at PH 7.2. To obtain the test solution,

1. Take 1 liter of distilled water in a glass flask.

2. Mix the required amount of chemicals into the distilled water and agitate the

69

m)

μ z(

λ 2R

x (μm)

Figure 4.5: Sample fit of a swimming track (black dots) to a sine wave (gray line) along with the parameters employed.

70 mixtures until all chemicals were completely dissolved.

3. Adjust PH by adding NaOH or HCl until the PH of the solution reaches ∼ 7.2.

Usually, the PH changes in several hours due to unsolved chemicals. Keep checking the PH for few days and just before experiments.

For an experiment, Paramecia were adapted to the test solution for about 2 hours.

Paramecia showed increased swimming speed just after exposed to the test solution.

However, this behavior disappeared around 2 hours later.

4.4.2 Methyl Cellulose solution

We created viscous solutions by adding 400 cP Methyl Cellulose (Sigma Aldrich, St.

Louis, MO) to test solution. We chose Methyl Cellulose to make clear connections with previous experiments on Paramecia [38, 79]. Also, the density of the Methyl

Cellulose solutions was more comparable to water than, for example, Ficoll solutions of the same viscosity. We found that the swimming speed in Ficoll solutions vs. viscosity is nearly identical to the Methyl Cellulose results (see chapter 6).

The Methyl Cellulose solutions were made in several steps. To make a high- concentration Methyl Cellulose solution, we first heated about 1/3 of the required volume of test solution to at least 80◦C and added the Methyl Cellulose powder to the hot water with agitation. Then we agitated the mixture until the powder was thoroughly wetted and evenly dispersed. For complete solubilization, the reminder of the water was added as cold water or ice to lower the temperature of the dispersion.

Once the dispersion reached the temperature at which that particular Methyl Cellu-

71 Table 4.1: Recommended viscosity range for each model and modified viscometer constant at 20◦C. Model Range Viscometer Constant at 20◦C

50 0.8 to 4 (cP) 0.004416 (mm2/s2)

75 1.6 to 8 (cP) 0.007487 (mm2/s2)

100 3 to 15 (cP) 0.014120 (mm2/s2)

150 7 to 35 (cP) 0.034090 (mm2/s2)

lose product became water soluble, the powder began to hydrate and the viscosity increased. Solution were cooled to about 0◦C ∼ 5◦C for 30 mins. We continued agi- tation for at least 30 mins after the proper temperature was reached. Then the high concentration Methyl Cellulose solution was diluted with test solution to produce a solution with the reduced viscosity. The dilutions were done by inverting the mixing tubes about 10 times.

4.4.3 Viscosity measurement

Viscosities were measured at constant flow using glass capillary viscometers (Cannon,

State College, PA). The final experimental solutions had Methyl Cellulose concentra- tions of 0%, 0.2%, 0.33%, and 0.5% and ηw of 1.0, 2.3, 4.1, and 6.9, respectively. To measure various values of the viscosity, we used flow viscometers with four different viscosity range as shown in Table 4.1 and Figure 4.6. The viscosity of a solution was measured in several steps. First, a viscometer was cleaned using chromic acid or non-chromium cleaning solution and by passing clean, dry, and filtered air through

72 Figure 4.6: Viscometer for transparent solutions. the viscometer to remove the final traces of solvents. Periodically, traces of organic deposits had be removed with the cleaning solution. To fill a viscometer, it was in- verted to immerse tube N in the solution, and suction was applied to tube L. Once

filled to F , the viscometer was returned to its original vertical position. For the mea- surement, the viscometer was placed into a holder and aligned vertically. Suction was appled to tube N to draw the liquid slightly above mark E. To measure the efflux time, the solution was allowed to flow freely down past mark F, while measuring the time for the meniscus to pass from mark E to F. Finally, the kinematic viscosity of

73 the sample was calculated by multiplying the efflux time in seconds by the viscome- ter constant in Table 4.1. For simplicity, we assumed the viscometer constant to be linearly proportional to temperature.

In addition, we used a cone and plate rheometer (AR2000, TA Instruments, New

Castle, DE) provided by Anubhav Tripathi to measure the rheology of the highest concentration Methyl Cellulose solution that we employed in our work. The viscosity measured at a shear rate of 10 Hz, which is the estimated shear rate associated with ciliary beating, agreed to within 10% of the value measured with a flow viscometer.

We estimated the shear rate as the speed of the tips of the cilia during the power stroke relative to a Paramecium body, 100 µm/s over the length of a cilium, 10 µm.

Over a shear rate of 3 Hz to 40 Hz, the measured viscosity varied by ± 6%. Thus, the solution does not exhibit shear thinning over the range of frequencies relevant to ciliary motion. Also, we performed oscillatory measurements to get G’ and G” as a function of frequency where the storage modulus, G’, describes the elastic properties and the loss modulus, G”, describes the viscous properties. In the 10 Hz range, G’ was below the sensitivity of the rheometer. We estimate G’ to be more than a factor of 25 smaller than G” at 10 Hz based on the smallest G’ that we were able to measure.

On the basis of the above measurements, the Methyl Cellulose solutions employed in the current experiments behave like Newtonian fluids for swimming Paramecia.

At the Methyl Cellulose concentration of 0.5%, the density of the solution was

0.2% larger than pure test solution. The molecular weight of the Methyl Cellulose employed was 41 kDa, which corresponds to a radius of gyration of about 25 nm [80].

The areal density of the cilia is ∼ 1/(300 nm)2. Thus, the radius of gyration is about

74 1/10th the spacing between cilia. At a concentration of 0.5% the inter molecular

spacing is about 25 nm.

4.5 Paramecia culturing

4.5.1 General procedure

Two types of the species of protozoans, Paramecium caudatum and Paramecium

tetraurelia (Carolina Biological Supply, Burlington, NC), were cultured in our labo-

ratory. Once they were shipped, we first carefully opened the container and aerated

the Paramecia using a pipet. To avoid contamination, we used a different pipet for

each culture. We kept them at room temperature for 30 mins for them to settle. After

the settlement, we removed other microorganisms in the new culturing. To clean the

new cultures, we carefully took a few mL of the culturing medium using a pipet and diluted it with test solution in a petridish. Using another pipet, we took a Parame- cium from the petridish and moved it into a new petridish filled with test solution.

This procedure was repeated 3 more times. We prepared 5 sets of the cleaned sin- gle Paramecium medium and moved each of them into a jar with culturing medium containing food (see the next section). Some of the new culturing medium failed due to contamination, but most of them successfully multiplied after one or two weeks.

We subcultured old medium using the same method after the parent culture reached maximum population or before we started a new experiment.

75 4.5.2 Bacteria culturing

Paramecia must prey on other microorganisms to provide their energy. To culture these protozoans successfully, you must provide them with food by setting up a food chain. When a new culture was inoculated with Paramecia, small amounts of bacteria and Chilomonas were included in the medium. The bacteria multiplied around wheat grains in the medium, the Chilomonas fed on the bacteria, and the Paramecia fed on the increased numbers of Chilomonas. However, the Paramecia increased rapidly and consumed all their food in a few weeks because they had no predators. Therefore, we had to keep adding the food for Paramecia to survive for long periods.

We purchased bacteria, Enterobacter aerogenes (Carolina Biological Supply, Burling- ton, NC), grown on a tube with agar gel. To transfer the new bacteria to a petri dish,

1. Clean all work surfaces as much as possible to avoid contamination.

2. Mix 4.6 g agar with 200 mL distilled water and boil them for one min.

3. Cool down the medium (45◦ ∼ 50◦) and pour it into a petri dish.

4. Quickly cover its lid and cool down more to room temperature.

5. Store them in a fridge upside down to prevent condensation contaminating the agar gel.

6. To culture bacteria, take a few of the agar dishes from the fridge and wait until they reach room temperature.

7. Flame an inoculating loop and the mouth of the tube containing the bacteria.

8. Insert the inoculation loop into the tube and take a small quantity of the agar

76 culture.

9. Open the covers of the petri dishes and touch the loop to the top of the dishes and streak from side to side all the way to the bottom edge. The finished plates will have zigzag patterns from edge to edge.

10. Store the petri dishes upside down in a incubator at 32◦. After 1-2 days the zigzag pattern will be thicker and small dotted areas of bacteria will be apparent.

11. Store them upside down in the fridge to prevent condensed water from dripping onto the agar and causing the colonies to run together.

4.5.3 Culturing medium

Paramecia are best cultured under conditions of moderate light, a neutral or slightly alkaline pH, and temperature of 20 to 21◦C. To maintain Paramecia culture, several culturing techniques have been developed [81]. In this study, we combined protozoan pellets and wheat grains (Carolina Biological Supply, Burlington, NC) to grow the culture. We kept our cultures in jars provided from the suppliers.

1. Boil 1 liter of distilled water in a flask and cool down to room temperature.

2. Add one protozoan pellet into the boiled water and let it dissolve. Do not filter or remove residue.

3. Divide the liquid into smaller flasks (200 mL).

4. Add ten boiled wheat grains to each container.

5. Cover the top of the flasks with aluminum foil and autoclave them (see the next section).

77 4.5.4 Autoclaving

Autoclaving was performed using a electric pressure steam sterilizer (Model 25×, All

American, Manitowoc, WI) for about an hour. To autoclave the culturing media in the previous section,

1. Apply a high vacuum grease to the edges where the cover and the body meet.

Only a thin film is required. Excess amounts may cause leaking.

2. Place distilled water in the bottom of the sterilizer directly over the heating element.

3. Place the inner container rack into the bottom of the container with the lip or edge side downward. The purpose of the inner container rack is to provide an air space in the bottom of the container so that air may circulate freely.

4. Then place culturing media covered by aluminum foil into the sterilizer.

5. Pour in cold water and then turning on the unit so that the water is getting warmed prior to begin a sterilization procedure.

6. Place the sterilizer cover on the unit. Make sure that the index alignment arrow on the cover aligns with index line on the side of the bottom.

7. Open the control valve by placing the valve lever in an upright position. It is important that the steam should be permitted to escape from the unit for at least

5 mins. The main cause of sterilization failure is the trapping of air in the material being sterilized. It is imperative that all trapped air be exhausted.

8. Increase pressure inside the sterilizer by changing the position of the control valve to the horizontal position. The pressure will be indicated on the pressure gauge.

78 9. Adjust the heat knob to reach the operating pressure of about 20 psi and sterilize for 50 mins.

10. At the end of the sterilization period, turn off the switch and move the lever on the control valve to an upright position so that the steam is permitted to escape.

11. Remove the cover. When removing the cover, always tilt and angle the cover away from yourself or any other people in the area to prevent injury from the hot steam.

12. Cool down the culturing media to room temperature and inoculate.

4.5.5 Inoculation

After the autoclaving, we could inoculate the culturing medium with the bacteria we cultured on the agar plates. To inoculate,

1. Take one agar plate containing bacteria from the fridge.

2. Flame an inoculating loop and use it to take small about of bacteria from the agar plate.

3. Mix the bacteria into the flask containing the autoclaved culturing medium.

4. Store the flask at 32◦ for a day.

5. Then add about 50 mL of the old medium with Paramecia into the 200 mL of the culturing medium. The volume of 50 mL can be adjusted depending on the density of the old medium. However, 50 mL was enough.

6. Cover the flask and keep it at room temperature. Transfer the culturing medium to small containers for experiments.

79 4.5.6 Maintaining Cultures

A week after the inoculation, the number of Paramecia will grow exponentially. After

2-3 weeks, however, the number of the Paramecia will decrease because their food has depleted. At this point, we need to reculture them. To start new cultures for an experiment, just repeat the procedures above. The culture should be about 7 days old to be in its logarithmic phase of growth for experiments. However, just to keep

Paramecia in the lab, they could be easily multiplied by removing old wheat grains and adding a few new wheat grains.

80 Chapter 5

Two Ciliary Motors in a Single

Microorganism

81 5.1 Introduction

There are thousands of cilia covering Paramecia that beat to both propel and feed them [81]. Two spatially contiguous groups accomplish these tasks. The cilia in the oral-groove that leads toward the gullet control flows near the gullet [31]. These flows provide nourishment while sampling the chemistry of the nearby environment. The cilia that cover the outermost surfaces largely control the swimming. In addition to propelling Paramecia along their nominal helical trajectories, these body cilia can beat to turn, reverse direction, or accelerate in response to environmental cues [81, 82].

It is remarkable that these two structurally similar groups, which share a common intracellular space, can effectively perform such disparate functions.

Here, we have investigated whether these groups of cilia can be differentiated by their so-called motor characteristics. A motor characteristic is the relation between the force exerted by and the speed at which a motor operates [32, 83, 84, 85, 86, 87].

For biological motors, the characteristic is dictated by a network of mechanical and chemical elements. In the case of Paramecia, this network includes elements that enable the body cilia to respond to mechanical perturbations. For example, a jab to the posterior causes swimming acceleration or a decrease in buoyancy causes greater upward propulsion [1, 88]. It seems likely that the oral-groove and body cilia networks have evolved to produce motor characteristics optimized for their different functions.

Organisms that swim at low Reynolds number (Re) employ drag forces for propul-

Most of this chapter is from the article, Evidence for two extremes of ciliary motor response in a single swimming microorganism, published in Biophysical Journal, 106, 106 (2014).

82 sion while simultaneously overcoming drag forces that resist their motion [89]. Prop- erties of the propulsion system can be revealed by investigating how changes in vis- cosity of the surroundings affect their swimming characteristics [38, 90]. Therefore, an established method for probing the characteristics of small biological motors is to observe their operation in media with a range of viscosities. This approach works well for systems operating in the low Reynold’s number Stokes limit since the forces on the moving elements, which produce the load on the motor, are directly proportional to the viscosity [83, 85]. For microorganisms like Paramecia the propulsion results from the coordinated beating of thousands of cilia [81]. Both the beating and the coordination can depend on viscosity [91]. Theoretical models [17, 92, 93] have been developed that provide clear predictions for how effectively cilia propel in different viscosity fluids. Here we report investigations of how the swimming trajectories of a ciliated protist, Paramecium caudatum, varies with viscosity. We also compare these measurements to predictions of a simple model.

Paramecium swimming can be described by low Re hydrodynamics where Re is

ηuL/ρ with η solution viscosity, u a characteristic flow velocity, L a characteristic length scale of the flow, and ρ the fluid density. These approximately 250 µm long organisms swim at speeds 1 mm/s, which corresponds to Re ∼ 0.2. The propelling cilia, which are about 10 µm long, beat at a frequency of 30/s [88] corresponding to Re < .01. Under normal conditions, Paramecia swim along a left-handed helical trajectory with an average pitch of 700 µm and radius of 50 µm [75, 94]. The long axis of a Paramecium (approximating it as a prolate ellipsoid [95, 96]) is oriented tangent to the trajectory. It rotates around its long axis so that its oral-groove is

83 always facing the axis of the helix. This helical motion results from the beating of two distinct groups of cilia, the body cilia and the oral-groove as indicated earlier.

The body cilia have a power stroke that is directed mainly toward the posterior and slightly to the side of the anterior-posterior axis [38]. The oral-groove cilia beat to draw fluid toward the oral-groove. They beat harder than the body cilia [10].

Viscosity effects on ciliary systems operating in a variety of organisms [97, 98, 99,

100, 101] have been investigated. Information about Paramecia ciliary motor charac- teristics, however, has been somewhat mixed. Pigon, Tawada, and Yagi [79, 102, 103] reported that the swimming speed decreases approximately inversely with viscosity implying a constant force vs. speed propulsion motor characteristic. Sleigh [104] re- ported that the ciliary beat frequency also decreased with viscosity suggesting that the swimming speed decrease was directly related to slowing of the cilia. Subsequently, however, Machemer [38] showed that the ciliary beat frequency varied significantly over the body. Moreover, he presented measurements showing that the oral-groove cilia beat frequency decreased only about 20% for a factor of 10 change in viscosity.

He presumed that the body cilia experienced a similarly small frequency decrease implying that their ciliary motors operate at nearly constant speed.

We have combined high speed and magnification video investigations of ciliary motion with lower magnification studies of swimming trajectories and a phenomeno- logical model to acquire a clearer detailed picture of Paramecia’s ciliary motor char- acteristics. Uniquely, we have used the variation of the helical radius, helical pitch, and speed of the motion to infer how the speed and direction of the body cilia beat- ing changes with viscosity. Our results, for 1 < ηw < 7 where ηw is the viscosity

84 normalized to water, recapitulate findings from the previous investigations like the

slowing of Paramecia’s linear speed with increasing viscosity and the variation of cil- iary beat frequency over the body. In addition, we have found that the ciliary beat frequency variation with viscosity depends on the region of the surface. In particular, the beat frequency in the oral-groove barely changes while the beat frequency in the mid-body region decreases almost inversely with viscosity. We also found that the rotation rate of a swimming Paramecium decreases at nearly the same rate as the

linear speed indicating that the propelling torques have the same viscosity depen-

dence as the propulsion force. Data analysis with a phenomenological model implies

that these swimming speed changes largely result from the mid-body cilia decreasing

their beat speed without changing their beat angle. The model also shows that the

oral-groove cilia beating contributes to the rotational motion of the swimmers. The

results imply that the mid-body cilia have a constant force motor characteristic. This

characteristic agrees well with currently available numerical models of ciliary motion

[105, 106, 107]. The oral-groove cilia, on the other hand, exhibit a roughly constant

speed motor characteristic. We believe that the observation of two such distinct mo-

tor characteristics in a single cell is unique. We point out that this finding suggests

further Paramecia studies for insight into factors controlling ciliary motor response.

85 5.2 Materials and methods

5.2.1 Culturing Paramecium caudatum

Paramecium caudatum (Carolina Biological Supply, Burlington, NC) was cultured with its food, Enterobacter aerogenes (Carolina Biological Supply, Burlington, NC).

At the stationary phase of their growth, they were collected by using their gravi- taxis [2]. The collected Paramecia were suspended in test solution containing 1 mM

CaCl2, 1 mM KCL, 0.1 mM MgSO4, and 1.5 mM MOPS at PH ∼ 7.2 and various concentrations of Methyl Cellulose and magnetic impurities for at least 2 hours prior to putting them in experimental chambers.

5.2.2 Viscous solution

To obtain viscous solution, we added 400 cP Methyl Cellulose (Sigma Aldrich, St.

Louis, MO) to test solution as reported from previous experiments on Paramecia

[38, 79]. The density of the Methyl Cellulose solutions was more comparable to water than Ficoll solutions of the same viscosity. The swimming speeds in Ficoll solutions were nearly identical to those of the Methyl Cellulose as described in the previous chapter. The Methyl Cellulose solutions were made in two steps. First, a high- concentration Methyl Cellulose solution was made by mixing Methyl Cellulose with test solution. Second, the high concentration Methyl Cellulose solution was diluted with test solution to produce a solution with the reduced viscosity. The dilutions were done by inverting the mixing tubes about 10 times. Viscosities were measured at constant flow using glass capillary viscometers (Cannon, State College, PA). The

86 experimental solutions had Methyl Cellulose concentration of 0%, 0.2%, 0.33%, and

0.5% and ηw of 1.0, 2.3, 4.1, and 6.9, respectively. A cone and plate rheometer

(AR2000, TA Instruments, New Castle, DE) was used to determine the elastic modu-

lus of a concentrated solution with ηw = 7.9. It was more than 25 times smaller than the viscous term at 10 Hz. The density of the solution was 0.2% larger than pure test solution at 0.5% Methyl Cellulose concentration. The molecular weight of the Methyl

Cellulose was about 41 kDa corresponding to a radius of gyration of about 25 nm

[80]. The density of the cilia is ∼1/(300 nm)2. Thus, the radius of gyration is about

1/10th the spacing between cilia. At a concentration of 0.5% the inter molecular spacing is about 25 nm. Magnetic susceptibilities of the solutions for Magnetic force buoyancy variation (MFBV) were enhanced by doping with paramagnetic impurities,

Gd-DTPA (Sigma Aldrich, St. Louis, MO) [108, 109].

5.2.3 Swimming trajectory experimental setup

For trajectory measurements, populations of Paramecia swam in rectangular cham- bers of dimensions 2 mm×16 mm×16 mm (see Figure 5.1). The chambers were formed with a plexiglass frame sandwiched by two coverslips. The coverslips were sealed to the frame with VALAP (1:1:1, vaseline:lanolin:paraffin) which is nontoxic for microorganisms. A glass strip bisected the chambers to see both up and down swimmers at the same time. The Paramecia-containing solutions were carefully in- jected by a syringe through a hole in the chamber frame. The temperature of the swimming-chamber assembly was held to 23 ± 1◦C by using a water circulating bath.

87 Swimming Paramecia, which were allowed to acclimate to the experimental chamber

for at least 10 minutes, were recorded using a 90 degree sideview borescope (123008,

ITI, Westfield, MA) and a CCD camera (XCD-SX 90, SONY, Tokyo, Japan) at 7.5

frames per second. A green LED array (TBL 1×1, Metaphase Technologies, Ben- salem, PA) backlit the samples.

B(z)

5mm

Figure 5.1: Experimental chamber. The small dots indicate Paramecia. The solid line

in the middle of the chamber is a glass strip that separates the 2 mm×16 mm×16 mm

chamber into two chambers. The fact that there are two chambers is not important

for this work. The gray circle and black box regions inside the circle indicate field

of view of a camera and areas of interest, respectively, where swimming analysis was

done. Note that two holes were needed to inject Paramecia to avoid air bubbles. This

figure is adopted from Biophysical Journal, 106, 106 (2014).

88 5.2.4 Magnetic force buoyancy variation

Swimming trajectory parameters were determined by observing Paramecia swimming under simulated neutral buoyancy conditions to eliminate sedimentation and gravi- kinetic effects [1, 13]. Paramecia were subjected to neutral buoyancy conditions using the method of MFBV. In this method, the chambers were placed in the bore of a resistive magnet at the National High Magnetic Field Laboratory. This made the Paramecia swim in an intense and inhomogeneous magnetic field [4] in solutions doped with paramagnetic impurities, Gd-DTPA (Sigma Aldrich, St. Louse, MO)

[108, 109]. The magnetic force exerted on a Paramecium and the solution makes the apparent weight magnetic field dependent:

W  χ − χ B dB  = 1 − P S , (5.1) W1g ρP − ρS gµ0 dz

where W , ρP , ρS, χP , χS, z, g, and µ0 are the apprent weight, the densities and

magnetic susceptibilities of the Paramecia (P ) and solution(S), the vertical coordi-

nate, the acceleration of gravity, and the permeability of free space, respectively. The

swimming chambers were placed in a maximum magnetic force region of a 31 Tesla

maximum field resistive magnet at the National High Magnetic Field Laboratory

(Tallahassee, FL). Adjusting the current in the magnet alters B(dB/dz) to alter the

apparent weight. Notice that the apparent weight force can be inverted as well as

augmented or reduced. Data presented here were obtained for Paramecia swimming

with W = 0. To determine the correct megnetic field to levitate Paramecia, we im-

mobilized Paramecia using 0.5mM NiCl2. After they were immobilized, we measured

the magnetic field where W = 0 by increasing or decreasing magnetic field until

89 immobilized Paramecia did not sediment or rise.

More than 50 trajectories were analyzed for each η. The tracks of the swimming trajectories were obtained and analyzed using Image Pro Analyzer 7.0 (Media Cy- bernetics, Rockville, MD). We set the rectangular area of interest at least 10 body lengths away from the plexiglass frame and the cover glass strip as shown in Fig- ure 5.1. Paramecia that were very close to the top and the bottom cover glasses were filtered by focusing the borescope on the middle of the chamber. The depth of focus of the borescope was ∼ 1 mm. The individual 2D swimming tracks were

fit to x = R sin(2πz/λ + φ) where x, z, φ, R, and λ are the horizontal and ver-

tical coordinates, a phase constant, amplitude, and wavelength, respectively (see

Figure 4.5). Note that the z-axis points along the applied magnetic field, which

is the direction along which the trajectories tended to align [4]. Fits with reduced

2 Rreduced < 0.9 or of tracks shorter than one wavelength were discarded. Using the

parameters describing the 2D tracks, we obtained the angular frequency of the mo-

tion ω using ω = (2π/λ)(∆z/∆t) and the speed along the 3D helical trajectories, q v = ω R2 + (λ/2π)2 [78]. The components of ω were obtained using

2 2 2 ω = ωk + ω⊥ (5.2)

ω λ k = . (5.3) ω⊥ 2πR

5.2.5 Beating frequency measurement

For high speed video measurements of ciliary motion, Paramecia containing solutions

were placed in drops sandwiched between coverslips separated by parafilm (Sigma

90 Aldrich, St. Louis, MO) sheets of approximately 125µm thickness. This relatively constrained geometry caused their swimming trajectories to deviate from normal but seemed to exert little influence on their swimming speed. We inferred the beat frequencies by observing the periodic appearance of metachronal wave features in the form of crests or collective sweeps of groups of cilia at fixed positions along a

Paramecium’s surface (see videos V1-V4). Examples of crests are indicated in Figure

5.2(a). The period of time between crest appearances corresponds to the period of time between recovery strokes. The wavecrests appear sharper or more diffuse in the movies depending on the direction of the metachronal wave propagation or the location on the body. The samples were imaged using the phase contrast optics of an inverted light microscope (Nikon TE2000, Tokyo, Japan) and recorded at 500 frames per second using a fast camera (Fastcam PCI R2, Photron USA, San Diego, CA) and its associated software (Fastcam Viewer, Photron USA).

5.3 Results

5.3.1 High speed imaging of ciliary motion

The surface of a Paramecium was divided into four regions (anterior, mid-body, oral- groove, and posterior) for observations of ciliary motion Figure 5.2(a). We obtained the beat frequencies for Paramecia swimming at speeds comparable to their speeds far from walls. Consequently, we presumed that the beating frequencies obtained in the more constrained chambers were similar to those in the larger chambers. We mea-

91 (a) O A

P

B

(b)

O, Machemer

O

w

f A

P B

ηw

Figure 5.2: Viscosity dependence of ciliary beat frequencies. (a) Phase contrast

image of a Paramecium. The brackets define 4 observation regions: O–oral-groove,

A–anterior, B–mid-body, and P–posterior. Examples of 2 metachronal wave crests

can be seen within the dotted ellipse. The scale bar is 50 microns. (b) Ciliary

beat frequencies fw measured in the four regions (solid symbols) normalized to their

average value in standard test solution plotted versus ηw. Results obtained for the

oral-groove by Machemer (open circles) are shown for comparison. The lines are

guided to the eye and not fits. This figure is adopted from Biophysical Journal, 106,

106 (2014). 92 sured the beating frequencies by noting the time between appearances of collective

structures like the crests shown in Figure 5.2(a) at a fixed position on a Paramecium

(see videos V1-V4). The cilia forming these crests are executing their recovery stroke.

The crest motion along the surface is a manifestation of the anti-plectic metachronal

wave associated with the coordination of the ciliary beating [37]. As shown in Table

5.1, the frequency of ciliary beating varies over the surface at a fixed viscosity. It

is highest in the oral-groove and anterior regions and diminishes toward the poste-

rior. This anterior to posterior gradient was noted previously [38]. We also measured

the averaged distance between two crests in the body region in Table 5.2. The dis-

tances increased as viscosity increased. More detailed investigation will be needed to

find reasons for this change in the crest distance. Figure 5.2(b) shows that the beat

frequencies, fw, normalized to their water values respond differently to increasing vis-

cosity in the four regions. Here, and throughout, the w subscript denotes a quantity

normalized by its value for water. The body ciliary beat frequency changes the most

decreasing by a factor of 4 for the factor of 7 change in viscosity. In sharpest con-

trast, the oral-groove frequency changes little decreasing by only 20%. The anterior

and posterior frequencies show intermediate behavior with the posterior frequency

diminishing more.

These differences indicate that the ciliary motor regulation varies over the sur-

face, ranging between two extremes. The oral-groove cilia are regulated to beat at a

nearly constant frequency. The mid-body cilia, on the other hand, appear regulated

to beat with viscosity×frequency = constant. Because the drag forces in the Stokes

flow regime are proportional to the viscosity times the speed, this dependence sug-

93 Table 5.1: Averaged ciliary beat frequencies in the four different regions from five measurements. This table is adopted from Biophysical Journal, 106, 106 (2014).

ηw Oral-groove Anterior Body Posterior

1.0 35.5 ± 3.1 Hz 34.5 ± 3.4 Hz 31.4 ± 8.3 Hz 15.2 ± 2.3 Hz

2.3 34.3 ± 1.0 Hz 30.4 ± 4.5 Hz 15.7 ± 2.1 Hz 12.6 ± 1.6 Hz

4.1 31.0 ± 3.4 Hz 22.4 ± 4.3 Hz 11.3 ± 1.6 Hz 10.5 ± 2.0 Hz

6.9 29.0 ± 3.6 Hz 20.6 ± 4.8 Hz 7.8 ± 1.0 Hz 6.3 ± 1.3 Hz

Table 5.2: Averaged distances between crests in the mid-body region from three measurements.

ηw Distances

1.0 16.1 ± 1.1 µm

2.3 18.0 ± 2.0 µm

4.1 19.0 ± 3.6 µm

6.9 21.5 ± 0.9 µm

94 gests nearly constant force regulation. In the next section, we present an analysis of swimming trajectories that strongly supports this assertion.

5.3.2 Swimming trajectory investigations

To characterize the ciliary motor for propulsion, we performed a series of measure- ments of Paramecia swimming trajectories in different viscosities. This approach follows that of a number of experiments [32, 83, 84, 85, 86] on bacteria that have varied viscosity to obtain force-velocity curves of their flagella motors that in turn have been used to test models of the motors. The relatively straightforward separa- tion of the part of the organism responsible for propulsion—the flagellum—from the part of the organism that experiences drag alone—the body—simplifies the analysis

[84, 110]. Correspondingly, some experiments characterized bacterial flagella motors by measuring the rotation rates of the flagella of bacteria with their heads fixed to a surface [111]. Others used simultaneous measurements of the head rotation rate and swimming speed with a model of force and torque balance to calculate the motor characteristic [86, 112, 113]. For Paramecium caudatum, the propulsion results from the coordinated metachronal beating of cilia. The propulsion and drag both depend on the details of the beating of the individual cilia as well as the coordination among cilia and the body shape [81].

Machemer’s detailed characterization [38] of ciliary beating as a function of viscos- ity showed that the geometry and rate of the individual beating and the metachronal coordination change systematically. The specifics of how to translate Machemer’s

95 ciliary observations to the propulsive forces they generate, however, represent a sig-

nificant challenge in fluid dynamics modelling. Extensions of current numerical theo-

retical models that predict how effectively a small number of cilia and flagella propel

in fluids of varying viscosity are required [17, 92, 93, 99, 105, 106, 107]. By measuring

the trajectories, we are able to determine the net forces and torques exerted by the

cilia that are responsible for their motion. This approach allows us to get the motor

characteristics of the propulsion system as a whole.

A neutrally buoyant Paramecium normally swims along a helical trajectory with

its body aligned tangent to its path and its oral-groove facing the axis of the helix

(see Figure 5.3) [10, 16, 114]. This motion decomposes into a linear translation

characterized by a speed v and a precession about an axis parallel to the axis of the

helix [115]. The speed v is the velocity of a Paramecium relative to the fluid far

from the Paramecium. This velocity is directed along the anterior-posterior (A-P)

axis. It comes from the component of the beating of the body cilia along the A-P

axis. The precession angular velocity has components along two orthogonal body

axes, ωk and ω⊥. ωk is the angular velocity of rotation about the long axis. This

angular velocity comes from the axis component of the ciliary beating transverse to

the A-P axis. ω⊥ is the angular velocity of rotation about an axis perpendicular to the A-P axis. This angular velocity comes from the beating of the cilia in the oral-groove and the beating of the body cilia behind the oral-groove. These two groups of cilia beat differently leading to an asymmetry that produces the torque necessary for ω⊥. Experimentally, ωk and ω⊥ are calculated from measurements of the maximum speed and the wavelength and the amplitude of the sinusoidal form of

96 (b)

(a) A

P

Figure 5.3: (a) Sketch of a Paramecium indicating the translational velocity vector

~v and the rotational velocity vector ~ω that produce its motion along a left handed helical trajectory as shown in (b). A and P denote the anterior and posterior regions, respectively. The lightly shaded patch corresponds to the oral-groove. This figure is adopted from Biophysical Journal, 106, 106 (2014).

97 the 2D projection of the helix. The decomposition of the motion only works in the absence of sedimentation because the long axis of a sedimenting swimmer does not align tangent to the helix. We have employed the method of MFBV [1, 3] to make the Paramecia neutrally buoyant. This technique employs paramagnetic impurities and magnetic force rather than fluid density adjustment to alter the buoyancy.

The most apparent effect of increasing the viscosity is readily observed in the videos V5-V8. The Paramecia swim ever more slowly with increasing viscosity. We measured the average rate, which are displayed in Figure 5.4, from fits to individual

2D tracks of swimming. At each viscosity, more than 50 swimming trajectories were analyzed. The swimming speed distributions in Figure 5.4(a)-(d) shift to lower speeds and narrow with increasing viscosity η. Each fits well to a Gaussian, allowing us to characterize each speed distribution with a mean v and variance δv as shown in

Figure 5.4(e)-(f). These quantities decrease at similar rates with increasing η so that

δv/v varies only ± 25% for a factor of seven change in η. The average pitch of the trajectories does not change discernibly (Figure 5.4(g)), while the radius increases

(Figure 5.4(h)) with increasing viscosity.

The average values of ωk and ω⊥ decrease monotonically with viscosity as shown in Figure 5.5. This result together with the result above that the standard deviations of the speed distributions decrease monotonically at roughly the same rate as the speeds gives us confidence that the changes in medium viscosity influence swimmers throughout the speed distribution similarly. It is important to establish whether this is a purely viscous response as Methyl Cellulose solutions are known to exhibit visco- elastic effects at high concentrations. To check, we measured the speed as a function

98 (a) (e)

(b) (f)

(c) (g)

(d) (h)

Figure 5.4: (a)-(d) The swimming speed distributions for different η. The bin size is

the SD of the swimming distributions. (e)-(f) The mean swimming speeds and δv/v as a function of ηw. (g)-(h) Swimming trajectory parameters, radius and pitch. Each bar represents the SD. 99 Figure 5.5: The averages of ωk and ω⊥ derived from analysis of more than 50 tra- jectories at each viscosity. The bars give the standard deviation of the populations.

This figure is adopted from Biophysical Journal, 106, 106 (2014). of viscosity of Paramecia in Ficoll solutions which are known to be Newtonian. The nearly identical v(η) characteristics in Figure 5.6 imply that non-Newtonian effects exert a negligible influence on the swimming. Furthermore, the Re is estimated to be

< 0.2 for the lowest viscosity investigated here [37]. This implies that inertial effects on the flows are small.

In the Stokes regime and under conditions of neutral buoyancy, the propulsion forces and torques are proportional to these speeds and the viscosity. They are decoupled provided the body is axi–symmetric. For example, we decompose the

fluid forces into a forward propulsive force P~ and a drag force D~ which sum to zero,

~ ~ P + D = 0. At low Re, D ∼ ηv. We plot the normalized propulsion force pw = ηwvw and torques, τk,w = ηwωk,w and τ⊥,w = ηwω⊥,w, as a function of ηw in Figure 5.7.

100 Figure 5.6: Comparison of Paramecia swimming speeds as a function of viscosity in Methyl Cellulose (square) solution and Ficoll solution (circle). Paramecia show the nearly identical swimming speeds in both solutions. The bars give the standard deviation of the populations. The lines are guides to the eye. This figure is adopted from Biophysical Journal, 106, 106 (2014).

All three exhibit a weak dependence on viscosity. For example, τ⊥,w which changes the most only increases by about 60% for the factor of 7 change in viscosity. The propulsive force varies less than ± 30%, which implies that v ∼ η−1 as found earlier

[102, 79]. In other words, the ciliary motors driving translation and rotation exhibit

nearly constant force (p, τk, τ⊥) vs. speed (v, ωk, ω⊥) characteristics.

The similarity of the propulsive force and torques motor characteristics provides

insight into how the forces exerted by the cilia vary over the body. Each motor

101

2 (a) w

v

w 1 η 0

2 (b)

║,w

ω 1 w

η 0

2 (c)

w ,

┴ ω

1

w η

0 2 4 6 8 ηw

Figure 5.7: Viscosity dependence of propulsive forces and torques. (a)-(c) Products of the viscosity with each of the rates shown in Figure 5.4(e) and Figure 5.5. The bars give the standard deviation of the populations. This figure is adopted from

Biophysical Journal, 106, 106 (2014).

102 characteristic depends differently on the spatial distribution of forces as well as the

variations in force magnitudes and directions. The propulsion force depends on the

vector sum of the forces exerted by the different regions of the surface but does not

depend on the spatial distribution of the forces. The torques, on the other hand,

depend on the spatial force distribution because it determines the moment arms

through which the torques act and depend on different components of the force. For

example, τ⊥ is more strongly influenced by forces close to the anterior and posterior

regions than the cilia in the mid-body region while the opposite holds true for τk.

Moreover, forces directed from A to P exert the primary influence on τ⊥ while forces in the azimuthal direction exert the primary influence on τk. Thus, a simple conclusion

from the data is that the magnitudes and directions of the forces exerted by the

different regions do not change with viscosity.

5.3.3 Phenomenological model of ciliary propulsion

We use a phenomenological model to relate the force exerted by the ciliary beating

of an average region on the body and in the oral-groove to the trajectory parameters.

One of the unique features of the current model is that it predicts correlations between

trajectory parameters that provide an internal consistency check. The presumption

that an average region provides a meaningful description is supported by the discus-

sion above as well as the observations of the highly coordinated metachronal ciliary

beating waves that travel coherently along the body [38]. It has been invoked in pre-

vious qualitative descriptions of the relation between ciliary activity and propulsion

103 [10] and other quantitative analyses of gravity dependent swimming of Paramecia

[16, 114]. Following Machemer and Baba, we characterize the time-averaged force exerted by an average region using two parameters: a speed uB, which could reflect the average speed of the cilia relative to the cell body and an oblique angle θ mea- sured relative to a vector pointing from P to A along the local longitude. This angle could reflect the direction of the trajectory of the body cilia (blue bars in Figure 5.8) tips during the power stroke. We also include the influence of the oral-groove cilia

(green bars in Figure 5.8) on the rotation of the swimmers as qualitatively described by Jennings [10]. The oral-groove cilia are presumed to beat perpendicular to the propulsion with a distinct speed uO. This beating contributes to the torque τ⊥ that gives rise to ω⊥, but not to the propulsion.

Thus, we write the propulsive force and the two torques, which are proportional to η, as

p = −AηuB cos θ (5.4)

τk = −NkηuB sin θ (5.5)

τ⊥ = −N⊥ηuB cos θ + MηuO. (5.6)

The constants A, Nk, N⊥, and M increase with the number, length and density of the cilia and on the size and shape of the body. The N’s and M are also proportional to the effective moment arm through which the forces act. N⊥ originates from the asymmetry in the body cilia distribution due to the existence of the oral-groove. Note that A has units of length, and M and the N’s have units of length-squared. All are taken to be positive.

104 1 3

2 2 3 1

Figure 5.8: Ciliary beaing of a Paramecium. The body cilia (blue) of Paramecium beat with an oblique angle while the oral groove cilia (green) of the Paramecium beat perpendicular to the direction of its propulsion. The numbers indicate the positions of the cilia as a function of time.

105 To relate the force and torques to the kinematics, we assume low Reynold’s number dynamics and that the body has the symmetric shape of a prolate ellipsoid so that there is no coupling between the linear and rotational velocities in the resistance matrix [17]. The propulsive force p balances the drag force, leading to speed v = p/(ηCL), where CL is the body’s longitudinal drag coefficient. The torques lead to rotations ωk = τk/(ηγk) and ω⊥ = τ⊥/(ηγ⊥), where γk and γ⊥ are the rotational drag coefficients. Thus, A v = − uB cos θ (5.7) CL

Nk ωk = − uB sin θ (5.8) γk

N⊥ M ω⊥ = − uB cos θ + uO. (5.9) γ⊥ γ⊥

We can isolate the θ-dependence with the ratio

ω k = G tan θ, (5.10) v 1

where G1 = (Nk/γk)(CL/A) depends on the number of body cilia and the shape of a

Paramecium. This model also predicts a correlation between v and ω⊥, according to

Equation (5.7) and (5.9), in the form

ω⊥ = G2v + G3uO, (5.11)

where G2 = (N⊥/γ⊥)(CL/A) and G3 = M/γ⊥ are constants depending on the number of cilia and the shape of a Paramecium. They are independent of η.

With the above relations in hand, we turn to the swimming data. The decrease in v with η implies that uB cos θ decreases with η presuming A and CL are independent of viscosity. To determine whether this decrease results from changes in either uB

106

)

3 (a) (b)

10 10 3

2 (rad/s)

5 ┴ (rad/m

ω 1

/v /v ║

ω 0 2 4 6 8 0 5 10 -4 ηw v (m/s 10 )

Figure 5.9: Insights from the phenomenological model. (a) Plot of ωk/v vs. ηw to compare to Equation (5.10). This ratio is proportional to tan θ, where θ gives the direction of the force exerted by the average patch of cilia. (b) ω⊥ vs. v to compare to Equation (5.11) at different viscosities. The intercept gives the oral-groove cilia contribution to the torque producing ω⊥. The line is a fit that tests the expected linear dependence of ω⊥ on v. The bars give the standard deviation of the populations. This

figure is adopted from Biophysical Journal, 106, 106 (2014).

107 or θ or both, we plotted ωk/v as a function of ηw in Figure 5.9(a) to compare to

Equation (5.10). The plots show ωk/v, which corresponds to tan θ decreases by about

30%. This behavior qualitatively agrees with Machemer’s observation of a clockwise rotation of the power stroke direction with increasing viscosity. Using θ = π/6 at

ηw = 1, corresponding to the power stroke direction measured by Machemer, we estimate θ at ηw = 6.9 to be, 22 degrees. This corresponds to only a 7% increase in cos θ.

Consequently, the large decrease in v with η corresponds to a similar decrease in uB. The dependence of ω⊥ on v shown in Figure 5.9(b) supports the presumption that the constants depending on the spatial distribution of beating cilia, like A, are inde- pendent of viscosity. ω⊥ grows nearly linearly with v implying that G2 and G3uO (see

Equation (5.11)) depend very little on η. In addition, the small intercept indicates that the oral-groove beating contributes only about 10% to the rotation at normal viscosity and about 30% at ηw = 6.9. This contribution is small enough that the ap- proximately 15% decrease in oral-groove beat frequency is consistent with uO being constant in this viscosity range. To summarize, the analysis with this phenomeno- logical model indicates that the decrease in v, ωk and ω⊥ with η results primarily from a decrease in uB. And, the spatial distribution of beating cilia embodied in the constants, A, N’s and M does not depend on viscosity. We suggest that this model may be helpful for interpreting swimming trajectory changes induced by other parameters like temperature [79] and gravity [1, 13, 116, 117].

108 (a) (b)

O O

4 4

B B Force Force 2 2

Pw Pw

0 2 4 6 8 0 0.5 1 ηw Speed

Figure 5.10: Comparisons of motor characteristics. Propulsive force from these exper- iments pw and force predictions based on the measured beat frequencies, mid-body,

B and oral-groove, O, shown in Figure 5.2(b) and numerical model calculations. The forces are the product of the viscosity and the rate (i.e. speed or frequency). (a)

Forces as a function of viscosity. (b) Forces as a function of rates. All forces and rates are normalized to their values in water. The lines are guided to the eye and not

fits. This figure is adopted from Biophysical Journal, 106, 106 (2014).

5.4 Discussion

The phenomenological model analysis implies that the mid-body cilia produce a con- stant propulsion force by reducing their beating speed with viscosity. This result agrees well with the direct measurements of the mid-body cilia beat frequency, which show it to decrease nearly inversely proportional to viscosity (Figure 5.2(b)). It con- trasts sharply with the oral-groove beat frequency, which barely changes.

Models of ciliary beating [105, 106, 107] produce motor characteristics like the

109 mid-body cilia. These models include the elastic properties of the cilia, the internal force generation by molecular motors and the interaction of the cilia with their sur- rounding fluid. They predict the shape of the stroke, the coordination among cilia, and the rate of beating. Nearly uniformly, they show that the frequency of beating decreases while the geometry of the beating varies little with viscosity. This behav- ior implies that the effective propulsive force ∝ ηf. As shown in Figure 5.10(a), the model predictions more closely follow the viscosity dependence of the mid-body ciliary beat frequency and the propulsion force than the oral-groove cilia. Figure

5.10(b) translates this comparison into the motor characteristics. The force versus speed motor characteristics are horizontal for the ciliary swimming motors and the models but vertical for the oral-groove ciliary motor.

Whether simple modifications of the models will enable them to capture the oral- groove constant speed characteristics requires further work. Experiments on Parame- cia that compare the responses of the mid-body and oral-groove cilia to chemical and mechanical perturbations have the potential to give insight into the factors that differ- entiate their motors and chemical attractant pathways [118]. The open architecture of cortical sheet preparations like those used in recent studies of dynein and cyclic

AMP interactions involved in ciliary beating [119] may be particularly attractive for such investigations.

110 Chapter 6

Trapping Microorganisms under

Varying Buoyancy

111 6.1 Introduction

Swimming organisms interact with surfaces as they negotiate their environs [20, 32,

33, 94, 120]. Higher organisms, such as fish, use whiskers, eyes, and other means to detect obstacles that they actively avoid by adjusting their swimming. Lower or- ganisms, such as bacteria, appear to navigate around obstacles much more passively

[21, 121, 122, 123, 124]. Studies of the tendency of bacteria to accumulate near sur- faces, for example, indicate that the hydrodynamic and contact interactions between swimmers and surfaces are sufficient to drive accumulation [20, 124]. That is, the interactions closely related with collective bacterial behavior such as biofilm forma- tion [125] and swarming [126] do not appear to lead to a change in the propulsion that could be considered an active response. Here we present investigations of surface interactions for another swimming unicellular organism, Paramecium, which is known to actively respond to mechanical perturbations [68]. When prodded in the posterior they accelerate away and when prodded in the anterior they perform an avoiding reaction, backing up to swivel off in a new swimming direction [10, 12]. We address whether their collisions with nearly planar surfaces evoke similar active changes in propulsion.

Paramecia are among the many microorganisms that are small enough that their motion occurs at low Re but large enough that their apparent weight, ~w, influences their swimming [14, 57, 127, 128, 129, 130, 131, 132, 133, 134, 135]. This combi-

Most of this chapter is from the article, Trapping of swimming microorganisms at lower surface by increasing buoyancy, published in Physical Review Letters, 113, 218101 (2014).

112 nation indicates that surface collision forces might be large enough to provoke an

active response. We demonstrate this by first estimating the scale of the collision

forces. A freely swimming Paramecium experiences forces that decompose typically

into propulsion, P , and drag, D. These sum to zero at low Re [89, 17]. At the moment

of a head on collision, the drag force drops out and the normal force, N, exerted by

the surface grows to balance P . Consequently, the propulsive force sets the collision

force scale. In the presence of gravity, Paramecia must swim hard enough that P > w

in order for them to stay suspended where w indicates apparent weight. Altogether

the collision force is expected to exceed the apparent weight force.

The influence of the apparent weight force on Paramecia swimming has been

categorized into two phenomena. They show negative gravitaxis, which describes

their tendency to orient their swimming direction antiparallel with the gravity vector

[2, 55, 136]. This behavior causes populations of Paramecia to collect at the top

surfaces of containers. They also exhibit negative gravikinesis, which describes their

tendency to exert a stronger propulsive force when swimming against their apparent

weight [13, 39, 42]. Both of these behaviors disappear when Paramecia are neutrally

buoyant [1, 136]. The gravitactic response appears to result from a passive response

to a mechanical torque arising from the asymmetry of the Paramecium body [2].

It drives them to align passively while swimming. The gravikinetic response, on

the other hand, appears to be active. How Paramecia transduce the very small

apparent weight force (∼ 100 pN) remains unclear, although it has been the subject of many investigations [1, 12, 13, 55]. The main proposal attributes the response to the mechanical activation of ion channels in the cell membrane that produce changes

113 in membrane potential, which controls ciliary beating [44]. Thus, Paramecia are suspected of actively responding to a force that is smaller than forces involved in collisions.

Here, we introduce a new model presenting a novel method for driving the surface accumulation of microorganisms. We show that applying forces and torques compara- ble to the propulsive force influence Paramecium’s interactions with upper and lower surfaces. These studies were motivated by unexpected observations of two species of

Paramecia becoming trapped at lower surfaces when they were buoyant. We used a magnet based technique, Magnetic Force Buoyancy Variation (MFBV), for adjusting

~w in situ [1]. The magnetic field also exerted a torque that aligned Paramecia to swim along the vertical magnetic field lines. The torque arises from an intrinsic magnetic susceptibility anisotropy, ∆χ = (χk − χ⊥), there χk and χ⊥ are the susceptibilities parallel to and perpendicular to a Paramecium long axis. We observed that these imposed forces and torques control the surface trapping by enhancing or inhibiting the turning of Paramecia that allows them to escape from surfaces. We present a passive mechanical model that qualitatively and quantitatively accounts for the range of observed phenomena. In particular, the model explains the counterintuitive result that forces directed away from a surface drive trapping at the surface. The two species of Paramecia behave slightly differently from one another in a manner that can be qualitatively captured by adjusting ∆χ, ~w, and P .

114 6.2 Materials and methods

6.2.1 Paramecium tetraurelia and caudatum

The data presented were obtained for single populations of P. tetraurelia and P. cau- datum (Carolina Biological Supply, Burlington, NC). The paramagnetic susceptibility of the swimming medium was enhanced by the addition of 3.2 mM (P. tetraurelia) and

4.0 mM (P. caudatum) Gd-DTPA (Sigma Aldrich, St. Louis, MO) and the swimming chambers were maintained at a constant temperature to ±0.1 degrees using a water

circulating bath. The Paramecia were cultured with Enterobacter aerogenes. At their

stationary phase of growth, they were collected using their gravi-taxis, suspended in

their test solutions containing 1 mM CaCl2, 1 mM KCl, 0.1 mM MgSO4, and 1.5

mM MOPS at PH ∼ 7.2 for at least 2 hours and then placed in their observation

chambers. They were allowed to acclimate for 10 minutes prior to observation.

For P. tetraurelia, chambers (2 mm×16 mm×16 mm) were formed with a plex-

iglass frame sandwiched by two coverslips. The coverslips were sealed to the frame

with VALAP (1:1:1, vaseline:lanolin:paraffin) which is nontoxic for microorganisms.

The chambers only allowed imaging of a single surface. Paramecia were recorded

with 90 degree sideview borescope (123008, ITI, Westfield, MA) and a CCD cam-

era (XCD-SX 90, SONY, Tokyo, Japan) at 7.5 frame rate. A green LED backlight

(TBL 1 × 1, Metaphase Technologies, Bensalem, PA) was used as a light source. The

swimming images were analyzed using Image Pro Analyzer 7.0 (Media Cybernetics,

Rockville, MD).

For P. caudatum, borosilicate rectangular glass tubes (2 mm×4 mm×10 mm)

115 which were sealed with arcylic caps were used and enabled viewing both upper and lower surfaces. The swimming of Paramecia was observed using 90 degree side view borescope (123006, ITI, Westfield, MA) and a CCD camera (XC-33, SONY, Tokyo,

Japan). The images were recorded with a videocassette recorder. LED green light

(Luxeon Vstar LED, Lumileds Lighting, San Jose, CA) was used to illuminate the borescope. For digitizing and analyzing images, a frame grabber (EPIX, Buffalo

Grove, IL) and an analyzing program (XCAP, EPIX, Buffalo Grove, IL) were used, respectively.

6.2.2 Magnetic force buoyancy variation

We tuned the apparent weight force magnitude and direction using the magnetic force buoyancy variation technique. In this method, the Paramecia swim in chambers that are placed in the bore of a high field resistive magnet at the National High Magnetic

Field Laboratory. Adjusting the magnet current alters the magnetic force acting on the Paramecia and the solution to alter the apparent weight of the Paramecia. The apparent weight force is given by:

w (χ − χ ) B(z) dB(z) = 1 − P s (6.1) w1g (ρP − ρs) gµ0 dz

where ρP , ρs, χP , and χs are the densities and magnetic susceptibilities of the Parame- cia (P) and solution(s), respectively. g and z are the earth’s gravitational force per mass and the coordinate along the axis of the bore, respectively. Notice that the apparent weight force can be inverted as well as augmented or reduced depending on three distinct positions shown schematically in Figure 6.1. Negative apparent weights

116 Z B

Levitation position

Center

w = w1g Sedimentation position

Figure 6.1: Schematic of the magnet. For Paramecia placed above the center of the magnet, the levitation position, magnetic forces are upward (negative) while swimmers below the center of the magnet, the sedimentation position, are exposed to downward (positive) magnetic forces. At the center of the magnet where dB/dz is zero, there is no magnetic force. This figure is adopted from Physical Review Letters,

113, 218101 (2014).

are set by placing the Paramecia above the center of the magnet, where the product

B(dB/dz) is most negative. We deem this the levitation position. The large positive apparent weight conditions are set by placing the Paramecia at the equivalent posi- tion below the center of the magnet. We deem this the sedimentation position. In the third position, the center, the magnetic field is maximum but the magnetic force is zero because the field is homogeneous (dB/dz = 0).

117 Equation 6.1 can be rewritten as (see chapter 3 for the derivation):

w  B 2 = 1 ± (6.2) w1g Bneut

where w1g and Bneut are the apparent weight at 1g and the magnetic field at neutral

buoyancy, respectively. The − and + signs are for the levitation and sedimentation

positions, respectively. Bneut was measured as the field at which immobilized Parame-

cia ceased to sediment or rise in solution. In our experiment, Bneut was 10 T for P. tetraurelia with 3.2 mM Gd-DTPA concentration and 7.8 T for P. caudatum 4.0 mM at the center of the magnet. Note that a negative apparent weight force is directed upward.

6.2.3 Magnetic torques on swimmers

The magnetic field also exerts a torque,

1 2 τB = − ∆χB sin 2θ (6.3) 2µ0

where ∆χ = (χk − χ⊥)V , V is the volumn, and χk and χ⊥ are the susceptibilities

parallel to and perpendicular to the long axis of a Paramecium, respectively. The sign

of the anisotropy ∆χ > 0 implies that the magnetic field aligns Paramecia to swim

along the magnetic field lines. The torque is strong enough in these experiments to

ensure that the vast majority of Paramecia essentially swim either up or down [4].

118 ioildsrbto.Ti gr saotdfrom (2014). adopted 218101 is figure This followed measurements distribution. the binomial presuming a estimated uncertainty the give points and the probability on the both where denotes line dotted The and surface ( bottom ( force weight of Trapping (d) steps. top. in the to bottom the from of is STP direction (c) force the that indicates force negative The surface. the of along STP swam they (b) as horizontal from away canted were bodies their of axes 3 showing Frame of (STP) probability trapping surface The (a) 6.2: Figure (d) (a) O Bottom

ufc.()SPof STP (f) surface. ) w / / w w .tetraurelia P. w / / w w Top .tetraurelia P. w/w w/w

1g 1g 1 = = g .tetraurelia P. 1 - - 2 safnto ftm.Tesldln ak h hne in changes the marks line solid The time. of function a as g 6

= − and )i ple.()SPof STP (e) applied. is 6) ertebto ufc as surface bottom the near (e) .caudatum P. w/w

STP (%) 100 STP (%) 100

20 40 60 80 20 40 60 80 0 0 ttebto ufc when surface bottom the at

1 1 .caudatum P.

1 - (b) g 6 6 sldln)a ucino time. of function a as line) (solid -

3 3

0 0 w / / w w w / / w w

0 0 119 ttebto ufc.Uwr apparent Upward surface. bottom the at

3 3 1g 1g -

1 1 6 6

9 ttetpsraemnstebottom the minus surface top the at

- 2

(f) .caudatum P.

-

Top – Bot (%) STP (%) 100 100 100

hsclRve Letters Review Physical - w/w 20 40 60 80 50 50 0 0

.tetraurelia P.

0 10 0 10 0 20 80 20 40 60 0 w/w (c) 1 g

w/w Time (min) hne rm1to 1 from changes Time (min) 1 g r eo h bars The zero. are ttetp( top the at 1 g = w ttebottom. the at

−

20 a changed was .Telong The 2.

- - 1 0 - - 0 5 10 1 2

10 5

N

w/w w / w1g w / w1g , and ) 113 − 1 2. g , . 6.3 Results

6.3.1 Accumulation of Paramecium tetraurelia

Varying the apparent weight of swimming P. tetraurelia (120-150 µm) using MFBV

led to changes in their surface trapping. For simplicity, we will study the accumulation

of Paramecia as a function of normalized apparent weight, w(B)/w1g. P. tetraurelia

tended to collect near and swim adjacent to surfaces when the net apparent weight

force ~w exerted on them was directed away from rather than toward the surface as

shown in Figure 6.2. Individual swimmers in video 9 swam into a surface at a speed

constant to within a few %, rotated toward parallel, and then swam canted at a

constant angle along the surface at a constant speed with their anterior in contact.

The video 9 also shows that some of the trapped Paramecia swam in circles. The

circles appear predominantly left handed, which is the same handedness of the helix

executed by a free swimmer. Occasionally P. tetraurelia assumed a static vertical

orientation. By contrast, P. tetraurelia that swam toward the surface with ~w only

briefly made contact as they reversed their swimming direction (see Video 10).

We found that trapped Paramecia escaped primarily after collisions with other

Paramecia, the front and back coverslips of the swimming chamber, or dust particles.

Random variations in the orientation of the Paramecia due to rotational diffusion

could not be discerned. The regular wobble associated with the helical trajectory

probably masked them. A rough scaling argument shows that rotational diffusion is

expected to be more than 100,000 times smaller for the much larger Paramecia than bacteria. The constancy of the approaching speed, swimming speed along surfaces,

120 and canted angle (a few percent) also indicates that hydrodynamic interactions do not

influence the initial collision and subsequent swimming speed. Moreover, it indicates

that the swimming motor does not exhibit stochastic variations over time scales of a

second or more.

The probability for P. tetraurelia to accumulate at the bottom surface varied systematically with w(B)/w1g as shown in Figure 6.2(b). To obtain the Surface

Trapping Probability (STP), we observed Paramecia that swam into contact with a surface and determined the fraction that subsequently resided on the surface for more than the time it would take to swim a few body lengths. We chose 3 seconds.

We observed 20 total events for each data point and measured the STP every 30 seconds. The STP increases from near zero at w(B)/w1g = 1 to about 80% at w(B)/w1g = −2. The STP closely tracks changes in w(B)/w1g as shown by Figure

6.2(c). The correlation between ~w and the probability is strong evidence that the changes in ~w caused the changes in the STP.

6.3.2 Accumulation of Paramecium caudatum

We also observed accumulation of a second larger species P. caudatum that is ap-

proximately 180-250 µm long or twice as long as P. tetraurelia. Comparison of its

behavior with P. tetraurelia gives insight into various parameters driving accumula-

tion. P. caudatum also showed ~w dependent surface trapping. Like P. tetraurelia, they approached the surface, made an incomplete turn toward parallel, and swam along it at a constant speed and canting angle. Their escape appeared to be dictated

121 by collisions. Trapped P. caudatum were usually oriented perpendicular to a surface

(see Figure 6.2(d) and Video 11) while a few were observed, usually at lower apparent weights, swimming canted relative to the surface as in the case of P. tetraurelia (see

Video 12). The P. caudatum oriented perpendicular to the surface often swayed with a period of about 1 second with an amplitude of about 10 degrees.

Figure 6.2(e) shows the STP for P. caudatum on both the top and bottom surfaces as w/w1g varies from −6 to 10. Very little trapping occurred for −2 ≤ w/w1g ≤ 2. At

the most negative apparent weights, the bottom STP approached 100% and the top

STP remained low. At the most positive apparent weights these behaviors exchanged

(Video 11). The difference between the top and bottom STPs closely tracks the

changes in w/w1g (Figure 6.2(f)). The videos indicate that when the STP on either

surface is low the P. caudatum easily turned at the surface to swim away (Video 12).

Thus, like P. tetraurelia, trapping occurred primarily for P. caudatum swimming

against ~w on their way to a surface.

In addition, P. caudatum became trapped by intense homogeneous magnetic fields

at fixed apparent weight. Figure 6.3 shows the STP at the top and bottom surfaces

for magnetic fields up to 29 Tesla with the chamber at the center position of the

magnet. At the center, w(B)/w1g = 1 at any magnetic field. The STPs at both

surfaces grow with magnetic field. The top SAP is slightly higher at each field. The

trapped P. caudatum were primarily oriented vertically on both surfaces.

122 100

80

60 (%)

40 STP 20

0 0 10 20 30 Magnetic Field (T)

Figure 6.3: STP at the top (tringle) and bottom (downward triangle) surface as a function of homogeneous magnetic field at constant apparent weight of w/w1g = 1.

The bars on the points give the uncertainty estimated presuming the measurements followed a binomial distribution. This figure is adopted from Physical Review Letters,

113, 218101 (2014).

123 6.3.3 Summary of observations

The observations suggest that Paramecia become trapped at a surface by failing to execute a full turn. In some cases, they did not turn at all and maintained an orien- tation normal to the surface. In other cases, they turned partially and subsequently swam along the surface in a canted orientation. Surprisingly, this trapping, as mea- sured by the STP, grew most when the apparent weight force was directed away from the surface and increased. Orienting the apparent weight force toward a surface, on the other hand, facilitated turning.

The videos show that once a Paramecium became parallel it usually escaped. In most cases the wobble in their normal helical trajectory appeared to tilt them off the surface sufficiently for the magnetic torque to turn them to swim away from the surface [4]. In some cases they bumped into another swimmer, which drove them off the surface. A small fraction, when a strong apparent weight force was directed at the surface, turned parallel with the surface and remained. This behavior could be a sign that the apparent weight force exceeded their propulsion force to pin them to the surface.

6.3.4 Force and torque balance model

The strong w and B dependence of the STP led us to consider a model of mechanical torques that contribute to turning at surfaces. The model in Figure 6.4(a) presumes a passive swimmer with a prolate ellipsoid shaped body with major axis, L, that swims

124 (a) w θ B N D L

P Coverslip surface (b)

I(a) I(b) II III IV

π/2

(rad)

max max

θ π/4

0

2 0 -2 -4 -6 -8 -10 -12 -14 w / w1g

Figure 6.4: (a) A sketch of a canted Paramecium in contact with and swimming along the bottom surface. In this figure, the Paramecium swims against ~w. (b) Phase diagram of P. caudatum behavior at the bottom surface. The upper four frames give schematics of the swimming behavior in the magnetic field regions I–IV. The gray region specifies the range of θmax expected for a typical population of swimmers. This

figure is adopted from Physical Review Letters, 113, 218101 (2014).

125 in a plane and approaches the surface at normal incidence. The apparent weight force ~w is directed away from the surface in Figure 6.4(a). Paramecia beat their cilia to exert a propulsive force P~ oriented along their long axis. It ignores the helical motion, which causes it to wobble as a first approximation. The body interacts with the surface through a normal force N~ . ~w, D~ , and P~ , which is oriented along the long axis, act at the center. P is presumed constant since Paramecia approach and swim along the surfaces at constant speeds. Correspondingly, the model neglects stochas- tic variations in the propulsive mechanism, which can influence smaller organisms

[21]. Also, it neglects long-range hydrodynamic interaction between Paramecia as phenomena like dancing Volvox [137] were not apparent. This indicated to us that surface mediated hydrodynamic interactions between Paramecia did not influence the trapping.

At low Re number, P~ + N~ + D~ + ~w = 0. The perpendicular components satisfy

N = P cos θ − ~w ·nˆ where θ is the angle between the long axis and the surface normal, nˆ. The torques turning the body about its center also sum to zero, ~τN + ~τB + ~τD = 0, where L L τ = N sin θ = (P cos θ − ~w · nˆ) sin θ (6.4) N 2 2 is the normal force torque, τB is the magnetic torque shown in Equation (6.3), and τD is the rotational drag torque. µ0 is the permeability of free space. τD is proportional to θ˙. Since Paramecia stop turning to become trapped, we consider the condition,

θ˙ = 0, to compare to the observations. Since we will consider only cases in which

˙ Paramecia have stopped turning, we ignore τR which is proportional to θ. Because

126 ∆χ > 0, τB aligns Paramecia to swim along the magnetic field. ∆χ is large enough to orient the vast majority of Paramecia essentially to swim vertically in magnetic

fields as low as 3 T [4].

It is noteworthy that this model leaves out a hydrodynamic force dipole image torque that appears to align bacteria parallel to surfaces to trap them [21] and the gravitactic torque [2]. If the dipole torque played a role, then Paramecia that turned to parallel would stay trapped, which is contrary to the observations. A plausible explanation for this different behavior is that the force dipole, P l, is relatively smaller for Paramecia than for bacteria. Here l is the distance between the center of drag and propulsion. For bacteria like Escherichia coli (E. coli), l ≈ L since the center of drag is near the body center and the center of propulsion is near the center of the

flagella bundle. Consequently, the torque due to propulsion, which scales as PL is comparable to the dipole torque for E. coli. For Paramecia, however, P l << P L, as the propulsion involves the beating of cilia that uniformly cover the body, which makes the centers of drag and propulsion nealy coincide. Thus, it is reasonable to omit this hydrodynamic dipole image torque. The gravitactic torque, which can be written as wc where c is a body asymmetry length, is estimated to be less than 5% of L which makes its contribution to turning at a surface negligible.

We define the torque that drives turning as τdrive = τN + τB. When τdrive > 0, the long axis of a Paramecium rotates toward parallel to the surface. τD simply opposes this motion. Swimmers rotate up to a maximum angle θmax, where τdrive(θmax) = 0:

 ~w · nˆ  θ = cos−1 . (6.5) max P − 2 ∆χB2 µ0L

127 They swim along the surface canted at θmax.

This model predicts four distinct behaviors as a function of w/w1g. They are

depicted for a lower surface in Figure 6.4(b) with a plot of θmax appropriate for

P. caudatum (see below). At the most positive w/w1g, regions I(a) and I(b), τdrive rotates the swimmers to θmax & π/2. Because their helical motion causes them to

◦ wobble by an angle θw ≈ 15 , we presume that swimmers rotated by more than

π/2 − θw are able to wobble their way through a complete turn. In region II, where

π/2 − θw > θmax > 0, τdrive rotates them to swim at a canted angle θmax. In region

III, the magnetic torque dominates to prevent the rotation of the swimmers to leave

them vertical. Finally, in region IV, the force stalls the swimmers before they reach

the surface. These behaviors are evident in the videos.

The model also quantitatively accounts for the data. The points in Figure 6.4(b)

were plotted using,

−1  −w/w1g  θmax = cos 2 (6.6) P 2∆χBneut ± (1 − w/w1g) w1g µ0w1gL

where the plus and minus signs are for Paramecia swimming into the bottom surface

along and against forces, respectively. Note that the positive direction of w is defined

as the normal gravity direction. The measured average parameters P , w1g, ∆χ,

−23 3 Bneut, µ0, and L for P. caudatum are 1000 pN, 100 pN, 6.7 × 10 m [4], 7.8 T ,

4π × 10−7 Hm−1, and 215 µm, respectively. Thus the model predicts that average

Paramecia turn successfully for w/w1g > −2, become trapped for w/w1g < −2, and become vertically oriented at w/w1g ' −7.5 in accord with Figure 6.2(e) and the video observations. The range of behaviors exhibited at a fixed value of w/w1g can

128 be attributed to the variation in P , w1g, and ∆χ in the population. The gray band in

Figure 6.4(b) is the expected spread in θmax if we presume that P varies by ±25% [34].

This band implies, for example, that when w/w1g ' −7.5, some trapped swimmers are predicted to be canted while others are oriented normal.

While P. tetraurelia and P. caudatum behave very similarly, the model predicts differences that are observed. P. tetraurelia has a smaller ratio of w1g/P , which re- duces the field scale over which trapping is observed (Figure 6.2). In addition, P. tetraurelia is a factor of two shorter than P. caudatum which leads to a weaker mag- netic susceptibility anisotropy [4] and less of a tendency for them to orient vertically while trapped at a surface.

6.4 Discussion

Overall, this passive model appears to capture the MFBV driven trapping phenomena.

This passive interpretation of trapping is in accord with studies of a few species of bacteria. Berke and coworkers [33] produced a hydrodynamic model to account for the accumulation of Escherichia coli at surfaces. Li and Tang [20] modeled the swimming of Caulobacter Crescentus near a surface considering the balance of forces and torques due to contact with a surface that tend to align them while the rotational Brownian motion can enable them to turn from a surface to swim away. Further evidence that microorganisms interact passively with surfaces comes from experiments showing that surfaces can guide swimmers. This guiding, which can be exploited to make microorganism ratchets, can be attributed to mechanical forces alone [21, 121, 122,

129 123, 124].

Closer inspection of Paramecia swimming into surfaces with higher spatial and

temporal resolution than employed here are required to eliminate the contribution of

active responses. There are some indications that the Paramecia swimming with the

apparent weight into surfaces bounce on impact at the highest w/w1g. This effect

might be a slight avoiding reaction and thus a sign of an active response. The results

presented here, however, imply that Paramecium navigation around smooth obstacles is dominated by simple mechanics involving short-range forces.

130 Chapter 7

Conclusion and Applications

131 This study has been motivated by the fact that when Paramecia are exposed to

external forces such as earth’s gravity they respond to it. These observations clearly

indicate that the cells have a very special ability to sense a tiny force because their

apparent weight under gravity is less than 100 pN. The gravisensitivity in Paramecia

inspired us to investigate how they respond to altered mechanical drag forces. The

results in chapter 5 revealed that single ciliate such as Paramecium can have two

extremely different ciliary motor characteristics. The two responses to the mechanical

drag forces in a Paramecium implied that cilia with a similar structure can function differently. Additional investigations will be useful to understand what underlying mechanisms determine different motor characteristics of cilia with a similar structure because these motor characteristics are ubiquitous. Like the body cilia in Paramecia, the beat frequency of the lateral cilia in the mussel Mytilus edulis rapidly changes by about 40% as viscosity changes from 1 cP to 1.7 cP [101]. In contrast, like the oral-groove cilia, the beat frequencies of cilia in rabbit trachea [97], Planaria [98], and frog esophagus [99] change only by 10’s of percent for many fold increases in viscosity above the water value. For example, rabbit trachea reduces their beat rate by only

30% as viscosity increases from 1 cP to 15 cP. In addition, the coexistence of the two motor characteristics may present potential motivations to other investigations related to ciliary function. For example, experiments such as stimulus-dependent variation in cilia [118] and ciliary beaing efficiency [138] will provide more evidence to describe the different mechanisms underlying the two motor characteristics.

The avoiding reaction in chapter 2 indicates that Paramecia actively respond to mechanical perturbations. In chapter 7, we tested their mechanical sensitivities under

132 the different type of physical stimulation such as collisions with flat surfaces. During

the experiment, we observed a counterintuitive surface trapping of swimming Parame-

cia. The passive model indicated that the torque balance between the magnetic torque

and the normal force torque could explain the strange behavior. In particular, the re-

sult showed that contrary to their normal active responses to mechanical stimulations

such as a sharp needle, swimming Paramecia can passively interact with flat surfaces.

This result may subsequently imply that membrane potential response mechanisms

in Paramecium to a physical stimulation is very complicated. Additional experiments such as the investigation of cells’ simultaneous responses to flat and sharp stimulations will be interesting. These experiments can be conducted by observing cells’ behav- iors after colliding with a rough surface. The cells may show both passive or active responses depending on the roughness of the surface. The resultant ciliary beating may describe the mechanisms of each passive or active response to the external forces.

In addition, the experiment can provide an inspiration to other biophysicists or bi- ologists. For example, manually tuning surface accumulation using inhomogeneous magnetic field may provide a new method to control or investigate collective microor- ganism behaviors such as biofilm formation [139], collective cell migration of tumor cells [140], and microbial fuel cells [141].

The static magnetic field effects on biological materials have been studied for a long time. It can affect human lung cell proliferation [142], human lung cell cy- cle distribution [143], bacteria mutation [144], metabolic activity in human leukemic cells [145], and microorganism swimming orientation [4, 51, 146]. In particular, we could manipulate swimming Paramecia using the strong static magnetic fields. The

133 controlled straight swimming trajectories provide extremely useful advantages to in- vestigate swimming microorganisms. It allows experimentalists to analyze statistical data by orienting the swimming trajectories of a population microorganism. The magnetic field induced cell alignment of swimming microorganisms can be used as a tool to investigate cell responses to other fields and perturbations [13, 75]. The

MFBV enabled us to apply variable apparent weights on swimming Paramecia. Dur- ing our experiments, we obtained the variable apparent weights from −8 to 10 times its normal value. This broad adjustable apparent weight range can provide potential applications of the MFBV as an alternative to current gravity simulation. It may be used for the investigation of physiological changes in human or animal life such as muscle atrophy, blood volume, bone loss, and immune system [147, 148, 149]. In addition, the simulated microgravity can affect microorganism characteristics such as gene expression, pathogenicity, and growth rate [150, 151, 152].

In particular, the MFBV can be further applied to understand gravikinesis. Fig- ure 7.1 shows our preliminary results in gravikinesis in Paramecium under viscosity changes. These results indicate that the gravikinesis factor in Paramecia is roughly inversely proportional to the viscosity. The investigation in body ciliary beating dependence on viscosity based on the phenomenological model in chapter 5 may pro- vide crucial ideas to describe gravikinesis. Measurements of swimming parameters may also yield reliable results in how cells change their ciliary beating direction and frequency by variations in their apparent weight. However, the phenomenological model assumed that Paramecia swim under neutrally buoyant environment. Based on the model in Figure 3.4, additional measurements such as different cell trajectory

134 (a) (b) (c)

w / w1g w / w1g w / w1g

Figure 7.1: The gravikinesis under different viscosity of (a) 1.0 cP , (b) 2.3 cP , and

(c) 4.1 cP . The trendlines correspond to the gravikinesis of (a) −54 µm/s, (b) −27

µm/s, and (c) −18 µm/s, respectively. (Bars) Standard deviation of the population. directions relative to the vertical may be required.

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