Measuring Mitotic Spindle Dynamics in Budding Yeast

Measuring mitotic spindle dynamics in budding yeast

Kemp Plumb

B.Sc.

Department of Physics

McGill University

Montreal,Qu´ ebec´ July, 2009

A thesis submitted to McGill University in partial fulﬁllment of the requirements of the degree of Master of Science c Kemp Plumb, 2009

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ACKNOWLEDGEMENTS

First and foremost I would like to thank my supervisor, Dr. Maria Kilfoil, whose patience and guidance were invaluable at every step of this thesis project. I am grateful to Dr. Kilfoil for introducing me to the field of Biophysics and for giving me the opportunity to work on this exciting project. She has been a continuous source of support and guidance throughout my time here at McGill. This work would not have been possible without the help and support of Dr. Jackie Vogel, in the department of Biology. I would like to thank Dr. Vogel for sharing her biological expertise and for allowing me to work in her lab. I acknowledge the work Dr. Vogel has done preparing budding yeast strains for fluorescence microscopy and performing genetic analysis of those yeast strains. Dr. Vogel has helped me to appreciate and better understand the field of cell biology, an area of science that I had not been exposed to prior to my graduate work. I am extremely grateful to Dr. Vincent Pelletier and would like to acknowledge his work writing the majority of the feature finding and volume reconstruction code. I would also like to thank Dr. Pelletier for introducing me to the optical equipment in the Kilfoil lab and for continually providing helpful advice whenever a problem arose.

I extend my gratitude to past and present members of the Kilfoil group: Dr. Yongx- iang Gao, Elvis Prandzic, Stefan Nicolau, and Dr. Stephanie Deboeuf for providing an intellectually stimulating and pleasant working atmosphere. In particular, I would like to thank Stefan Nicolau for implementing extensive error checking in the analysis

ii code, which made my life much easier. I am additionally grateful to Dr. Deboeuf for translating the abstract to French. I would also like to thank Dr. Susi Kaitna and Elena Nazarova for their help with biological aspects of this thesis. Dr. Kaitna has put much work into developing motor- mutant budding yeast strains. Both Dr. Kaitna and Elena were always available to help when I had trouble culturing and preparing yeast cells for microscopy, and to elucidate any aspects of the biology that were unclear to me. Finally, thank you to Courtney Ross for editing this thesis, for her endless support, and for just generally putting up with me throughout the writing of this thesis. I could not have done this without her.

iii ABSTRACT

In order to carry out its life cycle and produce viable progeny through cell division, a cell must successfully coordinate and execute a number of complex processes with high ﬁdelity, in an environment dominated by thermal noise. One important example of such a process is the assembly and positioning of the mitotic spindle prior to chromosome segregation. The mitotic spindle is a modular structure composed of two spindle pole bodies, separated in space and spanned by ﬁlamentous proteins called microtubules, along which the genetic material of the cell is held. The spindle is responsible for alignment and subsequent segregation of chromosomes into two equal parts; proper spindle positioning and timing ensure that genetic material is appropriately divided amongst mother and daughter cells.

In this thesis, I describe fluorescence confocal microscopy and automated image analysis algorithms, which I have used to observe and analyze the real space dynamics of the mitotic spindle in budding yeast. The software can locate structures in three spatial dimensions and track their movement in time. By selecting fluorescent proteins which specifically label the spindle poles and cell periphery, mitotic spindle dynamics have been measured in a coordinate system relevant to the cell division. I describe how I have characterised the accuracy and precision of the algorithms by simulating fluorescence data for both spindle poles and the budding yeast cell surface. In this thesis I also describe the construction of a microfluidic apparatus that allows for the measurement of long time-scale dynamics of individual cells and the development of a cell population. The tools developed in this thesis work will facilitate in-depth quantitative analysis of the non-equilibrium processes in living cells.

iv ABREG´ E´

La regulation´ du cycle cellulaire et la proliferation´ de gen´ erations´ viables requierent` a` la fois la reproductibilite´ et la coordination des differents´ processus complexes en jeu dans un environnement toutefois domine´ par l’agitation thermique. Un exemple essentiel est l’assemblage et la migration du fuseau mitotique qui doivent avoir lieu correctement avant mme la segr´ egation´ des chromosomes. Le fuseau mitotique est une structure transitoire composee´ de deux polesˆ separ´ es´ par des filaments de proteines´ appeles´ les microtubules, auxquelles est rattache´ le materiel´ gen´ etique´ de la cellule. Le fuseau mitotique est notamment implique´ dans l’alignement des chromosomes, puis leur segr´ egation´ vers des polesˆ opposes´ de la cellule ; d’ou` la necessit´ e´ d’un positionnement spatial precis´ et temporellement coordonne´ du fuseau mitotique pour assurer la division correcte du materiel´ gen´ etique´ entre les cellules mere` et fille. Dans ce memoire,´ je decrirai´ les techniques de microscopie confocale par fluorescence ainsi que les algorithmes automatises´ d’analyse d’image, que j’ai mis en oeuvre pour observer et analyser la dynamique spatiale en temps reel´ du fuseau mitotique chez la levure bourgeonnante. Les programmes developp´ es´ permettent de localiser dans l’espace tridimensionnel des structures subcellulaires et de detecter´ leurs deplacements´ au cours du temps.

Le marquage par proteines´ ﬂuorescentes des polesˆ du fuseau mitotique et de la membrane cellulaire a permis de quantiﬁer la dynamique du fuseau mitotique dans un systeme` de coordonnees´ pertinent pour la division cellulaire.

Je decrirai´ des simulations numeriques´ de signaux fluorescents de ces structures subcellulaires qui m’ont permis de caracteriser´ la fiabilite´ et de quantifier la precision´

v des programmes d’analyse. Je terminerai ce memoire´ par la description d’un dispositif microﬂuidique permettant a` la fois la culture de cellules et la caracterisation´ de leur dynamique a` l’echelle´ individuelle, et ce sur de longues echelles´ de temps.

Les outils developp´ es´ au cours de cette these` et present´ es´ dans ce memoire´ offrent la possibilite´ d’analyses quantitatives nouvelles des processus hors equilibre´ en jeu chez les cellules vivantes en gen´ eral.´

vi TABLE OF CONTENTS

ACKNOWLEDGEMENTS ...... ii ABSTRACT ...... iv ABREG´ E...... ´ v

LIST OF TABLES ...... ix LIST OF FIGURES ...... x

1 The cell cycle and the mitotic spindle ...... 1 1.1 The cell cycle control system ...... 1 1.2 Mitosis ...... 3 1.2.1 Budding yeast: a model system for study of mitosis ...... 4 1.3 Structure of the mitotic spindle in budding yeast ...... 5 1.3.1 Microtubules ...... 6 1.3.2 Spindle poles ...... 8 1.3.3 Chromatin ...... 8 1.3.4 Kinetochores ...... 9 1.3.5 Mitotic motors ...... 10 1.4 Spindle assembly and positioning ...... 12 1.4.1 Measurements of spindle dynamics ...... 13 1.4.2 Modelling spindle assembly ...... 16

2 Microscopy ...... 19 2.1 Overview of optical microscopy ...... 19 2.1.1 Fluorescence microscopy ...... 20 2.1.2 The confocal microscope ...... 24 2.2 Optical resolution limits for an objective lens ...... 28 2.3 Image formation in a confocal microscope ...... 31 2.3.1 Fluorescence microscope as a linear translation invariant system 31 2.3.2 The point spread function in a confocal ﬂuorescence microscope 34

vii 2.3.3 CCD array for image detection ...... 37 2.3.4 Noise sources ...... 38 3 Feature finding and tracking ...... 41 3.1 Overview of feature localization and tracking in biology ...... 41 3.2 Automated feature finding in a cell population ...... 50 3.2.1 Three-dimensional tracking of point-like features ...... 51 3.2.2 Automated surface finding ...... 55 4 Evaluating the accuracy of feature localization ...... 61

4.1 Characterizing the imaging system ...... 61 4.2 Simulation of spindle poles ...... 66 4.2.1 Propagation of localization error ...... 76 4.3 Simulation of the cell periphery ...... 78 5 In vivo study of spindle positioning ...... 84

5.1 Microscopy and sample preparation ...... 84 5.2 Results ...... 87 5.2.1 Spindle orientation and positioning in wild-type cells . . . . . 87 5.2.2 Spindle orientation and positioning in mitotic kinesin motor mutants ...... 96 5.2.3 Effect of imaging media on spindle dynamics ...... 101

6 Extension to long-time dynamics: a microﬂuidic chamber ...... 109 6.1 Device fabrication ...... 111 6.2 Preliminary tests ...... 113 7 Conclusions and future directions ...... 114

REFERENCES ...... 119

viii LIST OF TABLES Table page

5–1 Mean and standard deviation of cell cavity volumes ...... 99

ix LIST OF FIGURES Figure page

1–1 Phases of the cell cycle ...... 1 1–2 The cell cycle of budding yeast ...... 5 1–3 The mitotic spindle ...... 6

2–1 Simplified fluorescence Jablonski diagram ...... 21 2–2 General fluorescence microscope ...... 22

2–3 The light path in a general confocal microscope ...... 25 2–4 The Yokogawa spinning disk confocal microscope ...... 27

2–5 Information ﬂow in a confocal microscope ...... 34 3–1 Coordinate system for spindle poles as deﬁned by the cell periphery . . . 51 3–2 General principle of point localization ...... 52

3–3 Filtered and thinned surface proﬁle ...... 57 3–4 Semi-automated input of surface parameters ...... 59

3–5 Ellipsoid ﬁts to mother and bud cavities ...... 60 4–1 The measured point spread function ...... 62 4–2 Measured background signal for CCD detector ...... 65

4–3 Image formation for a diffraction-limited spot ...... 67 4–4 Simulation of pixelation and noise ...... 68

4–5 Localization error versus signal-to-noise ...... 70 4–6 Localization error versus point-to-point displacement ...... 72

x 4–7 Measured separation versus actual separation ...... 73

4–8 Localization error in one dimension ...... 74 4–9 Intensity distribution for closely spaced diffraction-limited spots . . . . . 75

4–10 Comparison of measured error with propagated error for point-to-point separation ...... 77 4–11 Simulation of signal from cell periphery ...... 79

4–12 Accuracy of volume determination ...... 80 4–13 Error of ellipsoid centroid estimation ...... 82

5–1 Chromatic shift between GFP and CFP ﬂuorescence channels ...... 86 5–2 Time evolution of spindle pole separation for wild-type population . . . . 88 5–3 Spindle length versus orientation for wild-type population ...... 91

5–4 Spindle separation and positioning ﬂuctuations for wild-type population . 92 5–5 Mean spindle length versus distance to neck plane for wild-type population 94

5–6 Mother and bud cavity volumes for wild-type population ...... 95 5–7 Mean spindle length versus volume ratio for wild-type population . . . . . 96

5–8 Spindle dynamics in mitotic kinesin deletion strains ...... 97 5–9 Mean distance to neck plane versus mean spindle length for mitotic kinesin deletion strains ...... 98 5–10 Cell volume distribution of mitotic kinesin deletion strains ...... 99

5–11 Ratio of bud to mother volume versus mean spindle length in mitotic kinesin deletion strains ...... 101 5–12 Spindle pole separation versus time for Wild-type cells cultured and imaged in SC media...... 103

5–13 Spindle pole separation versus time for mitotic kinesin deletion strains, cultured and imaged in SC media...... 104

xi 5–14 Fluctuations in mean spindle length versus mean spindle length for cells cultured and imaged in SC media...... 105 6–1 Schematic of microﬂuidic chamber ...... 110 6–2 Fluidic apparatus assembly ...... 113

xii CHAPTER 1 The cell cycle and the mitotic spindle

1.1 The cell cycle control system

Fundamental to all life is the process of cell reproduction. The sequence of events through which a cell duplicates all of the DNA in its chromosomes, doubles the mass of inter-cellular proteins and organelles, and segregates a copy of each chromosome to produce two daughter cells, is termed the cell cycle. The cell cycle can be divided into a series of distinct and highly regulated events which are grouped into phases. A schematic representation of the traditional view of the cell cycle is shown below in Figure 1–1. During synthesis, or S-phase, the DNA is replicated and chromosomes are duplicated to

S G2

M G1 mitosis

cytokinesis

Figure 1–1: S phase is the initial stage of the cell cycle, where DNA duplication occurs. In M phase, the chromosomes are segregated during mitosis. During cytokinesis, cytoplasmic division occurs. During the gap phases, G1 and G2, the cell monitors internal and external conditions. form sister chromatid pairs. Segregation of the sister chromatids into individual daughter

1 cells occurs during mitosis or M-phase. Cytokinesis, the physical division of the cell into two viable cells, is the ﬁnal stage of the cell cycle. The gap phases G1 and G2 allow time for cell growth as well as the production of key proteins and organelles. Gap phases also serve as regulatory transition periods during which the cell can monitor its external and internal environment. The cell will only progress from the gap phases to the critical

S-phase or M-phase when conditions are suitable and resources are available [1, 2]. Cell division is a highly coordinated process which relies on feedback between mechanical and biochemical signals, and cues related to spatial localization of key components. Temporal coordination of events is controlled by the so-called cell cycle control system. The control system is based on a family of protein kinases, termed cyclin dependent kinases, or Cdks, which phosphorylate proteins to initiate signalling cascades that control the major events of the cell cycle. The activity of Cdks is dependent on an array of regulatory proteins and enzymes termed cyclins. Cdks have no protein kinase activity unless they are tightly bound to a cyclin. Throughout the cell cycle, the concentration of Cdks is roughly constant. Cdk activities, and hence cell cycle events, are modulated by oscillations in the levels of the cyclins [1, 2]. Progression is monitored at a series of checkpoints where the cell cycle arrests if particular previous events have not been initiated or completed. Checkpoints are generally negative regulatory signals that require the completion of a process to switch off. When the conditions of a particular checkpoint are satisﬁed, its signalling activity is de-activated and the cell cycle can proceed. For example, the spindle assembly checkpoint (SAC) acts to delay chromosome segregation until all chromosomes are properly attached to the mitotic spindle.

2 Beyond ensuring high ﬁdelity chromosome segregation into daughter cells, the cell cycle must coordinate with cell growth. For unicellular organisms, if cell size is to remain constant across divisions, the duration of the cell cycle must match the time it takes a cell to double in size. Cell growth also depends on external conditions that cannot be controlled, such as the availability of nutrients, and thus the cell cycle must accommodate this [1, 3]. 1.2 Mitosis

In eukaryotic cells, the organization and segregation of chromosomes during mitosis is directed by an intricate and robust molecular machine called the mitotic spindle. The spindle is composed of two pole bodies separated in space and spanned by an array of microtubules along which the chromosomes are held.

Conventionally, mitosis is thought of as a series of discrete events which occur in sequential order. In the initial stage of mitosis, called prophase, replicated chromosomes condense and assembly of the mitotic spindle begins. Prometaphase is characterized by the breakdown of the nuclear envelope. Next, during metaphase, microtubules attach sister chromatids to the centrosome or spindle pole bodies, and the sister chromatids are arranged on the spindle. These stages of mitosis are not a general description for all eukaryotic cells; for example, budding yeast undergo a closed cell division, meaning that nuclear envelope breakdown does not occur, and in this case the distinction between prophase and prometaphase is unclear and rarely made. Furthermore, the spindle is in close proximity to the DNA, and sister chromatids can become attached to the spindle before the completion of DNA replication, blurring the division between S-phase and

M-phase [4]. In anaphase, sister chromatids are physically segregated into daughter cells.

3 There are two mechanistically distinct stages of anaphase. The ﬁrst, anaphase A, involves the movement of sister chromatids within the spindle towards adjacent spindle poles. The second, anaphase B, involves the movements of the poles themselves towards the distal ends of the daughter cells, resulting in spindle elongation. 1.2.1 Budding yeast: a model system for study of mitosis

Budding yeast, Saccharomyces cerevisiae, is a powerful system for studying the eukaryotic cell cycle. Its greatest experimental advantage is its genetic tractability: the complete genomic sequence of budding yeast is available and budding yeast can proliferate in a haploid state wherein only a single copy of each chromosome present in the cell. In the haploid state, it is relatively simple to inactivate certain genes, replace or modify them with deﬁned mutations, or express a gene under the control of an environmental or chemical promoter [1, 2]. Additionally, most of the biochemical and mechanical apparatus of cell division are often evolutionarily conserved, but are less complex for budding yeast than for higher eukaryotes [1, 2]. A schematic representation of the budding yeast cell cycle is shown below in Figure 1–2. Budding yeast undergo an asymetric cell division, requiring a directed displacement of the spindle from the centre of the mother cell to span the plane of cytokinesis, so that the spindle is oriented along the mother-daughter axis. In this case, the spindle contains the apparatus enabling it to orient itself and to separate sister chromosomes. The spindle positioning and chromosome segregation machinery function together to ensure high-ﬁdelity partitioning of genetic material amongst the mother and daughter cells.

4 spindle formation

SPB duplication S G2

M G1 bud mitosis emergence

chromosome segregation

cytokinesis

Figure 1–2: The cell cycle in budding yeast. Budding yeast undergo an asymmetric cell division. The position of the spindle pole bodies and nuclear envelope throughout the division are shown.

1.3 Structure of the mitotic spindle in budding yeast

Budding yeast cells contain 16 chromosomes in the haploid state and 32 in the diploid state. The mitotic spindle of budding yeast is composed of two mictrotubule organisation centres, known as spindle pole bodies (SPB’s), which are embedded in the nuclear envelope. Microtubules span the spindle pole bodies, connecting to sister chromatids at the spindle mid-zone. As well, cytoplasmic microtubules extend radially from the spindle poles toward the cell cortex. The length of the Saccharomyces spindle can range from 1-2 µm at the initiation of prophase, to 7-9 µm in late anaphase. ∼ ∼ Figure 1–3 illustrates the mitotic spindle apparatus in budding yeast; including the

5 molecular motors, which apply forces to the spindle, aiding in spindle assembly and positioning.

mother

nuclear envelope

neck bud + nMT + + kMT _ aMT _ + + ______+ + _ _ + + + SPB +

Kar3 Kip1/Cin8 Kip3 Dynein Cik1 sister chromatids Kar9 Kar3 & kinetochore Kip2 Dynactin Bni1 Vik1

Figure 1–3: The mitotic spindle is a complex cytoskeletal machine, the key components are: microtubules, spindle pole bodies, kinetochores, chromosomes, and an array of motor proteins. This ﬁgure is adapted from [5].

1.3.1 Microtubules

Microtubules are dynamic, noncovalent polymers composed of the globular protein tubulin. The tubulin monomers of microtubules are heterodimers of closely related α and

β tubulin. Tubulin heterodimers assemble “head-to-tail” to form linear protoﬁlaments of alternating α and β tubulins. Microtubules are comprised of 13 such protoﬁlaments arranged in parallel, with like subunits aligned laterally, to form a tubular structure of

25 nm outer diameter [1, 6]. This creates a structural and biochemical polarization with

6 α subunits at one end and β subunits at the other. The two ends of a microtubule ﬁlament are also dynamically distinct: one end is highly dynamic and is called the plus end, while the other end is more stable and is called the minus end. An important property of microtubules is that they alternate between persistent phases of growth and shortening, a process known as dynamic instability [7]. The switch from growth to shrinkage is called catastrophe and from shrinkage to growth is called rescue. Dynamic instability depends on hydrolysis of guanosine triphosphate (GTP) bound to tubulin, as well as on the pool of subunits available for polymerization [1, 7]. When coupled to a rigid structure, microtubules have the ability to exert forces by rectifying thermal ﬂuctuations [6]. Microtubule dynamics can be affected by intercellular factors [1, 2]. Various microtubule associated proteins (MAPs), plus tip interacting proteins, and motor proteins can act to stabilize or destabilize microtubules [1]. Thus, the cell cycle control system can regulate directly the key structural components of the mitotic spindle through phosphorylation events. Growth of microtubules is nucleation-limited. In the cell, microtubule nucleation is controlled by a central microtubule organisation centre (MTOC). In yeast this is the spindle pole body. MTOC’s contain γ-tubulin, a globular protein that aids in microtubule nucleation. At the MTOC, microtubules are nucleated from their minus ends, with plus ends extending outwards.

Within the mitotic spindle, there are three classes of microtubules. Cytoplasmic, or astral microtubules (aMT’s) have plus ends which extend radially from the spindle poles towards the cell cortex. Interpolar, or overlap microtubules (nMT’s) extend from each spindle pole towards the other, and interdigitate with one another to form an anti-parallel

7 array at the spindle mid-zone. Kinetochore microtubules (kMT’s) connect the spindle pole bodies to kinetochores on sister chromatids. The budding yeast spindle is comprised of: one kMT attached to each chromosome, four nMT’s from each spindle, and two to three cytoplasmic MT’s [4, 8]. Budding yeast spindle microtubules are only dynamic at the plus ends, with minus ends remaining stably attached to SPB’s [9].

1.3.2 Spindle poles

In budding yeast, the spindle pole body is a cylindrical structure between 80-110 nm in diameter and composed of three plate-like elements called plaques [8, 10]. The outer plaque is exposed to the cytoplasm and nucleates cytoplasmic microtubules. The inner plaque is exposed to the nucleoplasm and nucleates both kinetochore and interpolar microtubules. The central plaque is enclosed in the nuclear membrane. The SPB remains embedded in the nuclear membrane throughout the cell cycle.

SPB duplication occurs early in the G1 stage, in a region of the nuclear envelope associated with the central plaque called the half bridge. The half bridge elongates in

G1, accumulating material to form a satellite plaque located in the cell cytoplasm. This satellite plaque is eventually moved into the nuclear envelope where it matures [11]. Immediately following duplication, the SPB’s are adjacent to one another. At the initiation of metaphase, the poles move apart to form a 1-2 µm spindle [12]. This movement is dependent on microtubules, motor proteins, and other MAP’s. 1.3.3 Chromatin

Beyond encoding the information essential to the proliferation of life, DNA is a mechanical element that plays an active role in spindle structure and function. The compaction of DNA into chromosomes results in a soft, elastic gel-like structure [4]. In

8 the mitotic spindle, a protein complex called cohesin binds sister chromatids together.

The sister chromatids along with cohesin contribute to the elastic properties of the spindle. When a sister chromatid pair is correctly attached to the poles of the spindle by kMT’s, the linkage between chromatid pairs is strained. This strain appears to be highly localized to the region at the middle of each chromosome, called the centromere, indicating that the centromere may have unique elastic properties [4, 13]. Observations of centromere strain are interpreted as a tension that is transduced by the spindle in a signal indicating proper chromosome bi-orientation to the spindle assembly checkpoint [14, 15].

When all sister chromatids are correctly attached to the spindle, a protease cleaves the cohesin complex, allowing sister chromatids to separate and anaphase to proceed.

1.3.4 Kinetochores

The kinetochore is a multi-protein complex which binds the plus ends of microtubules to chromatid centromeres. Kinetochores play an active role in the spindle, directing and monitoring the movements of chromosomes during mitosis [4]. It is not clear how kinetochores attach to MT plus ends since, when pulling the chromosome towards the pole, this attachment must remain stable while the MT plus ends undergo polymerization and depolymerization events [16, 17]. The putative mechanism of kinetochore microtubule attachment is the formation of a protein complex collar around microtubules. This ring structure is thought to be anchored to a long rod-like complex [18], and to generate pulling force by coupling to MT depolymerization [1, 18].

Kinetochores have been implicated as the active site for tension transduction in the spindle assembly checkpoint [14, 19, 20], although the mechanism of such a transduction is not yet understood.

9 1.3.5 Mitotic motors

Motor proteins convert the energy of ATP hydrolysis into mechanical work. Within an organism, many different types of motor proteins exist, each performing a specialized task; examples are: the transport of membrane-enclosed organelles, cell crawling, muscle contraction, and cell division. Cytoskeletal motor proteins associate with polarized cytoskeletal filaments either actin or microtubules and move in a directed stepping motion along the filament track. Motor proteins generate this directed motion and corresponding force by utilizing chemical energy to rectify thermal fluctuations [21]: as a motor protein undergoes thermal diffusion along its track, the energy from ATP hydrolysis induces a conformational change in the protein, which greatly biases the motion in a particular direction. Individual motor proteins can apply forces typically on the order of picoNewtons and move at velocities in the µm /s range [1, 22]. Cytoskeletal motors are broadly categorized based on the filament with which they interact. Those that interact with actin to move along actin filaments are termed myosins. Kinesins and dyneins interact with tubulin to move along microtubules, and are further classified as either plus end directed or minus end directed, depending on the direction they move relative to the direction of microtubule polarization. During mitosis, microtubule motors act cooperatively and play a number of essential roles: cross-linking of microtubules; application of forces to direct spindle assembly and elongation; control of spindle positioning via interactions with the cell cortex; and regulation of microtubule dynamics [1]. The budding yeast genome encodes seven microtubule motor proteins comprised of six kinesin motors: cin8p, kip1p, kip3p, kar3p, kip2p, sym1p; and one cytoplasmic

10 dynein heavy chain, dyn1p. All but sym1p posses identiﬁed functional roles in mitosis

[5]. Microtubule motors are essential for mitotic spindle formation, elongation, and positioning. As a result of functional redundancies, no individual motor is essential for viable division; rather, proper bipolar spindle assembly requires the presence of antagonistically-acting motor proteins [5, 23–25].

Cin8p and kip1p are localized to the mitotic spindle and are plus end directed motors predominantly responsible for anaphase spindle elongation. Cin8p and kip1p are thought to facilitate spindle pole separation by cross-linking anti-parallel nuclear microtubules and sliding them past one another. Cells which do not express both cin8p or kip1p cannot separate duplicated spindle poles to form a bipolar structure; however, cells which do not express one or the other of cin8p or kip1p can form and maintain a bipolar spindle structure [24, 25]. Despite their apparent redundancy, cin8p and kip1p have been observed to have some functions that are distinct during spindle elongation. Cin8p plays a major role in the rapid stage of spindle elongation, which occurs at the onset of anaphase B, while kip1p is predominantly active in the succeeding stage of slow elongation [24]. Kar3p is a minus end directed motor which localizes to the spindle as well as to the cytoplasmic microtubules. Kar3p forms heterodimers with accessory proteins vik1p or cik1p. The behaviour and localization of the kar3p/cik1p complex is distinct from that of the kar3p/vik1p complex, and each are responsible for different functions [5]. In the mitotic spindle, kar3p antagonizes the actions of cin8p and kip1p, acting to pull the poles together [25], and has been shown to affect microtubule dynamics [26].

11 The plus end directed motor kip3p localizes to both the cytoplasmic and spindle microtubules at different stages of the cell cycle [27]. During spindle elongation, kip3p opposes the growth of the metaphase spindle, an outcome thought to be a result of the microtubule destabilizing ability of kip3p [24]. In cells that do not express kip3p, the duration of anaphase is extended, and telophase spindle microtubules are much longer than for wild-type cells [24]. Kip3p also functions in spindle positioning [23, 28] and has been implicated as an important regulator of cytoplasmic microtubule length [27–29]. Functional overlap has been identiﬁed between kip3p and kar3p and cells that are unable to produce both of these motors are not able to form a bipolar spindle [28]. Kip2p is involved in spindle positioning and is localized to cytoplasmic microtubules [23]. Kip2p acts to stabilize microtubules, and functions cooperatively with dynein to position the mitotic spindle at the bud-neck [27, 30].

Dynein is a minus end directed motor which interacts with the cell cortex through the protein complex dynactin and acts on the spindle via cytoplasmic microtubules. Dynein contributes to both spindle positioning and spindle elongation, and has been implicated as the major driving force for nuclear migration to the bud upon anaphase onset [29]. During spindle elongation, dynein acts cooperatively with cin8p and kip1p

[31]. In the conventional model, dynein is thought to function like a winch; anchored at the cell cortex, dynein pulls on the cytoplasmic microtubules to help position the spindle and thereby contribute to spindle elongation [23, 29, 32]. 1.4 Spindle assembly and positioning

While the basic structure of the mitotic spindle is universal to all organisms [1, 2], many aspects of spindle self-assembly and positioning are not clearly understood. How

12 is the bipolar array of MT’s constructed? How are sister chromatid pairs attached to the array with the correct bi-orientation? What factors control the size of spindles? For asymmetric cell division, how is the spindle positioned in coordination with its assembly? What is clear is that assembly and positioning of a stable but dynamic spindle depends on the actions of antagonistic and cooperatively-acting elements. A brief overview of the current understanding of mitotic spindle dynamics is presented below. 1.4.1 Measurements of spindle dynamics

In budding yeast, antagonistic roles for molecular motors have been demonstrated as essential for mitotic spindle formation [23, 24, 31]. Spindle positioning is also dependent on cooperative action of motor proteins with microtubule polymerization and depolymerization events [32].

Functional roles in the mitotic spindle are largely interrogated using genetic manipulations combined with fluorescence microscopy. In an early investigation of spindle dynamics, Straight et al. [33] visualized spindle and chromosome movements by expressing two different protein fusions of green fluorescent protein that label both the chromosomes and spindle MT’s. Using a wide-field fluorescence microscope, they collected vertical stacks of images of synchronized cells undergoing mitosis. Each image was subject to an iterative deconvolution procedure to remove distortions resulting from the point spread function of the microscope (the point spread function is discussed in Chapter 2). Since the chromosomes appear as spots and the microtubules as

ﬁlaments, the ﬂuorescent signals belonging to each structure were readily distinguished. Chromosome positions were measured over a time-series of images by selecting the brightest pixel from the GFP intensity distribution. While the spatial resolution of

13 this study was therefore limited, the authors were able to observe many interesting characteristics of budding yeast mitosis. Prior to the onset of anaphase, the chromosomes failed to align along a metaphase plate, as is common in higher eukaryotes. Instead,

Straight et al. observed many cells in which the centromeres oscillated back and forth along the axis of the spindle. These oscillations spanned the length of the spindle and continued until anaphase onset. The two kinetically distinct phases of anaphase A and anaphase B were also observed by these authors. During anaphase A, sister chromatids are pulled apart at their centromeres along the spindle axis. Spindle elongation during this stage is very slow. Anaphase B begins with the physical separation of sister chromatids, associated with a short, rapid phase of spindle elongation. Following the rapid phase, spindle elongation continues throughout anaphase B.

In a later related work, Straight et al. [24] studied the contributions of cin8, kip1, and kip3 to anaphase dynamics. Applying the same methods as in their previous work [33] to cells that had been mutated to lack one or more of kip1, kip3, and cin8, the roles of individual mitotic motors were determined. Both cin8 and kip1 were observed to contribute to anaphase spindle elongation; however, cin8 alone was found to be predominantly responsible for the rapid phase of anaphase B, while kip1 was the main contributor to anaphase B spindle elongation. Cells with kip3 deletions were observed to have a prolonged anaphase, and the spindle continued to elongate to the point that it was constrained by the cell wall and was bent. Thus, by investigating single cell dynamics, these authors were able to determine subtle differences in the roles of molecular motors that would not have been apparent in bulk biochemical tests.

14 The role of tension in the spindle assembly checkpoint was ﬁrst investigated by Nicklas et al. in a series of works [14, 19, 20]. Using direct micromanipulation experiments, these authors showed that the spindle assembly checkpoint detects the absence of tension, and that tension alters the phosphorylation of kinetochore proteins in praying mantid spermatocytes. Using a micromanipulation needle to physically pull on unpaired sex chromosomes in a number of cells caused anaphase to proceed in these cells well in advance of anaphase in sister cells with an unpaired chromosome that was not manipulated. The micormanipulation was combined with a staining that is sensitive to phosphorylation of kinetochore proteins. In this assay, chromosomes that were attached to the spindle did not stain brightly, indicating that protein dephosphorylation had occurred. The bright staining returned once the chromosomes were detached with a microneedle. Furthermore, when tension was applied to improperly attached chromosomes using a micromanipulation needle, protein dephosphorylation resulted. This provided direct evidence that, for insect spermatocytes, mechanical force can speciﬁcally alter kinetochore chemistry. Nicklas and colleagues proposed that the dephosphorylation of kinetochore proteins, as a result of tension, signals the spindle assembly checkpoint to turn off. Investigation of the link between mechanics and chemistry at the spindle assembly checkpoint is an active area of current research [15, 34, 35]. It is still not clear that it is a mechanical tension, and not simply an attachment or displacement that is transduced. Physical models of spindle assembly have been integrated directly into experiments in an interesting series of investigations that use “model convolution microscopy”

[36–38]. Because they are densely spaced on the mitotic spindle, individual kinetochores

15 cannot be observed in vivo. Instead, fluorescently-labelled kinetochores appear as a localized but broad distribution of intensity. To test physical models for stable spindle formation, Gardner et al. [36] simulated the spatial distribution of kinetochores on the mitotic spindle, assuming a number of different dynamical models for the underlying behaviour. Fluorescence images were then created in silico by convolution of the simulated kinetochores with the microscope point spread function. Intensity distributions from the simulated images were compared with fluorescence microscopy measurements, and statistically tested to rule out a number of candidate models. Models for spindle assembly that were found to be consistent with the observed spatial distribution of fluorescently labelled kinetochores relied on two physical mechanisms. First, tension between sister kinetochores modulated kMT plus end rescue frequency. Second, the kinetochores sense a stable gradient between poles by some unknown mechanism, to control catastrophe frequency [37]. Extending this investigation further, Gardener et al. found that cin8 and kip1 mediate chromosome movement by suppressing kMT plus end assembly of longer kMT’s. Cin8 and kip1 are proposed as the putative generators of the spatial catastrophe gradient [38]. 1.4.2 Modelling spindle assembly

Current mechanistic models of spindle assembly are essentially force balance relations, where a low Reynolds number approximation is applied so that F ∝ ds/dt, where s is a spatial coordinate describing spindle length. The forces are related to microtubule polymerization dynamics through kinematic equations for s. Commonly, models include forces related to some or all of the following: antagonistic motors which act to slide MT’s past one another; a tensile Hooke’s law term that represents spindle

16 elasticity; and a Brownian ratchet force based on MT depolymerization, which can be biased to pull kinetochores toward the poles or push poles apart [37, 39–43]. Computer simulations of simple systems have been used to demonstrate the minimum requirements for spindle self-assembly. In a system consisting of two microtubule asters and motor complexes that can bind up to two microtubules, it was shown that no combination of motor complexes in which both motor heads move in the same direction, were capable of forming a bipolar spindle. However, complexes with two motors that move in different directions, i.e. antagonistic motor proteins, can form stable structures with overlapping anti-parallel microtubules [39]. In another study, it was shown that in an assembly of anastral MT’s, which nucleate within the spindle, only two motors are necessary to form a stable bipolar structure [43]. In this model, one motor acts to slide anti-parallel MT’s past one another and the other acts to bundle parallel MT’s.

Clearly, the dynamics of the mitotic spindle assembly and its components are complex and rich. Genetic and biochemical assays, as well as electron and ﬂuorescence microscopy, have been used to develop a full picture of the molecular architecture of the budding yeast spindle [4,8–13,23–33,36,44–47]. Redundant roles of spindle components may serve as fail-safe mechanisms to ensure that mitosis is robust across a range of environmental conditions. However, it is entirely possible that individual motors play more subtle roles that have not yet been observed. From a biophysicial perspective, the development of a general mechanical framework for mitosis that is applicable to cells in higher organisms is an ultimate goal. The development of this framework can only proceed through a detailed understanding of the speciﬁc function of individual elements

17 on the mitotic spindle and the cellular mechanisms for coordinating elements during mitosis. It is believed that precise and accurate measurements of the dynamics of spindle assembly and positioning at the single cell level will elucidate fundamental connections between biochemical signalling pathways and mechanical control during cell division. In this thesis work, experimental tools for quantifying real space mitotic spindle dynamics in vivo have been developed. The tools are designed for parallel high throughput analysis of a large number of cells to capture statistical variation inherent in cellular processes.

Spindle dynamics in living cells can be observed with a minimal amount of perturbation to the cells using ﬂuorescence confocal microscopy. The length-scales of mitotic spindle dynamics range from micrometers to tens of nanometers, the limit of achievable resolution with current optical instrumentation. The operating principle and limitations of modern microscope systems are discussed in detail in the following chapter.

18 CHAPTER 2 Microscopy

2.1 Overview of optical microscopy

In the most general terms, an optical microscope is an instrument that manipulates light to produce a magniﬁed image of a specimen. As a probe of the structure of matter, optical microscopes are fundamentally limited in resolution by the wavelength of visible light. The limit of spatial resolution, referred to as the diffraction limit, is roughly

λ/2, where λ is the wavelength, which for visible light is between 400 and 750 nm. While instruments such as electron microscopes can achieve signiﬁcantly higher spatial resolution, light microscopes are particularly advantageous for studying biological systems because visible light is relatively non-destructive. It can be argued that the ﬁeld of cell biology began with the introduction of the light microscope [1], and recent developments in instrumentation and analysis techniques have established the light microscope as an essential tool for quantitative study of cell structure and function. Quantitative analysis of signals measured with microscopy necessitates an understanding of the functioning and limitations of the microscope system to accurately reproduce characteristics of the object under study.

The microscope, illumination source, and light detector together comprise an optical system. The components of an optical system, including the object under study, act to manipulate the amplitude and phase of electromagnetic waves in a deterministic manner, which can be measured and interpreted using a light detector. It is important to point out

19 that light detectors and sensors are not sensitive to the amplitude or phase directly and instead measure the intensity of incident radiation over the period of measurement:

I = U(x,t) 2 , (2.1) h| | i where U is the energy of the incident radiation, x is a spatial coordinate on the detector, and the angle brackets indicate a temporal average. The image formed by the light detector of an optical microscope represents the intensity distribution of light in the specimen plane. When an object is illuminated by some external light source and visualized by a light microscope, the image formed depends on differences in optical absorption and index of refraction over the object. According to the Abbe theory, the object acts as a diffraction grating, and an image is formed by the interference between diffracted rays in the image plane [48]. Contrast in conventional light microscopy is limited by the optical properties of a given specimen: if the index of refraction does not vary sufficiently over different regions of an object then those regions cannot be distinguished in an image. Furthermore, no information can be obtained for objects which are smaller than the diffraction limit. Understanding the mechanisms of cellular processes on the molecular scale is greatly aided by the ability to visualize the spatial distribution of specific molecules, with dimensions below the diffraction limit, in a given specimen. This can be accomplished using fluorescence confocal microscopy. 2.1.1 Fluorescence microscopy

Fluorescence microscopy depends on the ability of molecules to ﬂuoresce in the presence of an optical excitation. When radiation of sufﬁcient energy is impingent upon

20 a fluorescent molecule, the absorption of a photon by the molecule excites an electron to a state in the first singlet band. Figure 2–1 depicts a simplified energy level diagram of the process. An electron excited to some level in the first singlet band relaxes very

Excited singlet vibrational relaxation states

γex γem absorbtion uorescence Ground state

Figure 2–1: A simplified Jablonski diagram showing energy levels in a fluorescent molecule. An electron is excited to the first singlet band from the ground state by a photon γex. The excited electron relaxes within the first singlet band by a vibrational process. A fluorescent photon, γem, is emitted when the electron relaxes to the ground state. quickly to the lowest level of the band without emitting a photon [49, 50]. Fluorescence emission occurs when the electron subsequently relaxes to the initial ground state.

Energy dissipated during the initial relaxation within the first singlet band results in a shift between excitation and emission wavelengths, termed the Stokes shift. Since the absorption of a photon by a single fluorescent molecule will result in fluorescent emission, objects with a size below the diffraction limit will produce an observable signal. Using this technique it is possible to monitor the spatial distribution of single molecular species within a specimen. A schematic diagram of the light path in a typical fluorescence microscope is shown in Figure 2–2 below. Fluorescence microscopes are usually configured so

21 Figure 2–2: A fluorescence microscope in epi-illumination configuration uses filter cubes matched to the emission and excitation wavelengths of the fluorophore. Images are captured by an electronic detector such as a CCD camera. Lasers or broadband sources can be used for excitation, as long as appropriate filter sets are employed. that the objective lens also acts as the condenser; this arrangement is termed epi- illumination. Filtering elements are used to separate the excitation and emission light based on their wavelengths. The filter set consists of three main components: excitation filter, emission filter, and a dichroic mirror. A narrow band excitation filter passes wavelengths in the excitation spectrum of the fluorophore. The dichroic mirror reflects the excitation wavelength and transmits emission wavelengths. The emission filter is a narrow band filter that passes the fluorescence wavelengths. It is possible to use a broadband source, such as a mercury or xenon arc lamp, with the appropriate excitation filters or monochromatic laser sources for fluorescent excitation.

The accuracy of fluorescence measurements depends on the detectable fluorescent signal and is therefore limited by the efficiency of fluorescence detection and by the photochemisty of the particular fluorophore employed. A given specimen will contain

22 a population of fluorescent molecules that can be excited, with the number of excited molecules dependent on the incident photon flux, or optical power, delivered by the excitation source. For low excitation intensities, fluorescence emission is roughly proportional to the excitation intensity [49]. This linear relation no longer holds at high excitation intensities because a significant fraction of the fluorophore population can become excited, so that the fluorescence emission rate of the population is limited by excited state saturation [49, 50]. Furthermore, the total number of available photons is limited by photobleaching, the irreversible chemical change of a fluorescent molecule to a non-fluorescent species. In live cell imaging, the excitation illumination can be deleterious to the cell, resulting in harmful phototoxicity or photodamage. When

fluorescence measurements are performed, the illumination intensity must be carefully adjusted to maximize the signal, while at the same time minimizing photobleaching and any toxic effects of the light. The use of fluorescence microscopy for measurements of processes in living cells has been revolutionized by the introduction of green fluorescent protein (GFP) and its spectral variants. The 2008 Nobel prize in chemistry was awarded to Osamu Shimomura, Martin Chalfie, and Roger Y. Tsien for the discovery of GFP. GFP can be encoded within the genome of an organism so that it is expressed as a fluorescent tag on a specific protein structure. Expression and fluorescence of GFP in the organism can then be directly observed as a marker for the expression and spatial organisation of the particular protein it is associated with. GFP and its derivatives can be used to follow nearly any protein produced in the cell. Moreover, several spectral variants can be combined for observation of many different structures near-simultaneously.

23 2.1.2 The confocal microscope

In a conventional wide-ﬁeld ﬂuorescence microscope, objects outside of the focal plane produce light that is collected by the objective lens and that ultimately reduces the contrast of the in-focus signal. Confocal microscopes eliminate this out-of-focus light to produce thin and clean optical sections. The confocal microscope was conceived in 1957 by Marvin Minskey, while he was a postdoctoral fellow at Harvard University, and the

ﬁrst commercial instrument appeared in 1987 [51]. In the confocal geometry shown in Figure 2–3, a point on the specimen is illuminated with a focused diffraction-limited spot. A pinhole aperture is placed in front of a detector, in the plane conjugate to the image plane that contains the spot in the specimen, so that only light originating from the spot being scanned is passed through to the detector. To construct a two-dimensional image of the sample, the illumination spot is raster- scanned over the image plane. A three-dimensional reconstruction is also possible via movement of the objective lens in the axial direction to collect a series of images of different focal planes in the sample. Many different conﬁgurations exist for performing the lateral scan, the most common arrangement being the confocal laser scanning microscope (CLSM), where the illuminating laser beam is scanned over the objective lens using optical elements. The signal recorded by a photon detector in a scanning confocal microscope is a set of intensity values for each voxel in some sub-volume of the sample. Depending on how the scan was performed, this set of intensity values does not necessarily represent an image of the sample. The confocal scanning microscope is a serial sampling instrument, rather than a parallel imaging instrument. The scan can be

24 Figure 2–3: Schematic of the light path in a general confocal microscope. In this type of microscope, a pinhole aperture is placed in the image plane in front of a photon detector, so that ﬂuorescent light, which is focused to a point at the aperture conjugate to the focal plane, is transmitted to the detector. The aperture eliminates signals from sources above and below the focal plane. controlled to measure the signal from an individual point many times, a sub-volume of the sample, or any conﬁguration of points within the sample. Investigation of dynamics of molecules and organelles in living cells at high spatial and temporal resolution places strict requirements on the speed of data collection and limitations on light exposure. To accurately measure the dynamics of complex intercellular organisational processes, it is often necessary to collect three-dimensional images of the process over time, at a collection rate commensurate with the rate of the process. It is also often necessary to collect data over the full time duration of the process. Thus, photodamage and photobleaching must be minimized. To meet these

25 requirements, it is possible to parallelize the process of confocal sampling by scanning the object with many points simultaneously using a spinning Nipkow disk. A Nipkow disk contains thousands of pinholes arranged in a series of nested spirals. The disk is positioned in a conjugate image plane so that the pinholes perform both the point illumination and point detection necessary for confocal imaging. A wide ﬁeld of pinholes are illuminated at once so that rotation of the disk scans the specimen with many beams of light that cover the entire image plane. An image is formed on a detector array by summing the intensity distributions from diffraction-limited spots across the ﬁeld of view simultaneously. Early Nipkow disk scanning microscopes had very low light transmission, with

90% to 99% of the incident illumination reﬂecting from the disk [52]. Beyond reducing the signal, reﬂection from the disk contributes unwanted high background. To increase light transmission, pinholes must be closely spaced. In such an arrangement, light from structures outside of the focal plane of a given pinhole may pass through adjacent pinholes. This out-of-focus light limits the z-resolution of the Nipkow disk scope compared with conventional confocal arrangements. These problems have been largely solved in modern tandem spinning disk scopes, which employ the Yokogawa design shown in Figure 2–4. In the Yokogawa design, a microlens array is used to focus light into each of the pinholes. This can increase light transmission up to 40 % [52] without ∼ increasing the size or number of pinholes. The principal advantage of disk scanning confocal microscopy is that multiple points on the sample are illuminated and detected in parallel, resulting in improved temporal resolution. There is also less photobleaching because sampling time, as well

26 Excitation Beam Microlens Array

CCD Detector lens

Pinhole Array Emission Path

Dichroic

Objective Lens

Sample

Figure 2–4: The Yokogawa spinning disk confocal microscope. In this arrangement, the specimen is scanned with thousands of points simultaneously using tandem spinning Nipkow disks. The Yokogawa design uses two disks, each with 20 000 to 200 000 pinholes that rotate as a single unit [52]. The upper disk contains a microlens array that focuses the illumination onto its corresponding pinhole. The pinholes are usually arranged in a series of nested Archimedian spirals, spaced densely enough so the entire sample is scanned once by a partial rotation of the disk. Images are acquired with a camera that is synchronized with the rotation of the disk. as local excitation intensities, are reduced. Additionally, parallel confocal sampling effectively generates a real image that can be detected with a CCD camera. State of the art CCD detectors can be incorporated and upgraded as needed without modifying the confocal optics. An important limitation of disk scanning instruments is that the pinhole size is ﬁxed, whereas laser scanning confocal arrangements allow a user to select one of many

27 different pinhole sizes. In the case of a disk scanning instrument, the pinhole is often optimized for one objective lens and it is not possible to make adjustments for other magniﬁcations.

2.2 Optical resolution limits for an objective lens

The fundamental components of any light microscope are: the objective lens, which collects light from the object and forms a magnified real image; and an illumination component comprised of a light source and a condenser lens, which distributes the illumination over the object. The resolution limit of an optical microscope is determined by the properties of both its objective and condenser lenses, as well as the wavelength of light used [48]. The resolving power of an objective lens depends on its ability to capture light over a large solid angle, expressed in terms of the numerical aperture (NA) of the objective, NA = nsinα, where n is the refractive index of the medium between object and objective, and α is the half angle of collected rays from the object point. For an objective lens free of optical aberrations, the image of an infinitesimal, monochromatic, self-luminous point in the specimen plane is a diffraction-limited spot formed at the intermediate image plane. Here, the well known result for the three- dimensional distribution of light near the focal plane of a well corrected lens is used. Detailed derivations of these results are available in [53]. These results are relevant for incoherent illumination, which is the case in fluorescence microscopy. The light distribution in the focal plane of a well-corrected lens arises essentially from Fraunhofer diffraction on the aperture of the lens. The intensity distribution is the

28 Airy diffraction pattern formed by the exit pupil of the objective lens:

2 2J1(ν) I(0,ν) = I0 , (2.2) ν where J1(ν) is the first-order Bessel function. The parameter ν is a dimensionless coordinate in the focal plane given by ν = x0 NA2π/λ, where x0 is a distance in the object space, and λ is the wavelength of light. The central bright peak is known as the Airy disk. Its radius defines the spatial resolution of an imaging system in the specimen plane [54]: 1.22λ r = . (2.3) Airy 2NA The Rayleigh criterion for optical resolution defines the minimum spacing between two adjacent objects in the specimen plane at which the objects can be individually resolved. This minimum separation is when the centroid of their respective diffraction spots are spaced by at least the radius of the Airy disk. The Rayleigh criterion represents an ideal scenario and in practice, even in the absence of lens aberrations, the objects usually differ in intensities making it more difficult to distinguish between them. Further complications arise because real images are corrupted by noise and background signals which reduce contrast. Hence, the minimum resolvable distance is often larger than rAiry. Along the axial direction, the intensity distribution is given by:

sinu/4 2 I(u,0) = I , (2.4) u/4 0 where u is a dimensionless coordinate along the optical axis given by u = NA2 z2π/λ. Axial resolution is deﬁned as the distance from the centre of the diffraction pattern to the

29 ﬁrst axial minimum, so that the minimum resolvable distance in the axial direction is:

2λn zmin = . (2.5) (NA)2

Axial resolution is related to the depth of focus, defined as the axial distance in the image space on each side of the image plane within which the image remains acceptably sharp. The depth of field is the axial distance in the object space that appears to be in focus for one setting of the microscope focus adjustment. For an ideal detector, the diffraction-limited depth of field is taken to be one-quarter of the distance to the first axial minimum above and below the focal plane [54]. In wide-field fluorescence microscopy, a significant portion of the fluorescence signal from the cone of light striking objects above and below the focal plane is collected by the objective lens and passed to the detector. This ultimately reduces the contrast from the signal that is in focus. As will be discussed below in Section 2.3.2, confocal imaging eliminates this out of focus light, so that only the objects that lie within the axial resolution of the microscope are visible in the image. Knowledge of the intensity distribution of a diffraction-limited spot can be used to localize the spot to a precision much higher than the spatial resolution of the microscope. Moreover, for single isolated spots of sufficient brightness, the centre of the intensity distribution can be localized to nanometer resolution (Chapter 3, this thesis).

By applying linear system theory to a ﬂuorescent microscope, it can be shown that the intensity distribution from a single diffraction-limited spot deﬁnes the imaging characteristics of the microscope and can be used to describe the formation of an image for any arbitrary object. This is presented in the following section.

30 2.3 Image formation in a confocal microscope

2.3.1 Fluorescence microscope as a linear translation invariant system

An optical system, such as a light microscope, takes an input in the form of an illumination wavefront and maps it to an output to be detected by a sensor. The microscope therefore can be represented as some functional operator which maps an input function in object space to an output function in the image space. A particularly powerful tool for analyzing systems such as this is linear system theory. In order to use linear system theory to describe image formation, it is necessary to understand the conditions under which a ﬂuorescent microscope can be approximated as a linear, translation-invariant system. A linear system must obey the principle of superposition. If the actions of the system are represented in terms of an operator Λ, the action of Λ on function p(x) in input space maps it to another function q(x) in output space. For the principle of superposition to hold, for any value of the constants a and b we must have:

Λ ap(x) + bq(x) = aΛq(x) + bΛq(x). (2.6) { }

In the case of an optical system, recalling that light detectors measure intensity as given by equation 2.1, linear behaviour implies:

aU(x ) + bU(x ) 2 = a U(x ) 2 + b U(x ) 2 , (2.7) h| 1 2 | i h| 1 | i h| 2 | i which requires:

U(x )∗U(x ) = U(x )U(x )∗ = 0. (2.8) h 1 2 i h 1 2 i

31 Thus, the linear assumption requires that wavefronts from different sources in the object be incoherent. In fluorescence microscopy, no phase relationship exists between light emitted from the molecules that compose the object, so that the fluorescence can be considered incoherent. In an optical system, translation invariance requires that the response to a point source must change only in location and not in form, for all translations of the point source in the image plane. That is, the image of a point source is the same no matter where the point source is located in the field-of-view. In a real microscope, this assumption is difficult to justify for the entire field-of-view, since image plane aberrations, such as curvature of field and inconsistencies in illumination, affect different regions of the

ﬁeld-of-view differently. However, it is usually possible to divide a ﬁeld-of-view into a number of sub-regions over which the system is translation invariant [55].

For a linear system, the response to an arbitrary input can be expressed in terms of a sum of the suitably-weighted responses to a simple input. This can be shown by writing a general object function, g1, in terms of the δ-function [56]:

∞ g1(r1) = g1(ξ)δ(r1 ξ)dξ, (2.9) Z ∞ − − with the function g1(ξ) as a weighting factor. If we apply the system operator Λ to the general object function, recalling the linearity condition in equation 2.6, then:

∞ g2(r2) = Λ g(ξ)δ(r1 ξ)dξ , Z ∞ − ∞ − g2(r2) = g(ξ)Λδ(r1 ξ)dξ. (2.10) Z ∞ − −

32 Now the response of the system at the coordinate r2 of the output space to an input at the coordinate ξ can be written as:

h(r ;ξ) = Λδ(r ξ), (2.11) 2 1 − so that the system input and output can be related by the so called superposition integral:

∞ g2(r2) = g(ξ)h(r2;ξ)dξ. (2.12) Z ∞ − This is a fundamental result of linear system theory, which shows that for a linear system, the output is completely characterized by its response to a unit impulse. The function h(r2;ξ) is called the impulse response function. In optics, h(r2;ξ) has a speciﬁc name and is termed the point-spread function (PSF). The PSF for an aberration-free objective lens is the intensity distribution of a diffraction-limited spot [48, 54], described above in Section 2.2.

For a translation-invariant system, the response depends only on the distances between the excitation point and the response point, so that h(r ;ξ) = h(r ξ), and 2 2 − consequently the superposition integral can be written as [56]:

∞ g2(r2) = g(ξ)h(r2 ξ)dξ. (2.13) Z ∞ − − This is simply the convolution of the impulse response, or PSF, with the object function. Thus, under the assumptions of linearity and translation invariance, the formation of an image in a microscope can be modelled as the convolution of the object with the point spread function of the optical system:

I = g h. (2.14) ⊗

33 2.3.2 The point spread function in a confocal ﬂuorescence microscope

In an ideal confocal microscope, light from a small illuminated aperture is focused to a diffraction-limited spot in the specimen, producing a light distribution equivalent to the wide-field PSF. Light emerging from the object is spatially filtered through a detection aperture, located in the plane conjugate to the focal plane. Effectively, this applies the wide-field PSF twice [55]. Figure 2–5 is a schematic of the confocal arrangement, drawn to highlight the flow of information in a confocal microscope from illumination to detection. To derive the PSF of a confocal fluorescence microscope, the

Aex(x1) Objective Collector Aem(x2)

O(xo) Detector

∫ dx2 Scan, x I (x ) Scan, s ex o Iem(xo) λ λ x ex em s

Figure 2–5: Information ﬂow in a confocal ﬂuorescence microscope. The excitation path depends on the excitation aperture Aex and the objective lens. The emission path depends on the collector lens and the emission aperture Aem. In conventional confocal microscopes, epi-illumination is used and the same aperture is used for excitation and emission. excitation and emission paths are considered as distinct optical paths, with the excitation path comprising the illumination source, excitation aperture Aex, and the objective lens.

The collector lens, emission aperture Aem, and the detector together form the emission path. Both apertures have a characteristic size a. The excitation intensity Iex and emission intensity Iem are related by the ﬂuorophore distribution in the object plane, described by the object function O(x0). For simplicity, we consider a ﬂuorescent microscope that is

34 equipped with perfect band-pass ﬁlters in the excitation and emission paths. In this case, the excitation and emission light will be monochromatic:

I (λ) = I δ(λ λ ), ex ex − ex I (λ) = I δ(λ λ ). (2.15) em em − em

In the following, only one spatial coordinate will be used for notational simplicity; however, all arguments presented are readily extended to three dimensions. The excitation intensity in the object plane is the convolution of Aex with the PSF of the objective lens at the excitation wavelength hex [57]:

∞ Iex(x0) = hex(x0 x1)Aex(x1)dx1. (2.16) Z ∞ − − The image formed in the conjugate plane at the detection aperture is the convolution of the emission intensity in the object plane with the collector lens PSF. The detector measures the intensity in the conjugate plane integrated over the detector aperture:

∞ Idet = Aem(x2) hem(x2 x0)Iem(x0)dx0 dx2. (2.17) Z Z ∞ − x < a − | 2| | | For a fluorescent object under sufficiently low excitation intensity that saturation has not occurred, the emission intensity is proportional to the excitation intensity [49], as has been mentioned earlier. The constant of proportionality is related to the fluorophore distribution in the object O(x0). The relation between the signal at the detector and the illumination intensity can be described by writing the emission intensity, equation 2.17,

35 as the product of the excitation intensity, equation 2.16, with the object:

∞ ∞ Idet = Aem(x2) hem(x2 x0)O(x0) hex(x0 x1)Aex(x1)dx1 dx0 dx2. (2.18) Z Z ∞ − Z ∞ − x < a − − | 2| | | In an ideal confocal geometry, the excitation and emission apertures are inﬁnitesimal pinholes that can be modelled as delta functions. An image is formed by scanning the apertures over the points xs, corresponding to translating the delta functions:

∞ ∞ Idet = δ(x2 xs) hem(x2 x0)O(x0) hex(x0 x1)δ(x1 xs)dx1 dx0 dx2, Z − Z ∞ − Z ∞ − − x < a − − | 2| | | ∞ = δ(x2 xs) hem(x2 x0)O(x0)hex(x0 xs)dx0dx2, Z − Z ∞ − − x < a − | 2| | | ∞ = hem(xs x0)O(x0)hex(x0 xs)dx0, (2.19) Z ∞ − − − which is the convolution:

I = [h h ] O. (2.20) det em ex ⊗ Thus, the PSF of a confocal microscope, with an ideally small pinhole and negligible

Stokes shift λ λ , is the square of the wide-field PSF [57]. Under ideal confocal ex ≈ em behaviour with pinhole excitation and emission apertures, the radius at half maximum of the Airy disk is reduced by a factor of 1.36. This increases the resolution limit of an ideal confocal microscope by a factor of √2 over the Rayleigh limit for a wide-field microscope [54]. However in practice, a pinhole with dimensions smaller than the Airy disk drastically reduces the signal level, and is usually not suitable for obtaining sufficient image contrast [58]. Pinhole size is adjusted to optimize both the optical

36 sectioning capability and the signal-to-noise; thus, the optimal size ultimately depends on the particular application. 2.3.3 CCD array for image detection

In modern microscopes, images are recorded digitally usually using a photomulti- plier tube (PMT) or a CCD array. Typically, wide field and disk scanning microscopes utilize a CCD array. When images are acquired using a CCD array, a continuous intensity distribution is recorded as a set of discrete intensity values over equally-spaced regions on a grid, defined by each of the CCD elements, or pixels. This process results in the quantization of the spatial distribution of intensities in the image. Spatial quantization depends on the size of the pixel elements and on the magnification of the image in the detector plane.

To reproduce the intensity distribution in the image plane accurately, a digital image must reproduce the intensity at all spatial frequencies that comprise the object, with high

fidelity. When an object is imaged in a microscope, the PSF acts as a low-pass filter, so that spatial frequencies in the object that are above a characteristic value are greatly attenuated in the image. This characteristic spatial frequency is termed the “cut-off frequency” and is defined by the Fourier transform of the PSF, termed the optical transfer function (OTF). The Nyquist criterion defines the frequency at which a continuous signal must be sampled so that all information contained in the signal be reproduced. Sampling must occur at at a minimum of twice the highest frequency contained in the continuous signal. For an imaging system, this means that the image must be sampled by the detector at a frequency of at least twice the cut-off frequency of the system’s

OTF. In the spatial domain, the Nyquist criterion dictates that the size of a pixel must be

37 sufﬁciently small that at least four pixels are contained within the area of the Airy disk for the imaging system [59]. In order to meet the Rayleigh limit for resolution, images must be acquired at a spatial sampling that meets or exceeds the Nyquist criterion.

The signal from each CCD element (each pixel) is a voltage that is proportional to the number of photons absorbed over that element during the exposure time. The probability that a photon striking a detector element will produce a signal electron is represented by the quantum efﬁciency of the detector, QE. QE is always less than 1, but in modern, cooled EM-CCD cameras that are used in most research-grade confocal microscopes, QE can approach 1 [60]. The photoelectron signal is ampliﬁed and sent to an analog-to-digital converter (ADC). Intensities are reported by the computer in terms of grey scale or analog-digital units (ADU’s), which represent the smallest digital interval on the ADC. The range of intensities reported is limited by the bit-depth of the

ADC, as well as by the noise floor for the detection process (discussed in the following section). CCD cameras have a specified gain setting that defines the number of electrons per ADU. This value is set by the manufacturer of the CCD to be close to the RMS noise value of the detector, so that changes in grey level represent meaningful changes in the signal [59].

2.3.4 Noise sources

Images recorded by a microscope are subject to many sources of noise. At the most fundamental level, the photon detection process is stochastic, and has an intrinsic statistical noise. In addition, the CCD detector and its associated electronics are subject to dark current and thermal noise. Other noise sources include light scattering in the sample [49, 50, 58] and background signals introduced by autoﬂuorescence of the

38 cements and coatings used in optical components, or of endogenous ﬂuorophores in biological samples. At the detector, light which comprises the image should be thought of in terms of photons, each with an associated energy hν, where h is Planks constant and ν is the frequency of the light. A beam of light of power P that strikes a detector element over a time interval t contains np photons, given by: np = Pt λ/(hc), where c is the speed of light. The detection of an individual photon is a quantum mechanical event with a probability governed by Poisson statistics. If, for each of a number of trials, np photons impinge on a single detector element having quantum efﬁciency QE, the detector will register a mean of Qpnp photons with a variance of Qpnp. Photon counting statistics p intrinsically limits the precision to which the intensity distribution can be reproduced, a fundamental limit which cannot be surpassed.

In the absence of illumination, dark current is produced in the detector by electrons that are thermally generated in the depletion region and collected by the bias voltage. Dark current generation is a statistical process with a variation also governed by Poisson statistics [59]. Readout noise is introduced by the amplifying electronics associated with the CCD and ADC. Both dark current and readout noise are drastically reduced by cooling of the detector [58]. While readout noise and dark noise are often extremely small in absolute terms, on the order of 1 electron in modern cooled CCD detectors [60], it is important to note that in live cell applications it is common for the signal to be very weak. In disk scanning microscopes, the signal can be on the order of 10 photons, ∼ over a typical integration time of 100 ms, so that readout noise and dark current can be signiﬁcant relative to the signal. A further uncertainty is introduced to the detection

39 process by the non-zero dimensions of the pixels. Although not technically a source of noise, this uncertainty is often referred to as pixelation noise, and results from the inability to know exactly where a photon struck, within an individual detector element.

Light scattering in the sample-containing medium can contribute to fluorescence- like signals, that are proportional to laser power but do not saturate as the fluorophore signal does [50]. Consequently, excessive laser power will decrease the contrast between signal and background [49]. Background signals cannot be eliminated. They are minimized in practice by careful sample preparation and the use of appropriate narrow band-pass excitation and emission filters. Using the results from linear system theory and an understanding of the quantization and noise introduced by digitally sampling an image, a general model for the formation of an image is developed:

I(x,y,z) = Λ[g h + η]; (2.21) ⊗ where g is a function representing the distribution of ﬂuorophores in the object, h is the point spread function of the imaging system, η is a stochastic term that accounts for all of the intrinsic and extrinsic noise sources, and Λ is an operator that describes the spatial sampling of the intensity distribution on the CCD detector.

40 CHAPTER 3 Feature ﬁnding and tracking

3.1 Overview of feature localization and tracking in biology

Feature finding algorithms are designed to automatically localize the intensity centroid of a specific feature in an image. In biological systems, the features of interest are often fluorescently labelled proteins and organelles, with dimensions below the diffraction limit; the intensity distributions to be localized are therefore diffraction limited spots. Feature finding algorithms consist of two main steps to automatically detect and then localize all features of interest in a given image. The first step is an initial detection of local intensity maxima over the entire image, identifying candidate feature locations to pixel resolution. In the second step, the full intensity distribution of the candidate features, which spans many pixels, is used to localize the centroid of each intensity distribution to sub-pixel resolution. To extract the real space dynamics of a feature from a time-series of images, tracking algorithms establish the correspondence between features in successive image frames, forming a set of single-particle trajectories. In general, feature correspondence is determined by minimizing the displacement of localized features between frames. In some implementations of feature finding algorithms, constraints are set for the tracking by assuming a particular model for the underlying dynamics of the features to be tracked, i.e. assuming Brownian motion. The ability of tracking algorithms to reconstruct feature

41 trajectories with high fidelity is critically dependent on the ability of the localization algorithm to determine the particle positions accurately. In this chapter, a number of articles which assess the limits of accuracy and precision of particle localization algorithms are discussed. As well, two recent approaches for high fidelity tracking in dense systems where particle trajectories may overlap are presented. This discussion serves as a brief overview of the current state and limitations of real space particle tracking in cells. This is followed by a discussion of the methods developed in this thesis work to track budding yeast spindle pole bodies in three-dimensions and reconstruct the cell surface from fluorescence image data. Cheesum et al. [61] quantitatively compare algorithms for localizing fluorescently labelled objects in two-dimensions. These authors assert that a relevant assessment of the accuracy of a localization algorithm requires knowledge of the actual position of the object being tracked. To this end, images of objects over a range of sizes were simulated and the performance of localization algorithms assessed for varying imaging conditions. Four common two-dimensional implementations were compared:

i. Centre-of-mass algorithms estimate the position of a particle to be the centre-of- mass of the intensity distribution associated with the particle. For an intensity

distribution in an image, the centre-of-mass is deﬁned as the average of pixel positions in some neighbourhood of the object, weighted by their intensities.

Accurate calculation of the centre-of-mass requires bright objects which can be readily discriminated from the image background. To exclude the background intensity, a threshold operation is usually employed by setting pixels with an

42 intensity below the threshold value to zero. Performance of such centroid algorithms

for point objects is critically dependent on this threshold value. ii. Direct Gaussian ﬁtting exploits the shape of the intensity distribution for a

diffraction-limited spot. In this method, a two-dimensional Gaussian function is ﬁt directly to pixels in some neighbourhood of the object. The centre of the Gaussian

function deﬁnes the position of the object. iii. Cross-correlation algorithms compare an image to the region in a successive image that contains the object being tracked. This region is termed the kernel. The kernel

is shifted iteratively at pixel increments with respect to the image, and the cross- correlation calculated at each step. The kernel position where cross-correlation is

maximized is taken as the object position. iv. Sum-absolute difference methods are similar to cross-correlation algorithms. In

this case, the translation between the image and the kernel, that minimizes the sum- absolute difference of intensities between overlapping pixels is taken as the position of the object.

Typically, centre-of-mass and direct Gaussian ﬁtting methods obtain sub-pixel resolution directly. In contrast, both the cross-correlation and sum-absolute difference methods determine the object’s position to only pixel resolution. To obtain sub-pixel resolution, the cross-correlation or sum-absolute difference matrices must be interpolated.

Cheesum et al. measured accuracy as the mean difference between the localized position and actual position of the simulated object over a large number of trials. The precision of each algorithm was the standard deviation of the localization measurements for a stationary object. The direct Gaussian ﬁt was determined to be the superior

43 algorithm for localizing point sources. At a signal-to-noise ratio of less than 10, typical for single particle tracking in cells, the accuracy of the Gaussian algorithm was reported to be 10 nm; however, accuracy degraded rapidly as the signal-to-noise ratio dropped ∼ below 4. Thompson et al. examine the factors that limit the precision of centroid localization

[62]. Using a framework of least-squares ﬁtting, a simple relation to describe localization precision was derived. The authors considered two limiting cases where each data point was weighted by a noise term in the least-squares ﬁtting procedure. When a large number

N of signal photons are collected, the data is considered photon shot-noise limited. In the limit of small N, when the signal level is close to the background level b, the image is considered background noise limited. The theoretical limiting precision for one spatial dimension, x, was obtained by considering each of these limits independently, and then approximating the regime of intermediate N by linear interpolation:

s2 + a2/12 8π s4 b2 (∆x)2 = + , (3.1) h i N a2 N2 where s is the size of the diffraction-limited spot, in this case represented by the standard deviation of a Gaussian PSF; a is the pixel size; and the angle brackets indicate the average over many measurements. This prediction was compared with an experimentally measured precision using immobile fluorescent beads, as well as with Monte-Carlo generated data. The direct Gaussian fitting method was used to localize particles via a simplified least-squares fit that did not account for noise. The functional form of the precision, Equation 3.1, was in agreement with experiment over a range of detected photon numbers and pixel sizes. However, the measured precision was systematically

44 30% higher than the predicted precisions, over the range of signal levels and pixel sizes tested. This discrepancy was attributed to three sources: interpolation between the limit of high and low signal photon counts, the truncation of higher-order terms in 1/N when deriving Equation 3.1, and an implicit assumption of vanishing pixel size. The precision of the algorithm might have also been improved by incorporating noise models into the

Gaussian fit used for localization. Although it contains many simplifications, Equation 3.1 is nevertheless a useful relation, explicitly highlighting factors that limit the precision of feature finding in experimental applications.

In a more general treatment, Ober et al. have used a Fisher information matrix to determine the limit of localization precision for single-molecule microscopy [63]. The

Fisher information matrix plays a central role in the analysis of estimation algorithms. Its inverse is the so-called Cramer-Rao´ lower bound for the variance of any estimation that has a mean equal to the true value; that is, for any unbiased estimator. The use of a Fisher information matrix is a particularly powerful assessment because its prediction depends only on the statistical model of data generation. For a diffraction-limited spot in an image, elements of the information matrix depend only on the PSF and noise models. The inverse of the Fisher information matrix is independent of the particular localization procedure. However, importantly, Ober et al. assumed the localization procedure would provide an unbiased estimate of the object location. This assumption is not justiﬁed for all localization routines, as was shown in [61] and by Gao and Kilfoil [64]. The ultimate limit of localization precision, as a function of the emission wavelength λem, photon

45 emission rate A, objective lens numerical aperture NA, and acquisition time t, was:

λem (∆x)2 = (∆y)2 = , (3.2) h i h i 2π NA√γ At where γ is the optical system efficiency, defined as the fraction of photons leaving the object that reach the detector. This is a fundamental limit of localization and does not include the effects of pixelation or noise. When these effects are included, the relation is much more complicated, and the limiting precision depends on the particular imaging conditions and detector specifications. Knowledge of the expected intensity distribution for a diffraction-limited spot is used in feature localization routines to obtain the spatial coordinates of an object with high precision and accuracy. This knowledge can also be used to segregate two spots that are closely spaced. Thomann et al. [65] developed a method for automatic detection of diffraction-limited spots separated by less than the Rayleigh limit in three-dimensional image stacks. The key assumption in this work is that each spot detected in an image is comprised of a finite number of superimposed PSF’s. Under this assumption, a fitting procedure that minimizes χ2 values for so-called “candidate mixture models” was developed. The candidate mixture models were superpositions of Gaussian functions, and the free parameters in the fitting routine were: the number of Gaussian kernels, the centre position and amplitudes of each Gaussian, and a constant background value. The standard deviation of each Gaussian function was constrained to a value predicted from the model of the true PSF, i.e. the radius of the Airy disk in x and y, given by equation

2.3, and the axial-resolution limit in the z direction, given by equation 2.5. The ﬁtting

46 routine was initialized with a small number of kernels, and a sophisticated procedure determined whether n or n + 1 kernels were necessary, based on statistical signiﬁcance. Using simulated data, Thomann et al. [65] found that their algorithm can resolve points at sub-Rayleigh separation. The localization accuracy approached the nanometer range for signal-to-noise greater than 12 dB, and was sub-20 nm for lower signal-to- noise ratios. High accuracy was only maintained for points separated by at least the Rayleigh limit, and depended on the relative brightness of the two points. Spots of equal brightness could be distinguished at a distance of half the Rayleigh limit. For cases where the spots differ in brightness, the resolution limit decreased by up to 50 %. To construct trajectories, Thomann et al. implemented a common tracking algorithm that minimizes the sum of weighted squared displacements between successive frames for all localized points. Because they attempted to track spots that became separated by distances less than the Rayleigh limit in one or more frames, it was possible for two spots to merge, or for a spot that appeared to represent a single feature to split into two. The tracking algorithm considered all such scenarios, and minimization was based on all possible separation or fusion events. However, the expected number of spots was always known, simplifying the treatment of merging or separation events. Dense systems that have a large but unknown number of particles require more sophisticated tracking algorithms to obtain the most useful information from single particle tracking.

For systems with a large number of particles, given the particle positions in each frame of a time-series of images, the most accurate method of forming trajectories is to construct all possible particle paths within constraints of the expected behaviour and over the entire duration of the image sequence. The best solution is then the largest

47 non-conflicting ensemble of paths. This method is termed “multiple hypothesis testing” and is the optimal solution; however, it is extremely computationally intensive, even for small ensembles of particles. Two recently published works [66, 67] proposed algorithms that were approximations to the multiple hypothesis testing solution. The algorithms were designed for dense systems in two-dimensions, where an individual particle may disappear or appear by moving out of or into the focal plane, or when two or more particles approach one another to within a distance smaller than the Rayleigh limit. Both methods detected particles using an iterative method. In a given image, an initial set of particles was detected by searching for local maxima and subsequently fitting the intensity maxima with Gaussian functions. To detect particles whose intensity maxima may have been obscured by neighbours of higher intensity, the approximate PSF of each particle detected in the initial pass was then subtracted from the image. This process, termed “deflation,” was applied iteratively with a localization routine to detect all particles. Serge´ et al. [66] developed a single particle tracking tool which they termed

“multiple-target tracing”. The construction of particle trajectories relied on a minimum hypothesis for the underlying dynamics. The authors assumed each particle to be tracked undergoes Brownian motion with a specific diffusion coefficient. To test the algorithm, quantum dots were used as fluorescent markers. Quantum dots require special consideration in single particle tracking applications because of their “blinking” property. Depending on the timescale of on/off events, an individual particle may disappear for an entire time frame, or have a reduced intensity for an individual frame.

For each localized particle, the algorithm searched for the corresponding particle

48 detected in the subsequent frame, taking into consideration the effects of blinking.

Possible trajectories were constrained by the dynamical model assumed, deﬁning a “reconnection domain” for each particle. In dense systems, the reconnection domains were permitted to overlap. When this occured, the algorithm evaluated the probability for each possible combination, including blinking. Evaluation of the probabilities was dependent on several frames of a given trajectory. The algorithm was also applied to observe the dynamics of a membrane bound protein. These authors were able to develop a global picture of the dynamics of the membrane protien, observing evidence of both diffusion and conﬁnement and these results were used as a proof of principle of their method.

In a related work, Jaqaman et al. [67] developed a single particle tracking algorithm using the framework of the “linear assignment problem”. The authors considered the particular scenarios of two particles that could merge physically or a single particle that could split into two. After robust localization, tracking was performed in two steps. First, particles detected in consecutive frames were linked into trajectory segments. Trajectory segments were formed in an initial pass by minimizing particle displacements between frames, with the condition of a one-to-one particle correspondence in each frame. A second step linked trajectory segments, capturing merging and splitting events. In the linear assignment framework, each potential trajectory was characterized by a cost matrix. Each element in the cost matrix was assigned a speciﬁc weighting term for each possible event. Solving the linear assignment problem is the minimization of the sum of all costs. The cost functions must be tailored to the speciﬁc tracking application, and therefore this requires a level of physical intuition of the process under investigation.

49 Costs for a speciﬁc event can be functions of the imaging conditions, such as intensities.

For example, the likelihood of a merging event was determined by comparing the sum of each of the intensities of the tracks to be merged with the intensity of the merged track, or the converse for splitting events. As one demonstration of the merits of their particular tracking implementations, Jaqaman et al. investigated the dynamics of a transmembrane receptor protein known to form aggregates on the membrane, although details of the aggregation are not fully understood. The ability of their algorithm to resolve merging and splitting events enabled them to study how individual receptors associate and disassociate with the aggregates. 3.2 Automated feature ﬁnding in a cell population

In this thesis work, the goal of feature finding and tracking was to investigate budding yeast mitotic spindle dynamics at a high temporal resolution, over the entire cell cycle. Experiments were performed at a temperature of 25 ◦C. At this temperature the budding yeast cell cycle is 2 hours. Photobleaching restricts the window of time ∼ that a single cell can be observed to 5 min. Thus, to capture spindle dynamics over the ∼ entire cell division, populations of unsynchronized cells were observed, with each cell in a field-of-view acting as a sample of a specific temporal window of the cell cycle. For this and for statistical reasons, it is necessary to collect data for a large number of cells in order to develop a complete picture of the division. The feature finding algorithms are semi-automated to maximize the number of cells that can be analyzed with a minimal amount of subjective manual input. Both the spindle poles and cell periphery were fluorescently labelled. The spindle pole trajectories and mother and bud cavity surfaces were reconstructed from fluorescence data.

50 x

r 2 z

Figure 3–1: The coordinate system for spindle poles as deﬁned by the cell periphery for budded cells. The z coordinate is positive in the mother cavity and negative in the bud.

The ﬁnal result of pole and surface ﬁnding is an ensemble of spindle pole trajectories, with each trajectory described in a coordinate system relevant to the cell division for that cell. The geometry resulting from this analysis for a single cell is plotted in

Figure 3–1. In this section, the algorithms used for extracting spindle pole dynamics and cell surface morphology are described. This speciﬁc implementation was originally developed in the Kilfoil lab by Dr. Vincent Pelletier and further extended in this thesis work. All implementations were written in MATLAB (Mathworks, Natick MA). 3.2.1 Three-dimensional tracking of point-like features

As was stated in Section 2.2, knowledge of the expected intensity distribution for a diffraction-limited spot can be used to automatically localize ﬂuorescently labelled objects that are smaller than the diffraction-limit. The general principle of the localization routine is depicted in Figure 3–2. By ﬁtting the full intensity distribution of the object to a three-dimensional Gaussian function, the centroid of the spot can be determined to

51 sub-pixel resolution in all three dimensions. To construct spindle pole trajectories from

Figure 3–2: The general principle of point localization. The intensity distribution of a diffraction-limited spot is fit to a three-dimensional Gaussian function. Localized spindle poles are shown on the right. a time series of confocal fluorescence image stacks, an automated three-dimensional feature-finding and tracking algorithm was used. The algorithm was originally developed by Maria Kilfoil and Yongxiang Gao for applications in colloidal systems [64], and adapted for application in living cells by Maria Kilfoil and Vincent Pelletier. Feature-

ﬁnding locates the centroids of the spindle pole intensity distributions, termed features, to sub-pixel resolution in a series of successive confocal image stacks acquired at controlled time intervals. The tracking algorithm links localized features in sequential image stacks and outputs a time series of coordinates for each feature. The resolution of the three- dimensional localization depends on the signal-to-noise ratio of the images, as well as the magnitude and direction of the displacement vector spanning the spindle poles, as will be discussed in detail in Section 4.2. For this study, the localization accuracy is between 10 and 40 nm.

52 The automated feature finding consists of three main steps: filtering, initial estimation of feature positions, and refinement of position estimates. To suppress image artifacts and noise, which exhibit high spatial frequencies (small length scale), the images are initially filtered using a three-dimensional band-pass Gaussian kernel. By choosing a spatial cut-off for the filter kernel close to the characteristic size of a diffraction-limited spot, high frequency noise is attenuated to a level far below that of true features. Once the image has been filtered, pixels that are local intensity maxima can be used as initial estimates for feature positions. To further distinguish real features from noise, the integrated intensity is calculated for a sub-volume surrounding each of the local maxima. Features for which the integrated intensity is below a cut-off value are excluded from further processing. At this point, the feature position estimates have a spatial resolution equal to the pixel size, typically 180 nm inx ˆ andy ˆ and 300 nm inz ˆ.

To refine the feature position, a three-dimensional Gaussian function is fit to the intensity distribution within the sub-volume surrounding the initial estimate, by non-linear least squares. The final sub-pixel estimation of the feature location is obtained by a second iteration of Gaussian fitting. The second iteration involves first shifting the sub-volume so that it is distributed around the centroid of the initial Gaussian fit, folled by a second least squares Gaussian function fit, using the outputs of the initial fit as parameter estimates for the second fit.

The band-pass length scale, the dimensions of the sub-volume used for fitting, the integrated intensity cut-off value, and a characteristic size of the object used to define a region around the candidate feature in which a second feature cannot be located are defined by the user. The last parameter is adjusted to allow or reject any overlap between

53 two or more features of interest. These four parameters are set to optimize the fidelity of feature localization across a large number of cells contained in a single field-of-view. Since photobleaching degrades the signal of a given feature as it is observed over time, the integrated intensity cut-off is automatically reduced by a user-defined constant factor for each timepoint in the series of confocal stacks.

The purpose of tracking algorithms is to connect the feature positions localized in consecutive frames into a time series. The algorithm used here is based on an implementation of Crocker et al. [68] for two dimensional data, that was later extended to three-dimensions by Maria Kilfoil and Naama Gal. Feature coordinates are linked in time by minimizing the total displacement of each of the identiﬁed features between successive frames. The search for corresponding features in consecutive frames is constrained to a maximum likely displacement over the timescale between frames.

This constraint is termed the cut-off distance in the tracking algorithm. Accurate and robust tracking is obtained by tailoring the timescale of image acquisition and the cut- off distance to the particular dynamics of the features being tracked. In particular, it is essential that each of the features of interest move less than the typical separation between features during the time interval between frames. A spindle pole may disappear when the intensity distributions of the two poles overlap closely. For closely-spaced features, ﬂuctuations in their positions between time-steps can move the features to a separation at which they cannot be resolved individually. High ﬁdelity tracking is ensured by applying successive iterations of the tracking algorithm to each data set. Initially, strict constraints are applied to each candidate trajectory, and with each iteration the constraints are relaxed. In the initial pass, only

54 the candidate trajectories that persist for the entire time-series are accepted by the algorithm and the cut-off distance is set to its optimal, conservative value. In each successive pass, features that have been tracked are removed from the data set, and the algorithm is reapplied. During each iteration, features are permitted to skip one or more non-consecutive frames, with the trajectory persisting when the feature reappears. For each subsequent iteration, the cut-off distance is held constant and the constraint on trajectory length reduced. This method of successive iterations ensures that the majority of trajectories over a cell population are captured, and results in data sets biased for trajectories that persist to long times [64]. 3.2.2 Automated surface ﬁnding

The cell periphery can be visualized using fluorescent reporters that localize to the cell cortex. If the reporter is distributed uniformly over the entire surface, the fluorescence intensity distribution in a confocal image stack can be used to reconstruct the cell surface. Key features of the surface morphology that are important to cell division, such as the bud neck in budding yeast, may then be localized. In the case of budding yeast, both the mother and bud cavity shapes are closely approximated by ellipsoids. The surface reconstruction procedure described here was originally developed in the Kilfoil group by Dr. Vincent Pelletier. Surface reconstruction provides the best estimate of the true cell surface from the intensity distribution in the confocal image stack. Noise and imaging artifacts are attenuated in the confocal stack by first spatially filtering the image stack with a Gaussian kernel, as described above for the spindle poles.

In this case, the characteristic size of the ﬁlter corresponds to the surface thickness.

55 However, because of the complex topography of the surface compared to that of a single diffraction-limited spot, the choice of filtering parameters is not as straightforward as for the poles. Generally, it is necessary to iterate through many different values to obtain the optimum filter parameters for a single stack of images. At this point, the filtered image stack is reduced to a binary image by application of an intensity threshold. The thresholding procedure is designed to maximize the proportion of the true surface included in further processing, while minimizing clusters of intensity not contiguous with the surface as well as continuous regions of intensity spanning opposite sides of the cavity. Small unconnected clusters that survive the threshold are removed by setting a minimum size constraint on sets of connected voxels in the binary image. Typically, the true surface data is comprised of thousands of connected voxels, while small clusters span only a few hundred, providing an effective means for this method of rejection. A thinning operation is then applied to the binary image, which reduces the surviving data to the minimum set of connected points that conserve the general shape. Thinning is performed using the bwmorph tool in MATLAB’s Image Processing Toolbox. The algorithm can only be applied to two-dimensional images; therefore, in the implementation developed here, thinning is performed individually on each (x,y)-plane in the z-stack. Near the equator of the cell, the image of the surface resembles a ring; however, the intensity distributions in (x,y)-planes at the top and bottom of the cell resemble gently curved, filled surfaces. Thinning tends to reduce these surfaces to a single point. To correct for this, the binary image is also sliced vertically to produce sets of (x,z)-planes and (y,z)-planes, which are processed with the thinning algorithm. The

56 minimal set of points that describes the surface is then deﬁned to be all of the points that appear in the intersection of at least two of the (x,y), (x,z), and (y,z) sets. The resulting three-dimensional “thinned” version of the threshold image stack is composed of discrete surface points centred approximately in the middle of the original thick surface appearing in the image stack. A single focal plane of processed surface data, following the ﬁltering, thesholding, and thinning steps, is displayed in Figure 3–3.

2 μm 2 μm

Figure 3–3: On the left is a central confocal plane of the processed surface data after ﬁltering. On the right, light grey pixels are the data after applying the intensity threshold, white pixels are the result of thinning.

The simple shape of the budding yeast cell cavity is exploited by fitting a general ellipsoid to the surface. Ellipsoid fitting may be carried out on either the filtered image stacks or the filtered and thinned image stacks. The results of both types of fits were compared in the development of the algorithm. A non-linear least-squares fitting algorithm is used to minimize the distance between each data point describing the surface and the surface of a general ellipsoid. Two iterations of fitting are used to obtain the best parameters. In the first iteration, the mother and bud cavities are fit simultaneously to two ellipsoids, each with variable centre position (x0,y0,z0), semi- major axes lengths (axˆ,byˆ,czˆ), and orientations of ellipsoidx ˆ axis with respect to the

57 imaging frame x axis, deﬁned by the angle φ. This comprises a total of 14 parameters that must be simultaneously optimized by the non-linear least-squares algorithm. The non-linear least-squares minimization routine requires an initial estimate of ellipsoid

fit parameters. The estimation of all 14 required parameters for each cell analyzed has been semi-automated through the use of a modified version of MATLAB’s ginput function. As part of this process, an image of the sum of all focal planes along the z direction, superimposed with the filtered data points, is displayed for the user. The use of ginput facilitates user entry of in-plane parameter estimates by clicking with the mouse cursor at the location of the centre and subsequently at two points on the periphery of each cavity, which together define the magnitude and orientation of each of the semi- major axis (axˆ,byˆ) initial estimates. Initial estimates of z0 and magnitude of czˆ are set automatically by locating the data points with the maximum and minimum z coordinates in the neighbourhood of the (x0,y0) estimates. The result of the initial fit is two ellipsoids that are reasonable estimates of the mother and bud cavities. This initial result is used to segregate data points into subsets defining the mother cavity and the bud cavity. In a second iteration, an ellipsoid is fit to each of these subsets independently, using the output of the first iteration for initial parameter estimates. When fitting is performed on data that has not been thinned, each surface point is weighted in the fitting algorithm by its intensity. The results of surface ellipsoid fitting for a single cell are shown in Figure 3–5. Mother and bud cavity volumes are calculated directly from the ellipsoid fits via

V = 4πabc/3. The neck position is estimated as the point on the line joining the ellipsoid centres that is equidistant to both ellipsoid surfaces. Data points that “belong” to the

58 10

30 Y (pixels) 40

10 20 30 40 X (pixels)

Figure 3–4: A modified version of MATLAB’s ginput function is used for initial estimates of surface parameters. All data is projected into a single z-plane and the thinned data is overlayed in blue on a filtered image. neck are segregated based on their distance from the estimated neck position. The orientation of the neck plane is taken to be perpendicular to the line joining the ellipsoid centres, which is termed the mother-bud axis. The neck area is estimated by fitting an ellipse to the data points belonging to the neck. This ellipse is depicted in green in Figure 3–5. As can be shown, this automated procedure results in faithful ellipsoid fitting to the data describing the two cavities, as well as faithful description of the orientation and position of the neck plane, between the mother and bud cavities.

The accuracy of feature finding and tracking in cells is influenced by many factors. Signal-to-noise must be maximized with well defined intensity distributions for each feature, spanning multiple voxels so that the Nyquist criterion is satisfied. The hardware, signal-to-noise, and sampling requirements limit the achievable temporal resolution. Investigation of fast dynamics in the cell requires some trade-off between temporal and

59 Figure 3–5: Ellipsoid fits to mother and bud cavities are shown in blue. The grey-scaled dots are the filtered fluorescent signal. Red points are the filtered and thinned data. The bud neck is shown as a green ring, with the view here side on to the bud neck plane. spatial resolution. A detailed assessment of the factors influencing the performance of localization algorithms to determine the limits of the spatial resolution greatly aids in the design and interpretation of feature finding experiments.

60 CHAPTER 4 Evaluating the accuracy of feature localization

4.1 Characterizing the imaging system

A simulation of diffraction-limited spots that is relevant to the experimental conditions requires knowledge of the true PSF of the optical system used for the experiment. The theoretical description of a confocal PSF is only relevant in the ideal aberration-free scenario, and it is necessary to measure the real PSF of the microscope. Measurement of the PSF will reveal any aberrations that may be present in the optical system. The image of an object that is smaller than the diffraction-limit closely approxi- mates the true PSF of the microscope. In an optical system, a diffraction-limited object acts as a low-pass spatial ﬁlter with a high cut-off frequency relative to that of the OTF.

The cut-off frequency increases as the inverse of the object size. The true PSF is the image of an infinitesimal object, i.e. one with an infinite bandwidth. However, in practical imaging conditions, background and noise signals often mask any contribution of high frequency components to the PSF, so that the effect of a small but non-zero object size on the PSF is negligible. Therefore subresolution fluorescent beads may be used to measure the PSF. Details of the microscope system used are described in Section 5.1. To measure the true PSF of the microscope, sub-resolution 0.100 µm diameter beads (TetraSpec, Molecular Probes, Eugene, OR) were imaged using the CFP (434 nm) and GFP (491

61 nm) laser lines at 60 ms exposure time. Coverslips were coated with 0.1 mg/ml poly-

D-lysine to adhere beads to the coverslip. The beads remained immobile throughout the experiment. The concentration of beads was sufficiently low that they were spaced far apart, and only 15 to 20 beads were visible in a given field-of-view (90nm 90nm). × For five different sets of beads, i.e. fields-of-view, a set of confocal image stacks was acquired at a spacing of 0.2 µm between confocal planes. Typically, 40 stacks were acquired per set. To estimate the noise-free PSF, the intensity distribution of each bead in each of the image stacks was localized to sub-pixel resolution using the feature finding algorithm described in Chapter 3. All intensity distributions for a single bead in a set were then averaged with their centroids aligned. Since noise is not temporally or spatially coherent, this averaging dramatically reduces the noise signal. xy and xz profiles of the PSF obtained in this manner for the GFP channel are shown in Figure 4–1. The measured point spread function was symmetric about the x, y and

1 1

0.8 0.8 0.6

0.6 ° ° I/I I/I 0.4 0.4 0.2 0.2 0 0 1.5 0.9 3 1.8 0.9 4.5 0.9 2.7 1.8 1.8 y (µm) 2.7 z (µm) 6 2.7 3.6 3.6 x (µm) 7.5 3.6 x (µm)

Figure 4–1: The centre xy and xz planes of the three-dimensional point spread function measured by acquiring and averaging images of 100 nm diameter ﬂuorescently labelled beads.

62 z axes, and showed no significant optical aberrations. It has been shown that the point spread function of a spinning disk microscope can be approximated accurately by a Gaussian distribution [69], which is much more tractable mathematically than the Airy function. The measured PSF’s were fit to a three-dimensional Gaussian distribution: 2 2 2 (x µx) (y µy) (z µz) − − − Aexp 2 2 + 2 2 2 2 , to obtain a model PSF for simulations. Reported h− σx σy σz i values are the mean, plus or minus the standard deviation, of the Gaussian fit parameters, over all five different fields-of-view. The standard deviations of the Gaussian PSF obtained for the GFP channel were 1.105 0.007 pixels, 1.104 0.018 pixels, and ± ± 2.838 0.016 pixels in the x, y, and z directions, respectively. For the CFP channel, ± the measured standard deviations of the Gaussian PSF were 1.002 0.012 pixels in x, ± 0.992 0.011 pixels in y, and 2.522 0.020 pixels in z. ± ± To simulate all aspects of image acquisition realistically, the background or bias signal and the readout noise from the CCD camera were also measured. The readout noise cannot properly be determined by calculating a histogram for all pixel intensities across a single image because “fixed-pattern” noise sources add additional spread to the intensity distribution. “Fixed-pattern” noise arises from pixel-to-pixel variations in dark current, as well as non-uniformities in photoresponse [59].

The background signal in the absence of ﬁxed-pattern noise was obtained by using the intensity recorded for a single pixel over many measurements. To accurately reproduce the background present when imaging live cells, a sample of lactate medium was placed in the microscope during the measurements. The objective lens was focused to a point in the media just beyond the coverslip, and 3000 images of a single focal plane were acquired using the shortest possible exposure time ( 30 ms). This was performed ∼

63 under three different conditions. In the dark ﬁeld condition, all shutters were open but no illumination source was used. For CFP and GFP illumination conditions, the 434 nm and 491 nm excitation lasers were respectively turned on. Since no ﬂorescent reporter is present in the sample, the measured intensity distributions represent an estimate of the detector background signal including scattered light effects, as well as CCD readout and dark current noise. Because the relevant units for describing noise in the photon detector are electrons, the signal is converted from ADU’s to electrons in the CCD by multiplying by the CCD gain factor of 5.8 electrons/ADU and dividing by the EM gain factor of

1200.

Detector noise from dark current is expected to vary as σD = √DT, where D is the dark current and T is the exposure time. For the camera used, D = 0.01 electrons/pixel/sec [60], so σD = 0.02 electrons, and the noise from dark current is expected to be negligible. Readout noise should be Gaussian distributed with a variance of 1 ∼ electron [60]. Histograms were calculated for a single pixel at different regions over the field-of-view for each illumination condition, and these are shown in Figure 4–2. For the dark field and 491 nm illumination, no appreciable change was observed between the histograms from different regions across the field of view. The intensity distribution for each of these conditions follows a Gaussian profile, but with a small exponential tail. This exponential tail may be explained by multiplicative noise in the gain-register of the EM-CCD, which will produce an exponential distribution of pulse heights with many small pulses and few large ones [59]. For the 434 nm illumination, the intensity distribution near the edge of the field was drastically different from that observed near the centre of the field. As shown in Figure 4–2, near the outside edge of the field-of-view,

64 0.5 Dark Field 0.45 489 nm 0.4 440 nm center of field 440 nm edge of field 0.35

0.3

0.25

0.2

0.15 Normalized Counts (N=3000) 0.1

0.05

0 8 8.5 9 9.5 10 10.5 11 11.5 12 Signal (electrons)

Figure 4–2: Images of the background were collected for a number of different conditions. At each condition 3000 identical images were acquired. From each set of images the values for a few pixels over the set, at different locations in the field of view, are collected and converted from digital units to electrons. the detector response follows a Gaussian profile with an exponential tail; however, over the centre of the field-of-view, the intensity spread resembles a Rayleigh distribution. The origin of the Rayleigh distribution around the centre of the field-of-view is not clear; it may represent excessive scattering at this wavelength, a problem with the filter sets, or a combination of different effects. To isolate potential sources of this anomalous behaviour, a number of different CFP emission filters have been tested to identify whether

65 they are sources, and the 434 nm wavelength laser line alignment was checked. Ulti- mately, the cause of the aberrant background signal observed for the CFP illumination has not yet been isolated.

4.2 Simulation of spindle poles

In order to determine the accuracy and precision associated with the feature localization algorithms used in this thesis work, the localization algorithms were applied to computer generated confocal images of SPB’s. The image of a diffraction-limited spot was constructed by ﬁrst generating a sphere of 150 nm diameter in a high resolution object space. The object space containing the sphere is a three-dimensional grid of voxels with dimensions of 29 nm in x and y, and 50 nm in z. The choice of object space pixel size is a compromise between approximation of a continuous space, which requires small pixel sizes, and available computer memory, which limits the size of arrays that can be convolved in a practical time frame. A Gaussian approximation to the microscope

PSF was also generated in the object space using the parameters found earlier. The Gaussian function used to generate the PSF was not normalized and the amplitude was set to 1 unit. Convolution of the object and PSF was performed by: fast Fourier transform of each of the arrays, multiplication in the frequency domain, and ﬁnally, inverse Fourier transformation.

Convolution of the object with the PSF results in the intensity distribution of a diffraction-limited spot in the continuous image space, the magnitude of which is interpreted in terms of a number of photons. To generate a realistic image that is comparable to experimentally acquired images, it is necessary to sample the continuous image space intensity distribution into a discrete distribution collected by the CCD

66 m

290 nm 290 nm 500 nm Object PSF Image

Figure 4–3: Convolution of the object and PSF was performed by fast Fourier transform and multiplication in Fourier space. The object and PSF arrays were zero-padded prior to Fourier transfomation to avoid circular convolution. detector. To do this, the image space intensity distribution is resampled to a pixel grid, with each pixel representing an individual CCD element. To simulate the spindle poles, two diffraction-limited spots are produced in the image. The CCD grid has voxel dimensions of 174 nm in x and y, and 300 nm in z, to mimic the experimental conditions. To construct the CCD voxel distribution, the continuous image space distribution was sampled at a number of focal planes in the z-stack. Finite depth-of-ﬁeld was explicitly accounted for at each z-slice by weighting the image space matrix by a Gaussian function with an amplitude of 1 and a standard deviation of 2.8 pixels, centred on the particular z-slice. The intensity distribution in the focal plane was then constructed by integrating the depth-of-ﬁeld-weighted image space matrix along z. Each CCD pixel in the focal plane was formed by integrating the focal plane over square regions corresponding to the CCD pixels. The resulting matrix represents the distribution of photons impingent upon each of the CCD elements for each focal plane collected. This matrix is adjusted by a scaling factor to control the signal-to-noise. Sub-pixel displacements of the object were

67 produced by shifting the image space grid by one or more of its elements, with respect to the overlying CCD grid.

image detector

500 nm 11 μm 1 μm

RS/N = 11 dB Sample image Add noise and to pixel space scale intensity

Figure 4–4: Image is sampled from the continuous image space to discrete pixel space on the CCD grid.

Detector noise and background offset were added to each image stack at intensity levels corresponding to experimental values. To represent the photon detection process accurately, Poisson noise was simulated for each pixel by random sampling from a Poisson distribution with a mean of N, where N represents the number of photons impinging on each pixel. The intensity units, which represent numbers of photons, were converted to grey scale units by multiplying by the quantum efﬁciency and by the gain factors of the CCD, and dividing by the photo-electron/ADU conversion factor provided in the camera manual [60]. These values could then be directly converted to integers, that represent grey scale levels in the camera’s dynamic range (16 bit). The signal-to-noise ratio was calculated as the mean pixel value above background of the diffraction-limited spot divided by the standard deviation of the background pixel levels. The mean signal

2 level µsignal, mean background level µbkgrd, and variance σbkgrnd were calculated in terms

68 of numbers of photons, and the signal-to-noise ratio is expressed as a decibel quantity:

µsignal µbkgrd RS/N = 10log10 − . (4.1) σbkgrd

For each simulation run, an image was generated at a given signal level and with a given displacement between the two diffraction-limited spots in a prescribed direction in three-dimensional space. Noise was added to the image and the feature localization algorithm was applied. At each signal level and spot separation tested, 500 iterations of feature ﬁnding were performed with a new noise proﬁle generated for each iteration. All simulations were performed in MATLAB. Feature localization parameters were set to the values used for live cell trials. The feature coordinates, as determined by the localization algorithm, were then compared to the real coordinates to obtain an estimate of the localization error over a range of signal-to-noise values and spot separations.

Knowledge of the true spot locations allows for an estimate of both precision and accuracy of a given measurement. Plots that display the deviation between the actual position and the localized position, for a number of point-to-point displacements and signal-to-noise levels, are shown in Figure 4–5. At a given signal-to-noise ratio, the localization error is highly dependent on both the magnitude and orientation of the pole- to-pole displacement vector. Dependence on orientation occurs because of the large pixel spacing in z compared to the imaging plane pixel size, and because of the anisotropy of the PSF. When the points are sufﬁciently separated such that there is no overlap between their respective PSF’s, there is always a larger error in the localized z-coordinate than that in the x or y coordinates. When the intensity distributions from the two points begin to overlap, the localization error is most strongly affected along the direction of

69 80 60 σ σ x 55 x 70 σ σ y y 50 σ σ z 60 z 45 40 50 L= 2.25 µm L= 3.9 µm 35 40 30 25 30 20

Localization error (nm) 20 Localization error (nm) 15 10 10 5 0 0 4 6 8 10 12 14 16 4 6 8 10 12 14 16 R (dB) R (dB) S/N S/N

50 50 σ σ x 45 x 45 σ σ y y 40 σ 40 σ z z 35 35 L= 1.72 µm 30 L= 1.05 µm 30 25 25 20 20 15 15 Localization error (nm) 10 10 Localization accuracy (nm) 5 5 0 0 4 6 8 10 12 14 16 4 6 8 10 12 14 16 R (dB) R (dB) S/N S/N

Figure 4–5: Point localization error versus signal-to-noise for a number of different separations and displacement vector orientations. In the top two figures, the displacement vector between the two points is oriented in thez ˆ direction. In the bottom two figures, the displacement vector is oriented in thex ˆ direction, which is equivalent to they ˆ direction. Each data point represents the mean localization error over 500 iterations. The error bars are the variance, which represents the precision of the measurement. the point-to-point displacement. In order to quantify carefully how overlap of intensity distributions affects the fidelity of position localization, an arbitrary critical separation between points can be defined for the Gaussian PSF model as the distance along the point-to-point displacement vector at which the PSF magnitude is equal to 1% of its peak

70 value. For a displacement vector oriented along direction d, this critical separation is r2 = 2σ 2 ln(0.01), where σ is the standard deviation of the Gaussian PSF. When the crit − d d features are separated by a distance of less than 2rcrit for their particular displacement vector, overlap between the intensity distributions from each feature greatly affects the Gaussian curve fit. Since the centroid reported by the Gaussian curve fit is taken as the feature position, intensity overlap compromises the fidelity of position localization. Figure 4–6 shows the dependence of localization accuracy on the displacement vector between the points. The error is smallest when the displacement is entirely in a single plane, and largest for displacements entirely in the axial direction. The values of rcrit for the two scenarios were 5.16 and 3.34 pixels for a displacement vector oriented along the zˆ andx ˆ directions respectively. The PSF is circularly symmetric about thez ˆ axis, thus rcrit is the same for all displacement orientations when the two points lie within the same imaging plane. In Figure 4–6 it is apparent that the localization error is not a monotonic function of the point-to-point separation. As the displacement decreases below 2rcrit, there is a large increase in the localization error. At some separation below 2rcrit, the error decreases, and then upon further reduction in separation rapidly diverges as the two features become difficult to distinguish. In order to examine this effect more closely, the measured point-to-point separation is compared to the true point-to-point separation in

Figure 4–7. Figure 4–7 shows that the initial increase in error, at 7-9 pixels separation, is due ∼ to a systematic over-estimation of the point-to-point separation. The overlap between the intensity distributions forces the least-squares ﬁtting routine to localize the centroid

71 50 zˆ xˆ xˆ +y ˆ 40

Localization error (nm) 10

0 4 6 8 10 12 14 Separation (pixels)

Figure 4–6: Error in point localization versus displacement between points, for a number of different displacement vector orientations. Each data point is the average of 500 iterations. Error bars are the variance, which reports the precision of the measurement. RS/N is 10.5 dB. of each intensity distribution away from the overlap region, resulting in overestimation of the point-to-point separation. As overlap increases, the centroids of each feature are forced back towards the overlap region, initially decreasing the estimated separation, until signiﬁcant overlap has occurred, at 5 pixels, and the two spots are difﬁcult to ∼ distinguish. To investigate this behaviour of the localization error at close separations in greater detail, a simulation was performed wherein the features were localized in only one- dimension. Three-dimensional images of point features were created as described above.

The features were displaced from each other along thez ˆ direction by a known amount

72 11 Separation inx ˆ +y ˆ 10 Separation inz ˆ 9 Separation inx ˆ

Measured Separation (pixels) 4

3 4 6 8 10 Actual Separation (pixels)

Figure 4–7: Point-to-point separation measured by the localization algorithm plotted against the true separation for a number of different displacement vector orientations. The dashed line indicates where measured separation equals the actual separation, for reference. Each data point is the average of 500 iterations. RS/N is 10.5 dB. and the z coordinate of each point was localized by fitting each intensity distribution independently to a one-dimensional Gaussian function. The centroids were displaced at fractional pixel values along the z -axis and the integer pixel locations of the centroids were used as initial parameter estimates for the Gaussian fits. Since the intensity distribution is influenced by pixelation, noise, overlap, and filtering, simulations were performed with and without noise and under varying filter kernel parameters. The zˆ-direction was chosen because localization of the z-coordinate has the largest error for closely spaced points. To localize features with limited spatial filtering, i.e. the extreme values of f ∞ and f = 1, the noise had to be reduced to unrealistically small c → c 73 values, mimicking unattainably perfect imaging conditions. For suitable comparison, all simulations in Figure 4–8 were performed with R ∞. S/N →

70 f → ∞ c 60 f = 1 c f = 1/2 50 c f = 1/3 c 40

20 Localization error (nm) 10

0 5 6 7 8 9 10 11 12 Separation (pixels)

Figure 4–8: The error in one-dimensional localization for a displacement vector oriented in thez ˆ direction. Results for different values of the band-pass filter cut-off frequencies fc, are shown. As fc increases, the effect of the filter is reduced, and fc ∞ represents the case of no band-pass filtering. Each data point is the average of 500 iterations.→

These results reveal the underlying source of the small overestimation of the separations at slight overlapping of the intensity distributions. Filtering removes high spatial frequency components from the image, predominantly noise; however, the operation also smooths components of the real intensity distribution that vary rapidly in space. For closely spaced features, the filtering operation modifies the summed intensity distribution of the two features such that the distance between the two is overestimated by fitting each feature independently to a Gaussian function. The one-dimensional

74 simulations verify the behaviour of the localization error found in the three-dimensional simulations. Plots of the intensity distributions before and after ﬁltering for two different spacings are shown in Figure 4–9. The origin of the behaviour of the localization error

1 1

0.8 0.8

0.6 0.6 ° ° I/I I/I 0.4 0.4

0.2 0.2

0 0 10 15 20 25 30 10 15 20 25 z (pixels) z (pixels)

Figure 4–9: The intensity distributions along the axial, orz ˆ direction, for closely spaced diffraction-limited spots. In the left figure, the spots are centred at 16.25 and 23.75 pixels. In the right figure, the spots are centred at 16.42 and 21.75 pixels. Light grey dashed lines represent the intensity distributions from the individual points. Solid black lines represent the sum of the individual intensity distributions. Dashed red lines are the filtered intensity distribution. The effect of the band-pass filter can be seen in the region between the two spots and on the outside edges of the spots. for features that are closely spaced is verified upon examining the effect of the band-pass filter. The effect of the filter is most dramatic when the distance between points is smaller than 2rcrit but larger than an undetermined value greater than rcrit. For this range of separations, spatial band-pass filtering attenuates the summed intensity distribution in the region between the two points. The effect of this is that the mean squared error in the curve-fit is minimized by locating the centroid of each Gaussian in the direction away from the region between the two points. At some separation close to rcrit, the effect of filtering in the overlap region becomes negligible compared to the effect of the overlap

75 itself. This is the point of crossover of the measured separation with the actual separation in Figure 4–7. For automated feature detection with live cell image data, spatial band- pass filtering is necessary. Real data is subject to many sources of noise, and accurate automated segregation of the real features of interest from noise is not possible without spatial filtering. Therefore, the systematic bias demonstrated here will always contribute to the localization error for closely spaced features. Although small, this effect must be considered in the interpretation of the results of a feature finding experiment. 4.2.1 Propagation of localization error

a b The error for any arbitrary quantity ε(xi ,xi ) that has been calculated from the set of coordinates for both features xa,xb can be estimated by propagating the errors { i i } associated with each of the coordinates. For notational simplicity, the coordinates representing the positions of each of the features are written as the set x , where { i} i = 1,2,...,6 . The errors for all the coordinates are written generally as σ ,σ , and { } { xi xi j } the covariance terms σxi j are given by:

1 σ 2 = (x µ ) x µ . (4.2) xi j N ∑ i − xi j − x j i, j The set of values µ is the set of most probable estimates of the true coordinates, { xi } which is exactly known for the simulations. Frequently when uncertainties are estimated, the covariant terms are assumed negligible. However, for parameters obtained from ﬁtting a curve to data, the covariant terms can contribute signiﬁcantly to uncertainties

[70]. The appropriate relation for estimation of uncertainty in ε(xi) is:

2 2 2 ∂ε ∂ε σε ∑σxi + ∑ σxi j . (4.3) ≃ i ∂xi i= j ∂xi j 6

76 The error in point-to-point distance can be determined directly during a simulation by comparing the real separation with the measured separation. This error can also be determined from the uncertainties of each coordinate via equation 4.3, since L2 = 3 a b 2 ∑ (xi x j ) . To test this, a simulation was performed with the displacement vector i=1 − between the points oriented in an arbitrary direction inx ˆ,y ˆ,andz ˆ, and the error in L determined over a range of separations. The results of this simulation are displayed in Figure 4–10.

60 estimated error 50 estimated error, covariance =0 measured error

30 (nm) L σ 20

0 0.8 1 1.2 1.4 1.6 1.8 2 L (µm)

Figure 4–10: Error in point-to-point separation determined directly from simulation, compared with the error calculated by propagating through the errors in coordinate localization. Light grey dashed line is propagated error neglecting covariance terms. Black dashed line is the error propagated with equation 4.3. RS/N for this simulation is 10.5 dB.

It can be seen that for large separations, the covariant terms are negligible. As the two features approach one another, the covariant terms become signiﬁcant and they must

77 be included for an accurate estimate of the error. By simulating point-to-point separation trajectories for each of the directions in the set xˆ,zˆ,xˆ+ yˆ+ zˆ the form of all σ ,σ , { } { xi xi j } can be obtained. Using relation 4.3, the errors for all quantities determined from feature

ﬁnding can be estimated. 4.3 Simulation of the cell periphery

To obtain an estimate for the error associated with the surface reconstruction described in Section 3.2.2, the budding yeast cell surface was simulated. The exact distribution of fluorophores in the cell cortex is not known, therefore for simulation, it was assumed that fluorophore labelled G-proteins distribute randomly over the cell surface. The mother and bud cavities were modelled by generating a set of points randomly distributed over the surface of an ellipsoid in the object space. The three-dimensional image of a cell surface was constructed in the same manner as diffraction limited spots were generated in Section 4.2, by convolution in the object space and subsequent spatial sampling of the detector space. Cavities were created with fixed dimensions: a = 1.71 µm, b = 1.8 µm, c = 1.58 µm for the mother, and a = 1.35 µm, b = 1.17 µm, c = 1.12 µm for the bud. These dimensions are typical of cavity dimensions observed in experiment. Voxel dimensions of the object space grid were set to 90 nm in x and y, and 75 nm in z. The detector space grid voxel dimensions were set to 180 nm in x and y, and 300 nm in z to reproduce the pixel size in experiments. Object space grid size, and consequently mother and bud cavity dimensions, were limited by available computer memory and by the practical duration of a single simulation run. To simulate experimental data collection using the GFP channel, a background with

Gaussian-exponential distributed noise was added to generated images. To account for

78 photon shot-noise, Poisson distributed noise was simulated for each pixel in the images.

Prior to adding noise, image intensity distributions were multiplied by a scale factor to adjust signal levels. The signal level is the mean pixel value over the periphery of the ellipsoids.

object image detector

1 μm

1 μm 2 μm RS/N = 12 dB

Convolve with PSF Resample to pixel space and add noise.

Figure 4–11: Fluorophore distribution on the cell periphery was modelled as a uniform random distribution over the surface of an ellipsoid. Image intensity distributions were constructed from the object in the same manner as for diffraction-limited spots.

At each of a number of values of signal-to-noise tested, 500 iterations of surface reconstruction were performed. Ellipsoids were fit to simulated data that had been processed by either band-pass filtering alone, or band-pass filtering combined with the thinning operations discussed in Section 3.2.2. Filter parameters and threshold values were similar to those used for real data. As the signal level was increased during the simulation, the threshold level was adjusted accordingly. To estimate accuracy, the mean volume and centroid position determined from surface reconstruction for each ellipsoid at a given signal-to-noise was compared with the known volume and centroid simulated.

In Figure 4–12, the error of the mother and bud volume measurements are plotted as

79 a function of signal-to-noise levels. The error bars in Figure 4–12 are the standard

6 Mother, thinned 5.5 Mother, filter Bud, thinned 5 Bud, filter 4.5 ) 3 4 m

µ 3.5 ( 3

volume 2.5 σ 2 1.5 1 0.5 0 6 7 8 9 10 11 12 R (dB) S/N

Figure 4–12: Error in volumes obtained from ellipsoid fits versus signal-to-noise. Ellip- soids were fit to data that had been band-pass filtered alone as well as to data that had been band-pass filtered and thinned. Each data point is the average over 500 iterations; error bars are the variance. deviation in volume determinations over 500 iterations, and represent the precision with which the volume may be measured. The thinned data provide a more accurate estimation of volumes at all signal-to-noise levels tested for both the mother and bud cavities. While the absolute error displayed in Figure 4–12 is smaller for the bud cavity volume than for the mother cavity volume, the relative error at R > 10 is 17% for S/N ∼ the bud and 8.5% for the mother. ∼ At all RS/N values tested, the measured centroid values had a large variance compared with the accuracy of each estimate. In Figure 4–13 top, the localization error

80 can be seen to increase slightly as signal-to-noise is increased; however, the variance decreases with increasing signal-to-noise as shown in Figure 4–13 bottom. This indicates a systematic bias in estimates of the centre of the cell obtained from ellipsoid fits to the cell periphery. Since the effect is greatest in the smaller cavity, it is likely that this bias is a result of overlapping intensities in the cavity interior. Overlap increases the apparent thickness of the cell periphery towards the centre of the cavity. The extent of overlap depends on both the shape and size of the cavity. Areas of high curvature will have a larger overlap. In a small cavity, the overlap may be sufficiently large so that bright pixels belonging to the real cell periphery cannot be distinguished from neighbouring bright pixels whose intensities represent the summed contribution from the overlapping tails of many PSF’s. In contrast to the error associated with volume determination, the estimate of the true ellipsoid centre provided by fitting to the filtered data is better in terms of both accuracy and precision. The previous simulation is a rough estimate of the error in surface reconstruction, and likely represents a lower bound. Although ellipsoids are a suitable qualitative representation of the yeast periphery, it is not clear that the shape of the budding yeast surface, constrained under a microscope slide, is best described by an ellipsoid.

Furthermore, the actual distribution of fluorophores in the cell cortex is not known, and may differ between the bud cavity, neck, and mother cavity peripheries. A yet more detailed treatment of the error associated with surface reconstruction should investigate the effect of both the form and density of fluorophore distribution, and in that case could allow for differences between the two cavities. Limitations imposed by the object space grid size are a potential problem with this simulation. It is possible that the fluorophore

200 Mother, thinned Bud, filter 150 Bud, thinned Mother, filter

100

Centroid localization error (nm) −50

6 7 8 9 10 11 12 R (dB) S/N

100 Mother, thinned 90 Mother, filter 80 Bud, filter Bud, thinned 70 60 50 40 30 20 10 Centroid localization precision (nm) 0 6 7 8 9 10 11 12 R (dB) S/N

Figure 4–13: The error in centroid estimation from ellipsoid fits to the cell periphery for data that has been filtered alone as well as data that has been filtered and thinned. At top, centroid localization accuracy is plotted versus signal-to-noise. Each data point is the average of 500 iterations. Error bars in the top figure are the variance. At bottom, magnitude of localization precision as determined from the variance of 500 iterations is plotted versus signal-to-noise.

82 distribution simulated in the object space (90 nm) may be too thick. Given more time and computer memory, the object space grid could be adjusted to create a finer fluorophore distribution at the cortex. Another potential area of extension of this simulation is the handling of the bud neck region. The mother and bud cavities were simulated as two continuous ellipsoids, ignoring the neck opening, a region of special relevance to the morphology of the cell during division. Future work should investigate the influence, if any, of the morphology and fluorphore distribution within the bud neck region on the fidelity of surface reconstruction.

The simulations of diffraction-limited spots and cell periphery presented in this chapter are of greater value than merely a means to obtain an error estimate. These novel methods can be extended to any object geometry and used to investigate the effects of the majority of experimental parameters associated with image acquisition, an invaluable aid for both the design of new experiments and the interpretation of results.

83 CHAPTER 5 In vivo study of spindle positioning

Budding yeast have been used as a model system in which to study mitotic spindle dynamics. Budding yeast strains expressing fluorescent reporters for both the SPB’s and the cell cortex have been constructed by Dr. Susi Kaitna, in the Vogel lab, in the department of Biology at McGill University. Using the methodology described in Chapter 3, fluorescently-labelled SPB’s were localized and tracked, and cell surfaces were reconstructed from fluorescence confocal images for large numbers of wild-type cells. The methods were also applied to large numbers of cells with mitotic kinesin deletions. 5.1 Microscopy and sample preparation

Live unsynchronized cell populations expressing the SPB reporter spc42-CFP and the surface reporter gpa1-EGFP were imaged by confocal fluorescence microscopy. SPB’s labelled with spc42-CFP are point-like features which were localized and tracked using the methodology described in Chapter 3 Section 3.2.1. Surface data was analyzed using the surface fitting described in Chapter 3 Section 3.2.2. All yeast strains used were derived from strain BY4741 [71]. Media used for yeast culture for most cells (rich medium YPAD and low fluorescence lactate medium) and yeast genetic manipulations are described in [72]. Synthetic complete (SC)medium used for cell culturing and imaging to study the effect of the culturing medium is described in Section 5.2.3.

84 For microscopy, yeast strains were incubated in YPAD (or SC) at 25◦C until early logarithmic phase (0.3 0.05 normalized optical density at 600 nm). 1 ml of suspended ± cells were pelleted for 15 s at 500 g, washed twice with 1ml of lactate medium, then × re-suspended by vortexing for 30 s in 100 µl of lactate medium and imaged immediately. All measurements were made using a spinning disk confocal microscope system mounted on a Leica DMI6000 B, equipped with a customized Yokogawa CSU10 confocal head, Nanodrive piezo stage (ASI), 434 nm and 491 nm solid state lasers (Spectral), and a 63X 1.4 NA objective lens. Images were acquired with a C9100-13 BT

EM/CCD camera (Hamamatsu Corp.) using Metamorph software (Molecular Devices).

The pixel size in x and y was 0.178 µm. A time series of image stacks was collected for live cell data. Typically, 15 focal planes per stack at 0.3 µm spacing in z were acquired, which span the entirety of the cells in the axial direction. GFP and CFP excitation laser exposure times were 70 ms. A confocal stack of images with CFP laser line (434 nm) and filters was acquired for every timepoint. At every tenth timepoint, CFP acquisition was immediately followed by the collection of another image stack with GFP laser line (491 nm) and filters. The acquisition rate was held at 5 seconds per stack and data was acquired for 5 minutes before the signal degraded by photobleaching beyond an acceptable level for accurate and completely reliable feature finding. Images of living cells were acquired at 25 ◦C. For all analysis, data was collected from cell populations representing four independently-derived yeast strains. MATLAB (The Mathworks, Natick, MA) was used for all offline analysis. Chromatic shift was measured and corrected for prior to combining results from imaging cells using two different wavelengths. The chromatic shift in x, y, and z between

85 the GFP and CFP acquisition channels was determined by imaging approximately 200

2.5 µm-diameter InSpeck beads (Invitrogen) immobilized on a poly-K-coated cover glass. The intensity distribution of each of the beads in a field-of-view was fit to a three- dimensional Gaussian function to find the centroid of every bead in each channel. GFP and CFP centroid positions were then compared for each of the immobilized beads.

Figure 5–1 displays the measured chromatic shift for all beads, plotted as a histogram. The z position in the CFP channel was systematically higher than it was in the GFP channel by 221nm for the 63X objective lens. The in-plane chromatic shift between the

CFP and GFP channels was 64nm and 31nm, in x and y dimensions, respectively. This − is corrected for in all succeeding analysis.

80 x; mean 64 nm y; mean −31 nm z; mean 221 nm 60

40 count

0 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Shift between CFP and GFP (µm)

Figure 5–1: Chromatic shift between GFP and CFP fluorescence channels, measured by comparing the localized coordinates from green fluorescence and cyan fluorescence channels for immobile beads.

86 5.2 Results

5.2.1 Spindle orientation and positioning in wild-type cells

The SPB’s were tracked in time in three-dimensions. For each timepoint from this data, the displacement between the two poles was computed to construct the time evolution of spindle lengths for all cells. For cells in S-phase and early M-phase, the separation between SPB’s corresponds to the spindle length. For cells in late M-Phase, the elongated spindles may have some curvature, so that the SPB separation does not report the true curvilinear length of the spindle. Within a given ﬁeld-of-view, only cells for which two spindle poles could be detected within our resolution limits were analyzed.

Signal-to-noise levels were typically between 9 and 11 dB at the initial timepoint, and close to 4 dB at the ﬁnal timepoint. At the top left of Figure 5–2, the z-stack projection of a representative confocal ﬁeld-of-view is displayed. The GFP and CFP channels are overlaid in false colour. At the top right of Figure 5–2, spindle pole trajectories within two cells are shown. The bottom of Figure 5–2 displays the instantaneous spindle separation as a function of time for the full wild-type data set, comprised of 160 cells. Each cell is plotted as a different colour/symbol combination. The instantaneous SPB separations plotted as a function of time in Figure 5–2 are clustered in two regions: those spindles less than 2 µm long (on average 1.5 µm), and a population of longer ∼ spindles with separations spanning 3 µm to 8 µm. Since asynchronous populations ∼ of cells were observed, the gap between the two populations could arise from a rapid phase of spindle elongation so rapid that it could not be captured at the 5 second time resolution. However, literaure reported values for the rapid phase of anaphase B in budding yeast are much slower than this, and the transition from short to long spindle,

87 1 μm

10 μm 1 μm

2 SPB separation (μm) 0 0 50 100 150 200 250 300 time (s)

Figure 5–2: The time evolution of spindle pole separations for a characteristic ﬁeld-of- view. At the top left, a z-stack projection of a representative ﬁeld-of-view is displayed with GFP (green) and CFP (red) channels overlaid in false colour. The spindle pole trajectories for two cells are shown at top right. At bottom, the instantaneous spindle separations are plotted as a function of time for the full wild-type data set of 160 cells. Each cell is plotted as a different colour/symbol combination. even if instantaneous, was not observed for a single cell in the data set of 160 cells. This discrepancy with the expected biology remained a troubling puzzle. Investigations of the effect of the imaging media have revealed that the presence of this gap is a result of a

88 photoxic effect that occurs when the cells are imaged in lactate medium. The photoxicity arrests anaphase cells. These ﬁndings are presented below in section 5.2.3. For most cells observed, the average SPB separation was stable in time, and any

fluctuations in the SPB separations were well above the resolution limits of feature finding of 10 40 nm since the signal to noise level over the imaging timescale remained − at a level such that tracking fidelity was not compromised (Chapter 4). In future work, signal-to-noise level monitoring and the error propagation methods developed in the previous chapter may be integrated directly into the tracking algorithms so that quantitatively exact statements regarding the feature finding accuracy could be made for each timepoint, if desired.

Combining surface reconstruction with spindle pole tracking allowed for characterization of spindle dynamics in a coordinate system deﬁned by the plane of cell division.

Using the mother and bud cavity fits for each cell, the neck plane position and orientation was calculated for t = 0; that is, the initial timepoint; and assumed fixed over the five minute time window. This assumption was made to minimize computation time, after

ﬁrst having been veriﬁed by comparing the neck plane orientation and position at the initial timepoint with the position after 5 minutes. Each cell is monitored for a limited

ﬁve minute time window; this observation period is of a shorter timescale than any major changes in the bud development in any single cell. Together, the spindle position and cavity sizes were used to identify the mother and bud cavities in each cell, as follows. Initially, the positions of both SPB’s with respect to each cavity were determined. If both SPB’s were contained in a single cavity, then that cavity was deﬁned as the mother. If

89 each cavity contained a single SPB, then the volumes of the two cavities were compared, and the largest cavity was defined as the mother. The orientation of the neck plane normal vector~n was defined by the vector spanning from the mother cavity centre to the bud cavity centre, which defines the mother-bud axis. A spindle vector~s was defined as the vector spanning from the pole distal to the bud to the pole proximal to the bud. Spindle orientation is then parametrized by the angle Ψ between~s and~n, given by~s ~n = ~s ~n cosΨ, where 0 < ψ < 90 . As a · | || | ◦ cell enters anaphase, the spindle aligns with the neck, and Ψ approaches 0. For each cell, the mean spindle length and orientation were calculated over the five minute observation window. The results are plotted in Figure 5–3 for 160 wild-type cells studied. Each data point represents a single wild-type cell at some random point in the cell cycle. Figure 5–3 shows that cells with short spindles have a larger range of orientational freedom than do long spindles. In fact, in these cells, short spindles span all possible values of Ψ from 0 to 90 , while in cells with spindles between 3 µm and 8 µm h i ◦ ∼ in length, the spindles are constrained to within 20◦of alignment with the mother-bud axis. This is consistent with the expected behaviour of the mitotic spindle in budding yeast. Spindle pole separation and spindle orientation parametrize two kinetically distinct processes: spindle elongation and spindle positioning. Spindle elongation is thought to depend largely on motor activity and MT dynamics within the spindle itself, while spindle positioning is thought to depend on the action of motors on astral microtubules [4, 28, 29, 45]. To properly distribute genetic material, the spindle must be aligned with the mother-bud axis prior to chromosome segregation, which occurs as the spindle elongates in anaphase.

90 90

(degrees) 40 〈Ψ〉 30

0 0 2 4 6 8 〈L〉 (µm)

Figure 5–3: Mean spindle length versus mean spindle orientation over a ﬁve minute time window. Each data point represents an individual wild-type cell.

SPB separation fluctuations and orientational fluctuations were calculated for each cell over the time series. At left in Figure 5–4, the spindle length fluctuations are plotted versus mean spindle length. Below the figure, the distribution of mean spindle lengths is plotted. At right in Figure 5–4, the spindle orientational fluctuations are plotted versus mean spindle orientation, with the distribution of mean spindle orientations plotted below. At the left of Figure 5–4, it is apparent that cells with short spindles (< 3µm) undergo much larger fluctuations in spindle pole separation than do those with larger spindles. Initially, a maximum spindle length criterion was used to segregate the pre-anaphase population from the anaphase population, motivated by the gap observed between cells with short and long spindles in the time-dependent

91 0.1 200 ) 2 ) 2 0.08

m 150 µ ( 〉 0.06 2 (degrees 〉 L

2 100 〈∆

− 0.04 2 〈∆Ψ 〉 − L

2 50

〈∆ 0.02 〈∆Ψ〉 0 0 0 2 4 6 8 0 15 30 45 60 75 90 〈L〉 (µm) 〈Ψ〉 (degrees) 0 0 10 10 20 Counts Counts 30 20

Figure 5–4: At the top left, fluctuations in spindle length versus the mean spindle length are plotted. At the top right, the fluctuations in spindle orientation are plotted versus mean spindle orientation. Cells in the pre-anaphase population are plotted as circles and those in the anaphase population are plotted as triangles. The distributions of mean spindle length and mean spindle orientation are also plotted for each population: red bars are pre-anaphase population, blue bars the anaphase population. instantaneous spindle lengths shown in Figure 5–2. This cut-off length was chosen by first constructing a histogram of the mean spindle length distribution for the full population. The combined distribution is well represented by a bi-modal Gaussian, as shown in the lower left histogram in Figure 5–4. The position between the two peaks where the bi-modal Gaussian is a minimum was taken as the cut-off distance. For this wild-type cell population, the cut-off distance was 2.3 µm.

92 One hypothesis for spindles with large fluctuations in Ψ is that these cells may be exploring spindle configurations until proper alignment with the mother-bud axis is obtained. When the pre-anaphase and anaphase populations are segregated in the plot at the right of Figure 5–4, it can be seen that only the short, pre-anaphase spindles undergo large orientational fluctuations, while the longer, anaphase spindles maintain relatively stable alignment with the mother-bud axis. The distance of the spindle poles from the neck plane was computed by constructing the vectors rxn, representing the displacement of each SPB located at point xn from the neck centre at point x . The distance, given by d = ~n ~r , is positive for all x in the 0 · xn n mother cavity, and negative for all xn in the bud cavity. The pole closest to or inside the bud is termed the bud-proximal pole. When d for the bud-proximal pole is plotted h i in Figure 5–5 against the mean distance between the two SPB’s, the distribution of pre-anaphase and anaphase spindles becomes apparent. The population of cells in pre- anaphase have d > 0, while cells in anaphase have d < 0. As shown in Figure 5–5, h i h i the parameter d for the bud-proximal pole serves as a complementary metric to spindle h i length for segregating anaphase from pre-anaphase cell populations. Pre-anaphase cells have spindles shorter than the cut-off length, and have both poles in the mother: d > 0. h i Anaphase cells have spindles of greater separation than the cut-off length, and have one pole residing in the bud: d < 0. These two criteria are not entirely universal. In a small h i proportion of wild-type cells (< 3%), the spindle is aligned along the mother-bud axis and one pole passes into the bud before the spindle has elongated; this corresponds to d < 0 while L is less than the cut-off separation. In the mean spindle length and h i h i orientation distributions plotted at the bottom of Figure 5–4, anaphase and pre-anaphase

93 3

0 m) µ ( 〉 d

〈 −1

−2

−3

−4 0 2 4 6 8 10 〈L〉 (µm)

Figure 5–5: Mean spindle length versus distance of the bud-proximal pole to the neck plane. The neck plane is located at d =0. The bud-proximal pole is in the mother cavity for all positive values of d , and inh thei bud cavity for all negative values of d . h i h i populations (shown as blue and red respectively) were segregated based on the position of poles with respect to the bud. Cell cavity volumes were calculated from the ﬁtted ellipsoid parameters using the thinned data. At left in Figure 5–6, the bud cavity volume is plotted versus the mother cavity volume. Again each data point corresponds to a separate wild-type cell.

As expected, the bud cavity was always smaller than the mother cavity [3]. At right in Figure 5–6, the mother and bud volume distributions over the cell population have been plotted. Each of the mother and bud volume distributions were ﬁt to a Gaussian function.

The mean and standard deviation of mother cavity volumes over the population was

94 50 15

10 )

3 30 m µ ( bud

V 20

Number of cells 5

0 0 0 10 20 30 40 50 0 10 20 30 40 50 3 V (µm3) Volume (µm ) mother

Figure 5–6: Mother and bud cavity volumes determined from ellipsoid ﬁts. At left: each bud volume is plotted against the volume of its mother cavity. The line at Vbud = Vmother is drawn for reference. At right: distribution of mother and bud cavity volumes across the population.

3 3 µmother = 30.46 µm and σmother = 4.33 µm . The mean and standard deviation of bud 3 3 cavity volumes was µbud = 10.91 µm and σbud = 6.05 µm . Since cell growth is coordinated with the cell cycle [3], the bud-to-mother volume ratio serves as a complementary metric for cell cycle progression that is apparently spatially-independent from the spindle. In Figure 5–7, this ratio is plotted against the mean SBP separation. For pre-anaphase cells, the bud-to-mother volume ratio is roughly proportional to the mean SPB separation, indicating a direct dependency between cell growth and mitotic progression. For cells that have passed through the metaphase to anaphase transition, there is no apparent relation between mother and bud cavity volumes. The coupling between spindle length and cell growth appears to be a fundamental distinction between anaphase and pre-anaphase cell populations.

95 0.8

0.7

0.6

0.5

mother 0.4 /V bud

V 0.3

0.2

0.1

0 0 2 4 6 8 〈L〉 (µm)

Figure 5–7: Mean spindle length versus bud-to-mother volume ratio for wild-type population of 160 cells.

5.2.2 Spindle orientation and positioning in mitotic kinesin motor mutants

To investigate the specific contributions of mitotic motors to spindle dynamics, genetic perturbations of the mitotic motors were induced in budding yeast, and analysis of spindle dynamics was carried out. This was performed for two populations with deletion of the mitotic motors kip1 and cin8, respectively. Motor deletion strains were constructed in the Vogel lab by Dr. Susi Kaitana. The spindle pole dynamics in the metrics quantified above did not differ appreciably from wild-type for both kip1∆ and cin8∆ motor deletion strains. The spindle length fluctuations versus mean spindle length and mean spindle orientation versus mean spindle length for these motor deletion strains are displayed in Figure 5–8, superimposed

96 on the results for wild-type cells. At left in Figure 5–8, it can be seen that the pre-

0.1 90 WT kip1∆ cin8∆ 80 cin8∆ kip1∆ WT 0.08 70 ) 2 60 m

µ 0.06 ( 2

〉 50 L 〈∆ (degrees) − 40 〉

2 0.04 〈Ψ〉 L

〈∆ 30

0.02 20 10

0 0 0 2 4 6 8 10 0 2 4 6 8 〈L〉 (µm) 〈L〉 (µm)

Figure 5–8: Spindle dynamics in mitotic kinesin deletion strains. At left, mean spindle length versus spindle pole separation fluctuations. At right, mean spindle length versus mean spindle orientation. anaphase spindle fluctuations in the cin8∆ population appear suppressed relative to those for the wild-type and kip1∆ populations. The mean distance to the neck plane of the bud-proximal pole is plotted versus mean spindle length for the wild-type and motor deletion populations in Figure 5–9. The results for this metric of cell cycle progression did not differ appreciably between the wild-type and kip1∆ populations. However, for the cin8∆ population, the distinction between pre-anaphase and anaphase sub-populations based on d was not as clear as h i in wild-type cells. The cin8∆ population had many cells with a short spindle yet with only one pole in the mother cavity and the other having progressed into the bud. This difference from wild-type behaviour may be due to differences in spindle dynamics; however, interpretation is complicated because in motor deletion populations it is not possible to distinguish the mother and bud cavities unequivocally. For the wild-type cell population, definitions of the mother and bud cavities based on spindle position

97 3 kip1∆ 2 WT cin8∆ 1

0 m) µ

( −1 〉 d 〈 −2

−3

−4

−5 0 2 4 6 8 10 〈L〉 (µm)

Figure 5–9: Mean distance to neck plane versus mean spindle length for wild-type cells and for cells with mitotic kinesin deletions. The bud-proximal pole is in the mother cavity for all positive values of d , and in the bud cavity for all negative values of d . h i h i and cavity volumes were entirely consistent with each other. However, in the cin8∆ population, approximately 5% of the cells had both poles contained in the smaller cavity, and in 10% of the population, the two cavities were of near equal volumes. In these ∼ cases, it is possible that either the spindle had transversed entirely into the bud, which would be prevented in normal cell cycle progression, or the bud cavity had in fact grown larger than the mother cavity. Cell volumes for the motor deletion and wild-type populations are shown in

Figure 5–10. In the distribution of cavity volumes at right in Figure 5–10, it is clear that volumes in the cin8∆ population are larger, for both the mother and bud cavities, than in

98 20

15 90 10 kip1∆ 80 WT 5 Number of Cells 70 cin8∆ 0 0 20 40 60 80 Volume (µm3) 60 20 ) 3 15

m 50 µ ( 10

bud 40 V 5

30 Number of Cells 0 0 20 40 60 80 20 Volume (µm3) 20 10 15 0 0 20 40 60 80 10 V (µm3) mother 5 Number of Cells 0 0 20 40 60 80 Volume (µm3)

Figure 5–10: Distribution of mother and bud cell volumes for wild-type cells and for cells with mitotic kinesin deletions. The volumes were determined from ellipsoids that had been fit to filtered and thinned data. On the left, the bud volume is plotted against the mother volume. The line at Vbud = Vmother is drawn for reference. On the right, the distributions of mother and bud cavity volumes for each population are displayed. wild-type and kip1∆ populations. The distribution of cin8∆ volumes is also spread over a much larger range, blurring the distinction between the two cavities. The mean and standard deviations of mother and bud cavity volumes obtained from Gaussian fits to the distributions are tabulated below in table 5–1. Table 5–1: Mean µ and standard deviation σ of mother and bud volumes over a population of N cells for wild-type cells and for cells with mitotic kinesin deletions.

3 3 3 3 Population µmother (µm ) σmother (µm ) µbud (µm ) σbud (µm ) N wild-type 30.46 4.33 10.91 6.05 160 kip1∆ 27.94 5.14 11.05 6.30 199 cin8∆ 45.05 10.97 32.96 14.92 124

99 The most striking difference observed between wild-type and motor deletion populations are the large mother and bud cavity volumes for the cin8∆ population. This indicates a potential link between the control mechanism for chromosome segregation and that for cell morphology in the cell cycle. It is possible that the large volumes observed in the cin8∆ population were a result of a delay in the cell cycle for this population during a stage of cell growth. If chromosome segregation in the mitotic spindle is delayed, the cell may continue to produce and partition membrane and mass to the bud. The result of this is a bud much larger than normal, and successively, a second generation of cells that are larger than the ﬁrst generation. As the population develops, the cell volume is most probably still limited by the cell cycle control system. However, removal of the cin8 motor may perturb components of the control system, altering the coupling between chromosome segregation and cell morphology control. This can help to explain how the bud cavity may even become larger than the mother cavity. To test this hypothesis, some method that can unambiguously distinguish the mother cavity from bud cavity is necessary.

In Figure 5–11, the bud-to-mother volume ratio is plotted versus mean spindle length for the wild-type and motor deletion populations. The bud-to-mother volume ratio versus mean spindle length for the motor deletion populations follows the same trend as for wild-type cells. The bud-to-mother volume ratio is proportional to L for h i cells in pre-anaphase, while cells that have passed the metaphase to anaphase transition, there is no apparent relation between spindle length and bud-to-mother volume ratio. For the pre-anaphase cin8∆ population, it is possible that in some cells, the bud has grown larger than the mother, and the values of the bud-to-mother volume ratio greater than

100 2 kip1∆ WT cin8∆ 1.5

mother 1 /V bud V

0.5

0 0 2 4 6 8 10 〈L〉 (µm)

Figure 5–11: Ratio of bud-to-mother cavity volume versus mean spindle length for for wild-type cells and for cells with mitotic kinesin deletions. one, obtained with the deﬁnition outlined for wild-type cells in section 5.2.1, are robust.

However, it is also possible that both poles have passed into the mother and the true bud-to-mother volume ratio is never exceeds one, i.e. the mother is always larger than the bud.

5.2.3 Effect of imaging media on spindle dynamics

It was discovered that for yeast cells imaged in the lactate medium, exposure to CFP laser irradiation at 5 second intervals results in a photoxic effect which causes cells to arrest at anaphase. This effect was conﬁrmed by observing cells in different media types, and varying exposure times for prolonged periods under sealed coverslips.

101 Wild-type yeast strains expressing the spindle pole reporter spc42-CFP were prepared for microscopy as discussed above using lactate medium. A small population of asynchronous cells were imaged under a sealed coverslip for 30 minutes at 20 second exposure intervals. Six different fields of view were examined with approximately ten cells per field of view. No qualitative evidence of anaphase spindle elongation was observable in any of the cells. Another population of cells was cultured, washed and imaged in Synthetic Complete (SC) medium. Four different fields of view were examined with approximately ten cells per field of view. The cells were imaged for

30 minutes at 20 second exposure intervals. In this population, ﬁve cells displayed visible anaphase spindle elongation. Furthermore, the spc42 labelled spindle poles were observed to maintain signal intensity longer when imaged in SC media than in the lactate media, indicative of fewer oxygen free radicals being created from absorption of photons.

Following these initial qualitative studies, the spindle dynamics of cells incubated and imaged in SC medium were investigated in more detail, using the protocol for quantitative image acquisition as discussed for the prior studies. A population of unsynchronized wild-type cells expressing the spc42-CFP reporter was incubated, washed, and imaged in SC medium. To minimize photoxic effects, the acquisition interval was increased from 5 to 10 seconds and the total acquisition length increased to 20 min. A confocal stack was collected with the CFP laser line for every time point, with the CFP exposure time set to 70 ms. The measured spindle pole separation versus time obtained following analysis of the resulting images is plotted in ﬁgure 5–12. A representative subset of the total imaged population is displayed so that individual trajectories can be resolved across the entire imaging time. This plot of

102 9 8 m) µ 7 6 5 4 3 2

Spindle pole separation ( 1 0 0 200 400 600 800 1000 1200 time (s)

Figure 5–12: Spindle pole separation versus time for wild-type cells cultured and imaged in SC media. Results are displayed for 16 cells from a total population of 75 analyzed cells. spindle pole separation versus time shows a number of anaphase spindle elongation events, conﬁrming that the cells are passing through the cell cycle when imaged in SC medium. Compared with the cells imaged in lactate medium, cells imaged in SC medium show a larger range of dynamical behaviour. There are rapid elongation events corresponding to the cell passing through anaphase, and rapid decreases in spindle length from 7-8 µm to between 4 and 5 µm corresponding to the stage of spindle breakdown. Many spindles that are longer than 5 µm undergo rapid elongation and shrinking events. This may be due to spindle breakdown, or to a defect in anaphase spindle positioning.

103 Cells in which mitotic motor kip1∆ or cin8∆ have been deleted were also cultured and imaged in the SC medium. The spindle pole separation versus time for a subset of the total population imaged is plotted in ﬁgure 5–13. This ﬁgure shows many of the cells

8 9

7 8 m) m)

µ µ 7 6 6 5 5 4 4 3 3 2 2 Spinlde pole separation ( Spindle pole separation ( 1 1

0 0 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 time (s) time (s)

Figure 5–13: Spindle pole separation versus time for mitotic kinesin deletion strains cultured and imaged in SC media. On the left the spindle pole separation versus time is plotted for the kip1∆ strain, results are displayed for 18 cells from the total analyzed population of 51. On the right the spindle pole separation versus time is plotted for a subpopulation the cin8∆ strain, results are displayed for 20 cells from the total analyzed population of 80. passing through anaphase spindle elongation, as well as other cells undergoing spindle breakdown, in both the kip1 and cin8 deletion strains. Measured spindle pole length fluctuations versus mean spindle length for wild-type and, for comparison, the mitotic motor deletion strains, are plotted in figure 5–14. The spindle length fluctuations displayed in figure 5–14 do not differ appreciably between wild-type and mitotic motor mutant populations. In contrast to the results presented above for cells in lactate medium, cells with an average spindle length greater than 3

µm display a larger range of ﬂuctuations in spindle pole separation than do cells with average spindle length less than 3 µm. Consequently, the anaphase and pre-anaphase cell

104 0.14 cin8∆ 0.12 kip1∆ WT

) 0.1 2 m µ

( 0.08 2 〉 L 〈∆

− 0.06 〉 2 L

〈∆ 0.04

0.02

0 0 2 4 6 8 〈L〉 (µm)

Figure 5–14: Fluctuations in mean spindle length versus mean spindle length for cells cultured and imaged in SC media. populations are no longer clearly distinguished by their relative magnitudes of spindle pole separation ﬂuctuations.

Cells imaged in SC medium were observed to undergo stages of cell division that were not observed for cells imaged in lactate medium. Drastically different dynamical behaviour is observable over the imaging time across spindles in cells at different stages of their life-cycles. Moreover, over the longer time window of image acquisition, individual cells can exhibit different phases of spindle dynamics. Cells in pre-anaphase, with spindle lengths > 2 µm , generally maintain a near-constant spindle length over the imaging time. Cells entering anaphase display distinct regions of rapid and slower spindle elongation. Cells with spindles of lengths between 4 and 6 µm , likely at later

105 stages of anaphase, maintain a roughly constant spindle elongation rate which is less than that for the shorter spindles of cells in early anaphase. A large proportion of the spindles that are longer than 5 µm undergo rapid increases and decreases in length. These variations may be attributed to either defects in spindle elongation, or to the spindle breakdown stage. Accurate segregation of aberrant spindle behaviour from normal cell processes during spindle breakdown requires knowledge of the spindle size and position relative to the mother and bud cavities, achievable in cells in SC medium with both poles and surfaces labelled. This can be included in future measurements.

Over a 20 minute temporal window of total image acquisition for a given ﬁeld of view, a single spindle can exhibit many distinct dynamical phases. Therefore, future analysis of the spindle dynamics should carefully distinguish between these different phases for each cell analyzed, i.e. if such a longer temporal window is used, the average spindle length over the time window is no longer a meaningful metric of spindle pole separation for spindles that are growing. Conversely, distinguishing and assignment of dynamic phases may be done by manual segregation of individual traces, or by automated segregation based on biologically relevant criteria such as, for example, differences in rate of spindle pole separation between the rapid and slow phases of anaphase spindle elongation. The results presented here serve as a proof of principle of the methods developed to track spindle pole bodies and reconstruct budding yeast cell surfaces. Measuring the dynamics of the mitotic spindle in a coordinate space deﬁned by the plane of cell division, over a population of cells, reveals dynamical and morphological cues related to stages of the cell cycle not detectable by any other method. In a population

106 of asynchronous wild-type cells, imaged in lactate medium, the sub-populations of pre-anaphase and anaphase cells are clearly distinguished via metrics deﬁned in terms of both SPB dynamics and size of bud relative to the mother. In those cells, for wild-type cells, the bud size and spindle dynamical metrics were observed were observed to be internally consistent. In mitotic kinesin motor deletion cell populations, the distinction between anaphase and pre-anaphase cells based on spindle position and cell size metrics becomes blurred. Most striking is the effect of cin8 deletions on cell volume. It is expected that in wild-type cells imaged in SC medium, the sub-populations of pre- anaphase and anaphase cells can be equally clearly distinguished via metrics deﬁned in terms of both SPB dynamics and relative bud and mother sizes. It is hypothesized that the results for the cin8∆ population in lactate medium reveal a connection between the spindle dynamics and the partitioning of mass from the mother to its bud during cell cycle progression. This could not be investigated further with the current experimental setup as it is not possible to determine unequivocally which cavity is the mother and which is the bud.

Observation over the entire cell cycle in a controlled environment would enable accurate determination of the mother and bud cavities. If the cell progeny could be followed over the full course of their development, the origin of any trends in the cell morphology that appear in the population could be determined. Furthermore, gene expression may be controlled by exposing the cell population to small-molecule inducers that control promoters for speciﬁc genes of interest - here, kip1 and cin8. Monitoring the development of a cell population in a controlled environment would enable a detailed investigation of the relationship between chromosome segregation and cell size control.

107 Moreover, mechanisms of signal-transduction and gene-induction resulting from such signalling could be studied by performing similar small-molecule inducible-promoter control of signalling molecule gene expression.

108 CHAPTER 6 Extension to long-time dynamics: a microﬂuidic chamber

In this chapter, the construction of an apparatus for observing the development of a cell population in a controlled environment, starting from a small number of cells, is described. The criteria used to choose and design the device were as follows. To facilitate long term imaging and analysis of an entire population, dividing cells are best confined to a plane so that they do not “pile-up” on top of each other, nor have their growth inhibited in the lateral dimensions. The colony of cells must remain stationary over the full period of observation, so that individual cells and their progeny can be identified and monitored by quantitative metrics via imaging. Nutrients must be supplied and distributed evenly across the cell population to ensure unperturbed growth. To investigate the response of cells to time-dependent stimuli, the apparatus must have some means of switching the growth media. Integration of the apparatus with a confocal fluorescence microscope would permit investigation of the response of cells to time- dependent perturbations, as well as analysis of the relationship between spindle dynamics in a single cell and the behaviour of the population as a whole, using the methods descibed in Chapter 3.

Following the design of Charvin et al. [73], a microfluidic apparatus that uses a dialysis membrane to separate the cell colony from media flow [73, 74] has been constructed. In this implementation, budding yeast cells are confined to grow between a

109 polydimethylsiloxane (PDMS)-coated glass coverslip and a cellulose dialysis membrane. Media ﬂow is directed over top of the membrane by a PDMS ﬂow chamber and distributed across the cell population by diffusion through the cellulose membrane.

The cellulose membrane ensures an even distribution of nutrients, and mechanically constrains cells to bud horizontally in a single plane [73, 74]. A schematic drawing of the device is displayed in Figure 6–1. The PDMS coating on the coverslip serves two

media in media out

PDMS !ow cell Cellulose di"usion of media membrane

PDMS layer Glass coverslip

Objective lens

Figure 6–1: Schematic diagram of microfluidic chamber, adapted from [73]. Budding yeast are constrained between a cellulose membrane and PDMS coated coverslip. Media flow is directed over the top of the membrane and nutrients diffuse through the membrane. purposes: first, the PDMS is much softer than budding yeast cells which have a stiff outer cell wall [75, 76]; second, budding yeast cells adhere to the hydrophobic surface of the PDMS [75]. The PDMS coating and coverslip must have a total thickness less than the free working distance of the objective lens, which is 210 µm for the 63X 1.4 NA ∼ 110 objective lens used in this work. The media flow channel is 80 µm deep, 40 mm long, and 600 µm wide. The flow channel was connected to media reservoirs, which are sterile syringes, using plastic tubing, and media flow is controlled with a syringe pump.

6.1 Device fabrication

The ﬂow channel was constructed from PDMS using standard soft lithography methods [75, 77, 78]. A silicon/photoresist master mould was fabricated in the McGill microfabrication facilities. The ﬂow channel layout was drawn using computer-aided design (CAD) software and printed as a chrome photomask (Fineline Imaging). An

80 µm thick layer of SU-8 negative photoresist was spin-coated on a silicon wafer, which was then exposed to UV under the photo mask and subsequently developed in order to dissolve unexposed regions of photoresist. The resulting structure served as a master mould for casting PDMS devices. The surface of the silicon/photoresist master was treated with ﬂuorinated silanes to prevent irreversible bonding to PDMS.

To cast the ﬂow cell, the PDMS elastomer and curing agent were mixed in the manufacture-recommended 10:1 ratio and poured over the mould. All air bubbles were removed by placing the PDMS mould into a vacuum for 1 hour prior to curing. The ∼ PDMS was cured at 70◦C for 1.5 hours. Inlet and outlet holes were bored with blunt-tip 18 gauge needles. PDMS-coated glass coverslips were fabricated by spin coating of

PDMS onto 22mm 50mm No. 1.5 (170 µm - 180 µm thickness) glass coverslips to × create a layer between 20 µm and 40 µm thick. Cellulose membranes were prepared from dialysis tubing (Sigma Aldrich, D9627), with pore size deﬁned by an upper molecular weight cut-off of 14 kDa. Following the preparation of Charvin et al. [73], ∼ the dialysis tubing was cut into pieces having the same dimensions as the coverslips, and

111 then soaked in distilled water for one hour to separate the two halves of the membrane.

The membrane pieces were then boiled in 2% w./v. sodium carbonate solution for 30 minutes to remove sulfur and glycerin residues. Finally, the membrane was boiled in TE buffer (pH 8) for 30 minutes. To assemble the device, a solution of cells (0.4 0.05 normalized optical density at ± 600 nm) was pipetted onto the PDMS-coated coverslip and the membrane was carefully laid over the cells. The wet membrane was allowed to dry for approximately 15 minutes before the flow chamber was placed on top. The condition of the membrane prior to placement of the flow chamber is critical. If the upper surface of the membrane was wet, the PDMS flow chamber did not seal with the membrane and the chamber leaked. If the membrane was allowed to dry for too long, it buckled, forming ridges across its surface that prevented the PDMS flow chamber from forming a seal with the membrane and resulted in leaks. Once assembled, the entire device was clamped lightly to a temperature controlled aluminum plate with a transparent acrylic cover. Temperature was controlled by heating the aluminum plate with two resistive elements. The microscope objective lens was also heated to control its temperature. The temperature of the flow chamber was monitored by a thermistor and held constant at 25 ◦C. Temperature could be varied and held at a stable value between 25◦C and 37◦C. The flow channel was connected to a media source and a sink by plastic tubing, and media flow was controlled by two syringe pumps. Electric control valves were used to switch between two sources, allowing for rapid change of media. Media flow rate, valve operation, and temperature were all controlled using MATLAB. A picture of the microfluidic assembly mounted on the stage of an inverted microscope is shown in Figure 6–2.

112 out !ow in !ow

!ow chamber

resistive heaters

Figure 6–2: Microﬂuidic apparatus assembly. The ﬂow chamber is mounted in a temperature controlled aluminum plate and held in place by a transparent acrylic clamp. Plastic tubing is used to connect to the inlet and outlet. The aluminum plate and microscope objective lens are heated with resistive heaters and temperature is monitored with a thermistor.

6.2 Preliminary tests

At present, a working device has been constructed and tested with live cells. The temperature control and media switching apparatus are operational. Live cells have been observed to proliferate for a period of up to 12 hours; however, no quantitative tests to measure the rate of cell growth have been performed to date.

113 CHAPTER 7 Conclusions and future directions

This thesis work focused on the development and characterization of tools to allow for the analysis of dynamical process in single cells. Algorithms designed to locate and track spindle pole bodies and the cell periphery of budding yeast, in a time-series of confocal fluorescence image stacks, were initially developed in the Kilfoil lab by Dr. Vincent Pelletier, in collaboration with Dr. Maria Kilfoil. In this thesis work, that software has been extended in its capability for automated large scale analysis, and to increase tracking robustness. The software has been applied to wild-type budding yeast cell populations and yeast cell populations lacking specific mitotic kinesin motor proteins. Moreover, a microfluidic chamber that will allow for investigation of the long time dynamics within individual cells and for observation of dynamical quantities during the development of a cell colony across many generations, has been constructed. Fitting ellipsoids to the budding yeast surface provides a direct measurement of cell morphology. Determination of the bud neck plane and orientation enables characterization of spindle pole dynamics in a coordinate frame relevant to the cell division. By using a biologically relevant coordinate system, single cell spindle dynamics, measured in many different cells that may vary in size and shape in addition to orientation in the imaging plane, can be compared directly. Since quantitative single-cell biology depends on the obtaining of data from many similar cells, this method allows for such dynamical metrics to be measured and compared across different populations of cells.

114 In this thesis, I have investigated the precision and accuracy of both the point

finding and surface fitting algorithms in detail by simulating fluorescence images. For point features, it was found that the accuracy of feature finding depends on imaging conditions, the extent of image post processing, and the configuration of features with respect to each other within a three-dimensional image stack; and these dependencies were quantified. The error associated with point localization was found to be always greater in the axial (or,z ˆ) direction due to anisotropy in the point spread function and to the larger pixel spacing in z. The capacity of the feature finding algorithm to localize two closely-spaced features was affected by both the degree of overlap between intensity distributions for each feature and band-pass filtering. Filtering tends to attenuate intensities in the overlap region between two features that are separated by a distance less than a critical value 2rcrit but larger than some undetermined distance greater than rcrit. This attenuation results in a systematic overestimation of the distance between the features. As the feature separation approaches rcrit, overlap becomes significant and the distance between the two features is underestimated. When the features are separated by a distance less than rcrit, individual features are difficult to distinguish in the summed intensity distribution, and the localization error rapidly diverges with decreasing separation. For typical signal-to-noise levels between 6 and 11 dB, the three-dimensional localization error ranges between 10 nm and 40 nm, depending on the relative positions of the two features. Performance of the surface reconstruction was found to depend on signal-to-noise ratio and on the size of the cavity. Fitting an ellipsoid to data that has been processed by band-pass filtering and by a “thinning” operation provides a more accurate estimate of

115 cavity volumes than fitting to data that has been filtered only. Conversely, the ellipsoid centroid values were estimated with greater accuracy when fitting was performed on data that had been only band-pass filtered. Overlap between intensity distributions from the point-emitters that form the cavity surface results in an apparent thickening of the cavity wall interior in the direction of the centre of the cavity. This effect is greatest for small cavities and surfaces of high curvature. Surface reconstruction simulations could therefore be extended to explore the effects of varying cavity shape and fluorophore distribution, to investigate this effect. In addition to determining rigorous error estimates, image simulations are a complementary tool to feature finding and tracking algorithms, which can aid in the design of new experiments and the interpretation of results.

In this thesis, the utility of feature ﬁnding and surface reconstruction algorithms have been demonstrated in live cell trials. In wild-type cells, an apparent correlation between dynamical behaviour of the mitotic spindle and stages in the cell life cycle were observed. These dynamics could not have been observed without the three-dimensional feature ﬁnding and surface reconstruction algorithms presented in this thesis. Spindle pole separation, position with respect to the bud, and mother-to-bud volume ratio were found to be internally-consistent metrics for cell cycle progress in wild-type cells.

These same spindle dynamical metrics were measured in two mitotic kinesin motor deletion strains: kip1∆ and cin8∆. The observed single-cell spindle dynamics in these populations were found to be similar to those of wild-type strains. However, metrics based on spindle dynamics together with bud-to-mother volume ratio did not provide the clear distinction between anaphase and pre-anaphase populations that was observed for wild-type cells. The cin8∆ population was observed to have mother and bud cavity

116 volumes much greater than those for the wild-type population. This could indicate a decoupling of spindle positioning from cell growth in the cin8∆ population. However, this could not be investigated further in this work, as mother and bud cavities could not be identified unequivocally. The observed long time scale behaviour in wild-type cells was not as expected during normal cell division. In particular, no cells were observed to pass through anaphase. In response to this, the effect of phototoxicity was explored by varying the timescale of image acquisition as well as the imaging media. When synthetic complete media was used in place of lactate media and the rate of image acquisition was decreased from five to ten seconds per stack, a significant number of cells were observed to pass through anaphase spindle elongation. Compared with cells that were imaged in lactate media, cells imaged in synthetic complete media displayed a larger range of dynamical behaviour over the timescale of image acquisition. Rapid spindle elongation events, corresponding to the cell passing through anaphase, as well as rapid decreases in spindle length, corresponding to the stage of spindle breakdown, were observed. An accurate investigation of these processes requires knowledge of the size and position of the spindle relative to the mother and bud cavities. Future measurements may achieve this by imaging cells in synthetic completer medium with both poles and surfaces labelled. Modifying both the imaging media and timescale reduced photobleaching considerably, and resulted in cells that exhibit the expected spindle dynamical behaviour during division; therefore it is concluded that the faster imaging rate combined with the lactate medium caused the cells to arrest in anaphase.

117 To allow for monitoring of an entire population of cells in a controlled environment, and perform small molecule inducible-promoter control of gene expression, a microfluidic chamber was constructed. The chamber will allow for the measurement of spindle dynamics over the entire cell life cycle. The design enables the rapid switching of media in the flow cell so that the response of cells to time-dependent genetic or chemical perturbations can be investigated. Wild-type cells have been observed to proliferate normally for up to 12 hours in the device. The work presented in this thesis represents a starting point for a deeper investigation into the connections between biochemical signalling and mechanical cues during the cell cycle. The utility of the three dimensional feature finding and tracking algorithms presented here is not limited to the SPB’s in budding yeast; rather, any protein structure that is fluorescently labelled can be tracked in three-dimensions, provided its local intensity may be distinguished from background and spectrally or spatially separated from neighbouring labelled structures. It is also possible to correlate spindle dynamics with biochemical signalling by simultaneously labelling components on the spindle and utilizing other fluorescence reporters for the expression of a particular protein of interest. Towards this goal, work has begun in the Kilfoil lab to coordinate cyclin levels with spindle dynamics and cell growth. Much insight may be gained by supplementing experimental observations of spindle dynamics and cell morphology with dynamical modelling of the mitotic spindle and cell growth. In the future, models may be integrated with the simulations of fluorescence images presented in this thesis to aid in connecting observations with model predictions.

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