Mechanistic Mathematical Modeling of Spatiotemporal Microtubule Dynamics and Regulation in Vivo

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Doctoral Thesis

Mechanistic mathematical modeling of spatiotemporal microtubule dynamics and regulation in vivo

Author(s): Widmer, Lukas A.

Publication Date: 2018

Permanent Link: https://doi.org/10.3929/ethz-b-000328562

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ETH Library diss. eth no. 25588

MECHANISTICMATHEMATICAL MODELINGOFSPATIOTEMPORAL MICROTUBULEDYNAMICSAND REGULATION INVIVO

A thesis submitted to attain the degree of

DOCTOR OF SCIENCES of ETH ZURICH (dr. sc. eth zurich)

presented by LUKASANDREASWIDMER msc. eth cbb born on 11. 03. 1987 citizen of luzern and ruswil lu, switzerland

accepted on the recommendation of Prof. Dr. Jörg Stelling, examiner Prof. Dr. Yves Barral, co-examiner Prof. Dr. François Nédélec, co-examiner Prof. Dr. Linda Petzold, co-examiner

We are all much more than the sum of our work, and there is a great many whom I would like to thank for their support and encouragement, without which this thesis would not exist. I would like to thank my supervisor, Prof. Jörg Stelling, for giving me the opportunity to conduct my PhD research in his group. Jörg, you have been a great scientific mentor, and the scientific freedom you give your students is something I enjoyed a lot – you made it possible for me to develop my own theories, and put them to the test. I thank you for the trust you put into me, giving me a challenge to rise up to, and for always having an open door, whether in times of excitement or despair. It was a great pleasure to work with my comrades-in-arms on the TubeX project. I thank Xiuzhen Chen and her supervisor Prof. Yves Barral at D-BIOL for teaching me a lot about yeast in general and its microtubule cytoskeleton in particular, their openness in sharing data, intense and stimulating discussions, and keeping an open mind regarding modeling results. I thank Marcel Stangier and Prof. Michel Steinmetz from the Paul Scherrer Institute for providing valuable structural insights on the many proteins interacting to give rise to microtubule dynamics, and stimulating discussions during our meetings. I would also like to thank the former TubeX members Mathias Bayer, Jette Lengefeld and Denis Samuylov for fruitful discussions. My committee members also contributed valuable insights, and I would like to thank Prof. François Nédélec for organizing the EMBO Practical Course on Modeling Cellular Processes in Space and Time, during which I learnt a lot all while having a great time, as well as for stimulating discussions during the bi-annual microtubule symposia at EMBL. I thank Prof. Linda Petzold for valuable discussions regarding coordinate-aware reaction-diffusion master equation models during the BSSE department review. My parents Andreas and Evelyne awoke my curiosity, encouraged me to choose my own path, and their support in my pursuit of science was invaluable – dad, I will always remember your motto: “but where is the data?” Without my siblings Felix, David and Jasmin, the past years would have indeed been much duller, and I am very grateful for the time we get to spend together discussing, playing games, skiing, snowboarding, and traveling – you guys are awesome! I thank the entire current and past Stelling group – Alina, Asli, Andreas, Charlotte, Claude, Diana, Eleni, Ellis, Eve, Fabian, Hans-Michael, Jana, Julia, Kristina, Lekshmi, Mattia, Mikolaj, Moritz, Mikael, Pencho, Robert, Sotiris and Thomas – for discussions, being great colleagues, and all the fun we had during our retreats! I would especially like to thank Moritz for valuable discussions and being a great friend, if it had not been for you, I might not have embarked on this scientific adventure in the first place. Mathias was a great help while getting started. Jana is the best office mate one could wish for, I thank her for all the scientific and emotional support, and I wish her all the best in finishing up her PhD thesis. I would also like to thank Kristina and Andreas for introducing me to yeast lab and microscopy basics, and my master student Alain and project student Natalia for their contributions to modeling the yeast signaling pathways involved in mitosis.

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Next, my thanks go to my shared Master student Harun, who performed excellent work on GapMiner, a joint project with Sabine, as well as preparatory work in his lab rotation. I also had the privilege of tutoring our iGEM students of 2013, 2014 and 2015 – you taught me a lot, and I hope I was able to teach you some skills you still find useful as well. I also would like to extend thanks to the ITSC team for keeping our cluster happy and science happening – a special thank you goes to John Ryan and Martin Fox for their excellent support and sharing their sysadmin experience over the last few years – keeping everything running smoothly was not always easy. Having good friends along for the ride during my PhD studies also was of great help – I always enjoyed time with Andreas Ritter, from planning weddings to discussing beer brewing and other non-work-related topics. I also enjoyed a lot of board and card game nights with Lukas Beyeler, Susana Posada Cespedes, and Marek Pikulski – we definitely saved the world from overwhelming epidemics more often than we succumbed to rapidly-mutating strains. I want to thank my better half, Sabine Österle, for her unconditional support through a time that was filled with joy, tears, sweat, and excitement. Doing challenging research together and co-supervising students was a fun experience – it was a privilege to be able to work on a true “in-house” collaboration. Finally, financial support by the Swiss Initiative for Systems Biology (SystemsX.ch, project TubeX) evaluated by the Swiss National Science Foundation is gratefully acknowl- edged. ABSTRACT

The microtubule cytoskeleton is a key component of eukaryotic life: during cell division it forms the mitotic spindle, an apparatus that is responsible for faithfully segregating genetic material into the daughter cells. For this purpose, cells form microtubule asters on opposite poles in the spindle – in the budding yeast Saccharomyces cerevisiae these are termed spindle pole bodies (SPBs), the equivalent of the mammalian centrosomes. In this thesis, we are concerned with the basic mechanisms of function of the microtubule dynamics that allow for spindle assembly, alignment, and positioning in vivo, using budding yeast as a model organism. We aim to further our mechanistic understanding by mathematically modeling them in space, time, and on the appropriate molecular abundance scales: from microscopic representations of chemical reactions up to continuum dynamics. We highlight recent progress in bridging model classes and outline current challenges in such multi-scale models. We then choose a modeling approach with an appropriate resolution to cover stochastic intracellular microtubule dynamics as well as spatiotemporal regulatory networks that modulate microtubule dynamics through microtubule-associated proteins (MAPs) in vivo. We developed a simulation software in C++ that can simulate models of the chosen reaction-diffusion master equation type in conjunction with microtubule dynamics, and apply it to simulate autonucleated microtubules in Xenopus laevis egg extract in conjunction with a stochastic guanosine triphosphate cap model for each microtubule. To form a quantitative basis for our modeling efforts in yeast, we next investigate the abundance and basic polymerization properties of tubulin in vitro and in vivo. Our meta- analysis of data on tubulin abundance in S. cerevisiae and Schizosaccharomyces pombe cells, as well as in their respective spindles, suggests that there is a necessity for microtubule polymerases in vivo due to the low amount of free tubulin. Furthermore, we illustrate that asymmetric microtubule dynamics require regulation beyond controlling tubulin assembly via the global tubulin concentration. Finally, we establish an automated pipeline to analyze manually-annotated traces along microtubules imaged in vivo using ﬂuorescence microscopy. This pipeline allowed us to visualize mean motor density along microtubules of varying lengths on the example of the kinesin motor Kip2. We develop a spatiotemporal stochastic model that describes motor binding and movement on the microtubule, and a ﬂuorescence imaging measurement model corresponding to our in vivo analysis pipeline. Combining the in vivo data from this pipeline, analytical and computational predictions from the stochastic and measurement models, as well as structural information, we conclusively show that Kip2 is predominantly loaded at the SPBs in vivo, acting as a remote control for microtubule plus end dynamics from the SPB at the minus end. This remote control is enabled by preventing Kip2 from binding on the microtubule lattice, a feature that can be disabled by preventing Kip2 from being phosphorylated, and the high processivity and movement speed of the Kip2 motor. Asymmetric Kip2 binding at the two SPBs can thus explain the asymmetry in microtubule length between the microtubules in the mother and the bud

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during mitosis. This binding depends on the SPB outer plaque component Spc72, and is modulated by phosphorylation by the Polo-like kinase Cdc5. In the future, we envision a spatiotemporal model that integrates the mitotic exit network that Cdc5 (and opposing phosphatases) are a part of, as well as microtubule dynamics at the two SPBs in mother and bud with the associated MAPs – quantitatively capturing our current state of knowledge in silico, helping to discriminate between biological hypotheses and serving as a tool to drive new experiments. ZUSAMMENFASSUNG

Das Mikrotubuli-Zytoskelett ist ein grundlegender Bestandteil des eukaryotischen Le- bens: Während der Zellteilung bildet es den mitotischen Spindelapparat, der für die genaue Aufteilung des Erbgutes in die Tochterzellen verantwortlich ist. Zu diesem Zweck bilden die Zellen Mikrotubuli, welche von Zentren an entgegengesetzten Polen in der Spindel ausgehen – in der knospenden Hefe Saccharomyces cerevisiae werden diese als Spindelpolkörper (SPBs) bezeichnet, das Äquivalent zu den Zentrosomen der Säugetiere. In dieser Arbeit beschäftigen wir uns mit den grundlegenden Funktionsmechanismen der Mikrotubuli-Dynamik, welche die Ausbildung der mitotischen Spindel, sowie ihre Ausrichtung und Positionierung in vivo ermöglichen – wobei die Knosphefe als Modell- organismus dient. Um unser Verständnis dieser Mechanismen zu vertiefen, modellieren wir sie mathematisch in Raum, Zeit und der korrekten Anzahl: von mikroskopischen che- mischen Reaktionen bis hin zu kontinuierlichen makroskopischen Konzentrationsfeldern. Wir beleuchten die jüngsten Fortschritte bei der Überbrückung solcher Modellklassen und erläutern die aktuellen Herausforderungen in diesen Skala-übergreifenden Modellen. Als Nächstes wählen wir ein Modellierungsverfahren mit einer geeigneten Auﬂö- sung, um die stochastische intrazelluläre Mikrotubuli-Dynamik sowie raumzeitliche regulatorische Netzwerke abzudecken, die die Mikrotubuli-Dynamik durch Mikrotubuli- assoziierte Proteine (MAPs) in vivo modulieren. Wir haben eine Simulationssoftware in C++ entwickelt, die Modelle des gewählten Reaktions-Diffusions-Master-Gleichungstyps zusammen mit Mikrotubuli-Dynamik simulieren kann, und verwenden diese am Beispiel selbst-nukleierender Mikrotubuli in Xenopus laevis Eizell-Extrakt in Verbindung mit einem stochastischen Guanosintriphosphat-Kappen-Modell für jedes Mikrotubulum. Um eine quantitative Grundlage für unsere Modellierungsbemühungen in Hefezellen zu schaffen, untersuchen wir zunächst die grundlegenden Polymerisationseigenschaf- ten von Tubulin in vitro und die verfügbaren Tubulin-Konzentrationen in vivo. Unsere Meta-Analyse verfügbarer Daten über die Tubulinkonzentration in S. cerevisiae und Schizosaccharomyces pombe Zellen, sowie in ihren jeweiligen Spindeln, deutet darauf hin, dass Mikrotubuli-Polymerasen in vivo aufgrund der geringen freien Tubulinkonzentra- tionen notwendig sind. Darüber hinaus veranschaulichen wir, dass die asymmetrische Mikrotubuli-Dynamik und -Polymerisation in Knosphefen eine Regulierung erfordert, die über die Kontrolle der globalen Tubulinkonzentration hinausgeht. Zum Schluss etablieren wir eine automatisierte Analysesoftware für manuell annotierte Fluoreszenz-Mikroskopiedaten von in vivo Mikrotubuli. Diese Software ermöglichte es uns, die durchschnittliche Motordichte entlang unterschiedlich langer Mikrotubuli am Beispiel des Kinesin-Motors Kip2 zu visualisieren. Wir entwickelten ein raum-zeitliches stochastisches Modell welches die Motor-Bindung und -Bewegung auf den Mikrotubuli beschreibt, sowie ein Fluoreszenz-Mess-Modell, welches in silico Fluoreszenzdaten aus Simulationen des stochastischen Modells erzeugen kann. Durch die Kombination der mit unserer Software generierten in vivo Daten, analytischer und rechnerischer Vorhersagen aus den stochastischen und Mess-Modellen sowie Informationen bezüglich der Kip2 Proteinstruktur zeigen wir abschliessend, dass Kip2 in vivo überwiegend via die SPBs

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auf Mikrotubuli geladen wird, und als Fernsteuerung der Mikrotubuli-Dynamik am Plus-Ende vom SPB am Minus-Ende aus dient. Diese Fernsteuerung wird durch die Verhinderung der Bindung des Kip2-Motors an das Mikrotubulum, sowie die hohe Prozessivität und Bewegungsgeschwindigkeit dieses Motors ermöglicht. Wir zeigen dass die Kip2-Bindung auf dem Mikrotubulum durch die Phosphorylierung von Kip2 verhindert wird, welche wiederum von der Polo- ähnlichen Kinase Cdc5 moduliert wird, während die Bindung via SPB von der äusseren SPB-Plaquekomponente Spc72 abhängt. Auf Basis unserer gesammelten Resultate stellen wir uns ein raum-zeit-liches Modell vor, welches sowohl Signalmoleküle wie die Kinase Cdc5 und entgegengesetzte Phos- phatasen beschreibt, wie auch ihren Effekt auf die Mikro-Tubuli-Dynamik an den beiden Spindelpolkörpern via MAPs. Ein solches Modell würde quantitativ unseren aktuellen Wissensstand in silico erfassen, und als wertvolles Werkzeug zur Unterscheidung zwischen biologischen Hypothesen, sowie zur Motivation neuer Experimente dienen. CONTENTS

1 introduction1 1.1 A brief overview of the cytoskeleton ...... 1 1.2 The microtubule cytoskeleton ...... 2 1.3 Intrinsic microtubule dynamics ...... 5 1.3.1 Microtubule catastrophe ...... 5 1.3.2 Microtubule rescue ...... 7 1.4 Tubulin and microtubules in yeast cells ...... 8 1.5 Microtubule-associated proteins (MAPs) in yeast ...... 9 1.5.1 EB plus tip tracking proteins ...... 9 1.5.2 CLIP family ...... 10 1.5.3 XMAP polymerases ...... 11 1.5.4 CLASPs ...... 12 1.5.5 Kinesins ...... 13 1.5.6 MAP4 / Mhp1 ...... 15 1.6 Contribution of this thesis ...... 15 2 bridging intracellular scales by mechanistic computational models 17 2.1 Abstract ...... 17 2.2 Introduction ...... 18 2.3 Reaction, diffusion, and crowding ...... 20 2.4 Spatial stochastic effects in cell signaling ...... 22 2.5 Challenge: Active transport ...... 24 2.6 Challenge: Dynamic cellular geometries ...... 24 2.7 Challenge: Model calibration and inference ...... 25 2.8 Conclusions ...... 25 2.9 Acknowledgements ...... 26 3 stochastic simulation of microtubules with the coordinate- aware c++ reaction-diffusion master equation simulator rdmecpp. 27 3.1 Abstract ...... 27 3.2 Introduction ...... 28 3.3 Materials and methods ...... 29 3.3.1 Geometry, meshing and matrix assembly ...... 29 3.3.2 Point-to-subvolume mapping ...... 29 3.3.3 Microtubule dynamics across subvolume boundaries ...... 30 3.3.4 Data analysis and visualization in MATLAB ...... 31 3.4 Results and discussion ...... 32 3.4.1 Stochastic simulation engine ...... 32 3.4.2 Simulation performance ...... 32 3.4.3 Deterministic model of autocatalytic microtubule nucleation . . . 32 3.4.4 Stochastic model of autocatalytic microtubule nucleation ...... 34 3.4.5 Steady state solution and dynamics ...... 36 3.5 Conclusion ...... 41

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3.5.1 Outlook ...... 41 3.6 Acknowledgements ...... 41 3.7 Author contributions ...... 42 3.8 Supporting information ...... 42 3.8.1 Continuous reaction-diffusion problem ...... 42 3.8.2 Mass Matrix Diagonalization ...... 45 3.8.3 Conversion to molecule numbers ...... 46 3.8.4 Deterministic simulation ...... 46 3.8.5 Reaction terms ...... 46 3.8.6 Documentation ...... 46 3.8.7 URDME compatibility ...... 46 4 on the essentiality of microtubule (de-)polymerases – why is tubulin not enough? 49 4.1 Highlights ...... 49 4.2 Summary ...... 49 4.3 Introduction ...... 50 4.4 Results and Discussion ...... 51 4.4.1 Estimating the total tubulin concentrations in S. cerevisiae and S. pombe 51 4.4.2 During mitosis, a substantial fraction of tubulin is bound in the spindle ...... 52 4.4.3 Free tubulin concentrations during mitosis are in the hundred- nanomolar range ...... 54 4.4.4 Validating tubulin estimates required for spindle assembly in S. cerevisiae using knockout data ...... 54 4.4.5 Free concentrations in vivo are below the critical concentrations for yeast tubulin assembly in vitro ...... 56 4.4.6 Concentration requirements for microtubule nucleation are higher than for microtubule growth ...... 57 4.4.7 Our tubulin estimates and microtubule nucleation / polymerization model explain the lethality of microtubule polymerase knockouts 60 4.4.8 In vivo microtubule dynamics mandate regulated nucleation, polymerization, and depolymerization mechanisms beyond the capabilities of free tubulin pool regulation ...... 62 4.4.9 Conclusions ...... 63 4.5 Acknowledgements ...... 63 4.6 Author contributions ...... 63 4.7 Supplementary Information ...... 64 4.7.1 Supplementary Figures ...... 64 5 a mechanism for the control of microtubule plus-end dynamics from the minus-end 65 5.1 Abstract ...... 65 5.1.1 One Sentence Summary ...... 65 5.2 Main Text ...... 66 5.3 Acknowledgements ...... 73 5.4 Author contributions ...... 73 5.5 Data and materials availability ...... 73 contents xi

5.6 Supplementary materials ...... 74 5.6.1 Materials and methods ...... 74 5.6.2 Supplementary Text ...... 76 5.6.3 Supplementary Figures ...... 85 5.6.4 Supplementary Tables ...... 93 6 concluding remarks and outlook 97 6.1 Towards an integrated in silico model of the microtubule cytoskeleton and its regulation ...... 98 6.2 Quantitative modeling of tubulin allocation and availability: testing our predictions further ...... 99 6.3 SPBs exert more control over the microtubules they nucleate than previously believed ...... 100 6.4 Future uses for the Kip2 data analysis pipeline ...... 101 Bibliography 103

INTRODUCTION

In this thesis, we are concerned with two of the most basic ingredients of life: cell organization, and cell division. Without either, life as we know it would not exist: the ∼ 3 × 1013 cells in our bodies are organized in a myriad of different ways, and have a wide diversity of shapes and dimensions (Sender et al., 2016). Over a human lifetime, there are on the order of 1016 division events (BNID 100379, Milo et al., 2009) that have to be correctly executed – a truly amazing feat. To further our understanding of the underlying principles that govern life from single-cellular organisms all the way up to mammalians, science has turned to model organisms. Model organisms are biological species that have been extensively studied – these days, usually their complete genome is available and substantial efforts have been put into annotating it. Many of them have not only been studied, but also been the target of substantial engineering efforts – enabling introduction of genomic material of our choice, or knocking out existing material. These engineering efforts have not halted at the organisms themselves: the development of many quantification techniques has been driven by a scientific need arising from studying model organisms. Examples include fluorescent light microscopy, and associated tools, such as cell cultures and microfluidic chambers that allow for long-term cultivation and quantification. Budding yeast is a small single-cellular organism – as its latin name Saccharomyces cerevisiae suggests, it is used in beer brewing, but other strains are also used in winemak- ing, both practices dating back far in human history (Legras et al., 2007). Apart from its everyday use, due to the ease of culturing it, and its – some would say, surprising – complexity, it also has been a model organism for cell biology: S. cerevisiae has many genetic tools available, and was the first eukaryotic organism to have its complete genome sequenced (Goffeau et al., 1996). It faithfully segregates its chromosomes inside a spatial compartment – the nucleus – as it divides, and is able to break symmetry and polarize upon detecting external spatial stimuli, for example by responding to a pheromone of a cell of the opposite mating type and projecting a shmoo towards it. The compartmentalization of the genome, its ability to establish an asymmetric shmoo towards another cell, and the structure that segregates the chromosomes – the mitotic spindle – are all made possible by the yeast cytoskeleton, which will be the prime target of investigation in this thesis.

1.1 a brief overview of the cytoskeleton

In eukaryotic cells, the cytoskeleton is made up of three broad types of filaments: actin filaments, microtubules, and intermediate filaments. Each of these filament types is assembled from different protein subunits (Alberts et al., 2014) – however, since yeasts do not have intermediate filaments (Herrmann and Strelkov, 2011), we will not further discuss them here.

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Actin filaments are essential for establishing polarity and enabling motility, providing guidance to the spindle during cell division (Figure 1.2), and for endocytosis (matter intake through cell membrane invagination). They are made up of actin protein – Act1 in S. cerevisiae – and during mitosis, organize into cortical actin patches, polarized actin cables, and a cytokinetic actin ring (Moseley and Goode, 2006). Actin patches localize to sites of polarized cell growth, move along the cortex, and then disappear again – during mitosis, they especially localize to the emerging bud. The polar actin cables emanate from the bud into the mother cell, serving as a track for myosin motors that transport cargos to the bud, and align the mitotic spindle along the mother-bud axis (Moseley and Goode, 2006). The cytokinetic actomyosin ring forms between the mother and bud cell during mitosis, and it is responsible for generating the force to enable cytokinesis (Mishra et al., 2014). In budding yeast, interestingly, it is positioned between two septin rings that serve as cortical diffusion barriers (Dobbelaere and Barral, 2004). Microtubules in eukaryotic cells are essential in orchestrating cell division, and faithful segregation of genetic material into daughter cells. For this purpose, cells undergoing division build up the mitotic spindle, an elaborate apparatus that forms two poles, from which microtubules originate. These microtubules then attach to the duplicating chromosomes at an attachment point called the kinetochore, and segregate them towards their respective pole of origin (reviewed in detail in Winey and Bloom, 2012). Microtubules are not just used as chromosome transport systems inside the nucleus, but also are involved in cytoplasmic transport. Examples (reviewed in Welte, 2004 and Gross, 2004) include the transport of entire organelles like mitochondria in axons, which grow too long for diffusive transport to be effective (Maday et al., 2014), or the transport of various cargoes between nucleus and the cell periphery. This system is even exploited by viruses (Dodding and Way, 2011) – for example, influenza hijacks these microtubules to get transported towards the nucleus (Lakadamyali et al., 2003). Furthermore, microtubules are used as molecular scaffolds in centrioles (which nucleate the formation of the centrosome, the mammalian equivalent of the yeast spindle pole body), and enable propulsion in cilia and flagella (Carvalho-Santos et al., 2011). Microtubules are composed of tubulin “building blocks”, which themselves are heterodimers of α- and β-tubulin proteins (Bryan and Wilson, 1971), and will be the major subject of study in this thesis.

1.2 the microtubule cytoskeleton

As the name implies, microtubules are microscopically small cylindrical tubes that can grow to lengths of up to ∼ 100 µm (in axons; Bray and Bunge, 1981). The microtubules in cilia go far back into single-cellular eukaryotic history (according to Mitchell, 2016, approximately one billion years back). Tubulin heterodimers probably arose from the protein FtsZ which is part of the bacterial cytokinesis machinery (Erickson, 2017). Each hollow microtubule is made out of a varying number of protofilaments – 11 to 17 were reported in vitro (Pierson et al., 1978) – which form its walls. Microtubules in vivo usually have a more controlled number of protofilaments, for example, by nucleating off a template that has a well-defined symmetry, and in S. cerevisiae and other organisms this usually leads to 13-protofilament microtubules (Tilney et al., 1973; Chretien et al., 1992; Figure 1.1). In addition, microtubules-associated proteins (MAPs) can also favor 13-protofilament microtubules, for example, Mal3 (Georges et al., 2008; Section 1.5.1). 3

A B Plus End (+) 9 7 8 5 6 10 4 10 11 3 2 12 1 9 13 13 12 11 Heterodimer 8 r GTP- or R β 1 GDP-bound 7 α 2 6 3 5 4

Minus End (-)

Figure 1.1 – 13-protofilament microtubule geometry. (A) Top view of a 13-protofilament microtubule, numbered in the standard clockwise tip-to-base manner (Linck and Stephens, 2007). Each protofilament has a radius r ≈ 3 nm, and the microtubule thus has a central radius of R = 9.5 nm. The inner diameter is thus 2(R − r) ≈ 13 nm, and the outer diameter is 2(R + r) ≈ 25 nm (Tani and Ametani, 1970). (B) Side view of a 13-protofilament microtubule. Each protofilament is composed of αβ-tubulin heterodimers, with the α-tubulin (blue) end oriented towards the minus end of the microtubule, and β-tubulin (purple) towards the plus end, respectively. The exposed GTP/GDP binding site of β-tubulin at its E-site can change its state from GDP-bound to GTP-bound when the heterodimer is free, and the β-tubulin GTP stochastically hydrolyzes after the heterodimer is incorporated into the microtubule lattice. The thirteen protofilaments form a three-start helix (i.e., for one turn, the helix rises by three monomers), in a B-lattice (when following the lattice sideways, we have α-α-. . . -α, not an A-lattice of α-β-α-. . . ), with a seam (red dashed line) between protofilaments 1 and 13 (Mandelkow et al., 1986). 4

Growth

SPB duplication Bud outgrowth G1 Start M SPB Segregation S G2

Figure 1.2 – Spindle pole body (SPB) dynamics during the mitotic cell cycle of S. cerevisiae. Budding yeast cells start off in G1 phase, and grow until they reach the “Start” decision point. After start, bud outgrowth begins, and SPB duplication (pre-existing SPB: dark red, new SPB: light red) as well as DNA replication are initiated. The SPBs then build up the mitotic spindle (black) and move to opposite sides of the nucleus. Next, the mitotic spindle extends and aligns along the mother-bud axis, and one of the SPBs (usually the pre-existing one) is segregated into the bud. Finally, cytokinesis occurs: the cell divides asymmetrically, and the newly-separated cells continue growth and repeat the cycle. A variety of checkpoints blocks progression in case a step is not completed before proceeding.

Each protofilament is in turn composed of polar tubulin heterodimers. If viewed from the end of the microtubule where nucleation usually starts (the minus end), each heterodimer starts with an α-tubulin protein which is connected to a β-tubulin protein – this interface then either marks the current end (the plus end) of the microtubule, or can be connected to the α-tubulin of the next heterodimer (Figure 1.1B). In vivo, microtubules are usually nucleated from templates consisting out of γ-tubulin (Tub4 in S. cerevisiae), and the minus end remains firmly attached to this template. These templates are mostly found in microtubule-organizing centers (MTOCs), with the dynamic microtubule plus ends free to explore the surrounding space. Basal bodies are MTOCs that organize flagella and cilia, and centrosomes organize the mitotic and meiotic spindles, which faithfully distribute genetic material during cell division. Yeasts like S. cerevisiae and S. pombe have spindle pole bodies (SPBs), which are the equivalent of the mammalian centrosomes. SPBs are located in the nuclear envelope, and are duplicated during mitosis while genome replication occurs, and then segregated such that each cell has one SPB – and one copy of the attached chromosomes – each (Figure 1.2). Microtubule dynamics are what enable this amazing feat. In the following, I will give an overview of the current state of knowledge on how this microtubule self-assembly system works, and how it is modulated and regulated by a variety of molecules in vivo. 5

1.3 intrinsic microtubule dynamics

Once αβ-tubulin heterodimers could be puriﬁed, a remarkable property was discovered: given a mixture of sufﬁciently-concentrated αβ-tubulin heterodimers with GTP in an appropriate buffer, microtubules can self-assemble from their heterodimer building blocks. Moreover, they do not just self-assemble, but this growth phase is stochastically interrupted by a phase of rapid shrinkage – a transition named “catastrophe”, whereas the backward transition from rapid shrinkage to growth was termed “rescue” (Mitchison and Kirschner, 1984). These non-equilibrium dynamics have since been summarized under the title of the aforementioned landmark publication: “dynamic instability”. These dynamics are powered by GTP hydrolysis – the GTP in the E-site of the β- tubulin subunit in each heterodimer can be hydrolyzed after its incorporation into the microtubule lattice. This process starts as soon as another heterodimer “sits” on top, as non-polymerizing tubulin heterodimers on their own do not readily hydrolyze GTP (David-Pfeuty et al., 1977; Vandecandelaere et al., 1999). Upon dissociating – for example due to microtubule catastrophe – free β-tubulin heterodimers allow for exchange of their GDP at the E-site back to GTP (thus “E-site”), “reloading” the heterodimer for a new round of polymerizing activity. While α-tubulin also contains a GTP binding site – the N-site – the GTP there is non-exchangeable and non-hydrolyzable (Nogales et al., 1998). Our current knowledge on intrinsic microtubule dynamics is summarized graphically in Figure 1.3.

1.3.1 Microtubule catastrophe

When β-tubulin hydrolyzes its E-site-bound GTP to GDP, it tends to bend outwards – making the lattice structure less stable. Since the stochastic hydrolysis “timer” starts upon incorporation of the tubulin heterodimer on top of the β-tubulin in question, the microtubule plus tip tends to contain GTP-tubulin, and the majority of the lattice will contain GDP-tubulin. This has led to the GTP-tubulin stabilizing cap model proposed by Mitchison and Kirschner, 1984: if the stabilizing GTP-tubulin plus end cap gets lost, the microtubule catastrophes by rapidly losing subunits from its plus end (Figure 1.3). One prediction that can be made from this model is that, if hydrolysis was drastically slowed down, catastrophe should no longer occur. The discovery of GTP analogues enabled testing of this hypothesis in vitro by substituting GTP with the slowly-hydrolyzable GTP analog guanylyl-(α,β)-methylene-diphosphonate (GMPCPP) – and indeed, catastrophe was no longer observed (Hyman et al., 1992). Furthermore, GDP-tubulin microtubules could also be stabilized using GMPCPP tubulin, indicating that the plus end cap can indeed stop a microtubule from depolymerizing (Mickey and Howard, 1995). Directly observing the GTP-cap has however been quite an elusive endeavor, as this would require a sensor for the hydrolysis state of each tubulin heterodimer (Desai and Mitchison, 1997). While the existence of the GTP-tubulin cap is no longer a debated issue, there is no consensus on its nature, more speciﬁcally its extent, yet (Bowne-Anderson et al., 2013). One way to measure GTP cap extent is from dilution experiments – that is, experiments in which microtubules are polymerized from GTP-tubulin in solution, after which the GTP tubulin solution is rapidly replaced by buffer. In recent experiments, slow microtubule shrinkage akin to GMPCPP microtubules can be observed shortly after 6

GTP-Tubulin Hydrolysis

- +

Growing Microtubule

Reloading Rescue Catastrophe

Shrinking Microtubule

- + GDP-Tubulin

Figure 1.3 – Microtubule dynamic instability. Microtubules grow by incorporating GTP-tubulin heterodimers (blue-pink) at their plus (+) ends. Once another tubulin heterodimer binds on top, the GTP-tubulin stochastically hydrolyzes over time, releasing its phosphate to become a less stable GDP-tubulin heterodimer (blue-purple). Once the GTP-tubulin density at the microtubule plus end gets too low, catastrophe occurs, i.e., the microtubule transitions into a rapidly shrinking phase. The opposite transition from rapidly shrinking to growing is termed rescue. In the cytoplasm (in vivo) / solution (in vitro), the hydrolyzed GDP at the β-tubulin E-site gets reloaded with GTP, and this GTP-tubulin heterodimer is free to bind to a microtubule again. 7 dilution, probably due to GTP-tubulin both dissociating and hydrolyzing. After this brief period of slow shrinkage, catastrophe occurs and the microtubule depolymerizes rapidly (Duellberg et al., 2016). Another way of measuring GTP-tubulin content at the plus tip and in the lattice is by fluorescently tagging and monitoring the EB class of microtubule-associated proteins (detailed in section 1.5.1), which bind close to the E-site (Zanic et al., 2009; Maurer et al., 2011; Maurer et al., 2014; Zhang et al., 2015). They thereby can detect the nucleotide state of tubulin heterodimers, as demonstrated with the GTPγS GTP analogue: EBs bind to GTPγS-tubulin, but do not bind GDP-tubulin – if we accept GTPγS-tubulin as mimicking GTP-tubulin, we can use EBs as GTP-tubulin sensors. Interestingly, the choice of GTP analogue has a profound impact: EBs do not bind GMPCPP-tubulin, possibly because GMPCPP-tubulin has stronger lateral contacts (Maurer et al., 2011). One wrinkle in using EBs to detect nucleotide state is that they also change microtubule dynamics (specifically, they accelerate hydrolysis, see Maurer et al., 2014), and are as such not exactly the perfect sensors either. Nevertheless, this approach did lead to useful estimates on the GTP-cap size: Seetapun et al., 2012 estimated a cap size of approximately 60 layers in vivo in LLCPK1 epithelial cells. Bowne-Anderson et al., 2013, hypothesized that the part of the GTP cap that is actually required for stabilizing the microtubule plus tip may be substantially smaller than the overall GTP cap. Since then, the GTP cap size has been shown to stochastically fluctuate, and while the cap can grow to a considerable size, what matters for preventing catastrophe is that the GTP-tubulin density at the microtubule plus tip (the last 1-10 layers) on average is above ∼ 29% (Duellberg et al., 2016). If the GTP cap is longer and sufficiently dense, fluctuations at the plus end that lead to catastrophe can be rapidly rescued. However, in vivo, rescue also occurs further to the back of microtubules, likely out of reach of the GTP-cap. The following section will cover the current state of knowledge on microtubule rescue mechanisms.

1.3.2 Microtubule rescue

In vitro, in assays using purified bovine brain and other tubulins without micro-tubule- associated proteins, rescue is extremely infrequent. This has made studying microtubule rescue much more difficult than microtubule catastrophe, and relatively few data are available. However, to understand the mechanistic basis of rescue, we must first understand these infrequent in vitro events – adding microtubule-associated proteins only adds to the complexity of the intrinsic tubulin-driven microtubule dynamics. The classic study on tubulin polymerization by Walker et al., 1988 measured both catastrophe and rescue frequencies as a function of bovine brain tubulin concentration. While catastrophe frequency decreased rapidly with increasing tubulin concentration – consistent with increasing growth rate by polymerases in S. pombe tubulin (Hussmann et al., 2016) and S. cerevisiae tubulin (Hibbel et al., 2015) – rescue frequency only increased little. One explanation that emerged both theoretically (Antal et al., 2007) and shortly after in vivo, using a GTP-tubulin-recognizing antibody (Dimitrov et al., 2008), is that “GTP-tubulin islands” or “GTP remnants” in the microtubule lattice could explain sites of rescue. Indeed, artificially generating such GTP islands by introducing mixed GTP/GDP/GMPCPP regions into microtubules in vitro induced rescues at these sites 8

(Tropini et al., 2012). What is less clear however, is how these GTP-tubulin islands are generated in the first place. A first hypothesis could be that hydrolysis is modulated in order to generate these GTP islands. In human cells, pVHL (von Hippel Landau tumor suppressor) was determined to reduce the hydrolysis rate of tubulin in microtubules, thus leading to a higher GTP- content (Thoma et al., 2010) – however, yeasts have no known homolog of this protein. Another candidate for modifying the hydrolysis rate of tubulin hetero-dimers is β- tubulin (as it hosts the E-site). Since a variety of β-tubulin isotypes are expressed in mammalian cells, expressing different ones could change the intrinsic catastrophe and rescue frequencies – in vitro experimental data from both brain and HeLa cell tubulin (which express different β-tubulin isotypes) suggest this is a possibility (Newton et al., 2002). Conversely, β5-tubulin seems to be accelerating GTP hydrolysis, leading to earlier and more persistent catastrophe, and few rescues (Bhattacharya et al., 2011). S. cerevisiae however only has a single isotype of β-tubulin (the same holds for S. pombe). Nevertheless, there is evidence suggesting a difference in rescue frequencies for the Tub3 α-tubulin isotype versus Tub1 or the wild-type mix (Bode et al., 2003) – one could speculate that this is also due to a difference in hydrolysis rate for the Tub1/3-Tub2 α-β heterodimer isotypes. A conceptually different way of generating GTP islands is to generate them de novo, rather than “conserving” GTP-tubulin in the microtubule lattice. This presents the difficulty of exchanging the GDP in tubulin heterodimers built into the microtubule lattice to GTP. One way of achieving this is introducing damage, for example using a laser, into microtubules while keeping GTP-tubulin heterodimers in solution readily available, which are able to integrate into the damaged sites. These newly-generated GTP-tubulin patches are sufficient to enable rescue once a catastrophing microtubule reaches them (Aumeier et al., 2016). Damage can also be introduced mechanically, for example by making microtubules collide and cross in vitro (Forges et al., 2016) – these crossing sites are able to incorporate GTP-tubulin and become GTP islands. It is likely that in vivo, microtubule crossings and / or collisions with other organelles would activate the same repair mechanism. Finally, lattice damage can also be introduced by microtubule-severing enzymes such as spastin and katanin (both of which do not have homologs in yeast), which catalyze tubulin exchange at the site of damage (Vemu et al., 2018). In vivo, a variety of MAPs act on top of these intrinsic dynamics. These MAPs either promote rescue via one of the above intrinsic mechanisms, or contribute mechanisms of their own (Section 1.5.4 on CLASPs). Thus, the next section will cover microtubule dynamics in vivo in budding yeast, and review the zoo of MAPs available.

1.4 tubulin and microtubules in yeast cells

For in vivo studies, there is a major advantage to working with S. cerevisiae cells: during mitosis, cytoplasmic microtubules are nucleated at the (already somewhat separated) SPBs, and there are usually only few: 0 to 3, typically a single microtubule. This makes budding yeast cytoplasmic microtubules and their regulators an ideal system to study by ﬂuorescence microscopy in vivo, as tracking them using ﬂuorescently-tagged proteins typically leads to non-overlapping, separated traces – in stark contrast to, for example, Xenopus laevis (African clawed frog) egg extract microtubule asters. 9

After identifying the S. cerevisiae tubulin heterodimer (Baum et al., 1978), the individual α- and β-tubulin genes were found. S. cerevisiae expresses two α-tubulin proteins, Tub1 and Tub3 (Schatz et al., 1986b; Schatz et al., 1986a), and single β-tubulin protein, Tub2 (Neff et al., 1983). Tub1 and Tub3 were reported make up total α-tubulin in approximately a 90 % / 10 % ratio (Bode et al., 2003), though more recent research put it closer to 60 % / 40 % (Keren et al., 2016). There are two isotypes of tubulin heterodimer that can be built from these components: the Tub1-Tub2 heterodimer, and the Tub3-Tub2 heterodimer. They give rise to different microtubule dynamics, especially in terms of catastrophe rate and shrinkage rate during catastrophe phases (Bode et al., 2003). In S. pombe, there is also two α-tubulin genes – Nda2 and Atb2 – and a single β-tubulin gene – Nda3 (Hagan, 1998). While intrinsic microtubule dynamics form a solid basis for exploring intracellular space, in vivo, the microtubule cytoskeleton has to fulfill a variety of functions, and intrinsic microtubule dynamics are insufficient to, for example, construct a mitotic spindle. As we will discuss in the following section, the key to adapt microtubule dynamics to specific functions in vivo is to combine microtubules with a variety of additional molecules.

1.5 microtubule-associated proteins (maps) in yeast

Intrinsic microtubule dynamics can be modified by MAPs to suit specific needs. Since the minus end of microtubules is usually anchored to a nucleating site, it is the plus end dynamics that are the main target of regulation (Akhmanova and Steinmetz, 2008). The class of MAPs that act at microtubule plus tips were important enough to warrant their own name: plus-end-tracking proteins or +TIPs (Schuyler and Pellman, 2001). +TIPs are able to bind to plus tips of growing and / or shrinking microtubules, can modify their dynamics and enable attachment of other proteins to the plus tips (so-called hitchhiking, Carvalho et al., 2003). This implies that this class of proteins can distinguish the microtubule plus tip from the minus end and the microtubule lattice – a remarkable feat. Furthermore, some +TIPs only bind to and track growing microtubules, while others even remain attached to shrinking microtubule ends – all while tubulin subunits are dynamically added or removed, respectively. This is possible because, as previously discussed, growing and shrinking microtubule plus ends, the minus end, and the lattice all have distinct structures, that can in principle be recognized by appropriately-evolved molecules. In the following, I will briefly introduce the major MAPs, starting with +TIPs, continuing with kinesins, and ending with other, less well-characterized MAPs.

1.5.1 EB plus tip tracking proteins

The class of end binding (EB) +TIPs is probably the most extensively-studied one. EB1 was described first in mammalian cells in the context of cancer (Su et al., 1995), at which point it was already speculated that budding yeast contains a homolog: the still unnamed gene YER016W. The yeast gene YER016W was confirmed to encode a microtubule-binding protein using the, at the time newly-discovered, green fluorescent protein (GFP) two years later (Schwartz et al., 1997), and has been discussed under the name Bim1 (Binding to microtubules) ever since. Later, the microtubule-binding property was also confirmed 10

in mammalian cells, and more homologs to EB1 were found: EB1, EB2 and EB3 (Su and Qi, 2001). EB proteins generally consist of three domains: the N-terminal calponin homology domain (CH domain), a linker, a coiled-coil domain that is essential for homodimerization, and at the C-terminus, the EB homology domain (EBH domain) followed by a conserved EEY/F motif (Slep et al., 2005; Honnappa et al., 2005). The CH domain speciﬁcally binds and tracks growing microtubule plus ends by recognizing the GTP state of the plus tip β-tubulin (Slep and Vale, 2007). It achieves this by binding between two protoﬁlaments and heterodimers, thus making contact with four tubulin heterodimers (Maurer et al., 2012; Zhang et al., 2015). The binding of EBs both regularizes the growing microtubule lattice (Maurer et al., 2012; Zhang et al., 2015), and accelerates GTP hydrolysis (Maurer et al., 2014), causing an increase in microtubule growth rate, but also in catastrophe frequency. The C-terminal EBH domain and EEY-/F motif is a major hub for other proteins to bind to, and “hitchhike” along the plus tip (a term coined in Carvalho et al., 2003), building on top of the EBs’ tip tracking feature. Since this concept was suggested, a complex network of interacting plus tip proteins been uncovered (as recently reviewed by Akhmanova and Steinmetz, 2015). So far, there are two linear motifs that were determined to enable hitchhiking of other proteins on the EBH domain: SxIP (Honnappa et al., 2009), and LxxPTPh (Kumar et al., 2017) – with both motifs competing for the same binding site. In addition to EBH domain binding, the CAP-Gly domains found in the CLIP family (and others) can bind to the C-terminal EEY-/F motif of EB (Weisbrich et al., 2007). SxIP motif proteins in budding yeast include Kip2 and Kar9, though Kar9 also includes a LxxPTPh motif (Akhmanova and Steinmetz, 2015). Proteins that interact with the EEY-/F EB (and α-tubulin) C-terminal via CAP-Gly domains include the CLIP170 family, with the Bik1 homolog in budding yeast.

1.5.2 CLIP family

Cytoplasmic Linker Protein CLIP170 was the first of its family to be discovered to link vesicles to microtubules (Pierre et al., 1992). The orthologs bilateral karyogamy defect (Bik1) in S. cerevisiae (Trueheart et al., 1987; Berlin et al., 1990) and Tip1 in S. pombe (Brunner and Nurse, 2000) were also found to interact with microtubules, and these interactions were confirmed to be mediated via their respective N-terminal CAP-Gly domains (Carvalho et al., 2004). After the N-terminal CAP-Gly domain, both Bik1 and Tip1 contain a coiled-coil domain that allows them to homo-dimerize, and a C-terminal unstructured domain. The CLIPs interact with microtubules by recognizing their α- tubulin EEY/F tails, but for plus-tip specificity, they require EBs – CLIP170 requires EB1 (Dixit et al., 2009), and Bik1 requires Bim1 (Caudron et al., 2008; Stangier et al., 2018) to localize to microtubule plus tips. There is still conflicting reports regarding the functions of CLIP-170, Bik1 and Tip1: in vivo, loss of CLIP-170 leads to lower rescue frequencies (Komarova et al., 2002). In vitro, H2, the N-terminal fragment of CLIP-170, was shown to strongly promote rescue, and slightly increase microtubule growth rate while decreasing catastrophe rate (Arnal et al., 2004). CLIP170 in vitro seems to recognize repaired sites that contain GTP-tubulin, rather than causing rescue by themselves (Forges et al., 2016). The S. pombe protein Tip1 seems 11 to spatially regulate catastrophe, preventing it when microtubules hit the cortex (Brunner and Nurse, 2000). It is transported towards microtubule plus ends by the kinesin Tea2 (Busch et al., 2004). The S. cerevisiae protein Bik1 was reported to increase catastrophe (Blake-Hodek et al., 2010) – however, this analysis was performed using bovine brain tubulin, which in the case of Stu2 already caused even qualitatively different dynamics than on yeast tubulin (Podolski et al., 2014). In vivo, cytoplasmic microtubules in bik1∆ strains show lower growth speeds compared to wt (Wolyniak et al., 2006), suggesting it is involved in microtubule polymerization. Bik1 was also shown to bind to SPBs via Stu2, and travels towards the plus end on the Kip2 kinesin motor (Carvalho et al., 2004; Roberts et al., 2014). In summary, the function of CLIP family proteins is not entirely clear, and warrants further research.

1.5.3 XMAP polymerases

In contrast to the CLIPs, the XMAP family of proteins does not (solely) depend on the EBs for binding to microtubule plus ends, but has its own way of recognizing them. The 215 kDa Xenopus microtubule assembly protein (XMAP215) was described first – in Xenopus laevis egg extract (Gard and Kirschner, 1987). In this in vitro system, XMAP215 promoted microtubule growth speed by a factor of ∼ 10. It turned out that the XMAP family is highly conserved across species. Its homologs include Stu2 (suppressor of tubulin) in S. cerevisiae, Alp14 and Dis1 in S. pombe, the mammalian ch-TOG / CKAP5, ZYG-9 in C. elegans, and many more (reviewed by Ohkura et al., 2001). All of them interact with microtubules and tubulin through a varying number of N-terminal “tumor overexpressed gene” (TOG) domains, a term coined when the human homolog “colonic and hepatic tumor over-expressed gene” (ch-TOG) was identified (Charrasse et al., 1995). Rather than relying on plus tip targeting through EBs, XMAP215 and homologs use their TOG domains to preferably recognize curved tubulin heterodimers at the microtubule plus tip – rather than the straight heterodimer conformation in the microtubule lattice (Ayaz et al., 2012). The S. cerevisiae homolog Stu2 first had been shown to be an essential part of the SPBs, independently of microtubules (Wang and Huffaker, 1997). Starting at the N-terminus, it contains two TOG domains that are spaced by a linker, followed by a basic linker region, and a C-terminal coiled-coil domain for dimerization (Ohkura et al., 2001; Geyer et al., 2018). This results in a dimer that has four TOG domains in total, of which at least two (in any combination) are necessary for its function (Geyer et al., 2018). The multiple TOG domains are suggested to concentrate tubulin heterodimers in the vicinity of the microtubule plus end by a tethering mechanism (Ayaz et al., 2014). In vivo in budding yeast, tubulin FRAP (fluorescence recovery after photobleaching) experiments on mitotic half-spindles demonstrated that Stu2 depletion leads to a massive reduction in spindle microtubule dynamics (Kosco et al., 2001; Wolyniak et al., 2006). Basically, without Stu2, spindles are “dead”, with no fluorescence recovery, in line with Stu2 being an essential polymerase. However, polymerization speed in the cytoplasm seems to be associated with Bik1 rather than with Stu2 (Wolyniak et al., 2006). While an initial in vitro study on Stu2 in bovine brain tubulin indicated that Stu2 is a depolymerase (Van Breugel et al., 2003; Popov and Karsenti, 2003), more recent work 12

demonstrated that with tubulin puriﬁed from S. cerevisiae, Stu2 does in fact function as a polymerase (Podolski et al., 2014). In vitro, the polymerization activity of wild-type Stu2 on S. cerevisiae tubulin increases microtubule growth speeds up to 6×, with a measured KM between 23 nm (Podolski et al., 2014) and 62 nm (Geyer et al., 2018). In line with these measurements, recent data using the C. elegans homolog ZYG-9 (which contains three TOG domains, Ohkura et al., 2001) added to a condensate of the centrosomal component SPD-5 indicates that at the centrosome, ZYG-9 can increase tubulin concentration by approximately factor four (Woodruff et al., 2017). S. pombe has two XMAP215 homologs with two TOG domains, each, that function as microtubule polymerases: Dis1 and Alp14 (Garcia et al., 2001). Similar to Stu2, Dis1 has also ﬁrst been shown to bind to spindle pole bodies and microtubules (Nabeshima et al., 1995), before its function as a microtubule polymerase could be demonstrated (Matsuo et al., 2016). Alp14 has also been shown to bind to SPBs and microtubules (Garcia et al., 2001), and subsequently, function as a polymerase (Al-Bassam et al., 2012). Alp14 is required at high temperatures: alp14∆ is temperature-sensitive and does not grow at 36 ◦C. Dis1 is required at cold temperatures: dis1∆ is cold-sensitive (Garcia et al., 2001). The double deletion alp14∆ dis1∆ is lethal (Garcia et al., 2002). There is however one organism where the XMAP215 homolog does not seem to be essential: deleting AlpA in Aspergillus Nidulans only causes slight temperature sensitivity (Enke et al., 2007). However, it could be that Aspergillus nidulans expresses a paralog in addition to AlpA, analogous to the situation with Alp14 and Dis1 in S. pombe.

1.5.4 CLASPs

Stu1 was initially described as an essential component of the mitotic spindle (Pasqualone and Huffaker, 1994) before it was assigned to the family of CLASPs (cytoplasmic linker- associated proteins – note that Stu1 predominantly localizes to the nucleus, not the cytoplasm). Spindles lacking Stu1 are unable to assemble and elongate normally, leading to cell death (Yin et al., 2002). This elongation is driven by Stu1 re-localizing from kinetochores – where it assists microtubule capture – to the spindle midzone once all the kinetochores have been captured by microtubules. This mechanism prevents the spindle from extending before all the kinetochores are captured (Ortiz et al., 2009). At kinetochores, Stu1 is required for Stu2 recruitment, which in turn promotes nucleation of microtubules at kinetochores (Vasileva et al., 2017). Once Stu2 is localized to the kinetochores, Stu1 is dispensible for microtubule generation there. Similar to Stu2 in terms of microtubule binding, Stu1 contains two TOGL (TOG-like) domains, but in contrast to Stu2, Stu1 was not hypothesized to drive polymerization, but rescue (Al-Bassam and Chang, 2011). Mechanistically, the ﬁrst of the two Stu1 TOG domains is required for binding kinetochores, explaining why Stu1 localizes to spindle midzones, and only the second TOG domain binds to tubulin (Funk et al., 2014). Recently, the mechanism of rescue promotion by Stu1 has been elucidated: while the ﬁrst TOG domain of Stu1 only shows very weak binding to unpolymerized tubulin and microtubules, the second one binds to both (Majumdar et al., 2018). This second TOG domain strongly suppresses catastrophe and massively promotes rescue, largely without changing growth or shrinkage rate on S. cerevisiae tubulin (Majumdar et al., 2018). 13

While Stu1 can explain the rescue activity in the nucleus, it has not been observed on cytoplasmic / astral microtubules, so rescue there is probably not caused by Stu1.

1.5.5 Kinesins

Kinesins are a class of motile and non-motile molecular motors (from the Greek kinein, to move, Vale et al., 1985). They convert chemical energy in the form of ATP into mechanical energy by undergoing a conformational change when hydrolyzing ATP to ADP + P on microtubules – that is, they are microtubule-activated ATPases (Kuznetsov and Gelfand, 1986). Most kinesins share this mode of movement, but many have since been demonstrated to exhibit additional functions, for example driving microtubule plus end polymerization or depolymerization, cross-linking microtubules, or enabling microtubule sliding. Most kinesins move towards the microtubule plus ends, with a few notable exceptions: a minority of kinesins is non-motile, or even minus-end directed (a comprehensive review of mitotic kinesins can be found in Cross and McAinsh, 2014). In mammalians, the current count for the number of kinesin super-family proteins (KIFs) is at least 45. KIFs are not just relevant during mitosis, but also for transport in cilia and neurons, and KIF-related malfunctions are linked to a variety of diseases (Lawrence et al., 2004; Hirokawa and Tanaka, 2015). Here, the focus will be on kinesins that inﬂuence microtubule dynamics in yeast, and how they interact (Drummond, 2011 provides a comprehensive cross-organism review). In S. cerevisiae, only 6 of the 45 families are present, and 5 of these are motile kinesins: Cin8, Kip1, Kip2 and Kar3 were discovered early (Roof et al., 1991; Roof et al., 1992; Saunders and Hoyt, 1992), with Kip3 (DeZwaan et al., 1997; Cottingham and Hoyt, 1997) and Smy1 joining later (Hildebrandt and Hoyt, 2000). Note that Smy1 is non-motile and involved in actin assembly rather than microtubule dynamics (Eskin et al., 2016).

1.5.5.1 Kinesin-8 / Kip3 The only Kinesin-8 family member to be described in S. cerevisiae is the homolog Kip3 (kinesin related protein 3). Deleting it induces spindle positioning defects toward the mother cell in metaphase (DeZwaan et al., 1997; Cottingham and Hoyt, 1997). Kip3 was first hypothesized to be involved in depolymerization (Wordeman, 2005), and confirmed to be a depolymerizing kinesin shortly after (Gupta et al., 2006; Varga et al., 2006). Kip3 consists of an N-terminal motor domain, followed by a coiled-coil domain and a C-terminal tail domain, and functions as a dimer. It increases catastrophe rate, but also rescue frequency, possibly because of its tail’s ability to interact both with microtubules, but also with free tubulin heterodimers (Su et al., 2011). Kip3 bound to microtubules walks towards plus ends highly processively – with an average run length of about 11 µm – moving with a velocity of about 32 nm s−1 (Su et al., 2013). On a microtubule, this high processivity in combination with unidirectional movement causes more and more Kip3 to accumulate towards the plus end – a concept termed “antenna model” (Varga et al., 2006). This implies that the depolymerizing action of Kip3 increases with microtubule length (Varga et al., 2006; Varga et al., 2009). The mode of depolymerization has recently been investigated more in detail: Kip3 has a high affinity towards curved tubulin heterodimers, and once it binds in this conformation, it ceases to hydrolyze ATP and dwells at the microtubule plus end, destabilizing it 14

(Arellano-Santoyo et al., 2017). The authors also show that motility is dispensable for the de-polymerizing action of Kip3 – as expected from the lack of ATP hydrolysis. Even more recently, Kip3 has been shown to be able to side-step from protofilament to protofilament (except at the microtubule seam), potentially enabling it to circumvent obstacles on individual microtubule protofilaments in order to induce catastrophe at the microtubule plus tip (Bugiel and Schäffer, 2018). Side-stepping increases when ATP-conditions are limiting, a feature that seems to depend on the length of the linker region between the two motor domains (Mitra et al., 2018). Its catastrophe-inducing capability is likely also the reason why Kip3 is involved in spindle disassembly (Woodruff et al., 2010; Woodruff et al., 2012). On top of its well-described catastrophe-inducing function, Kip3 has – somewhat paradoxically – also been reported to have microtubule-stabilizing functions. Deleting Kip3 in vivo consistently leads to a lower rescue frequency for cytoplasmic microtubules (Gupta et al., 2006; Su et al., 2011; Fukuda et al., 2014). This conundrum has very recently been resolved: the motor domain seems to be responsible for the catastrophe activity, while the proximal region of the Kip3 tail is responsible for mediating rescue on cytoplasmic / astral microtubules, and the distal region temporally regulates catastrophe induction inside the nucleus, controlling spindle disassembly (Dave et al., 2018). In S. pombe, there are two homologs of Kip3: Klp5 and Klp6. In vivo, Klp5 and Klp6 queue behind Tea2 at the plus end on growing microtubules – when the microtubule no longer grows, the Tea2 signal decreases and Klp5/Klp6 is able to move to the plus tip, inducing catastrophe (Meadows et al., 2018). Klp5 and Klp6 require Mcp1 in order to induce catastrophe at cell ends (Meadows et al., 2018). Interestingly, Mcp1 does not seem to be required in S. cerevisiae for this function (Gupta et al., 2006), since it does not have a homolog.

1.5.5.2 Kinesin-14 / Kar3

Kar3 (after karyogamy – nuclear fusion) is another non-essential kinesin that has been implicated in having a depolymerizing function. Rather than moving towards the microtubule plus end, it “walks” towards the minus end of microtubules (Endow et al., 1994; Middleton and Carbon, 1994). Kar3 can form heterodimers either with Cik1 (chromosome instability and karyogamy) or its paralog Vik1 (vegetative interaction with Kar3; Mackey et al., 2004). Cik1 and Vik1 are involved in targeting Kar3 to cytoplasmic or nuclear microtubules, primarily to enable karyogamy (Mieck et al., 2015) – however during mitosis, Kar3 also promotes kinetochore attachment to chromosomes (Jin et al., 2012).

1.5.5.3 Kinesin-5 / Kip1 and Cin8

In initial studies, at least one of the S. cerevisiae kinesins Cin8 (chromo-some instability) and Kip1 (kinesin-related protein 1) – both part of the Kinesin-5 family – was required for bipolar spindle assembly from the duplicated spindle poles (Figure 1.2, Roof et al., 1991; Roof et al., 1992; Saunders and Hoyt, 1992). Cin8 deletion however has a more severe phenotype, as it is synthetically lethal with inactivation of the spindle assembly checkpoint (Geiser et al., 1997). Kar3 contracts the spindle, while Cin8 and Kip1 push the spindle poles apart. Later, both Kip1 (Fridman et al., 2013) and Cin8 (Gerson-Gurwitz 15 et al., 2011; Roostalu et al., 2011) were found to be bi-directional spindle motors, localized to the spindle midzone (Nannas et al., 2014).

1.5.5.4 Kinesin-7 / Kip2

The kinesin Kip2 has originally been found to have no detectable phenotype upon deletion in vivo (Roof et al., 1992), but this has since then been revised for an opposite effect compared to the microtubule-destabilizing motors Kip3 and Kar3 – as assayed by deletion (Cottingham and Hoyt, 1997; Huyett et al., 1998). This suggests a microtubule- stabilizing function, initially proposed to be driven by moving Bik1 to microtubule plus tips (Carvalho et al., 2004). Later however, in vitro, Kip2 was shown to act as a polymerase itself once it reaches the plus tip, both on bovine brain tubulin and yeast tubulin, with a movement speed of between 68 nm s−1 (at 24.5 ◦C) and 83 nm s−1 (at 28 ◦C), and a processivity of about 4.1 µm (Hibbel et al., 2015). The relationship of Kip2 with Bik1 is not entirely clear, as Kip2 also has been shown to become more processive (4.8 µm) in conjunction with Bik1 and Bim1 (Roberts et al., 2014), but the base processivity of Kip2 without Bik1 and Bim1 in these experiments was substantially lower at 1.2 µm. It is unclear how Bik1 and Kip2 quantitatively function together, and what the mechanism of Kip2’s polymerase function is – in contrast to the polymerizing action of Stu2. Tea2 in S. pombe is the homolog of Kip2 (Browning et al., 2000), and also is assisted in binding to microtubules by the Bim1 homolog Mal3 (Browning and Hackney, 2005). Interestingly, the homologs of Kip2, Bik1 and Bim1 – Tea2, Tip1 and Mal3 – were also used in an in vitro reconstitution experiment that indicated all three components to be necessary for their respective tip tracking (Bieling et al., 2007) and stabilizing function (Meadows et al., 2018).

1.5.6 MAP4 / Mhp1

While yeasts do not share a tau protein, S. cerevisiae does have a homolog to the mammalian MAP4: MAP-homologous protein 1 (Mhp1, Irminger-Finger et al., 1996; Irminger- Finger and Mathis, 1998). It is essential, and was reported to increase microtubule stability using immunoﬂuorescence methods. MAP4 is similar to tau and MAP2, so an attractive hypothesis would be that Mhp1 stabilizes microtubules through lateral binding, though very recent results indicate MAP4 could also interfere with the movement of kinesins (Shigematsu et al., 2018). However, since 1998, no new data on the Mhp1 protein has become available, and how it functions is still largely unknown – even though it is an essential protein – and the Yeast Resource Center Public Image Repository primarily shows cytoplasmic localization (Rifﬂe and Davis, 2010). It would seem that Mhp1 is still an open mystery and could potentially contribute to rescue activity.

1.6 contribution of this thesis

The aim of my PhD work is to investigate the interplay between spindle pole body segregation and microtubule dynamics in S. cerevisiae, including regulation of microtubule dynamics and the involved molecules. Since the cytoplasmic microtubules in budding yeast and their interactors / MAPs can be tracked in vivo, there is hope to establish 16

comprehensive conceptual and computational models that are predictive for various perturbations, for example knockouts, temperature-sensitive mutants, or tagging proteins with degradation tags. On the opposite end, such computational models can be calibrated bottom-up from in vitro reconstitution data. In Chapter 2, we give an overview of modeling techniques for single-cellular organisms that are applicable to spatially-varying dynamic systems. Cytoplasmic microtubule dynamics in budding yeast are a particular application for such modeling techniques, since they are spatially regulated in vivo. Therefore, a technique that allows for em- bedding stochastic microtubule dynamics in a reaction-diffusion model that describes spatiotemporal regulation and signalling is desirable. One such technique is reaction-diffusion master equation (RDME) based modeling, which subdivides the cell into spatially separate reaction volumes that exchange diffusive events with their neighboring ones. Simulating arbitrarily-located and dynamically nucleating, growing and catastrophing microtubules with a GTP-cap in such models has however not been attempted before. In Chapter 3, we describe a computational pipeline that can simulate such models while maintaining or even surpassing the performance of state-of-the-art RDME simulation engines. As an example application that does depend on a complex signaling network, but can be validated against an existing deterministic model, it is then applied to microtubule-nucleated microtubule dynamics in X. laevis egg extract, combining a GTP-cap model for microtubule catastrophe dynamics with microtubule autonucleation. While nucleation in Xenopus laevis egg extract is microtubule-dependent, cytoplasmic microtubules during mitosis in the budding yeast S. cerevisiae are nucleated from SPBs. As detailed in the introduction, there is a wealth of knowledge on microtubule dynamics and associated proteins in budding yeast. This allows us to analyze in detail how in vitro characterizations of microtubule dynamics translate into budding yeast by quantitatively defining the in vivo environment. In Chapter 4, we perform a meta-analysis of available data, starting from requirements for microtubule nucleation and polymerization. This characterization will give an explanation as to why microtubule polymerases are essential in vivo, and can explain a variety of knockout phenotypes. It also forms a basis for future microtubule modeling in budding yeast and fission yeast, as it is the first analysis that quantitatively defines tubulin abundance and allocation in vivo. Finally, in Chapter 5, we investigate the mechanistic basis for the asymmetry of cytoplasmic microtubule dynamics between the two SPBs in S. cerevisiae mitosis. We show that this difference depends on the in vivo function and activity of the Kip2 kinesin motor, which was recently discovered to drive microtubule polymerization in vitro. Using mathematical modeling, fluorescent imaging and structural data, we investigate how this motor is regulated and where it is recruited in vivo. We determined that Kip2 is differentially recruited by the two SPBs, and regulated in a way that prevents its binding to the microtubule lattice – acting as a novel SPB-based “remote control” for microtubule plus end dynamics. 2

BRIDGINGINTRACELLULARSCALESBYMECHANISTIC COMPUTATIONALMODELS

This chapter is published in

Lukas A. Widmer and Jörg Stelling (2018). „Bridging intracellular scales by mechanistic computational models.“ In: Current Opinion in Biotechnology 52, pp. 17–24. doi: 10.1016/ j.copbio.2018.02.005

2.1 abstract

The impact of intracellular spatial organization beyond classical compartments on processes such as cell signaling is increasingly recognized. A quantitative, mechanistic understanding of cellular systems therefore needs to account for different scales in at least three coordinates: time, molecular abundances, and space. Mechanistic mathematical models may span all these scales, but corresponding multi-scale models need to resolve mechanistic details on small scales while maintaining computational tractability for larger ones. This typically results in models that combine different levels of description: from a microscopic representation of chemical reactions up to continuum dynamics in space and time. We highlight recent progress in bridging these model classes and outline current challenges in multi-scale models such as active transport and dynamic geometries.

Cytoplasm Cell Dynamic geometry

Nucleus Active transport Crowding

Graphical Abstract

17 18

2.2 introduction

Rapid developments of-primarily-imaging technologies have revealed the structural and dynamic richness of life at the single-cell level. Structural organization inside cells reaches substantially beyond classical membrane-delimited compartments and stochastic dynamics resulting from low copy numbers of molecules are increasingly recognized as an important contributor to cellular functions such as information processing (Lee et al., 2017). Correspondingly, current cell biology seeks to answer a variety of questions considering drastically different scales, and covering a variety of levels of detail: from the atomistic up to entire cells in space, from picoseconds of enzyme structure fluctuations up to years of population dynamics in time, and from single molecules to crowded macromolecular environments in terms of molecular abundances. While experimental methods such as multi-scale imaging (Follain et al., 2017) may eventually bridge some of these scales, we argue that computational or mathematical modeling is in a unique position to integrate knowledge of varying levels of detail and a variety of heterogeneous experimental data. Using cell signaling as an example, systems biology models have been very successful in achieving a holistic and quantitative understanding of the “signaling ballet in space and time” (Kholodenko et al., 2010). Typical systems biology models in this domain, however, use classical systems of ordinary differential equations (ODEs) that ignore spatial distributions of molecules and assume that all chemical species are abundant enough to warrant treatment as a concentration field (Figure 2.1; Hasenauer et al., 2015; Yu and Bagheri, 2016). To increase spatial resolution, classical partial differential equation (PDE) modeling can describe the evolution of concentrations in time and space; it leads to a purely deterministic model on a spatial domain that is discretized by a mesh for which the temporal evolution of the solution is computed (Figure 2.1; Edelstein-Keshet, 2005). To increase molecular resolution, stochastic models based on solving the well-mixed chemical master equation (CME) usually employ simulation algorithms to infer noise statistics from sampled trajectories (Gillespie et al., 2013). Importantly, efficient simulation algorithms exist for all three model classes, for instance a stochastic simulation algorithm with a runtime that is independent of the number of possible reactions (Thanh et al., 2017). However, while the most prevalent type of model is single-class (using only one of the modeling approaches above) and single-scale (covering only a single spatial, temporal, and abundance scale) (Moran et al., 2010), biological complexity often requires analyzing phenomena on different temporal and spatial scales as discussed above (Yu and Bagheri, 2016). The main challenge of multi-scale models is to resolve the necessary mechanistic details on short time-, length-, and low abundance-scales while maintaining computational tractability for longer and higher ones. This typically results in multi-class models that combine different levels of description. For example, low-abundance species may co-exist with high-abundance species in the same model, making purely stochastic simulation computationally expensive and inefficient. Fully deterministic treatment, in contrast, can lead to inaccurate results due to the low abundance species. High spatial resolution–which leads to low molecule counts per compartment and therefore increases stochasticity–further increases demand for hybrid methods, that is, multi-class approaches that combine stochastic and deterministic representations of cell states. 19

Reaction Rate Chemical Equation Master Equation [C] N

t t S p Mean Concentrations Molecule Counts a t i Reaction Propensities a Reaction Rates l r e s o l u t i Partial Differential Reaction-Diffusion Smoluchowski Brownian o n Equation Master Equation Dynamics Dynamics

1 0

2 6

Local Concentrations Molecule Counts Particle Positions Reaction Rates Reaction Propensities Interaction Radii Diffusion Rates Diffusion Propensities Diffusive Movement per node

per particle Collision Radii per subvolume

Molecular resolution

Figure 2.1 – Spatio-temporal modeling approaches. Reaction rate equation (RRE) models (top left), the most common model class in systems biology that relies on ordinary differential equations (ODEs), disregards space and tracks only the mean of the concentrations (blue) over time. To introduce space in deterministic approaches, partial differential equation (PDE) models track continuous concentrations on mesh nodes and use basis functions to interpolate in the interior, coarse-graining the discrete nature of reacting and diffusing molecules. To increase molecular resolution compared to RRE models requires a stochastic approach; in a system that is assumed to be well-mixed, the chemical master equation (CME) captures how molecules react with reaction propensities depending on molecule counts. Combining increased molecular and spatial resolution is achieved by discretized-geometry, subvolume-based stochastic approaches. The Reaction- Diffusion Master Equation (RDME) partitions space into discrete subvolumes in which molecule counts (blue) are tracked. In these subvolumes, molecules react stochastically, and they diffuse with corresponding diffusion propensities for jumps of molecules from one subvolume to an adjacent one (violet). Continuous- geometry, particle-based stochastic approaches yield the highest resolution. In Brownian Dynamics, each chemical species is represented as a hard-shell sphere with an explicit position (blue), an interaction radius (orange dotted) at which a molecule reacts (reacting particles in color), and a collision radius (black dotted) at which molecules collide when moving by diffusion (violet). When the simulation advances in time, the particles diffuse, collide and react accordingly. Smoluchowski Dynamics considers only the reacting particles (orange reaction radii) and it idealizes particles as points (blue), leading to much faster simulations because collisions of non-reacting particles are neglected. 20

Over the past years, substantial effort was put into bridging modeling approaches with the purpose of both elucidating the mechanistic basis of more coarse-grained modeling approaches, and enabling multi-scale models that use the most adequate and computationally efﬁcient approach for each scale considered. For example, efﬁcient hybrid methods that maintain the accuracy of the solution have recently been developed both for simulating spatiotemporally-resolved (Kim et al., 2017) and well-mixed (Herajy et al., 2017; Marchetti et al., 2016) models. Here, we summarize recent developments of dynamic models based on the underlying physics and chemistry of the biological process considered, which will eventually enable the simulation of large-scale models that accurately account for microscopic phenomena. While statistical and machine learning models (Hastie et al., 2009) are undoubtedly useful and rising in popularity, they fall outside our scope. We will restrict ourselves to recent applications and methodological progress covering the scales from molecular detail, via protein complexes and sub-cellular compartments, up to the cellular scale. Such models have shed light on a variety of phenomena such as cytoplasmic crowding, intracellular transport, intracellular signaling, and control of cell polarization and movement.

2.3 reaction, diffusion, and crowding

A widespread issue in in vivo modeling is accounting for the effects of macromolecular crowding agents (Rivas and Minton, 2016) on reactions and diffusion of molecules of interest in the cytoplasm or the cell membrane (Bressloff and Newby, 2013), especially when comparing to in vitro data, which are usually not generated in a crowded environment. This is a major bottleneck for analyzing atomistic models of the cytoplasm as they were recently developed for the bacterium Mycoplasma genitalum (Feig et al., 2015; Yu et al., 2016). Here, one must consider the collisions between molecules of interest and molecular crowders on nanosecond time-scales (Figure 2.2A). Recently, model-based analysis addressed the effects of crowders of various size (such as they occur in the cytoplasm) on the diffusion of macro-molecules (Kondrat et al., 2015), on their binding, and on simple enzymatic reactions (Gomez and Klumpp, 2015). However, reactions between molecules of interest on a cellular scale typically happen on time-scales that are orders of magnitude longer (Kekenes-Huskey et al., 2016), requiring efﬁcient multi-scale treatment. At the molecular scale, Brownian Dynamics (BD) models molecules as hard spheres with a collision radius and an interaction radius, at which they react (Schöneberg and Noé, 2013; Figure 2.1). The state of the model encompasses the positions and types of all molecules in a continuous (ﬁxed) geometry as they evolve over time. Molecules diffuse by random walks, where inertia is usually disregarded due to the high viscosity and low mass of the cytoplasm (Purcell, 1977). This approach of diffusing and colliding spheres makes it possible to analyze, for example, the effects of molecular crowding on reactions and diffusion of molecules of interest (Kondrat et al., 2015; Gomez and Klumpp, 2015; Meinecke and Eriksson, 2017). However, model simulations are very computationally expensive because all molecule collisions–even with crowders that do not participate in any reactions–must be resolved. Simulation time steps are at the nanosecond scale (depending on the diffusion constant), such that BD models capture dynamics only up to seconds. 21

A Ca2+ Pumps B Voltage-Gated Ca2+ Crowders Ca2+ Channels

Target Active Zone Buffer Protein B

Complex Protein A

C Subunits D Actin Filaments

Microtubule Motors

Pheromone Cdc42 Crowders Filament Nucleation Cdc42 (bound)

Figure 2.2 – Recent applications of dynamic intra-cellular multi-scale modeling. (A) Diffusion of proteins of interest (green and orange) in the cytoplasm of cells is hindered by macromolecular crowders (grey), which can be modelled by colliding and reacting spheres in the Brownian Dynamics framework (Kondrat et al., 2015) (Figure 2.1). Coarse-graining the dynamics leads to length- and time-scale dependent diffusion coefficients: they are high for very short diffusive trajectories as no crowders impede the path of the diffusing molecule, and lower for longer length- and time-scales due to collisions with crowders (Bressloff and Newby, 2013). Crowders also influence binding equilibria by slowing down unbinding reactions, and they can have opposing effects on reaction- and diffusion-limited reaction rates (reaction radii: dotted lines) (Gomez and Klumpp, 2015). These effects depend on crowder size, necessitating multi-scale simulation to analyze larger systems. (B) In neurons, calcium ions flow into the cell by voltage-gated channels when an action potential arrives. Calcium ions bind to buffer molecules and can release vesicles containing neurotransmitter from the pre-synaptic terminal by binding to their neck and causing vesicle fusion (target, in red) (Guerrier and Holcman, 2016). Since binding to the target is a rare event, the full Brownian dynamics are too expensive to simulate, necessitating a multi-scale model. One proposed model uses reaction rate equations for the bulk dynamics of calcium-buffer (un-)binding and calcium export by pumps, and a Markov model for binding to the target (Guerrier and Holcman, 2017). (C) Yeast cell polarization during mating has been modelled in a coarse-grained manner via the interaction of membrane-bound Cdc42 and pheromone, inducing localized actin filament nucleation, and locally increasing Cdc42 binding by active transport on actin filaments (Muller et al., 2016). A more mechanistic description of the interplay between the signaling pathway starting with pheromone binding to Ste2 and the cytoskeleton and transport systems requires integrating phenomena between microscopic and cellular scale. (D) Self-assembly systems such as microtubules assemble from subunits after nucleation. While subunits are usually relatively abundant, directed transport often relies on low-abundance motors. Filaments stochastically switch between growth and shrinkage phases, and over prolonged time periods they can assembly into complex structures such as the mitotic spindle, spanning the domain of a multitude of modeling techniques (Thomas and Schwartz, 2017). 22

To attain longer timescales, one needs to coarse-grain the molecular detail of collisions in the crowded cytoplasm. For example, one can compute an effective cytoplasmic diffusion coefficient that is valid for length-scales greater than micrometers and time- scales greater than seconds (Bressloff and Newby, 2013), but this will underestimate the local diffusivity of molecules. Such coarse-graining allows for neglecting the molecules’ spatial extent. It yields a point particle description, where only the interactions of the reacting molecules are considered, and the crowding macromolecules are abstracted away into a modified diffusion constant. This is the framework of Smoluchowski Dynamics, where point particles interact when they come within a given radius of each other (Figure 2.1). Interaction dynamics in these systems can be simulated exactly, for example, using the Green’s Function Reaction Dynamics (GFRD) approach (Schöneberg et al., 2014), where the time of the next interaction is computed and the algorithm proceeds in a particle-interaction-event-driven fashion. However, simulation engines commonly sacrifice exactness for computational efficiency. For example, the popular SmolDyn software (Andrews, 2017) uses fixed time steps on the order of tenths of milliseconds to evaluate which reactions occur. This enables simulations of processes up to the cellular scale, and for time scales of up to hours. Recent applications of the approach included analyses of the Min systems in Escherichia coli and Synechococcus elongates to elucidate mechanisms of spatio-temporal coordination in bacterial cell division, enabling the investigation of their robustness with respect to diffusive obstacles such as internal membranes in cyanobacteria (MacCready et al., 2017). In any case, it is computationally challenging to include details such as crowding in longer time-scale models, and coarse-graining the microscopic dynamics remains non-trivial. Brownian and Smoluchowski Dynamics operate on a continuous geometry to which particles are confined (Figure 2.1). Coarse-graining by using discretized (spatial) subdomains is a common approach for multi-scaling, but it can lead to significant artifacts (Meinecke and Eriksson, 2017). Only recently, efficient multi-scale simulation of crowding in larger-scale models has been successful, for example, through a modified crowder-free BD algorithm (Smith and Grima, 2017) and a subvolume-based coarse-graining (Cianci et al., 2017). To mechanistically model processes such as allosteric inhibition, more detailed molecule representations than spheres—and thereby even more computational effort—may be necessary (Michalski and Loew, 2016). In addition, because diffusion is a stochastic process, many or long simulations may be necessary to gather the required statistics. A point in case are distributed catalytic activities. For example, in Ras-SOS signaling. Many inactive cytoplasmic SOS molecules need to be tracked as they are recruited to the membrane via a conformational change of a membrane receptor upon ligand binding, but only few highly active SOS molecules effectively catalyze exchange of Ras-GDP with Ras-GTP, ultimately leading to downstream signaling in the nucleus (Iversen et al., 2014).

2.4 spatial stochastic effects in cell signaling

An example of a system recurring in biology is the diffusion and binding of molecules to small target sites that trigger downstream signaling. For instance, in neurons, predicting the timing of intracellular release of neurotransmitters after an influx of Ca2+ ions through voltage-gated calcium channels requires tracking the diffusion of calcium ions to 23 a small target region at the vesicle neck, where the arrival of sufficiently many Ca2+ ions triggers vesicle fusion and neurotransmitter release (Figure 2.2B). However, simulating the full reaction-diffusion system of Ca2+ ions and buffers (and even worse, the crowded cytoplasm) as hard-shell particles long enough to capture the binding to the small domain that is the vesicle neck in the active zone is computationally prohibitively expensive. This is due to the small time- and length-scales at which the particles move, and the long ones required to reach and hit the target on a subcellular scale. Such processes can be described by the Narrow Escape Theory (NET) (Bressloff and Newby, 2013; Holcman and Schuss, 2014). The NET coarse-grains the Brownian Dynamics into a Markovian jump process with a binding rate that is the total flux on the small target (Bressloff and Newby, 2013; Holcman and Schuss, 2014). A multi-scale model for the system combines the NET coarse-graining for the Ca2+ diffusion and binding to the vesicle neck with an ordinary differential equation (ODE) model for the large-scale dynamics of the Ca2+ import/export and binding to buffers (Guerrier and Holcman, 2016; Guerrier and Holcman, 2017). This achieves a crucial speedup over full BD simulations, making predictions of the time required to trigger vesicle fusion computationally feasible. To consider large numbers of molecules (more than approximately 10’000) at the expense of spatial resolution, subvolume-based approaches operate on a discretized version of the spatial domain; they eliminate the explicit description of particles and their positions. The most commonly used subvolume-based method in systems biology relies on the Reaction-Diffusion Master Equation (RDME). It sub-divides the geometry into discrete compartments. Within each compartment, molecules are assumed to be well-mixed. Therefore, the state of the system is represented as molecule numbers within each subvolume (Figure 2.1). Molecules react according to the chemical master equation (CME), that is, the probability of a reaction occurring depends on the current state of the subvolume. To implement diffusion between compartments, molecules stochastically jump from one subvolume to an adjacent one. The rate of these jump reactions can either be inferred directly from the diffusion constant in case of a uniform discretization, or by discretizing the diffusion PDE on a non-uniform (for example, tetrahedral) mesh. Recent methodological developments have clarified the issue of very fine discretizations leading to a loss of all bimolecular reactions (Hellander et al., 2015; Hellander and Petzold, 2017), finally giving this method a mechanistically sound theoretical basis. Indeed, one can refine the discretization up to a critical point, such that the solution of the RDME converges toward the solution given by the microscopic Smoluchowski Dynamics. If the dynamics of the system need to be resolved close to this level of detail, however, RDME approaches become computationally inefficient. Multi-class methods that integrate both subvolume-based and particle-based simulation can tackle this problem. They are now readily available (Robinson et al., 2015) and a scheme with automatic partitioning of the system has also recently been proposed (Hellander et al., 2017). If single simulation speed (and not generation of an ensemble of trajectories) is a bottleneck, parallelization of the discrete event simulation can provide a speedup, linking the classic CME stochastic simulation field with the one of parallel discrete event simulation (Lindén et al., 2017), at the expense of ensemble simulation efficiency. In recent applications to cell signaling, the RDME framework was used to analyze stochastic variability in cell cycle timing in Caulobacter crescentus (Li et al., 2016) and ribosome biogenesis during cell division of E. coli (Earnest et al., 2016). An analysis of 24

the Hes1 system, which is responsible for timing of somite segmentation in embryonic development, illustrates that spatial organization – and models that capture it – can have profound impact on the mechanistic interpretation of biological systems. Negative, time-delayed Hes1 feedback causes Hes1 protein oscillations, which has classically been modelled using non-spatial approaches with explicit time delay. Only recently, the spatial diffusion-driven origin of this temporal delay and its effects on oscillatory behaviors of gene regulatory networks (GRNs) were elucidated (Macnamara and Chaplain, 2016).

2.5 challenge: active transport

So far, we assumed that molecules move purely by diffusion, but intracellular transport can be mediated by the actin and microtubule cytoskeleton in an active, directed manner by molecular motors. This requires energy (ATP) consumption – meaning that this type of transport cannot occur at thermodynamic equilibrium. Cells need active transport because purely diffusive transport can take too long to reach far distances, for instance in neurons, and diffusion makes it difficult to target specific molecules to specific areas of the cell (Bressloff and Newby, 2013). Indeed, microtubule dynamics (Barsegov et al., 2017) have also been suggested as a source of oscillation in GRNs, where the time delay required to sustain oscillations such as in the Hes1 example can be explained by the spatial dynamics of transcribed mRNA. mRNA is exported out of the nucleus, and then transported along microtubules before translation occurs in the cytoplasm; the negative feedback is closed by the protein re-entering the nucleus and blocking its own transcription (Szymańskaet al., 2014). As another example, cell polarization, the process of establishing an intracellular spatial axis (Rappel and Edelstein-Keshet, 2017) enables directed motion of the cell for chemotaxis in E. coli, and budding yeast mating. Here, cells must re-distribute internal components to respond to a non-uniform external stimulus. Typically, active transport is modelled using PDEs (Bressloff and Newby, 2013), for example, for molecular motors moving on microtubules by a totally asymmetric exclusion process (TASEP). In yeast, it recently became possible to predict cell polarization due to pheromone gradients during mating using a stochastic PDE model (Muller et al., 2016). This model applies coarse-graining of the microscopic transport of the small Rho GTPase Cdc42 by diffusion and induction of actin-cable-mediated directed transport by Cdc42-pheromone co-localization (Figure 2.2C). It predicts the dynamics and direction of polarization on the cellular scale successfully, but, as the authors concede, the model is phenomenological. It does not explicitly consider mechanistic details of the interplay between signaling molecules such as Cdc42 and their feedback onto cytoskeleton and transport dynamics. To account for them, one needs to integrate short time-scale dynamics of pheromone arrival and sensing, downstream signaling, longer time-scale modification of the cytoskeleton, and ultimately, cell polarization.

2.6 challenge: dynamic cellular geometries

Cell polarization is illustrative of another challenge, namely that cellular (here: microtubule) geometries are dynamic. When modeling self-assembly systems such as the actin and microtubule cytoskeletons or virus capsid self-assembly, the respective dynamics are affected by transport on an ever-changing geometry, diffusion, subcellular compartmen- 25 talization, chemical reactions, and molecular crowding (Figure 2.2D). Integrating all these phenomena into a coherent model requires efficiently coupling the multiple relevant temporal and spatial scales (Thomas and Schwartz, 2017). Since there may be more possible intermediate states of assembly than molecules available – similar to the problem of exponential state explosion when modeling combinations of post-translational modifications in intracellular signaling (Münzner et al., 2017) – the space of possible states cannot be enumerated explicitly for many systems, which presents a problem for ODE and PDE models. Such systems have to be captured by a rule-based approach, only modeling the limited number of molecules in the system. Doing this in a spatially-resolved manner using Smoluchowski and Brownian Dynamics approaches (Figure 2.1), however, only recently gained traction (Yu et al., 2016; Schöneberg et al., 2014; Thomas and Schwartz, 2017). Dynamic geometries also gained renewed attention because they arise from the creation and destruction of intracellular compartments such as P-granules and nucleoli by phase separation, where their constituting proteins locally self-organize into a liquid phase. Importantly, formation of these sub-cellular structures can be regulated by its heterogeneous components in vivo, for example, by post-translational modifications, and their mis-regulation has been associated with disease (Shin and Brangwynne, 2017). Modeling efforts for such structures have only recently started, and currently require specialized simulation techniques (Jacobs and Frenkel, 2017).

2.7 challenge: model calibration and inference

Once correctness and accuracy of a simulation is established, usually, parameters must be estimated by minimizing the discrepancy between model predictions and experimental data (Hasenauer et al., 2015; Heinemann and Raue, 2016). This has become feasible for large-scale ODE models (Fröhlich et al., 2017; Penas et al., 2017), and to a lesser extent, for stochastic models (Schnoerr et al., 2017). However, the calibration of multi-scale, multi-class models remains difﬁcult in practice (Babtie and Stumpf, 2017); it is mostly achieved by roughly calibrating the parameters manually, measuring them, or estimating them by using sub-modules (Lang and Stelling, 2016) or approximations of the model (Hasenauer et al., 2015; Lan et al., 2016). Similarly, quantifying parametric uncertainty after estimation with, for example, sampled parameter distributions from Markov Chain Monte Carlo techniques (Ballnus et al., 2017), remains computationally too expensive to apply in multi-scale models. Further research is needed to make these tools useable (Tuncer et al., 2016). Finally, once a model is properly calibrated, we might want to address model selection problems, for example, test different model versions to ﬁnd out which mechanisms help describing the data best, while maintaining a reasonable model complexity (Geris and Gomez-Cabrero, 2016). Model selection already requires heuristics for stochastic and ODE models due to the combinatorial explosion of model variants when testing different combinations of hypotheses, and it is an open problem for multi-scale models (Jagiella et al., 2017).

2.8 conclusions

Simulation and analysis methods for multi-class and multi-scale models have recently made major leaps forward, but determining the relevant scales for a particular problem 26

and coupling them in a computationally efficient manner remains challenging (Yu and Bagheri, 2016). Correspondingly, there is a lot of room for development of novel methods, especially in the cases where the model cannot be simplified or reduced. The challenges of bridging scales in cellular biology have already lead to computational solutions not found in other scientific domains. For example, reactive multiparticle collision (MPC) dynamics models reactants as explicit particles, but diffusion as a random rotation of the particle velocities, which has allowed to efficiently model diffusion-influenced signaling pathways such as the CheY/Z chemotaxis system in E. coli (Strehl and Rohlf, 2016). As another example, the spatio-temporal chemical master equation (ST-CME), a specific type of RDME, considers metastable sub-compartments (Winkelmann and Schütte, 2016), which could be tailored to analyze phase separation phenomena in cell biology. As a missing methodological component, the uncertainty in solutions of multi-scale models resulting from numerical errors or biological uncertainty is usually not quantified. Multi- scale modeling could benefit from the renewed attention to uncertainty quantification for differential equations (Conrad et al., 2017) to balance numerical accuracy with (problem- specific) biological variability. As illustrated by the three challenges discussed in detail above, it is a long way ahead to mechanistic, physics-based whole-cell models that, among others, account for cytoskeletal architecture, macromolecular crowding based on the cell’s known composition, and the evolution of reactions given this complex environment. Such models, however, could really bridge the world between in vitro and in vivo experiments. They may also allow for targeting cell behaviors in novel ways by exploiting spatial information (such as effectors of nuclear import) and by more precisely interfering with transport processes mediated by the cytoskeleton.

2.9 acknowledgements

We thank Sabine Österle for feedback on the manuscript. We acknowledge ﬁnancial support by the SystemsX.ch RTD Grant #2012/192 TubeX of the Swiss National Science Foundation. Conﬂict of interest: none declared. 3

STOCHASTICSIMULATIONOFMICROTUBULESWITHTHE COORDINATE-AWARE C++ REACTION-DIFFUSION MASTER EQUATIONSIMULATORRDMECPP.

This chapter is to be submitted as

Lukas A. Widmer and Jörg Stelling. “Stochastic simulation of microtubule dynamics with the coordinate-aware C++ reaction-diffusion master equation simulator RDMEcpp.”

3.1 abstract

Modelling microtubule dynamics in cells together with regulatory networks requires integrating the spatiotemporal evolution of the regulating molecules with stochastically growing and shrinking microtubules. Such stochastic spatiotemporal models of cells can, in principle, be simulated using the reaction-diffusion master equation (RDME)- type framework. Since there is no existing software that simulates RDME models with embedded ﬁlaments that, themselves, evolve according to a stochastic model, we developed RDMEcpp – a simulation engine that can handle such models. RDMEcpp is a high-performance, extensible, and cross-platform solution in modern C++ for simulating 1D-3D RDME-type models that require subvolume coordinate lookup at runtime, e.g., to determine concentrations along dynamic microtubules. RDMEcpp exhibits similar or better performance compared to the state-of-the art URDME on the MinD oscillation model from Escherichia coli. We further demonstrate its capabilities by simulating 1D Xenopus laevis egg extract spindle autonucleation, which so far was only modelled deterministically. Explicit microtubules in this model evolve according to a more realistic stochastic microtubule tip model, which determines the time until catastrophe occurs. We contrast the results of the RDME model with the deterministic, partial differential equation model, and test how microtubule tip evolution impacts aster microtubule density. We achieve good agreement between the two models, with the exception of the nucleation rate. In future work, we hope to extend this model to 2D and 3D asters, incorporating nucleation angle measurements between microtubules.

27 28

3.2 introduction

Stochastic simulation of well-mixed systems is a standard method to analyze the effect of intrinsic noise in biological systems (Gillespie et al., 2013). However, spatially-resolved stochastic models are a more recent development. For a comprehensive overview on both well-mixed and spatially-resolved stochastic chemical kinetics as well as simulation algorithms, we refer to (Widmer and Stelling, 2018). The two major approaches for modeling stochastic cellular processes in time and space are the reaction-diffusion master equation (RDME), as implemented, e.g., in URDME (Drawert et al., 2012), and stochastic point-particle reaction-diffusion, as implemented, e.g., in Smoldyn (Andrews, 2017). They have been applied to great effect in the past, e.g., for describing noise-induced phenomena in the MinD system during cell division in E. coli (Fange and Elf, 2006).

Our goal is to be able to model stochastically growing and shrinking microtubules in cells together with spatiotemporal regulatory networks that can interact with these dynamic microtubules via microtubule-associated proteins. As an example of spatiotemporally regulated microtubule dynamics, we investigate the fundamental biological problem of size regulation on the mitotic spindle (Goehring and Hyman, 2012). In X. laevis eggs extract, the size of the mitotic spindle scales with droplet size (Good et al., 2013), which suggests the size of the mitotic spindle is regulated by the amount of material present. This mode of regulation can work well for single organelles, but does not allow for tight size regulation in the case of many regulated entities (Suarez et al., 2017; Mohapatra et al., 2017) – such as in our case, microtubules. Indeed, it was recently uncovered that the size of microtubule asters in X. laevis is not limited by the amount of tubulin available (Decker et al., 2018), meaning that the amount of building blocks that make up the structure is not limiting. The size of the resulting aster (Figure 3.1) is rather a consequence of autoregulated microtubule nucleation: aster size depends on the amount of active nucleator, which in turn both depends on the total amount of nucleator available, and the amount of it that is activated at the aster. We extend the deterministic model for autocatalytic microtubule nucleation developed in (Decker et al., 2018) by interfacing it with a more realistic stochastic microtubule tip model (Brun et al., 2009) that accounts for microtubule lifetime increasing with growth velocity (Walker et al., 1988) as well as their non-exponential lifetime (Odde et al., 1995; Gardner et al., 2011; Bowne-Anderson et al., 2013), and investigate how this changes predictions.

Here, we present a cross-platform, high-performance simulation framework that interfaces with COMSOL and MATLAB, with import/export facilities for (Py-)URDME (Drawert et al., 2012) models. In contrast to the method presented by Drawert et al., 2012, the RDMEcpp solver is spatially aware – the edges (1D), triangles (2D), and tetrahedra (3D) that make up the primal mesh are exported from the mesh generator and passed into the RDME solver, such that binding sites on arbitrarily-oriented ﬁlaments such as microtubules can be mapped between spatial coordinates and RDME subvolumes. In addition to the standard RDME propensity and reaction interfaces, which only depend on and modify the local subvolume, RDMEcpp provides extended reaction functions that enable simulation of microtubules growing and shrinking across subvolumes. This allowed us to simulate the autonucleated microtubule dynamics centered around a monopole / aster from X. laevis. 29

A B 160 140 120 100 80 Intensity (a.u.) 60 0 5 10 15 20 25 Distance from center of aster (μm)

Figure 3.1 – X. laevis egg extract monopole. (A) Aster reproduced from (Decker et al., 2018) under the CC-BY license. Scale bar corresponds to 20 µm, yellow circle indicates region quantified for generation of radial profile (radius 25.9 µm). (B) Radial profile of monopole, quantified using the “Radial Profile Extended“ plugin of Fiji (Schindelin et al., 2012).

3.3 materials and methods

3.3.1 Geometry, meshing and matrix assembly

For specification of the model geometry we use COMSOL Multiphysics, as it has an interface that easily allows for generating complex geometries, and has successfully been applied to biologically-relevant imaging-based geometries (Iber et al., 2016). Diffusion and mass matrix are then automatically assembled by COMSOL and imported into RDMEcpp, similar to URDME (Drawert et al., 2012). For a derivation of the assembly method used in URDME, see Engblom et al., 2009. We re-derive the diffusion and mass matrices from the finite element formulation in the supporting information, section 3.8.1. The diffusion matrix is subsequently converted to a jump matrix with non-negative coefficients. Several methods for this conversion exist (Meinecke et al., 2016), therefore we provide a modular interface which allows for swapping out and comparing methods easily (Figure 3.2). Note that if a completely open-source pipeline is preferred, we provide a converter for PyURDME models that enables them to run with RDMEcpp and converts the results back into PyURDME format for further analysis using that pipeline (Drawert et al., 2016). However, this interface does not support spatially-aware models, and no custom reactions that can update arbitrary subvolumes, which are required for subvolume-crossing microtubule growth / shrinkage.

3.3.2 Point-to-subvolume mapping

To enable spatially-aware simulation at runtime, the solver binary (Figure 3.2) must be aware of the simulation geometry, and be able to map points in 1D, 2D and 3D to the appropriate RDME subvolume. To achieve this, the geometry (1D: edges, 2D: triangles, 3D: tetrahedra) is passed as part of the HDF5 simulation data structure into RDMEcpp. Since the linear basis functions chosen for the ﬁnite element model discretization are barycentric, it is convenient to test a point p for membership within a primal mesh element by computing the local coordinates li, and testing for

0 ≤ li ≤ 1 ∀i, i ∈ [1, ..., 1 + D], 30

Geometry and Meshing Modular Matrix Converters Model File Native Code Data Analysis and Visualization Diffusion Coeff. Simulation Engine

Stoichiometric Mass Sub- and Dependency Matrix Volumes Matrices HDF

Discretized

HDF Geometry Area of rectangle

#include 4.5 μ m using namespace std;

const double pi = 3.14159;

int main() { float length, width, area;

cout << "Enter The Length Of The Rectangle: "; cin >> length; cout << "Enter The Width Of Rectangle: "; cin >> width; Initial State area = length*width;

Diffusion Jump cout <<"The area of the rectangle is : "<< area << endl;

return 0; Time Steps }

Matrix Matrix 1 μm Custom Reactions and Propensities MATLAB COMSOL MATLAB RDMEcpp COMSOL

Figure 3.2 – Schematic of the RDMEcpp modeling pipeline for the example of the MinD system in E. coli. Items highlighted in red correspond to steps where manual input of the modeller is required. The geometry, chemical species and their diffusion coefficients are specified in COMSOL, which also serves as a mesh and diffusion matrix generator (in principle, any FEM suite that is able to generate these can be used here, in Section 3.8.1 we describe their conversion to jump matrices and dual mesh subvolumes). These are then exported to MATLAB, where the stochiometric and propensity update dependency matrices, as well as initial state and time steps are specified. All data required to start a simulation is packaged into a HDF5 file, which is loaded by the C++ simulation binary. In order to enable high-performance simulation, the C++ code for the propensities and reactions is compiled into this binary. Finally, the results are written into an HDF5 file again, which can be loaded into MATLAB (and subsequently, COMSOL), as well as Python, for further analysis, visualization, parameter optimization, or other custom workflows like the custom microtubule visualization in Figure 3.8, and comparison to analytical solutions in Figure 3.5A.

where D is the dimension of the model. Once the primal mesh element p belongs to is established, the dual mesh element / node j which p belongs to can be established by ﬁnding

i = arg max li, i and mapping the local node index i to the global node index j.

3.3.3 Microtubule dynamics across subvolume boundaries

Microtubule dynamics can cross subvolume boundaries, i.e., a microtubule growing in an arbitrary direction will, at some point, cross from one subvolume to the next. This violates the assumption that reactions only directly act on the subvolume they occur in, and the effect on other subvolumes is only via diffusion. Thus, the propensities in the Next Subvolume Method (NSM) simulation algorithm cannot be updated with a dependency matrix that only covers intra-subvolume reactions. To remedy this, RDMEcpp supports an extended reaction interface, where the propensities and subvolumes that need to be updated after each reaction can be specified by the reaction implementation dynamically. For example, the hydrolysis reaction for a microtubule can either just lead to hydrolysis at the tip, in which case only the hydrolysis propensity must be updated, or subsequently to catastrophe. In the latter case, all the nucleators bound to the microtubule dissociate, which means all the reaction propensities of the subvolumes spanned by the microtubule must be updated. Efficiently dissociating all nucleators bound to a specific microtubule also requires a data structure that allows for fast lookup of all nucleators bound to that microtubule, and of all subvolumes the microtubule spans. For this purpose, we leverage the Boost 31 multi-index container with a hashed index for each lookup combination we need to be able to perform. To verify the correctness of reactions implemented in this way, in debug mode, RDMEcpp checks the consistency of all the subvolume propensities versus de-novo computed ones, and aborts if an inconsistency is detected.

3.3.4 Data analysis and visualization in MATLAB

Results are saved in the HDF5 file format and thus can be directly imported into MATLAB via drag-and-drop or the load command, or imported into Python using the h5py package. Data analysis and visualization can then be performed in MATLAB and Python, and result import into COMSOL is available via MATLAB. RDMEcpp provides functions for mesh visualization, as well as surface and volume visualization over the time course of a spatiotemporal stochastic simulation. Time course animations can be exported as movies on all supported platforms. For example, the time course data can further be used to perform in silico fluorescence imaging experiments, as we previously did (Samuylov et al., 2015). For visualizing model output in MATLAB, several plotting options are available and summarized in Figure 3.3. In contrast to URDME (Drawert et al., 2012), the model does not need to be re-imported into COMSOL for visualization – this can directly be performed in MATLAB. All concentration visualization plots can also be viewed as movies interactively in MATLAB, or exported to movie files.

A B C D

4 4 4 4

3 3 3 3

2 2 2 2 z (µm) 1 1 1 1

0 0 0 0

0.5 0.5 0.5 0.5 y (µm) 0 0.5 0 0.5 0 0.5 0 0.5 0 0 0 0 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 x (µm) Figure 3.3 – Overview of visualization options available in RDMEcpp for the 3D E. coli example model. (A) Primal mesh nodes colored by subdomains: cytoplasm (blue), and membrane (red). (B) Surface concentration of MinD bound to membrane, with semitransparent visualization of primal mesh. (C) Concentration of membrane-bound MinD in the E. coli cell in the tetrahedral primal mesh elements. (D) Same concentration displayed via coloring the primal mesh node corresponding to the respective subvolume. Concentrations in (B–D) are indicated low to high – blue to red. 32

3.4 results and discussion

3.4.1 Stochastic simulation engine

The simulation engine currently supports our high-performance implementation of the next subvolume method (NSM, detailed in Fange and Elf, 2006). It is developed against the C++11 standard, using header-only libraries whenever possible, such as Eigen for vectorized matrix algebra and Boost for data structures. In contrast to PyURDME (Drawert et al., 2016), no manual memory management is required, and many common matrix operations are available, simplifying implementation of new algorithms substantially. The only binary dependency is libhdf5 through matFileCpp, our minimal library for reading/writing v7.3 MATLAB .mat ﬁles to/from Eigen matrices (available from https: //gitlab.com/csb.ethz/matFileCpp). libhdf5 is available for all tested systems either as a package or as a binary installer, as well as source code.

3.4.2 Simulation performance

Even though RDMEcpp uses a higher-level language and is geometry-aware, a speedup of up to 12% compared to the already highly-optimized implementation in URDME (Drawert et al., 2012) could be achieved (Table 3.1) by optimizing random number generation, the binary heap data structure used in the next subvolume method simulation algorithm, as well as optimizing for (execution-time-dominating) diffusion events.

Table 3.1 – Simulation performance of RDMEcpp versus state-of-the-art. Simulation performance of RDMEcpp and URDME Drawert et al., 2012 on the E. coli MinD example model with 1555 subvolumes as shown in Figure 3.3, with 5 species and 5 reaction channels. Indicated times are the mean simulation runtime and standard deviation over 20 runs. Compiled with Eigen 3.3.2 and Boost 1.6.3 on x64, respectively, on an Intel Core i7-6600U for Ubuntu and Windows, and an Intel Core i5-4570S for OS X, with 16 GB RAM each.

Operating system Compiler version URDME (s) RDMEcpp (s)

Windows 10 MSVC 2015 N/A 157 ± 3 Ubuntu 16.04 GCC 5.4 199 ± 6 178 ± 5 Ubuntu 16.04 Clang 3.8 193 ± 3 172 ± 3 OS X 10.11.3 AppleClang 7.3.0.7030031 158 ± 3 156 ± 4

3.4.3 Deterministic model of autocatalytic microtubule nucleation

Our deterministic model for autonucleated microtubule asters is based on (Decker et al., 2018) and is summarized in Figure 3.4A. We consider three types of molecules, and thus concentration variables: active, unbound nucleator nu, bound nucleator nb, and the microtubule density ρ. The unbound nucleators nu evolve according to Equations (3.1) and (3.4): they diffuse with diffusion coefficient D, bind to microtubules ρ with rate kblb, and unbind either from a microtubule at rate ku, or, in contrast to (Decker et al., 2018), when microtubule turnover at rate Θ happens. They also inactivate at rate k0, and active unbound nucleator flows in at x = 0 with flow Γ. 33

n (x): active, unbound nucleator A u D Γ n (x): active, bound nucleator vp k b ρ(x): ρ(0) bra Aster microtubule density ku kblb Θ ∅ ∅ k0 deterministic model x=0 B stochastic model D Γ v k p ρ(0) bra,s Aster kblb ku

∅ k0

C vstoch GTP hstoch,1 GTP GTP hstoch,2 GDP vstoch GDP GTP catastrophe ∅ GTP GDP GDP GDP catastrophe ∅

Ntip=1 Ntip=2 stochastic microtubule tip models

Figure 3.4 – Overview of deterministic and stochastic microtubule aster autonucleation models. (A) Deterministic model based on (Decker et al., 2018): active, unbound nucleator ﬂows into the simulation domain from the center of the microtubule aster at x = 0 with rate Γ (for summary of parameters, see Table 3.2). Unbound nucleators diffuse with diffusion constant D, get inactivated at rate k0, and bind to a continuous microtubule density ρ at rate kblb. Bound nucleators nb no longer diffuse, and dissociate with rate ku. New microtubule density ρ is nucleated from bound nucleators with rate kbra. Microtubules then move away from the aster with a growth velocity of vp, and turn over at rate Θ.(B) Stochastic microtubule aster model: in contrast to the deterministic model, microtubules are modelled as protoﬁlaments with distinct 8 nm sites (corresponding to tubulin heterodimers). Each site can either be free, and a nucleator can bind to it, or occupied, in which case binding is not possible. Rather than creating a continuous density, each nucleator can nucleate a discrete microtubule with rate kbra,s. In addition, each nucleator can only nucleate one microtubule at a time – the assumption being that the nucleator contains some kind of template for the microtubule lattice, though this may not be required (Roostalu and Surrey, 2017). (C) Stochastic microtubule tip model: microtubules grow in 8 nm increments with a rate vstoch, and have a hydrolysis rate of hstoch. Each tip tubulin heterodimer can either be GTP-bound, or GDP-bound. As a minimal model than can reproduce existing data (Bowne-Anderson et al., 2013), we use the model presented in (Brun et al., 2009). For Ntip = 1, catastrophe immediately occurs upon GTP hydrolysis, and for Ntip = 2, both sites must hydrolyze for catastrophe to occur. Upon catastrophe, the microtubule is assumed to instantly depolymerize, and all bound nucleators unbind and become free active nucleators again. 34

The bound nucleators nb evolve according to Equation (3.2): unbound nucleators arrive with rate kblb, and bound nucleators dissociate by unbinding with rate ku, or due to microtubule turnover with rate Θ. Finally, the microtubule density ρ evolves according to Equations (3.3) and (3.5): the microtubule density in the center of the aster is constant, and microtubule tips move away from the center of the aster with speed vp, while bound nucleators create new microtubules at rate kbra, and microtubules turn over with rate Θ. This description results in the following equation system: ∂n (x, t) u = D∇2n − k l n ρ + (k + δΘ) n − k n (3.1) ∂t u b b u u b 0 u ∂n (x, t) b = k l n ρ − (k + δΘ) n (3.2) ∂t b b u u b ∂ρ(x, t) = −v · ∇ρ + k n − Θρ (3.3) ∂t p bra b ∂n (x, t) −D u = Γ (3.4) ∂x x=0 ρ(0, t) = ρ∗(0) (3.5) The original model does not account for bound nucleator dissociating during microtubule depolymerization, i.e., it is equivalent to Equations (3.1)–(3.5) with δ = 0. Here, by setting δ = 1, we can include a ﬂux of −δΘnb for the time derivative of bound nucleator nb in Equation (3.2) – this describes the unbinding of nucleator due to microtubule turnover. We then assume these nucleators are available to bind another microtubule and thus add to the unbound nucleators nu in Equation (3.1). The parameters for this model, and the following stochastic model, are summarized in Table 3.2. We can solve for the steady state solution of Equations (3.1)–(3.5), which is given by √ ∗ Γ − k0/Dx nu(x) = √ e Dk0 = n∗ (0)e−x/lu ,(3.6) u √ − k /Dx k k l Γ k k l Γe 0 +xθk (ku+δΘ) ∗ ∗ bra b b − bra b b 0 ρ (x) = ρ (0)e k0(ku+δΘ)vp e k0(ku+δΘ)vp − −x/lu − θ x = ρ∗(0)eα(1 e )e vp ,(3.7) √ − k /Dx ∗ √ k k l Γ k k l Γe 0 +xθk (ku+δΘ) kblbΓρ (0) bra b b − bra b b 0 ∗ − k0/Dx k0(ku+δΘ)vp k0(ku+δΘ)vp nb(x) = √ e e e (ku + δΘ) Dk0 −x/lu − θ x ∗ −x/lu α(1−e ) vp = nb(0)e e e ,(3.8) where we deﬁne auxiliary variables s ∗ Γ D Γl0kbra lbkb nu(0) := √ , lu := , α := , l0 := , Dk0 k0 vpk0 ku + δΘ ∗ ∗ ∗ nb(0) := nu(0)l0ρ (0).

3.4.4 Stochastic model of autocatalytic microtubule nucleation

For the stochastic microtubule model, we make the following changes (Figure 3.4B): each microtubule is now modelled as a ﬁlament with 8 nm binding sites, and grows at rate 35

Table 3.2 – Parameter Overview. Parameters used for the deterministic and stochastic autocatalytic microtubule nucleation models in this study.

Parameter Value Description & Notes

11 −1 −1 kb 1.35 × 10 s µm Nucleator binding rate, arbitrary ratio with lb, l0 3 lb 10 µm Nucleator binding length scale −1 ku 1 min Nucleator unbinding rate −1 kbra 5/3 Θ = 0.083 s Branching nucleation rate, example in Decker et al., 2018, Figure 4 Supplement 1. −1 k0 0.097 s Nucleator deactivation rate, inferred from D and lu −1 vp 21 µm min Microtubule growth rate, measured from monopole (Decker et al., 2018) Θ 0.05 s−1 Microtubule turnover rate, measured in Decker et al., 2018 Γ 2.24 µm µm s−1 Active (unbound) nucleator inﬂux at centrosome D 45 µm2/s Nucleator diffusion constant – assumed to be same as tubulin (Krouglova et al., 2004)

∗ √ nu(0) 4.96 µm = Γ/ Dk0, unbound nucleator concentration at center ρ∗(0) 5.4 µm Polymerized tubulin concentration at center ∗ ∗ ∗ nb(0) 2 µm = nu(0)l0ρ (0), bound nucleator concentration at center √ lu 24.9 µm = D/k0, model ﬁt by Decker et al., 2018 3 l0 81 m /m = lbkb/ku α 2.38 = Γlbkbkbra = Γl0kbra , model ﬁt by Decker et al., 2018 vpk0ku vpk0

−1 vstoch 43.75 s Subunit attachment rate for the stochastic model (Brun et al., 2009), converted from vp using a subunit length of 8 nm. −1 hstoch,1 0.05 s Hydrolysis rate for the stochastic model (Brun et al., 2009) with N = 1, equal to Θ from Decker et al., 2018. −1 hstoch,2 0.85 s Hydrolysis rate for the stochastic model (Brun et al., 2009) with N = 2, 2 converted using T ≈ vstoch/(3hstoch), and a microtubule mean life time of T = 20 s from Decker et al., 2018.

vstoch. The microtubule number at x = 0 is fixed – this is implemented by having a fixed number of microtubules with their minus ends nucleated at x = 0, and immediately re-nucleating them if they catastrophe. Nucleators can bind to the growing microtubules, and each binding site can at most contain one nucleator. Each nucleator can then nucleate a discrete, new microtubule at rate kbra,s – note that in contrast to the deterministic model, this nucleation reaction does not increase with the local nucleator concentration. The nucleator is then occupied by this microtubule until it catastrophes, and only then can it nucleate a new microtubule. Catastrophe occurs according to the microtubule plus tip model described in (Brun et al., 2009) and illustrated in Figure 3.4C. The basic idea of this model is that upon hydrolysis of the stabilizing GTP-tubulin cap at the growing microtubule plus end (reviewed by (Bowne-Anderson et al., 2013)), the microtubule catastrophes. This tip has either one layer (Ntip = 1) or two layers (Ntip = 2). Upon catastrophe, the microtubule instantly depolymerizes (since its shrinkage speed is much faster than growth speed), and all attached nucleators dissociate. If a nucleator dissociates, any possible microtubule that it nucleates remains until it experiences catastrophe at its respective plus tip. We discretize the model uniformly, and include a special microtubule dynamics subvolume without a valid spatial coordinate, as well as a node at x = 0 that has an influx reaction (FigureS 3.1). 36

3.4.5 Steady state solution and dynamics

We first check the accuracy of the analytical solution of the deterministic model detailed in Figure 3.4A by numerically simulating Equations (3.1)–(3.5), and comparing against the analytical steady state given in Equations (3.6)–(3.8). The numeric simulation results are in excellent agreement with the analytical solution where no nucleator unbinding due to microtubule turnover is considered (δ = 0, Figure 3.5A). When unbinding of nucleators due to microtubule turnover is considered (δ = 1, Figure 3.5B), the concentration of bound nucleators is lower, as expected. This in turn leads to a more rapid decay of the microtubule density with increasing distance from the monopole – the density ρ(0) at x = 0 is fixed, thus there is no effect at the monopole itself. Next, we check how well this analytical solution matches the stochastic model detailed in Figure 3.4B. First, we fix the microtubule density by instantiating two fixed microtubules in the simulation domain (Figure 3.6A). Here, the limited number of binding sites for nucleators on the microtubules decreases the amount of bound nucleator with respect to the deterministic model, which does not have a notion of binding sites. Having established nucleator influx, binding and unbinding, we next instantiate two microtubules that have growth, hydrolysis and catastrophe determined by the tip model illustrated in Figure 3.4C, but do not allow for autonucleation (kbra,s = 0). We run both the tip model with exponentially-distributed time-to-catastrophe (Ntip = 1) and a more realistic one with non-exponentially-distributed time-to-catastrophe (Ntip = 2, Brun et al., 2009). Both models behave very similarly (Figure 3.6BC), indicating that the exponential lifetime approximation for individual microtubules is inconsequential for radial microtubule density in the aster. For both stochastic models, the improved deterministic model that accounts for nucleator unbinding upon microtubule catastrophe quite accurately tracks the simulation mean (Figure 3.6BC). Finally, we manually fit k0 and kbra,s to experimental monopole data from (Decker et al., 2018), where, as far as we could ascertain, the central part where the monopole is located and microtubule density is lower, was not used for the fit. The nucleation process in the stochastic model is conceptually different from the deterministic model: in the deterministic case, the amount of microtubule density created is determined by the concentration of bound nucleator alone. In the stochastic model, each bound nucleator can at maximum create one microtubule. We deem this model more reasonable given that a microtubule – if it indeed stays bound to the nucleator – will likely bind to a substantial area on the nucleators, and the nucleators must be able to diffuse reasonably quickly, and thus cannot be very large. We could achieve a reasonable qualitative agreement between intensities (Figure 3.7) with kbra,s approximately 100× lower than the deterministic parameter kbra. An example time course of microtubule dynamics is shown in Figure 3.8. While the fit could be optimized further, a more realistic model will have to take into account the three- dimensional nature of the monopoles, and the angle between nucleating and nucleated microtubules, which will change the resulting parameters. 37

A B

Figure 3.5 – Steady state of deterministic autonucleated microtubule aster models. Bound and unbound nucleator concentration, as well as microtubule density, are shown as a function of distance from the center of the aster at x = 0 µm.(A) Results obtained by simulating equations (3.1)–(3.5) with the parameters in Table 3.2, and δ = 0. Numerical results are plotted as solid lines, and the analytical solution given by Equations (3.6)–(3.8) is overlayed in square markers, respectively. (B) Steady state of original model, which does not account for nucleator unbinding due to microtubule turnover (solid lines, δ = 0), and the model that does (dotted lines, δ = 1) – note that the steady state of the unbound nucleator is exactly the same since the respective terms cancel in Equation (3.1). 38

A B

Unbound Nucleator Unbound Nucleator 5 5 Bound Nucleator Bound Nucleator Microtubule Density Microtubule Density 4.5 4.5

4 4

3.5 3.5

3 3

2.5 2.5

2 2 Concentration ( µ M) Concentration ( µ M) 1.5 1.5

1 1

0.5 0.5

0 0 0 50 100 150 0 50 100 150 Distance from monopole (µm) Distance from monopole (µm) C

Unbound Nucleator 5 Bound Nucleator Microtubule Density 4.5

3.5

2.5

2 Concentration ( µ M) 1.5

0.5

0 0 50 100 150 Distance from monopole (µm)

Figure 3.6 – Steady state mean concentrations (temporal averages) of stochastic microtubule aster models for different tip models without autonucleation. (A) Steady state of stochastic model shown in Figure 3.4B, with two fixed-length microtubules that span the entire domain (kbra = 0, hstoch = 0, solid lines), and the analytical solution of the deterministic model (Figure 3.4A) with δ = 0, kbra = 0, θ = 0, i.e., ρ(x) = ρ(0) (boxes). Due to the limited amount of binding sites for nucleators, close to the aster at x = 0, the bound nucleator concentration is lower than in the analytical solution that disregards this effect. (B) Mean concentration at steady state (temporal average) for Ntip = 1, without autonucleation (kbra = 0, solid lines), versus analytical solution for kbra = 0, δ = 1 (filled boxes). Except for x = 0, where the effect of limited binding sites is visible, the stochastic model closely tracks the analytic solution of our modified deterministic model with δ = 1.(C) shows the same for Ntip = 2 – there is a slight shift of the microtubule density away from the monopole. Sampled time steps for all panels were t = [0, . . . , 500] s, and samples for t ≥ 100 s were used for computing temporal averages. 39

0.9

0.8

0.7

0.6

0.5

0.4

0.3 Normalized intensity (a.u.)

0.2

0.1

0 0 5 10 15 20 25 Distance from monopole (µm) B C

7 7 Unbound Nucleator Unbound Nucleator Bound Nucleator Bound Nucleator 6 Microtubule Density 6 Microtubule Density

5 5

4 4

3 3 Concentration ( µ M) Concentration ( µ M) 2 2

1 1

0 0 0 5 10 15 20 25 0 5 10 15 20 25 Distance from monopole (µm) Distance from monopole (µm)

Figure 3.7 – Steady state mean concentrations (temporal averages) of stochastic microtubule aster models for different tip models with autonucleation. Bound and unbound nucleator concentration, as well as microtubule density as a function of distance from the center of the aster at x = 0 µm.(A) shows the microtubule fluorescence intensity profile from Figure 3.1, normalized between zero (background) and one (maximum intensity). (B) and (C) show the results obtained from the analytical solution (3.6)–(3.8) with the parameters in Table 3.2, but with δ = 1, k˜0 = 6k0 (filled boxes). Temporal averages from stochastic simulations are plotted as solid lines (averaged over 2 h simulation time, sampled in 0.5 s intervals). (B) −4 −1 shows the stochastic model with Ntip = 1, and kbra,s = 1.9 × 10 s , and (C) shows the model with −4 −1 Ntip = 2, and kbra,s = 1.5 × 10 s . 40

Time = 3710 seconds Time = 3715 seconds

10 10

5 5 Microtubule index Microtubule index 0 0 0 5 10 15 20 25 0 5 10 15 20 25 x (µm) x (µm) Time = 3720 seconds Time = 3725 seconds

10 10

5 5 Microtubule index Microtubule index 0 0 0 5 10 15 20 25 0 5 10 15 20 25 x (µm) x (µm) Time = 3730 seconds Time = 3735 seconds

10 10

5 5 Microtubule index Microtubule index 0 0 0 5 10 15 20 25 0 5 10 15 20 25 x (µm) x (µm) Time = 3740 seconds Time = 3745 seconds

10 10

5 5 Microtubule index Microtubule index 0 0 0 5 10 15 20 25 0 5 10 15 20 25 x (µm) x (µm) Time = 3750 seconds Time = 3755 seconds

10 10

5 5 Microtubule index Microtubule index 0 0 0 5 10 15 20 25 0 5 10 15 20 25 x (µm) x (µm) Time = 3760 seconds Time = 3765 seconds

10 10

5 5 Microtubule index Microtubule index 0 0 0 5 10 15 20 25 0 5 10 15 20 25 x (µm) x (µm)

Figure 3.8 – Stochastic microtubule dynamics. Custom visualization for the stochastic microtubule dynamics model in Figure 3.4BC, with Ntip = 2. Microtubules #1 and #2 are re-nucleated upon catastrophe to enforce the constant-concentration boundary condition at x = 0, while the other microtubules vanish upon catastrophe. 41

3.5 conclusion

RDMEcpp provides a simulation engine for spatial stochastic models on unstructured meshes with state-of-the-art performance, and runs on any platform where support for the HDF5 data format and a modern C++ compiler is available. It enables simulation of stochastic microtubule models in arbitrary directions, as we demonstrated by simulating a recent model for aster generation by autonucleated microtubule dynamics. To our knowledge, we are the ﬁrst to combine a stochastic microtubule tip model with the RDME framework. We showed that accounting for nucleator unbinding due to microtubule depolymerization both in the deterministic and stochastic models leads to a lower amount of bound nucleator overall. In addition, the limited number of nucleator binding sites reduces the amount of nucleators in saturating conditions.

3.5.1 Outlook

Future work will involve implementing a more efficient subvolume lookup algorithm, i.e., replacing the current linear lookup by a better heuristic developed by Hellander et al., 2012, making simulation of models with a large number of subvolumes, for example a 2D/3D geometry of a metaphase yeast cell, more efficient. In addition, for 3D models, better methods for avoiding negative diffusion matrix entries, such as the ones developed by (Meinecke et al., 2016), could be plugged into the existing interfaces of RDMEcpp. An interesting future application regarding microtubule dynamics will be integrating regulatory networks such as the mitotic exit network signaling pathway (Hotz and Barral, 2014) in S. cerevisiae with stochastic microtubule dynamics. However, as is the case with the exact nature of the nucleator for the autonucleated microtubule aster model in Xenopus, further work is needed to clarify the mechanism that regulates microtubule dynamics downstream of the mitotic exit network (MEN), before a comprehensive understanding of this system can be achieved. In Xenopus, a more quantitatively accurate model could incorporate data on the angles between newly nucleated microtubules and pre-existing microtubules, which were quantified in (Petry et al., 2013), and simulate the resulting system in 2D. Very recent research indicates that it is the augmin protein can target the γ-tubulin ring complexes (γ-TuRCs) that act as nucleators to the microtubules in X. laevis egg extract by acting as an adaptor between microtubules and the γ-TuRCs (Song et al., 2018). This new knowledge can be used to improve the mechanistic accuracy of the stochastic autonucleation model presented here, and will be part of future work.

3.6 acknowledgements

We thank Dr. Franziska Decker for valuable discussions, and Sabine Österle for feedback on the manuscript. We acknowledge ﬁnancial support by the SystemsX.ch RTD Grant #2012/192 TubeX of the Swiss National Science Foundation. Conﬂict of interest: none declared. 42

3.7 author contributions

J. S. obtained the funding. L. A. W. and J. S. designed the study concept. L. A. W. developed software and analyzed the data. L. A. W. and J. S. wrote the manuscript. J. S. supervised the study. Authors declare no competing interests.

3.8 supporting information

3.8.1 Continuous reaction-diffusion problem

The continuous reaction-diffusion problem we are solving is given by

∂u(x, t, p) = ∇ · (D(u, x)∇u(x, t)) + R(u(x, t), x, t) .(3.9) ∂t | {z } | {z } Diffusion term Reaction term

u(x, t, p) is the concentration vector of the Ns chemical species   u1(x, t, p)  .  u(x, t, p) =  .  ,(3.10)  . 

uNs (x, t, p)

in moles per cubic meter, where x is the spatial location in meters, t is time in seconds, and p are the parameters for the reactions inﬂuencing the concentrations of the chemical ∂ species over a spatial domain of interest, Ω, and ∇ = ∂x .

Finite element discretization

(e) We subdivide the domain Ω into into Ne elements Ω , e = 1, . . . , Ne, such that

N [e Ω(i) = Ω, and Ω(i) ∩ Ω(j) = ∅ for i 6= j. i=1

Nodes 3

1 NaN 0 50 100 150 200 Subdomains x (μm)

Figure S3.1 – Spatial discretization of the stochastic microtubule aster model. Each point corresponds to a primal mesh element. In total, the model contains three types of nodes: 1. the node at x = 0, which corresponds to the node with the inﬂow of active, unbound nucleators nu. 2. all other nodes in the simulation domain, into which unbound nucleators can diffuse, and microtubules can grow. 3. a special “microtubule” node without a valid spatial coordinate, and no diffusion. This subvolume contains the propensities for all microtubule plus tips, i.e., the growth and hydrolysis propensities. Hydrolysis then gives rise to catastrophe, as deﬁned by the microtubule plus tip model (Figure 3.4). The y and z dimension of the model is assumed to be 1 µm, respectively, leading to a model volume of 200 µm × 1 µm × 1 µm = 200 µm3. 43

3.8.1.1 Basis functions and discretized concentrations

We assume linear, barycentric basis functions ϕj(x) for approximating the concentrations u(x) between Nn nodes at xj, j = 1, . . . , Nn, at which the concentration is uˆ j. The test function w(x) is an arbitrary function that is described by ψj(x) . In the Galerkin method, ψ are assumed to be the same as the basis functions ϕ (Zienkiewicz et al., 2013; Smith et al., 2014). Thus we approximate coordinates x, the concentrations u(x), and the test function w(x) with the same basis functions:

x ≈ ∑ ϕi(x)xi, i u(x) ≈ ∑ ϕi(x)uˆ i, i w(x) ≈ ∑ ψi(x)wˆ i = ∑ ϕi(x)wˆ i. i i

After discretization (for example, the one given in FigureS 3.1), uˆj(t, p) is the concentration vector (3.10) at node j with location xj, giving rise to the system state matrix Uˆ ,     uˆ1,j(t, p) u1,1ˆ (t, p) ... uˆ1,Nn (t, p)  .   . .  uˆ (t, p) =  .  , Uˆ (t, p) =  . .  . j  .   . . 

uˆNs,j(t, p) uNˆs,1(t, p) ... uˆNs,Nn (t, p)

If only one node is available, such as in conventional ODE models or stochastic models of a well-mixed reaction volume, the index i is omitted, and Uˆ is simply the vector uˆ 1. Since we are not interested in parameter sensitivities here, we drop the explicit dependency on the parameters p for the rest of the derivation, and rewrite matrix Uˆ as a (column-major) column vectoru ˆ for all the state variables of the system:

| u( ) = ˆ t uˆ1,1(t) ... uˆNs,1(t) ... uˆ1,Nn (t) ... uˆNs,Nn (t)

We note that in the implementation, this ensures storage of all the species in the same subvolume in a continuous block of memory, making local updates to propensities cache-efﬁcient to compute.

Description in Terms of Molecule Numbers

We can then describe the molecule numbers zj at each node using the relation

zj(t) = uˆ j(t) NAvolj, Z(t) = Uˆ (t) NAdiag(vol),(3.11) | {z } | {z } Ξj Ξ where volj is some volume associated with node j, and NA is Avogadro’s Number. 44

3.8.1.2 Weak Form of the Continuous Formulation

To derive the weak form of the continuous reaction-diffusion equation (3.9), we multiply by the test functions w(x) (Zienkiewicz et al., 2013) and integrate over the domain Ω:   Z ∂u(x, t) ∇ · (D(u, x)∇u(x, t)) + R(u(x, t), x, t) −  w(x)dΩ = 0. Ω | {z } | {z } ∂t Diffusion term Reaction term

We then apply multivariate integration by parts to the diffusion term to reduce higher- order derivatives in space: Z Z (∇ · (D(u, x)∇u(x, t))) w(x)dΩ = D(u, x)∇u(x, t)w(x) · ∂Ω Ω ∂Ω Z − D(u, x)∇u(x, t) · ∇w(x)dΩ. Ω Since we assume natural (i.e., Neumann / zero flux) boundary conditions, the flux in direction of the boundary normal ∂Ω must be 0, thus above equation simplifies to Z Z (∇ · D(u, x)∇u(x, t)) w(x)dΩ = − D(u, x)∇u(x, t) · ∇w(x)dΩ. Ω Ω Thus we can assemble the weak form of the reaction-diffusion equation: Z Z − D(u, x)∇u(x, t) · ∇w(x)dΩ + R(u(x, t), x, t)w(x)dΩ Ω Ω Z ∂u(x, t) − w(x)dΩ = 0. (3.12) Ω ∂t Next, we insert the basis function Ansatz for the concentrations u and the test function w, and assume a constant (and isotropic) diffusion coefficient D: Z Z D∇u(x, t) · ∇w(x)dΩ = D(u, x)∇u(x, t) · ∇w(x)dΩ Ω Ω Z ! ! ≈ D ∇ ∑ ϕi(x)uˆ i · ∇ ∑ ϕi(x)wˆ i dΩ Ω i i Z ! ! = D ∑ uˆ i∇ϕi(x) · ∇ ∑ ϕi(x)wˆ i dΩ Ω i i Z = D uˆ ∇ϕ(x) · ∇ϕ(x)wˆ dΩ Ω Z = uˆ D ∇ϕ(x) · ∇ϕ(x)dΩ wˆ . Ω | {z } Diffusion Matrix D

Inserting the Ansatz into the time derivative, we get

Z ∂u(x, t, p) Z ∂uϕˆ (x) w(x)dΩ ≈ φdΩ w Ω ∂t Ω ∂t ∂uˆ Z = ϕ(x)ϕ(x)dΩ wˆ , ∂t Ω | {z } Mass Matrix M 45 and for the reaction term, Z Z R(u(x, t), x, t)w(x)dΩ ≈ R(uˆ (x, t), x, t)ϕ(x)dΩ wˆ . Ω Ω | {z } Rϕ

Note that Rϕ is the change in concentration over the reaction volume Ω, i.e., Rϕ is in moles per second. The equation now reads ∂uˆ (x, t, p) −uDˆ wˆ + R wˆ − Mw = 0 ϕ ∂t ∂uˆ ⇔ −uDˆ + R − M wˆ = 0 ϕ ∂t ∂uˆ ⇔ −uDˆ + R = M ϕ ∂t Therefore, ∂uˆ = uˆ (−D)M −1 + R M −1. ∂t ϕ

Note that this now is a coupled ODE system in time, over the ﬁnite elements.

3.8.2 Mass Matrix Diagonalization

The mass matrix M (Zienkiewicz et al., 2013) is deﬁned as Z Z T M = ψϕ dΩ, Mm,n = ψm ϕndΩ Ω Ω and for the Galerkin method, ψ = ϕ, thus Z Z T M = ϕϕ dΩ, Mm,n = ϕm ϕndΩ. Ω Ω

The row sum method then computes mΣ: Z mi,Σ = ∑ ϕi(x)ϕj(x)dΩ j Ω Z = ϕi(x) ∑ ϕj(x) dΩ Ω j | {z } =1 Z = ϕi(x)dΩ Ω Z = ∑ ϕi(x)dΩe e Ωe Z = ∑ ϕi(x)dΩe Ωe e adjacent to xi = voli,(3.13) where voli is the volume of the barycentric dual mesh volume centered at node i. Here we exploit that the sum of all basis polynomials is one, which is, e.g., the case for Lagrange basis functions, and that the basis functions are zero outside the previously-mentioned shape functions of the elements e adjacent to the node at xi, by construction. This allows us to deﬁne the diagonal matrix MΣ = diag(mΣ). 46

3.8.3 Conversion to molecule numbers

We can the rewrite equation (3.9) in terms of molecule numbers z using relation (3.11):

MΣ z }| { zj(t) = uˆ j(t) NAvolj, Z(t) = Uˆ (t) NA diag(vol), | {z } | {z } Ξj Ξ

Uˆ (t) = Z(t)Ξ−1,

∂u M = u(−D) + R ∂t ϕ ∂z M −1N−1M = zM −1N−1(−D) + R ∂t A A ϕ ∂z −1 = z M (−D) + Rϕ NA ∂t | {z } | {z } D˜ R˜ ∂z = zD˜ + R˜ ∂t

3.8.4 Deterministic simulation

COMSOL Multiphysics was used to numerically simulate the deterministic PDE system, as well as to compare the numerical against the analytical solution.

3.8.5 Reaction terms

Usually, the entries Ri of the reaction term R(u(x, t), x, t) are additive terms governed by mass action kinetics, which specify the reaction rates (Table 3.3). Other possibilities include, e.g., using various simpliﬁcations such as the stochastic Michaelis-Menten approximation, but care must be taken to ensure its validity as properties that hold for single well-mixed systems (e.g., Widmer et al., 2013) do not necessarily translate with changing discretization element size (Lawson et al., 2015) in systems formulated using the RDME framework. Assuming concentrations in, for example, Moles per cubic meter, and stochastic simulation operating on absolute molecule numbers, we can map these to propensities (Table 3.3).

3.8.6 Documentation

For help on any MATLAB function contained in RDMEcpp, help inside MATLAB is available. E.g., call help RDMEcpp_load, or press F1 when highlighting the function name in MATLAB.

3.8.7 URDME compatibility

URDME .mat ﬁles can be imported using the RDMEcpp_urdme_import function. Note that the propensity functions must be speciﬁed in a slightly different form in RDMEcppModel.hpp 47

Table 3.3 – Mass action reaction rates vi and propensities ai. For each reaction, k is the deterministic reaction rate, and [X] is shorthand for the concentration of X at node j, uX,j, in m. c is the stochastic rate −1 constant, which always has a unit of s . Ξ is shorthand for Ξj, i.e., the volume volj associated with this −1 node, times Avogadro’s Number NA, thus the unit of Ξ is m . nX corresponds to the number of molecules of type X in the dual subvolume spanned by node j, i.e., zX,j = uX,jΞj. The reaction term for all species in a subvolume is then given by N · v.

Change Vectors

Reaction vi Unit of k c ai Si Pi Ni       0 1 1 k −1       A k m s kΞ c 0 0 0 0 0 0       1 0 −1 k −1       A k[A] s k c 0 0  0  0 0 0       1 0 −1 k −1       A B k[A] s k cnA 0 1  1  0 0 0       1 0 −1 k −1 −1 −1       A + B C k[A][B] m s kΞ cnAnB 1 0 −1 0 1 1       2 0 −2 k ( − ) [ ]2 −1 −1 −1 nA nA 1       A + A B k A m s 2kΞ c 2 0 1  1  0 0 0 for the E. coli MinD oscillation example in RDMEcpp, and mincde.c in URDME (Draw- ert et al., 2012), but conversion is straightforward (see the MinD example for detailed information). Simulation results can then be saved back into URDME .mat file format using the RDMEcpp_urdme_export function in MATLAB. For further information and an example, see the urdmeImportExport folder in the example folder of RDMEcpp. We designed our .mat file structure to be as close to the one of URDME as possible in order to facilitate model transfer – the details on the URDME file format can be found in the URDME manual (Bauer et al., 2012).

ONTHEESSENTIALITYOFMICROTUBULE(DE-)POLYMERASES– WHYISTUBULINNOTENOUGH?

This chapter is to be submitted as

Lukas A. Widmer, Xiuzhen Chen, Yves Barral, Jörg Stelling*. “On the essentiality of microtubule (de-)polymerases – why is tubulin not enough?” * corresponding author

4.1 highlights

• We present the ﬁrst in vivo tubulin concentration meta-estimates in yeast.

• Feasible concentrations for microtubule nucleation and polymerization are determined.

• Critical concentrations can only be achieved using microtubule-associated proteins.

• Both critical concentration and regulatory requirements mandate (de)-polymerases.

4.2 summary

The microtubule cytoskeleton is an essential ingredient in eukaryotic life: it forms the mitotic and meiotic spindle, distributing the duplicating chromosomes accurately into the daughter cells during cell division. Our estimates of the allocation of tubulin heterodimers – computed by integrating in vivo data on protein abundance and spindle tubulin content – suggest that overall tubulin concentrations in Saccharomyces cerevisiae and Schizosaccharomyces pombe are in the low micromolar range. Moreover, the free tubulin concentration during mitosis is only around 500 nm – too low for tubulin alone to polymerize into microtubules. We deﬁne the critical concentrations relevant for microtubule growth and nucleation. Applying them suggests that microtubule polymerases are essential in vivo to nucleate and polymerize microtubules. In addition to this quantitative argument, we provide examples that indicate both polymerases and depolymerases are required to tune the basic regulation provided by a single free tubulin pool.

49 50

4.3 introduction

Microtubules are polymers assembled from the tubulin protein, and are found in all dividing eukaryotic cells. They are an essential ingredient in eukaryotic life: they form both the mitotic and meiotic spindles – large arrays of microtubules whose function is to faithfully distribute genomic material during cell division. To be able to fulﬁll this function, the microtubule cytoskeleton must therefore be highly regulated, for example, to ensure that each duplicated chromosome is transported to the opposing end of the dividing cell by nuclear microtubules in the mitotic spindle. In budding yeast, these nuclear microtubules are anchored in one of the two opposing spindle pole bodies (SPBs, the equivalents of the mammalian centrosomes) in the nuclear envelope (Yeh et al., 1995). On the cytoplasmic side of the SPBs, astral microtubules ensure proper orientation of the mitotic spindle, and position it correctly, such that one of the SPBs remains in the mother cell, and the other one gets segregated into the bud (Palmer et al., 1992).

Microtubules are polar hollow tubes made of (usually) 13 protoﬁlaments, which in turn are composed of tubulin heterodimers. Each heterodimer starts with an α-tubulin protein which is connected to a β-tubulin protein – this asymmetry determines the polarity of the microtubule: the α-tubulin end of the microtubule is termed the minus-end, and the more dynamic β-tubulin end, the plus-end (Mitchison and Kirschner, 1984). β-tubulin contains a guanosine triphosphate (GTP) which can hydrolyze, and change the preferred conformation of the α-β-tubulin heterodimer to bend outwards (Zhang et al., 2015), which makes the microtubule less stable. Typically, microtubules start to polymerize from γ-tubulin ring complexes that act as templates in vivo, or stabilized microtubule seeds in vitro. Such microtubules polymerize by adding GTP-tubulin heterodimers at their plus ends, leading to a GTP-tubulin cap at the microtubule plus tip, and a zone that stochastically hydrolyzes into GDP-tubulin behind it. If polymerization is too slow, or too many GTP-tubulin heterodimers dissociate at the plus tip, the GTP-tubulin density at the plus end can get too low, leading to rapid microtubule de-polymerization (catastrophe).

In this work, we derive quantitative estimates for the tubulin content and allocation during mitosis in the model organism yeast species Saccharomyces cerevisiae and Schizosac- charomyces pombe. In mammalian cells, tubulin concentrations are typically in the 10 µm range (Wieczorek et al., 2013) – is the situation in yeast any different? According to one account (Winey and Bloom, 2012), S. cerevisiae contains 30 µm tubulin – even more than mammalian cells. To ﬁnd reliable estimates, we will base our new estimates on a meta-analysis of available protein abundance data.

Next, we will deﬁne critical free tubulin heterodimer concentrations that need to be exceeded to enable microtubule growth and microtubule nucleation, and then compare them against our previously-obtained estimates. We will discuss how microtubule- associated proteins (MAPs) can change critical tubulin concentrations, regulating microtubule polymerization in vivo. Finally, we discuss whether microtubule nucleation and polymerization in vivo can be regulated by a single free tubulin pool, and we discuss the implications for spindle assembly and disassembly. 51

4.4 results and discussion

4.4.1 Estimating the total tubulin concentrations in S. cerevisiae and S. pombe

To obtain a quantitative estimate for the amount of α- and β-tubulin, as well as the amount of tubulin heterodimers, we used the protein abundance data for the two α- tubulin isoforms and β-tubulin in S. cerevisiae and S. pombe from Pax-DB (Wang et al., 2015). We compute intracellular concentrations in µm from the data given in parts per million (ppm, Figure 4.1). This puts the medians of the total α- and β-tubulin concentrations in both yeast species in the low micromolar regime.

A B

S. cerevisiae proteins 1 S. pombe proteins 101 10

3 103 10 M) M) 0 100 10

2 102 10

-1 -1 Abundance (ppm) 10 Abundance (ppm) 10 Concentration ( Concentration ( 1 101 10 Mean Mean PaxDB weighted mean PaxDB weighted mean -2 10-2 10

-Tubulin -Tubulin

Total Tubulin Dimers -Tubulin -Tubulin(Nda2)Total (Atb2) -Tubulin Tubulin(Nda3) Dimers -Tubulin-Tubulin (Tub1p) (Tub3p)-Tubulin (Tub2p)

Figure 4.1 – Concentration and abundance estimates for tubulin heterodimers in Saccharomyces. Ag- gregated protein abundance data was downloaded from PaxDB (Wang et al., 2015), and converted to concentrations using a density value for yeast of 2.5 × 106 proteins/µm3 (Milo, 2013). Distribution quartiles (medians) are indicated with blue boxes (red lines), with means indicated as red circles, and the PaxDB integrated abundance (Wang et al., 2015) indicated as a red cross. (A) S. cerevisiae estimates: each data point (black ﬁlled circle) corresponds to data reported in the studies by Ghaemmaghami et al., 2003; Serikawa et al., 2003; Craig et al., 2004; Newman et al., 2006; Piening et al., 2006; Lu et al., 2007; Godoy et al., 2008; Deutsch et al., 2008; Nagaraj et al., 2012; Breker et al., 2013; Denervaud et al., 2013.(B) S. pombe estimates: each data point corresponds to data reported in the studies by Craig et al., 2004; Mancuso et al., 2012; Marguerat et al., 2012; Gunaratne et al., 2013.

Heterodimer dissociation is unfavorable, with the most recently reported KD being 84 nm (Montecinos-Franjola et al., 2016), though previously reported values were between 2 nm and 2 µm (Montecinos-Franjola et al., 2016), with one study reporting a KD of 10 pm (Caplow and Fee, 2002). Tubulin concentrations are certainly signiﬁcantly higher than this, so one can expect almost all available α- and β-tubulin to be bound to each other. This assumption is corroborated by an in vivo study by Laceﬁeld et al., 2006 in S. cerevisiae, which found 93 % of tubulin to be bound in heterodimers. Furthermore, this study reports that there is more unbound β-tubulin (Tub2) than unbound α-tubulin (Tub1/3), consistent with it being more abundant than total α-tubulin (Tub1 and Tub3, Figure 4.1). 52

Since excess β-tubulin forms toxic aggregates in S. cerevisiae (Burke et al., 1989; Bollag et al., 1990), it is likely the non-heterodimeric β-tubulin is bound to Rbl2, a protein in the tubulin folding pathway that prevents this aggregation, if it is available in sufﬁcient quantities (Archer et al., 1995). Indeed, Rbl2 has been demonstrated to sequester surplus β-tubulin in the cell (Archer et al., 1998). Concerning α-tubulin, our estimates for the two isotypes in S. cerevisiae give rise to a somewhat different ratio of Tub1 to Tub3 than the ones previously reported: Bode et al., 2003 measured a ratio of 90 % Tub1 and 10 % Tub3, our estimates give 57 % and 43 % (Table 4.1), respectively – this ratio is however quite consistent with the more recently reported 60 % and 40 % (Keren et al., 2016). For S. pombe, we found no previously- published quantitative data – we predict that Nda2 and Atb2 make up approximately 63 % and 37 % of the total α-tubulin, respectively. Considering these observations, it is reasonable to assume that the heterodimer concentration is the minimum of the α- and β-tubulin concentrations, which results in an estimated median tubulin heterodimer concentration of 1.4 µm for both S. cerevisiae and S. pombe. This value is substantially lower than a previous estimate of 30 µm to 35 µm (Winey and Bloom, 2012), which depends on a single study on tubulin abundance. Our median estimate in Figure 4.1 is backed by many independent studies, and is in line with recent experimentally-determined tubulin heterodimer concentrations of 1 µm to 2 µm in S. pombe, measured by puriﬁcation from an asynchronous culture of cells (Douglas Drummond, private communication). Therefore, we will continue our analysis with this value, and later further validate our new estimate with knockout data in Section 4.4.4.

4.4.2 During mitosis, a substantial fraction of tubulin is bound in the spindle

The primary function of the microtubule cytoskeleton in yeast is to build the mitotic and meiotic spindles, and ensure accurate chromosome segregation (Winsor and Schiebel, 1997; Hagan, 1998; Cavanaugh and Jaspersen, 2017). Now that we have established an estimate for the overall tubulin content in mitotic S. cerevisiae and S. pombe cells, we should check that this estimate is consistent with the number of tubulin heterodimers built into their respective mitotic spindles, that is, there should be sufﬁcient tubulin available to build them. Therefore, we acquired data from studies that performed electron microscopy on mitotic spindles in both organisms and computed the total number of heterodimers in each spindle reported for S. cerevisiae (Winey et al., 1995, Figure 4.2A) and S. pombe (Ding et al., 1993, Figure 4.2B). Note that spindle length and total tubulin do not perfectly correlate because even though the mean microtubule length increases with length, the number of microtubules in the spindle decreases with spindle length (Winey et al., 1995). In S. cerevisiae, the mitotic spindles reported are comprised of 1.6 × 104 to 5.2 × 104 tubulin heterodimers, while the longer spindles in S. pombe require between 7.6 × 104 and 1.2 × 105 heterodimers. To be able to compare the amount of spindle tubulin heterodimers to the overall available tubulin, we have to derive an absolute number of tubulin heterodimers per cell – this requires determining a value for the respective cell volumes. To minimize the impact of outliers, we proceed with the median spindle tubulin content and median spindle length in each organism (Table 4.1). From the median spindle length we can approximately determine the corresponding cell volume: the median S. cerevisiae spindle 53

A B 104 104 6 12

5 11

4 10

3 9

2 8 Tubulin Heterodimers (count) Tubulin Heterodimers (count) 1 7

0.5 1 2 4 8 2 4 8 16 Spindle Length ( m) Spindle Length ( m) C D Spc42-mCherry, Kip3-3xsfGFP GFP-Atb2, Cut11-GFP, Sid2-Tom R = 2 μ m Spindle h = 9.15 μm r = 1.25 μ m R = 2.16 μ m h = 3.1 μm Spindle

V = 50 μm3 V = 148.5 μm3

Figure 4.2 – Architecture of yeast cells and their spindles during mitosis. (A, B) Number of tubulin heterodimers bound in the spindle (black circles) as a function of spindle length, computed from the number of microtubules and mean microtubule length of (A) S. cerevisiae spindles (n = 15) reported by Winey et al., 1995, and the total MT length for (B) S. pombe spindles (n = 7) reported by Ding et al., 1993. In both studies, spindles were analyzed by electron microscopy after cryofixation. Quartiles (median and mean) for spindle length and the number of tubulin heterodimers are indicated by blue boxes (red lines and red circles). For conversion from microtubule length to number of heterodimers, microtubules were assumed to have 13 protofilaments, and heterodimers to have a length of 8 nm.(C, D) Cell geometries corresponding to median spindle lengths in (A, B). (C) S. cerevisiae cell geometry and volume corresponding to median spindle length in (A). Top panel: metaphase cell with fluorescently tagged kinesin Kip3 and spindle pole component Spc42. Scale bar: 2 µm. Bottom panel: schematic corresponding to top panel. A culture of haploid S. cerevisiae cells growing on YPD at 25 ◦C in metaphase (spindle still located in the mother cell) has a volume of approximately 50 µm3 (Uchida et al., 2011). (D) S. pombe cell geometry and volume corresponding to median spindle length in (B). Top panel: anaphase cell with fluoescently tagged alpha tubulin Atb2, nuclear envelope trans-membrane protein Cut11, and the septin initiation network kinase Sin2 as a SPB marker. Scale bar: 5 µm. Image reproduced from Lucena et al., 2015 under CC BY-NC-SA 3.0 license. Bottom panel: The cells have a diameter of approximately 2 µm (Wu and Pollard, 2005; Piel and Tran, 2009), and cell length at median spindle length is already maximal (Selhuber-Unkel et al., 2009). Given a cell volume of 148.5 µm3 (Mitchison, 1957), the cell length is approximately 13.15 µm. 54

length is 1.5 µm (Figure 4.2A), corresponding to a metaphase cell, where the spindle is usually still in the mother cell and in the process of aligning towards the bud (Figure 4.2C), and the cell volume is approximately 50 µm3 (Uchida et al., 2011). In contrast, the median spindle length reported for S. pombe is 8.1 µm, corresponding to a cell in anaphase (Figure 4.2D) with a mean volume of around 148.5 µm3 (Mitchison, 1957). With the cell volumes determined, we can now ﬁnally compute the number of α- and β-tubulins as well as the number of heterodimers per cell. For S. cerevisiae, we get a median of 4.2 × 104 α-tubulins per cell, and 7.3 × 104 β-tubulins per cell, leading to a median total number of tubulin heterodimers per cell in metaphase of approximately 4.2 × 104. For S. pombe, we get 1.3 × 105 (Table 4.1). As expected, the total amount of tubulin heterodimers is higher than the amount required to build the respective spindles, fulﬁlling this requirement for spindle assembly. Nevertheless, a substantial amount of tubulin is bound in the mitotic spindle: ≈ 67 % in S. cerevisiae, and ≈ 65 % in S. pombe.

4.4.3 Free tubulin concentrations during mitosis are in the hundred-nanomolar range

The next step is determining the free tubulin heterodimer concentration in the cytoplasm that is available for binding to microtubule plus tips, since this is the concentration that matters for characterizing microtubule growth rate. We can compute this concentration by subtracting the number of heterodimers in the mitotic spindle from the total number of tubulin heterodimers, and accounting for the cell volume. For S. cerevisiae, this leaves 1.4 × 104 tubulin heterodimers, corresponding to a free tubulin concentration of 0.47 µm. For S. pombe, 4.3 × 104 tubulin heterodimers are free, giving rise to a very similar concentration of 0.48 µm (Table 4.1). The estimates for tubulin heterodimer concentration and absolute abundance are likely not too high, as spindle assembly would otherwise be impossible. In fact, these concentrations are substantially lower than tubulin concentrations in mammalian cells, which is typically in the 10-micromolar range (Wieczorek et al., 2013).

4.4.4 Validating tubulin estimates required for spindle assembly in S. cerevisiae using knockout data

To ensure that the estimated tubulin concentration is not too low, we can challenge our estimate of S. cerevisiae tubulin concentration with knockout data reported in the pac10∆ plp1∆ yap4∆ strain (Laceﬁeld et al., 2006). In this strain, the folded tubulin heterodimer levels were reported to be 19 % of wild type, due to the lower amount of functional β-tubulin Tub2. This leads to a spindle phenotype in pac10∆ plp1∆ yap4∆ cells: they exhibit shorter spindles (6.40 ± 1.13 µm instead of 10.02 ± 1.29 µm in wild type). If our estimates are reasonable, the amount of tubulin available in this mutant should be close to the amount required to build the mitotic spindle, as the spindle is unable to grow further in this mutant. We can estimate the tubulin content of the wild type large-budded cells in anaphase from the volume and a tubulin concentration of 1.4 µm (Table 4.1). To compute the volume of these large-budded cells, we assume the spindle spans from one end of the mother cell to the other of the daughter cell, and the cells’ architecture to correspond to two spheres of equal radius. The radius is thus 2.5 µm, yielding a (quite large) volume of 131 µm3, 55

Table 4.1 – Tubulin heterodimer concentrations and abundances in yeast cells. Median tubulin concentrations in cells are reported as shown in Figure 4.1, and compared against spindle tubulin (Figure 4.2AB). This allows for computing an approximation of the free tubulin concentration during mitosis. Abundances are computed using the cell volumes reported in Figure 4.2CD, for S. cerevisiae and S. pombe. Total tubulin heterodimer counts are computed by taking the median of the minimum of α- and β-tubulin subunit counts for each abundance study in Figure 4.1, respectively.

S. cerevisiae S. pombe

Total Protein Concen- Abundance Protein Concen- Abundance tubulin tration (molecules ×104) tration (molecules ×104) (µm) (µm)

α-Tubulin Tub1 1.4 4.2 Nda2 2.2 19 (total) + Tub3 + Atb2 Major α Tub1 0.85 2.6 Nda2 1.7 15 isoform Minor α Tub3 0.63 1.9 Atb2 1.0 8.8 isoform β-Tubulin Tub2 2.4 7.3 Nda3 0.81 7.2 (total) Tubulin 1.4 4.2 1.4 13 heterodimer (total)

Spindle 0.92 2.8 0.95 8.5 tubulin

Free 0.47 1.4 0.48 4.3 tubulin

and around 1.1 × 105 tubulin heterodimers. For the pac10∆ plp1∆ yap4∆ strain, only 19 % of this tubulin is available, i.e., 2.1 × 104 heterodimers. The two closest reported spindles in Winey et al., 1995 require 1.6 × 104 and 2.4 × 104 tubulin heterodimers (Figure 4.2A) – very close to the total amount of tubulin heterodimers we would expect in this mutant. Therefore, if we assume that tubulin is mostly produced during cell growth (Spellman et al., 1998), we would expect these cells to have to grow further than wild type cell in order to produce the necessary amount of tubulin to assemble the spindle.

Indeed, in pac10∆ plp1∆ yap4∆, mitotic progression is arrested by the spindle assembly checkpoint (SAC) as assessed by deletion of mad2, time in G2/M phases is increased, and the fraction of large-budded cells increases from 25 % to 48 % (Laceﬁeld et al., 2006) – though the spindle position checkpoint (SPOC) is not required. According to our estimates, this phenotype can be explained by further growth being required for more functional tubulin to be expressed, surpassing the minimum heterodimer count necessary for spindle assembly (∼ 1.6 × 104 heterodimers, Figure 4.2A) only at larger bud volumes. Although spindles show a nuclear positioning defect – in anaphase, the spindle no longer extends normally – the spindle still centers at the bud neck with one SPB on either side (Laceﬁeld et al., 2006). This is further evidence for the amount of functional tubulin heterodimers being a limiting factor for spindle elongation in pac10∆ plp1∆ yap4∆ cells. Therefore, we believe our tubulin heterodimer estimates for S. cerevisiae are neither substantially too high, nor substantially too low. 56

4.4.5 Free concentrations in vivo are below the critical concentrations for yeast tubulin assembly in vitro

Having to polymerize microtubules from a free pool that has such a low concentration raises an interesting problem: tubulin concentrations required for polymerizing microtubules in vitro are higher than 500 nm. For S. cerevisiae tubulin, the lowest concentration where polymerization was observed in vitro was 1.8 µm (Bode et al., 2003), while for S. pombe, it was even higher, at 3.1 µm (Hussmann et al., 2016; Figure 4.3A). The net growth rate rg ([T]) of the microtubule plus tip (in individual subunits per second) is a balance of the incidence of tubulin heterodimers (ron, subunits per second), as well as detachment of tubulin heterodimers (koff, subunits per second):

rg ([T]) = kon[T] −koff.(4.1) | {z } ron This equation gives rise to a linear dependency of growth rate on tubulin concentration [T], and good ﬁts to in vitro data were achieved both for S. pombe tubulin (Hussmann et al., 2016, Figure 4.3A), and for older experiments using bovine brain tubulin (Walker et al., 1988, Figure 4.3B). We, again, assume the in vivo microtubules have 13 protoﬁlaments (Tilney et al., 1973; Kollman et al., 2010) – though in vitro, other conformations are possible (Pierson et al., 1978) – and each subunit has a length of 8 nm (Grimstone and Klug, 1966). Then, it follows that by attaching a single subunit, on average, the microtubule thus grows by δ = 8/13 nm, and we can convert the growth rate (4.1) from subunits per second to the microtubule growth velocity vg in nanometers per second:

vg ([T]) = δrg ([T]) .(4.2)

Using the growth velocity vg, we can deﬁne two critical concentrations. First we deﬁne −1 the zero-growth critical concentration [T]0, for which it holds that vg ([T]0) = 0 nm s :

koff [T]0 = .(4.3) kon Note that this is not the critical concentration in the equilibrium sense of classical polymer physics, since microtubules undergo transitions from growth to catastrophe (and this equation only holds during the growth phase), and vice versa – and though the deﬁnition is identical, the interpretation is not (Mitchison and Kirschner, 1987). Microtubule growth is not directly observable at this concentration (Figure 4.3AB). For tubulin concentrations

above [T]0, microtubules grow, below, they shrink. Shrinking microtubules that are not in the process of catastrophing are not a completely hypothetical scenario, since microtubules in tubulin dilution experiments, directly after dilution, can shrink on the order of 100 nm before they catastrophe due to loss of the protective GTP-tubulin cap (Duellberg et al., 2016). Using the ﬁt linear net growth rate model for S. pombe data and extrapolating at the 500 nm free tubulin concentration given by our estimates above, one would expect existing microtubules to shrink or catastrophe (Figure 4.3A), and no new microtubules to nucleate – cells are obviously able to overcome this problem. For this reason, we deﬁne a second critical concentration: the lowest experimentally observable concentration that enables microtubule nucleation and polymerization, [T]N. It was documented to be 57

A B 20 80 S. pombe Brain tubulin ) ) -1 15 (Hussmann et al., 2016) -1 60 (Walker et al., 1988) S. cerevisiae 10 (Bode et al., 2003) 40 5 20 0 0 -5 -20 Growth rate (nm s Growth rate (nm s -10 -40 10 2 3 4 5 6 0 5 10 15 Free tubulin concentration (µM) Free tubulin concentration (µM) C D

) 20 -1 15 Tubulin origin [T]0 (µm) [T]N (µm)

(nm s (nm 10 rg > h a g S. cerevisiae N/A 1.8 5 S. pombeb 2.0 3.1 0 < rg < h 0 Bovine brainc 4.8 6.8 a -5 rg < 0 Bode et al., 2003 b Hussmann et al., 2016 Growth rate v rate Growth -10 c Walker et al., 1988 0 1 2 3 4 5 6 Free tubulin concentration (µM)

Figure 4.3 – Tubulin critical concentrations. (A) Growth rate of microtubules polymerized from wildtype (two-isoform) heterodimers in S. pombe (Hussmann et al., 2016) and S. cerevisiae (Bode et al., 2003), respectively. The tubulin concentration in Bode et al., 2003 was converted from 0.18 mg mL−1 to µm using an atomic mass of 100 kDa for the heterodimer, which in turn was computed as the sum of atomic masses of Tub1 (49 kDa) and Tub2 (51 kDa) as reported in the Saccharomyces Genome Database (SGD, Cherry et al., 2012). Dotted lines indicate the experimentally measured critical concentration [T]N, below which no polymerization could be observed (Robert Cross and Douglas Drummond, private communication). Dashed lines indicate the (extrapolated, since unobservable) zero-growth critical concentration [T]0.(B) ditto, but for microtubules polymerizing from bovine brain tubulin (Walker et al., 1988). (C) Regions of growth velocity vg as described by Equation (4.2), classiﬁed by growth rate rg (Equation (4.1)) into a net-shrinking zone (red), a zone where hydrolysis is, on average, faster, than growth (light red), and zone where microtubule growth is possible as growth outruns hydrolysis. The in vivo estimate for the free tubulin concentration is displayed as a dash-dotted line (green) at 500 nm. Black dashed / dotted lines equivalent to ones speciﬁed in (A) for S. pombe.(D) Critical concentrations corresponding to dotted ([T]N) / dashed ([T]0) lines in (A, B). higher than the zero-growth tubulin concentration, around 1.8 µm for S. cerevisiae tubulin (Bode et al., 2003) and 3.1 µm for S. pombe (Hussmann et al., 2016). Note that [T]N > [T]0 also holds for bovine brain tubulin as measured by Walker et al., 1988, three decades ago, and it is likely that this property is conserved across more organisms.

4.4.6 Concentration requirements for microtubule nucleation are higher than for microtubule growth

In the following we will try to answer the following two questions:

1. Why do we not observe microtubule nucleation and polymerization below the experimental critical concentration [T]N, closer to the zero-growth concentration [T]0? 58

2. In S. cerevisiae and S. pombe cells, [T]free < [T]N, and in S. pombe, it even holds that [T]free < [T]0 – how can these cells have any microtubules at all, if they should be shrinking, even if they are not undergoing catastrophe?

To address the first question, we proceed in (temporal) order: we first address nucleation, and then polymerization. We assume a nucleation template such as the γ-TuRC (in vivo, Cavanaugh and Jaspersen, 2017) or a GMPCPP stabilized microtubule template (in vitro, Hussmann et al., 2016) exists to nucleate a microtubule from. Then, in the first step of nucleating a microtubule, and to prevent an immediate catastrophe, the stabilizing GTP-tubulin cap (Bowne-Anderson et al., 2013) must be created from scratch (even if nucleating from a template). For this to work, the tubulin attachment rate rg (4.1) must be sufficiently high to outrun tubulin hydrolysis (at rate h, in tubulin heterodimers per second), i.e.,

rg ([T]) > h.(4.4)

Conversely, given the hydrolysis rate h, we can now solve for the critical concentration

[T]N, at which hydrolysis rate and growth rate are equal:

h + koff [T]N = (4.5) kon

Thus, for the GTP-tubulin stabilizing cap to exist (on average), the apparent tubulin

concentration [T] at the nucleation site and plus tip must be greater or equal to [T]N. This explains why both for S. pombe tubulin (Figure 4.3A) and for bovine brain tubulin

(Figure 4.3B), [T]N > [T]0, as illustrated in Figure 4.3C. The GTP-tubulin stabilizing cap can then still be lost due to the stochasticity of tubulin incorporation, dissociation and hydrolysis (Brun et al., 2009; Bowne-Anderson et al., 2013), leading to the catastrophe of the (now extended) microtubule. Next, we address the second question – how can S. cerevisiae and S. pombe cells nucleate and polymerize microtubules, if the free tubulin concentration [T]free during mitosis is below [T]N, at around 500 nm? While it has been hypothesized that microtubule assembly might be a diffusion-limited process (Odde, 1997), with our concentration estimates we now have more evidence that this is the case in vivo. We begin our analysis by re- examining Equations (4.1) and (4.2), and investigate which parameters can be modiﬁed to enable microtubule nucleation, and growth. There are three angles of attack that could close the distance between [T]N and [T]free:

1. Increasing the influx of tubulin heterodimers, ron = kon [T], at the plus end: We expect the on-rate constant kon of purified tubulin heterodimers in vitro to be similar to the one in vivo, barring buffer conditions and temperature (Kuchnir Fygenson et al., 1995). In vivo, microtubules are usually anchored in a template at the minus end – and this minus end structure could locally concentrate tubulin, leading to an available tubulin concentration of [T]free,− [T]free, thus increasing ron. The prime candidates for this function are microtubule polymerases of the XMAP215 family, which can bind tubulin heterodimers with their TOG domains (Al-Bassam et al., 2007). These flexibly-connected domains could be used to create a local higher-density cloud of tubulin, increasing [T]free,−, and the on rate ron locally (Ayaz et al., 2014). 59

Indeed, the S. cerevisiae XMAP215 homolog Stu2 has been shown to bind to the spindle poles independent of microtubules (Wang and Huffaker, 1997), making it a prime candidate for mediating this local concentration increase. While an initial in vitro study on Stu2 in bovine brain tubulin indicated that Stu2 is a depolymerase (Van Breugel et al., 2003; Popov and Karsenti, 2003), more recent work demonstrated that with tubulin puriﬁed from S. cerevisiae, Stu2 does in fact function as a polymerase (Podolski et al., 2014). Stu2 was recently shown to be able to polymerize S. cerevisiae tubulin down to a concentration of 500 nm in vitro (Geyer et al., 2018) – highly consistent with our in vivo free tubulin concentration estimate. In addition, Stu2 tip-tracks (Kosco et al., 2001), thus being able to “carry” along a local tubulin heterodimer supply with the growing microtubule end, driving its further growth. S. pombe has two XMAP215 homologs that function as microtubule polymerases: Dis1 and Alp14 (Garcia et al., 2001). Similar to Stu2, Dis1 has also ﬁrst been shown to bind to spindle pole bodies and microtubules (Nabeshima et al., 1995), before its function as a microtubule polymerase could be demonstrated (Matsuo et al., 2016). Alp14 has also been shown to bind to SPBs and microtubules (Garcia et al., 2001), and subsequently, function as a polymerase (Al-Bassam et al., 2012). Importantly, by titrating Alp14 into a solution containing 500 nm S. pombe tubulin “in the low nanomolar range” (Robert Cross, private communication), also this polymerase enabled microtubule nucleation and growth from GMPCPP-stabilized microtubule seeds (Hussmann et al., 2016), in line with our free tubulin heterodimer concentration estimate for S. pombe. Other XMAP215 homologs and their binding partners are important enough to warrant a more in-depth analysis in Section 4.4.7.

2. Decreasing the off-rate koff: Different isotypes of α-β heterodimers, for example, Tub1-Tub2 and Tub1-Tub3 in S. cerevisiae, or Nda2-Nda3 and Atb2-Nda3 in S. pombe, seem to have different off rates. For example, the off speed −vg (0 µm) = δkoff, is clearly lower for single-isotype Nda2-Nda3 compared to the wild-type mix of Nda2- Nda3 and Atb2-Nda3 isotypes (4.1 nm s−1 vs. 9.3 nm s−1, Hussmann et al., 2016). In vitro, decreasing the off rate can be achieved by using tubulin that incorporates the GTP-analogue GMPCPP (Hyman et al., 1992), which dissociates less readily. Indeed, the linear fit in the data on polymerization velocity in relation to GMPCPP-tubulin concentration in Hyman et al., 1992 goes through the origin, indicating a very low off rate. Another possible explanation for a lower off rate in cells is molecular crowding in the cytoplasm and nucleus, which can contribute to lowering off rates in this simplified model. More precisely, for transition-state-limited oligomerization reactions (Minton, 2000; Minton, 2005), the transition state is stabilized, leading (in this simplified model) to a slightly lower off-rate, and a higher on rate (Minton, 1983). In vitro, emulating cytoplasmic crowding by increasing viscosity to 1 cP to 2 cP (centi-Poise, reviewed in Seksek et al., 1997) has been successful at increasing polymerization velocity (Herzog and Weber, 1978; Wieczorek et al., 2013), giving a strong hint that crowding at the microtubule plus tip in vivo may also play a role in lowering the free concentration necessary to achieve microtubule nucleation and polymerization. However, from Wieczorek et al., 2013, it is also evident that 60

crowding alone can neither account for the relatively fast polymerization velocities observed in vivo, nor enable polymerization at the low free tubulin concentrations in vivo. Molecular crowding rather further contributes to increasing the polymerization rate, on top of the effect the MAPs mentioned before have on the on rate. There is one more – possibly underappreciated – factor that could decrease the off-rate koff in S. cerevisiae: the essential protein Mhp1. It was reported to stabilize microtubules in vivo, and is likely a homolog of the mammalian MAP4 (Irminger- Finger et al., 1996; Irminger-Finger and Mathis, 1998). However, as far as we can tell, it has received little attention after its initial discovery, and might warrant further research.

3. Slowing down the hydrolysis rate h: In vivo, this is the least plausible hypothesis, since the hydrolysis rate is likely inherent to the tubulin heterodimer. If a different isotype of tubulin heterodimer in S. pombe had a different hydrolysis rate, this should lead to microtubules that can nucleate at a lower growth rate – which does not seem to be the case (Hussmann et al., 2016). However, this possibility cannot be excluded for S. cerevisiae tubulin, as only experimental critical concentrations [T]N have been reported so far (Bode et al., 2003), but not the zero-growth critical concentrations [T]0. Specifically nucleating microtubules with a slower-hydrolyzing isotype of tubulin would lower the initial velocity required to outrun hydrolysis and build up the GTP stabilizing cap, however, this hypothesis is pure speculation at this point. In contrast, mammalian tubulins seem to exhibit a difference in hydrolysis rate. There are many α- and β-isotypes, which can be combined into even more α-β heterodimer isotypes, and these isotypes are differentially expressed (Hagan, 1998; Ludueña and Banerjee, 2008). αβIII tubulin specifically has a lower hydrolysis rate (Lu et al., 1999; Ludueña and Banerjee, 2008) and according to Equation (4.5), should thus also have a lower experimental critical concentration [T]N – this is also consistent with newer experiments that show the same growth rate, but lower catastrophe rate for αβIII tubulin (Vemu et al., 2017). The reality in vivo thus is likely a combination of all three possibilities mentioned above, with the most drastic effect being due to microtubule polymerases, and cytoplasmic crowding and tubulin isotypes playing a secondary, but also important, role. In Caenorhab- ditis elegans, recent studies confirm that nucleation is possible by locally concentrating tubulin at the centrosome using the XMAP215 homolog ZYG-9 (Woodruff et al., 2017; Rale et al., 2018).

4.4.7 Our tubulin estimates and microtubule nucleation / polymerization model explain the lethality of microtubule polymerase knockouts

From the model discussed in the last section, it also follows that microtubule polymerases in yeast must be essential, as it is otherwise impossible to nucleate and grow a sufﬁcient amount of microtubules during mitosis with the low concentration of free tubulin in vivo. In S. cerevisiae, there are two microtubule polymerases known. Stu2 was discovered as an essential SPB component (Wang and Huffaker, 1997) that is homologous to the classic XMAP215 protein originally discovered in Xenopus laevis (Gard and Kirschner, 61

1987), which only later was described as a tubulin polymerase (Brouhard et al., 2008). In addition to Stu2, a second polymerase was more recently discovered: Kip2, a plus-end directed microtubule motor kinesin was shown to also polymerize microtubules (Hibbel et al., 2015). In contrast to Stu2, Kip2 is only active on cytoplasmic microtubules, delivering dynein to microtubule plus ends with the help of the Bik1 and Bim1 proteins (Roberts et al., 2014). Stu2 nuclear export is tightly controlled (Vaart et al., 2017), enabling regulation of spindle size by controlling microtubule polymerization speed through Stu2 availability in the nucleus (analogous to XMAP215 regulating spindle length in Xenopus, see Reber et al., 2013). Interfering with Stu2 nuclear export did not change cytoplasmic microtubule length (Vaart et al., 2017), leaving the cytoplasmic polymerase function mostly to Kip2. Since Kip2 only acts on cytoplasmic microtubules, the combination of both proteins allows for separate control of microtubule polymerization in the nucleus and the cytoplasm. In an attempt to create a strain of S. cerevisiae where all microtubules could be depoly- merized, we tried to combine the stu2-13 mutation with the (non-lethal) kip2∆ genotype. However, even at the permissive temperature of stu2-13 this combination turned out to be synthetically lethal – indicating that even attenuation of Stu2 function in a kip2∆ background cannot be tolerated (FigureS 4.1). Our analysis indicates that tubulin polymerases must be essential due to the low free concentration of tubulin. To investigate whether this point could also apply to other organisms with a microtubule cytoskeleton, we investigated whether these polymerases also are essential there. As previously mentioned, S. pombe has two homologs to the Stu2 protein, Alp14 and Dis1. Alp14 is required at high temperatures: alp14∆ is temperature-sensitive and does not grow at 36 ◦C. Dis1 is required at lower temperatures: dis1∆ is cold-sensitive (Garcia et al., 2001). As we would expect, the double deletion alp14∆ dis1∆ is lethal (Garcia et al., 2002). According to the Orthologous MAtrix (OMA, Altenhoff et al., 2018), Alp14 is the ortholog of Stu2. In conjunction with Alp7, Alp14 cooperatively polymerizes microtubules more efﬁciently (Al-Bassam et al., 2012; Tang et al., 2014; Hussmann et al., 2016), while Dis1 acts in conjunction with Mal3 (the S. pombe EB homolog, Matsuo et al., 2016). This cooperativity also seems to be a pattern that is conserved across species: the prototypical microtubule polymerase XMAP215 in Xenopus laevis acts together with the plus-tip binding protein EB1 (Zanic et al., 2013) – it would be interesting to see if this cooperativity also applies to Bim1, Bik1 and Stu2 in S. cerevisiae. Though it is unclear if deleting XMAP215 is lethal, it is highly likely (Gard et al., 2004). In Caenorhabditis elegans the Stu2 homolog ZYG-9 is essential as well – loss-of-function mutants were described as “zygote defective” (Bellanger et al., 2007), and loss of the plant homolog MOR1/GEM1 is also lethal (Twell et al., 2002). The mammalian homolog of Stu2, CKAP5, is essential in mice (Barbarese et al., 2013), as well as in tumor cells (Martens-de Kemp et al., 2013), and its overexpression has been associated with non-small cell lung cancer (Schneider et al., 2017). This leads us to believe that keeping a sub-critical free tubulin pool that is made critical by microtubule polymerases (and associated proteins that cooperatively assist in polymerization) is quite universal. There is however one organism where the Stu2 homolog does not seem to be essential: AlpA in Aspergillus nidulans – deleting it only causes slight temperature sensitivity (Enke et al., 2007). Possibly Aspergillus Nidulans has more abundant tubulin, or like S. pombe, expresses a paralog in addition to AlpA. 62

4.4.8 In vivo microtubule dynamics mandate regulated nucleation, polymerization, and depolymerization mechanisms beyond the capabilities of free tubulin pool regulation

Microtubule polymerases are not only required due to a relatively low free tubulin concentration in vivo, but there are regulatory requirements that make such a design highly desirable. First, if tubulin concentrations in vivo were higher than the critical concentration, microtubules could spontaneously nucleate. A review by Desai and Mitchison, 1997, put it succinctly: “The kinetic barrier to nucleation plays a fundamental role in the function and intracellular dynamics of microtubules by inhibiting spontaneous polymerization of microtubules in the cytoplasm. Without this barrier, the spatial organization of microtubules would be random”. Indeed, overexpression of both α- and β-tubulin in S. cerevisiae leads to ﬁlamentous arrays at the cell membrane that are not observed in wild type cells (Bollag et al., 1990). In a highly-organized environment such as the mitotic spindle, randomizing microtubule organization does not seem like a particularly good idea, either. It rather makes sense to restrict microtubule nucleation to sites that have their nucleation potential regulated, for example, by post-translational modiﬁcations such as phosphorylation, and keep the nucleation potential elsewhere as low as possible.

If we consider the cytoplasmic side of the two segregating SPBs in S. cerevisiae, the one that is closer to the bud should create a longer microtubule in order to attach its plus tip to actin cables emanating from the bud (Hotz and Barral, 2014), while the one remaining in the mother cell should create no microtubules, or only short ones. Given that both cytoplasmic SPB plaques are connected to the same cytoplasmic pool, tubulin concentrations must be the same. Since the nucleation potential and microtubule length are different, they must be regulated differentially on both sides, which cannot be accomplished by modulating tubulin concentrations in the cytoplasm.

After the spindle is assembled, positioned correctly and elongated, it is split when cytokinesis occurs and divides the cells (Woodruff et al., 2010). If during mitosis, the free tubulin pool was increased to the point that nucleation was favorable without polymerase assistance (about factor four to six, Figure 4.3C), disassembling the spindle halves after cytokinesis would require degrading or inactivating substantial amounts of tubulin. S. cerevisiae does not seem to do this, since deletion of the microtubule-depolymerizing kinesin Kip3 in addition to the APC cofactor Cdh1, which is responsible for ubiquination of spindle linkers, causes spindle halves to persist for over 80 min (Woodruff et al., 2010). A Kip3- and APC-dependent disassembly mechanism would allow for recycling the existing tubulin (from the last division), and for only producing the amount of tubulin necessary to reach the same level again after division. This would only require a set concentration – making the number of tubulins increase proportionally to cell volume, and approximately halving upon division – or a set amount of tubulin to produced after committing to mitosis. There is some indication for the latter to be the case: preventing disassembly of spindle halves (using a td-kip3 doc1∆ strain at restrictive temperature) causes newly-produced spindles to collapse and is lethal. Both inducing production of α- and β-tubulin or spindle depolymerization by adding nocodazole in such a situation rescues the phenotype and enables progression through the cell cycle (Woodruff et al., 2012). Finally, this further validates the upper bound of our concentration estimates 63 as detailed in section 4.4.3, as there by default does not seem to be sufﬁcient tubulin available to build two spindles in S. cerevisiae (Table 4.1).

4.4.9 Conclusions

To our knowledge, we are the first to present quantitative in vivo tubulin concentration and absolute abundance estimates assembled from a meta-analysis of measurements in S. cerevisiae and S. pombe cells during mitosis. We showed that our estimates are sufficiently high to assemble the mitotic spindles reported in literature (Winey et al., 1995; Ding et al., 1993). They are also consistent with the spindle phenotype in the pac10∆ plp1∆ yap4∆ knockout strain, and with phenotypes observed in td-kip3 doc1∆ cells that cannot disassemble their spindles. Therefore, we believe these estimates are a major quantitative step forward from the ones reported previously, which were higher than in mammalian cells (Winey and Bloom, 2012; Wieczorek et al., 2013). Our simple mathematical model for microtubule nucleation and polymerization can explain which concentrations must be reached to first enable nucleation by creating a stabilizing GTP-cap de novo, and then grow the nascent microtubule. It also explains why microtubule polymerases are required for microtubule growth, and even more so for microtubule nucleation in vivo in S. cerevisiae and S. pombe. Building on top of these quantitative estimates combined with theory, we envision a quantitative computational model of microtubule nucleation and polymerization in vivo that integrates all the above factors. On top of the intrinsic dynamics instability of microtubules, it will have to account for the limited pool of tubulin in a cell with a crowded cytoplasm, the even more limited pool of microtubule-associated proteins that may be regulated differentially in the nucleus and the cytoplasm, and their (potentially cooperative or competing) effects on polymerization and microtubule stability. Such a model could truly act as a bridge between in vitro reconstitutions of microtubule dynamics, and in vivo knockout data.

4.5 acknowledgements

We thank Don W. Cleveland and Soni Lacefield for discussions regarding tubulin regulation in yeast, Robert Cross and Douglas Drummond for sharing unpublished S. pombe data, Mohan L. Gupta Jr for clarifications regarding critical concentrations for S. cerevisiae tubulin, and Sabine Österle for critical feedback on the manuscript. We acknowledge financial support by the SystemsX.ch RTD Grant #2012/192 TubeX of the Swiss National Science Foundation.

4.6 author contributions

J. S., and Y. B. obtained the funding. L. A. W. and J. S. designed the study concept. X. C. performed biological experiments and light microscopy. L. A. W. and J. S. analyzed and/or interpreted the data. L. A. W. and J. S. wrote the manuscript with inputs from all authors. J. S. supervised the study. Authors declare no competing interests. 64

4.7 supplementary information

4.7.1 Supplementary Figures

Stu2-13 kip2∆ Stu2-13, kip2∆

T1 T1

T2 T2

T3 T3

T4 T4

T5 T5

T6 T6

T7 T7

Figure S4.1 – Stu2-13 is synthetic lethal with kip2∆ even at permissive temperature. Tetrad analysis of the cross between Stu2-13 and kip2∆ strains. Colonies were grown on YPD (yeast extract peptone dextrose) at 25 ◦C for 3 d. Tetrads are aligned horizontally. Circles denote the spores containing kip2∆, boxes mark the spores containing the stu2-13 allele. 5

AMECHANISMFORTHECONTROLOFMICROTUBULEPLUS-END DYNAMICSFROMTHEMINUS-END

This chapter is submitted as

Xiuzhen Chen†, Lukas A. Widmer†, Marcel M. Stangier, Michel O. Steinmetz, Jörg Stelling*, Yves Barral*. “A mechanism for the control of microtubule plus-end dynamics from the minus-end.” † contributed equally * corresponding authors

5.1 abstract

To properly position their mitotic spindle and cleavage plane, many asymmetrically dividing cells form microtubule asters of distinct sizes on opposite spindle poles. However, how the length of astral microtubules is differentially controlled is poorly understood. Here, we show that yeast cells recruit the conserved kinesin and microtubule polymerase Kip2 to microtubules at spindle pole bodies (SPBs, centrosome equivalents). Phosphory- lation inhibits Kip2 recruitment on microtubule shafts. Moreover, the bud-directed SPB recruits Kip2 more efﬁciently and forms longer microtubules than the mother-bound SPB. Mutations perturbing the SPB-dependent recruitment of Kip2 or its asymmetry cause microtubules of both asters to accumulate similar Kip2 levels at their plus-ends and to become similarly long. Thus, Kip2 acts as a messenger kinesin that links microtubule growth to SPB identity.

5.1.1 One Sentence Summary

Centrosomes control microtubule growth remotely, using a molecular motor as a messenger.

65 66

5.2 maintext

Budding yeast cells use cytoplasmic microtubules to align the mitotic spindle with the cell’s axis of division. The minus-ends of these microtubules are anchored in the outer-plaque of either of the two SPBs, which are embedded in the nuclear envelope at the poles of the intranuclear, mitotic spindle. The cytoplasmic microtubules emanating from the SPB dedicated to the mother cell (m-SPB, m-microtubules) are shorter and less stable throughout metaphase than the microtubules emanating from the bud-directed SPB (b-SPB, b-microtubules) (Shaw et al., 1997; Lengefeld et al., 2018), which pull the daughter nucleus into the bud in anaphase (Adames and Cooper, 2000; Tang et al., 2012). These distinct microtubule behaviors are remarkable because all microtubules are composed of tubulin dimers drawn from the same cytoplasm. Thus, the SPBs located at microtubule minus-ends somehow communicate with microtubule plus-ends to regulate their dynamics. Understanding this communication mechanism may provide key insights into how eukaryotic cells control microtubule stability, length and organization. Among the many microtubule plus-end associated proteins identiﬁed over the last decades, microtubule-dependent motor proteins of the kinesin family have emerged as powerful regulators of microtubule dynamics (Howard and Hyman, 2007; Akhmanova and Steinmetz, 2015). The yeast kinesin-8 Kip3 promotes microtubule catastrophes (Varga et al., 2006; Arellano-Santoyo et al., 2017; Varga et al., 2009; Gupta et al., 2006), i.e., the transition from microtubule growth to shrinkage. In contrast, the kinesin-7 Kip2 inhibits catastrophes and promotes microtubule elongation (Hibbel et al., 2015; Cottingham and Hoyt, 1997; Huyett et al., 1998; Carvalho et al., 2004). Despite extensive experimental studies and mathematical modelling of kinesin behavior in vitro, their distribution and function in vivo has not been precisely investigated.

To characterize the distribution of Kip2 and Kip3 along cytoplasmic microtubules of metaphase yeast cells, we fused each protein to three copies of the super-folder green ﬂuorescent protein (3xsfGFP) on its C-terminus at their endogenous loci. The SPBs were visualized by fusing the SPB component Spc42 to mCherry. We aligned and averaged the signal obtained for each reporter for large collections of images (n ≥ 279 per strain) and computed the distribution of each kinesin along microtubules as a function of microtubule length.

In agreement with in vitro studies (Varga et al., 2006; Varga et al., 2009), Kip3-3xsfGFP intensity increased along metaphase microtubules, the protein being barely detectable at minus-ends and reaching maximal levels at plus-ends (Figure 5.1A). Accordingly, Kip3 amounts at plus-ends increased linearly with microtubule length (Figure 5.1A and S5.1A) as described and modelled in vitro (Varga et al., 2006; Varga et al., 2009). This distribution requires the kinesin to walk towards the microtubule plus-end over long distances without falling off, to walk faster than microtubules grow, and to land and start walking from any site on the microtubule lattice. Consequently, long microtubules collect Kip3 molecules over a greater surface than short ones (Varga et al., 2006). Since Kip3 triggers plus-end catastrophes in a concentration-dependent manner (Varga et al., 2006; Varga et al., 2009), our analysis suggests that Kip3 might be a length-dependent microtubule depolymerase in vivo as in vitro (Varga et al., 2006; Varga et al., 2009; Gupta et al., 2006). 67

A B Spc42-mCherry, Kip3-3xsfGFP Spc42-mCherry, Kip2-3xsfGFP

b-SPB 20 b-SPB 20 slope: 5.00 ± 0.32 a.u. μm-1 slope: 0.11 ± 0.18 a.u. μm-1 intercept: 1.68 ± 0.40 a.u intercept: 6.01 ± 0.29 a.u

15 15

10 10

5 5 GFP fluorescence along aMT (a.u.) 0 GFP fluorescence along aMT (a.u.) 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Distance from SPB (μm) Distance from SPB (μm)

Figure 5.1 – Kinesins Kip3 and Kip2 exhibit distinct localization patterns along microtubules in vivo. Representative images (top) and quantifications (bottom) of fluorescence intensities from endogenous Kip3- 3xsfGFP (A) or Kip2-3xsfGFP (B) along metaphase astral microtubules (aMTs; boxed areas). We aligned signals to b-SPBs using the peak of Spc42-mCherry and binned by microtubule length (2-pixel = 266.7 nm bin size). Colored lines show mean Kip2/3-3xsfGFP fluorescence per bin and shaded areas represent 95% confidence intervals for the mean. Grey dashed lines denote weighted linear regressions for the mean GFP fluorescence on plus-ends over all bins. The grey area in panel (A) marks Kip3-3xsfGFP fluorescence inside nuclei. Scale bars, 2 µm. 10 ≤ n ≤ 130 per bin.

Because in vitro Kip2 shows a similar run length and speed as Kip3 (Varga et al., 2006; Roberts et al., 2014), it is also expected to accumulate along microtubules and at their plus-ends as a function of microtubule length. Surprisingly, however, Kip2 distribution followed a completely different pattern: the signal for Kip2-3xsfGFP remained flat and non-zero from the microtubule minus-end up to the vicinity of the microtubule plus-end, where it peaked locally (Figure 5.1B andS 5.1B). Neither the level of the protein along the microtubule nor the height of its plus-end peak varied with microtubule length. This sharp discrepancy between the observed and expected distributions suggested that yeast cells actively control Kip2 distribution. To understand Kip2 behavior in vivo, we modelled the distribution profile of an idealized plus-end-directed motor as a function of four motor parameters: on- and off-rates for binding along the microtubule shaft, stepping speed, and rate of detachment from microtubule plus-ends (Figure 5.2A). This model easily explains the accumulation of Kip2 at microtubule plus-ends by a low plus-end detachment rate. It predicts a flat distribution in special cases, such as when the on-rate is zero (no motor on the microtubule lattice) or the motor speed is zero (inactive motors binding everywhere; see Supplementary Information for details). However, none of these cases is compatible with Kip2 moving processively and as fast as Kip3, both in vitro (Varga et al., 2009; Hibbel et al., 2015; Roberts et al., 2014) and in vivo (FigureS 5.3AB). The flat profile of Kip2 on microtubules resembles the average distribution of vehicles on a track between entry and exit sites at a constant speed and entry rate. We therefore included a minus-end entry site by adding a minus-end loading rate parameter. This allowed us to model flat motor distributions along the microtubule shaft and to fit the experimental concentration profile of Kip2 on microtubules of average length in 68

Figure 5.2 – Mathematical model predicts that Kip2 is predominantly loaded from microtubule minus- ends. (A) Schematic of the mathematical model. Free Kip2 (concentration [Kip2]free) binds to the microtubule minus-end anchored at the SPB with rate rin = kin[Kip2]free if the minus-end site is free, and to any free lattice site with rate ron = kon[Kip2]free. A bound motor can detach with rate koff, and it can advance with rate kstep if the next site towards the plus-end is free. At the plus end, the motor detaches with a different rate, kout.(B) Model fit to data for aMTs in the bin 1.33 µm to 1.60 µm (red), and predictions for other bins (blue) with respect to the mean (gray) and standard error (light gray) of the Kip2 wild type data in Figure 5.1B. Dashed lines past plus-end and SPB indicate model extrapolations without support by data. (C) Likelihood of estimated parameters: the maximum-likelihood solution (red x) corresponds to the red profile plotted in (B) used to estimate the parameters. Color-coded (gray) samples fall within (outside) the 95% confidence region. Left, out rate at plus-end as a function of maximum loading rate at SPB. Right, in rate at the minus-end inversely correlates with total Kip2.(D) Experimental kymograph (bottom) showing how Kip2-3xsfGFP speckles depart from the SPB (yellow arrows), marked with Spc72-GFP. Top kymograph, corresponding model predictions using the maximum-likelihood parameter set in Figure 5.2C. 69 vivo (Figure 5.2B). Importantly, with the same estimated parameter values, the model also predicted distribution profiles for long and short microtubules accurately, lending support to the model. The estimated parameters predicted minus-end recruitment and detachment rates of similar magnitude, an almost-zero off-rate, and low Kip2 recruitment on the lattice (Figure 5.2C). Specifically, the recruitment rate at minus-ends was 900 times higher than that at each lattice site (FigureS 5.9 and Supplementary Information for details). Thus, Kip2 distributions along microtubules in metaphase cells are consistent with Kip2 being recruited to microtubules predominantly at minus-ends-at SPBs. Three lines of evidence support this hypothesis. First and as predicted by the model, in all our Kip2-3xsfGFP kymographs (n=45) all trains of motors moving along microtubules started from SPBs (Figure 5.2D, Movie S1); in contrast, Kip3 started moving from random places along microtubules (FigureS 5.3A, Movie S2). Second, to visualize Kip2 loading sites on microtubules, we mutated a conserved glycine required for ATP hydrolysis in all kinesins (Kip2-G374A; FigureS 5.4). The ATPase deficient Kip2-G374A-3xsfGFP that cannot move from its loading site accumulated at or near SPBs (Figure 5.3A), consistent with Kip2 being nearly exclusively recruited to microtubule minus-ends. Accordingly, Kip2-G374A-3xsfGFP localization extensively overlapped with that of the γ-tubulin complex receptor on the SPB outer-plaque, Spc72 (Knop and Schiebel, 1998) (Figure S5.5A), a protein that might interact with Kip2 (Wang et al., 2012). In heterozygous diploid cells expressing both the wild type protein fused to mCherry (Kip2-mCherry) and Kip2-G374A-3xsfGFP, Kip2-mCherry localized to microtubule shafts and plus-ends, whereas the ATPase deficient mutant remained near SPBs (Figure 5.3DE). Third, inactivation of Spc72, using the spc72-2 temperature sensitive allele (Chen et al., 1998), impaired the recruitment of Kip2-G374A-3xsfGFP to SPBs significantly (Figure 5.3AC). Spc72 inactivation also caused the loss of wild type Kip2-3xsfGFP from cytoplasmic microtubules (FigureS 5.5B). Thus, Kip2 is primarily recruited at SPBs and starts walking along microtubules from their minus-ends. Besides SPB recruitment, our model of Kip2 distribution also predicts that loading of the motor is inhibited elsewhere along microtubule shafts. Notably, Kip2’s N-terminus is heavily phosphorylated in vivo in a GSK3-, Cdk1-, and Dbf2/20-dependent manner (Drechsler et al., 2015) (Figure 5.3B). Gsk3-dependent phosphorylation is primed through phosphorylation of serine 63, which is conserved across Kip2 orthologues in fungi (Figure S5.6) and falls into a consensus site for the mitotic kinases Cdk1 and Dbf2/20 (Dbf2/20 function in the yeast Hippo pathway (Hergovich et al., 2006)). Mutation of S63 to alanine (KIP2-S63A) abrogated Kip2 phosphorylation ((Drechsler et al., 2015) and Figure 3B). Strikingly, the Kip2-S63A-3xsfGFP protein showed a Kip3-3xsfGFP-like profile: its levels linearly increased from the minus- to the plus-ends of microtubules, and plus-end levels increased with microtubule length (Figure 5.3F, Movie S3). Additionally, instead of decorating only SPBs the ATPase-deficient Kip2-S63A-G374A-3xsfGFP mutant decorated microtubule shafts (Figure 5.3ADE). Thus, proper inhibition of Kip2 recruitment along microtubules depends at least in part on the phosphorylation of its N-terminus. Importantly, quantification of Kip2-G374A-3xsfGFP established that the b-SPB is around five times more active in recruiting Kip2 than the m-SPB (Figure 5.3C). The S63A mutation did not reduce this difference dramatically, although both SPBs recruited Kip2- S63A-G374A-3xsfGFP more efficiently (Figure 5.3C). In contrast, Kip2-G374A-3xsfGFP decorated both SPBs equally sparsely after inactivating the polo kinase Cdc5 (Figure 70

A B Kip2-G374A-3xsfGFP Spc42-mCherry Composite SPC72 Low-complexity Motor domain Coiled coil Kip2:N C consensus sites WT S63A SSTRSNS(63)PLR anti- HA Dbf2/Dbf20 unspecific Cdk1 GSK-3

30 ℃ 2 hrs spc72-2 p-Kip2-6HA Kip2-6HA

C b-SPB m-SPB

30 ℃ 2 hrs x 3.5 37 ℃ 50 min

200 ****

Kip2-G374A

150

x 4.8 x 5.0 **** **** 100

x 2.1

25 ℃ Kip2-S63A-G374A *** 50 x 1.1 x 3.1 n.s. * Normalized Kip2-G374A-3xsfGF P

fluorescence associated with SPBs (%) 0

WT WT WT cdc5-2 spc72-2 Kip2-S63A D Kip2-G374A-3xsfGFP Kip2-mCherry Composite Kip2-S63A-G374A-3xsfGFP Kip2-mCherry Composite

1 4 3

E F b-SPB slope: 3.06 ± 0.51 a.u. μm-1 200 130 intercept: 8.04 ± 1.12 a.u. 1 2 20

120 150 110 15 Fluorescence (%) Fluorescence (%) 100 100 0 1 2 3 0 1 2 3 10 200 130 3 4

120

150 Kip2-S63A-3xsfGFP 5 110 fluorescence along aMT (a.u.) Fluorescence (%) Fluorescence (%) 100 100 0 1 2 3 0 1 2 3 0 Distance from microtubule minus-ends (μm) -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Distance from SPB (μm)

Figure 5.3 – Recruitment of Kip2 takes place at SPBs and is inhibited along microtubule lattices. (A) Fluorescence microscopy images of metaphase cells with the ATPase deficient protein Kip2-G374A-3xsfGFP (green) and SPBs (magenta) in cells of indicated genotype. Scale bars, 2 µm.(B) Schematic of full-length Kip2 and its disordered N-terminus with residue serine 63 (red) and kinase consensus sites (blue). Lysates prepared from cycling cells of indicated genotype were blotted and analyzed with anti-HA antibodies. (C) Quantification of Kip2-G374A-3xsfGFP associated with b- and m-SPBs in cells of indicated genotype grown at indicated temperature; mean ± s.d., n ≥ 3 independent clones, > 150 cells per genotype. Statistical significances were calculated using one-way ANOVA. (D,E) Fluorescence microscopy images of heterozygous diploid cells with Kip2-mCherry (magenta) and Kip2-G374A-3xsfGFP (green) or Kip2-S63A-G374A-3xsfGFP (green). Intensity scans along the numbered microtubules are plotted in E. Scale bars, 2 µm.(F) Distribution of Kip2-S63A-3xsfGFP fluorescence along cytoplasmic microtubules as a function of microtubule length (29 ≤ n ≤ 55 per bin; as in Figure 5.1B). 71

B A 400 b-MT m-MT Kip2-3xsfGFP, Spc42-mCherry ****x 1.6 x** 1.2

300 ROI fluorescence

Kip2-S63A-3xsfGFP, Spc42-mCherry 200

ROI

100 Normalized GF P associated with aMT plus-ends (%)

Kip2-S63A

C D E

b-MT m-MT 37 ℃ 50 min b-MT m-MT 37 ℃ 50 min 5

Bik1-3xGFP, Spc72-GFP ) ** ** n.s. WT Kip2-S63A * * μm 100 4 **** 80

b-MT 3 60

spindle 40 WT cdc5-2 m-MT 2 20 aMT lifetime (s) aMT 1 0 3D maximum aMT length (

37 ℃ 50 min below detection limit −20 0

WT WT WT WT cdc5-2 cdc5-2 Kip2-S63A Kip2-S63A

Figure 5.4 – Kip2 phosphorylation and Polo kinase Cdc5 ensure faithful asymmetric patterning of cytoplasmic microtubules driven by SPBs. (A) Representative images of indicated proteins in indicated colors in metaphase cells with b- and m-microtubules. (B) Quantification of GFP fluorescence at plus-ends of b- and m-microtubules (mean ± s.d.; n = 115 for wild type (WT), 180 for Kip2-S63A). (C) Illustration and representative images of b- and m-microtubules using Bik1-3xGFP and Spc72-GFP as microtubule plus- and minus-end markers, respectively. (D, E) Microtubule lengths (D) and lifetimes (E) in cells with detectable b- and m- microtubules of the indicted genotypes at indicated temperatures (n > 233 cells from ≥ 3 independent clones; long (short) black bar: median (mean)). For all panels, **** p < 0.0001, *** p < 0.001, ** p < 0.01,* p < 0.05, n.s., not significant. Statistical significances were calculated by two-tailed Student’s t-test (B, D). Scale bars, 2 µm.

5.3C andS 5.7A), without reducing the recruitment of Spc72 to SPBs (FigureS 5.7B). Cdc5 localizes to b-SPBs during yeast metaphase where it phosphorylates different SPB components, including Spc72 (Keck et al., 2011; Snead et al., 2007). Thus, the SPBs’ differential Kip2 recruitment does not depend on Kip2 phosphorylation but more probably on the phosphorylation of some b-SPB components by Cdc5. Since Kip2 is a microtubule polymerase (Hibbel et al., 2015), the differential recruitment of Kip2 between SPBs raises the intriguing possibility that it links microtubule growth to SPB identity. Accordingly, while most cells formed a microtubule only on the b- SPB (FigureS 5.8D), cells harboring microtubules on both SPBs had 60% more Kip2 on the plus-ends of b- compared to m-microtubules (Figure 5.4B andS 5.2). As expected, these b-microtubules were on average longer and longer-lived than m-microtubules (Figure 5.4C-E). Allowing Kip2 recruitment anywhere along microtubules using the S63A mutation increased Kip2 levels on all microtubules and reduced their asymmetry between b- and m-microtubules (Figure 5.4B). In these cells, all microtubules were long 72

and the m-microtubules were not shorter or shorter-lived than b-microtubules (Figure 5.4DE). Furthermore, inactivation of Cdc5 erased the length and lifetime asymmetry between m- and b-microtubules (Figure 5.4C-E, TableS 5.1). Thus, the restriction of Kip2 recruitment to SPBs and the differential activities of the SPBs in this process are molecular determinants for microtubule patterning in vivo. Together, these data establish that the yeast kinesin Kip2 functions as a “messenger” between microtubule minus-ends at microtubule-organizing centers (MTOCs) and their distal plus-ends. Because Kip2 is also a microtubule polymerase, its messenger function enables yeast cells to control microtubule dynamics and length in an SPB-dependent manner. Future studies need to determine how SPBs enable Kip2 recruitment, but we assume that it involves binding sites for Kip2 on the SPB itself, as well as a phosphatase that restores Kip2’s afﬁnity for microtubule lattices. This simple and powerful mechanism is potentially conserved across various kinesins and across asymmetrically dividing cells that form asters of different sizes. If different MTOCs were to recruit kinesins with distinct activities on microtubule dynamics and cargo transport, this mechanism might have offered a rich playground to evolution for sophisticated microtubule patterning during mitosis and cell differentiation. 73

5.3 acknowledgements

We thank J. Hehl and the light microscopy center of ETH Zürich (ScopeM) for microscopy support; D. Panozzo for assistance on image analysis; H.-M. Kaltenbach for input to the measurement model, and A.-M. Farcas, S. Österle for comments on the manuscript.

5.4 author contributions

M. O. S., J. S., and Y. B. obtained the funding. X. C., L. A. W., J. S., and Y. B. designed the study concept. X. C. performed biological experiments, L. A. W. performed the computational modelling, M. M. S. and M. O. S. identiﬁed the critical motif for Kip2’s ATPase activity. X. C., L. A. W., J. S., and Y. B. acquired, analyzed, and/or interpreted the data. J. S. and Y. B. wrote the manuscript with inputs from all authors. J. S. and Y. B. supervised the study. Authors declare no competing interests.

5.5 data and materials availability

All data is available in the main text, the supplementary materials, or at https://gitlab. com/csb.ethz/Kip2-SPB-Profile-Manuscript. 74

5.6 supplementary materials

5.6.1 Materials and methods

5.6.1.1 Yeast strains

Yeast strains used in this study are listed in TableS 5.2. All strains are isogenic to S288C. Fluorescent or HA-tagged proteins were tagged at endogenous loci (Knop et al., 1999). Specific Kip2 mutations were introduced on a pRS314-Kip2-3xsfGFP:KanMX plasmid or a pRS304-Kip2 plasmid via site-directed mutagenesis (pfu-Turbo, Stratagene). Kip2 locus was then amplified and integrated in a kip2∆ strain and the correct integration was verified by PCR and sequencing.

5.6.1.2 Media and growth conditions

Cells were cultured in YEPD (yeast extract peptone, 2% dextrose) for collecting western blotting samples. For live cell imaging, overnight cultures in SC (synthetic medium, 2% dextrose) were diluted to OD600 0.15 and cultivated for 4 h more before being placed on an SC-medium agar patch for microscopy imaging. Overnight cultures of temperature ◦ sensitive mutant strains were diluted to OD600 0.15 and cultivated at 25 C for 4 h before shifting to 37 ◦C for 50 min or to 30 ◦C for 2 h as speciﬁed.

5.6.1.3 Fluorescence microscopy

A Nipkow spinning disk (Carl Zeiss) equipped with an incubator for temperature was employed. Time-lapse movies were acquired using a back-illuminated EM-CCD camera Evolve 512 (Photometrics, Inc.) mounted on the spinning disk microscope with a motorized piezo stage (ASI MS-2000) and 100x 1.46 NA alpha Plan Apochromat oil immersion objective, driven by Metamorph based software VisiVIEW (Visitron Systems). 17 Z-section images separated by 0.24 µm increments were captured with the exposure time of 30 ms each, the whole stack took 1.07 s. For imaging aMT dynamics, 80 continuous repetitions were taken. For imaging strains with both GFP and mCherry signals, the GFP channel was always set to 30 ms exposure time and the mCherry channel to 50 ms exposure time. For diploid cells expressing Kip2-mCherry, the exposure time for the mCherry channel was set at 100 ms. Images in ﬁgures represent sum ﬂuorescence intensities across Z-projections. Scale bars represent 2 µm.

5.6.1.4 Image and data analysis

Metaphase cells were collected based on the shape of cells and the size of spindles. For analyzing the profiles of GFP fusion proteins along astral microtubules (aMTs), the sum intensity projection of the images was used. A 5-pixel (666.7 nm) width line was used to scan aMTs from plus-ends toward SPBs both in the GFP and the mCherry channels using Fiji (Schindelin et al., 2012), and exported to CSV files. These profiles were then aggregated for further analysis using MATLAB (R2018a, Mathworks), and peak detection for the GFP and mCherry signals was performed, respectively. Profile length was defined as the peak-to-peak distance, and the profiles were then binned into length bins as detailed in the figure legends. 75

Kymographs were generated and analyzed using Fiji. Shortly, a 5-pixel width line was placed along metaphase cytoplasmic microtubules from SPBs towards plus-ends, and kymographs were created using the ‘Reslice’ function without interpolation. The speed of fluorescent speckles moving towards microtubule plus-ends was calculated by extracting the coordinates of the starting and terminal positions of each speckle. For fluorescence intensity, a Region Of Interest (ROI) was drawn around the area of interest (AOI) and the integrated density was quantified. An identically sized ROI was put next to the AOI to determine the background signal. The background intensity was subtracted from the ROI intensity to yield the fluorescence intensity (a.u.). For every experiment that was performed for quantification of fluorescence intensity (a.u.), corresponding wild type cells were imaged and analyzed for comparison to mutant cells. Average values of wild type cells of different experiments were used for normalization and comparison between experiments. To determine the length of aMTs, endogenously expressed Bik1-3xGFP and Spc72-GFP were used as the aMT plus-end marker and the minus-end marker, respectively. Three- dimensional coordinates of aMT plus-ends and the corresponding SPBs were extracted with the Low Light Tracking Tool ((Krull et al., 2014)). The tracking tool does make mistakes when microtubules depolymerize with a very high rate, or when microtubules pivot quickly with large angles. Therefore, all tracked trajectories were inspected by eye to find and correct those very rare mistakes. All of the time series tracking results were analyzed with custom functions written in Matlab (MathWorks). The distance between the b- and m- SPBs is the spindle length. Cells with spindles longer than 2 µm were excluded. The distance between an aMT plus-end and the corresponding SPB represents the length of the microtubule. Only microtubules longer than 5-pixels (666.7 µm) were considered detectable due to the limit of the microscope resolution. Using this criterion, the maximum length and lifetime of each microtubule within the recorded time window (85.6 s) were extracted. All catastrophe and rescue events, as well as growth and shrinkage phases, were annotated manually.

5.6.1.5 Western blot

For protein extraction, 2 OD600 log phase cell cultures were spun down and pellets were washed once with ice cold PBS, then lysed with Zirconia-Silicate beads in lysis buffer (50 mm Tris pH 7.5, 150 mm NaCl, 0.5 mm EDTA, 1 mm MgCl2, Roche Complete Protease and phosphatase inhibitors and 0.2% NP-40) on a FastPrep-24 homogenizer. Lysate was cleared by centrifugation at 5000 × g , 4 ◦C for 5 min. Samples were separated on a 6% SDS-polyacrylamide gel (SDS-PAGE), wet-transferred onto a polyvinylidene ﬂuoride (PVDF) membrane for western blotting. Antibodies used were primary antibody anti-HA (1:1000, mouse monoclonal, Covance Inc.) and secondary antibody goat anti-Mouse IgG conjugated to horseradish peroxidase (1:5000, Bio-Rad).

5.6.1.6 Statistics

Each experiment was repeated with three or more independent clones (biological repli- cates). The standard deviation (s.d.) of independent clones is shown in the graphs, or as indicated. n.s. (not signiﬁcant) or asterisks indicate P values from Student’s t-test or one-way ANOVA as indicated. 76

5.6.2 Supplementary Text

5.6.2.1 Model: Mathematical formulation We formulate the following ordinary differential equations for the binding probability of kinesin motors at the minus-end p1 (5.1), on the interior lattice pi (5.2), and at the plus-end pN (5.3) of a 1D protoﬁlament of length N: dp 1 = (1 − p ) r + (1 − p ) r − p k − p (1 − p ) k (5.1) dt 1 in 1 on 1 off 1 2 step dpi = p − (1 − p ) k + (1 − p ) r − p k − p (1 − p + ) k (5.2) dt i 1 i step i on i off i i 1 step dpN = p − (1 − p ) k + (1 − p ) r − p k (5.3) dt N 1 N step N on N out

Here, rin = kin[Kip2]free and ron = kon [Kip2]free, and the probabilities are subject to 0 ≤ p1 ≤ 1, 0 ≤ pi ≤ 1, 0 ≤ pN ≤ 1.

5.6.2.2 Stochastic Simulation To perform stochastic simulations, we use a direct-method Gillespie algorithm (Gillespie,

1977). Initially, all Kip2 is assumed to be free, i.e., [Kip2]free = [Kip2]total. The state x of the protoﬁlament is a binary vector of length N, with each entry describing a binding site xi. Each binding site can either be free, xi = 0, or occupied by a motor, xi = 1, i.e., ( 1 xi = 0 free (xi) := , and occ (xi) := 1 − free (xi) 0 otherwise

The state then evolves according to the propensities and state change vectors given in TableS 5.3.

5.6.2.3 Model Parameters

Lattice Loading Rate ron

The lattice loading rate is given by ron = kon[Kip2]free, where kon is the Kip2 motor on rate per binding site, per s, per nm Kip2. We start parameter search around the published in vitro on rate per microtubule, per minute, per nM Kip2 (Roberts et al., 2014). Since in vivo, both Bik1 and Bim1 are present, we use the value of 3.9 µm−1 min−1 nm−1, which, assuming 13 protofilaments and 125 binding sites per micron of protofilament, i.e., 8 nm per binding site, translates to 4 × 10−5 s−1 nm−1. Total Kip2 Concentration [Kip2]total Another unknown parameter is the Kip2 concentration [Kip2]total, which we specify in nm units. Estimating this value is more involved – we start by querying PaxDB (Wang et al., 2015), a database that aggregates protein abundances for different organisms in parts per million, for Kip2 abundance. From the experimental estimates in PaxDB (discarding the integrated model), we compute the median Kip2 abundance in ppm, 14.6 ppm. This number can then be converted to proteins per micron cubed by multiplying by 2.5 × 106 proteins µm−3 as derived for S. cerevisiae (Milo, 2013), to yield an estimated 36.5 Kip2 molecules per micron cubed. The volume of a yeast cell in metaphase is approximately 50 µm3 (Uchida et al., 2011). Thus, approximately 1825 Kip2 molecules 77 are available in a metaphase yeast cell, corresponding to a concentration of 61 nm. If we consider a single microtubule of 13 protofilaments, there are about 140 Kip2 molecules available per protofilament. We use this number as an initial value for parameter search in our single-protofilament model. Note that we do not fix this value since the standard deviation of the median estimator in PaxDB is greater than the median itself – we do not expect this estimate to be very accurate. Minus-End Loading Rate rin The minus-end loading rate was unknown at first. However, it is useful to compare the minus-end loading rate rin = kin[Kip2]free with the overall loading rate on the lattice

N N ron,total = ∑ ron = ∑ kon[Kip2]free = Nkon[Kip2]free i=1 i=1 where rin ron,total if and only if kin Nkon. Since koff ≈ 0, no motors can detach on the way, and we can distinguish between minus-end-dominated and lattice-dominated motor binding by comparing the in- and on-rates. If kin/kon N, then motor binding is minus-end-dominated, if kin/kon N, then motor binding is lattice-dominated. For initializing the parameter search, we can thus explore the regime around kin ≈ Nkon. Stepping rate kstep The stepping rate kstep can be inferred from the velocity the motor shows while not encountering any obstacle, if we assume 8 nm steps (corresponding to the length of a tubulin heterodimer). The free-stepping motor velocity in vitro was quantiﬁed in the presence of the proteins Bim1 and Bik1, Kip2 binding partners in vivo, in (Roberts et al., 2014). Since measurements of speckle speeds from our own kymographs were available, we used the measured stepping rate of 13.04 s−1 in the model. Lattice off rate koff The lattice off rate koff was previously determined in vitro (Roberts et al., 2014), both in the absence of Bim1 and Bik1 (0.473 s−1), as well as their presence (7.3 × 10−3 s−1). However, these experiments were performed without crowding agents, and kinesin motors tend to be more processive in crowded environments (Conway and Ross, 2014). The microtubules we consider are shorter than 4 µm, but, on average, a motor is expected to run around 8 µm without falling off, and potentially even further in the crowded cytoplasm in vivo. We therefore set the off rate in the model to 0 s−1. Plus-end off rate kout The plus-end off rate kout is conceptually different from the lattice off rate koff and the stepping rate kstep because the motor cannot simply step off the end, and it does not stay bound forever. An off rate of 2.27 × 10−2 s−1 for single motors has previously been determined in vitro (Hibbel et al., 2015). However, it is unclear if multiple motors at the plus-tip would detach more readily. In the limiting case of the plus-end off rate being equal to the step rate, the motors would simply step off past the microtubule with the same speed as they move on the lattice. Therefore, we investigated the parameter range between the single-motor off rate, and the stepping rate.

5.6.2.4 Conditions for Flat Kinesin Proﬁles

To derive a condition on the parameters required for a ﬂat mean motor occupancy proﬁle, we require that the probability of a motor being bound on the lattice is equal for all lattice 78

sites in the midzone, i.e., the zone between the microtubule minus- and plus-ends. The equations for such a system read as follows:

dp 1 = (1 − p ) r + (1 − p ) r − p k − p (1 − p ) k (5.4) dt 1 in 1 on 1 off 1 M step dp M = p (1 − p ) k + (1 − p ) r dt 1 M step M on − pMkoff − pM (1 − pM) kstep (5.5) dp M = p (1 − p ) k + (1 − p ) r dt M M step M on − pMkoff − pM (1 − pM) kstep (5.6) dp M = p (1 − p ) k + (1 − p ) r dt M M step M on − pMkoff − pM (1 − pN) kstep (5.7) dp N = p (1 − p ) k + (1 − p ) r − p k (5.8) dt M N step N on N out

where the probabilities are subject to 0 ≤ p1 ≤ 1, 0 ≤ pM ≤ 1, 0 ≤ pN ≤ 1. We are interested in the steady-state proﬁle generated. Therefore, we set the left-hand side ∗ of above equation system to zero and derive the steady-states pi , depending on the (0) parameters. pi refers to the initial state of pi.

Case 1: kstep = 0 If kstep = 0, it follows immediately that all the equations become uncoupled, i.e., each lattice binding site becomes an independent binding site. For the minus-end site, unless ∗ (0) rin = ron = koff = 0 and p1 = p1 (there are no dynamics),

∗ rin + ron p1 = . rin + ron + koff

∗ (0) For the inner lattice sites, unless ron = koff = 0 and pM = pM ,

∗ ron pM = . ron + koff

∗ (0) For the plus-end site, unless ron = kout = 0 and pN = pN ,

∗ ron pN = ron + kout

Case 2: kstep > 0, kout = 0 This scenario consists of a permanent roadblock at the plus end. From equation (5.8),

∗ ∗ ∗ ∗ pM (1 − pN) kstep + (1 − pN) ron − pNkout = 0 (5.9) | {z } =0

∗ ∗ it follows that (1 − pN) ron + pMkstep = 0. This implies

∗ • pN = 1, and/or 79

∗ ∗ (0) • ron = 0 and pM = 0. pN is then the same as the initial condition pN . From equation (5.5),

∗ ∗ ∗ ∗ ∗ ∗ p1 (1 − pM) kstep + (1 − pM) ron − pMkoff − pM (1 − pM) kstep = 0 | {z } | {z } | {z } | {z } ∗ =0 =0 = =p1 kstep 0 ∗ it follows that p1 = 0. Using equation (5.4), ∗ ∗ ∗ ∗ ∗ (1 − p1) rin + (1 − p1) ron − p1 koff − p1 (1 − pM) kstep = 0 | {z } | {z } | {z } | {z } =rin =0 =0 =0

immediately shows that rin = 0. ∗ If pN = 1, the plus-end site is fully occupied. Then equations (5.7) and (5.6), i.e., ∗ ∗ ∗ pM (1 − pM) kstep + (1 − pM) ron ∗ ∗ ∗ − pMkoff − pM (1 − pN) kstep = 0, and | {z } =0 ∗ ∗ ∗ pM (1 − pM) kstep + (1 − pM) ron ∗ ∗ ∗ − pMkoff − pM (1 − pM) kstep = 0 imply that either of two cases must hold:

∗ ∗ ∗ 1. pM = 0 and ron = 0. It then again follows that p1 (1 − pM) kstep = 0 and thus ∗ p1 = 0, which in turn implies that rin = 0. ∗ 2. pM = 1 and koff = 0. In this case, the motors arrive from the minus-end and get stuck on the lattice, never reaching the plus end. Then,

∗ ∗ ∗ ∗ ∗ (1 − p1) rin + (1 − p1) ron − p1koff − p1 (1 − pM) kstep = 0 | {z } | {z } =0 =0

∗ ∗ (0) either implies p1 = 1, or rin + ron = 0, in which case p1 = p1 .

Case 3: kstep > 0, kout > 0, kin = 0, kon = 0 In this case, the steady-state equation (5.4) ∗ for p1 reads ∗ ∗ ∗ ∗ ∗ (1 − p1) rin + (1 − p1) ron −p1koff − p1 (1 − pM) kstep = 0 | {z } | {z } =0 =0 ∗ ∗ ⇔ p1 koff + (1 − pM) kstep = 0

Two sub-cases fulﬁll this condition:

∗ ∗ (0) 1. pM and koff = 0, in which case p1 = p1 . This corresponds to case 2 above, where rin + ron = 0, and then

∗ ∗ ∗ ∗ ∗ ∗ pM (1 − pM) kstep + (1 − pM) ron − pMkoff − pM (1 − pN) kstep = 0 | {z } | {z } | {z } | {z } =0 =0 =0 ∗ =(1−pN )kstep

∗ implies that pN = 1. Thus, the lattice is fully occupied, and any motor bound at the minus-end site cannot move. 80

∗ 2. p1 = 0, i.e., the ﬁrst binding site is always empty. Then, ∗ ∗ ∗ ∗ ∗ ∗ p1 (1 − pM) kstep + (1 − pM) ron −pMkoff − pM (1 − pM) kstep = 0 | {z } | {z } =0 =0 ∗ ∗ ⇔ pM koff + (1 − pM) kstep = 0

Then, there are again two cases: ∗ a) pM = 0. Then, ∗ ∗ ∗ ∗ pM (1 − pN) kstep + (1 − pN) ron −pNkout = 0, | {z } | {z } =0 =0 ∗ which implies pN = 0, i.e., all the sites are empty. ∗ b) pM = 1 and koff = 0. Then ∗ ∗ ∗ ∗ ∗ ∗ pM (1 − pM) kstep + (1 − pM) ron − pMkoff − pM (1 − pN) kstep = 0, | {z } | {z } | {z } | {z } =0 =0 =0 ∗ =(1−pN )kstep

∗ implies that pN = 1, i.e., the ﬁrst site is empty, and the lattice and plus-end sites are occupied.

Case 4: kstep > 0, kout > 0, kin + kon > 0 From the steady-state equations

∗ ∗ ∗ ∗ ∗ ∗ pM (1 − pM) kstep + (1 − pM) ron − pMkoff − pM (1 − pM) kstep = 0 ∗ 2 ∗ ∗ ∗ ∗ ∗ 2 ⇔ −kstep pM + kstep pM + ron − pMron − pMkoff − pMkstep + pM kstep = 0 ∗ ∗ ⇔ ron − pMron − pMkoff = 0

If ron + koff > 0, then ∗ ron pM = ron + koff For the plus-end site, this leads to:

∗ ∗ ∗ ∗ ∗ ∗ 0 = pM (1 − pM) kstep + (1 − pM) ron − pMkoff − pM (1 − pN) kstep ∗ ∗ koff + ron ron ⇔ pN = pM + − ∗ kstep kstep pM ∗ ron koff + ron ron ron + koff ⇔ pN = + − ron + koff kstep kstep ron ∗ ron ⇔ pN = ron + koff From

∗ ∗ ∗ ∗ pM (1 − pN) kstep + (1 − pN) ron − pNkout = 0 ron koff koff ron kstep + ron − kout = 0 ron + koff ron + koff ron + koff ron + koff it follows that ron + koff + kstep kout = koff . ron + koff 81

Then, for the minus end site, ∗ ∗ ∗ ∗ ∗ ∗ p1 (1 − pM) kstep + (1 − pM) ron − pMkoff − pM (1 − pM) kstep = 0 ∗ 2 ∗ ∗ ∗ ⇔ kstep pM − kstep p1 + koff + kstep + ron pM + p1kstep + ron = 0 2 ron ∗ ron ⇔ kstep − kstep p1 + koff + kstep + ron ron + koff ron + koff ∗ + p1kstep + ron = 0 2 ∗ ⇔ kstepron − kstep p1 + koff + kstep + ron ron (ron + koff) ∗ 2 + p1kstep + ron (ron + koff) = 0 k k (k p∗ − r (1 − p∗)) ⇔ off step off 1 on 1 = 2 0 (koff + ron) This holds in two cases:

• koff = 0 and ron > 0, which implies kout = 0 and thus is covered in case 2, and / or • p∗ = ron . Then, from 1 ron+koff ∗ ∗ ∗ ∗ ∗ (1 − p1) rin + (1 − p1) ron − p1koff − p1 (1 − pM) kstep = 0 koff koff ron rin + ron − koff ron + koff ron + koff ron + koff ron koff − kstep = 0 ron + koff ron + koff ronkoffkstep koffrin + koffron − ronkoff − = 0 ron + koff ronkstep ⇔ = rin ron + koff

Thus, for ron + koff > 0, we have ∗ ∗ ∗ ∗ ron p = p1 = pM = pN = . ron + koff The flow rate of motors from one site to the next is accordingly r k ∗ ( − ∗) = on off p 1 p kstep 2 kstep (ron + koff) This is the same as the inflow at the minus-end ∗ ∗ koff ron (1 − p )(rin + ron) − p koff = (rin + ron) − koff ron + koff ron + koff ronkstep koff = ron+koff ron + koff r k = on off 2 kstep (ron + koff) In contrast, the outflow at the plus-end is

∗ ron ron + koff + kstep p kout = koff ron + koff ron + koff r k = on off + + 2 ron koff kstep (ron + koff) ∗ ∗ ∗ = p (1 − p ) kstep + (1 − p ) ron 82

If ron + koff = 0, it follows that rin > 0, and equations (5.4), (5.5), (5.7), and (5.8) then read

∗ ∗ ∗ 0 = (1 − p1) rin − p1 (1 − pM) kstep ∗ ∗ ∗ 0 = (p1 − pM)(1 − pM) kstep ∗ ∗ ∗ 0 = pM (pN − pM) kstep ∗ ∗ ∗ 0 = pM (1 − pN) kstep − pNkout

There are again multiple solutions to this system of equations:

∗ ∗ ∗ • p1 = pM = pN = 1, where kout = 0 and rin is arbitrary (but > 0 per assumption).

• p∗ = p∗ = p∗ = p∗ < 1. Then, we get p∗ = rin , r < k and k = k − k . 1 M N kstep in step out step in The ﬂow rate of motors from one site to the next then is ∗ ∗ rin kstep − rin p (1 − p ) kstep = kstep

The inﬂow at the minus end is ∗ rin kstep − rin (1 − p ) rin = , kstep

and the outﬂow at the plus end is ∗ rin kstep − rin pNkout = . kstep

This is the solution that is most consistent with the parameter values and observed behavior for the Kip2-3xsfGFP mean proﬁle in wildtype cells.

5.6.2.5 Measurement Model

The measurement model maps fluorescent Kip2 molecules on the protofilament lattice of the in silico model to in vivo fluorescence signals as measured by a confocal microscope. Since the underlying microtubule length l resulting in the minus-end-peak to plus-end- peak length lpeak is à priori unknown, we first simulate a model of length N = lpeak/8 nm 0 binding sites. Then, the actual plus-end peak location lactual < lpeak is determined by executing steps 1-5 and 7 of the algorithm below. Step 6 is omitted for the first estimation because the initial length of the model may be too short to create sufficient profiles to 0 properly compute lactual.

1. Stochastic simulation Simulate model as detailed in the stochastic simulation section for two hours of simulation time.

2. Temporal sampling Sampling of time points from simulation at 0.25 Hz (1 Hz for Figure 5.2B) for one hour of simulation time after initial burn-in of one hour simulation time. 83

3. Convolution of sampled proﬁles with 1D point-spread function (PSF) We use a 1D Gaussian PSF with λ σ = 0.21 NA where λ ≈ 509 nm is the emission wavelength of the used GFP ﬂuorophore, and NA = 1.46 is the numerical aperture of the objective lens used (Thomann et al., 2002).

4. Pixel sampling and binning Each pixel has a side length of 133 nm, which corresponds to approximately 17 motor binding sites of 8 nm on the protofilament. We compute all mappings from binding sites to pixels, with integer offsets of binding sites, by averaging the fluorescence values of the binding sites in each pixel, resulting in 17 possible samplings of each convolved profile.

5. Peak detection For detecting the location of the microtubule plus-end, as for the in vivo data, the microtubule plus-end was defined by finding the local fluorescence maximum at the microtubule plus-tip.

6. Restriction to profile lengths within bin Since the in vivo profiles were analyzed by binning their lengths from the minus-end to the plus-end with a ±1 pixel threshold, model profiles were binned analogously.

7. Centered within-bin alignment & mean computation Finally, all the proﬁles within a bin are aligned by centering them, and the bin mean is computed.

Then, we correct the protofilament length by the numerically-determined peak-to-plus- 0 0 0 end offset lactual to give rise to l = lpeak + lpeak − lactual . Now, step 6 in the algorithm is enabled (minimum and maximum peak-to-peak lengths are enforced), and the same estimation is performed to determine lactual. Now, the model length l is computed using l = lpeak + lpeak − lactual . However, since peak location is the result of a stochastic process, shorter / longer model lengths can still generate samples with appropriate ˜ peak lengths lactual. Therefore, we compute models with l = l + o · i, where i ∈ [−4, 4] and o = 88 nm is an offset chosen small enough such that increasing the number of averaged models of different lengths l˜ has negligible effects on the final model mean µmodel(xi,model, p) with sample locations xi,model and parameters p, which is computed by averaging all the profiles generated that pass step 6 according to the same algorithm detailed in step 7.

5.6.2.6 Likelihood function and conﬁdence region

We compare the mean proﬁle µdata (xi,data) with standard error SEM (xi,data) at locations xi,data obtained from in vivo data, with the mean proﬁle

µ˜model (xi,model, p, A, B) = A + B · µmodel (xi,model, p) with parameters p at locations xi,model, by linearly interpolating

µ˜model (xi,model, p, A, B) 84

at xi,data and thus estimating µ˜model (xi,data, p, A, B). A is given by the mean background ﬂuorescence, and B is the maximum mean ﬂuorescence corresponding to a fully occupied lattice, which can be estimated from the saturation of the S63A mutant at approximately 17 a.u. (and normalized between measurements using the wild type control, leading to 10.2 a.u. for the wild type data in Figure 5.1). We assume a normal error for the data, which gives rise to the likelihood function

2 (µ˜ (x ,p,A,B)−µ (x )) N N − model data,i data data,i 1 1 2 L (model|data) = · e 2 SEM(xdata,i) N/2 ∏ ( ) ∏ (2π) i=1 SEM xdata,i i=1 In order to ﬁnd the best-ﬁtting parameter set, rather than maximizing the likelihood directly, we minimize the negative log-likelihood:

N N −2LL = −2 − log(2π) − ∑ SEM(xdata,i) 2 i=1 ! N (µ˜ (x , p, A, B) − µ (x ))2 − model data,i data data,i ∑ 2 i=1 2 SEM(xdata,i) N = N log(2π) + 2 ∑ SEM(xdata,i) i=1 | {z } =const.=:C N (µ˜ (x , p, A, B) − µ (x ))2 + model data,i data data,i ∑ 2 i=1 SEM(xdata,i) | {z } =:SSE(p)

Since C is a constant, we can simply minimize the sum-of-squares term over all possible parameter sets p in order to ﬁnd the maximum likelihood parameter set pML:

pML := arg min (SSE(p)) p

In order to compute, e.g., 95% conﬁdence regions for the parameters p, we require that

(p) ≤ −1( − ) SSE Fχ2 0.95, nD nP

−1 − where Fχ2 is the inverse of the chi-square distribution with degrees of freedom, nD nP being the number of data points in the proﬁle (indicated in red in Figure 5.2B), and nP = 3 being the number of model parameters, respectively. 85

5.6.3 Supplementary Figures

A B slope = 58.79 ± 5.726 % μm-1 slope = 4.862 ± 4.566 % μm-1 y y t 300 t 300 s i Y-intercept = 7.178 ± 9.415 % s i Y-intercept = 93.19 ± 7.737 % e n e n t t n n %) %) i i ( ( P P F 200 F 200 3xsf G 3xsf G - -

100 100 on aMT plus-ends on aMT plus-ends

0 0 Normalized Kip3 0 1 2 3 4 Normalized Kip2 0 1 2 3 4 2D b-microtubule Length (μm) 2D b-microtubule Length (μm)

Figure S5.1 – Kip3, and not Kip2, accumulates on microtubule plus-ends in a length-dependent manner. (A, B) Scatterplots with fitted linear regression lines between two-dimensional (2D) b-MT length and normalized Kip3-3xsfGFP (A) or Kip2-3xsfGFP (B) intensity on b-MT plus-ends (%). Kip3 shows a significant positive correlation (p < 0.0001, n = 183 cells) in contrast to Kip2, which shows no significant correlation (p = 0.2883, n = 200 cells). In addition, Kip3-3xsfGFP is nearly absent on very short microtubules, whereas Kip2-3xsfGFP is equally abundant on microtubules of all lengths. 86

A b-SPB m-SPB 20 Bin (μm) Bin (μm) 0.80 - 1.06 (n = 108) 0.80 - 1.06 (n = 42) 1.06 - 1.33 (n = 111) 1.06 - 1.33 (n = 17) 1.33 - 1.60 (n = 81) 15 1.60 - 1.86 (n = 47) 1.86 - 2.13 (n = 23) Slope Intercept

2.13 - 2.40 (n = 10) (a.u. μm-1) (a.u.) b-SPB 5.00 ± 0 .32 1.68 ± 0.40 Kip3 10 m-SPB 1.35 5.54 b-SPB (w/o m-MTs) 4.99 ± 0.82 4.10 ± 1.67 Kip2-S63A b-SPB (w/ m-MTs) 1.04 ± 0.73 11.9 ± 1.7 m-SPB 2.20 ± 0.80 6.28 ± 1.88 along aMT (a.u.) 5 b-SPB (w/o m-MTs) 0.289 ± 0.137 5.44 ± 0.21 Kip2 b-SPB (w/ m-MTs) –0.136 ± 0.452 7.06 ± 0.73

Kip3-3xsfGFP fluorescence m-SPB –0.524 ± 0.264 5.45 ± 0.39 0

0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Distance from SPB (μm) Distance from SPB (μm)

B Metaphase cells without detectable m-MTs Metaphase cells with detectable m-MTs b-SPB b-SPB m-SPB 20 Bin (μm) Bin (μm) Bin (μm) 1.06 - 1.33 (n = 12) 1.06 - 1.33 (n = 21) 1.06 - 1.33 (n = 15) 1.33 - 1.60 (n = 17) 1.33 - 1.60 (n = 30) 1.33 - 1.60 (n = 17) 1.60 - 1.86 (n = 13) 1.60 - 1.86 (n = 29) 1.60 - 1.86 (n = 20) 15 1.86 - 2.13 (n = 17) 1.86 - 2.13 (n = 38) 1.86 - 2.13 (n = 23) 2.13 - 2.40 (n = 17) 2.13 - 2.40 (n = 30) 2.13 - 2.40 (n = 29) 2.40 - 2.66 (n = 15) 2.40 - 2.66 (n = 24) 2.40 - 2.66 (n = 27) 2.66 - 2.93 (n = 17) 2.66 - 2.93 (n = 20) 2.66 - 2.93 (n = 30) 10 2.93 - 3.20 (n = 15) 2.93 - 3.20 (n = 22) 2.93 - 3.20 (n = 34) 3.20 - 3.46 (n = 17) 3.20 - 3.46 (n = 12) 3.20 - 3.46 (n = 27)

along aMT (a.u.) 5

0 Kip2-S63A-3xsfGFP fluorescence -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Distance from SPB (μm) Distance from SPB (μm) Distance from SPB (μm)

C Metaphase cells without detectable m-MTs Metaphase cells with detectable m-MTs b-SPB b-SPB m-SPB 9 Bin (μm) Bin (μm) Bin (μm) 8 0.80 - 1.06 (n = 76) 0.80 - 1.06 (n = 39) 0.80 - 1.06 (n = 35) 1.06 - 1.33 (n = 89) 1.06 - 1.33 (n = 41) 1.06 - 1.33 (n = 32) 7 1.33 - 1.60 (n = 80) 1.33 - 1.60 (n = 34) 1.33 - 1.60 (n = 29) 1.60 - 1.86 (n = 82) 1.60 - 1.86 (n = 26) 1.60 - 1.86 (n = 23) 6 1.86 - 2.13 (n = 51) 1.86 - 2.13 (n = 17) 1.86 - 2.13 (n = 16) 2.13 - 2.40 (n = 39) 2.13 - 2.40 (n = 14) 5

3 along aMT (a.u.) 2

Kip2-3xsfGFP fluorescence 1

0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Distance from SPB (μm) Distance from SPB (μm) Distance from SPB (μm)

Figure S5.2 – Kinesin localization patterns along microtubules in vivo. Distribution of Kip3-3xsfGFP (A), Kip2-S63A-3xsfGFP (B), and Kip2-3xsfGFP (C) fluorescence (a.u.) along aMTs as a function of microtubule length (µm), of the SPB of origin (b-SPB or m-SPB), and of the presence of m-MTs. Colored lines show mean GFP fluorescence per length bin and shaded areas 95% confidence intervals for the mean. Grey dashed lines denote weighted linear regression for the mean GFP fluorescence on microtubule plus-ends over all bins, with slopes, intercepts, and standard errors as detailed in the top right. The numbers of analyzed metaphase cells are marked within the graph of each condition and these data were also used to generate Figure 5.1 and FigureS 5.3F. 87

A Time (s) Time (s) Time (s) Kip3-3xsfGFP, Spc72-GFP Kip2-3xsfGFP, Spc72-GFP Kip3-3xsfGFP, Spc72-GFP 10 s 10 s Kip2-S63A-3xsfGFP, Spc72-GFP 10 s 2 µm + end 2 µm + end 2 µm + end SPB SPB SPB

B C D

15 n.s.

) 16 **** -1 Bik1-3xGFP, Spc72-GFP min )

-1 8 (x2,y2,z2,t) d ( μ m 10 3D distance } 4 (x1,y1,z1,t)

5 3D Speed ( μm min

1 2D Speckle moving spee

0 0.5 80 time points at an interval of 1.07 s Kip3 Kip2 Growth Shrinkage Kip2-S63A

Figure S5.3 – Measurements of kinesin movement speeds, as well as the speeds of b-MT growth and shrinkage in living cells. (A) Representative kymographs and highlighted trajectories (red dashed lines) of Kip3-3xsfGFP (left), Kip2-3xsfGFP (middle), and Kip2-S63A-3xsfGFP (right) speckles moving along a metaphase b-MT. Unlike Kip2, Kip3-3xsfGFP rarely forms obvious speckles. (B) A collection of kymographs was generated to quantify the speed of speckles moving towards microtubule plus-ends for each mutant. Kip3-3xsfGFP speckles move at a speed of 3.03 ± 1.76 µm min−1 (n = 10 speckles), Kip2- 3xsfGFP at 6.26 ± 2.07 µm min−1 (n = 192), and Kip2-S63A-3xsfGFP at 6.03 ± 2.02 µm min−1 (n = 66). (C–D) The speeds of b-MT growth, 1.38 ± 1.05 µm min−1 (n = 250 phases) and shrinkage. 1.95 ± 1.41 µm min−1 (n = 409 phases) were derived from the 3D b-MT length over 85.6 s. Growth and shrinkage phases were annotated manually. All cells were cultivated and imaged at 25 ◦C. All values are reported as mean ± s.d.. Statistical signiﬁcances were calculated using two-tailed student t-test. For all panels, ∗ ∗ ∗ ∗ p < 0.0001, n.s., not signiﬁcant.

scKip2 --GTSRSSTLSLCDLAGSERATGQ---- 381 scKip3 --SQHTFATLSIIDLAGSERAAATR-NR 353 scKip1 NKNFVKIGKLNLVDLAGSENINRSG-AE 329 hsCENPE CEGSVKVSHLNLVDLAGSERAAQTG-AA 248 hsKif5a -----LSGKLYLVDLAGSEKVSKTG-AE 245 hsKif2C -----MHGKFSLVDLAGNERGADTSSAD 506

Figure S5.4 – Sequence alignment of various kinesin motor domains focused on the switch-2 motif (DxxGxE, highlighted in bold, Marx et al., 2009) that is essential for the ATP hydrolysis. The yeast (sc) kinesins Kip2 (UniProt ID:P 28743), Kip1 (UniProt ID:P 28742) and Kip3 (UniProt ID:P 53086) were aligned with the human (hs) kinesins CENPE (UniProt ID:Q 02224), Kif5a (UniProt ID:Q 12840) and Kif2C (UniProt ID:Q 99661) to illustrate conservation across species and kinesin families. The crucial glycine residue, which was mutated to alanine for Kip2, is highlighted in blue. 88

A Kip2-G374A-3xsfGFP Spc72-mCherry Composite

B Spc42-mCherry

Kip2-3xsfGFP, SPC72 spc72-2

Figure S5.5 – Spc72 is essential for the recruitment of Kip2-3xsfGFP to cytoplasmic microtubules emanating from SPBs. (A) Kip2-G374A-3xsfGFP colocalizes with Spc72-mCherry. (B) Representative images of cells expressing endogenous Kip2-3xsfGFP and Spc42-mCherry in the context of wildtype SPC72 or the temperature sensitive allele spc72-2 at the restrictive temperature (30 ◦C) for 2 h. For all panels, scale bar, 2 µm. All cells were cultivated at 25 ◦C and imaged at 25 ◦C or as speciﬁed. 89

RxxSPxR YPL155C_Scer 12---PSTRSSSGSSNIPQSPSVRST------SSFSNLTRNSIRSTSNSGSQSISASSTRSNSPLRSV 68 CAG62417_Cgla ---SPSPSLDSAVSISSYARSRSS------VTRGFRNGSSS------GSSSRSSSPLRPG AET38128_Ecd7 QTPGSSSMGLCIQNGA-IK--RSS------VLHH--PQATYITPLVKS----EGTYSRPSSPFRQ- AAS51371_Agos QTPGSRSFALGAHPGP-QK--RIG------GPAQ--GQTAFITPLVTP----DGIYSRPSSPYMQ- AGO11393_Ssaa QTPGSRSFPLGTHPGP-QK--RIG------GPAQ--GQTAFITPLVTP----DGIYSRPSSPYMQ- CEP60136_Llan ------MSRPSYQQVPQSPAIRSG------KLVSSVKRKNSVTPLSTPISVSSTSRNRHASPTRS- CAR22762_Ltc6 QTPVFQTGGGTFSSRR-RK--SSV------TPLT--TPLSTTSVRRR------SRQSSPSRSS CAH00423_Klac ---SSYNSPKRLMRPPSTPNLRTS------STVRPRSTS-----SSSSSQCSSPARS- CCF59650_Kac2 GSSGIPPPSPSLRTVSSFSNIRRS------SHMRSTSGS------SDVSSSSPPPS- CCE62768_Tpc4 ------VGSFNNLK------KGNRFGTTSTT------TTRSSSPIRS- EDO18336_Vpd7 ---TSSYSNLRKNRIGSTSSSTST------TRSSSPVR-- CCE89956_Tdel GSPTLRS-ASSLSSLR-KKYLRSG------SISTSSS------SRSASPTRS- CAR28585_Zrou NSY-RSQSPSARSSLS-FSNLRKG------RIRSGSLS------ASSSRSSSPIRSV CDH14079_Zbis -RPSFKSQSPSVRSTSSFSNLKRG------RIGSGSLS------ASSSRSSSPLRS- CDH08781_Zbis -RPSFKSQSPSVRSTSSFSNLKRG------RIGSGSLS------ASSSRSSSPLRS- EHN04287_Scxs ---PSTRSSSGSSNIPQSPSVRST------SSFSNLTRNSIRSTSNSGSQSISASSTRSNSPLRSV CCC69125_Ncc4 ---SSHVSAPSTPLPPSPSVRSTS------SFSNLRRPTRTNSICSSAASSRSSSPLRS- CCD25025_Ndc4 GSLRSTPSHTNNVSMQ-SPSIRST------SSFSNIRRNITRTSSNS------PANSRSSSPLRSV EGW32734_Spny SAMGMSLASLGTPSKP-PQ------LSDFKRPPSRY------HQSNSPSQ-- EMG49399_Cmxu ---STYNPSSSTMSST-PS--KNS------FMRPPSRI------SYYPKSSSPIQP- EER35154_Ctm3 SCNANT-----TMNST-PS--RNS------FRPPSRI------SMVPKSSSPIQP- CAX44217_Cdcd SCYANSSTT--TINST-PS--RNS------FRPPSRL------SMIPKSSSPIQP- EAK95283_Casc SCYANSSTT--SMNST-PS--RNS------FRPPSRL------SLIPRSSSPIQP- EAK95325_Casc SCYANSSTTTMNSTPS-----RNS------FRPPSRL------SLIPRSSSPIQP- CCG25901_Coc9 SSLGNSSSASVSSLST-PTRPSSS------VKSTFRPPSRP------TTIRSSSPVLQH EDK44343_Leny SSLGSGHHTTNATIAT-ATNTSME------PTGTGIKRMTYRPPSRR------ASTHRPASPAHPL CCE80363_Mfc7 SLMNISISNEKTPSVG-SAKSRSG------FTELRRPSTSR------MHRSTTPRGS- CCE81128_Mfc7 SLMNISISNEKTPSVG-SAKSRSG------FTELRRPSTSR------MHRSTTPRGS- EDK39482_Mga6 ---STTSHGSSTPSSANPP--RSE------MRRLERPPSRT------FRPASPRHPG EEQ38807_Cla4 SCSGLITCVLPLSERC-KHIFSYRLDTLFHTCQKTKKGPFPMRPSSSLS----GGPPARSSSQLSK- EGV61951_Cta1 NSPAISSSTLAASQNQ-TPKTRVS------VTDLRRPPSSR------SNRPASPRMSF ABN67231_Ssc6 ------RPPSRS------FRPSSPQVV- CAG86285_Dhcb SSLANFSTAGMSTPSQQTTKPRAT------LTELRRPPSSR------TMRPASPRMTT CCH40504_Wcif --HSALPSTPNLRPSSRPSSRANT------PSSAFRPPSTI------SRYERPFSPLR-- CAY71318_Kpas ---GRVARPQGHLRPP-SSASSSS------SQMSTSSYDSTKPRAQSA------LSGRTPTRR- EIF49528_Bbaw SVLSMSSPPPSNLSRLRPRSSMADIGKRYGSISRFTVPRTPSRLSSRS--VAKSSLRGXPQSTIRSV ESX02920_Opd1 ------CDK25677_Kcc1 SSQGLHHPPSTLRSRP-ASSLSIS------HSAG-----GVKTPNLR------RPMTPSTSN CCH60001_Tbc6 ---LRYNRLSSTSSMSSTSSAGST------CSISSFSSYEIED- CAG78742_Ylip QTPVLRR-PASTMSMR-SN--RMG------DISPLR-- KGK40411_Pkud NSRGTSERVPSYNSET-FHYISTS------SPKASMLGPRSRNPSRAP----SRQASRDIYPPPPR CCK72253_Knc8 SAKGTLISTPKSKSAPLSPLLSSR------SSLRRTPSQE------RSRAPST--

Figure S5.6 – Conservation of Kip2’s low complexity N-terminus. Protein sequence alignment of the Kip2 low complexity N-terminus and its homologs among Saccharomycetales. The tandem kinase consensus site [RxxSPxR, in which x represents any amino acid residue] covering Serine 63, as well as the richness in serine and threonine residues are highly conserved. Alignment was generated using MUSCLE (Edgar, 2004) and manually curated. 90

A Kip2-G374A-3xsfGFP Spc42-mCherry Composite

cdc5-2

B C Spc72-GFP Spc42-mCherry Composite WT

200 ***

150

cdc5-2 100

50 intensity on b-SPB (%) Normalized Spc72-GFP

0 WT cdc5-2

Figure S5.7 – Polo-like kinase Cdc5 regulates the recruitment of Kip2 at SPBs. (A) Representative images of Kip2-G374A-3xsfGFP distribution in the context of wildtype CDC5 and the temperature sensitive allele cdc5-2 at the restrictive temperature (37 ◦C) for 50 min. The corresponding quantification result is shown in Figure 5.3C. (B) Inactivation of Cdc5 resulted in elevated levels of Spc72-GFP at b-SPBs, quantification results are shown in (C)(n = 3 or 6 independent clones, with a total of > 139 cells per genotype analyzed). Thus, the reduction of Kip2-G374A-3xsfGFP recruitment at SPBs in cdc5-2 cells is not a consequence of reduced levels of Spc72. For all panels, scale bar, 2 µm. All cells were cultivated at 25 ◦C and shifted to 37 ◦C for 50 min before imaging at 37 ◦C for no longer than 20 min. One-way ANOVA was performed to test significance. *** p < 0.001. 91

Acquisition of z stacks A C ∆T = 1.07 s 3 0 1 2 3 80 T = 0 s T = 85.6 s 0 X X X 80

Y Y Y ∆Z = 3.84 µm ) ∆Z ∆Z step = 240 nm μm 17 images ( 2 h ng t e time (s) B Projection Z T1 Bik1-3xGFP, Spc72-GFP distance (µm) 10 s 1 b-MT 3D a MT l 1 µm m-MT

b- +end below detection limit: 667 nm

b-SPB 0 (x,y,z,t) 0 20 40 60 80 Time (s) m-SPB m- +end

150 D b-SPB m-SPB distance (μm) Kymograph X

2 µm 37 ℃ 50 min 100 10 s time (s)

50 emanating detectable aMTs (%) Percentage of metaphase SPBs

WT WT cdc5-2

10 s Kip2-S63A

1 µm Kymograph Y

Figure S5.8 – Quantification of the dynamics of metaphase astral microtubules in vivo. (A) Scheme of image acquisition. (B) Scheme of metaphase aMT organization and a representative metaphase cell expressing Bik1-3xGFP and Spc72-GFP, which are markers of aMT plus-ends and minus-ends, respectively. 3D coordinates of these markers over time were extracted with the low-light tracking tool (Krull et al., 2014). (C) The recorded 3D length of b-MT (blue) and m-MT (yellow) from the cell shown in (B) over time. Manually annotated catastrophe and rescue events are indicated with red and black arrows, respectively. Due to the resolution of the microscope, only microtubules longer than 5 pixels (666.7 nm) were considered detectable. Following this principle, the portions of b-SPBs and m-SPBs emanating detectable microtubules were quantified (%) and shown in (D)(n = 3 or 4 independent clones, with a total of > 233 cells per genotype analyzed). Scale bar, 2 µm. All cells were cultivated at 30 ◦C or as specified. 92

Figure S5.9 – Parameter likelihoods. Full likelihood landscape for the three model parameters ([Kip2]total, kin and kout ) fit to the data from the bin demarcated in red in Figure 5.2B. Colored samples are within the 95% confidence region, gray samples are outside. The maximum-likelihood estimate (and 95% confidence region) is indicated as a red cross (colored region) and corresponds to the parameter set [Kip2]total =108.9 nm −1 −1 −1 −1 −1 (95% CI: 41.0 nm to 236.2 nm), kin = 0.036 nm s (0.011 nm s to 0.336 nm s ), kout = 3.57 s (3.06 s to 4.56 s−1). 93

5.6.4 Supplementary Tables 94 al S Table eghadsed r eotda en±sd;nme fgot n hikg hssi nparentheses. in is phases shrinkage and growth of number s.d.; ± mean as reported a are speeds and Length (min min (µm iei hikg (s) shrinkage in Time (s) growth in Time ecefeuny(min frequency Rescue number Rescue frequency Catastrophe number Catastrophe min (µm speed Shrinkage speed Growth (µm) length MT MTs of Number diinlmve eeaaye o olcigmr -T,o hc -T eentquantiﬁed. not were b-MTs which of m-MTs, more collecting for analyzed were movies Additional − 1 5 ) . − 1 1 ) – unicto ftednmc fmtpaeata microtubules astral metaphase of dynamics the of Quantiﬁcation − 1 ) − 1 ) 3.0 1.90 2.08 ± 13312 11364 ( 1.21 bud 0.97 2.80 206 216 531 511 ± ± 1.31 0.62 ) ( 520 ) WT a 1.81 2.76 1.73 2547.2 ( ( mom 2286 2.99 4.59 127 175 110 169 122 ± ± ± 0.72 2.46 1.31 ) ) 30 ◦ C 2.65 2.07 2.26 7953.6 ( ( 7620 1.88 bud 2.72 210 248 266 346 266 ± ± ± 2.69 1.52 0.68 ) ) Kip 2 -S nvivo in 63 2.20 A 2.17 2.48 a ± 3514.2 ( mom 3998 3.21 3.23 . 188 115 183 215 1.75 ± ± 1.72 0.75 ) ( 54 ) 3.39 2.22 2.55 11241 11246 ( ( 1.54 bud 2.49 463 286 289 467 410 ± ± ± 2.04 1.55 0.87 ) ) WT 3.21 2.27 2.1 2749.6 ( ( mom 2438 2.49 3.74 114 152 109 112 168 ± ± ± 0.88 2.62 1.91 ) ) 37 ◦ ,5 min 50 C, 2.51 3.69 2.49 6238.4 ( ( 6440 1.78 bud 2.63 185 279 185 282 233 ± ± ± 0.95 2.27 1.91 ) ) cdc 5 - 2 2.79 2.01 2.48 4466.8 ( ( mom 4418 2.57 3.45 191 159 254 190 254 ± ± ± 2.04 1.30 0.86 ) ) 95

Table S5.2 – Strains used in this study. The background for all strains listed is S288C. ayYB.

Yeast Mat- Genotype Source strain ing numbera type

14590 a Kip3-3xsfGFP:KanMX Spc42-mCherry:NatMX ura3-52 his3∆200 leu2 this study lys2-801 trp1∆63 Ade2+ 15100 a Kip2-3xsfGFP:KanMX Spc42-mCherry:NatMX ura3-52 his3∆200 leu2 this study lys2-801 trp1∆63 Ade2+ 15352 a Kip2-3xsfGFP:KanMX Spc42-mCherry:NatMX spc72-2:HIS3 ura3-52 this study his3∆200 leu2 lys2-801 trp1∆63 Ade2+ 15102 a kip2∆::Kip2-S63A-3xsfGFP:KanMX Spc42-mCherry:NatMX ura3-52 this study his3∆200 leu2 lys2-801 trp1∆63 Ade2+ 10414 a Kip3-3xsfGFP:KanMX4 Spc72-GFP:His3MX ura3-52 his3∆200 leu2 this study lys2-801 trp1∆63 Ade2+ 9806 a Kip2-3xsfGFP:KanMX4 Spc72-GFP:His3MX ura3-52 his3∆200 leu2 this study lys2-801 trp1∆63 Ade2+ 10676 a kip2∆::Kip2-S63A-3xsfGFP:KanMX4 Spc72-GFP:His3MX ura3-52 his3∆200 this study leu2 lys2-801 trp1∆63 Ade2+ 15105 a kip2∆::Kip2-G374A-3xsfGFP:KanMX Spc42-mCherry:NatMX ura3-52 this study his3∆200 leu2 lys2-801 trp1∆63 Ade2+ 15107 a kip2∆::Kip2-S63A-G374A-3xsfGFP:KanMX Spc42-mCherry:NatMX this study ura3-52 his3∆200 leu2 lys2-801 trp1∆63 Ade2+ 15149 a kip2∆::Kip2-G374A-3xsfGFP:KanMX Spc42-mCherry:NatMX cdc5-2:URA this study ura3-52 his3∆200 leu2 lys2-801 trp1∆63 Ade2+ 15148 a kip2∆::Kip2-G374A-3xsfGFP:KanMX Spc42-mCherry:NatMX spc72-2:HIS3 this study ura3-52 his3∆200 leu2 lys2-801 trp1∆63 Ade2+ 15187 diploid Kip2-G374A-3xsfGFP:KanMX/Kip2-mCherry:hphNT1 ura3-52 his3∆200 this study leu2 lys2-801 trp1∆63 Ade2+ 15336 diploid Kip2-S63A-G374A-3xsfGFP:KanMX/Kip2-mCherry:hphNT1 ura3-52 this study his3∆200 leu2 lys2-801 trp1∆63 Ade2+ 14269 a kip2∆::Kip2-6HA:NatMX ura3-52 his3∆200 leu2 lys2-801 trp1∆63 Ade2+ this study 14405 a kip2∆::Kip2-S63A-6HA::NatMX ura3-52 his3∆200 leu2 lys2-801 trp1∆63 this study Ade2+ 11068 alpha Bik1-3xGFP:hyg Spc72-GFP:His3MX ura3-52 his3∆200 leu2 lys2-801 Lengefeld trp1∆63 Ade2+ et al., 2018 11069 a Bik1-3xGFP:hyg Spc72-GFP:His3MX ura3-52 his3∆200 leu2 lys2-801 Lengefeld trp1∆63 Ade2+ et al., 2018 11365 a Bik1-3xGFP:hyg Spc72-GFP:His3MX kip2∆::Kip2-S63A:TRP ura3-52 this study his3∆200 leu2 lys2-801 trp1∆63 Ade2+ 12564 a Bik1-3xGFP:hyg Spc72-GFP:His3MX kip2∆::Kip2-wt:TRP ura3-52 this study his3∆200 leu2 lys2-801 trp1∆63 Ade2+ 15166 a Bik1-3xGFP:hyg Spc72-GFP:HIS cdc5-2:URA ura3-52 his3∆200 leu2 this study lys2-801 trp1∆63 Ade2+ 15140 alpha Spc42-mCherry:NatMX Spc72-GFP:HIS3 ura3-52 his3∆200 leu2 lys2-801 this study trp1∆63 Ade2+ 15210 a Spc42-mCherry:NatMX Spc72-GFP:HIS3 cdc5-2:URA ura3-52 his3∆200 this study leu2 lys2-801 trp1∆63 Ade2+ 15176 a kip2∆::Kip2-G374A-3xsfGFP:KanMX Spc72-mCherry:KanMX ura3-52 this study his3∆200 leu2 lys2-801 trp1∆63 Ade2+ 96

Table S5.3 – Stochastic simulation propensities and change vectors.

Reaction Propensity Applies at State change

Minus-end binding free (x1) rin minus end site only x1 → 1 Lattice binding free (xi) ron all lattice sites xi → 1

Lattice dissociation occ (xi) koff all sites except plus end xi → 0 Stepping occ (xi) free (xi+1) kstep all sites except plus end xi → 0, xi+1 → 1 Plus-end dissociation occ (xN ) kout plus end site only xN → 0 6 CONCLUDINGREMARKSANDOUTLOOK

Microtubule dynamics are still a fascinating research topic: while we have developed a reasonable understanding of microtubules’ intrinsic dynamics over the last few years, quantitatively explaining the behavior of microtubules together with their associated proteins in vivo is still an open challenge. Over the course of this thesis, we considered a variety of approaches that can be used to create mathematical models that quantitatively capture intracellular microtubule dynamics in time and space. While a variety of modeling techniques exist that would conceptually allow for modeling microtubule dynamics in vivo, only few are fast enough to simulate both microtubule dynamics, as well as the regulatory and signaling networks that interact with microtubule-associated proteins (MAPs), which in turn regulate microtubule dynamics. In Chapter 2, we gave an overview of such techniques, as well as the methodological challenges that they face, especially in terms of dealing with active transport and time-varying geometries (both of which apply to microtubule dynamics), as well as with regards to calibrating such models to in vivo data. We further investigated a modeling framework that could allow us to both simulate stochastic microtubule dynamics in time and space – namely, reaction diffusion master equation (RDME)-type models. In Chapter 3, we developed the RDMEcpp simulation engine that is able to simulate RDME models with state-of-the-art performance – similar or better than an existing, lower-level solution written in C (Drawert et al., 2012). More importantly, RDMEcpp is coordinate-aware, meaning that it can be used to simulate microtubules nucleating from arbitrary locations, and growing in arbitrary directions, all while accounting for local interactions of microtubules with stochastically diffusing and reacting molecules. RDMEcpp allowed us to combine mechanistic microtubule plus tip models in growing and shrinking microtubules with diffusing nucleators that regulate the creation of new microtubules, thus simulating autonucleated microtubule regulation in Xenopus leavis egg extract as recently described experimentally (Decker et al., 2018). Since our ultimate target is to develop quantitative spatiotemporal models for microtubule dynamics in yeast, our next goal was to characterize the tubulin polymerization conditions and microtubule-associated proteins in Saccharomyces cerevisiae and Schizosac- charomyces pombe. In Chapter 4, we estimated how many tubulin heterodimers are actually in a mitotic S. cerevisiae or S. pombe cell, how many tubulins are in their respective spindles, and how many are free to interact with growing microtubule plus ends or nucleation sites. This exercise in “tubulin accounting” led to the deﬁnition of two critical concentrations and the realization that the free tubulin concentration in these cells is too low to drive microtubule polymerization – or even nucleation – on its own. This fact, along with previous evidence showing tubulin overexpression is toxic in these cells, can explain why microtubule polymerases are essential in yeast – and interestingly, this property is conserved even up to humans. In S. cerevisiae, there are two known microtubule polymerases, Stu2 and Kip2. While Stu2 has been subject of substantial research due to its homology with the XMAP215

97 98

polymerase, the polymerase function of Kip2 has only been discovered over the course of this thesis (Hibbel et al., 2015). Since it is possible to image single cytoplasmic microtubules that Kip2 binds to and walks on in vivo, together with our collaborators Xiuzhen Chen in the Barral group and Marcel Stangier in the Steinmetz group, we investigated the mechanistic function of Kip2 and found that unexpectedly, it predominantly binds to microtubules (and starts walking) at the minus end anchored in the spindle pole body (SPB). In Chapter 5, using in vivo imaging, stochastic spatiotemporal modeling and in silico microscopy, as well as structural modiﬁcations to the Kip2 motor itself, we showed conclusively that Kip2 predominantly binds at the spindle pole body (SPB), and that phosphorylation at S63 prevents Kip2 loading on the microtubule lattice. This information will be valuable for developing new models that integrate intrinsic microtubule dynamics, essential MAPs such as microtubule polymerases, and regulatory networks that control these MAPs. In the following sections, we will give concrete ideas that push the work we did in the individual chapters further, and more importantly, ideas on how to use them together to drive our quantitative understanding of the microtubule cytoskeleton forward.

6.1 towards an integrated in silico model of the microtubule cytoskeleton and its regulation

In Chapter 3, we developed RDMEcpp – a simulation pipeline and software that allows us to both simulate stochastic microtubule dynamics in time and space as well as spatiotemporal regulatory networks – in this case, microtubule nucleators – together in a single model. An obvious next step to improve the existing 1D model will be to add very recent results on the nature of these nucleators (Song et al., 2018) and data on the angular distribution of microtubule nucleation (Petry et al., 2013) to make this model two-dimensional, mechanistically motivate the nature of the nucleator binding sites on microtubules, and compare the results of this model with available imaging data. We did not use RDMEcpp to simulate the 1D Kip2 model in Chapter 5 due to the fact that implementing a totally asymmetric exclusion process (TASEP) model as a standard RDME model is not possible (if considering binding sites as subvolumes). However, RDMEcpp could actually be applied to such models, since its out-of-subvolume propensity update interface would, for example, allow reactions that move a motor from one subvolume to the next to also update the other subvolume’s propensities – in our case, ensuring only one motor can be bound to one binding site, at a time. A true next step then would be to couple this Kip2 model with a model for microtubule growth, catastrophe and rescue, making the Kip2 motor able to inﬂuence microtubule dynamics, and in turn get inﬂuenced by microtubule dynamics in silico. Since the Kip3 motor binds to the same binding sites on the microtubule, it would compete with the Kip2 motor when added, which given their opposing functions should lead to quite interesting dynamics – especially given Kip3’s behavior both as a catastrophe factor on the plus end, and as a rescue factor on the microtubule lattice (Dave et al., 2018). More long-term, other components such as Bim1, Bik1 and Stu2 could successively be added – though properly characterizing their combined effect on microtubule dynamics will most likely also require futher in vitro reconstitution experiments with microtubules polymerized from S. cerevisiae tubulin. 99

Further into the future, we suggest applying this strategy to more complex regulatory networks, such as cell cycle signaling cascades in S. cerevisiae, to investigate how the spatial information from components of the mitotic exit network is read out by the MAPs that interact with the microtubule cytoskeleton (Hotz and Barral, 2014). To make this computationally tractable, it might be necessary to implement hybrid simulation approaches in RDMEcpp. Speciﬁcally, we suggest looking into the algorithm devised by Haseltine and Rawl- ings, 2002, that was recently applied to simulate a (non-spatial) S. cerevisiae cell cycle hybrid model (Wang et al., 2016). Since we now both have a high-performance C++ implementation of a spatial stochastic simulation engine in RDMEcpp, and over the course of this thesis implemented a high-performance C++ implementation (odeSDcpp, unpublished) of our ODE solver odeSD (Gonnet et al., 2012), it would be tempting to combine both solvers to create a hybrid spatiotemporal simulation engine. Such an engine would require implementing the event detection mechanism from odeSD in odeSDcpp, so we can, for example, implement the model by Wang et al., 2016 and add a spatial component to it. Since hybrid solvers can run into issues with negative species and these are currently solved heuristically (Chen and Cao, 2018), it might also be worthwhile to attempt using event detection to switch all reactions concerning a low-abundance species from deterministic to stochastic. What we currently neglect, but for the purpose of elucidating the effect of both the Kip2 and Kip3 kinesin motors on mitotic spindle positioning via stabilizing / destabilizing cytoplasmic microtubules would be interesting to explore, is integrating the behavior of these motors into a simulation environment such as Cytosim (Nédélec and Foethke, 2007), which takes into account forces that are generated by microtubules and motors. Strikingly, in S. pombe, the combination of the Kip3 homologs Klp5/Klp6 together with the protein Mcp1 (which does not have a homolog in S. cerevisiae) are able to speciﬁcally make cytoplasmic microtubules catastrophe at the cell cortex. This behavior would also make a lot of sense in S. cerevisiae, as the cytoplasmic microtubules that guide the bSPB into the bud should be pulled on by the Bim1-Kar9-Myo2 complex moving on the actin cables originating in the bud. Pushing forces exerted by these microtubules would rather move the spindle back into the mother cell, and are therefore undesirable. However, this idea has never been tested quantitatively with measured microtubule polymerization stall forces (for example, revisiting the results by Gupta et al., 2006 and Tischer et al., 2009) – something that should ultimately be doable in silico.

6.2 quantitative modeling of tubulin allocation and availability: testing our predictions further

Our work in Chapter 4 established quantitative estimates for the tubulin content and allocation of mitotic S. cerevisiae and S. pombe cells. With these quantitative estimates we can explain the spindle phenotypes in the pac10∆ plp1∆ yap4∆ mutant, as well as the inability of td-kip3 doc1∆ strains to generate a stable spindle while half spindles from the previous division remain. Another prediction from this analysis is that, without polymerases, there should not be any microtubule dynamics in vivo. This could be directly tested by putting the two known microtubule polymerases Kip2 and Stu2 in S. cerevisiae, or the two polymerases 100

Alp14 and Dis1 in S. pombe, under the control of a degradation system. If our model is accurate, inducing polymerase degradation should eventually lead to dissociation of all microtubules. This loss of function should (at least partially) be recoverable by overexpressing both α- and β-tubulins simultaneously. Since the amount of tubulin per cell is finite, this also raises the question as to how this tubulin is partitioned between nucleus and cytoplasm. In at least one study, it was possible to tip the balance towards cytoplasmic tubulin and microtubule polymerization when overexpressing Bik1 (Berlin et al., 1990), which acts with Kip2 to elongate cytoplasmic microtubules – conversely, these cells only produce short spindles. Further analysis of the functions of Stu2 and Kip2 in the cytoplasm could also be enlightening: cytoplasmic microtubules in stu2cu do not grow any slower, but display reduced rescue rate (Wolyniak et al., 2006) – inconsistent with Stu2’s function as a polymerase. Consistent with Kip2 requiring Bik1 for driving polymerization in vivo however, growth speed decreases in bik1∆ strains. This could indicate that Stu2’s primary function is in the nucleus, and Kip2’s primary function is in the cytoplasm, and Stu2 and Kip2 are regulated such that they are each active in their own “zone”. While Stu2 probably acts as a microtubule nucleator (Geyer et al., 2018) both in the nucleus and in the cytoplasm due to its interaction with Spc72 (Gunzelmann et al., 2018), it might be downregulated in the cytoplasm and upregulated in the nucleus. Stu2 could be regulated by sumoylation (addition of small ubiquitin-like modifiers, Greenlee et al., 2018), or by phosphorylation at S603 (Humphrey et al., 2018). This would be consistent with data reported on mutants that lower nuclear export of Stu2 (Vaart et al., 2017), which show an increase in microtubule length in the nucleus, but little effect in the cytoplasm. In addition, even in S. cerevisiae, it is also not clear if we know all of the components interacting with microtubules yet. For example, the Mhp1 protein, which was hypothesized to be a homolog of the mammalian MAP4 protein (Irminger-Finger et al., 1996; Irminger-Finger and Mathis, 1998), has not been investigated with modern methods like fluorescent protein tagging yet, even though it is an essential protein.

6.3 spbs exert more control over the microtubules they nucleate than previously believed

In Chapter 5, we showed conclusively that Kip2 predominantly binds at the spindle pole body (SPB), and that phosphorylation at S63 prevents Kip2 loading on the microtubule lattice. We can already suggest a few avenues that will make the motor model we presented there more mechanistically accurate: first of all, from the Kip2-G374A variant (which is not motile), it is clear that there must be binding sites for Kip2 at the SPBs. Using an approach that was used to estimate the number of molecules of the spindle position checkpoint proteins Bfa1, Bub2 and Tem1 at SPBs (Caydasi et al., 2012), we could estimate how many Kip2 molecules bind at the two SPBs. Kip2-G374A also clearly illustrates that the binding sites are differentially regulated between mSPB and bSPB (Figure 5.3C), and that there is approximately 5× more Kip2-G374A binding at the bSPB compared to the mSPB. At the microtubule plus end, we also believe the accuracy of our model can be improved further: unpublished data from the bik1∆ strain (and others, as detailed in the thesis of Xiuzhen Chen, 2018), structural analyses (performed by Marcel Stangier), and our 101 model-based parameter estimates for the Kip2 out rate, indicate that Kip2 can only function as a polymerase efficiently if Bik1 is available at microtubule plus ends. While Kip2 in vitro does not rely on Bik1 for its polymerase function (Hibbel et al., 2015), it could be that polymerization is much more efficient in conjunction with Bik1, akin to the cooperativity shown between XMAP215 and EB1 (Zanic et al., 2013). In any case, it seems that Kip2 is switching binding modes at the plus end by binding to Bik1 with its tail and unbinding its motor domains from the microtubule such that they are free to bind tubulin heterodimers, thus accelerating microtubule polymerization. Bik1 has a high turnover rate at the plus end (unpublished data), and therefore a more accurate description than a simple out-rate for Kip2 would be another compartment that Kip2 binds to from the microtubule, and leaves with fast turnover. We then would expect microtubule polymerization to depend more on the Kip2 throughput, as a high plus tip concentration could both be due to a Kip2 traffic jam on the microtubule at the plus end – which would not contribute to microtubule polymerization – and Kip2 bound to Bik1 at the plus tip – which would. Even though Kip2 does not have orthologs in mammalians (Altenhoff et al., 2018), it might still be interesting to analyze if this remote control pattern also exists in other organisms.

6.4 future uses for the kip2 data analysis pipeline

We also have further plans for the data analysis pipeline developed for analyzing manually-annotated fluorescence line scans along microtubules in Chapter 5. We are using this pipeline to visualize how the microtubule binding and mean distribution of Kip2 changes in a bik1∆ background, as well as how Kip2-∆Tail (which removes Kip2’s binding interface with Bik1) changes these features, and the measurement model will be applied to infer motor parameters in these strains as well. It will be interesting to see if mutants that interfere with SPB loading of Kip2 (for example, dbf2∆, bfa1∆ bub2∆, or cdc5- 2) or enable its lattice loading (for example, Kip2-S63A) show any correlations between motor parameters and microtubule length, which would be indicative of regulation. A related next step would be to apply the pipeline to kinase mutants and investigate what the candidates for Kip2 binding sites at the SPB could be, and how they are regulated. There are also interesting questions that could be answered by applying this pipeline to other proteins. For example, one could visualize Bik1 intensity and compare it with Kip2 intensity, as these two interactors have repeatedly been shown to move together (Carvalho et al., 2004), and see if there is any change at microtubule plus tips, where presumably the interaction mode changes. It could also be applied in a strain where Kip3 is tagged with an orthogonal fluorophore, and successive line scans could be taken before catastrophe occurs (time-aligned to the point of microtubule catastrophe). This would allow us to answer the question if Kip2 dissociates and makes way for Kip3 to induce catastrophe in such events, and it would especially be interesting to investigate these events close to the cell cortex, as the Kip3 homologs Klp5/Klp6 (and the Kip2 homolog Tea2) have been demonstrated to specifically cause catastrophe there in S. pombe (Meadows et al., 2018). Short-term plans for this pipeline also include empowering the researchers that generate the imaging data by training them in applying and modifying the software. Future work could include automating the labor-intensive step of manually annotating line profiles 102

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