Force- and kinesin-8-dependent effects in the spatial regulation of fission yeast microtubule dynamics

EB3-informed dynamics of the microtubule stabilizing cap during stalled growth

Microtubule stability is known to be governed by a stabilizing GTP/GDP-Pi cap, but the exact relation between growth velocity, GTP hydrolysis, and catastrophes remains unclear. We investigate the dynamics of the stabilizing cap through in vitro reconstitution of microtubule dynamics in contact with microfabricated barriers, using the plus-end binding protein GFP-EB3 as a marker for the nucleotide state of the tip. The interaction of growing microtubules with steric objects is known to slow down microtubule growth and accelerate catastrophes. We show that the lifetime distributions of stalled microtubules, as well as the corresponding lifetime distributions of freely growing microtubules, can be fully described with a simple phenomenological 1D model based on noisy microtubule growth and a single EB3-dependent hydrolysis rate. This same model is furthermore capable of explaining both the previously reported mild catastrophe dependence on microtubule growth rates and the catastrophe statistics during tubulin washout experiments.

The motor domain of the kinesin Kip2 promotes microtubule polymerization at microtubule tips

Kinesins are microtubule-dependent motor proteins, some of which moonlight as microtubule polymerases, such as the yeast protein Kip2. Here, we show that the CLIP-170 ortholog Bik1 stabilizes Kip2 at microtubule ends where the motor domain of Kip2 promotes microtubule polymerization. Live-cell imaging and mathematical estimation of Kip2 dynamics reveal that disrupting the Kip2-Bik1 interaction aborts Kip2 dwelling at microtubule ends and abrogates its microtubule polymerization activity. Structural modeling and biochemical experiments identify a patch of positively charged residues that enables the motor domain to bind free tubulin dimers alternatively to the microtubule shaft. Neutralizing this patch abolished the ability of Kip2 to promote microtubule growth both in vivo and in vitro without affecting its ability to walk along microtubules. Our studies suggest that Kip2 utilizes Bik1 as a cofactor to track microtubule tips, where its motor domain then recruits free tubulin and catalyzes microtubule assembly.

Reliable and robust control of nucleus centering is contingent on nonequilibrium force patterns

Cell centers their division apparatus to ensure symmetric cell division, a challenging task when the governing dynamics is stochastic. Using fission yeast, we show that the patterning of nonequilibrium polymerization forces of microtubule (MT) bundles controls the precise localization of spindle pole body (SPB), and hence the division septum, at the onset of mitosis. We define two cellular objectives, reliability, the mean SPB position relative to the geometric center, and robustness, the variance of the SPB position, which are sensitive to genetic perturbations that change cell length, MT bundle number/orientation, and MT dynamics. We show that simultaneous control of reliability and robustness is required to minimize septum positioning error achieved by the wild type (WT). A stochastic model for the MT-based nucleus centering, with parameters measured directly or estimated using Bayesian inference, recapitulates the maximum fidelity of WT. Using this, we perform a sensitivity analysis of the parameters that control nuclear centering.

Fission yeast Dis1 is an unconventional TOG/XMAP215 that induces microtubule catastrophe to drive chromosome pulling

The shortening of microtubules attached to kinetochores is the driving force of chromosome movement during cell division. Specific kinesins are believed to shorten microtubules but are dispensable for viability in yeast, implying the existence of additional factors responsible for microtubule shortening. Here, we demonstrate that Dis1, a TOG/XMAP215 ortholog in fission yeast, promotes microtubule shortening to carry chromosomes. Although TOG/XMAP215 orthologs are generally accepted as microtubule polymerases, Dis1 promoted microtubule catastrophe in vitro and in vivo. Notably, microtubule catastrophe was promoted when the tip was attached to kinetochores, as they steadily anchored Dis1 at the kinetochore-microtubule interface. Engineered Dis1 oligomers artificially tethered at a chromosome arm region induced the shortening of microtubules in contact, frequently pulling the chromosome arm towards spindle poles. This effect was not brought by oligomerised Alp14. Thus, unlike Alp14 and other TOG/XMAP215 orthologs, Dis1 plays an unconventional role in promoting microtubule catastrophe, thereby driving chromosome movement.

Microtubule rescue at midzone edges promotes overlap stability and prevents spindle collapse during anaphase B

During anaphase B, molecular motors slide interpolar microtubules to elongate the mitotic spindle, contributing to the separation of chromosomes. However, sliding of antiparallel microtubules reduces their overlap, which may lead to spindle breakage, unless microtubules grow to compensate sliding. How sliding and growth are coordinated is still poorly understood. In this study, we have used the fission yeast S. pombe to measure microtubule dynamics during anaphase B. We report that the coordination of microtubule growth and sliding relies on promoting rescues at the midzone edges. This makes microtubules stable from pole to midzone, while their distal parts including the plus ends alternate between assembly and disassembly. Consequently, the midzone keeps a constant length throughout anaphase, enabling sustained sliding without the need for a precise regulation of microtubule growth speed. Additionally, we found that in S. pombe, which undergoes closed mitosis, microtubule growth speed decreases when the nuclear membrane wraps around the spindle midzone.

Stochastic model of T Cell repolarization during target elimination (II)

Cytotoxic T lymphocytes (T cells) and natural killer cells form a tight contact, the immunological synapse (IS), with target cells, where they release their lytic granules containing perforin/granzyme and cytokine-containing vesicles. During this process the cell repolarizes and moves the microtubule organizing center (MTOC) toward the IS. In the first part of our work we developed a computational model for the molecular-motor-driven motion of the microtubule cytoskeleton during T cell polarization and analyzed the effects of cortical-sliding and capture-shrinkage mechanisms. Here we use this model to analyze the dynamics of the MTOC repositioning in situations in which 1) the IS is in an arbitrary position with respect to the initial position of the MTOC and 2) the T cell has two IS at two arbitrary positions. In the case of one IS, we found that the initial position determines which mechanism is dominant and that the time of repositioning does not rise monotonously with the MTOC-IS distance. In the case of two IS, we observe several scenarios that have also been reported experimentally: the MTOC alternates stochastically (but with a well-defined average transition time) between the two IS; it wiggles in between the two IS without transiting to one of the two; or it is at some point pulled to one of the two IS and stays there. Our model allows one to predict which scenario emerges in dependency of the mechanisms in action and the number of dyneins present. We report that the presence of capture-shrinkage mechanism in at least one IS is necessary to assure the transitions in every cell configuration. Moreover, the frequency of transitions does not decrease with the distance between the two IS and is the highest when both mechanisms are present in both IS.

Single depolymerizing and transport kinesins stabilize microtubule ends

Microtubules are highly dynamic cellular filaments and an accurate control of their length is important for many intracellular processes like cell division. Among other factors, microtubule length is actively modulated by motors from the kinesin superfamily. For example, yeast kinesin-8, Kip3, motors depolymerize microtubules by a cooperative, force- and length-dependent mechanism. However, whether single motors can also depolymerize microtubules is unclear. Here, we measured how single kinesin motors influenced the stability of microtubules in an in vitro assay. Using label-free interference reflection microscopy, we determined the spontaneous microtubule depolymerization rate of stabilized microtubules in the presence of kinesins. Surprisingly, we found that both single Kip3 and nondepolymerizing kinesin-1 transport motors, used as a control, stabilized microtubules further. For Kip3, this behavior is contrary to the collective force-dependent depolymerization activity of multiple motors. Because of the control measurement, the finding may hint at a more general stabilization mechanism. The complex, concentration-dependent interaction with microtubule ends provides new insights into the molecular mechanism of kinesin-8 and its regulatory function of microtubule length.

Biomechanics of chromosome alignment at the spindle midplane

During metaphase, chromosomes are aligned in a lineup at the equatorial plane of the spindle to ensure synchronous poleward movement of chromatids in anaphase and proper nuclear reformation at the end of mitosis. Chromosome alignment relies on microtubules, several types of motor protein and numerous other microtubule-associated and regulatory proteins. Because of the multitude of players involved, the mechanisms of chromosome alignment are still under debate. Here, we discuss the current models of alignment based on poleward pulling forces exerted onto sister kinetochores by kinetochore microtubules, which show length-dependent dynamics and undergo poleward flux, and polar ejection forces that push the chromosome arms away from the pole. We link these models with the recent ideas based on mechanical coupling between bridging and kinetochore microtubules, where sliding of bridging microtubules promotes overlap length-dependent sliding of kinetochore fibers and thus the alignment of sister kinetochores at the spindle equator. Finally, we discuss theoretical models of forces acting on chromosomes during metaphase.

Anaphase A

Anaphase A is the motion of recently separated chromosomes to the spindle pole they face. It is accompanied by the shortening of kinetochore-attached microtubules. The requisite tubulin depolymerization may occur at kinetochores, at poles, or both, depending on the species and/or the time in mitosis. These depolymerization events are local and suggest that cells regulate microtubule dynamics in specific places, presumably by the localization of relevant enzymes and microtubule-associated proteins to specific loci, such as pericentriolar material and outer kinetochores. Motor enzymes can contribute to anaphase A, both by altering microtubule stability and by pushing or pulling microtubules through the cell. The generation of force on chromosomes requires couplings that can both withstand the considerable force that spindles can generate and simultaneously permit tubulin addition and loss. This chapter reviews literature on the molecules that regulate anaphase microtubule dynamics, couple dynamic microtubules to kinetochores and poles, and generate forces for microtubule and chromosome motion.

Microtubule-associated proteins and motors required for ectopic microtubule array formation in Saccharomyces cerevisiae

The mitotic spindle is resilient to perturbation due to the concerted, and sometimes redundant, action of motors and microtubule-associated proteins. Here, we utilize an inducible ectopic microtubule nucleation site in the nucleus of Saccharomyces cerevisiae to study three necessary steps in the formation of a bipolar array: the recruitment of the γ-tubulin complex, nucleation and elongation of microtubules (MTs), and the organization of MTs relative to each other. This novel tool, an Spc110 chimera, reveals previously unreported roles of the microtubule-associated proteins Stu2, Bim1, and Bik1, and the motors Vik1 and Kip3. We report that Stu2 and Bim1 are required for nucleation and that Bik1 and Kip3 promote nucleation at the ectopic site. Stu2, Bim1, and Kip3 join their homologs XMAP215, EB1 and kinesin-8 as promoters of microtubule nucleation, while Bik1 promotes MT nucleation indirectly via its role in SPB positioning. Furthermore, we find that the nucleation activity of Stu2 in vivo correlates with its polymerase activity in vitro. Finally, we provide the first evidence that Vik1, a subunit of Kar3/Vik1 kinesin-14, promotes microtubule minus end focusing at the ectopic site.

Optogenetic control of PRC1 reveals its role in chromosome alignment on the spindle by overlap length-dependent forces

During metaphase, chromosome position at the spindle equator is regulated by the forces exerted by kinetochore microtubules and polar ejection forces. However, the role of forces arising from mechanical coupling of sister kinetochore fibers with bridging fibers in chromosome alignment is unknown. Here, we develop an optogenetic approach for acute removal of PRC1 to partially disassemble bridging fibers and show that they promote chromosome alignment. Tracking of the plus-end protein EB3 revealed longer antiparallel overlaps of bridging microtubules upon PRC1 removal, which was accompanied by misaligned and lagging kinetochores. Kif4A/kinesin-4 and Kif18A/kinesin-8 were found within the bridging fiber and largely lost upon PRC1 removal, suggesting that these proteins regulate the overlap length of bridging microtubules. We propose that PRC1-mediated crosslinking of bridging microtubules and recruitment of kinesins to the bridging fiber promote chromosome alignment by overlap length-dependent forces transmitted to the associated kinetochore fibers.

Before cells divide to create copies of themselves, they need to duplicate their genetic material. To help split their DNA evenly, they build a machine called the mitotic spindle. The mitotic spindle is made of fine, tube-like structures called microtubules, which catch the chromosomes containing the genetic information and line them up at the center of the spindle.

Microtubules push and pull the chromosomes by elongating or shortening their tips. But it remains unclear how the microtubules know when the chromosomes have reached center point. One way to find out is to remove proteins that accumulate in the middle of the spindle during division, such as the protein PRC1, which helps to assemble a subset of microtubules called bridging fibers, and the proteins Kif4A and Kif18A, which work like molecular rulers, shortening long microtubules. Usually, scientists would delete one of these proteins to see what impact this has. However, these experiments take days, giving the cell enough time to adapt and thus making it difficult to study the role of each of the proteins.

Here, Jagrić, Risteski, Martinčić et al. used light to manipulate proteins at the exact moment of chromosome alignment and to move PRC1 from the spindle to the cell membrane. Consequently, Kif4A and Kif18A were removed from the spindle center. This caused the bridging fibers, which overlap with the microtubules that connect to the chromosomes, to become thinner. Jagrić et al. discovered that without the molecular ruler proteins, the bridging fibers were also too long. This increased the overlap between the microtubules in the center of the spindle, causing the chromosomes to migrate away from the center. This suggests that the alignment of chromosomes in the middle of the spindle depends on the bridging microtubules, which need to be of a certain length to effectively move and keep the chromosomes at the center. Thus, forces that move the chromosomes are generated both at the tips of the microtubules and along the wall of microtubules.

These results might inspire other researchers to reassess the role of bridging fibers in cell division. The optogenetic technique described here could also help to determine the parts other proteins have to play. Ultimately, this might allow researchers to identify all the proteins needed to align the chromosomes.

Mechanobiology of the Mitotic Spindle

The mitotic spindle is a microtubule-based assembly that separates the chromosomes during cell division. As the spindle is basically a mechanical micro machine, the understanding of its functioning is constantly motivating the development of experimental approaches based on mechanical perturbations, which are complementary to and work together with the classical genetics and biochemistry methods. Recent data emerging from these approaches in combination with theoretical modeling led to novel ideas and significant revisions of the basic concepts in the field. In this Perspective, we discuss the advances in the understanding of spindle mechanics, focusing on microtubule forces that control chromosome movements.

Kinesin-8 motors: regulation of microtubule dynamics and chromosome movements

Microtubules are essential for intracellular transport, cell motility, spindle assembly, and chromosome segregation during cell division. Microtubule dynamics regulate the proper spindle organization and thus contribute to chromosome congression and segregation. Accumulating studies suggest that kinesin-8 motors are emerging regulators of microtubule dynamics and organizations. In this review, we provide an overview of the studies focused on kinesin-8 motors in cell division. We discuss the structures and molecular kinetics of kinesin-8 motors. We highlight the essential roles and mechanisms of kinesin-8 in the regulation of microtubule dynamics and spindle organization. We also shed light on the functions of kinesin-8 motors in chromosome movement and the spindle assembly checkpoint during the cell cycle.

The concerted actions of Tip1/CLIP-170, Klp5/Kinesin-8, and Alp14/XMAP215 regulate microtubule catastrophe at the cell end

Spatial regulation of microtubule catastrophe is important for controlling microtubule length and consequently contributes to the proper establishment of cell polarity and cell growth. The +TIP proteins including Tip1/CLIP-170, Klp5/Kinesin-8, and Alp14/XMAP215 reside at microtubule plus ends to regulate microtubule dynamics. In the fission yeast Schizosaccharomyces pombe, Tip1 and Alp14 serve as microtubule-stabilizing factors, while Klp5 functions oppositely as a catastrophe-promoting factor. Despite that Tip1 has been shown to play a key role in restricting microtubule catastrophe to the cell end, how Tip1 fulfills the role remains to be determined. Employing live-cell microscopy, we showed that the absence of Tip1 impairs the localization of both Klp5 and Alp14 at microtubule plus ends, but the absence of Klp5 prolongs the residence time of Tip1 at microtubule plus ends. We further revealed that Klp5 accumulates behind Tip1 at microtubule plus ends in a Tip1-dependent manner. In addition, artificially tethering Klp5 to microtubule plus ends promotes premature microtubule catastrophe, while tethering Alp14 to microtubule plus ends in the cells lacking Tip1 rescues the phenotype of short microtubules. These findings establish that Tip1 restricts microtubule catastrophe to the cell end likely by spatially restricting the microtubule catastrophe activity of Klp5 and stabilizing Alp14 at microtubule plus ends. Thus, the work demonstrates the orchestration of Tip1, Alp14, and Klp5 in ensuring microtubule catastrophe at the cell end.

Structural effects of cap, crack, and intrinsic curvature on the microtubule catastrophe kinetics

Microtubules (MTs) experience an effect called "catastrophe," which is the transition from the MT growth to a sudden dramatic shrinkage in length. The straight guanosine triphosphate (GTP)-tubulin cap at the filament tip and the intrinsic curvature of guanosine diphosphate (GDP)-tubulins are known to be the key thermodynamic factors that determine MT catastrophe, while the hydrolysis of this GTP-cap acts as the kinetic control of the process. Although several theoretical models have been developed, assuming the catastrophe occurs when the GTP-cap shrinks to a minimal stabilizing size, the structural effect of the GTP-cap and GDP-curvature is not explicitly included; thus, their influence on catastrophe kinetics remains less understood. To investigate this structural effect, we apply a single-protofilament model with one GTP-cap while assuming a random hydrolysis mechanism and take the occurrence of a crack in the lateral bonds between neighboring protofilaments as the onset of the catastrophe. Therein, we find the effective potential of the tip along the peel-off direction and formulate the catastrophe kinetics as a mean first-passage time problem, subject to thermal fluctuations. We consider cases with and without a compressive force on the MT tip, both of which give a quadratic effective potential, making MT catastrophe an Ornstein-Uhlenbeck process in our formalism. In the free-standing case, the mean catastrophe time has a sensitive tubulin-concentration dependence, similar to a double-exponential function, and agrees well with the experiment. For a compressed MT, we find a modified exponential function of force that shortens the catastrophe time.

Kinesin-8 and Dis1/TOG collaborate to limit spindle elongation from prophase to anaphase A for proper chromosome segregation in fission yeast

High-fidelity chromosome segregation relies on proper microtubule regulation. Kinesin-8 has been shown to destabilise microtubules to reduce metaphase spindle length and chromosome movements in multiple species. XMAP215/chTOG polymerases catalyse microtubule growth for spindle assembly, elongation and kinetochore-microtubule attachment. Understanding of their biochemical activity has advanced, but little work directly addresses the functionality and interplay of these conserved factors. We utilised the synthetic lethality of fission yeast kinesin-8 (Klp5-Klp6) and XMAP215/chTOG (Dis1) to study their individual and overlapping roles. We found that the non-motor kinesin-8 tailbox is essential for mitotic function; mutation compromises plus-end-directed processivity. Klp5-Klp6 induces catastrophes to control microtubule length and, surprisingly, Dis1 collaborates with kinesin-8 to slow spindle elongation. Together, they enforce a maximum spindle length for a viable metaphase-anaphase transition and limit elongation during anaphase A to prevent lagging chromatids. Our work provides mechanistic insight into how kinesin-8 negatively regulates microtubules and how this functionally overlaps with Dis1 and highlights the importance of spindle length control in mitosis.

Pivoting of microtubules driven by minus-end-directed motors leads to spindle assembly

BACKGROUND

At the beginning of mitosis, the cell forms a spindle made of microtubules and associated proteins to segregate chromosomes. An important part of spindle architecture is a set of antiparallel microtubule bundles connecting the spindle poles. A key question is how microtubules extending at arbitrary angles form an antiparallel interpolar bundle.

RESULTS

Here, we show in fission yeast that microtubules meet at an oblique angle and subsequently rotate into antiparallel alignment. Our live-cell imaging approach provides a direct observation of interpolar bundle formation. By combining experiments with theory, we show that microtubules from each pole search for those from the opposite pole by performing random angular movement. Upon contact, two microtubules slide sideways along each other in a directed manner towards the antiparallel configuration. We introduce the contour length of microtubules as a measure of activity of motors that drive microtubule sliding, which we used together with observation of Cut7/kinesin-5 motors and our theory to reveal the minus-end-directed motility of this motor in vivo.

CONCLUSION

Random rotational motion helps microtubules from the opposite poles to find each other and subsequent accumulation of motors allows them to generate forces that drive interpolar bundle formation.

Association of mitochondria with microtubules inhibits mitochondrial fission by precluding assembly of the fission protein Dnm1

Mitochondria are organized as tubular networks in the cell and undergo fission and fusion. Although several of the molecular players involved in mediating mitochondrial dynamics have been identified, the precise cellular cues that initiate mitochondrial fission or fusion remain largely unknown. In fission yeast (Schizosaccharomyces pombe), mitochondria are organized along microtubule bundles. Here, we employed deletions of kinesin-like proteins to perturb microtubule dynamics and used high-resolution and time-lapse fluorescence microscopy, revealing that mitochondrial lengths mimic microtubule lengths. Furthermore, we determined that compared with WT cells, mutant cells with long microtubules exhibit fewer mitochondria, and mutant cells with short microtubules have an increased number of mitochondria because of reduced mitochondrial fission in the former and elevated fission in the latter. Correspondingly, upon onset of closed mitosis in fission yeast, wherein interphase microtubules assemble to form the spindle within the nucleus, we observed increased mitochondrial fission. We found that the consequent rise in the mitochondrial copy number is necessary to reduce partitioning errors during independent segregation of mitochondria between daughter cells. We also discovered that the association of mitochondria with microtubules physically impedes the assembly of the fission protein Dnm1 around mitochondria, resulting in inhibition of mitochondrial fission. Taken together, we demonstrate a mechanism for the regulation of mitochondrial fission that is dictated by the interaction between mitochondria and the microtubule cytoskeleton.

Opposing kinesin complexes queue at plus tips to ensure microtubule catastrophe at cell ends

In fission yeast, the lengths of interphase microtubule (iMT) arrays are adapted to cell length to maintain cell polarity and to help centre the nucleus and cell division ring. Here, we show that length regulation of iMTs is dictated by spatially regulated competition between MT‐stabilising Tea2/Tip1/Mal3 (Kinesin‐7) and MT‐destabilising Klp5/Klp6/Mcp1 (Kinesin‐8) complexes at iMT plus ends. During MT growth, the Tea2/Tip1/Mal3 complex remains bound to the plus ends of iMT bundles, thereby restricting access to the plus ends by Klp5/Klp6/Mcp1, which accumulate behind it. At cell ends, Klp5/Klp6/Mcp1 invades the space occupied by the Tea2/Tip1/Tea1 kinesin complex triggering its displacement from iMT plus ends and MT catastrophe. These data show that in vivo, whilst an iMT length‐dependent model for catastrophe factor accumulation has validity, length control of iMTs is an emergent property reflecting spatially regulated competition between distinct kinesin complexes at the MT plus tip.

Asymmetric simple exclusion process with position-dependent hopping rates: Phase diagram from boundary-layer analysis

In this paper, we study a one-dimensional totally asymmetric simple exclusion process with position-dependent hopping rates. Under open boundary conditions, this system exhibits boundary-induced phase transitions in the steady state. Similarly to totally asymmetric simple exclusion processes with uniform hopping, the phase diagram consists of low-density, high-density, and maximal-current phases. In various phases, the shape of the average particle density profile across the lattice including its boundary-layer parts changes significantly. Using the tools of boundary-layer analysis, we obtain explicit solutions for the density profile in different phases. A detailed analysis of these solutions under different boundary conditions helps us obtain the equations for various phase boundaries. Next, we show how the shape of the entire density profile including the location of the boundary layers can be predicted from the fixed points of the differential equation describing the boundary layers. We discuss this in detail through several examples of density profiles in various phases. The maximal-current phase appears to be an especially interesting phase where the boundary layer flows to a bifurcation point on the fixed-point diagram.

Metaphase kinetochore movements are regulated by kinesin-8 motors and microtubule dynamic instability

During metaphase, sister chromatids are connected to microtubules extending from the opposite spindle poles via kinetochores to protein complexes on the chromosome. Kinetochores congress to the equatorial plane of the spindle and oscillate around it, with kinesin-8 motors restricting these movements. Yet, the physical mechanism underlying kinetochore movements is unclear. We show that kinetochore movements in the fission yeast Schizosaccharomyces pombe are regulated by kinesin-8-promoted microtubule catastrophe, force-induced rescue, and microtubule dynamic instability. A candidate screen showed that among the selected motors only kinesin-8 motors Klp5/Klp6 are required for kinetochore centering. Kinesin-8 accumulates at the end of microtubules, where it promotes catastrophe. Laser ablation of the spindle resulted in kinetochore movement toward the intact spindle pole in wild-type and klp5Δ cells, suggesting that kinetochore movement is driven by pulling forces. Our theoretical model with Langevin description of microtubule dynamic instability shows that kinesin-8 motors are required for kinetochore centering, whereas sensitivity of rescue to force is necessary for the generation of oscillations. We found that irregular kinetochore movements occur for a broader range of parameters than regular oscillations. Thus, our work provides an explanation for how regulation of microtubule dynamic instability contributes to kinetochore congression and the accompanying movements around the spindle center.

Polymerisation force of a rigid filament bundle: diffusive interaction leads to sublinear force-number scaling

Polymerising filaments generate force against an obstacle, as in, e.g., microtubule-kinetochore interactions in the eukaryotic cell. Earlier studies of this problem have not included explicit three-dimensional monomer diffusion, and consequently, missed out on two important aspects: (i) the barrier, even when it is far from the polymers, affects free diffusion of monomers and reduces their adsorption at the tips, while (ii) parallel filaments could interact through the monomer density field ("diffusive coupling"), leading to negative interference between them. In our study, both these effects are included and their consequences investigated in detail. A mathematical treatment based on a set of continuum Fokker-Planck equations for combined filament-wall dynamics suggests that the barrier-induced monomer depletion reduces the growth velocity and also the stall force, while the total force produced by many filaments remains additive. However, Brownian dynamics simulations show that the linear force-number scaling holds only when the filaments are far apart; when they are arranged close together, forming a bundle, sublinear scaling of force with number appears, which could be attributed to diffusive interaction between the growing polymer tips.

Understanding force-generating microtubule systems through in vitro reconstitution

Microtubules switch between growing and shrinking states, a feature known as dynamic instability. The biochemical parameters underlying dynamic instability are modulated by a wide variety of microtubule-associated proteins that enable the strict control of microtubule dynamics in cells. The forces generated by controlled growth and shrinkage of microtubules drive a large range of processes, including organelle positioning, mitotic spindle assembly, and chromosome segregation. In the past decade, our understanding of microtubule dynamics and microtubule force generation has progressed significantly. Here, we review the microtubule-intrinsic process of dynamic instability, the effect of external factors on this process, and how the resulting forces act on various biological systems. Recently, reconstitution-based approaches have strongly benefited from extensive biochemical and biophysical characterization of individual components that are involved in regulating or transmitting microtubule-driven forces. We will focus on the current state of reconstituting increasingly complex biological systems and provide new directions for future developments.

Microtubule organization is determined by the shape of epithelial cells

Interphase microtubule organization is critical for cell function and tissue architecture. In general, physical mechanisms are sufficient to drive microtubule organization in single cells, whereas cells within tissues are thought to utilize signalling mechanisms. By improving the imaging and quantitation of microtubule alignment within developing Drosophila embryos, here we demonstrate that microtubule alignment underneath the apical surface of epithelial cells follows cell shape. During development, epidermal cell elongation and microtubule alignment occur simultaneously, but by perturbing cell shape, we discover that microtubule organization responds to cell shape, rather than the converse. A simple set of microtubule behaviour rules is sufficient for a computer model to mimic the observed responses to changes in cell surface geometry. Moreover, we show that microtubules colliding with cell boundaries zip-up or depolymerize in an angle-dependent manner, as predicted by the model. Finally, we show microtubule alignment responds to cell shape in diverse epithelia.

Contributions of Microtubule Dynamic Instability and Rotational Diffusion to Kinetochore Capture

Microtubule dynamic instability allows search and capture of kinetochores during spindle formation, an important process for accurate chromosome segregation during cell division. Recent work has found that microtubule rotational diffusion about minus-end attachment points contributes to kinetochore capture in fission yeast, but the relative contributions of dynamic instability and rotational diffusion are not well understood. We have developed a biophysical model of kinetochore capture in small fission-yeast nuclei using hybrid Brownian dynamics/kinetic Monte Carlo simulation techniques. With this model, we have studied the importance of dynamic instability and microtubule rotational diffusion for kinetochore capture, both to the lateral surface of a microtubule and at or near its end. Over a range of biologically relevant parameters, microtubule rotational diffusion decreased capture time, but made a relatively small contribution compared to dynamic instability. At most, rotational diffusion reduced capture time by 25%. Our results suggest that while microtubule rotational diffusion can speed up kinetochore capture, it is unlikely to be the dominant physical mechanism for typical conditions in fission yeast. In addition, we found that when microtubules undergo dynamic instability, lateral captures predominate even in the absence of rotational diffusion. Counterintuitively, adding rotational diffusion to a dynamic microtubule increases the probability of end-on capture.

Spatial regulation of astral microtubule dynamics by Kif18B in PtK cells

The spatial and temporal control of microtubule dynamics is fundamentally important for proper spindle assembly and chromosome segregation. This is achieved, in part, by the multitude of proteins that bind to and regulate spindle microtubules, including kinesin superfamily members, which act as microtubule-destabilizing enzymes. These fall into two general classes: the kinesin-13 proteins, which directly depolymerize microtubules, and the kinesin-8 proteins, which are plus end-directed motors that either destabilize microtubules or cap the microtubule plus ends. Here we analyze the contribution of a PtK kinesin-8 protein, Kif18B, in the control of mitotic microtubule dynamics. Knockdown of Kif18B causes defects in spindle microtubule organization and a dramatic increase in astral microtubules. Kif18B-knockdown cells had defects in chromosome alignment, but there were no defects in chromosome segregation. The long astral microtubules that occur in the absence of Kif18B are limited in length by the cell cortex. Using EB1 tracking, we show that Kif18B activity is spatially controlled, as loss of Kif18B has the most dramatic effect on the lifetimes of astral microtubules that extend toward the cell cortex. Together our studies provide new insight into how diverse kinesins contribute to spatial microtubule organization in the spindle.

Paired arrangement of kinetochores together with microtubule pivoting and dynamics drive kinetochore capture in meiosis I

Kinetochores are protein complexes on the chromosomes, whose function as linkers between spindle microtubules and chromosomes is crucial for proper cell division. The mechanisms that facilitate kinetochore capture by microtubules are still unclear. In the present study, we combine experiments and theory to explore the mechanisms of kinetochore capture at the onset of meiosis I in fission yeast. We show that kinetochores on homologous chromosomes move together, microtubules are dynamic and pivot around the spindle pole, and the average capture time is 3–4 minutes. Our theory describes paired kinetochores on homologous chromosomes as a single object, as well as angular movement of microtubules and their dynamics. For the experimentally measured parameters, the model reproduces the measured capture kinetics and shows that the paired configuration of kinetochores accelerates capture, whereas microtubule pivoting and dynamics have a smaller contribution. Kinetochore pairing may be a general feature that increases capture efficiency in meiotic cells.

Kinesin-8 effects on mitotic microtubule dynamics contribute to spindle function in fission yeast

Kinesin-8 motor proteins destabilize microtubules and increase chromosome loss in mitosis. In fission yeast, aberrant microtubule-driven kinetochore pushing movements, tripolar mitotic spindles, and fluctuations in metaphase spindle length occurred in kinesin-8–deletion mutants. A mathematical model can explain these results.

Kinesin-8 motor proteins destabilize microtubules. Their absence during cell division is associated with disorganized mitotic chromosome movements and chromosome loss. Despite recent work studying effects of kinesin-8s on microtubule dynamics, it remains unclear whether the kinesin-8 mitotic phenotypes are consequences of their effect on microtubule dynamics, their well-established motor activity, or additional, unknown functions. To better understand the role of kinesin-8 proteins in mitosis, we studied the effects of deletion of the fission yeast kinesin-8 proteins Klp5 and Klp6 on chromosome movements and spindle length dynamics. Aberrant microtubule-driven kinetochore pushing movements and tripolar mitotic spindles occurred in cells lacking Klp5 but not Klp6. Kinesin-8–deletion strains showed large fluctuations in metaphase spindle length, suggesting a disruption of spindle length stabilization. Comparison of our results from light microscopy with a mathematical model suggests that kinesin-8–induced effects on microtubule dynamics, kinetochore attachment stability, and sliding force in the spindle can explain the aberrant chromosome movements and spindle length fluctuations seen.

Chimera proteins with affinity for membranes and microtubule tips polarize in the membrane of fission yeast cells

Cell polarity refers to a functional spatial organization of proteins that is crucial for the control of essential cellular processes such as growth and division. To establish polarity, cells rely on elaborate regulation networks that control the distribution of proteins at the cell membrane. In fission yeast cells, a microtubule-dependent network has been identified that polarizes the distribution of signaling proteins that restricts growth to cell ends and targets the cytokinetic machinery to the middle of the cell. Although many molecular components have been shown to play a role in this network, it remains unknown which molecular functionalities are minimally required to establish a polarized protein distribution in this system. Here we show that a membrane-binding protein fragment, which distributes homogeneously in wild-type fission yeast cells, can be made to concentrate at cell ends by attaching it to a cytoplasmic microtubule end-binding protein. This concentration results in a polarized pattern of chimera proteins with a spatial extension that is very reminiscent of natural polarity patterns in fission yeast. However, chimera levels fluctuate in response to microtubule dynamics, and disruption of microtubules leads to disappearance of the pattern. Numerical simulations confirm that the combined functionality of membrane anchoring and microtubule tip affinity is in principle sufficient to create polarized patterns. Our chimera protein may thus represent a simple molecular functionality that is able to polarize the membrane, onto which additional layers of molecular complexity may be built to provide the temporal robustness that is typical of natural polarity patterns.

Microtubule encounter-based catastrophe in Arabidopsis cortical microtubule arrays

BACKGROUND

The cortical microtubules (CMTs) that line the plasma membrane of interphase plant cells are extensively studied owing to their importance in forming cell walls, and their usefulness as a model system for the study of MT dynamic instability and acentrosomal MT organization. CMTs influence the orientation and structure of cellulose microfibrils in the cell wall by cooperatively forming arrays of varied patterns from parallel to netted. These CMT patterns are controlled by the combined activities of MT dynamic instability and MT-MT interactions. However, it is an open question as to how CMT patterns may feedback to influence CMT dynamics and interactions.

RESULTS

To address this question, we investigated the effects of CMT array patterning on encounter-based CMT catastrophe, which occurs when one CMT grows into another and is unable to cross over. We hypothesized that the varied CMT angles present in disordered (mixed CMTs) arrays will create more opportunities for MT-MT interactions, and thus increase encounter-based catastrophe rates and distribution. Using live-cell imaging of Arabidopsis cotyledon and leaf epidermal cells, we found that roughly 87% of catastrophes occur via the encounter-based mechanism, with the remainder occurring without encounter (free). When comparing ordered (parallel) and disordered (mixed orientation) CMT arrays, we found that disordered configurations show higher proportions of encounter-based catastrophe relative to free. Similarly, disordered CMT arrays have more catastrophes in general than ordered arrays. Encounter-based catastrophes were associated with frequent and sustained periods of pause prior to depolymerization, and CMTs with tight anchoring to the plasma membrane were more prone to undergo encounter-based catastrophe than weakly-attached ones. This suggests that encounter-based catastrophe has a mechanical basis, wherein MTs form physical barriers to one another. Lastly, we show that the commonly used measure of catastrophe frequencies (Fcat) can also be influenced by CMT ordering and plasma membrane anchoring.

CONCLUSIONS

Our observations add a new layer of complexity to our current understanding of MT organization in plants, showing that not only do individual CMT dynamics influence CMT array organization, but that CMT organization itself has a strong effect on the behavior of individual MTs.

A Review of Fluorescent Proteins for Use in Yeast

The field of fluorescent proteins (FPs) is constantly developing. The use of FPs changed the field of life sciences completely, starting a new era of direct observation and quantification of cellular processes. The broad spectrum of FPs (see Fig. 1) with a wide range of characteristics allows their use in many different experiments. This review discusses the use of FPs for imaging in budding yeast (Saccharomyces cerevisiae) and fission yeast Schizosaccharomyces pombe). The information included in this review is relevant for both species unless stated otherwise.

An Optically Driven Bistable Janus Rotor with Patterned Metal Coatings

Bistable rotation is realized for a gold-coated Janus colloidal particle in an infrared optical trap. The metal coating on the Janus particles are patterned by sputtering gold on a monolayer of closely packed polystyrene particles. The Janus particle is observed to stably rotate in an optical trap. Both the direction and the rate of rotation can be experimentally controlled. Numerical calculations reveal that the bistable rotation is the result of spontaneous symmetry breaking induced by the uneven curvature of the coating patterns on the Janus sphere. Our results thus provide a simple method to construct large quantities of fully functional rotary motors for nano- or microdevices.

PKA antagonizes CLASP-dependent microtubule stabilization to re-localize Pom1 and buffer cell size upon glucose limitation

Cells couple growth with division and regulate size in response to nutrient availability. In rod-shaped fission yeast, cell-size control occurs at mitotic commitment. An important regulator is the DYRK-family kinase Pom1, which forms gradients from cell poles and inhibits the mitotic activator Cdr2, itself localized at the medial cortex. Where and when Pom1 modulates Cdr2 activity is unclear as Pom1 medial cortical levels remain constant during cell elongation. Here we show that Pom1 re-localizes to cell sides upon environmental glucose limitation, where it strongly delays mitosis. This re-localization is caused by severe microtubule destabilization upon glucose starvation, with microtubules undergoing catastrophe and depositing the Pom1 gradient nucleator Tea4 at cell sides. Microtubule destabilization requires PKA/Pka1 activity, which negatively regulates the microtubule rescue factor CLASP/Cls1/Peg1, reducing CLASP's ability to stabilize microtubules. Thus, PKA signalling tunes CLASP's activity to promote Pom1 cell side localization and buffer cell size upon glucose starvation.

Fission yeast kinesin-8 controls chromosome congression independently of oscillations

In higher eukaryotes, efficient chromosome congression relies, among other players, on the activity of chromokinesins. Here, we provide a quantitative analysis of kinetochore oscillations and positioning in Schizosaccharomyces pombe, a model organism lacking chromokinesins. In wild-type cells, chromosomes align during prophase and, while oscillating, maintain this alignment throughout metaphase. Chromosome oscillations are dispensable both for kinetochore congression and stable kinetochore alignment during metaphase. In higher eukaryotes, kinesin-8 family members control chromosome congression by regulating their oscillations. By contrast, here, we demonstrate that fission yeast kinesin-8 controls chromosome congression by an alternative mechanism. We propose that kinesin-8 aligns chromosomes by controlling pulling forces in a length-dependent manner. A coarse-grained model of chromosome segregation implemented with a length-dependent process that controls the force at kinetochores is necessary and sufficient to mimic kinetochore alignment, and prevents the appearance of lagging chromosomes. Taken together, these data illustrate how the local action of a motor protein at kinetochores provides spatial cues within the spindle to align chromosomes and to prevent aneuploidy.

Highlighted Article: Quantitative analysis in S. pombe reveals that chromosome oscillations are dispensable for kinetochore congression in mitosis. Kinesin-8 controls chromosome congression through length-dependent pulling forces.

A stable microtubule array drives fission yeast polarity reestablishment upon quiescence exit

Cells perpetually face the decision to proliferate or to stay quiescent. Here we show that upon quiescence establishment, Schizosaccharomyces pombe cells drastically rearrange both their actin and microtubule (MT) cytoskeletons and lose their polarity. Indeed, while polarity markers are lost from cell extremities, actin patches and cables are reorganized into actin bodies, which are stable actin filament-containing structures. Astonishingly, MTs are also stabilized and rearranged into a novel antiparallel bundle associated with the spindle pole body, named Q-MT bundle. We have identified proteins involved in this process and propose a molecular model for Q-MT bundle formation. Finally and importantly, we reveal that Q-MT bundle elongation is involved in polarity reestablishment upon quiescence exit and thereby the efficient return to the proliferative state. Our work demonstrates that quiescent S. pombe cells assemble specific cytoskeleton structures that improve the swiftness of the transition back to proliferation.

Effects of aging in catastrophe on the steady state and dynamics of a microtubule population

Several independent observations have suggested that the catastrophe transition in microtubules is not a first-order process, as is usually assumed. Recent in vitro observations by Gardner et al. [M. K. Gardner et al., Cell 147, 1092 (2011)] showed that microtubule catastrophe takes place via multiple steps and the frequency increases with the age of the filament. Here we investigate, via numerical simulations and mathematical calculations, some of the consequences of the age dependence of catastrophe on the dynamics of microtubules as a function of the aging rate, for two different models of aging: exponential growth, but saturating asymptotically, and purely linear growth. The boundary demarcating the steady-state and non-steady-state regimes in the dynamics is derived analytically in both cases. Numerical simulations, supported by analytical calculations in the linear model, show that aging leads to nonexponential length distributions in steady state. More importantly, oscillations ensue in microtubule length and velocity. The regularity of oscillations, as characterized by the negative dip in the autocorrelation function, is reduced by increasing the frequency of rescue events. Our study shows that the age dependence of catastrophe could function as an intrinsic mechanism to generate oscillatory dynamics in a microtubule population, distinct from hitherto identified ones.

Kinesin-8 motors improve nuclear centering by promoting microtubule catastrophe

In fission yeast, microtubules push against the cell edge, thereby positioning the nucleus in the cell center. Kinesin-8 motors regulate microtubule catastrophe; however, their role in nuclear positioning is not known. Here we develop a physical model that describes how kinesin-8 motors affect nuclear centering by promoting a microtubule catastrophe. Our model predicts the improved centering of the nucleus in the presence of motors, which we confirmed experimentally in living cells. The model also predicts a characteristic time for the recentering of a displaced nucleus, which is supported by our experiments where we displaced the nucleus using optical tweezers.

Molecular mechanisms for microtubule length regulation by kinesin-8 and XMAP215 proteins

The cytoskeleton is regulated by a plethora of enzymes that influence the stability and dynamics of cytoskeletal filaments. How microtubules (MTs) are controlled is of particular importance for mitosis, during which dynamic MTs are responsible for proper segregation of chromosomes. Molecular motors of the kinesin-8 protein family have been shown to depolymerize MTs in a length-dependent manner, and recent experimental and theoretical evidence suggests a possible role for kinesin-8 in the dynamic regulation of MTs. However, so far the detailed molecular mechanisms of how these molecular motors interact with the growing MT tip remain elusive. Here we show that two distinct scenarios for the interactions of kinesin-8 with the MT tip lead to qualitatively different MT dynamics, including accurate length control as well as intermittent dynamics. We give a comprehensive analysis of the regimes where length regulation is possible and characterize how the stationary length depends on the biochemical rates and the bulk concentrations of the various proteins. For a neutral scenario, where MTs grow irrespective of whether the MT tip is occupied by a molecular motor, length regulation is possible only for a narrow range of biochemical rates, and, in particular, limited to small polymerization rates. By contrast, for an inhibition scenario, where the presence of a motor at the MT tip inhibits MT growth, the regime where length regulation is possible is extremely broad and includes high growth rates. These results also apply to situations where a polymerizing enzyme like XMAP215 and kinesin-8 mutually exclude each other from the MT tip. Moreover, we characterize the differences in the stochastic length dynamics between the two scenarios. While for the neutral scenario length is tightly controlled, length dynamics is intermittent for the inhibition scenario and exhibits extended periods of MT growth and shrinkage. On a broader perspective, the set of models established in this work quite generally suggest that mutual exclusion of molecules at the ends of cytoskeletal filaments is an important factor for filament dynamics and regulation.

Spatial control of microtubule length and lifetime by opposing stabilizing and destabilizing functions of Kinesin-8

BACKGROUND

To function in diverse cellular processes, the dynamic behavior of microtubules (MTs) must be differentially regulated within the cell. In budding yeast, the spindle position checkpoint (SPOC) inhibits mitotic exit in response to mispositioned spindles. To maintain SPOC-mediated anaphase arrest, astral MTs must maintain persistent interactions with and/or extend through the bud neck. However, the molecular mechanisms that ensure the stability of these interactions are not known.

RESULTS

The presence of an MT extending through and/or interacting with the bud neck is maintained by spatial control of catastrophe and rescue, which extends MT lifetime >25-fold and controls the length of dynamic MTs within the bud compartment. Moreover, the single kinesin-8 motor Kip3 alternately mediates both catastrophe and rescue of the bud MT. Kip3 accumulates in a length-dependent manner along the lattice of MTs within the bud, yet induces catastrophe spatially near the bud tip. Rather, this accumulation of Kip3 facilitates its association with depolymerizing MT plus ends, where Kip3 promotes rescue before MTs exit the bud. MT rescue within the bud requires the tail domain of Kip3, whereas the motor domain mediates catastrophe at the bud tip. In vitro, Kip3 exerts both stabilizing and destabilizing effects on reconstituted yeast MTs.

CONCLUSIONS

The kinesin-8 Kip3 is a multifunctional regulator that differentially stabilizes and destabilizes specific MTs. Control over MT catastrophe and rescue by Kip3 defines the length and lifetime of MTs within the bud compartment of cells with mispositioned spindles. This subcellular regulation of MT dynamics is critical to maintaining mitotic arrest in response to mispositioned spindles.

Budding yeast kinetochore proteins, Chl4 and Ctf19, are required to maintain SPB-centromere proximity during G1 and late anaphase

In the budding yeast, centromeres stay clustered near the spindle pole bodies (SPBs) through most of the cell cycle. This SPB-centromere proximity requires microtubules and functional kinetochores, which are protein complexes formed on the centromeres and capable of binding microtubules. The clustering is suggested by earlier studies to depend also on protein-protein interactions between SPB and kinetochore components. Previously it has been shown that the absence of non-essential kinetochore proteins of the Ctf19 complex weakens kinetochore-microtubule interaction, but whether this compromised interaction affects centromere/kinetochore positioning inside the nucleus is unknown. We found that in G1 and in late anaphase, SPB-centromere proximity was disturbed in mutant cells lacking Ctf19 complex members,Chl4p and/or Ctf19p, whose centromeres lay further away from their SPBs than those of the wild-type cells. We unequivocally show that the SPB-centromere proximity and distances are not dependent on physical interactions between SPB and kinetochore components, but involve microtubule-dependent forces only. Further insight on the positional difference between wild-type and mutant kinetochores was gained by generating computational models governed by (1) independently regulated, but constant kinetochore microtubule (kMT) dynamics, (2) poleward tension on kinetochore and the antagonistic polar ejection force and (3) length and force dependent kMT dynamics. Numerical data obtained from the third model concurs with experimental results and suggests that the absence of Chl4p and/or Ctf19p increases the penetration depth of a growing kMT inside the kinetochore and increases the rescue frequency of a depolymerizing kMT. Both the processes result in increased distance between SPB and centromere.

Role and regulation of kinesin-8 motors through the cell cycle

Members of the kinesin-8 motor family play a central role in controlling microtubule length throughout the eukaryotic cell cycle. Inactivation of kinesin-8 causes defects in cell polarity during interphase and astral and mitotic spindle length, metaphase chromosome alignment, timing of anaphase onset and accuracy of chromosome segregation. Although the biophysical mechanism by which kinesin-8 molecules influence microtubule dynamics has been studied extensively in a variety of species, a consensus view has yet to emerge. One reason for this might be that some members of the kinesin-8 family can associate to other microtubule-associated proteins, cell cycle regulatory proteins and other kinesin family members. In this review we consider how cell cycle specific modification and its association to other regulatory proteins may modulate the function of kinesin-8 to enable it to function as a master regulator of microtubule dynamics.

The kinesin-8 Kip3 scales anaphase spindle length by suppression of midzone microtubule polymerization

Mitotic spindle function is critical for cell division and genomic stability. During anaphase, the elongating spindle physically segregates the sister chromatids. However, the molecular mechanisms that determine the extent of anaphase spindle elongation remain largely unclear. In a screen of yeast mutants with altered spindle length, we identified the kinesin-8 Kip3 as essential to scale spindle length with cell size. Kip3 is a multifunctional motor protein with microtubule depolymerase, plus-end motility, and antiparallel sliding activities. Here we demonstrate that the depolymerase activity is indispensable to control spindle length, whereas the motility and sliding activities are not sufficient. Furthermore, the microtubule-destabilizing activity is required to counteract Stu2/XMAP215-mediated microtubule polymerization so that spindle elongation terminates once spindles reach the appropriate final length. Our data support a model where Kip3 directly suppresses spindle microtubule polymerization, limiting midzone length. As a result, sliding forces within the midzone cannot buckle spindle microtubules, which allows the cell boundary to define the extent of spindle elongation.

Model of fission yeast cell shape driven by membrane-bound growth factors and the cytoskeleton

Fission yeast serves as a model for how cellular polarization machinery consisting of signaling molecules and the actin and microtubule cytoskeleton regulates cell shape. In this work, we develop mathematical models to investigate how these cells maintain a tubular shape of approximately constant diameter. Many studies identify active Cdc42, found in a cap at the inner membrane of growing cell tips, as an important regulator of local cell wall remodeling, likely through control of exocyst tethering and the targeting of other polarity-enhancing structures. First, we show that a computational model with Cdc42-dependent local cell wall remodeling under turgor pressure predicts a relationship between spatial extent of growth signal and cell diameter that is in agreement with prior experiments. Second, we model the consequences of feedback between cell shape and distribution of Cdc42 growth signal at cell tips. We show that stability of cell diameter over successive cell divisions places restrictions on their mutual dependence. We argue that simple models where the spatial extent of the tip growth signal relies solely on geometrical alignment of confined microtubules might lead to unstable width regulation. Third, we study a computational model that combines a growth signal distributed over a characteristic length scale (as, for example, by a reaction-diffusion mechanism) with an axis-sensing microtubules system that places landmarks at positions where microtubule tips touch the cortex. A two-dimensional implementation of this model leads to stable cell diameter for a wide range of parameters. Changes to the parameters of this model reproduce straight, bent, and bulged cell shapes, and we discuss how this model is consistent with other observed cell shapes in mutants. Our work provides an initial quantitative framework for understanding the regulation of cell shape in fission yeast, and a scaffold for understanding this process on a more molecular level in the future.

Fission yeast is a rod-shaped organism that is studied, in part, as a model for how cells develop and regulate their shape. Despite extensive work identifying effects of genetic mutations and pharmacological treatments on the shape of these cells, there is a lack of mathematical and computational models examining how internal cell signals and the cytoskeleton organize to remodel the cell wall, direct growth at cell tips, and maintain tubular shape. In this work we describe how the spatial distribution of regulatory protein signal at growing cell tips relates to cell diameter. Further, we describe the consequences of this signal depending on the shape of the cell, namely its length and diameter. Finally, we propose a computational model for understanding growth and shape that includes an axis-sensing microtubule system, landmarks delivered to cell tips along those microtubules, and a growth zone signal that moves around but is attracted to the landmarks. This picture explains a large number of reported abnormal shapes in terms of only a few modular components.

Microtubule catastrophe from protofilament dynamics

The disappearance of the guanosine triphosphate- (GTP) tubulin cap is widely believed to be the forerunner event for the growth-shrinkage transition ("catastrophe") in microtubule filaments in eukaryotic cells. We study a discrete version of a stochastic model of the GTP cap dynamics, originally proposed by Flyvbjerg, Holy, and Leibler [Phys. Rev. Lett. 73, 2372 (1994)]. Our model includes both spontaneous and vectorial hydrolysis, as well as dissociation of a nonhydrolyzed dimer from the filament after incorporation. In the first part of the paper, we apply this model to a single protofilament of a microtubule. A catastrophe transition is defined for each protofilament, similarly to the earlier one-dimensional models, the frequency of occurrence of which is then calculated under various conditions but without explicit assumption of steady-state conditions. Using a perturbative approach, we show that the leading asymptotic behavior of the protofilament catastrophe in the limit of large growth velocities is remarkably similar across different models. In the second part of the paper, we extend our analysis to the entire filament by making a conjecture that a minimum number of such transitions are required to occur for the onset of microtubule catastrophe. The frequency of microtubule catastrophe is then determined using numerical simulations and compared with analytical and semianalytical estimates made under steady-state and quasi-steady-state assumptions, respectively, for the protofilament dynamics. A few relevant experimental results are analyzed in detail and compared with predictions from the model. Our results indicate that loss of GTP cap in two to three protofilaments is necessary to trigger catastrophe in a microtubule.

The multiple talents of kinesin-8

Multiple activities cooperate to determine the architecture of the mitotic spindle. Kip3 is a kinesin-8 motor protein in budding yeast that acts as a microtubule depolymerase. Now Kip3 is shown to also crosslink and slide antiparallel microtubules, providing additional insights into how kinesin-8 motors control spindle integrity.

Biophysics of filament length regulation by molecular motors

Regulating physical size is an essential problem that biological organisms must solve from the subcellular to the organismal scales, but it is not well understood what physical principles and mechanisms organisms use to sense and regulate their size. Any biophysical size-regulation scheme operates in a noisy environment and must be robust to other cellular dynamics and fluctuations. This work develops theory of filament length regulation inspired by recent experiments on kinesin-8 motor proteins, which move with directional bias on microtubule filaments and alter microtubule dynamics. Purified kinesin-8 motors can depolymerize chemically-stabilized microtubules. In the length-dependent depolymerization model, the rate of depolymerization tends to increase with filament length, because long filaments accumulate more motors at their tips and therefore shorten more quickly. When balanced with a constant filament growth rate, this mechanism can lead to a fixed polymer length. However, the mechanism by which kinesin-8 motors affect the length of dynamic microtubules in cells is less clear. We study the more biologically realistic problem of microtubule dynamic instability modulated by a motor-dependent increase in the filament catastrophe frequency. This leads to a significant decrease in the mean filament length and a narrowing of the filament length distribution. The results improve our understanding of the biophysics of length regulation in cells.

Microtubule catastrophe and rescue

Microtubules are long cylindrical polymers composed of tubulin subunits. In cells, microtubules play an essential role in architecture and motility. For example, microtubules give shape to cells, serve as intracellular transport tracks, and act as key elements in important cellular structures such as axonemes and mitotic spindles. To accomplish these varied functions, networks of microtubules in cells are very dynamic, continuously remodeling through stochastic length fluctuations at the ends of individual microtubules. The dynamic behavior at the end of an individual microtubule is termed 'dynamic instability'. This behavior manifests itself by periods of persistent microtubule growth interrupted by occasional switching to rapid shrinkage (called microtubule 'catastrophe'), and then by switching back from shrinkage to growth (called microtubule 'rescue'). In this review, we summarize recent findings which provide new insights into the mechanisms of microtubule catastrophe and rescue, and discuss the impact of these findings in regards to the role of microtubule dynamics inside of cells.