Proliferative advantage of specific aneuploid cells drives evolution of tumor karyotypes.

Most tumors have abnormal karyotypes, which arise from mistakes during mitotic division of healthy euploid cells and evolve through numerous complex mechanisms. In a recent mouse model with increased chromosome missegregation, chromosome gains dominate over losses both in pretumor and tumor tissues, whereas T-cell lymphomas are characterized by gains of chromosomes 14 and 15. However, the quantitative understanding of clonal selection leading to tumor karyotype evolution remains unknown. Here we show, by introducing a mathematical model based on a concept of a macro-karyotype, that tumor karyotypes can be explained by proliferation-driven evolution of aneuploid cells. In pretumor cells, increased apoptosis and slower proliferation of cells with monosomies lead to predominant chromosome gains over losses. Tumor karyotypes with gain of one chromosome can be explained by karyotype-dependent proliferation, whereas, for those with two chromosomes, an interplay with karyotype-dependent apoptosis is an additional possible pathway. Thus, evolution of tumor-specific karyotypes requires proliferative advantage of specific aneuploid karyotypes.

Kinetochore- and chromosome-driven transition of microtubules into bundles promotes spindle assembly.

Mitotic spindle assembly is crucial for chromosome segregation and relies on bundles of microtubules that extend from the poles and overlap in the middle. However, how these structures form remains poorly understood. Here we show that overlap bundles arise through a network-to-bundles transition driven by kinetochores and chromosomes. STED super-resolution microscopy reveals that PRC1-crosslinked microtubules initially form loose arrays, which become rearranged into bundles. Kinetochores promote microtubule bundling by lateral binding via CENP-E/kinesin-7 in an Aurora B-regulated manner. Steric interactions between the bundle-associated chromosomes at the spindle midplane drive bundle separation and spindle widening. In agreement with experiments, theoretical modeling suggests that bundles arise through competing attractive and repulsive mechanisms. Finally, perturbation of overlap bundles leads to inefficient correction of erroneous kinetochore-microtubule attachments. Thus, kinetochores and chromosomes drive coarsening of a uniform microtubule array into overlap bundles, which promote not only spindle formation but also chromosome segregation fidelity.

Length-dependent poleward flux of sister kinetochore fibers promotes chromosome alignment.

Chromosome alignment at the spindle equator promotes proper chromosome segregation and depends on pulling forces exerted at kinetochore fiber tips together with polar ejection forces. However, kinetochore fibers are also subjected to forces driving their poleward flux. Here we introduce a flux-driven centering model that relies on flux generated by forces within the overlaps of bridging and kinetochore fibers. This centering mechanism works so that the longer kinetochore fiber fluxes faster than the shorter one, moving the kinetochores toward the center. We develop speckle microscopy in human spindles and confirm the key prediction that kinetochore fiber flux is length dependent. Kinetochores are better centered when overlaps are shorter and the kinetochore fiber flux slower than the bridging fiber flux. We identify Kif18A and Kif4A as overlap and flux regulators and NuMA as a fiber coupler. Thus, length-dependent sliding forces exerted by the bridging fiber onto kinetochore fibers support chromosome alignment.

Oblique circle method for measuring the curvature and twist of mitotic spindle microtubule bundles.

The highly ordered spatial organization of microtubule bundles in the mitotic spindle is crucial for its proper functioning. The recent discovery of twisted shapes of microtubule bundles and spindle chirality suggests that the bundles extend along curved paths in three dimensions, rather than being confined to a plane. This, in turn, implies that rotational forces, i.e., torques, exist in the spindle in addition to the widely studied linear forces. However, studies of spindle architecture and forces are impeded by a lack of a robust method for the geometric quantification of microtubule bundles in the spindle. In this work, we describe a simple method for measuring and evaluating the shapes of microtubule bundles by characterizing them in terms of their curvature and twist. By using confocal microscopy, we obtain three-dimensional images of spindles, which allows us to trace the entire microtubule bundle. For each traced bundle, we first fit a plane and then fit a circle lying in that plane. With this robust method, we extract the curvature and twist, which represent the geometric information characteristic for each bundle. As the bundle shapes reflect the forces within them, this method is valuable for the understanding of forces that act on chromosomes during mitosis.

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.

Mitotic spindle: lessons from theoretical modeling.

Cell biology is immensely complex. To understand how cells work, we try to find patterns and suggest hypotheses to identify underlying mechanisms. However, it is not always easy to create a coherent picture from a huge amount of experimental data on biological systems, where the main players have multiple interactions or act in redundant pathways. In such situations, when a hypothesis does not lead to a conclusion in a direct way, theoretical modeling is a powerful tool because it allows us to formulate hypotheses in a quantitative manner and understand their consequences. A successful model should not only reproduce the basic features of the system but also provide exciting predictions, motivating new experiments. Much is learned when a model based on generally accepted knowledge cannot explain experiments of interest, as this indicates that the original hypothesis needs to be revised. In this Perspective, we discuss these points using our experiences in combining experiments with theory in the field of mitotic spindle mechanics.

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.

Pivot-and-bond model explains microtubule bundle formation.

During mitosis, microtubules form a spindle, which is responsible for proper segregation of the genetic material. A common structural element in a mitotic spindle is a parallel bundle, consisting of two or more microtubules growing from the same origin and held together by cross-linking proteins. An interesting question is what are the physical principles underlying the formation and stability of such microtubule bundles. Here we show, by introducing the pivot-and-bond model, that random angular movement of microtubules around the spindle pole and forces exerted by cross-linking proteins can explain the formation of microtubule bundles as observed in our experiments. The model predicts that stable parallel bundles can form in the presence of either passive crosslinkers or plus-end directed motors, but not minus-end directed motors. In the cases where bundles form, the time needed for their formation depends mainly on the concentration of cross-linking proteins and the angular diffusion of the microtubule. In conclusion, the angular motion drives the alignment of microtubules, which in turn allows the cross-linking proteins to connect the microtubules into a stable bundle.

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.

Helical Twist and Rotational Forces in the Mitotic Spindle.

The mitotic spindle segregates chromosomes into two daughter cells during cell division. This process relies on the precise regulation of forces acting on chromosomes as the cell progresses through mitosis. The forces in the spindle are difficult to directly measure using the available experimental techniques. Here, we review the ideas and recent advances of how forces can be determined from the spindle shape. By using these approaches, it has been shown that tension and compression coexist along a single kinetochore fiber, which are balanced by a bridging fiber between sister kinetochore fibers. An extension of this approach to three dimensions revealed that microtubule bundles have rich shapes, and extend not simply like meridians on the Earth's surface but, rather, twisted in a helical manner. Such complex shapes are due to rotational forces, which, in addition to linear forces, act in the spindle and may be generated by motor proteins such as kinesin-5. These findings open new questions for future studies, to understand the mechanisms of rotational forces and reveal their biological roles in cells.

The mitotic spindle is chiral due to torques within microtubule bundles.

Mitosis relies on forces generated in the spindle, a micro-machine composed of microtubules and associated proteins. Forces are required for the congression of chromosomes to the metaphase plate and their separation in anaphase. However, besides forces, torques may exist in the spindle, yet they have not been investigated. Here we show that the spindle is chiral. Chirality is evident from the finding that microtubule bundles in human spindles follow a left-handed helical path, which cannot be explained by forces but rather by torques. Kinesin-5 (Kif11/Eg5) inactivation abolishes spindle chirality. Our theoretical model predicts that bending and twisting moments may generate curved shapes of bundles. We found that bundles turn by about -2 deg µm-1 around the spindle axis, which we explain by a twisting moment of roughly -10 pNµm. We conclude that torques, in addition to forces, exist in the spindle and determine its chiral architecture.

Spindle Dynamics Model Explains Chromosome Loss Rates in Yeast Polyploid Cells.

Faithful chromosome segregation, driven by the mitotic spindle, is essential for organismal survival. Neopolyploid cells from diverse species exhibit a significant increase in mitotic errors relative to their diploid progenitors, resulting in chromosome nondisjunction. In the model system Saccharomyces cerevisiae, the rate of chromosome loss in haploid and diploid cells is measured to be one thousand times lower than the rate of loss in isogenic tetraploid cells. Currently it is unknown what constrains the number of chromosomes that can be segregated with high fidelity in an organism. Here we developed a simple mathematical model to study how different rates of chromosome loss in cells with different ploidy can arise from changes in (1) spindle dynamics and (2) a maximum duration of mitotic arrest, after which cells enter anaphase. We apply this model to S. cerevisiae to show that this model can explain the observed rates of chromosome loss in S. cerevisiae cells of different ploidy. Our model describes how small increases in spindle assembly time can result in dramatic differences in the rate of chromosomes loss between cells of increasing ploidy and predicts the maximum duration of mitotic arrest.

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.

Microtubule Sliding within the Bridging Fiber Pushes Kinetochore Fibers Apart to Segregate Chromosomes.

During cell division, mitotic spindle microtubules segregate chromosomes by exerting forces on kinetochores. What forces drive chromosome segregation in anaphase remains a central question. The current model for anaphase in human cells includes shortening of kinetochore fibers and separation of spindle poles. Both processes require kinetochores to be linked with the poles. Here we show, by combining laser ablation, photoactivation, and theoretical modeling, that kinetochores can separate without any attachment to one spindle pole. This separation requires the bridging fiber, a microtubule bundle that connects sister kinetochore fibers. Bridging fiber microtubules in intact spindles slide apart with kinetochore fibers, indicating strong crosslinks between them. We conclude that sliding of microtubules within the bridging fibers drives pole separation and pushes kinetochore fibers poleward by the friction of passive crosslinks between these fibers. Thus, sliding within the bridging fiber works together with the shortening of kinetochore fibers to segregate chromosomes.

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.

Self-Organization and Forces in the Mitotic Spindle.

At the onset of division, the cell forms a spindle, a precise self-constructed micromachine composed of microtubules and the associated proteins, which divides the chromosomes between the two nascent daughter cells. The spindle arises from self-organization of microtubules and chromosomes, whose different types of motion help them explore the space and eventually approach and interact with each other. Once the interactions between the chromosomes and the microtubules have been established, the chromosomes are moved to the equatorial plane of the spindle and ultimately toward the opposite spindle poles. These transport processes rely on directed forces that are precisely regulated in space and time. In this review, we discuss how microtubule dynamics and their rotational movement drive spindle self-organization, as well as how the forces acting in the spindle are generated, balanced, and regulated.

Laser microsurgery reveals conserved viscoelastic behavior of the kinetochore.

Accurate chromosome segregation depends on proper kinetochore-microtubule attachment. Upon microtubule interaction, kinetochores are subjected to forces generated by the microtubules. In this work, we used laser ablation to sever microtubules attached to a merotelic kinetochore, which is laterally stretched by opposing pulling forces exerted by microtubules, and inferred the mechanical response of the kinetochore from its length change. In both mammalian PtK1 cells and in the fission yeast Schizosaccharomyces pombe, kinetochores shortened after microtubule severing. Interestingly, the inner kinetochore-centromere relaxed faster than the outer kinetochore. Whereas in fission yeast all kinetochores relaxed to a similar length, in PtK1 cells the more stretched kinetochores remained more stretched. Simple models suggest that these differences arise because the mechanical structure of the mammalian kinetochore is more complex. Our study establishes merotelic kinetochores as an experimental model for studying the mechanical response of the kinetochore in live cells and reveals a viscoelastic behavior of the kinetochore that is conserved in yeast and mammalian cells.

Overlap microtubules link sister k-fibres and balance the forces on bi-oriented kinetochores.

During metaphase, forces on kinetochores are exerted by k-fibres, bundles of microtubules that end at the kinetochore. Interestingly, non-kinetochore microtubules have been observed between sister kinetochores, but their function is unknown. Here we show by laser-cutting of a k-fibre in HeLa and PtK1 cells that a bundle of non-kinetochore microtubules, which we term 'bridging fibre', bridges sister k-fibres and balances the interkinetochore tension. We found PRC1 and EB3 in the bridging fibre, suggesting that it consists of antiparallel dynamic microtubules. By using a theoretical model that includes a bridging fibre, we show that the forces at the pole and at the kinetochore depend on the bridging fibre thickness. Moreover, our theory and experiments show larger relaxation of the interkinetochore distance for cuts closer to kinetochores. We conclude that the bridging fibre, by linking sister k-fibres, withstands the tension between sister kinetochores and enables the spindle to obtain a curved shape.