Elastic and damping forces generated by confined arrays of dynamic microtubules

Relaxation and Noise-Driven Oscillations in a Model of Mitotic Spindle Dynamics

During cell division, the mitotic spindle moves dynamically through the cell to position the chromosomes and determine the ultimate spatial position of the two daughter cells. These movements have been attributed to the action of cortical force generators which pull on the astral microtubules to position the spindle, as well as pushing events by these same microtubules against the cell cortex and plasma membrane. Attachment and detachment of cortical force generators working antagonistically against centring forces of microtubules have been modelled previously (Grill et al. in Phys Rev Lett 94:108104, 2005) via stochastic simulations and mean-field Fokker-Planck equations (describing random motion of force generators) to predict oscillations of a spindle pole in one spatial dimension. Using systematic asymptotic methods, we reduce the Fokker-Planck system to a set of ordinary differential equations (ODEs), consistent with a set proposed by Grill et al., which can provide accurate predictions of the conditions for the Fokker-Planck system to exhibit oscillations. In the limit of small restoring forces, we derive an algebraic prediction of the amplitude of spindle-pole oscillations and demonstrate the relaxation structure of nonlinear oscillations. We also show how noise-induced oscillations can arise in stochastic simulations for conditions in which the mean-field Fokker-Planck system predicts stability, but for which the period can be estimated directly by the ODE model and the amplitude by a related stochastic differential equation that incorporates random binding kinetics.

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.

Contribution of cytoplasm viscoelastic properties to mitotic spindle positioning

The regulation of mitotic spindle positioning is a key process for tissue architecture, embryo development, and stem cells. To date, most models have assumed that spindles are positioned by forces exerted by polar cytoskeleton networks, like microtubule asters or actomyosin bundles. Here, using in situ magnetic tweezers to apply calibrated forces and torques to mitotic spindles in live dividing sea urchin cells, we found that the viscoelastic properties of the cytoplasm medium in which spindles are embedded can hold spindles in place and move them back if their original position is perturbed. These viscoelastic forces are large and may significantly participate in the force balance that position and orient mitotic spindles in many cell types.

Cells are filled with macromolecules and polymer networks that set scale-dependent viscous and elastic properties to the cytoplasm. Although the role of these parameters in molecular diffusion, reaction kinetics, and cellular biochemistry is being increasingly recognized, their contributions to the motion and positioning of larger organelles, such as mitotic spindles for cell division, remain unknown. Here, using magnetic tweezers to displace and rotate mitotic spindles in living embryos, we uncovered that the cytoplasm can impart viscoelastic reactive forces that move spindles, or passive objects with similar size, back to their original positions. These forces are independent of cytoskeletal force generators yet reach hundreds of piconewtons and scale with cytoplasm crowding. Spindle motion shears and fluidizes the cytoplasm, dissipating elastic energy and limiting spindle recoils with functional implications for asymmetric and oriented divisions. These findings suggest that bulk cytoplasm material properties may constitute important control elements for the regulation of division positioning and cellular organization.

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.

Mechanics of microtubule organizing center clustering and spindle positioning in budding yeast Cryptococcus neoformans

The dynamic process of mitotic spindle assembly depends on multitudes of inter-dependent interactions involving kinetochores (KTs), microtubules (MTs), spindle pole bodies (SPBs), and molecular motors. Before forming the mitotic spindle, multiple visible microtubule organizing centers (MTOCs) coalesce into a single focus to serve as an SPB in the pathogenic budding yeast, Cryptococcus neoformans. To explain this unusual phenomenon in the fungal kingdom, we propose a "search and capture" model, in which cytoplasmic MTs (cMTs) nucleated by MTOCs grow and capture each other to promote MTOC clustering. Our quantitative modeling identifies multiple redundant mechanisms mediated by a combination of cMT-cell cortex interactions and inter-cMT coupling to facilitate MTOC clustering within the physiological time limit as determined by time-lapse live-cell microscopy. Besides, we screen various possible mechanisms by computational modeling and propose optimal conditions that favor proper spindle positioning-a critical determinant for timely chromosome segregation. These analyses also reveal that a combined effect of MT buckling, dynein pull, and cortical push maintains spatiotemporal spindle localization.

The coordination of spindle-positioning forces during the asymmetric division of the Caenorhabditis elegans zygote

In Caenorhabditis elegans zygote, astral microtubules generate forces essential to position the mitotic spindle, by pushing against and pulling from the cortex. Measuring microtubule dynamics there, we revealed the presence of two populations, corresponding to pulling and pushing events. It offers a unique opportunity to study, under physiological conditions, the variations of both spindle-positioning forces along space and time. We propose a threefold control of pulling force, by polarity, spindle position and mitotic progression. We showed that the sole anteroposterior asymmetry in dynein on-rate, encoding pulling force imbalance, is sufficient to cause posterior spindle displacement. The positional regulation, reflecting the number of microtubule contacts in the posterior-most region, reinforces this imbalance only in late anaphase. Furthermore, we exhibited the first direct proof that dynein processivity increases along mitosis. It reflects the temporal control of pulling forces, which strengthens at anaphase onset following mitotic progression and independently from chromatid separation. In contrast, the pushing force remains constant and symmetric and contributes to maintaining the spindle at the cell centre during metaphase.

Stoichiometric interactions explain spindle dynamics and scaling across 100 million years of nematode evolution

The spindle shows remarkable diversity, and changes in an integrated fashion, as cells vary over evolution. Here, we provide a mechanistic explanation for variations in the first mitotic spindle in nematodes. We used a combination of quantitative genetics and biophysics to rule out broad classes of models of the regulation of spindle length and dynamics, and to establish the importance of a balance of cortical pulling forces acting in different directions. These experiments led us to construct a model of cortical pulling forces in which the stoichiometric interactions of microtubules and force generators (each force generator can bind only one microtubule), is key to explaining the dynamics of spindle positioning and elongation, and spindle final length and scaling with cell size. This model accounts for variations in all the spindle traits we studied here, both within species and across nematode species spanning over 100 million years of evolution.

Cell shape and intercellular adhesion regulate mitotic spindle orientation

Cell division orientation plays an essential role in tissue morphogenesis and cell fate decision. Recent studies showed that either cell shape or adhesion geometry can regulate the orientation of mitotic spindles and thereby the cell division orientation. However, how they together regulate the spindle orientation remains largely unclear. In this work, we use a general computational model to investigate the competitive mechanism of determining the spindle orientation between cell shape and intercellular adhesion in epithelial cells. We find the spindle orientation is dominated by the intercellular adhesion when the cell shape anisotropy is small, but dominated by the cell shape when the shape anisotropy is large. A strong adhesion and moderate adhesive size can ensure the planar division of epithelial cells with large apico-basal elongation. We also find the spindle orientation could be perpendicular to the adhesive region when only one side of the cell is adhered to an E-cadherin-coated matrix. But after the cell is compressed, the spindle orientation is governed by the cell shape and the spindle will be parallel to the adhesive region when the cell shape anisotropy is large. Finally, we demonstrate the competition between cell shape and tricellular junctions can also effectively regulate the spindle orientation.

Mechanistic three-dimensional model to study centrosome positioning in the interphase cell

During the interphase in mammalian cells, the position of the centrosome is actively maintained at a small but finite distance away from the nucleus. The perinuclear positioning of the centrosome is crucial for cellular trafficking and progression into mitosis. Although the literature suggests that the contributions of the microtubule-associated forces bring the centrosome to the center of the cell, the position of the centrosome was merely investigated in the absence of the nucleus. Upon performing a coarse-grained simulation study with mathematical analysis, we show that the combined effect of the forces due to the cell cortex and the nucleus facilitate the centrosome positioning. Our study also demonstrates that in the absence of nucleus-based forces, the centrosome collapses on the nucleus due to cortical forces. Depending upon the magnitudes of the cortical forces and the nucleus-based forces, the centrosome appears to stay at various distances away from the nucleus. Such null force regions are found to be stable as well as unstable fixed points. This study uncovers a set of redundant schemes that the cell may adopt to produce the required cortical and nucleus-based forces stabilizing the centrosome at a finite distance away from the nucleus.

Microtubule Feedback and LET-99-Dependent Control of Pulling Forces Ensure Robust Spindle Position

During asymmetric division of the Caenorhabditis elegans zygote, to properly distribute cell fate determinants, the mitotic spindle is asymmetrically localized by a combination of centering and cortical-pulling microtubule-mediated forces, the dynamics of the latter being regulated by mitotic progression. Here, we show a, to our knowledge, novel and additional regulation of these forces by spindle position itself. For that, we observed the onset of transverse spindle oscillations, which reflects the burst of anaphase pulling forces. After delaying anaphase onset, we found that the position at which the spindle starts to oscillate was unchanged compared to control embryos and uncorrelated to anaphase onset. In mapping the cortical microtubule dynamics, we measured a steep increase in microtubule contact density after the posterior centrosome reached the critical position of 70% of embryo length, strongly suggesting the presence of a positional switch for spindle oscillations. Expanding a previous model based on a force-generator temporal control, we implemented this positional switch and observed that the large increase in microtubule density accounted for the pulling force burst. Thus, we propose that the spindle position influences the cortical availability of microtubules on which the active force generators, controlled by cell cycle progression, can pull. Importantly, we found that this positional control relies on the polarity-dependent LET-99 cortical band, the boundary of which could be probed by microtubules. This dual positional and temporal control well accounted for our observation that the oscillation onset position resists changes in cellular geometry and moderate variations in the active force generator number. Finally, our model suggests that spindle position at mitosis end is more sensitive to the polarity factor LET-99, which restricts the region of active force generators to a posterior-most region, than to microtubule number or force generator number/activity. Overall, we show that robustness in spindle positioning originates in cell mechanics rather than biochemical networks.

How cells sense their own shape - mechanisms to probe cell geometry and their implications in cellular organization and function

Cells come in a variety of shapes that most often underlie their functions. Regulation of cell morphogenesis implies that there are mechanisms for shape sensing that still remain poorly appreciated. Global and local cell geometry features, such as aspect ratio, size or membrane curvature, may be probed by intracellular modules, such as the cytoskeleton, reaction-diffusion systems or molecular complexes. In multicellular tissues, cell shape emerges as an important means to transduce tissue-inherent chemical and mechanical cues into intracellular organization. One emergent paradigm is that cell-shape sensing is most often based upon mechanisms of self-organization, rather than determinism. Here, we review relevant work that has elucidated some of the core principles of how cellular geometry may be conveyed into spatial information to guide processes, such as polarity, signaling, morphogenesis and division-plane positioning.

Uncovering the balance of forces driving microtubule aster migration in C. elegans zygotes

Microtubule asters must be positioned precisely within cells. How forces generated by molecular motors such as dynein are integrated in space and time to enable such positioning remains unclear. In particular, whereas aster movements depend on the drag caused by cytoplasm viscosity, in vivo drag measurements are lacking, precluding a thorough understanding of the mechanisms governing aster positioning. Here, we investigate this fundamental question during the migration of asters and pronuclei in C. elegans zygotes, a process essential for the mixing of parental genomes. Detailed quantification of these movements using the female pronucleus as an in vivo probe establish that the drag coefficient of the male-asters complex is approximately five times that of the female pronucleus. Further analysis of embryos lacking cortical dynein, the connection between asters and male pronucleus, or the male pronucleus altogether, uncovers the balance of dynein-driven forces that accurately position microtubule asters in C. elegans zygotes.

Microtubule asters are positioned precisely within cells by forces generated by molecular motors, but it is unclear how these are integrated in space and time. Here the authors perform in vivo drag measurements and genetic manipulations to determine the balance of forces that position microtubule asters in C. elegans zygotes.

Regulating positioning and orientation of mitotic spindles via cell size and shape

Proper location of the mitotic spindle is critical for chromosome segregation and the selection of the cell division plane. However, how mitotic spindles sense cell size and shape to regulate their own position and orientation is still largely unclear. To investigate this question systematically, we used a general model by considering chromosomes, microtubule dynamics, and forces of various molecular motors. Our results show that in cells of various sizes and shapes, spindles can always be centered and oriented along the long axis robustly in the absence of other specified mechanisms. We found that the characteristic time of positioning and orientation processes increases with cell size. Spindles sense the cell size mainly by the cortical force in small cells and by the cytoplasmic force in large cells. In addition to the cell size, the cell shape mainly influences the orientation process. We found that more slender cells have a faster orientation process, and the final orientation is not necessarily along the longest axis but is determined by the radial profile and the symmetry of the cell shape. Finally, our model also reproduces the separation and repositioning of the spindle poles during the anaphase. Therefore, our work provides a general tool for studying the mitotic spindle across the whole mitotic phase.

Physical Limits on the Precision of Mitotic Spindle Positioning by Microtubule Pushing forces: Mechanics of mitotic spindle positioning

Tissues are shaped and patterned by mechanical and chemical processes. A key mechanical process is the positioning of the mitotic spindle, which determines the size and location of the daughter cells within the tissue. Recent force and position-fluctuation measurements indicate that pushing forces, mediated by the polymerization of astral microtubules against- the cell cortex, maintain the mitotic spindle at the cell center in Caenorhabditis elegans embryos. The magnitude of the centering forces suggests that the physical limit on the accuracy and precision of this centering mechanism is determined by the number of pushing microtubules rather than by thermally driven fluctuations. In cells that divide asymmetrically, anti-centering, pulling forces generated by cortically located dyneins, in conjunction with microtubule depolymerization, oppose the pushing forces to drive spindle displacements away from the center. Thus, a balance of centering pushing forces and anti-centering pulling forces localize the mitotic spindles within dividing C. elegans cells.

A Dynamic Biochemomechanical Model of Geometry-Confined Cell Spreading

Cell spreading is involved in many physiological and pathological processes. The spreading behavior of a cell significantly depends on its microenvironment, but the biochemomechanical mechanisms of geometry-confined cell spreading remain unclear. A dynamic model is here established to investigate the spreading of cells confined in a finite region with different geometries, e.g., rectangle, ellipse, triangle, and L-shape. This model incorporates both biophysical and biochemical mechanisms, including actin polymerization, integrin-mediated binding, plasma viscoelasticity, and the elasticity of membranes and microtubules. We simulate the dynamic configurational evolution of a cell under different geometric microenvironments, including the angular distribution of microtubule forces and the deformation of the nucleus. The results indicate that the positioning of the cell-division plane is affected by its boundary confinement: a cell divides in a plane perpendicular to its minimal principal axis of inertia of area. In addition, the effects of such physical factors as the adhesive bond density, membrane tension, and microtubule number are examined on the cell spreading dynamics. The theoretical predictions show a good agreement with relevant experimental results. This work sheds light on the geometry-confined spreading dynamics of cells and holds potential applications in regulating cell division and designing cell-based sensors.

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.

Geometric Asymmetry Induces Upper Limit of Mitotic Spindle Size

Proper organelle size is critical for many cell functions. However, how cells sense and control their organelle size remains elusive. Here, we develop a general model to study the size control of mitotic spindles by considering both extrinsic and intrinsic factors, such as the limited number of building blocks of the spindle, the interaction between the spindle and cell boundary, the DNA content, the forces generated by various molecular motors, and the dynamics of microtubules. We show that multiple pairs of chromatids, two centrosomes, and microtubules can self-assemble to form a mitotic spindle robustly. We also show that the boundary-sensing and volume-sensing mechanisms coexist in small cells, but both break down in large cells. Strikingly, we find that the upper limit of spindle length naturally arises from the geometric asymmetry of the spindle structure. Thus, our findings reveal, to our knowledge, a novel intrinsic mechanism that limits the organelle size.

Characterization of microtubule buckling in living cells

Microtubules are filamentous biopolymers involved in essential biological processes. They form key structures in eukaryotic cells, and thus it is very important to determine the mechanisms involved in the formation and maintenance of the microtubule network. Microtubule bucklings are transient and localized events commonly observed in living cells and characterized by a fast bending and its posterior relaxation. Active forces provided by molecular motors have been indicated as responsible for most of these rapid deformations. However, the factors that control the shape amplitude and the time scales of the rising and release stages remain unexplored. In this work, we study microtubule buckling in living cells using Xenopus laevis melanophores as a model system. We tracked single fluorescent microtubules from high temporal resolution (0.3-2 s) confocal movies. We recovered the center coordinates of the filaments with 10-nm precision and analyzed the amplitude of the deformation as a function of time. Using numerical simulations, we explored different force mechanisms resulting in microtubule bending. The simulated events reproduce many features observed for microtubules, suggesting that a mechanistic model captures the essential processes underlying microtubule buckling. Also, we studied the interplay between actively transported vesicles and the microtubule network using a two-color technique. Our results suggest that microtubules may affect transport indirectly besides serving as tracks of motor-driven organelles. For example, they could obstruct organelles at microtubule intersections or push them during filament mechanical relaxation.

Physical determinants of bipolar mitotic spindle assembly and stability in fission yeast

Mitotic spindles use an elegant bipolar architecture to segregate duplicated chromosomes with high fidelity. Bipolar spindles form from a monopolar initial condition; this is the most fundamental construction problem that the spindle must solve. Microtubules, motors, and cross-linkers are important for bipolarity, but the mechanisms necessary and sufficient for spindle assembly remain unknown. We describe a physical model that exhibits de novo bipolar spindle formation. We began with physical properties of fission-yeast spindle pole body size and microtubule number, kinesin-5 motors, kinesin-14 motors, and passive cross-linkers. Our model results agree quantitatively with our experiments in fission yeast, thereby establishing a minimal system with which to interrogate collective self-assembly. By varying the features of our model, we identify a set of functions essential for the generation and stability of spindle bipolarity. When kinesin-5 motors are present, their bidirectionality is essential, but spindles can form in the presence of passive cross-linkers alone. We also identify characteristic failed states of spindle assembly-the persistent monopole, X spindle, separated asters, and short spindle, which are avoided by the creation and maintenance of antiparallel microtubule overlaps. Our model can guide the identification of new, multifaceted strategies to induce mitotic catastrophes; these would constitute novel strategies for cancer chemotherapy.

Forces positioning the mitotic spindle: Theories, and now experiments

The position of the spindle determines the position of the cleavage plane, and is thus crucial for cell division. Although spindle positioning has been extensively studied, the underlying forces ultimately responsible for moving the spindle remain poorly understood. A recent pioneering study by Garzon-Coral et al. uses magnetic tweezers to perform the first direct measurements of the forces involved in positioning the mitotic spindle. Combining this with molecular perturbations and geometrical effects, they use their data to argue that the forces that keep the spindle in its proper position for cell division arise from astral microtubules growing and pushing against the cell's cortex. Here, we review these ground-breaking experiments, the various biomechanical models for spindle positioning that they seek to differentiate, and discuss new questions raised by these measurements.

The Mitotic Spindle in the One-Cell C. elegans Embryo Is Positioned with High Precision and Stability

Precise positioning of the mitotic spindle is important for specifying the plane of cell division, which in turn determines how the cytoplasmic contents of the mother cell are partitioned into the daughter cells, and how the daughters are positioned within the tissue. During metaphase in the early Caenorhabditis elegans embryo, the spindle is aligned and centered on the anterior-posterior axis by a microtubule-dependent machinery that exerts restoring forces when the spindle is displaced from the center. To investigate the accuracy and stability of centering, we tracked the position and orientation of the mitotic spindle during the first cell division with high temporal and spatial resolution. We found that the precision is remarkably high: the cell-to-cell variation in the transverse position of the center of the spindle during metaphase, as measured by the standard deviation, was only 1.5% of the length of the short axis of the cell. Spindle position is also very stable: the standard deviation of the fluctuations in transverse spindle position during metaphase was only 0.5% of the short axis of the cell. Assuming that stability is limited by fluctuations in the number of independent motor elements such as microtubules or dyneins underlying the centering machinery, we infer that the number is ∼1000, consistent with the several thousand of astral microtubules in these cells. Astral microtubules grow out from the two spindle poles, make contact with the cell cortex, and then shrink back shortly thereafter. The high stability of centering can be accounted for quantitatively if, while making contact with the cortex, the astral microtubules buckle as they exert compressive, pushing forces. We thus propose that the large number of microtubules in the asters provides a highly precise mechanism for positioning the spindle during metaphase while assembly is completed before the onset of anaphase.

A force-generating machinery maintains the spindle at the cell center during mitosis

The position and orientation of the mitotic spindle is precisely regulated to ensure the accurate partition of the cytoplasm between daughter cells and the correct localization of the daughters within growing tissue. Using magnetic tweezers to perturb the position of the spindle in intact cells, we discovered a force-generating machinery that maintains the spindle at the cell center during metaphase and anaphase in one- and two-cell Caenorhabditis elegans embryos. The forces increase with the number of microtubules and are larger in smaller cells. The machinery is rigid enough to suppress thermal fluctuations to ensure precise localization of the mitotic spindle, yet compliant enough to allow molecular force generators to fine-tune the position of the mitotic spindle to facilitate asymmetric division.

Buckling behavior of individual and bundled microtubules

As the major structural constituent of the cytoskeleton, microtubules (MTs) serve a variety of biological functions that range from facilitating organelle transport to maintaining the mechanical integrity of the cell. Neuronal MTs exhibit a distinct configuration, hexagonally packed bundles of MT filaments, interconnected by MT-associated protein (MAP) tau. Building on our previous work on mechanical response of axonal MT bundles under uniaxial tension, this study is focused on exploring the compression scenarios. Intracellular MTs carry a large fraction of the compressive loads sensed by the cell and therefore, like any other column-like structure, are prone to substantial bending and buckling. Various biological activities, e.g., actomyosin contractility and many pathological conditions are driven or followed by bending, looping, and buckling of MT filaments. The coarse-grained model previously developed in our lab has been used to study the mechanical behavior of individual and bundled in vivo MT filaments under uniaxial compression. Both configurations show tip-localized, decaying, and short-wavelength buckling. This behavior highlights the role of the surrounding cytoplasm and MAP tau on MT buckling behavior, which allows MT filaments to bear much larger compressive forces. It is observed that MAP tau interconnections improve this effect by a factor of two. The enhanced ability of MT bundles to damp buckling waves relative to individual MT filaments, may be interpreted as a self-defense mechanism because it helps axonal MTs to endure harsher environments while maintaining their function. The results indicate that MT filaments in a bundle do not buckle simultaneously implying that the applied stress is not equally shared among the MT filaments, that is a consequence of the nonuniform distribution of MAP tau proteins along the bundle length. Furthermore, from a pathological perspective, it is observed that axonal MT bundles are more vulnerable to failure in compression than tension.

Microtubules self-repair in response to mechanical stress

Microtubules--which define the shape of axons, cilia and flagella, and provide tracks for intracellular transport--can be highly bent by intracellular forces, and microtubule structure and stiffness are thought to be affected by physical constraints. Yet how microtubules tolerate the vast forces exerted on them remains unknown. Here, by using a microfluidic device, we show that microtubule stiffness decreases incrementally with each cycle of bending and release. Similar to other cases of material fatigue, the concentration of mechanical stresses on pre-existing defects in the microtubule lattice is responsible for the generation of more extensive damage, which further decreases microtubule stiffness. Strikingly, damaged microtubules were able to incorporate new tubulin dimers into their lattice and recover their initial stiffness. Our findings demonstrate that microtubules are ductile materials with self-healing properties, that their dynamics does not exclusively occur at their ends, and that their lattice plasticity enables the microtubules' adaptation to mechanical stresses.

Cell size modulates oscillation, positioning and length of mitotic spindles

Mitotic spindle is the main subcellular structure that accomplishes the chromosome segregation between daughter cells during cell division. However, how mitotic spindles sense and control their size, position and movement inside the cell still remains unclear. In this paper, we focus on the size effects of mitotic spindles, i.e., how cell size controls various interesting phenomena in the metaphase, such as oscillation, positioning and size limit of mitotic spindles. We systematically studied the frequency doubling phenomenon during chromosome oscillation and found that cell size can regulate the period and amplitude of chromosome oscillation. We found that the relaxation time of the positioning process increases exponentially with cell size. We also showed that the stabler microtubule-kinetochore attachments alone can directly lead to an upper limit of spindle size. Our work not only explains the existing experimental observations, but also provides some interesting predictions that can be verified or rejected by further experiments.

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.

Autophagy and microtubules - new story, old players

Both at a basal level and after induction (especially in response to nutrient starvation), the function of autophagy is to allow cells to degrade and recycle damaged organelles, proteins and other biological constituents. Here, we focus on the role microtubules have in autophagosome formation, autophagosome transport across the cytoplasm and in the formation of autolysosomes. Recent insights into the exact relationship between autophagy and microtubules now point to the importance of microtubule dynamics, tubulin post-translational modifications and microtubule motors in the autophagy process. Such factors regulate signaling pathways that converge to stimulate autophagosome formation. They also orchestrate the movements of pre-autophagosomal structures and autophagosomes or more globally organize and localize immature and mature autophagosomes and lysosomes. Most of the factors that now appear to link microtubules to autophagosome formation or to autophagosome dynamics and fate were identified initially without the notion that sequestration, recruitment and/or interaction with microtubules contribute to their function. Spatial and temporal coordination of many stages in the life of autophagosomes thus underlines the integrative role of microtubules and progressively reveals hidden parts of the autophagy machinery.

Centrosome polarization in T cells: a task for formins

T-cell antigen receptor (TCR) engagement triggers the rapid reorientation of the centrosome, which is associated with the secretory machinery, toward the immunological synapse (IS) for polarized protein trafficking. Recent evidence indicates that upon TCR triggering the INF2 formin, together with the formins DIA1 and FMNL1, promotes the formation of a specialized array of stable detyrosinated MTs that breaks the symmetrical organization of the T-cell microtubule (MT) cytoskeleton. The detyrosinated MT array and TCR-induced tyrosine phosphorylation should coincide for centrosome polarization. We propose that the pushing forces produced by the detyrosinated MT array, which modify the position of the centrosome, in concert with Src kinase dependent TCR signaling, which provide the reference frame with respect to which the centrosome reorients, result in the repositioning of the centrosome to the IS.

Cargo transport by cytoplasmic Dynein can center embryonic centrosomes

To complete meiosis II in animal cells, the male DNA material needs to meet the female DNA material contained in the female pronucleus at the egg center, but it is not known how the male pronucleus, deposited by the sperm at the periphery of the cell, finds the cell center in large eggs. Pronucleus centering is an active process that appears to involve microtubules and molecular motors. For small and medium-sized cells, the force required to move the centrosome can arise from either microtubule pushing on the cortex, or cortically-attached dynein pulling on microtubules. However, in large cells, such as the fertilized Xenopus laevis embryo, where microtubules are too long to support pushing forces or they do not reach all boundaries before centrosome centering begins, a different force generating mechanism must exist. Here, we present a centrosome positioning model in which the cytosolic drag experienced by cargoes hauled by cytoplasmic dynein on the sperm aster microtubules can move the centrosome towards the cell's center. We find that small, fast cargoes (diameter ∼100 nm, cargo velocity ∼2 µm/s) are sufficient to move the centrosome in the geometry of the Xenopus laevis embryo within the experimentally observed length and time scales.

Emergent complexity of the cytoskeleton: from single filaments to tissue

Despite their overwhelming complexity, living cells display a high degree of internal mechanical and functional organization which can largely be attributed to the intracellular biopolymer scaffold, the cytoskeleton. Being a very complex system far from thermodynamic equilibrium, the cytoskeleton's ability to organize is at the same time challenging and fascinating. The extensive amounts of frequently interacting cellular building blocks and their inherent multifunctionality permits highly adaptive behavior and obstructs a purely reductionist approach. Nevertheless (and despite the field's relative novelty), the physics approach has already proved to be extremely successful in revealing very fundamental concepts of cytoskeleton organization and behavior. This review aims at introducing the physics of the cytoskeleton ranging from single biopolymer filaments to multicellular organisms. Throughout this wide range of phenomena, the focus is set on the intertwined nature of the different physical scales (levels of complexity) that give rise to numerous emergent properties by means of self-organization or self-assembly.

Cytoplasmic dynein: tension generation on microtubules and the nucleus

Cytoplasmic dynein is a microtubule dependent motor protein that is central to vesicle transport, cell division and organelle positioning. Recent studies suggest that dynein can generate significant pulling forces on intracellular structures as it motors along microtubules. In this review, we discuss how dynein-generated pulling forces position the nucleus and the centrosome.

'Poking' microtubules bring about nuclear wriggling to position nuclei

Nuclei wriggle in the cells of the follicle epithelium of the Drosophila pre-vitellogenic egg primordia. Although similar phenomena have been reported for a number of cultured cell types and some neurons in the zebrafish embryo, the mechanism and importance of the process have remained unexplained. Wriggling involves successive sudden and random minor turns of the nuclei, approximately three twists per minute with roughly 12° per twist, one of which lasts typically for 14 seconds. Wriggling is generated by the growing microtubules seeded throughout the cell cortex, which, while poking the nuclei, buckle and exert 5-40 piconewtons over ∼16 seconds. While wriggling, the nuclei drift ∼5 µm in a day in the immensely growing follicle cells along the apical-basal axis from the apical to the basal cell region. A >2-fold excess of the microtubules nucleated in the apical cell region, as compared with those seeded in the basal cell cortex, makes the nuclei drift along the apical-basal axis. Nuclear wriggling and positioning appear to be tightly related processes: they cease simultaneously when the nuclei become anchored by the actin cytoskeleton; moreover, colchicine or taxol treatment eliminates both nuclear wriggling and positioning. We propose that the wriggling nuclei reveal a thus far undescribed nuclear positioning mechanism.

End-on microtubule-dynein interactions and pulling-based positioning of microtubule organizing centers

During important cellular processes such as centrosome and spindle positioning, dynein at the cortex interacts with dynamic microtubules in an apparent "end-on" fashion. It is well-established that dynein can generate forces by moving laterally along the microtubule lattice, but much less is known about dynein's interaction with dynamic microtubule ends. In this paper, we review recent in vitro experiments that show that dynein, attached to an artificial cortex, is able to capture microtubule ends, regulate microtubule dynamics and mediate the generation of pulling forces on shrinking microtubules. We further review existing ideas on the involvement of dynein-mediated cortical pulling forces in the positioning of microtubule organizing centers such as centrosomes. Recent in vitro experiments have demonstrated that cortical pulling forces in combination with pushing forces can lead to reliable centering of microtubule asters in quasi two-dimensional microfabricated chambers. In these experiments, pushing leads to slipping of microtubule ends along the chamber boundaries, resulting in an anisotropic distribution of cortical microtubule contacts that favors centering, once pulling force generators become engaged. This effect is predicted to be strongly geometry-dependent, and we therefore finally discuss ongoing efforts to repeat these experiments in three-dimensional, spherical and deformable geometries.

Growth, interaction, and positioning of microtubule asters in extremely large vertebrate embryo cells

Ray Rappaport spent many years studying microtubule asters, and how they induce cleavage furrows. Here, we review recent progress on aster structure and dynamics in zygotes and early blastomeres of Xenopus laevis and Zebrafish, where cells are extremely large. Mitotic and interphase asters differ markedly in size, and only interphase asters span the cell. Growth of interphase asters occurs by a mechanism that allows microtubule density at the aster periphery to remain approximately constant as radius increases. We discuss models for aster growth, and favor a branching nucleation process. Neighboring asters that grow into each other interact to block further growth at the shared boundary. We compare the morphology of interaction zones formed between pairs of asters that grow out from the poles of the same mitotic spindle (sister asters) and between pairs not related by mitosis (non-sister asters) that meet following polyspermic fertilization. We argue growing asters recognize each other by interaction between antiparallel microtubules at the mutual boundary, and discuss models for molecular organization of interaction zones. Finally, we discuss models for how asters, and the centrosomes within them, are positioned by dynein-mediated pulling forces so as to generate stereotyped cleavage patterns. Studying these problems in extremely large cells is starting to reveal how general principles of cell organization scale with cell size.

Investigation of in vivo microtubule and stress fiber mechanics with laser ablation

Laser ablation has emerged as a useful technique to study the mechanical properties of the cytoskeleton in living cells. Laser ablation perturbs the force balance in the cytoskeleton, resulting in a dynamic response which can be imaged. Quantitative measurement of the dynamic response allows the testing of mechanical theories of the cytoskeleton in living cells. This review discusses recent work in applying laser ablation to study stress fiber and microtubule mechanics in living endothelial cells. These studies reveal that molecular motors are major determinants of the mechanical properties of the cytoskeleton in cells.

Cortical dynein controls microtubule dynamics to generate pulling forces that position microtubule asters

Dynein at the cortex contributes to microtubule-based positioning processes such as spindle positioning during embryonic cell division and centrosome positioning during fibroblast migration. To investigate how cortical dynein interacts with microtubule ends to generate force and how this functional association impacts positioning, we have reconstituted the 'cortical' interaction between dynein and dynamic microtubule ends in an in vitro system using microfabricated barriers. We show that barrier-attached dynein captures microtubule ends, inhibits growth, and triggers microtubule catastrophes, thereby controlling microtubule length. The subsequent interaction with shrinking microtubule ends generates pulling forces up to several pN. By combining experiments in microchambers with a theoretical description of aster mechanics, we show that dynein-mediated pulling forces lead to the reliable centering of microtubule asters in simple confining geometries. Our results demonstrate the intrinsic ability of cortical microtubule-dynein interactions to regulate microtubule dynamics and drive positioning processes in living cells.

Growth and shortening of microtubules: a two-state model approach

In this study, a two-state mechanochemical model is presented to describe the dynamic instability of microtubules (MTs) in cells. The MT switches between two states, the assembly and disassembly states. In assembly state, the growth of MTs includes two processes: free GTP-tubulin binding to the tip of protofilament (PF) and conformation change of PF, during which the first tubulin unit that curls outwards is rearranged onto the MT surface, using the energy released from the hydrolysis of GTP in the penultimate tubulin unit. In the disassembly state, the shortening of MTs also includes two processes, the release of GDP-tubulin from the tip of PF and the curling of one new tubulin unit out of the MT surface. Switches between these two states, which are usually called rescue and catastrophe, happen stochastically with external force-dependent rates. Using this two-state model with parameters obtained by fitting the recent experimental data, detailed properties of MT growth are obtained. I find that MT is mainly in the assembly state, its mean growth velocity increases with both the external force and the GTP-tubulin concentration, and an MT will shorten on average without an external force. To know more about the external force and GTP-tubulin concentration-dependent properties of MT growth, and for future experimental verification of this two-state model, 11 critical forces are defined and discussed numerically.

Effects of dynein on microtubule mechanics and centrosome positioning

When microtubules are severed by laser ablation, newly created minus ends increase in curvature, but they straighten when dynein is inhibited. It is found that cytoplasmic dynein generates tension and friction along microtubule lengths and that these forces govern the dynamics of centrosome centering.

To determine forces on intracellular microtubules, we measured shape changes of individual microtubules following laser severing in bovine capillary endothelial cells. Surprisingly, regions near newly created minus ends increased in curvature following severing, whereas regions near new microtubule plus ends depolymerized without any observable change in shape. With dynein inhibited, regions near severed minus ends straightened rapidly following severing. These observations suggest that dynein exerts a pulling force on the microtubule that buckles the newly created minus end. Moreover, the lack of any observable straightening suggests that dynein prevents lateral motion of microtubules. To explain these results, we developed a model for intracellular microtubule mechanics that predicts the enhanced buckling at the minus end of a severed microtubule. Our results show that microtubule shapes reflect a dynamic force balance in which dynein motor and friction forces dominate elastic forces arising from bending moments. A centrosomal array of microtubules subjected to dynein pulling forces and resisted by dynein friction is predicted to center on the experimentally observed time scale, with or without the pushing forces derived from microtubule buckling at the cell periphery.

How dynein and microtubules rotate the nucleus

In living cells, a fluctuating torque is exerted on the nuclear surface but the origin of the torque is unclear. In this study, we found that the nuclear rotation angle is directionally persistent on a time scale of tens of minutes, but rotationally diffusive on longer time scales. Rotation required the activity of the microtubule motor dynein. We formulated a model based on microtubules undergoing dynamic instability, with tensional forces between a stationary centrosome and the nuclear surface mediated by dynein. Model simulations suggest that the persistence in rotation angle is due to the transient asymmetric configuration of microtubules exerting a net torque in one direction until the configuration is again randomized by dynamic instability. The model predicts that the rotational magnitude must depend on the distance between the nucleus and the centrosome. To test this prediction, rotation was quantified in patterned cells in which the cell's centrosome was close to the projected nuclear centroid. Consistent with the prediction, the angular displacement was found to decrease in these cells relative to unpatterned cells. This work provides the first mechanistic explanation for how nuclear dynein interactions with discrete microtubules emanating from a stationary centrosome cause rotational torque on the nucleus.

Influence of cell geometry on division-plane positioning

The spatial organization of cells depends on their ability to sense their own shape and size. Here, we investigate how cell shape affects the positioning of the nucleus, spindle and subsequent cell division plane. To manipulate geometrical parameters in a systematic manner, we place individual sea urchin eggs into microfabricated chambers of defined geometry (e.g., triangles, rectangles, and ellipses). In each shape, the nucleus is positioned at the center of mass and is stretched by microtubules along an axis maintained through mitosis and predictive of the future division plane. We develop a simple computational model that posits that microtubules sense cell geometry by probing cellular space and orient the nucleus by exerting pulling forces that scale to microtubule length. This model quantitatively predicts division-axis orientation probability for a wide variety of cell shapes, even in multicellular contexts, and estimates scaling exponents for length-dependent microtubule forces.

Symmetry, stability, and reversibility properties of idealized confined microtubule cytoskeletons

Many cell cytoskeletons include an aster of microtubules, with the centrosome serving as the focal point. The position of the centrosome within the cell is important in such directional activities as wound closure and interactions of immune cells. Here we analyzed the centrosome positioning as it is dictated by microtubule elasticity alone in a mechanical model of an intrinsically fully symmetric microtubule aster. We demonstrate that the symmetry and the central position of the centrosome are unstable. The equilibrium deviation of the centrosome from the center is approximately proportional to the difference of the microtubule length and cell radius. The proportionality coefficient is 1 in flat cells and 2 in three-dimensional cells. The loss of symmetry is irreversible, and in general, the equilibrium form of the aster exhibits memory of past perturbations. The equilibrium position of the centrosome as a function of the microtubule length exhibits hysteresis, and the history of the length variation is reflected in the aster form. These properties of the simple aster of elastic microtubules must be taken into account in the analysis of more comprehensive theoretical models, and in the design and interpretation of experiments addressing the complex process of cytoskeleton morphogenesis.

Towards a quantitative understanding of mitotic spindle assembly and mechanics

The 'simple' view of the mitotic spindle is that it self-assembles as a result of microtubules (MTs) randomly searching for chromosomes, after which the spindle length is maintained by a balance of outward tension exerted by molecular motors on the MTs connecting centrosomes and chromosomes, and compression generated by other motors on the MTs connecting the spindle poles. This picture is being challenged now by mounting evidence indicating that spindle assembly and maintenance rely on much more complex interconnected networks of microtubules, molecular motors, chromosomes and regulatory proteins. From an engineering point of view, three design principles of this molecular machine are especially important: the spindle assembles quickly, it assembles accurately, and it is mechanically robust--yet malleable. How is this design achieved with randomly interacting and impermanent molecular parts? Here, we review recent interdisciplinary studies that have started to shed light on this question. We discuss cooperative mechanisms of spindle self-assembly, error correction and maintenance of its mechanical properties, speculate on analogy between spindle and lamellipodial dynamics, and highlight the role of quantitative approaches in understanding the mitotic spindle design.

Growth, fluctuation and switching at microtubule plus ends

Recent experiments suggest that microtubules do not grow steadily but instead elongate at a rate that varies in time. We argue that this variation might arise from fluctuations in the length of a dynamic GTP-tubulin cap at the microtubule end. We propose that these fluctuations can lead to a switch in the dynamics of a microtubule end between growth and shrinkage, and provide insight into how the sensitivity of this switch can be changed by microtubule polymerases, such as XMAP215, and tensile forces, through the stabilization of initial contacts in the cap.

Self-organization of dynein motors generates meiotic nuclear oscillations

Meiotic nuclear oscillations in the fission yeast Schizosaccharomyces pombe are crucial for proper chromosome pairing and recombination. We report a mechanism of these oscillations on the basis of collective behavior of dynein motors linking the cell cortex and dynamic microtubules that extend from the spindle pole body in opposite directions. By combining quantitative live cell imaging and laser ablation with a theoretical description, we show that dynein dynamically redistributes in the cell in response to load forces, resulting in more dynein attached to the leading than to the trailing microtubules. The redistribution of motors introduces an asymmetry of motor forces pulling in opposite directions, leading to the generation of oscillations. Our work provides the first direct in vivo observation of self-organized dynamic dynein distributions, which, owing to the intrinsic motor properties, generate regular large-scale movements in the cell.

The elasticity of motor-microtubule bundles and shape of the mitotic spindle

In the process of cell division, chromosomes are segregated by mitotic spindles -- bipolar microtubule arrays that have a characteristic fusiform shape. Mitotic spindle function is based on motor-generated forces of hundreds of piconewtons. These forces have to deform the spindle, yet the role of microtubule elastic deformations in the spindle remains unclear. Here we solve equations of elasticity theory for spindle microtubules, compare the solutions with shapes of early Drosophila embryo spindles and discuss the biophysical and cell biological implications of this analysis. The model suggests that microtubule bundles in the spindle behave like effective compressed springs with stiffness of the order of tens of piconewtons per micron, that microtubule elasticity limits the motors' power, and that clamping and cross-linking of microtubules are needed to transduce the motors' forces in the spindle. Some data are hard to reconcile with the model predictions, suggesting that cytoskeletal structures laterally reinforce the spindle and/or that rapid microtubule turnover relieves the elastic stresses.

An experimental and computational study of effects of microtubule stabilization on T-cell polarity

T-killer cells eliminate infected and cancerous cells with precision by positioning their centrosome near the interface (immunological synapse) with the target cell. The mechanism of centrosome positioning has remained controversial, in particular the role of microtubule dynamics in it. We re-examined the issue in the experimental model of Jurkat cells presented with a T cell receptor-binding artificial substrate, which permits controlled stimulation and reproducible measurements. Neither 1-µM taxol nor 100-nM nocodazole inhibited the centrosome positioning at the “synapse” with the biomimetic substrate. At the same time, in micromolar taxol but not in nanomolar nocodazole the centrosome adopted a distinct peripheral rather than the normally central position within the synapse. This effect was reproduced in a computational energy-minimization model that assumed no microtubule dynamics, but only a taxol-induced increase in the length of the microtubules. Together, the experimental and computational results indicate that microtubule dynamics are not essential for the centrosome positioning, but that the fit of the microtubule array in the deformed body of the conjugated T cell is a major factor. The possibility of modulating the T-cell centrosome position with well-studied drugs and of predicting their effects in silico appears attractive for designing anti-cancer and antiviral therapies.