Biophysical model of self-organized spindle formation patterns without centrosomes and kinetochores

Spatiotemporal control of structure and dynamics in a polar active fluid

We apply optimal control theory to a model of a polar active fluid (the Toner-Tu model), with the objective of driving the system into particular emergent dynamical behaviors or programming switching between states on demand. We use the effective self-propulsion speed as the control parameter (i.e. the means of external actuation). We identify control protocols that achieve outcomes such as relocating asters to targeted positions, forcing propagating solitary waves to reorient to a particular direction, and switching between stationary asters and propagating fronts. We analyze the solutions to identify generic principles for controlling polar active fluids. Our findings have implications for achieving spatiotemporal control of active polar systems in experiments, particularly in vitro cytoskeletal systems. Additionally, this research paves the way for leveraging optimal control methods to engineer the structure and dynamics of active fluids more broadly.

Mechanisms by Which Kinesin-5 Motors Perform Their Multiple Intracellular Functions

Bipolar kinesin-5 motor proteins perform multiple intracellular functions, mainly during mitotic cell division. Their specialized structural characteristics enable these motors to perform their essential functions by crosslinking and sliding apart antiparallel microtubules (MTs). In this review, we discuss the specialized structural features of kinesin-5 motors, and the mechanisms by which these features relate to kinesin-5 functions and motile properties. In addition, we discuss the multiple roles of the kinesin-5 motors in dividing as well as in non-dividing cells, and examine their roles in pathogenetic conditions. We describe the recently discovered bidirectional motility in fungi kinesin-5 motors, and discuss its possible physiological relevance. Finally, we also focus on the multiple mechanisms of regulation of these unique motor proteins.

Mechanics of the spindle apparatus

During mitosis microtubules self-organize to form a bipolar mitotic spindle structure, which positions the sister chromatids on the spindle mid-plane and separates them afterwards. Previous studies have identified many spindle associated proteins. Yet, we do not fully understand how these nanoscopic proteins lead to force generation through interactions of individual microtubules, motor proteins and chromosomes, and how a large number of these local interactions ultimately determine the structure and mechanics of the spindle in micron scale. Here we review the current understanding and open questions related to the structure and mechanics of the mitotic spindle. We then discuss how a combination of electron microscopy and computational modeling can be used to tackle some of these open questions.

A computational model of the early stages of acentriolar meiotic spindle assembly

The mitotic spindle is an ensemble of microtubules responsible for the repartition of the chromosomal content between the two daughter cells during division. In metazoans, spindle assembly is a gradual process involving dynamic microtubules and recruitment of numerous associated proteins and motors. During mitosis, centrosomes organize and nucleate the majority of spindle microtubules. In contrast, oocytes lack canonical centrosomes but are still able to form bipolar spindles, starting from an initial ball that self-organizes in several hours. Interfering with early steps of meiotic spindle assembly can lead to erroneous chromosome segregation. Although not fully elucidated, this process is known to rely on antagonistic activities of plus end- and minus end-directed motors. We developed a model of early meiotic spindle assembly in mouse oocytes, including key factors such as microtubule dynamics and chromosome movement. We explored how the balance between plus end- and minus end-directed motors, as well as the influence of microtubule nucleation, impacts spindle morphology. In a refined model, we added spatial regulation of microtubule stability and minus-end clustering. We could reproduce the features of early stages of spindle assembly from 12 different experimental perturbations and predict eight additional perturbations. With its ability to characterize and predict chromosome individualization, this model can help deepen our understanding of spindle assembly.

Proteomic Profiling of Microtubule Self-organization in M-phase

Microtubules (MTs) and associated proteins can self-organize into complex structures such as the bipolar spindle, a process in which RanGTP plays a major role. Addition of RanGTP to M-phase Xenopus egg extracts promotes the nucleation and self-organization of MTs into asters and bipolar-like structures in the absence of centrosomes or chromosomes. We show here that the complex proteome of these RanGTP-induced MT assemblies is similar to that of mitotic spindles. Using proteomic profiling we show that MT self-organization in the M-phase cytoplasm involves the non-linear and non-stoichiometric recruitment of proteins from specific functional groups. Our study provides for the first time a temporal understanding of the protein dynamics driving MT self-organization in M-phase.

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.

From the cell membrane to the nucleus: unearthing transport mechanisms for dynein

Mutations in the motor protein cytoplasmic dynein have been found to cause Charcot-Marie-Tooth disease, spinal muscular atrophy, and severe intellectual disabilities in humans. In mouse models, neurodegeneration is observed. We sought to develop a novel model which could incorporate the effects of mutations on distance travelled and velocity. A mechanical model for the dynein mediated transport of endosomes is derived from first principles and solved numerically. The effects of variations in model parameter values are analysed to find those that have a significant impact on velocity and distance travelled. The model successfully describes the processivity of dynein and matches qualitatively the velocity profiles observed in experiments.

Dynamical self-regulation in self-propelled particle flows

We study a continuum model of overdamped self-propelled particles with aligning interactions in two dimensions. Combining analytical theory and computations, we map out the phase diagram for the parameter space covered by the model. We find that the system self-organizes into two robust structures in different regions of parameter space: solitary waves composed of ordered swarms moving through a low density disordered background, and stationary radially symmetric asters. The self-regulating nature of the flow yields phase separation, ubiquitous in this class of systems, and controls the formation of solitary waves. Self-propulsion and the associated active convection play a crucial role in aster formation.

Biophysics of mitosis

Mitosis is the process by which eukaryotic cells organize and segregate their chromosomes in preparation for cell division. It is accomplished by a cellular machine composed largely of microtubules (MTs) and their associated proteins. This article reviews literature on mitosis from a biophysical point of view, drawing attention to the assembly and motility processes required to do this complex job with precision. Work from both the recent and the older literature is integrated into a description of relevant biological events and the experiments that probe their mechanisms. Theoretical work on specific subprocesses is also reviewed. Our goal is to provide a document that will expose biophysicists to the fascination of this quite amazing process and provide them with a good background from which they can pursue their own research interests in the subject.

The emergence of sarcomeric, graded-polarity and spindle-like patterns in bundles of short cytoskeletal polymers and two opposite molecular motors

We use linear stability analysis and numerical solutions of partial differential equations to investigate pattern formation in the one-dimensional system of short dynamic polymers and one (plus-end directed) or two (one is plus-end, another minus-end directed) molecular motors. If polymer sliding and motor gliding rates are slow and/or the polymer turnover rate is fast, then the polymer-motor bundle has mixed polarity and homogeneous motor distribution. However, if motor gliding is fast, a sarcomeric pattern with periodic bands of alternating polymer polarity separated by motor aggregates evolves. On the other hand, if polymer sliding is fast, a graded-polarity bundle with motors at the center emerges. In the presence of the second, minus-end directed motor, the sarcomeric pattern is more ubiquitous, while the graded-polarity pattern is destabilized. However, if the minus-end motor is weaker than the plus-end directed one, and/or polymer nucleation is autocatalytic, and/or long polymers are present in the bundle, then a spindle-like architecture with a sorted-out polarity emerges with the plus-end motors at the center and minus-end motors at the edges. We discuss modeling implications for actin-myosin fibers and in vitro and meiotic spindles.

50 ways to build a spindle: the complexity of microtubule generation during mitosis

The accurate segregation of duplicated chromosomes, essential for the development and viability of a eukaryotic organism, requires the formation of a robust microtubule (MT)-based spindle apparatus. Entry into mitosis or meiosis precipitates a cascade of signalling events which result in the activation of pathways responsible for a dramatic reorganisation of the MT cytoskeleton: through changes in the properties of MT-associated proteins, local concentrations of free tubulin dimer and through enhanced MT nucleation. The latter is generally thought to be driven by localisation and activation of γ-tubulin-containing complexes (γ-TuSC and γ-TuRC) at specific subcellular locations. For example, upon entering mitosis, animal cells concentrate γ-tubulin at centrosomes to tenfold the normal level during interphase, resulting in an aster-driven search and capture of chromosomes and bipolar mitotic spindle formation. Thus, in these cells, centrosomes have traditionally been perceived as the primary microtubule organising centre during spindle formation. However, studies in meiotic cells, plants and cell-free extracts have revealed the existence of complementary mechanisms of spindle formation, mitotic chromatin, kinetochores and nucleation from existing MTs or the cytoplasm can all contribute to a bipolar spindle apparatus. Here, we outline the individual known mechanisms responsible for spindle formation and formulate ideas regarding the relationship between them in assembling a functional spindle apparatus.

Mitotic membrane helps to focus and stabilize the mitotic spindle

During mitosis, microtubules (MTs), aided by motors and associated proteins, assemble into a mitotic spindle. Recent evidence supports the notion that a membranous spindle matrix aids spindle formation; however, the mechanisms by which the matrix may contribute to spindle assembly are unknown. To search for a mechanism by which the presence of a mitotic membrane might help spindle morphology, we built a computational model that explores the interactions between these components. We show that an elastic membrane around the mitotic apparatus helps to focus MT minus ends and provides a resistive force that acts antagonistically to plus-end-directed MT motors such as Eg5.

Centriole movements in mammalian epithelial cells during cytokinesis

Background

In cytokinesis, when the cleavage furrow has been formed, the two centrioles in each daughter cell separate. It has been suggested that the centrioles facilitate and regulate cytokinesis to some extent. It has been postulated that termination of cytokinesis (abscission) depends on the migration of a centriole to the intercellular bridge and then back to the cell center. To investigate the involvement of centrioles in cytokinesis, we monitored the movements of centrioles in three mammalian epithelial cell lines, HeLa, MCF 10A, and the p53-deficient mouse mammary tumor cell line KP-7.7, by time-lapse imaging. Centrin1-EGFP and α-Tubulin-mCherry were co-expressed in the cells to visualize respectively the centrioles and microtubules.

Results

Here we report that separated centrioles that migrate from the cell pole are very mobile during cytokinesis and their movements can be characterized as 1) along the nuclear envelope, 2) irregular, and 3) along microtubules forming the spindle axis. Centriole movement towards the intercellular bridge was only seen occasionally and was highly cell-line dependent.

Conclusions

These findings show that centrioles are highly mobile during cytokinesis and suggest that the repositioning of a centriole to the intercellular bridge is not essential for controlling abscission. We suggest that centriole movements are microtubule dependent and that abscission is more dependent on other mechanisms than positioning of centrioles.

Anastral spindle assembly: a mathematical model

Assembly of an anastral spindle was modeled as a two-stage process: first, the aggregation of microtubule foci or asters around the chromosomes, and second, the elongation of cross-linked microtubules and onset of bipolarity. Several possibilities involving diffusion and transport were investigated for the first stage, and the most feasible was found to be binding of the asters to cytoskeletal filaments and directed transport toward the chromosomes. For the second stage, a differential-equation model was formulated and solved numerically; it involves cross-linking of microtubules with those aligned with the spindle axis and between microtubules bound to different chromosomes, and sliding of microtubules along the spindle axis to elongate the spindle. Ncd was postulated to perform both functions. The model shows that spindle formation begins with rapid cross-linking of microtubules, followed by elongation, which continues until the population of microtubules aligned with the spindle axis is depleted and microtubules cross-linking different chromosomes dominate. It also shows that when sliding is inhibited, short bipolar spindles still form, and if clustering is enhanced, normal-length spindles can form, although requiring longer assembly time. These findings are consistent with spindle assembly in live wild-type and ncd mutant Drosophila oocytes.

Spatial regulation improves antiparallel microtubule overlap during mitotic spindle assembly

The mitotic spindle plays an essential role in chromosome segregation during cell division. Spindle formation and proper function require that microtubules with opposite polarity overlap and interact. Previous computational simulations have demonstrated that these antiparallel interactions could be created by complexes combining plus- and minus-end-directed motors. The resulting spindles, however, exhibit sparse antiparallel microtubule overlap with motor complexes linking only a nominal number of antiparallel microtubules. Here we investigate the role that spatial differences in the regulation of microtubule interactions can have on spindle morphology. We show that the spatial regulation of microtubule catastrophe parameters can lead to significantly better spindle morphology and spindles with greater antiparallel MT overlap. We also demonstrate that antiparallel microtubule overlap can be increased by having new microtubules nucleated along the length of existing astral microtubules, but this increase negatively affects spindle morphology. Finally, we show that limiting the diffusion of motor complexes within the spindle region increases antiparallel microtubule interaction.

Slide-and-Cluster Models for Spindle Assembly

BACKGROUND

Mitotic and meiotic spindles are assemblies of microtubules (MTs) that form during cell division to physically separate sister chromosomes. How the various components of spindles act together to establish and maintain the dynamic bipolar structure of spindles is not understood. Interactions between MTs and motors have been studied both experimentally and theoretically in many contexts, including the self-organization of arrays of MTs by motors and the competition between different classes of motors to move a single load. This work demonstrates how the interplay between two types of motors together with continual nucleation of MTs by chromosomes could organize the MTs into spindles.

RESULTS

We propose a slide-and-cluster model based on four known molecular activities: MT nucleation near chromosomes, the sliding of MTs by a plus-end-directed motor, the clustering of their minus ends by a minus-end-directed motor, and the loss of MTs by dynamic instability. Our model applies to overlapping, nonkinetochore MTs in anastral spindles, and perhaps also to interpolar MTs in astral spindles. We show mathematically that the slide-and-cluster mechanism robustly forms bipolar spindles with sharp poles and a stable steady-state length. This model accounts for several experimental observations that were difficult to explain with existing models. Three new predictions of the model were tested and verified in Xenopus egg extracts.

CONCLUSIONS

We show that a simple two-motor model could create stable, bipolar spindles under a wide range of physical parameters. Our model is the first self-contained model for anastral spindle assembly and MT sliding (known as poleward flux). Our experimental results support the slide-and-cluster scenario; most significantly, we find that MT sliding slows near spindle poles, confirming the model's primary prediction.

Modelling microtubule patterns

The cellular cytoskeleton is well studied in terms of its biological and physical properties, making it an attractive subject for systems approaches. Here, we describe the experimental and theoretical strategies used to study the collective behaviour of microtubules and motors. We illustrate how this led to the beginning of an understanding of dynamic cellular patterns that have precise functions.