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

Coordination, cooperation, competition, crowding and congestion of molecular motors: Theoretical models and computer simulations

Cytoskeletal motor proteins are biological nanomachines that convert chemical energy into mechanical work to carry out various functions such as cell division, cell motility, cargo transport, muscle contraction, beating of cilia and flagella, and ciliogenesis. Most of these processes are driven by the collective operation of several motors in the crowded viscous intracellular environment. Imaging and manipulation of the motors with powerful experimental probes have been complemented by mathematical analysis and computer simulations of the corresponding theoretical models. In this article, we illustrate some of the key theoretical approaches used to understand how coordination, cooperation and competition of multiple motors in the crowded intra-cellular environment drive the processes that are essential for biological function of a cell. In spite of the focus on theory, experimentalists will also find this article as an useful summary of the progress made so far in understanding multiple motor systems.

A model of microtubule depolymerization by kinesin-8 motor proteins

The dimeric kinesin-8 motors have the biological function of depolymerizing microtubules (MTs) from the plus end. However, the molecular mechanism of the depolymerization promoted by the kinesin-8 motors is still undetermined. Here, a model is proposed for the MT depolymerization by the kinesin-8 motors. Based on the model, the dynamics of depolymerization in the presence of the single motor at the MT plus end under no load and under load on the motor is studied theoretically. The dynamics of depolymerization in the presence of multiple motors at the MT plus end is also analyzed. The theoretical results explain well the available experimental data. The studies can also be applicable to other families of kinesin motors such as kinesin-13 mitotic centromere-associated kinesin motors that have the ability to depolymerize MTs.

A model for the catalytic activity of microtubule polymerases

A XMAP215/Stu2/Alp14 polymerase can catalyze processively the tubulin addition to the microtubule (MT) plus end. In this work, a model is proposed for the underlying molecular mechanism of the polymerase activity, where the polymerase can not only catalyze processively the tubulin addition to but also promote the tubulin removal from the MT plus end. Based on the model the dynamics of both the wild-type and mutant polymerases is studied theoretically, explaining consistently and well various available experimental data. To further test the model, predicted results are provided.

Multivalent interactions facilitate motor-dependent protein accumulation at growing microtubule plus-ends

Growing microtubule ends organize end-tracking proteins into comets of mixed composition. Here using a reconstituted fission yeast system consisting of end-binding protein Mal3, kinesin Tea2 and cargo Tip1, we found that these proteins can be driven into liquid-phase droplets both in solution and at microtubule ends under crowding conditions. In the absence of crowding agents, cryo-electron tomography revealed that motor-dependent comets consist of disordered networks where multivalent interactions may facilitate non-stoichiometric accumulation of cargo Tip1. We found that two disordered protein regions in Mal3 are required for the formation of droplets and motor-dependent accumulation of Tip1, while autonomous Mal3 comet formation requires only one of them. Using theoretical modelling, we explore possible mechanisms by which motor activity and multivalent interactions may lead to the observed enrichment of Tip1 at microtubule ends. We conclude that microtubule ends may act as platforms where multivalent interactions condense microtubule-associated proteins into large multi-protein complexes.

Length Regulation Drives Self-Organization in Filament-Motor Mixtures

Cytoskeletal networks form complex intracellular structures. Here we investigate a minimal model for filament-motor mixtures in which motors act as depolymerases and thereby regulate filament length. Combining agent-based simulations and hydrodynamic equations, we show that resource-limited length regulation drives the formation of filament clusters despite the absence of mechanical interactions between filaments. Even though the orientation of individual remains fixed, collective filament orientation emerges in the clusters, aligned orthogonal to their interfaces.

Motor guidance by long-range communication on the microtubule highway

Coupling of motor proteins within arrays drives muscle contraction, flagellar beating, chromosome segregation, and other biological processes. Current models of motor coupling invoke either direct mechanical linkage or protein crowding, which rely on short-range motor-motor interactions. In contrast, coupling mechanisms that act at longer length scales remain largely unexplored. Here we report that microtubules can physically couple motor movement in the absence of detectable short-range interactions. The human kinesin-4 Kif4A changes the run length and velocity of other motors on the same microtubule in the dilute binding limit, when approximately 10-nm-sized motors are much farther apart than the motor size. This effect does not depend on specific motor-motor interactions because similar changes in Kif4A motility are induced by kinesin-1 motors. A micrometer-scale attractive interaction potential between motors is sufficient to recreate the experimental results in a biophysical model. Unexpectedly, our theory suggests that long-range microtubule-mediated coupling affects not only binding kinetics but also motor mechanochemistry. Therefore, the model predicts that motors can sense and respond to motors bound several micrometers away on a microtubule. Our results are consistent with a paradigm in which long-range motor interactions along the microtubule enable additional forms of collective motor behavior, possibly due to changes in the microtubule lattice.

Length regulation of multiple flagella that self-assemble from a shared pool of components

The single-celled green algae Chlamydomonas reinhardtii with its two flagella-microtubule-based structures of equal and constant lengths-is the canonical model organism for studying size control of organelles. Experiments have identified motor-driven transport of tubulin to the flagella tips as a key component of their length control. Here we consider a class of models whose key assumption is that proteins responsible for the intraflagellar transport (IFT) of tubulin are present in limiting amounts. We show that the limiting-pool assumption is insufficient to describe the results of severing experiments, in which a flagellum is regenerated after it has been severed. Next, we consider an extension of the limiting-pool model that incorporates proteins that depolymerize microtubules. We show that this 'active disassembly' model of flagellar length control explains in quantitative detail the results of severing experiments and use it to make predictions that can be tested in experiments.

Cooperative motor action to regulate microtubule length dynamics

Motivated by the recent experimental observations on motor induced cooperative mechanism controlling the length dynamics of microtubules (MTs), we examine how plus-end-targeted proteins of the kinesin family regulate MT polymerization and depolymerization routines. Here, we study a stochastic mathematical model capturing the unusual form of collective motor interaction on MT dynamics originating due to the molecular traffic near the MT tip. We provide an extensive analysis of the joint effect of motor impelled MT polymerization and complete depolymerization. The effect of the cooperative action is included by modifying the intrinsic depolymerization rate. We analyze the model within the framework of continuum mean-field theory and the resultant steady-state analytic solution is expressed in terms of Lambert W functions. Four distinct steady-state phases including a shock phase have been reported. The significant features of the shock including its position and height have been analyzed. Theoretical outcomes are supported by extensive Monte Carlo simulations. To explore the system alterations between the regime of growth and shrinkage phase, we consider kymographs of the microtubule along with the length distributions. Finally, we investigated the dependence of MT length kinetics both on modifying factor of depolymerization rate and motor concentration. The overall extensive study reveals that the flux of molecular traffic at the microtubule plus end initiates a cooperative mechanism, resulting in significant change in MT growth and shrinkage regime as also observed experimentally.

Self-organized system-size oscillation of a stochastic lattice-gas model

The totally asymmetric simple exclusion process (TASEP) is a paradigmatic stochastic model for nonequilibrium physics, and has been successfully applied to describe active transport of molecular motors along cytoskeletal filaments. Building on this simple model, we consider a two-lane lattice-gas model that couples directed transport (TASEP) to diffusive motion in a semiclosed geometry, and simultaneously accounts for spontaneous growth and particle-induced shrinkage of the system's size. This particular extension of the TASEP is motivated by the question of how active transport and diffusion might influence length regulation in confined systems. Surprisingly, we find that the size of our intrinsically stochastic system exhibits robust temporal patterns over a broad range of growth rates. More specifically, when particle diffusion is slow relative to the shrinkage dynamics, we observe quasiperiodic changes in length. We provide an intuitive explanation for the occurrence of these self-organized temporal patterns, which is based on the imbalance between the diffusion and shrinkage speed in the confined geometry. Finally, we formulate an effective theory for the oscillatory regime, which explains the origin of the oscillations and correctly predicts the dependence of key quantities, such as the oscillation frequency, on the growth rate.

Limited Resources Induce Bistability in Microtubule Length Regulation

The availability of protein is an important factor for the determination of the size of the mitotic spindle. Involved in spindle-size regulation is kinesin-8, a molecular motor and microtubule (MT) depolymerase, which is known to tightly control MT length. Here, we propose and analyze a theoretical model in which kinesin-induced MT depolymerization competes with spontaneous polymerization while supplies of both tubulin and kinesin are limited. In contrast to previous studies where resources were unconstrained, we find that, for a wide range of concentrations, MT length regulation is bistable. We test our predictions by conducting in vitro experiments and find that the bistable behavior manifests in a bimodal MT length distribution.

Dynamics of microtubules: highlights of recent computational and experimental investigations

Microtubules are found in most eukaryotic cells, with homologs in eubacteria and archea, and they have functional roles in mitosis, cell motility, intracellular transport, and the maintenance of cell shape. Numerous efforts have been expended over the last two decades to characterize the interactions between microtubules and the wide variety of microtubule associated proteins that control their dynamic behavior in cells resulting in microtubules being assembled and disassembled where and when they are required by the cell. We present the main findings regarding microtubule polymerization and depolymerization and review recent work about the molecular motors that modulate microtubule dynamics by inducing either microtubule depolymerization or severing. We also discuss the main experimental and computational approaches used to quantify the thermodynamics and mechanics of microtubule filaments.

Motor Protein Accumulation on Antiparallel Microtubule Overlaps

Biopolymers serve as one-dimensional tracks on which motor proteins move to perform their biological roles. Motor protein phenomena have inspired theoretical models of one-dimensional transport, crowding, and jamming. Experiments studying the motion of Xklp1 motors on reconstituted antiparallel microtubule overlaps demonstrated that motors recruited to the overlap walk toward the plus end of individual microtubules and frequently switch between filaments. We study a model of this system that couples the totally asymmetric simple exclusion process for motor motion with switches between antiparallel filaments and binding kinetics. We determine steady-state motor density profiles for fixed-length overlaps using exact and approximate solutions of the continuum differential equations and compare to kinetic Monte Carlo simulations. Overlap motor density profiles and motor trajectories resemble experimental measurements. The phase diagram of the model is similar to the single-filament case for low switching rate, while for high switching rate we find a new (to our knowledge) low density-high density-low density-high density phase. The overlap center region, far from the overlap ends, has a constant motor density as one would naïvely expect. However, rather than following a simple binding equilibrium, the center motor density depends on total overlap length, motor speed, and motor switching rate. The size of the crowded boundary layer near the overlap ends is also dependent on the overlap length and switching rate in addition to the motor speed and bulk concentration. The antiparallel microtubule overlap geometry may offer a previously unrecognized mechanism for biological regulation of protein concentration and consequent activity.

Generic Transport Mechanisms for Molecular Traffic in Cellular Protrusions

Transport of molecular motors along protein filaments in a half-closed geometry is a common feature of biologically relevant processes in cellular protrusions. Using a lattice-gas model we study how the interplay between active and diffusive transport and mass conservation leads to localized domain walls and tip localization of the motors. We identify a mechanism for task sharing between the active motors (maintaining a gradient) and the diffusive motion (transport to the tip), which ensures that energy consumption is low and motor exchange mostly happens at the tip. These features are attributed to strong nearest-neighbor correlations that lead to a strong reduction of active currents, which we calculate analytically using an exact moment identity, and might prove useful for the understanding of correlations and active transport also in more elaborate systems.

Altered nuclear dynamics in MDX myofibers

Duchenne muscular dystrophy (DMD) is a genetic disorder in which the absence of dystrophin leads to progressive muscle degeneration and weakness. Although the genetic basis is known, the pathophysiology of dystrophic skeletal muscle remains unclear. We examined nuclear movement in wild-type (WT) and muscular dystrophy mouse model for DMD (MDX) (dystrophin-null) mouse myofibers. We also examined expression of proteins in the linkers of nucleoskeleton and cytoskeleton (LINC) complex, as well as nuclear transcriptional activity via histone H3 acetylation and polyadenylate-binding nuclear protein-1. Because movement of nuclei is not only LINC dependent but also microtubule dependent, we analyzed microtubule density and organization in WT and MDX myofibers, including the application of a unique 3D tool to assess microtubule core structure. Nuclei in MDX myofibers were more mobile than in WT myofibers for both distance traveled and velocity. MDX muscle shows reduced expression and labeling intensity of nesprin-1, a LINC protein that attaches the nucleus to the microtubule and actin cytoskeleton. MDX nuclei also showed altered transcriptional activity. Previous studies established that microtubule structure at the cortex is disrupted in MDX myofibers; our analyses extend these findings by showing that microtubule structure in the core is also disrupted. In addition, we studied malformed MDX myofibers to better understand the role of altered myofiber morphology vs. microtubule architecture in the underlying susceptibility to injury seen in dystrophic muscles. We incorporated morphological and microtubule architectural concepts into a simplified finite element mathematical model of myofiber mechanics, which suggests a greater contribution of myofiber morphology than microtubule structure to muscle biomechanical performance.NEW & NOTEWORTHY Microtubules provide the means for nuclear movement but show altered organization in the muscular dystrophy mouse model (MDX) (dystrophin-null) muscle. Here, MDX myofibers show increased nuclear movement, altered transcriptional activity, and altered linkers of nucleoskeleton and cytoskeleton complex expression compared with healthy myofibers. Microtubule architecture was incorporated in finite element modeling of passive stretch, revealing a role of fiber malformation, commonly found in MDX muscle. The results suggest that alterations in microtubule architecture in MDX muscle affect nuclear movement, which is essential for muscle function.

Phase-plane analysis of the totally asymmetric simple exclusion process with binding kinetics and switching between antiparallel lanes

Motor protein motion on biopolymers can be described by models related to the totally asymmetric simple exclusion process (TASEP). Inspired by experiments on the motion of kinesin-4 motors on antiparallel microtubule overlaps, we analyze a model incorporating the TASEP on two antiparallel lanes with binding kinetics and lane switching. We determine the steady-state motor density profiles using phase-plane analysis of the steady-state mean field equations and kinetic Monte Carlo simulations. We focus on the density-density phase plane, where we find an analytic solution to the mean field model. By studying the phase-space flows, we determine the model's fixed points and their changes with parameters. Phases previously identified for the single-lane model occur for low switching rate between lanes. We predict a multiple coexistence phase due to additional fixed points that appear as the switching rate increases: switching moves motors from the higher-density to the lower-density lane, causing local jamming and creating multiple domain walls. We determine the phase diagram of the model for both symmetric and general boundary conditions.

Design Principles of Length Control of Cytoskeletal Structures

Cells contain elaborate and interconnected networks of protein polymers, which make up the cytoskeleton. The cytoskeleton governs the internal positioning and movement of vesicles and organelles and controls dynamic changes in cell polarity, shape, and movement. Many of these processes require tight control of the size and shape of cytoskeletal structures, which is achieved despite rapid turnover of their molecular components. Here we review mechanisms by which cells control the size of filamentous cytoskeletal structures, from the point of view of simple quantitative models that take into account stochastic dynamics of their assembly and disassembly. Significantly, these models make experimentally testable predictions that distinguish different mechanisms of length control. Although the primary focus of this review is on cytoskeletal structures, we believe that the broader principles and mechanisms discussed herein will apply to a range of other subcellular structures whose sizes are tightly controlled and are linked to their functions.