Theory of self-assembly of microtubules and motors

Mean-field theory approach to three-dimensional nematic phase transitions in microtubules

Microtubules are dynamic intracellular fibers that have been observed experimentally to undergo spontaneous self-alignment. We formulate a three-dimensional (3D) mean-field theory model to analyze the nematic phase transition of microtubules growing and interacting within a 3D space, then make a comparison with computational simulations. We identify a control parameter G_{eff} and predict a unique critical value G_{eff}=1.56 for which a phase transition can occur. Furthermore, we show both analytically and using simulations that this predicted critical value does not depend on the presence of zippering. The mean-field theory developed here provides an analytical estimate of microtubule patterning characteristics without running time-consuming simulations and is a step towards bridging scales from microtubule behavior to multicellular simulations.

Motor processivity and speed determine structure and dynamics of microtubule-motor assemblies

Active matter systems can generate highly ordered structures, avoiding equilibrium through the consumption of energy by individual constituents. How the microscopic parameters that characterize the active agents are translated to the observed mesoscopic properties of the assembly has remained an open question. These active systems are prevalent in living matter; for example, in cells, the cytoskeleton is organized into structures such as the mitotic spindle through the coordinated activity of many motor proteins walking along microtubules. Here, we investigate how the microscopic motor-microtubule interactions affect the coherent structures formed in a reconstituted motor-microtubule system. This question is of deeper evolutionary significance as we suspect motor and microtubule type contribute to the shape and size of resulting structures. We explore key parameters experimentally and theoretically, using a variety of motors with different speeds, processivities, and directionalities. We demonstrate that aster size depends on the motor used to create the aster, and develop a model for the distribution of motors and microtubules in steady-state asters that depends on parameters related to motor speed and processivity. Further, we show that network contraction rates scale linearly with the single-motor speed in quasi-one-dimensional contraction experiments. In all, this theoretical and experimental work helps elucidate how microscopic motor properties are translated to the much larger scale of collective motor-microtubule assemblies.

Homophily-Based Social Group Formation in a Spin Glass Self-Assembly Framework

Homophily, the tendency of humans to attract each other when sharing similar features, traits, or opinions, has been identified as one of the main driving forces behind the formation of structured societies. Here we ask to what extent homophily can explain the formation of social groups, particularly their size distribution. We propose a spin-glass-inspired framework of self-assembly, where opinions are represented as multidimensional spins that dynamically self-assemble into groups; individuals within a group tend to share similar opinions (intragroup homophily), and opinions between individuals belonging to different groups tend to be different (intergroup heterophily). We compute the associated nontrivial phase diagram by solving a self-consistency equation for "magnetization" (combined average opinion). Below a critical temperature, there exist two stable phases: one ordered with nonzero magnetization and large clusters, the other disordered with zero magnetization and no clusters. The system exhibits a first-order transition to the disordered phase. We analytically derive the group-size distribution that successfully matches empirical group-size distributions from online communities.

Metastability of Constant-Density Flocks

We study numerically the Toner-Tu field theory where the density field is maintained constant, a limit case of "Malthusian" flocks for which the asymptotic scaling of correlation functions in the ordered phase is known exactly. While we confirm these scaling laws, we also show that such constant-density flocks are metastable to the nucleation of a specific defect configuration, and are replaced by a globally disordered phase consisting of asters surrounded by shock lines that constantly evolves and remodels itself. We demonstrate that the main source of disorder lies along shock lines, rendering this active foam fundamentally different from the corresponding equilibrium system. We thus show that in the context of active matter also, a result obtained at all orders of perturbation theory can be superseded by nonperturbative effects, calling for a different approach.

Defect dynamics in active polar fluids vs. active nematics

Topological defects play a key role in two-dimensional active nematics, and a transient role in two-dimensional active polar fluids. Using a variational method, we study both the transient and long-time behavior of defects in two-dimensional active polar fluids in the limit of strong order and overdamped, compressible flow, and compare the defect dynamics with the corresponding active nematics model studied recently. One result is non-central interactions between defect pairs for active polar fluids, and by extending our analysis to allow orientation dynamics of defects, we find that the orientation of +1 defects, unlike that of ±1/2 defects in active nematics, is not locked to defect positions and relaxes to asters. Moreover, using a scaling argument, we explain the transient feature of active polar defects and show that in the steady state, active polar fluids are either devoid of defects or consist of a single aster. We argue that for contractile (extensile) active nematic systems, +1 vortices (asters) should emerge as bound states of a pair of +1/2 defects, which has been recently observed. Moreover, unlike the polar case, we show that for active nematics, a linear chain of equally spaced bound states of pairs of +1/2 defects can screen the activity term. A common feature in both models is the appearance of +1 defects (elementary in polar and composite in nematic) in the steady state.

Family of Asymptotic Solutions to the Two-Dimensional Kinetic Equation with a Nonlocal Cubic Nonlinearity

We apply the original semiclassical approach to the kinetic ionization equation with the nonlocal cubic nonlinearity in order to construct the family of its asymptotic solutions. The approach proposed relies on an auxiliary dynamical system of moments of the desired solution to the kinetic equation and the associated linear partial differential equation. The family of asymptotic solutions to the kinetic equation is constructed using the symmetry operators acting on functions concentrated in a neighborhood of a point determined by the dynamical system. Based on these solutions, we introduce the nonlinear superposition principle for the nonlinear kinetic equation. Our formalism based on the Maslov germ method is applied to the Cauchy problem for the specific two-dimensional kinetic equation. The evolution of the ion distribution in the kinetically enhanced metal vapor active medium is obtained as the nonlinear superposition using the numerical–analytical calculations.

Emergence and dynamics of unconfined self-organised vortices in active magnetic roller liquids

Actively driven colloids demonstrate complex out-of-equilibrium dynamics often rivaling self-organized patterns and collective behavior observed in living systems. Recent studies revealed the emergence of steady macroscopic states with multiple interacting vortices in an unconfined environment that emerge from the coupling between microscale particle rotation and translation. Yet, insights into the microscopic behavior during the vortex emergence, growth, and formation of a multi-vortical state remain lacking. Here, we investigate in experiments and simulations how the microscale magnetic roller behavior leads to the emergence of seed vortices, their aggregation or annihilation, and the formation of stable large-scale vortical structures. We reveal that the coupling of roller-induced hydrodynamic flows guides the local self-densifications and self-organization of the micro-rollers into seed vortices. The resulting multi-vortical state is sensitive to the external magnetic field amplitude and allows tuning the rollers' number density in a vortex and its characteristic size.

Thermodynamics of structure-forming systems

Structure-forming systems are ubiquitous in nature, ranging from atoms building molecules to self-assembly of colloidal amphibolic particles. The understanding of the underlying thermodynamics of such systems remains an important problem. Here, we derive the entropy for structure-forming systems that differs from Boltzmann-Gibbs entropy by a term that explicitly captures clustered states. For large systems and low concentrations the approach is equivalent to the grand-canonical ensemble; for small systems we find significant deviations. We derive the detailed fluctuation theorem and Crooks' work fluctuation theorem for structure-forming systems. The connection to the theory of particle self-assembly is discussed. We apply the results to several physical systems. We present the phase diagram for patchy particles described by the Kern-Frenkel potential. We show that the Curie-Weiss model with molecule structures exhibits a first-order phase transition.

Pattern formation in active model C with anchoring: bands, aster networks, and foams

We study the dynamics of pattern formation in a minimal model for active mixtures made of microtubules and molecular motors. We monitor the evolution of the (conserved) microtubule density and of the (non-conserved) nematic order parameter, focusing on the effects of an "anchoring" term that provides a direct coupling between the preferred microtubule direction and their density gradient. The key control parameter is the ratio between activity and elasticity. When elasticity dominates, the interplay between activity and anchoring leads to formation of banded structures that can undergo additional bending, rotational or splaying instabilities. When activity dominates, the nature of anchoring instead gives rise to a range of active cellular solids, including aster-like networks, disordered foams and spindle-like patterns. We speculate that the introduced "active model C" with anchoring is a minimal model to describe pattern formation in a biomimetic analogue of the microtubule cytoskeleton.

Emergence of self-organized multivortex states in flocks of active rollers

Active matter, both synthetic and biological, demonstrates complex spatiotemporal self-organization and the emergence of collective behavior. A coherent rotational motion, the vortex phase, is of great interest because of its ability to orchestrate well-organized motion of self-propelled particles over large distances. However, its generation without geometrical confinement has been a challenge. Here, we show by experiments and computational modeling that concentrated magnetic rollers self-organize into multivortex states in an unconfined environment. We find that the neighboring vortices more likely occur with the opposite sense of rotation. Our studies provide insights into the mechanism for the emergence of coherent collective motion on the macroscale from the coupling between microscale rotation and translation of individual active elements. These results may stimulate design strategies for self-assembled dynamic materials and microrobotics.

Defect dynamics in clusters of self-propelled rods in circular confinement

Rod-shaped active micro/nano-particles, such as bacterial and bipolar metallic micro/nano-motors, demonstrate novel collective phenomena far from the equilibrium state compared to passive particles. We apply a simulation approach --dissipative particle dynamics (DPD)-- to explore the collectively ordered states of self-propelled rods (SPRs). The SPRs are confined in a finite circular zone and repel each other when two rods touch each other. It is found that for a long enough rods system, the global vortex patterns, dynamic pattern oscillation between hedgehog pattern and vortex pattern, and hedgehog patterns are observed successively with increasing active force F_a. For the vortex pattern, the total interaction energy between the rods U is linear with active force F_a, i.e., U ∼ F_a . While the relation U ∼ F_a² is obtained for the hedgehog structure. It is observed that a new hedgehog pattern with one defect core is created by two ejections of polar cluster in opposite directions from the original hedgehog pattern, and then merges into one through the diffusion of the two aggregates, i.e., the creation and annihilation of topological charges.

Dry active turbulence in a model for microtubule-motor mixtures

We study the dynamics and phase behaviour of a dry suspension of microtubules and molecular motors. We obtain a set of continuum equations by rigorously coarse graining a microscopic model where motor-induced interactions lead to parallel or antiparallel ordering. Through numerical simulations, we show that this model generically creates either stable stripes, or a never-settling pattern where stripes periodically form, rotate and then split up. We derive a minimal model which displays the same instability as the full model, and clarifies the underlying physical mechanism. The necessary ingredients are an extensile flux arising from microtubule sliding and an interfacial torque favouring ordering along density gradients. We argue that our minimal model unifies various previous observations of chaotic behaviour in dry active matter into a general universality class.

From isolated structures to continuous networks: A categorization of cytoskeleton-based motile engineered biological microstructures

As technology at the small scale is advancing, motile engineered microstructures are becoming useful in drug delivery, biomedicine, and lab-on-a-chip devices. However, traditional engineering methods and materials can be inefficient or functionally inadequate for small-scale applications. Increasingly, researchers are turning to the biology of the cytoskeleton, including microtubules, actin filaments, kinesins, dyneins, myosins, and associated proteins, for both inspiration and solutions. They are engineering structures with components that range from being entirely biological to being entirely synthetic mimics of biology and on scales that range from isotropic continuous networks to single isolated structures. Motile biological microstructures trace their origins from the development of assays used to study the cytoskeleton to the array of structures currently available today. We define 12 types of motile biological microstructures, based on four categories: entirely biological, modular, hybrid, and synthetic, and three scales: networks, clusters, and isolated structures. We highlight some key examples, the unique functionalities, and the potential applications of each microstructure type, and we summarize the quantitative models that enable engineering them. By categorizing the diversity of motile biological microstructures in this way, we aim to establish a framework to classify these structures, define the gaps in current research, and spur ideas to fill those gaps. This article is categorized under: Nanotechnology Approaches to Biology > Nanoscale Systems in Biology Nanotechnology Approaches to Biology > Cells at the Nanoscale Biology-Inspired Nanomaterials > Protein and Virus-Based Structures Therapeutic Approaches and Drug Discovery > Emerging Technologies.

Chronology of motor-mediated microtubule streaming

We introduce a filament-based simulation model for coarse-grained, effective motor-mediated interaction between microtubule pairs to study the time-scales that compose cytoplasmic streaming. We characterise microtubule dynamics in two-dimensional systems by chronologically arranging five distinct processes of varying duration that make up streaming, from microtubule pairs to collective dynamics. The structures found were polarity sorted due to the propulsion of antialigned microtubules. This also gave rise to the formation of large polar-aligned domains, and streaming at the domain boundaries. Correlation functions, mean squared displacements, and velocity distributions reveal a cascade of processes ultimately leading to microtubule streaming and advection, spanning multiple microtubule lengths. The characteristic times for the processes extend over three orders of magnitude from fast single-microtubule processes to slow collective processes. Our approach can be used to directly test the importance of molecular components, such as motors and crosslinking proteins between microtubules, on the collective dynamics at cellular scale.

Kinetic theory of pattern formation in mixtures of microtubules and molecular motors

In this study we formulate a theoretical approach, based on a Boltzmann-like kinetic equation, to describe pattern formation in two-dimensional mixtures of microtubular filaments and molecular motors. Following the previous work by Aranson and Tsimring [Phys. Rev. E 74, 031915 (2006)PLEEE81539-375510.1103/PhysRevE.74.031915] we model the motor-induced reorientation of microtubules as collision rules, and devise a semianalytical method to calculate the corresponding interaction integrals. This procedure yields an infinite hierarchy of kinetic equations that we terminate by employing a well-established closure strategy, developed in the pattern-formation community and based on a power-counting argument. We thus arrive at a closed set of coupled equations for slowly varying local density and orientation of the microtubules, and study its behavior by performing a linear stability analysis and direct numerical simulations. By comparing our method with the work of Aranson and Tsimring, we assess the validity of the assumptions required to derive their and our theories. We demonstrate that our approximation-free evaluation of the interaction integrals and our choice of a systematic closure strategy result in a rather different dynamical behavior than was previously reported. Based on our theory, we discuss the ensuing phase diagram and the patterns observed.

State-space reduction and equivalence class sampling for a molecular self-assembly model

Direct simulation of a model with a large state space will generate enormous volumes of data, much of which is not relevant to the questions under study. In this paper, we consider a molecular self-assembly model as a typical example of a large state-space model, and present a method for selectively retrieving 'target information' from this model. This method partitions the state space into equivalence classes, as identified by an appropriate equivalence relation. The set of equivalence classes H, which serves as a reduced state space, contains none of the superfluous information of the original model. After construction and characterization of a Markov chain with state space H, the target information is efficiently retrieved via Markov chain Monte Carlo sampling. This approach represents a new breed of simulation techniques which are highly optimized for studying molecular self-assembly and, moreover, serves as a valuable guideline for analysis of other large state-space models.

Collective dynamics of active filament complexes

Networks of biofilaments are essential for the formation of cellular structures that support various biological functions. For the most part, previous studies have investigated the collective dynamics of rodlike biofilaments; however, the shapes of the actual subcellular components are often more elaborate. In this study, we considered an active object composed of two active filaments, which represents the progression from rodlike biofilaments to complex-shaped biofilaments. Specifically, we numerically assessed the collective behaviors of these active objects in two dimensions and observed several types of dynamics, depending on the density and the angle of the two filaments as shape parameters of the object. Among the observed collective dynamics, a moving density band that we named a "moving smectic" is introduced here for the first time. By analyzing the trajectories of individual objects and the interactions among them, this study demonstrated how interactions among active biofilaments with complex shapes could produce collective dynamics in a nontrivial manner.

Nonequilibrium phase transitions, fluctuations and correlations in an active contractile polar fluid

We study the patterning, fluctuations and correlations of an active polar fluid consisting of contractile polar filaments on a two-dimensional substrate, using a hydrodynamic description. The steady states generically consist of arrays of inward pointing asters and show a continuous transition from a moving lamellar phase, a moving aster street, to a stationary aster lattice with no net polar order. We next study the effect of spatio-temporal athermal noise, parametrized by an active temperature TA, on the stability of the ordered phases. In contrast to its equilibrium counterpart, we find that the active crystal shows true long range order at low TA. On increasing TA, the asters dynamically remodel, concomitantly we find novel phase transitions characterized by bond-orientational and polar order upon "heating".

Self-propelled rods exhibit a phase-separated state characterized by the presence of active stresses and the ejection of polar clusters

We study collections of self-propelled rods (SPR) moving in two dimensions for packing fractions less than or equal to 0.3. We find that in the thermodynamical limit the SPR undergo a phase transition between a disordered gas and a novel phase-separated system state. Interestingly, (global) orientational order patterns-contrary to what has been suggested-vanish in this limit. In the found novel state, the SPR self-organize into a highly dynamical, high-density, compact region-which we call aggregate-which is surrounded by a disordered gas. Active stresses build inside aggregates as a result of the combined effect of local orientational order and active forces. This leads to the most distinctive feature of these aggregates: constant ejection of polar clusters of SPR. This novel phase-separated state represents a novel state of matter characterized by large fluctuations in volume and shape, related to mass ejection, and exhibits positional as well as orientational local order. SPR systems display new physics unseen in other active matter systems.

Collective motion of self-propelled particles with memory

We show that memory, in the form of underdamped angular dynamics, is a crucial ingredient for the collective properties of self-propelled particles. Using Vicsek-style models with an Ornstein-Uhlenbeck process acting on angular velocity, we uncover a rich variety of collective phases not observed in usual overdamped systems, including vortex lattices and active foams. In a model with strictly nematic interactions the smectic arrangement of Vicsek waves giving rise to global polar order is observed. We also provide a calculation of the effective interaction between vortices in the case where a telegraphic noise process is at play, explaining thus the emergence and structure of the vortex lattices observed here and in motility assay experiments.

Large-scale chaos and fluctuations in active nematics

We show that dry active nematics, e.g., collections of shaken elongated granular particles, exhibit large-scale spatiotemporal chaos made of interacting dense, ordered, bandlike structures in a parameter region including the linear onset of nematic order. These results are obtained from the study of both the well-known (deterministic) hydrodynamic equations describing these systems and of the self-propelled particle model they were derived from. We prove, in particular, that the chaos stems from the generic instability of the band solution of the hydrodynamic equations. Revisiting the status of the strong fluctuations and long-range correlations in the particle model, we show that the giant number fluctuations observed in the chaotic phase are a trivial consequence of density segregation. However anomalous, curvature-driven number fluctuations are present in the homogeneous quasiordered nematic phase and characterized by a nontrivial scaling exponent.

Symmetry breaking of nematic umbilical defects through an amplitude equation

The existence, stability properties, and bifurcation diagram of the nematic umbilical defects is studied. Close to the Fréedericksz transition of nematic liquid crystals with negative anisotropic dielectric constant and homeotropic anchoring, an anisotropic Ginzburg-Landau equation for the amplitude of the tilt of the director away from the vertical axis is derived by taking the three-dimensional (3D) to 2D limit of the Frank-Oseen model. The anisotropic Ginzburg-Landau equation allows us to reveal the mechanism of symmetry breaking of nematic umbilical defects. The positive defect is fully characterized as a function of the anisotropy, while the negative defect is characterized perturbatively. Numerical simulations show quite good agreement with the analytical results.

Casimir effect in swimmer suspensions

We show that the Casimir effect can emerge in microswimmer suspensions. In principle, two effects conspire against the development of Casimir effects in swimmer suspensions. First, at low Reynolds number, the force on any closed volume vanishes, but here the relevant effect is the drag by the flow produced by the swimmers, which can be finite. Second, the fluid velocity and the pressure are linear on the swimmer force dipoles, and averaging over the swimmer orientations would lead to a vanishing effect. However, being that the suspension is a discrete system, the noise terms of the coarse-grained equations depend on the density, which itself fluctuates, resulting in effective nonlinear dynamics. Applying the tools developed for other nonequilibrium systems to general coarse-grained equations for swimmer suspensions, the Casimir drag is computed on immersed objects, and it is found to depend on the correlation function between the rescaled density and dipolar density fields. By introducing a model correlation function with medium-range order, explicit expressions are obtained for the Casimir drag on a body. When the correlation length is much larger than the microscopic cutoff, the average drag is independent of the correlation length, with a range that depends only on the size of the immersed bodies.

Brownian motion of tethered nanowires

Brownian motion of slender particles near a boundary is ubiquitous in biological systems and in nanomaterial assembly, but the complex hydrodynamic interaction in those systems is still poorly understood. Here, we report experimental and computational studies of the Brownian motion of silicon nanowires tethered on a substrate. An optical interference method enabled direct observation of microscopic rotations of the slender bodies in three dimensions with high angular and temporal resolutions. This quantitative observation revealed anisotropic and angle-dependent hydrodynamic wall effects: rotational diffusivity in inclined and azimuth directions follows different power laws as a function of the length, ∼ L(-2.5) and ∼ L(-3), respectively, and is more hindered for smaller inclined angles. In parallel, we developed an implicit simulation technique that takes the complex wire-wall hydrodynamic interactions into account efficiently, the result of which agreed well with the experimentally observed angle-dependent diffusion. The demonstrated techniques provide a platform for studying the microrheology of soft condensed matters, such as colloidal and biological systems near interfaces, and exploring the optimal self-assembly conditions of nanostructures.

Active polar fluid flow in finite droplets

We present a continuum level analytical model of a droplet of active contractile fluid consisting of filaments and motors. We calculate the steady state flows that result from a splayed polarisation of the filaments. We account for interaction with the external medium by imposing a viscous friction at the fixed droplet boundary. We then show that the droplet has non-zero force dipole and quadrupole moments, the latter of which is essential for self-propelled motion of the droplet at low Reynolds' number. Therefore, this calculation describes a simple mechanism for the motility of a droplet of active contractile fluid embedded in a three-dimensional environment, which is relevant to cell migration in confinement (for example, embedded within a gel or tissue). Our analytical results predict how the system depends on various parameters such as the effective friction coefficient, the phenomenological activity parameter and the splay of the imposed polarisation.

Directed self-assembly of 1D microtubule nano-arrays

The intrinsic properties of microtubule filaments (e.g., biological nano-rods) direct their self-assembly into one-dimensional nano-arrays over extended timescales.

Spontaneous motion and deformation of a self-propelled droplet

The time evolution equation of motion and shape are derived for a self-propelled droplet driven by a chemical reaction. The coupling between the chemical reaction and motion makes an inhomogeneous concentration distribution as well as a surrounding flow leading to the instability of a stationary state. The instability results in spontaneous motion by which the shape of the droplet deforms from a sphere. We found that the self-propelled droplet is elongated perpendicular to the direction of motion and is characterized as a pusher.

Continuum modeling of clustering of myxobacteria

In this paper we develop a continuum theory of clustering in ensembles of self-propelled inelastically colliding rods with applications to collective dynamics of common gliding bacteria Myxococcus Xanthus. A multiphase hydrodynamic model that couples densities of oriented and isotropic phases is described. This model is used for the analysis of an instability that leads to spontaneous formation of directionally moving dense clusters within initially dilute isotropic "gas" of myxobacteria. Numerical simulations of this model confirm the existence of stationary dense moving clusters and also elucidate the properties of their collisions. The results are shown to be in a qualitative agreement with experiments.

Nonlinear field equations for aligning self-propelled rods

We derive a set of minimal and well-behaved nonlinear field equations describing the collective properties of self-propelled rods from a simple microscopic starting point, the Vicsek model with nematic alignment. Analysis of their linear and nonlinear dynamics shows good agreement with the original microscopic model. In particular, we derive an explicit expression for density-segregated, banded solutions, allowing us to develop a more complete analytic picture of the problem at the nonlinear level.

Continuous theory of active matter systems with metric-free interactions

We derive a hydrodynamic description of metric-free active matter: starting from self-propelled particles aligning with neighbors defined by "topological" rules, not metric zones-a situation advocated recently to be relevant for bird flocks, fish schools, and crowds-we use a kinetic approach to obtain well-controlled nonlinear field equations. We show that the density-independent collision rate per particle characteristic of topological interactions suppresses the linear instability of the homogeneous ordered phase and the nonlinear density segregation generically present near threshold in metric models, in agreement with microscopic simulations.

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.

Model for the orientational ordering of the plant microtubule cortical array

The plant microtubule cortical array is a striking feature of all growing plant cells. It consists of a more or less homogeneously distributed array of highly aligned microtubules connected to the inner side of the plasma membrane and oriented transversely to the cell growth axis. Here, we formulate a continuum model to describe the origin of orientational order in such confined arrays of dynamical microtubules. The model is based on recent experimental observations that show that a growing cortical microtubule can interact through angle dependent collisions with pre-existing microtubules that can lead either to co-alignment of the growth, retraction through catastrophe induction or crossing over the encountered microtubule. We identify a single control parameter, which is fully determined by the nucleation rate and intrinsic dynamics of individual microtubules. We solve the model analytically in the stationary isotropic phase, discuss the limits of stability of this isotropic phase, and explicitly solve for the ordered stationary states in a simplified version of the model.

Survival of the aligned: ordering of the plant cortical microtubule array

The cortical array is a structure consisting of highly aligned microtubules which plays a crucial role in the characteristic uniaxial expansion of all growing plant cells. Recent experiments have shown polymerization-driven collisions between the membrane-bound cortical microtubules, suggesting a possible mechanism for their alignment. We present both a coarse-grained theoretical model and stochastic particle-based simulations of this mechanism, and we compare the results from these complementary approaches. Our results indicate that collisions that induce depolymerization are sufficient to generate the alignment of microtubules in the cortical array.

Coordination and collective properties of molecular motors: theory

Many cellular processes require molecular motors to produce motion and forces. Single molecule experiments have led to a precise description of how a motor works. Under most physiological conditions, however, molecular motors operate in groups. Interactions between motors yield collective behaviors that cannot be explained only from single molecule properties. The aim of this paper is to review the various theoretical descriptions that explain the emergence of collective effects in molecular motor assemblies. These include bidirectional motion, hysteretic behavior, spontaneous oscillations, and self-organization into dynamical structures. We discuss motors acting on the cytoskeleton both in a prescribed geometry such as in muscles or flagella and in the cytoplasm.

Collective alignment of polar filaments by molecular motors

We study the alignment of polar biofilaments, such as microtubules and actin, subject to the action of multiple molecular motors attached simultaneously to more than one filament. Focusing on a paradigm model of only two filaments interacting with multiple motors, we were able to investigate in detail the alignment dynamics. While almost no alignment occurs in the case of a single motor, the filaments become rapidly aligned due to the collective action of the motors. Our analysis shows that the alignment time is governed by the number of bound motors and the magnitude of the motors' stepping fluctuations. We predict that the time scale of alignment is in the order of seconds, much faster than that reported for passive crosslink-induced bundling. In vitro experiments on the alignment of microtubules by multiple-motor covered beads are in qualitative agreement. We also discuss another mode of fast alignment of filaments, namely the cooperation between motors and passive crosslinks.

Motor-mediated alignment of microtubules in semidilute mixtures

We propose and study a model of molecular motor-induced ordering in a cytoskeletal filament solution for the semidilute case. Motors attach to a pair of filaments and walk along the pair bringing them into closer alignment. In the semidilute regime multiple motors can bind a filament to several others and, for a critical motor density, induce a transition to an ordered phase with a nonzero mean orientation. The motors, on the one hand, cause closer filament alignment, and, on the other hand, induce fluctuations that are dependent on the relative orientation of the filaments to which the motors are attached. We develop a spatially homogenous, mean-field theory that explicitly accounts for a force-dependent detachment rate of motors, which in turn affects the mean and the fluctuations of the net force acting on a filament. This model considers each filament to be in motor contact with all other filaments in the solution. We show that the transition to the oriented state changes from second order to first order when the force-dependent detachment becomes important.

Instabilities in a two-dimensional polar-filament--motor system

The dynamical interaction between filaments and motor proteins is known for their propensity to self-organize into spatio-temporal patterns. Since the filaments are polar in the sense that motors define a direction of motion on them, the system can display a spatially homogeneous polar-filament orientation. We show that the latter anisotropic state itself may become unstable with respect to inhomogeneous fluctuations. This scenario shares similarities with instabilities in planarly aligned nematic liquid crystals: in both cases the wave vector of the instability may be oriented either parallel or oblique to the polarity axis. However, the encountered instabilities here are long-wave instead of short-wave and the destabilizing modes are drifting ones due to the polar symmetry. Additionally a nonpropagating transverse instability is possible. The stability diagrams related to the various wave vector orientations relative to the polarity axis are determined and discussed for a specific model of motor-filament interactions.

Simulation studies of self-organization of microtubules and molecular motors

We perform Monte Carlo type simulation studies of self-organization of microtubules interacting with molecular motors. We model microtubules as stiff polar rods of equal length exhibiting anisotropic diffusion in the plane. The molecular motors are implicitly introduced by specifying certain probabilistic collision rules resulting in realignment of the rods. This approximation of the complicated microtubule-motor interaction by a simple instant collision allows us to bypass the "computational bottlenecks" associated with the details of the diffusion and the dynamics of motors and the reorientation of microtubules. Consequently, we are able to perform simulations of large ensembles of microtubules and motors on a very large time scale. This simple model reproduces all important phenomenology observed in in vitro experiments: Formation of vortices for low motor density and raylike asters and bundles for higher motor density.

Swarming and swirling in self-propelled polar granular rods

Using experiments with anisotropic vibrated rods and quasi-2D numerical simulations, we show that shape plays an important role in the collective dynamics of self-propelled (SP) particles. We demonstrate that SP rods exhibit local ordering, aggregation at the side walls, and clustering absent in round SP particles. Furthermore, we find that at sufficiently strong excitation SP rods engage in a persistent swirling motion in which the velocity is strongly correlated with particle orientation.

Rheological and structural properties of dilute active filament solutions

The rheology and the structure of a dilute semiflexible biofilament solution, like F-actin, interacting via molecular motors is probed by molecular dynamics simulations. Oscillatory external shear is used to measure the storage and loss moduli as a function of motor activity in a range of frequencies and for low shear rates. The overall effect of the motor activity on the rheological properties is interpreted as an increase of the temperature, with the effective temperature proportional to the density of motors. However, the effect of motors on the structural properties of the solution, such as the orientation correlation function, is opposite: the motors drastically increase the orientation correlation length whereas thermal fluctuations decrease it.

Interactions of semiflexible filaments and molecular motors

This paper summarizes the results of numerical simulations of the interaction of a pair of biofilaments mediated by a molecular motor. The filaments are modeled as flexible rods, and the results are applicable to microtubules, which are relatively stiff, as well as to much softer filaments, such as actin. The results provide insight into the effects of flexibility on cytoskeleton formation and the rheology of semiflexible filament networks. The simulations are based on a nonlinear elasticity equation. The results show that flexibility enhances the tendency of filaments to align. The enhancement in turn favors the formation of large-scale structures in multifilament systems. Simulations for soft filaments show that the action of the motor can result in the formation of multiple loops of the filaments as a result of buckling, which can affect the structure of a cross-linked network and thereby its rheology. The estimate for the minimal buckling length as a function of the motor speed, the viscosity of the solvent, and the bending stiffness of the filament is derived analytically.

Model for dynamical coherence in thin films of self-propelled microorganisms

Concentrated bacterial suspensions spontaneously develop transient spatiotemporal patterns of coherent locomotion whose correlation lengths greatly exceed the size of individual organisms. Continuum models have indicated that a state of uniform swimming order is linearly unstable at finite wavelengths, but have not addressed the nonlinear dynamics of the coherent state, with its biological implications for mixing, transport, and intercellular communication. We investigate a specific model incorporating hydrodynamic interactions in thin-film geometries and show by numerical studies that it displays large scale persistently recurring vortices, as actually observed.