Spontaneous shear flow in confined cellular nematics.

In embryonic development or tumor evolution, cells often migrate collectively within confining tracks defined by their microenvironment 1,2. In some of these situations, the displacements within a cell strand are antiparallel 3, giving rise to shear flows. However, the mechanisms underlying these spontaneous flows remain poorly understood. Here, we show that an ensemble of spindle-shaped cells plated in a well-defined stripe spontaneously develop a shear flow whose characteristics depend on the width of the stripe. On wide stripes, the cells self-organize in a nematic phase with a director at a well-defined angle with the stripe's direction, and develop a shear flow close to the stripe's edges. However, on stripes narrower than a critical width, the cells perfectly align with the stripe's direction and the net flow vanishes. A hydrodynamic active gel theory provides an understanding of these observations and identifies the transition between the non-flowing phase oriented along the stripe and the tilted phase exhibiting shear flow as a Fréedericksz transition driven by the activity of the cells. This physical theory is grounded in the active nature of the cells and based on symmetries and conservation laws, providing a generic mechanism to interpret in vivo antiparallel cell displacements.

Bistability, Oscillations, and Bidirectional Motion of Ensemble of Hydrodynamically Coupled Molecular Motors.

We analyze the collective behavior of hydrodynamically coupled molecular motors. We show that the local fluxes induced by motor displacement can induce the experimentally observed bidirectional motion of cargoes and vesicles. By means of a mean-field approach we show that sustained oscillations as well as bistable collective motor motion arise even for very large collection of motors, when thermal noise is irrelevant. The analysis clarifies the physical mechanisms responsible for such dynamics by identifying the relevant coupling parameter and its dependence on the geometry of the hydrodynamic coupling as well as on system size. We quantify the phase diagram for the different phases that characterize the collective motion of hydrodynamically coupled motors and show that sustained oscillations can be reached for biologically relevant parameters, hence, demonstrating the relevance of hydrodynamic interactions in intracellular transport.

Nonequilibrium statistical mechanics of mixtures of particles in contact with different thermostats.

We introduce a novel type of locally driven systems made of two types of particles (or a polymer with two types of monomers) subject to a chaotic drive with approximately white noise spectrum, but different intensity; in other words, particles of different types are in contact with thermostats at different temperatures. We present complete systematic statistical mechanics treatment starting from first principles. Although we consider only corrections to the dilute limit due to pairwise collisions between particles, meaning we study a nonequilibrium analog of the second virial approximation, we find that the system exhibits a surprisingly rich behavior. In particular, pair correlation function of particles has an unusual quasi-Boltzmann structure governed by an effective temperature distinct from that of any of the two thermostats. We also show that at sufficiently strong drive the uniformly mixed system becomes unstable with respect to steady states consisting of phases enriched with different types of particles. In the second virial approximation, we define nonequilibrium "chemical potentials" whose gradients govern diffusion fluxes and a nonequilibrium "osmotic pressure," which governs the mechanical stability of the interface.

Growth, homeostatic regulation and stem cell dynamics in tissues.

The regulation of cell growth in animal tissues is a question of critical importance: most tissues contain different types of cells in interconversion and the fraction of each type has to be controlled in a precise way, by mechanisms that remain unclear. Here, we provide a theoretical framework for the homeostasis of stem-cell-containing epithelial tissues using mechanical equations, which describe the size of the tissue and kinetic equations, which describe the interconversions of the cell populations. We show that several features, such as the evolution of stem cell fractions during intestinal development, the shape of a developing intestinal wall, as well as the increase in the proliferative compartment in cancer initiation, can be studied and understood from generic modelling which does not rely on a particular regulatory mechanism. Finally, inspired by recent experiments, we propose a model where cell division rates are regulated by the mechanical stresses in the epithelial sheet. We show that pressure-controlled growth can, in addition to the previous features, also explain with few parameters the formation of stem cell compartments as well as the morphologies observed when a colonic crypt becomes cancerous. We also discuss optimal strategies of wound healing, in connection with experiments on the cornea.

The actin cortex as an active wetting layer.

Using active gel theory we study theoretically the properties of the cortical actin layer of animal cells. The cortical layer is described as a non-equilibrium wetting film on the cell membrane. The actin density is approximately constant in the layer and jumps to zero at its edge. The layer thickness is determined by the ratio of the polymerization velocity and the depolymerization rate of actin.

Tissue dynamics with permeation.

Animal tissues are complex assemblies of cells, extracellular matrix (ECM), and permeating interstitial fluid. Whereas key aspects of the multicellular dynamics can be captured by a one-component continuum description, cell division and apoptosis imply material turnover between different components that can lead to additional mechanical conditions on the tissue dynamics. We extend our previous description of tissues in order to account for a cell/ECM phase and the permeating interstitial fluid independently. In line with our earlier work, we consider the cell/ECM phase to behave as an elastic solid in the absence of cell division and apoptosis. In addition, we consider the interstitial fluid as ideal on the relevant length scales, i.e., we ignore viscous stresses in the interstitial fluid. Friction between the fluid and the cell/ECM phase leads to a Darcy-like relation for the interstitial fluid velocity and introduces a new characteristic length scale. We discuss the dynamics of a tissue confined in a chamber with a permeable piston close to the homeostatic state where cell division and apoptosis balance, and we calculate the rescaled effective diffusion coefficient for cells. For different mass densities of the cell/ECM component and the interstitial fluid, a treadmilling steady state due to gravitational forces can be found.

Bidirectional motion of motor assemblies and the weak-noise escape problem.

We present a detailed calculation that enables us to estimate the reversal time of a molecular motor assembly that displays bidirectional motion in the limit of weak noise. We derive a Fokker-Planck equation by taking a large volume expansion of a master equation, and we consider a simple choice of transition rates that enables us to reduce the number of variables to 2. We use the Wentzell-Freidlin theory to define an effective nonequilibrium potential and analytically estimate the reversal time. We also present the results of stochastic simulations that match very well our simulation results.

Instabilities of monolayered epithelia: shape and structure of villi and crypts.

We study theoretically the shapes of a dividing epithelial monolayer of cells lying on top of an elastic stroma. The negative tension created by cell division provokes a buckling instability at a finite wave vector leading to the formation of periodic arrays of villi and crypts. The instability is similar to the buckling of a metallic plate under compression. We use the results to rationalize the various structures of the intestinal lining observed in vivo. Taking into account the coupling between cell division and local curvature, we obtain different patterns of villi and crypts, which could explain the different morphologies of the small intestine and the colon.

Dynamical behavior of molecular motor assemblies in the rigid and crossbridge models.

We present a detailed analysis of the dynamical instabilities appearing in two kinetic theories for the collective behavior of molecular motors: the rigid two-state model and the two-state crossbridge (or power-stroke) model with continuous binding sites. We calculate force-velocity relations, discuss their stability, plot a diagram that summarizes the oscillation regimes, identify the location of the Hopf bifurcation with a memory effect, discuss the oscillation frequency and make a link with single-molecule experiments. We show that the instabilities present in these models naturally translate into non-linearities in force-displacement relations, and at linear order give forces that are similar to the delayed stretch activation observed in oscillating muscles. We also find that instabilities can appear for both apparent load-decelerated and load-accelerated detachment rates in a 3-state crossbridge model.

Motion reversal of molecular motor assemblies due to weak noise.

Bidirectional motion is an example of collective behavior of molecular motors. It occurs at finite noise level in a nonequilibrium system. We consider this problem as a first exit problem. We identify the noise strength by doing an expansion of a master equation and apply the Wentzell-Freidlin theory to define an effective nonequilibrium potential and provide analytical estimates of the reversal time. Our results match very well with the results of stochastic simulations.

Dynamic instabilities in assemblies of molecular motors with finite stiffness.

We propose a two-state "soft-motor" model for the collective behavior of molecular motors which takes into account both the internal motor stiffness and the periodic interaction with the filament. As in the Prandtl-Tomlinson model of tribology, the important parameter of the model is the pinning parameter, which compares the stiffness of the motors to the stiffness of the potential. The model predicts dynamic instabilities in two disconnected regions of parameter space. These parameter ranges correspond to two existing theories of motor assemblies, the rigid two-state model and the crossbridge model. The model also predicts a discontinuity of the slope of the force-velocity relation at small velocities.

Polarity patterns of stress fibers.

Stress fibers are contractile actomyosin bundles commonly observed in the cytoskeleton of metazoan cells. The spatial profile of the polarity of actin filaments inside contractile actomyosin bundles is either monotonic (graded) or periodic (alternating). In the framework of linear irreversible thermodynamics, we write the constitutive equations for a polar, active, elastic one-dimensional medium. An analysis of the resulting equations for the dynamics of polarity shows that the transition from graded to alternating polarity patterns is a nonequilibrium Lifshitz point. Active contractility is a necessary condition for the emergence of sarcomeric, alternating polarity patterns.

Spontaneous oscillations of a minimal actomyosin system under elastic loading.

Spontaneous mechanical oscillations occur in various types of biological systems where groups of motor molecules are elastically coupled to their environment. By using an optical trap to oppose the gliding motion of a single bead-tailed actin filament over a substrate densely coated with myosin motors, we mimicked this condition in vitro. We show that this minimal actomyosin system can oscillate spontaneously. Our finding accords quantitatively with a general theoretical framework where oscillatory instabilities emerge generically from the collective dynamics of molecular motors under load.

Generalized fluctuation-dissipation theorem for steady-state systems.

The fluctuation-dissipation theorem is a central result of statistical physics, which applies to any system at thermodynamic equilibrium. Its violation is a strong signature of nonequilibrium behavior. We show that for any system with Markovian dynamics, in a nonequilibrium steady state, a proper choice of observables restores a fluctuation-response theorem identical to a suitable version of the equilibrium fluctuation-dissipation theorem. This theorem applies to a broad class of dynamical systems. We illustrate it with linear stochastic dynamics and examples borrowed from the physics of molecular motors and Hopf bifurcations. Finally, we discuss general implications of the theorem.

Membrane tension lowering induced by protein activity.

Using videomicroscopy we present measurements of the fluctuation spectrum of giant vesicles containing bacteriorhodopsin pumps. When the pumps are activated, we observe a significant increase of the fluctuations in the low wave vector region, which we interpret as due to a lowering of the effective tension of the membrane.

Viscous-fingering-like instability of cell fragments.

We present a novel flow instability that can arise in thin films of cytoskeletal fluids if the friction with the substrate on which the film lies is sufficiently strong. We consider a two-dimensional, membrane-bound fragment containing actin filaments that polymerize at the edge and depolymerize in the fragment. Performing a linear stability analysis of the initial state due to perturbations of the fragment boundary, we find, in the limit of large friction, that the perturbed actin velocity and pressure fields obey the same laws governing the viscous fingering instability of an interface between immiscible fluids in a Hele-Shaw cell. A remarkable feature of this instability is that it is independent of the strength of the interaction between actin filaments and myosin motors.

Shape optimization in lipid nanotube networks.

Starting from a high surface free-energy state, lipid nanotube networks are capable to self-organize into tree-like structures with particular geometrical features. In this work we analyze the process of self-organization in such networks, and report a strong similarity to the Euclidian Steiner Tree Problem (ESTP). ESTP is a well-known NP-hard optimization problem of finding a network connecting a given set of terminal points on a plane, allowing addition of auxiliary points, with the overall objective to minimize the total network length. The present study shows that aggregate lipid structures self-organize into geometries that correspond to locally optimal solutions to such problems.

Collective dynamics of interacting molecular motors.

The collective dynamics of N interacting processive molecular motors are considered theoretically when an external force is applied to the leading motor. We show, using a discrete lattice model, that the force-velocity curves strongly depend on the effective dynamic interactions between motors and differ significantly from those of a simple approach where the motors equally share the force. Moreover, they become essentially independent of the number of motors if N is large enough (N> or approximately 5 for conventional kinesin). We show that a two-state ratchet model has a very similar behavior to that of the coarse-grained lattice model with effective interactions. The general picture is unaffected by motor attachment and detachment events.

Motor proteins transporting cargos.

Processive motor proteins such as kinesin and myosin-V are enzymes that use the energy of ATP hydrolysis to travel along polar cytoskeletal filaments. One of the functions of these proteins is the transport of vesicles and protein complexes that are linked to the light chains of the motors. Modeling the light chain by a linear elastic spring, and using the two-state model for one- and two-headed molecular motors, we study the influence of thermal fluctuations of the cargo on the motion of the motor-cargo complex. We solve numerically the Fokker-Planck equations of motor motion, and find that the mean velocity of the motor-cargo complex decreases monotonously as the spring becomes softer. This effect is due to the random force of thermal fluctuations of the cargo disrupting the operation of the motor. Increasing the size (thus, the friction coefficient) of the cargo also decreases the velocity. Surprisingly, we find that for a given size of the cargo, the velocity has a maximum for a certain friction of the motor. We explain this effect by the interplay between the characteristic length of thermal fluctuations of the cargo on a spring, the motor diffusion length, and the filament period. Our results may be relevant for the interpretation of single-molecule experiments with molecular motors (bead assays), where the motor motion is observed by tracking of a bead attached to the motor.