Membrane waves driven by actin and Myosin

We present a model which couples the membrane with the protrusive forces of actin polymerization and contractile forces of molecular motors, such as myosin. The actin polymerization at the membrane is activated by freely diffusing membrane proteins that have a spontaneous curvature. Molecular motors are recruited to the polymerizing actin filaments, from a constant reservoir, and produce a contractile force. All the forces and variables are treated in the linear limit. Our results show that for convex membrane proteins the myosin activity gives rise to robust transverse membrane waves, similar to those observed on different cells.     

Nonlinear study of symmetry breaking in actin gels: implications for cellular motility

Force generation by actin polymerization is an important step in cellular motility and can induce the motion of organelles or bacteria, which move inside their host cells by trailing an actin tail behind. Biomimetic experiments on beads and droplets have identified the biochemical ingredients to induce this motion, which requires a spontaneous symmetry breaking in the absence of external fields. We find that the symmetry breaking can be captured on the basis of elasticity theory and linear flux-force relationships. Furthermore, we develop a phase-field approach to study the fully nonlinear regime and show that actin-comet formation is a robust feature, triggered by growth and mechanical stresses. We discuss the implications of symmetry breaking for self-propulsion.     

Filament instability and rotational tissue anisotropy: A numerical study using detailed cardiac models

The role of cardiac tissue anisotropy in the breakup of vortex filaments is studied using two detailed cardiac models. In the Beeler-Reuter model, modified to produce stable spiral waves in two dimensions, we find that anisotropy can destabilize a vortex filament in a parallelepipedal slab of tissue. The mechanisms of the instability are similar to the ones reported in previous work on a simplified cardiac model by Fenton and Karma [Chaos 8, 20 (1998)]. In the Luo-Rudy model, also modified to produce stable spiral waves in two dimensions, we find that anisotropy does not destabilize filaments. A possible explanation for this model-dependent behavior based on spiral tip trajectories is offered. (c) 2001 American Institute of Physics.     

Activated dynamics of semiflexible polymers on structured substrates

We study the thermally activated motion of semiflexible polymers in double-well potentials using field-theoretic methods. Shape, energy, and effective diffusion constant of kink excitations are calculated, and their dependence on the bending rigidity of the semiflexible polymer is determined. For symmetric potentials, the kink motion is purely diffusive whereas kink motion becomes directed in the presence of a driving force. We determine the average velocity of the semiflexible polymer based on the kink dynamics. The Kramers escape over the potential barriers proceeds by nucleation and diffusive motion of kink-antikink pairs, the relaxation to the straight configuration by annihilation of kink-antikink pairs. We consider both uniform and point-like driving forces. For the case of point-like forces the polymer crosses the potential barrier only if the force exceeds a critical value. Our results apply to the activated motion of biopolymers such as DNA and actin filaments or of synthetic polyelectrolytes on structured substrates.     

Driven Diffusive Systems of Active Filament Bundles

The cytoskeleton is an important subsystem of cells that is involved for example in cell division and locomotion. It consists of filaments that are cross-linked by molecular motors that can induce relative sliding between filaments and generate stresses in the network. In order to study the effects of fluctuations on the dynamics of such a system we introduce here a new class of driven diffusive systems mimicking the dynamics of active filament bundles where the filaments are aligned with respect to a common axis. After introducing the model class we first analyze an exactly solvable case and find condensation. For the general case we perform a mean-field analysis and study the behavior on large length scales by coarse-graining. We determine conditions for condensation and establish a relation between the hopping rates and the tension generated in the bundle.     

Protrusion of a Virtual Model Lamellipodium by Actin Polymerization: A Coarse-Grained Langevin Dynamics Model

We report the development of a coarse-grained Langevin dynamics model of a lamellipodium featuring growing F-actin filaments in order to study the effect of stiffness of the F-actin filament, the G-actin monomer concentration, and the number of polymerization sites on lamellipodium protrusion. The virtual lamellipodium is modeled as a low-aspect-ratio doubly capped cylinder formed by triangulated particles on its surface. It is assumed that F-actin filaments are firmly attached to a lamellipodium surface where polymerization sites are located, and actin polymerization takes place by connecting a G-actin particle to a polymerization site and to the first particle of a growing F-actin filament. It is found that there is an optimal number of polymerization sites for rapid lamellipodium protrusion. The maximum speed of lamellipodium protrusion is related to competition between the number of polymerization sites and the number of available G-actin particles, and the degree of pulling and holding of the lamellipodium surface by non-polymerizing actin filaments. The lamellipodium protrusion by actin polymerization displays saltatory motion exhibiting pseudo-thermal equilibrium: the lamellipodium speed distribution is Maxwellian in two dimensions but the lamellipodium motion is biased so that the lamellipodium speed in the direction of the lamellipodium motion is much larger than that normal to the lamellipodium motion.     

Regular and alternant spiral waves of contractile motion on rat ventricle cell cultures

We demonstrate that meandering as well as regular spiral waves can form in a well-controlled culture layer of rat ventricle cells and that the meandering spiral wave, in particular, can generate an alternant rhythm. These observations are made possible by a newly developed, noninvasive phase contrast macro-optics that is simple but highly effective in visualizing the contractile motion of the populations of cardiac cells.     

A geometric theory for scroll wave filaments in anisotropic excitable media

Scroll waves are an important example of self-organisation in excitable media. In cardiac tissue, scroll waves of electrical activity underlie lethal ventricular arrhythmias and fibrillation. They rotate around a topological line defect which has been termed the filament. Numerical investigation has shown that anisotropy can substantially affect the dynamics of scroll waves. It has recently been hypothesised that stationary scroll wave filaments in cardiac tissue describe geodesics in a space whose metric is the inverse diffusion tensor. Several computational studies have validated this hypothesis, but until now no quantitative theory has been provided to study the effects of anisotropy on scroll wave filaments. Here, we review in detail the recently developed covariant formalism for scroll wave dynamics in general anisotropy and derive the equations of motion of filaments. These equations are fully covariant under general spatial coordinate transformations and describe the motion of filaments in a curved space whose metric tensor is the inverse diffusion tensor. Our dynamic equations are valid for thin filaments and for general anisotropy and we show that stationary filaments obey the geodesic equation. We extend previous work by allowing spatial variations in the determinant of the diffusion tensor and the reaction parameters, leading to drift of the filament.     

Three-dimensional spiral waves in an excitable reaction system: initiation and dynamics of scroll rings and scroll ring pairs

We report experimental results on spiral and scroll waves in the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction. The propagating concentration waves are detected by two-dimensional photometry and optical tomography. Wave pulses can disappear in front-to-front and front-to-back collisions. This anomaly causes the nucleation of vortices from collisions of three nonrotating waves. In three-dimensional systems, these vortices are scroll rings that rotate around initially circular filaments. Depending on reactant concentrations, the filaments shrink or expand indicating positive and negative filament tensions, respectively. Shrinkage results in vortex annihilation. Expansion is accompanied by filament buckling and bending, which is interpreted as developing Winfree turbulence. We also describe the initiation of scroll ring pairs in four-wave collisions. The two filaments are stacked on top of each other and their motion suggests filament repulsion.     

Study of motion of atoms in direct-current discharge of ring laser

Using a ring laser, we experimentally study the motion of atoms caused by their collisions with electrons and ions in a direct-current discharge. We find the dependences of the frequency difference between counterpropagating light waves on the discharge current and parameters of the gas mixture. It is found that, depending on the parameters of the discharge and excitation mechanism of energy levels, the sign of the frequency difference can be reversed. We show that recombination processes are mainly responsible for the occurrence of the frequency difference in the ring laser. The method of investigation makes it possible to detect at least 100 atoms, the velocity of the directed motion of which is comparable with the velocity of ions.     

Dynamics of microtubule aster formation by motor complexes

The polymerisation of filaments and their moving by motor proteins contribute to the organisation of the cytoskeleton of higher cells. What a mixture of such elements is capable of doing, is not yet understood. Using computer simulations, we study here in a simple system the kinetics of aster formation by motors and growing microtubules. We find that the system can be in three different regimes depending on the value of the microtubule polymerisation rate. We discuss that three types of motor links are present in the network and show how they contribute to each final pattern.     

Mechanical oscillations at the cellular scale

Active phenomena which involve force generation and motion play a key role in a number of phenomena in living cells such as cell motility, muscle contraction and the active transport of material and organelles. Here we discuss mechanical oscillations generated by active systems in cells. Examples are oscillatory regimes in muscles, the periodic beating of axonemal cilia and flagella and spontaneous oscillations of auditory hair cells which play a role in active amplification of weak sounds in hearing. As a prototype system for oscillation generation by proteins, we discuss a general mechanism by which many coupled active elements such as motor molecules can generate oscillations.     

Effective Potential of a Two-State Model for Molecular Motor

We study force generation and motion of molecular motors using a simple two-state model in the paper. Asymmetric and periodic potential is adopted to describe the interaction between motor proteins and filaments that are periodic and polar. The current and the slope of the effective potential as functions of the temperature and transition rates are calculated in the two-state model. The ratio of the slope of the effective potential to the current is also calculated. It is shown that the directed motion of motor proteins is relevant to the effective potential. The slope of the effective potential corresponds to an average force. The non-vanishing force therefore implies that detailed balance is broken in the process of transition between different states.     

Brownian ratchets driven by asymmetric nucleation of hydrolysis waves

We propose a stochastic process wherein molecular transport is mediated by asymmetric nucleation of domains on a one-dimensional substrate, in contrast with molecular motors that hydrolyze nucleotide triphosphates and undergo conformational change. We show that asymmetric nucleation of hydrolysis waves on a track can also result in directed motion of an attached particle. Asymmetrically cooperative kinetics between hydrolyzed and unhydrolyzed states on each lattice site generate moving domain walls that push a particle sitting on the track. We use a novel fluctuating-frame, finite-segment mean field theory to accurately compute steady-state velocities of the driven particle and to discover parameter regimes yielding maximal domain wall flux, leading to optimal particle drift.     

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.     

Dynamics and mechanics of motor-filament systems

Motivated by the cytoskeleton of eukaryotic cells, we develop a general framework for describing the large-scale dynamics of an active filament network. In the cytoskeleton, active cross-links are formed by motor proteins that are able to induce relative motion between filaments. Starting from pair-wise interactions of filaments via such active processes, our framework is based on momentum conservation and an analysis of the momentum flux. This allows us to calculate the stresses in the filament network generated by the action of motor proteins. We derive effective theories for the filament dynamics which can be related to continuum theories of active polar gels. As an example, we discuss the stability of homogenous isotropic filament distributions in two spatial dimensions.     

Random walks of cytoskeletal motors in open and closed compartments

Random walks of molecular motors, which bind to and unbind from cytoskeletal filaments, are studied theoretically. The bound and unbound motors undergo directed and nondirected motion, respectively. Motors in open compartments exhibit anomalous drift velocities. Motors in closed compartments generate stationary nonequilibrium states with spatially varying densities of the motor concentrations and currents. "Traffic jams" on the filaments lead to a maximum of the motor current at an optimal motor concentration. Quantitative estimates based on experimental data for bound motors indicate that these transport phenomena are accessible to experiments.     

Rotational effect in two-dimensional cooperative directed transport

In this review we investigate the rotation effect in the motion of coupled dimer in a two-dimensional asymmetric periodic potential. Free rotation does not generate directed transport in translational direction, while we find it plays an critical role in the motors motility when the dimer moves under the effect of asymmetry ratchet potential. In the presence of external force, we study the relation between the average current and the force numerically and theoretically. The numerical results show that only appropriate driving force could produce nonzero current and there are current transitions when the force is large enough. An analysis of stability analysis of limit cycles is applied to explain the occurrence of these transitions. Moreover, we numerically simulate the transport of this coupled dimer driven by the random fluctuations in the rotational direction. The existence of noise smooths the current transitions induced by the driving force and the resonance-like peaks which depend on the rod length emerge in small noise strength. Thanks to the noise in the rotational direction, autonomous motion emerges without the external force and large noise could make the current reversal happen. Eventually, the new mechanism to generate directed transport by the rotation is studied.     

Spontaneous sarcomere dynamics

Sarcomeres are the basic force generating units of striated muscles and consist of an interdigitating arrangement of actin and myosin filaments. While muscle contraction is usually triggered by neural signals, which eventually set myosin motors into motion, isolated sarcomeres can oscillate spontaneously between a contracted and a relaxed state. We analyze a model for sarcomere dynamics, which is based on a force-dependent detachment rate of myosin from actin. Our numerical bifurcation analysis of the spontaneous sarcomere dynamics reveals notably Hopf bifurcations, canard explosions, and gluing bifurcations. We discuss possible implications for experiments.