Motion reversal of molecular motor assemblies due to weak noise

Coarse-grained Stochastic Model of Myosin-Driven Vesicles into Dendritic Spines

We study the dynamics of membrane vesicle motor transport into dendritic spines, which are bulbous intracellular compartments in neurons that play a key role in transmitting signals between neurons. We consider the stochastic analog of the vesicle transport model in [Park and Fai, The Dynamics of Vesicles Driven Into Closed Constrictions by Molecular Motors. Bull. Math. Biol. 82, 141 (2020)]. The stochastic version, which may be considered as an agent-based model, relies mostly on the action of individual myosin motors to produce vesicle motion. To aid in our analysis, we coarse-grain this agent-based model using a master equation combined with a partial differential equation describing the probability of local motor positions. We confirm through convergence studies that the coarse-graining captures the essential features of bistability in velocity (observed in experiments) and waiting-time distributions to switch between steady-state velocities. Interestingly, these results allow us to reformulate the translocation problem in terms of conditional mean first passage times for a run-and-tumble particle moving on a finite domain with absorbing boundaries at the two ends. We conclude by presenting numerical and analytical calculations of vesicle translocation.

Dynamics of Vesicles Driven Into Closed Constrictions by Molecular Motors

We study the dynamics of a model of membrane vesicle transport into dendritic spines, which are bulbous intracellular compartments in neurons driven by molecular motors. We reduce the lubrication model proposed in Fai et al. (Phys Rev Fluids 2:113601, 2017) to a fast-slow system, yielding an analytically and numerically tractable equation equivalent to the original model in the overdamped limit. The model's key parameters include: (1) the ratio of motors that prefer to push toward the head of the dendritic spine to the motors that prefer to push in the opposite direction, and (2) the viscous drag exerted on the vesicle by the spine constriction. We perform a numerical bifurcation analysis in these parameters and find that steady-state vesicle velocities appear and disappear through several saddle-node bifurcations. This process allows us to identify the region of parameter space in which multiple stable velocities exist. We show by direct calculations that there can only be unidirectional motion for sufficiently close vesicle-to-spine diameter ratios. Our analysis predicts the critical vesicle-to-spine diameter ratio, at which there is a transition from unidirectional to bidirectional motion, consistent with experimental observations of vesicle trajectories in the literature.

Rectification of twitching bacteria through narrow channels: A numerical simulations study

Bacteria living on surfaces use different types of motility mechanisms to move on the surface in search of food or to form microcolonies. Twitching is one such form of motility employed by bacteria such as Neisseria gonorrhoeae, in which the polymeric extensions known as type IV pili mediate its movement. Pili extending from the cell body adhere to the surface and pull the bacteria by retraction. The bacterial movement is decided by the two-dimensional tug-of-war among the pili attached to the surface. Natural surfaces on which these microcrawlers dwell are generally spatially inhomogeneous and have varying surface properties. Their motility is known to be affected by the topography of the surfaces. Therefore, it is possible to control bacterial movement by designing structured surfaces which can be potentially utilized for controlling biofilm architecture. In this paper, we numerically investigate the twitching motility in a two-dimensional corrugated channel. The bacterial movement is simulated by two different models: (a) a detailed tug-of-war model which extensively describe the twitching motility of bacteria assisted by pili and (b) a coarse-grained run-and-tumble model which depicts the motion of wide-ranging self-propelled particles. The simulation of bacterial motion through asymmetric corrugated channels using the above models show rectification. The bacterial transport depends on the geometric parameters of the channel and inherent system parameters such as persistence length and self-propelled velocity. In particular, the variation of the particle current with the geometric parameters of the microchannels shows that one can optimize the particle current for specific values of these parameters.

Large and reversible myosin-dependent forces in rigidity sensing

Cells sense the rigidity of their environment through localized pinching, which occurs when myosin molecular motors generate contractions within actin filaments anchoring the cell to its surroundings. We present high-resolution experiments performed on these elementary contractile units in cells. Our experimental results challenge the current understanding of molecular motor force generation. Surprisingly, bipolar myosin filaments generate much larger forces per motor than measured in single molecule experiments. Further, contraction to a fixed distance, followed by relaxation at the same rate, is observed over a wide range of matrix rigidities. Lastly, step-wise displacements of the matrix contacts are apparent during both contraction and relaxation. Building upon a generic two-state model of molecular motor collections, we interpret these unexpected observations as spontaneously emerging features of a collective motor behavior. Our approach explains why, in the cellular context, collections of resilient and slow motors contract in a stepwise fashion while collections of weak and fast motors do not. We thus rationalize the specificity of motor contractions implied in rigidity sensing compared to previous in vitro observations.

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.

Path-integral formalism for stochastic resetting: Exactly solved examples and shortcuts to confinement

We study the dynamics of overdamped Brownian particles diffusing in conservative force fields and undergoing stochastic resetting to a given location at a generic space-dependent rate of resetting. We present a systematic approach involving path integrals and elements of renewal theory that allows us to derive analytical expressions for a variety of statistics of the dynamics such as (i) the propagator prior to first reset, (ii) the distribution of the first-reset time, and (iii) the spatial distribution of the particle at long times. We apply our approach to several representative and hitherto unexplored examples of resetting dynamics. A particularly interesting example for which we find analytical expressions for the statistics of resetting is that of a Brownian particle trapped in a harmonic potential with a rate of resetting that depends on the instantaneous energy of the particle. We find that using energy-dependent resetting processes is more effective in achieving spatial confinement of Brownian particles on a faster time scale than performing quenches of parameters of the harmonic potential.

Effects of molecular noise on bistable protein distributions in rod-shaped bacteria

The distributions of many proteins in rod-shaped bacteria are far from homogeneous. Often they accumulate at the cell poles or in the cell centre. At the same time, the copy number of proteins in a single cell is relatively small making the patterns noisy. To explore limits to protein patterns due to molecular noise, we studied a generic mechanism for spontaneous polar protein assemblies in rod-shaped bacteria, which are based on cooperative binding of proteins to the cytoplasmic membrane. For mono-polar assemblies, we find that the switching time between the two poles increases exponentially with the cell length and with the protein number. This feature could be beneficial to organelle maintenance in ageing bacteria.

Master equation approach for a cross-bridge power-stroke model with a finite number of motors

The cross-bridge power-stroke model has been widely used to describe the motion of large motor assemblies connected to a common rigid filament. In this paper, we go beyond the original velocity-ensemble approach and propose a master equation approach to account for the cooperative motion of a finite number of motors based on the cross-bridge model. By studying the force-velocity relationship for motors with strain-independent detachment rate, we show the convergence of our approach to the velocity-ensemble approach in the limit of large motor numbers. In the case that the detachment rate of motors is strain dependent, based on two assumptions for the strain distribution among motors, we show the occurrence of the bimodal distribution of the number of motors bound to the filament. This provides a new perspective to look at the instability of a multimotor system, which is essential for all the experimentally observed complex motions displayed by a group of motors, such as hysteresis, bidirectional motion, and spontaneous oscillation. By comparing the velocities calculated using the two assumptions with the stochastic simulation, it suggests that the coupling between motors via the common connection to the filament might facilitate the fast movement of filaments at small loading forces.

Effect of dynamic and static friction on an asymmetric granular piston

We investigate the influence of dry friction on an asymmetric, granular piston of mass M, composed of two materials, undergoing inelastic collisions with bath particles of mass m. Numerical simulations of the Boltzmann-Lorentz equation reveal the existence of two scaling regimes depending on the friction strength. In the large friction limit, we introduce an exact model giving the asymptotic behavior of the Boltzmann-Lorentz equation. For small friction and for large mass ratio M/m, we derive a Fokker-Planck equation for which the exact solution is also obtained. Static friction attenuates the motor effect and results in a discontinuous velocity distribution.

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.

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.