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

A Multiple Step Active Stiffness Integration Scheme to Couple a Stochastic Cross-Bridge Model and Continuum Mechanics for Uses in Both Basic Research and Clinical Applications of Heart Simulation

In a multiscale simulation of a beating heart, the very large difference in the time scales between rapid stochastic conformational changes of contractile proteins and deterministic macroscopic outcomes, such as the ventricular pressure and volume, have hampered the implementation of an efficient coupling algorithm for the two scales. Furthermore, the consideration of dynamic changes of muscle stiffness caused by the cross-bridge activity of motor proteins have not been well established in continuum mechanics. To overcome these issues, we propose a multiple time step scheme called the multiple step active stiffness integration scheme (MusAsi) for the coupling of Monte Carlo (MC) multiple steps and an implicit finite element (FE) time integration step. The method focuses on the active tension stiffness matrix, where the active tension derivatives concerning the current displacements in the FE model are correctly integrated into the total stiffness matrix to avoid instability. A sensitivity analysis of the number of samples used in the MC model and the combination of time step sizes confirmed the accuracy and robustness of MusAsi, and we concluded that the combination of a 1.25 ms FE time step and 0.005 ms MC multiple steps using a few hundred motor proteins in each finite element was appropriate in the tradeoff between accuracy and computational time. Furthermore, for a biventricular FE model consisting of 45,000 tetrahedral elements, one heartbeat could be computed within 1.5 h using 320 cores of a conventional parallel computer system. These results support the practicality of MusAsi for uses in both the basic research of the relationship between molecular mechanisms and cardiac outputs, and clinical applications of perioperative prediction.

Hierarchical modeling of force generation in cardiac muscle

Performing physiologically relevant simulations of the beating heart in clinical context requires to develop detailed models of the microscale force generation process. These models, however, may reveal difficult to implement in practice due to their high computational costs and complex calibration. We propose a hierarchy of three interconnected muscle contraction models-from the more refined to the more simplified-that are rigorously and systematically related to each other, offering a way to select, for a specific application, the model that yields a good trade-off between physiological fidelity, computational cost and calibration complexity. The three model families are compared to the same set of experimental data to systematically assess what physiological indicators can be reproduced or not and how these indicators constrain the model parameters. Finally, we discuss the applicability of these models for heart simulation.

Self-excited oscillation of the director field in cholesteric liquid crystalline droplets under a temperature gradient

In this study, we demonstrate a self-excited oscillation induced in cholesteric liquid crystalline droplets under a temperature gradient. At equilibrium, a winding Maltese cross pattern with a point defect was observed via polarised microscopy in the droplets dispersed in an isotropic solvent. When the temperature gradient was applied, the pattern was deformed owing to the Marangoni convection induced by the gradient. Here, when both the droplet size and temperature gradient were sufficiently large, the periodic movement of the defect together with the pattern deformation was observed, which demonstrated the self-excited oscillation of the director field. To describe this phenomenon, we theoretically analysed the flow and director fields by using Onsager's variational principle. This principle enabled the simplified description of the phenomenon; consequently, the time evolution of the director field could be expressed by the phenomenological equations for the two parameters characterising the field. These equations represented the van der Pol equation, which well expressed the mechanism of the self-excited oscillation.

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.

Embedding dual function into molecular motors through collective motion

Protein motors, such as kinesins and dyneins, bind to a microtubule and travel along it in a specific direction. Previously, it was thought that the directionality for a given motor was constant in the absence of an external force. However, the directionality of the kinesin-5 Cin8 was recently found to change as the number of motors that bind to the same microtubule is increased. Here, we introduce a simple mechanical model of a microtubule-sliding assay in which multiple motors interact with the filament. We show that, due to the collective phenomenon, the directionality of the motor changes (e.g., from minus- to plus- end directionality), depending on the number of motors. This is induced by a large diffusive component in the directional walk and by the subsequent frustrated motor configuration, in which multiple motors pull the filament in opposite directions, similar to a game of tug-of-war. A possible role of the dual-directional motors for the mitotic spindle formation is also discussed. Our framework provides a general mechanism to embed two conflicting tasks into a single molecular machine, which works context-dependently.

Statistical mechanics of the Huxley-Simmons model

The chemomechanical model of Huxley and Simmons (HS) [A. F. Huxley and R. M. Simmons, Nature 233, 533 (1971)NATUAS0028-083610.1038/233533a0] provides a paradigmatic description of mechanically induced collective conformational changes relevant in a variety of biological contexts, from muscles power stroke and hair cell gating to integrin binding and hairpin unzipping. We develop a statistical mechanical perspective on the HS model by exploiting a formal analogy with a paramagnetic Ising model. We first study the equilibrium HS model with a finite number of elements and compute explicitly its mechanical and thermal properties. To model kinetics, we derive a master equation and solve it for several loading protocols. The developed formalism is applicable to a broad range of allosteric systems with mean-field interactions.

Multiphysics and multiscale modelling, data-model fusion and integration of organ physiology in the clinic: ventricular cardiac mechanics

With heart and cardiovascular diseases continually challenging healthcare systems worldwide, translating basic research on cardiac (patho)physiology into clinical care is essential. Exacerbating this already extensive challenge is the complexity of the heart, relying on its hierarchical structure and function to maintain cardiovascular flow. Computational modelling has been proposed and actively pursued as a tool for accelerating research and translation. Allowing exploration of the relationships between physics, multiscale mechanisms and function, computational modelling provides a platform for improving our understanding of the heart. Further integration of experimental and clinical data through data assimilation and parameter estimation techniques is bringing computational models closer to use in routine clinical practice. This article reviews developments in computational cardiac modelling and how their integration with medical imaging data is providing new pathways for translational cardiac modelling.

Beam finite-element model of a molecular motor for the simulation of active fibre networks

Molecular motors are proteins that excessively increase the efficiency of subcellular transport processes. They allow for cell division, nutrient transport and even macroscopic muscle movement. In order to understand the effect of motors in large biopolymer networks, e.g. the cytoskeleton, we require a suitable model of a molecular motor. In this contribution, we present such a model based on a geometrically exact beam finite-element formulation. We discuss the numerical model of a non-processive motor such as myosin II, which interacts with actin filaments. Based on experimental data and inspired by the theoretical understanding offered by the power-stroke model and the swinging-cross-bridge model, we parametrize our numerical model in order to achieve the effect that a physiological motor has on its cargo. To this end, we introduce the mechanical and mathematical foundations of the model, then discuss its calibration, prove its usefulness by conducting finite-element simulations of actin-myosin motility assays and assess the influence of motors on the rheology of semi-flexible biopolymer networks.

Polarization of cells and soft objects driven by mechanical interactions: consequences for migration and chemotaxis

We study a generic model for the polarization and motility of self-propelled soft objects, biological cells, or biomimetic systems, interacting with a viscous substrate. The active forces generated by the cell on the substrate are modeled by means of oscillating force multipoles at the cell-substrate interface. Symmetry breaking and cell polarization for a range of cell sizes naturally "emerge" from long range mechanical interactions between oscillating units, mediated both by the intracellular medium and the substrate. However, the harnessing of cell polarization for motility requires substrate-mediated interactions. Motility can be optimized by adapting the oscillation frequency to the relaxation time of the system or when the substrate and cell viscosities match. Cellular noise can destroy mechanical coordination between force-generating elements within the cell, resulting in sudden changes of polarization. The persistence of the cell's motion is found to depend on the cell size and the substrate viscosity. Within such a model, chemotactic guidance of cell motion is obtained by directionally modulating the persistence of motion, rather than by modulating the instantaneous cell velocity, in a way that resembles the run and tumble chemotaxis of bacteria.

Active phase and amplitude fluctuations of flagellar beating

The eukaryotic flagellum beats periodically, driven by the oscillatory dynamics of molecular motors, to propel cells and pump fluids. Small but perceivable fluctuations in the beat of individual flagella have physiological implications for synchronization in collections of flagella as well as for hydrodynamic interactions between flagellated swimmers. Here, we characterize phase and amplitude fluctuations of flagellar bending waves using shape mode analysis and limit-cycle reconstruction. We report a quality factor of flagellar oscillations Q = 38.0 ± 16.7 (mean ± s.e.). Our analysis shows that flagellar fluctuations are dominantly of active origin. Using a minimal model of collective motor oscillations, we demonstrate how the stochastic dynamics of individual motors can give rise to active small-number fluctuations in motor-cytoskeleton systems.

Power-stroke-driven actomyosin contractility

In ratchet-based models describing actomyosin contraction the activity is usually associated with actin binding potential while the power-stroke mechanism, residing inside myosin heads, is viewed as passive. To show that contraction can be propelled directly through a conformational change, we propose an alternative model where the power stroke is the only active mechanism. The asymmetry, ensuring directional motion, resides in steric interaction between the externally driven power-stroke element and the passive nonpolar actin filament. The proposed model can reproduce all four discrete states of the minimal actomyosin catalytic cycle even though it is formulated in terms of continuous Langevin dynamics. We build a conceptual bridge between processive and nonprocessive molecular motors by demonstrating that not only the former but also the latter can use structural transformation as the main driving force.

Muscle as a metamaterial operating near a critical point

The passive mechanical response of skeletal muscles at fast time scales is dominated by long range interactions inducing cooperative behavior without breaking the detailed balance. This leads to such unusual "material properties" as negative equilibrium stiffness and different behavior in force and displacement controlled loading conditions. Our fitting of experimental data suggests that "muscle material" is finely tuned to perform close to a critical point which explains large fluctuations observed in muscles close to the stall force.

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.

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.