Models Of Molecular Motion Research Articles

Sufficient conditions for the additivity of stall forces generated by multiple filaments or motors.

Molecular motors and cytoskeletal filaments work collectively most of the time under opposing forces. This opposing force may be due to cargo carried by motors or resistance coming from the cell membrane pressing against the cytoskeletal filaments. Some recent studies have shown that the collective maximum force (stall force) generated by multiple cytoskeletal filaments or molecular motors may not always be just a simple sum of the stall forces of the individual filaments or motors. To understand this excess or deficit in the collective force, we study a broad class of models of both cytoskeletal filaments and molecular motors. We argue that the stall force generated by a group of filaments or motors is additive, that is, the stall force of N number of filaments (motors) is N times the stall force of one filament (motor), when the system is reversible at stall. Conversely, we show that this additive property typically does not hold true when the system is irreversible at stall. We thus present a novel and unified understanding of the existing models exhibiting such non-addivity, and generalise our arguments by developing new models that demonstrate this phenomena. We also propose a quantity similar to thermodynamic efficiency to easily predict this deviation from stall-force additivity for filament and motor collectives.

Dynamics of Methyl Groups in Membrane Proteins Studied by Deterium Solid State NMR Relaxation

NMR relaxation is a valuable tool to study membrane protein dynamics which are important for their functions. Dynamical parameters (correlation times and activation barriers) can provide important information about intra- and intermolecular interactions. Here we present our study of the dynamics of the rhodopsin-bound cofactor retinal and a high-affinity transducin peptide (GαCT2) bound to opsin. To observe how the NMR relaxation and molecular dynamics depend on the local environment, we introduced a deuterated methyl group at positions C5, C9, or C13 of retinal [1], and in GαCT2 (ILENLKDVGLF) at leucine positions 2, 5, and 10. Selectively labeled retinals were studied in the dark, preactive Meta-I, and active Meta-II states of rhodopsin. High-affinity peptides were studied as the dry powder, in water, and bound to the opsin apoprotein. We measured deuterium NMR relaxation rates of Zeeman and quadrupolar order (T1Z and T1Q) in a wide temperature range from −15 to −120 °C. The temperature dependences were fitted using two approaches: first, a model-free approach [2] and second by utilizing various models of molecular motion (3-site jump and rotational diffusion models). Jump rates (k) and rotational diffusion coefficients (corresponding to methyl group spinning and reorientation) showed site-specific methyl differences [1]. Plots of relaxation rates versus temperature for both retinal and the high-affinity GαCT2 peptide indicated that methyl group dynamics are mostly determined by intramolecular interactions. The protein environment can also affect the dynamics, either directly or through structural changes (e.g. changes of the torsion angle between the beta-ionone ring and the polyene chain) as shown by relaxation measurements. [1] A.V. Struts et al. (2011) PNAS 108, 8263. [2] X. Xu et al. (2014) eMagRes 3, 275-286.

Optimal control of overdamped systems.

Nonequilibrium physics encompasses a broad range of natural and synthetic small-scale systems. Optimizing transitions of such systems will be crucial for the development of nanoscale technologies and may reveal the physical principles underlying biological processes at the molecular level. Recent work has demonstrated that when a thermodynamic system is driven away from equilibrium then the space of controllable parameters has a Riemannian geometry induced by a generalized inverse diffusion tensor. We derive a simple, compact expression for the inverse diffusion tensor that depends solely on equilibrium information for a broad class of potentials. We use this formula to compute the minimal dissipation for two model systems relevant to small-scale information processing and biological molecular motors. In the first model, we optimally erase a single classical bit of information modeled by an overdamped particle in a smooth double-well potential. In the second model, we find the minimal dissipation of a simple molecular motor model coupled to an optical trap. In both models, we find that the minimal dissipation for the optimal protocol of duration τ is proportional to 1/τ, as expected, though the dissipation for the erasure model takes a different form than what we found previously for a similar system.

Hierarchical model of the actomyosin molecular motor based on ultrametric diffusion with drift

We discuss the approach to investigate molecular machines using systems of integro–differential ultrametric (p-adic) reaction–diffusion equations with drift. This approach combines the features of continuous and discrete dynamic models. We apply this model to investigation of actomyosin molecular motor. The introduced system of equations is solved analytically using p-adic wavelet theory. We find explicit stationary solutions and behavior in the relaxation regime.

Effective rates from thermodynamically consistent coarse-graining of models for molecular motors with probe particles.

Many single-molecule experiments for molecular motors comprise not only the motor but also large probe particles coupled to it. The theoretical analysis of these assays, however, often takes into account only the degrees of freedom representing the motor. We present a coarse-graining method that maps a model comprising two coupled degrees of freedom which represent motor and probe particle to such an effective one-particle model by eliminating the dynamics of the probe particle in a thermodynamically and dynamically consistent way. The coarse-grained rates obey a local detailed balance condition and reproduce the net currents. Moreover, the average entropy production as well as the thermodynamic efficiency is invariant under this coarse-graining procedure. Our analysis reveals that only by assuming unrealistically fast probe particles, the coarse-grained transition rates coincide with the transition rates of the traditionally used one-particle motor models. Additionally, we find that for multicyclic motors the stall force can depend on the probe size. We apply this coarse-graining method to specific case studies of the F(1)-ATPase and the kinesin motor.

Grab & Watch: Correlative Optical Tweezers-Fluorescence Microscopy (CTFM) as a Versatile Tool for Biological Studies

The combination of optical tweezers with confocal fluorescence microscopy and microfluidics is a promising approach to investigate biological system because of its unique ability to simultaneously manipulate, sense and visualize individual molecules with exquisite sensitivity and resolution. Here we present our efforts in validating this technology by developing a commercial-grade instrument that allows turn-key operation with ease of operation, without compromising performance. We call this instrument “Correlative optical Tweezers-Fluorescence Microscope” (CTFM). Using this instrument, we carry out a series of demonstration experiments, exploring applications of CTFM in the fields of DNA-protein interaction, nano-mechanics of intermediate filaments, synthetic models of signal transduction and molecular motors. These experiments show that the technological advances in hybrid single-molecule methods can be turned into an easy to use and stable instrument that has the ability to open up new venues in many research areas.

Director reorientation in a nematic liquid crystal with a photosensitive layer.

We use a molecular-motor model previously proposed for a nematic cell with an azo-dye monolayer to calculate the director orientation when light is normally impinged on the cell. We consider an initial planar configuration for which one of the surrounding plates, which we call the reference plate, is submitted to a hard-anchoring boundary condition. The other confining plate has a coating monolayer of azo-dye molecules such that the change of the orientation of azo-dye isomers, due to light, causes changes in the nematic director. The boundary conditions on both plates along with the optical field determine the director configuration in the bulk. The existence of periodic solutions for the density of isomers in trans and cis states, corresponding to weak optical fields, has been discussed in the literature. Using a similar approach, we find an approximate expression for the density of isomers, written in terms of the director angle, which allows us to close the equation for the director configuration on the boundary having a photosensitive plate. We decouple the director's angle and the isomer densities by assuming extremely different temporal time scales between them. We show for a given sample that switching times inversely depend on the trans-cis transition rate of photoexcitation whereas relaxation times do not depend on it. On the other hand, switching and relaxation times linearly depend on effective surface viscosity values. Our model allows us to estimate surface viscosity values.

Discrete Kinetic Models for Molecular Motors: Asymptotic Velocity and Gaussian Fluctuations

We consider random walks on quasi one dimensional lattices, as introduced in Faggionato and Silvestri (Random Walks on Quasi One Dimensional Lattices: Large Deviations and Fluctuation Theorems, 2014). This mathematical setting covers a large class of discrete kinetic models for non-cooperative molecular motors on periodic tracks. We derive general formulas for the asymptotic velocity and diffusion coefficient, and we show how to reduce their computation to suitable linear systems of the same degree of a single fundamental cell, with possible linear chain removals. We apply the above results to special families of kinetic models, also catching some errors in the biophysics literature.

How molecular motors work in the crowded environment of living cells: coexistence and efficiency of normal and anomalous transport.

Recent experiments reveal both passive subdiffusion of various nanoparticles and anomalous active transport of such particles by molecular motors in the molecularly crowded environment of living biological cells. Passive and active microrheology reveals that the origin of this anomalous dynamics is due to the viscoelasticity of the intracellular fluid. How do molecular motors perform in such a highly viscous, dissipative environment? Can we explain the observed co-existence of the anomalous transport of relatively large particles of 100 to 500 nm in size by kinesin motors with the normal transport of smaller particles by the same molecular motors? What is the efficiency of molecular motors in the anomalous transport regime? Here we answer these seemingly conflicting questions and consistently explain experimental findings in a generalization of the well-known continuous diffusion model for molecular motors with two conformational states in which viscoelastic effects are included.

Energetics and efficiency of a molecular motor model

The energetics and efficiency of a linear molecular motor model proposed by Mogilner et al are analyzed from an analytical point of view. The model, which is based on protein friction with a track, is described by coupled Langevin equations for the motion in combination with coupled master equations for the ATP hydrolysis. Here the energetics and efficiency of the motor are addressed using a many body scheme with focus on the efficiency at maximum power (EMP). It is found that the EMP is reduced from about 10% in a heuristic description of the motor to about 1 per mille when incorporating the full motor dynamics, owing to the strong dissipation associated with the motor action.

Stochastic thermodynamics of active Brownian particles

Examples of self-propulsion in strongly fluctuating environments are abundant in nature, e.g., molecular motors and pumps operating in living cells. Starting from the Langevin equation of motion, we develop a stochastic thermodynamic description of noninteracting self-propelled particles using simple models of velocity-dependent forces. We derive fluctuation theorems for entropy production and a modified fluctuation-dissipation relation, characterizing the linear response in nonequilibrium steady states. We study these notions in a simple model of molecular motors, and in the Rayleigh-Helmholtz and energy-depot models of self-propelled particles.

Superdiffusive transport by multivalent molecular walkers moving under load

We introduce a model for translational molecular motors to demonstrate that a multivalent catalytic walker with flexible, uncoordinated legs can transform the free energy of surface-bound substrate sites into mechanical work and undergo biased, superdiffusive motion, even in opposition to an external load force. The walker in the model lacks any inherent orientation of body or track, and its legs have no chemomechanical coupling other than the passive constraint imposed by their connection to a common body. Yet, under appropriate kinetic conditions, the walker's motion is biased in the direction of unvisited sites, which allows the walker to move nearly ballistically away from the origin as long as a local supply of unmodified substrate sites is available. The multivalent random walker model is mathematically formulated as a continuous-time Markov process and is studied numerically. We use Monte Carlo simulations to generate ensemble estimates of the mean squared displacement and mean work done for this nonergodic system. Our results show that a residence time bias between visited and unvisited sites leads to superdiffusive motion over significant times and distances. This mechanism can be used to adapt any enzyme-substrate system with appropriate kinetics for use as a functional chemical implementation of a molecular motor, without the need for structural anisotropy or conformationally mediated chemomechanical coordination.

Directed motion of periodically driven molecular motors: A graph-theoretical approach

We propose a numerical algorithm for calculation of quantized directed motion of a stochastic system of interacting particles induced by periodic changes of control parameters on the graph of microstates. As a main application, we consider models of catenane molecular motors, which demonstrated the possibility of a similar control of directed motion of molecular components. We show that our algorithm allows one to calculate the motion of a system in the space of its microstates even when the considered phase space is combinatorially large (~1 × 10(6) microscopic states). Several general observations are made about the structure of the phase diagram of the systems studied, which may be used for rational design and efficient control of new generations of molecular motors.

The single-direction energy transition model of molecular motor based on the control of adenosine triphosphate

The dynamic principle of molecular motor transport in overdamped solution remains unclear. Starting from the transport characteristics and phenomenon of the molecular motor system, the single-direction energy transition model is established, which conforms to the Langevin equation, and the stochastic dynamics of molecular motors is analyzed by Monte Carlo simulations. Results show that with the right transition energy, molecular motors could take a stable stepping motion and effective transport by means of the environment noise, and the load force can weaken material transportation of the molecular motor system. The potential field between a molecular motor and its orbit can affect the magnitude of the velocity of motor, but cannot change the direction of the velocity, the direction of motion of the molecular motor therefore is adjusted by the transition energy of the motor. In addition, although the average velocity is not zero for different noise intensities, the efficient transport of a molecular motor system indicates that the system is selective for the noise intensity.

Mechanochemical models of processive molecular motors

Motor proteins are the molecular engines powering the living cell. These nanometre-sized molecules convert chemical energy, both enthalpic and entropic, into useful mechanical work. High resolution single molecule experiments can now observe motor protein movement with increasing precision. The emerging data must be combined with structural and kinetic measurements to develop a quantitative mechanism. This article describes a modelling framework where quantitative understanding of motor behaviour can be developed based on the protein structure. The framework is applied to myosin motors, with emphasis on how synchrony between motor domains give rise to processive unidirectional movement. The modelling approach shows that the elasticity of protein domains are important in regulating motor function. Simple models of protein domain elasticity are presented. The framework can be generalized to other motor systems, or an ensemble of motors such as muscle contraction. Indeed, for hundreds of myosins, our framework can be reduced to the Huxely–Simmons description of muscle movement in the mean-field limit.

Modified fluctuation-dissipation and Einstein relation at nonequilibrium steady states

Starting from the pioneering work of Agarwal [G. S. Agarwal, Zeitschrift für Physik 252, 25 (1972)], we present a unified derivation of a number of modified fluctuation-dissipation relations (MFDR) that relate response to small perturbations around nonequilibrium steady states to steady-state correlations. Using this formalism we show the equivalence of velocity forms of MFDR derived using continuum Langevin and discrete master equation dynamics. The resulting additive correction to the Einstein relation is exemplified using a flashing ratchet model of molecular motors.

Improved Hidden Markov Models for Molecular Motors, Part 2: Extensions and Application to Experimental Data

Unbiased interpretation of noisy single molecular motor recordings remains a challenging task. To address this issue, we have developed robust algorithms based on hidden Markov models (HMMs) of motor proteins. The basic algorithm, called variable-stepsize HMM (VS-HMM), was introduced in the previous article. It improves on currently available Markov-model based techniques by allowing for arbitrary distributions of step sizes, and shows excellent convergence properties for the characterization of staircase motor timecourses in the presence of large measurement noise. In this article, we extend the VS-HMM framework for better performance with experimental data. The extended algorithm, variable-stepsize integrating-detector HMM (VSI-HMM) better models the data-acquisition process, and accounts for random baseline drifts. Further, as an extension, maximum a posteriori estimation is provided. When used as a blind step detector, the VSI-HMM outperforms conventional step detectors. The fidelity of the VSI-HMM is tested with simulations and is applied to in vitro myosin V data where a small 10 nm population of steps is identified. It is also applied to an in vivo recording of melanosome motion, where strong evidence is found for repeated, bidirectional steps smaller than 8 nm in size, implying that multiple motors simultaneously carry the cargo.

Two coarse-graining studies of stochastic models in molecular biology

We examine stochastic coarse-graining strategies for two biomolecular systems. First, we compute the large-scale transport properties of the basic flashing ratchet mathematical model for (Brownian) molecular motors and consider in this light whether the underlying continuous-space, continuous-time Markovian model can be coarse-grained as a discrete-state, continuous-time Markovian random walk model. Through careful computation of associated statistical signatures of Markovianity, we find that such a discrete coarse-graining is an excellent approximation over much but not all of the parameter regime. In particular, for the parameter values associated with the fastest transport by the flashing ratchet, the discretized model displays non-Markovian features such as waiting times between jumps which are not exponentially distributed. We provide a theoretical framework for understanding the conditions under which Markovianity is to be expected in the discretized model and two mechanisms by which the flashing ratchet model coarse-grains to a non-Markovian discretized model. Next we turn to a basic question of how the dynamics of water molecules near the surface of a solute can be represented by a simple drift-diffusion stochastic model. This question is of most interest for the purpose of accelerating molecular dynamics simulations of proteins, but for simplicity, we here examine the simple case where the solute is a C60 buckyball, which has a homogenous, roughly isotropic form. We compare the mathematical drift-diffusion framework with a statistical quantification of water dynamics near a solute discussed in the biophysical literature. A key concern is the choice of time interval on which to sample the molecular dynamics data to generate estimators for the drift and diffusivity. We use a simple mathematical toy model to establish insight and a strategy, but find for the actual molecular dynamics data, that the sampling times which produce the most faithful drift coefficient and the sampling times which produce the most faithful diffusion coefficient do not overlap, so that sacrifice of quality in one or the other parameter appears necessary.

A Mechanomolecular Model for Chromosome Movement During Mitosis

During mitosis chromosomes attach to dynamic microtubules in order to move to the cell equator. Kinetochores facilitate chromosome movement by maintaining a floating grip on inserted polymerizing or depolymerizing microtubules. In many vertebrates, chromosomes experience striking oscillatory movement both close to a pole and around the equator. Several chemical species that affect microtubule dynamics as well as proteins that bind to the microtubule lattice have been localized at kinetochores. Yet, there is no clear understanding of the force producing mechanism at microtubule attachment sites, and of how movement is controlled and coordinated at kinetochores so that a chromosome pinpoints the cell equator. Here we develop a mathematical model of chromosome motility that addresses the above two questions. First, we consider force production at kinetochores by developing a molecular motor model that describes molecular scale interactions between several kinetochore binders and an inserted microtubule. We find that for weak binding and low activation energies, the motor produces velocities that are fairly insensitive to loads and depend on the inserted polymer growth/shortening rates, in agreement with chromosome movement data. Then, we incorporate the molecular motor into a chromosome movement model in which motility results due to feedback between mechanical loads that arise from spindle-chromosome interactions and kinetochore chemical reactions that involve a force sensing kinase and a microtubule depolymerase.Numerical simulations show that kinetochore chemical control of movement is a robust mechanism for chromosome oscillations and centering at the equator. Further, we observe that proper transition between mitotic stages is strongly influenced by kinetochore reactions, which indicates that motility does not solely result from “tug of war” between opposing forces. We find that our proposed mechanism is sufficient to recreate chromosome movement from prometaphase to anaphase, in good agreement with experimental data.

Exact solution of the discrete (1+1)-dimensional RSOS model with field and surface interactions

We present the solution of a linear restricted solid-on-solid (RSOS) model in a field. Aside from the origins of this model in the context of describing the phase boundary in a magnet, interest also comes from more recent work on the steady state of non-equilibrium models of molecular motors. While similar to a previously solved (non-restricted) SOS model in its physical behaviour, mathematically the solution is more complex. Involving basic hypergeometric functions 3ϕ2, it introduces a new form of solution to the lexicon of directed lattice path generating functions.