Motor Protein Transport Along Inhomogeneous Microtubules.

Many cellular processes rely on the cell's ability to transport material to and from the nucleus. Networks consisting of many microtubules and actin filaments are key to this transport. Recently, the inhibition of intracellular transport has been implicated in neurodegenerative diseases such as Alzheimer's disease and Amyotrophic Lateral Sclerosis. Furthermore, microtubules may contain so-called defective regions where motor protein velocity is reduced due to accumulation of other motors and microtubule-associated proteins. In this work, we propose a new mathematical model describing the motion of motor proteins on microtubules which incorporate a defective region. We take a mean-field approach derived from a first principle lattice model to study motor protein dynamics and density profiles. In particular, given a set of model parameters we obtain a closed-form expression for the equilibrium density profile along a given microtubule. We then verify the analytic results using mathematical analysis on the discrete model and Monte Carlo simulations. This work will contribute to the fundamental understanding of inhomogeneous microtubules providing insight into microscopic interactions that may result in the onset of neurodegenerative diseases. Our results for inhomogeneous microtubules are consistent with prior work studying the homogeneous case.

Self-Organized Lane Formation in Bidirectional Transport by Molecular Motors.

Within cells, vesicles and proteins are actively transported several micrometers along the cytoskeletal filaments. The transport along microtubules is propelled by dynein and kinesin motors, which carry the cargo in opposite directions. Bidirectional intracellular transport is performed with great efficiency, even under strong confinement, as for example in the axon. For this kind of transport system, one would expect generically cluster formation. In this Letter, we discuss the effect of the recently observed self-enhanced binding affinity along the kinesin trajectories on the microtubule. We introduce a stochastic lattice-gas model, where the enhanced binding affinity is realized via a floor field. From Monte Carlo simulations and a mean-field analysis we show that this mechanism can lead to self-organized symmetry breaking and lane formation that indeed leads to efficient bidirectional transport in narrow environments.

Limited processivity of single motors improves overall transport flux of self-assembled motor-cargo complexes.

Single kinesin molecular motors can processively move along a microtubule (MT) a few micrometers on average before dissociating. However, cellular length scales over which transport occurs are several hundred microns and more. Why seemingly unreliable motors are used to transport cellular cargo remains poorly understood. We propose a theory for how low processivity, the average length of a single bout of directed motion, can enhance cellular transport when motors and cargos must first diffusively self-assemble into complexes. We employ stochastic modeling to determine the effect of processivity on overall cargo transport flux. We show that, under a wide range of physiologically relevant conditions, possessing "infinite" processivity does not maximize flux along MTs. Rather, we find that lowering processivity, i.e., weaker binding of motors to MTs, can improve transport flux. These results shed light on the relationship between processivity and transport efficiency and offer a theory for the physiological benefits of low motor processivity.

Bidirectional sliding of two parallel microtubules generated by multiple identical motors.

It is often assumed in biophysical studies that when multiple identical molecular motors interact with two parallel microtubules, the microtubules will be crosslinked and locked together. The aim of this study is to examine this assumption mathematically. We model the forces and movements generated by motors with a time-continuous Markov process and find that, counter-intuitively, a tug-of-war results from opposing actions of identical motors bound to different microtubules. The model shows that many motors bound to the same microtubule generate a great force applied to a smaller number of motors bound to another microtubule, which increases detachment rate for the motors in minority, stabilizing the directional sliding. However, stochastic effects cause occasional changes of the sliding direction, which has a profound effect on the character of the long-term microtubule motility, making it effectively diffusion-like. Here, we estimate the time between the rare events of switching direction and use them to estimate the effective diffusion coefficient for the microtubule pair. Our main result is that parallel microtubules interacting with multiple identical motors are not locked together, but rather slide bidirectionally. We find explicit formulae for the time between directional switching for various motor numbers.

Cooperative motor action to regulate microtubule length dynamics.

Motivated by the recent experimental observations on motor induced cooperative mechanism controlling the length dynamics of microtubules (MTs), we examine how plus-end-targeted proteins of the kinesin family regulate MT polymerization and depolymerization routines. Here, we study a stochastic mathematical model capturing the unusual form of collective motor interaction on MT dynamics originating due to the molecular traffic near the MT tip. We provide an extensive analysis of the joint effect of motor impelled MT polymerization and complete depolymerization. The effect of the cooperative action is included by modifying the intrinsic depolymerization rate. We analyze the model within the framework of continuum mean-field theory and the resultant steady-state analytic solution is expressed in terms of Lambert W functions. Four distinct steady-state phases including a shock phase have been reported. The significant features of the shock including its position and height have been analyzed. Theoretical outcomes are supported by extensive Monte Carlo simulations. To explore the system alterations between the regime of growth and shrinkage phase, we consider kymographs of the microtubule along with the length distributions. Finally, we investigated the dependence of MT length kinetics both on modifying factor of depolymerization rate and motor concentration. The overall extensive study reveals that the flux of molecular traffic at the microtubule plus end initiates a cooperative mechanism, resulting in significant change in MT growth and shrinkage regime as also observed experimentally.

Cargo diffusion shortens single-kinesin runs at low viscous drag.

Molecular motors such as kinesin-1 drive active, long-range transport of cargos along microtubules in cells. Thermal diffusion of the cargo can impose a randomly directed, fluctuating mechanical load on the motor carrying the cargo. Recent experiments highlighted a strong asymmetry in the sensitivity of single-kinesin run length to load direction, raising the intriguing possibility that cargo diffusion may non-trivially influence motor run length. To test this possibility, here we employed Monte Carlo-based simulations to evaluate the transport of cargo by a single kinesin. Our simulations included physiologically relevant viscous drag on the cargo and interrogated a large parameter space of cytoplasmic viscosities, cargo sizes, and motor velocities that captures their respective ranges in living cells. We found that cargo diffusion significantly shortens single-kinesin runs. This diffusion-based shortening is countered by viscous drag, leading to an unexpected, non-monotonic variation in run length as viscous drag increases. To our knowledge, this is the first identification of a significant effect of cargo diffusion on motor-based transport. Our study highlights the importance of cargo diffusion and load-detachment kinetics on single-motor functions under physiologically relevant conditions.

Perfect anomalous transport of subdiffusive cargos by molecular motors in viscoelastic cytosol.

Multiple experiments show that various submicron particles such as magnetosomes, RNA messengers, viruses, and even much smaller nanoparticles such as globular proteins diffuse anomalously slow in viscoelastic cytosol of living cells. Hence, their sufficiently fast directional transport by molecular motors such as kinesins is crucial for the cell operation. It has been shown recently that the traditional flashing Brownian ratchet models of molecular motors are capable to describe both normal and anomalous transport of such subdiffusing cargos by molecular motors with a very high efficiency. This work elucidates further an important role of mechanochemical coupling in such an anomalous transport. It shows a natural emergence of a perfect subdiffusive ratchet regime due to allosteric effects, where the random rotations of a "catalytic wheel" at the heart of the motor operation become perfectly synchronized with the random stepping of a heavily loaded motor, so that only one ATP molecule is consumed on average at each motor step along microtubule. However, the number of rotations made by the catalytic engine and the traveling distance both scale sublinearly in time. Nevertheless, this anomalous transport can be very fast in absolute terms.

Molecular motor translocation kinetics: Application of Monte Carlo computer simulations to determine microscopic kinetic parameters.

Methods for studying the translocation of motor proteins along a filament (e.g., nucleic acid and polypeptide) typically monitor the total production of ADP, the arrival/departure of the motor protein at/from a particular location (often one end of the filament), or the dissociation of the motor protein from the filament. The associated kinetic time courses are often analyzed using a simple sequential uniform n-step mechanism to estimate the macroscopic kinetic parameters (e.g., translocation rate and processivity) and the microscopic kinetic parameters (e.g., kinetic step-size and the rate constant for the rate-limiting step). These sequential uniform n-step mechanisms assume repetition of uniform and irreversible rate-limiting steps of forward motion along the filament. In order to determine how the presence of non-uniform motion (e.g., backward motion, random pauses, or jumping) affects the estimates of parameters obtained from such analyses, we evaluated computer simulated translocation time courses containing non-uniform motion using a simple sequential uniform n-step model. By comparing the kinetic parameters estimated from the analysis of the data generated by these simulations with the input parameters of the simulations, we were able to determine which of the kinetic parameters were likely to be over/under estimated due to non-uniform motion of the motor protein.

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.

Driven transport on open filaments with interfilament switching processes.

We study a two-filament driven lattice gas model with oppositely directed species of particles moving on two parallel filaments with filament-switching processes and particle inflow and outflow at filament ends. The filament-switching process is correlated with the occupation number of the adjacent site such that particles switch filaments with finite probability only when oppositely directed particles meet on the same filament. This model mimics some of the coarse-grained features observed in context of microtubule-(MT) based intracellular transport, wherein cellular cargo loaded and off-loaded at filament ends are transported on multiple parallel MT filaments and can switch between the parallel microtubule filaments. We focus on a regime where the filaments are weakly coupled, such that filament-switching rate of particles scale inversely as the length of the filament. We find that the interplay of (off-) loading processes at the boundaries and the filament-switching process of particles leads to some distinctive features of the system. These features includes occurrence of a variety of phases in the system with inhomogeneous density profiles including localized density shocks, density difference across the filaments, and bidirectional current flows in the system. We analyze the system by developing a mean field (MF) theory and comparing the results obtained from the MF theory with the Monte Carlo (MC) simulations of the dynamics of the system. We find that the steady-state density and current profiles of particles and the phase diagram obtained within the MF picture matches quite well with MC simulation results. These findings maybe useful for studying multifilament intracellular transport.

Phase-plane analysis of the totally asymmetric simple exclusion process with binding kinetics and switching between antiparallel lanes.

Motor protein motion on biopolymers can be described by models related to the totally asymmetric simple exclusion process (TASEP). Inspired by experiments on the motion of kinesin-4 motors on antiparallel microtubule overlaps, we analyze a model incorporating the TASEP on two antiparallel lanes with binding kinetics and lane switching. We determine the steady-state motor density profiles using phase-plane analysis of the steady-state mean field equations and kinetic Monte Carlo simulations. We focus on the density-density phase plane, where we find an analytic solution to the mean field model. By studying the phase-space flows, we determine the model's fixed points and their changes with parameters. Phases previously identified for the single-lane model occur for low switching rate between lanes. We predict a multiple coexistence phase due to additional fixed points that appear as the switching rate increases: switching moves motors from the higher-density to the lower-density lane, causing local jamming and creating multiple domain walls. We determine the phase diagram of the model for both symmetric and general boundary conditions.

Motor Protein Accumulation on Antiparallel Microtubule Overlaps.

Biopolymers serve as one-dimensional tracks on which motor proteins move to perform their biological roles. Motor protein phenomena have inspired theoretical models of one-dimensional transport, crowding, and jamming. Experiments studying the motion of Xklp1 motors on reconstituted antiparallel microtubule overlaps demonstrated that motors recruited to the overlap walk toward the plus end of individual microtubules and frequently switch between filaments. We study a model of this system that couples the totally asymmetric simple exclusion process for motor motion with switches between antiparallel filaments and binding kinetics. We determine steady-state motor density profiles for fixed-length overlaps using exact and approximate solutions of the continuum differential equations and compare to kinetic Monte Carlo simulations. Overlap motor density profiles and motor trajectories resemble experimental measurements. The phase diagram of the model is similar to the single-filament case for low switching rate, while for high switching rate we find a new (to our knowledge) low density-high density-low density-high density phase. The overlap center region, far from the overlap ends, has a constant motor density as one would naïvely expect. However, rather than following a simple binding equilibrium, the center motor density depends on total overlap length, motor speed, and motor switching rate. The size of the crowded boundary layer near the overlap ends is also dependent on the overlap length and switching rate in addition to the motor speed and bulk concentration. The antiparallel microtubule overlap geometry may offer a previously unrecognized mechanism for biological regulation of protein concentration and consequent activity.

Models for microtubule cargo transport coupling the Langevin equation to stochastic stepping motor dynamics: Caring about fluctuations.

One-dimensional models coupling a Langevin equation for the cargo position to stochastic stepping dynamics for the motors constitute a relevant framework for analyzing multiple-motor microtubule transport. In this work we explore the consistence of these models focusing on the effects of the thermal noise. We study how to define consistent stepping and detachment rates for the motors as functions of the local forces acting on them in such a way that the cargo velocity and run-time match previously specified functions of the external load, which are set on the base of experimental results. We show that due to the influence of the thermal fluctuations this is not a trivial problem, even for the single-motor case. As a solution, we propose a motor stepping dynamics which considers memory on the motor force. This model leads to better results for single-motor transport than the approaches previously considered in the literature. Moreover, it gives a much better prediction for the stall force of the two-motor case, highly compatible with the experimental findings. We also analyze the fast fluctuations of the cargo position and the influence of the viscosity, comparing the proposed model to the standard one, and we show how the differences on the single-motor dynamics propagate to the multiple motor situations. Finally, we find that the one-dimensional character of the models impede an appropriate description of the fast fluctuations of the cargo position at small loads. We show how this problem can be solved by considering two-dimensional models.

ATP hydrolysis assists phosphate release and promotes reaction ordering in F1-ATPase.

F1-ATPase (F1) is a rotary motor protein that can efficiently convert chemical energy to mechanical work of rotation via fine coordination of its conformational motions and reaction sequences. Compared with reactant binding and product release, the ATP hydrolysis has relatively little contributions to the torque and chemical energy generation. To scrutinize possible roles of ATP hydrolysis, we investigate the detailed statistics of the catalytic dwells from high-speed single wild-type F1 observations. Here we report a small rotation during the catalytic dwell triggered by the ATP hydrolysis that is indiscernible in previous studies. Moreover, we find in freely rotating F1 that ATP hydrolysis is followed by the release of inorganic phosphate with low synthesis rates. Finally, we propose functional roles of the ATP hydrolysis as a key to kinetically unlock the subsequent phosphate release and promote the correct reaction ordering.

Catch-slip bonds can be dispensable for motor force regulation during skeletal muscle contraction.

It is intriguing how multiple molecular motors can perform coordinated and synchronous functions, which is essential in various cellular processes. Recent studies on skeletal muscle might have shed light on this issue, where rather precise motor force regulation was partly attributed to the specific stochastic features of a single attached myosin motor. Though attached motors can randomly detach from actin filaments either through an adenosine triphosphate (ATP) hydrolysis cycle or through "catch-slip bond" breaking, their respective contribution in motor force regulation has not been clarified. Here, through simulating a mechanical model of sarcomere with a coupled Monte Carlo method and finite element method, we find that the stochastic features of an ATP hydrolysis cycle can be sufficient while those of catch-slip bonds can be dispensable for motor force regulation.

Thermodynamic uncertainty relation for biomolecular processes.

Biomolecular systems like molecular motors or pumps, transcription and translation machinery, and other enzymatic reactions, can be described as Markov processes on a suitable network. We show quite generally that, in a steady state, the dispersion of observables, like the number of consumed or produced molecules or the number of steps of a motor, is constrained by the thermodynamic cost of generating it. An uncertainty ε requires at least a cost of 2k(B)T/ε2 independent of the time required to generate the output.

Multiscale polar theory of microtubule and motor-protein assemblies.

Microtubules and motor proteins are building blocks of self-organized subcellular biological structures such as the mitotic spindle and the centrosomal microtubule array. These same ingredients can form new "bioactive" liquid-crystalline fluids that are intrinsically out of equilibrium and which display complex flows and defect dynamics. It is not yet well understood how microscopic activity, which involves polarity-dependent interactions between motor proteins and microtubules, yields such larger-scale dynamical structures. In our multiscale theory, Brownian dynamics simulations of polar microtubule ensembles driven by cross-linking motors allow us to study microscopic organization and stresses. Polarity sorting and cross-link relaxation emerge as two polar-specific sources of active destabilizing stress. On larger length scales, our continuum Doi-Onsager theory captures the hydrodynamic flows generated by polarity-dependent active stresses. The results connect local polar structure to flow structures and defect dynamics.

Muscle activation described with a differential equation model for large ensembles of locally coupled molecular motors.

Molecular motors, by turning chemical energy into mechanical work, are responsible for active cellular processes. Often groups of these motors work together to perform their biological role. Motors in an ensemble are coupled and exhibit complex emergent behavior. Although large motor ensembles can be modeled with partial differential equations (PDEs) by assuming that molecules function independently of their neighbors, this assumption is violated when motors are coupled locally. It is therefore unclear how to describe the ensemble behavior of the locally coupled motors responsible for biological processes such as calcium-dependent skeletal muscle activation. Here we develop a theory to describe locally coupled motor ensembles and apply the theory to skeletal muscle activation. The central idea is that a muscle filament can be divided into two phases: an active and an inactive phase. Dynamic changes in the relative size of these phases are described by a set of linear ordinary differential equations (ODEs). As the dynamics of the active phase are described by PDEs, muscle activation is governed by a set of coupled ODEs and PDEs, building on previous PDE models. With comparison to Monte Carlo simulations, we demonstrate that the theory captures the behavior of locally coupled ensembles. The theory also plausibly describes and predicts muscle experiments from molecular to whole muscle scales, suggesting that a micro- to macroscale muscle model is within reach.

Assessment of phosphorylation in Toxoplasma glideosome assembly and function.

Members of the phylum Apicomplexa possess a highly conserved molecular motor complex anchored in the parasite pellicle and associated with gliding motility, invasion and egress from infected cells. This machinery, called the glideosome, is structured around the acylated gliding-associated protein GAP45 that recruits the motor complex composed of myosin A and two associated myosin light chains (TgMLC1 and TgELC1). This motor is presumably firmly anchored to the inner membrane complex underneath the plasma membrane via an interaction with two integral membrane proteins, GAP50 and GAP40. To determine if the previously mapped phosphorylation sites on TgGAP45 and TgMLC1 have a direct significance for glideosome assembly and function, a series of phospho-mimetic and phospho-null mutants were generated. Neither the overexpression nor the allelic replacement of TgMLC1 with phospho-mutants impacted on glideosome assembly and parasite motility. TgGAP45 phosphorylation mutants were functionally investigated using a complementation strategy in a TgGAP45 inducible knockout background. The loss of interaction with TgGAP50 by one previously reported GAP45-mutant appeared to depend only on the presence of a remaining competing wild type copy of TgGAP45. Accordingly, this mutant displayed no phenotype in complementation experiments. Unexpectedly, GAP45 lacking the region encompassing the cluster of twelve phosphorylation sites did not impact on its dual function in motor recruitment and pellicle integrity. Despite the extensive phosphorylation of TgMLC1 and TgGAP45, this post-translational modification does not appear to be critical for the assembly and function of the glideosome.

Memory, bias, and correlations in bidirectional transport of molecular-motor-driven cargoes.

Molecular motors are specialized proteins that perform active, directed transport of cellular cargoes on cytoskeletal filaments. In many cases, cargo motion powered by motor proteins is found to be bidirectional, and may be viewed as a biased random walk with fast unidirectional runs interspersed with slow tug-of-war states. The statistical properties of this walk are not known in detail, and here, we study memory and bias, as well as directional correlations between successive runs in bidirectional transport. We show, based on a study of the direction-reversal probabilities of the cargo using a purely stochastic (tug-of-war) model, that bidirectional motion of cellular cargoes is, in general, a correlated random walk. In particular, while the motion of a cargo driven by two oppositely pulling motors is a Markovian random walk, memory of direction appears when multiple motors haul the cargo in one or both directions. In the latter case, the Markovian nature of the underlying single-motor processes is hidden by internal transitions between degenerate run and pause states of the cargo. Interestingly, memory is found to be a nonmonotonic function of the number of motors. Stochastic numerical simulations of the tug-of-war model support our mathematical results and extend them to biologically relevant situations.