Distinct transport regimes for two elastically coupled molecular motors

Cooperation and competition in the collective drive by motor proteins: mean active force, fluctuations, and self-load

We consider the dynamics of a bio-filament under the collective drive of motor proteins. They are attached irreversibly to a substrate and undergo stochastic attachment-detachment with the filament to produce a directed force on it. We establish the dependence of the mean directed force and force correlations on the parameters describing the individual motor proteins using analytical theory and direct numerical simulations. The effective Langevin description for the filament motion gives mean-squared displacement, asymptotic diffusion constant, and mobility leading to an effective temperature. Finally, we show how competition between motor protein extensions generates a self-load, describable in terms of the effective temperature, affecting the filament motion.

Tug-of-War and Coordination in Bidirectional Transport by Molecular Motors

Many cargoes in cells are transported in a bidirectional fashion by molecular motors pulling into opposite directions along a cytoskeletal filament, e.g., by kinesins and dyneins along microtubules. How opposite-polarity motors are coordinated has been under debate for a long time, with experimental evidence supporting both a tug-of-war between the motors as well as biochemical coordination mechanisms. Here we propose a model that extends a tug-of-war model by a mechanism of motor activation and inactivation and show that this model can explain some observations that are incompatible with a simple tug-of-war scenario, specifically long unidirectional runs and a directional memory after unbinding from the filament. Both features are present in two variants of the model in which motors are activated and inactivated individually and in opposite-direction pairs, respectively.

Cooperation and Competition Coexist in Bidirectional Transport by Motor Proteins

In intracellular transport, the cargo is usually simultaneously carried by two types of motor proteins that move oppositely, widely described as a "tug-of-war". We show theoretically that apart from the apparent competition, there is also a unintuitive cooperation between motors with opposite directionality. The model reproduces the in vivo experimental data with high accuracy. Under certain conditions, the cooperation can significantly increase the transport distance, rationalizing the choice of bidirectional over unidirectional transport in evolution. We further derive the exact analytical solution for the transport distance. Our results pave the road to understanding the physical nature of intracellular transport by motor proteins.

Cargo surface fluidity can reduce inter-motor mechanical interference, promote load-sharing and enhance processivity in teams of molecular motors

In cells, multiple molecular motors work together as teams to carry cargoes such as vesicles and organelles over long distances to their destinations by stepping along a network of cytoskeletal filaments. How motors that typically mechanically interfere with each other, work together as teams is unclear. Here we explored the possibility that purely physical mechanisms, such as cargo surface fluidity, may potentially enhance teamwork, both at the single motor and cargo level. To explore these mechanisms, we developed a three dimensional simulation of cargo transport along microtubules by teams of kinesin-1 motors. We accounted for cargo membrane fluidity by explicitly simulating the Brownian dynamics of motors on the cargo surface and considered both the load and ATP dependence of single motor functioning. Our simulations show that surface fluidity could lead to the reduction of negative mechanical interference between kinesins and enhanced load sharing thereby increasing the average duration of single motors on the filament. This, along with a cooperative increase in on-rates as more motors bind leads to enhanced collective processivity. At the cargo level, surface fluidity makes more motors available for binding, which can act synergistically with the above effects to further increase transport distances though this effect is significant only at low ATP or high motor density. Additionally, the fluid surface allows for the clustering of motors at a well defined location on the surface relative to the microtubule and the fluid-coupled motors can exert more collective force per motor against loads. Our work on understanding how teamwork arises in cargo-coupled motors allows us to connect single motor properties to overall transport, sheds new light on cellular processes, reconciles existing observations, encourages new experimental validation efforts and can also suggest new ways of improving the transport of artificial cargo powered by motor teams.

A change point analysis protocol for comparing intracellular transport by different molecular motor combinations

Intracellular transport by microtubule-based molecular motors is marked by qualitatively different behaviors. It is a long-standing and still-open challenge to accurately quantify the various individual-cargo behaviors and how they are affected by the presence or absence of particular motor families. In this work we introduce a protocol for analyzing change points in cargo trajectories that can be faithfully projected along the length of a (mostly) straight microtubule. Our protocol consists of automated identification of velocity change points, estimation of velocities during the behavior segments, and extrapolation to motor-specific velocity distributions. Using simulated data we show that our method compares favorably with existing methods. We then apply the technique to data sets in which quantum dots are transported by Kinesin-1, by Dynein-Dynactin-BicD2 (DDB), and by Kinesin-1/DDB pairs. In the end, we identify pausing behavior that is consistent with some tug-of-war model predictions, but also demonstrate that the simultaneous presence of antagonistic motors can lead to long processive runs that could contribute favorably to population-wide transport.

Dynamic catch-bonding generates the large stall forces of cytoplasmic dynein

Cytoplasmic dynein is an important molecular motor involved in the transport of vesicular and macromolecular cargo along microtubules in cells, often in conjunction with kinesin motors. Dynein is larger and more complex than kinesin and the mechanism and regulation of its movement is currently the subject of intense research. While it was believed for a long time that dynein motors are relatively weak in terms of the force they can generate, recent studies have shown that interactions with regulatory proteins confer large stall forces comparable to those of kinesin. This paper reports on a theoretical study which suggests that these large stall forces may be the result of an emergent, ATP-dependent, bistability resulting in a dynamic catch-bonding behavior that can cause the motor to switch between high and low load-force states.

Theoretical Analysis of Run Length Distributions for Coupled Motor Proteins

Motor proteins, also known as biological molecular motors, play important roles in various biological processes. In recent years, properties of single-motor proteins have been intensively investigated using multiple experimental and theoretical tools. However, in real cellular systems biological motors typically function in groups, but the mechanisms of their collective dynamics remain not well understood. Here we investigate theoretically distributions of run lengths for coupled motor proteins that move along linear tracks. Our approach utilizes a method of first-passage processes, which is supplemented by Monte Carlo computer simulations. Theoretical analysis allowed us to clarify several aspects of the cooperativity mechanisms for coupled biological molecular motors. It is found that the run length distribution for two motors, in contrast to single motors, is nonmonotonic. In addition, the transport efficiency of two-motor complexes might be strongly increased. We also argue that the degree of cooperativity is influenced by several characteristics of motor proteins such as the strength of intermolecular interactions, stall forces, dissociations constants, and the detachment forces. The application of our theoretical analysis for several motor proteins is also discussed.

Force dependence of unbinding rate of kinesin motor during its processive movement on microtubule

Kinesin is a biological molecular motor that can move continuously on microtubule until it unbinds. Here, we studied computationally the force dependence of the unbinding rate of the motor. Our results showed that while the unbinding rate under the forward load has the expected characteristic of "slip bond", with the unbinding rate increasing monotonically with the increase of the forward load, the unbinding rate under the backward load shows counterintuitive characteristic of "slip-catch-slip bond": as the backward load increases, the unbinding rate increases exponentially firstly, then drops rapidly and then increases again. Our calculated data are in agreement with the available single-molecule data from different research groups. The mechanism of the slip-catch-slip bond was revealed.

Force Generated by Two Kinesin Motors Depends on the Load Direction and Intermolecular Coupling

Kinesins are molecular motors that carry cellular cargoes. While the mechanics of single kinesins are well characterized experimentally, the behavior of multiple kinesins varies considerably among experiments. The basis for this variability is unknown. Here, we resolve single-motor force measurements into a vertical component, which accelerates kinesin detachment, and a horizontal component, which decelerates the detachment when resisting the motor. This directionality, when the different experimental geometries are considered, can account for much of the variation in multiple motor dynamics.

Force-Dependent Unbinding Rate of Molecular Motors from Stationary Optical Trap Data

Molecular motors walk along filaments until they detach stochastically with a force-dependent unbinding rate. Here, we show how this unbinding rate can be obtained from the analysis of experimental data of molecular motors moving in stationary optical traps. Two complementary methods are presented, based on the analysis of the distribution for the unbinding forces and of the motor's force traces. In the first method, analytically derived force distributions for slip bonds, slip-ideal bonds, and catch bonds are used to fit the cumulative distributions of the unbinding forces. The second method is based on the statistical analysis of the observed force traces. We validate both methods with stochastic simulations and apply them to experimental data for kinesin-1.

Dynamics of cooperative cargo transport by two elastically coupled kinesin motors

Intracellular transport is performed often by multiple motor proteins bound to the same cargo. Here, we study theoretically collective transport of the cargo by two kinesin motors. We propose that the motor has only the elastic interaction with the cargo via the linker connecting them and has no interaction with another motor. With parameters values for single motors from the available single-molecule data, we show that at linker's elastic strength [Formula: see text] pN/nm the theoretical data of both velocity and run length of the two-motor assembly under no load are identical to the available experimental data. The run length distribution is single exponential. The single-motor-bound state of the assembly dominates the transport. Both the force dependence of the velocity of the cargo driven by single load-bearing motor and that by two load-bearing motors in the assembly are consistent with the experimental data. The stall force of the assembly is larger than the sum of stall forces of two uncoupled motors. Moreover, we predict that the stall force increases with the increase of K and becomes saturated at large K, with the saturated value being 1.5-fold larger than the sum of stall forces of the two uncoupled motors.

Complex nearly immotile behaviour of enzymatically driven cargos

We report a minimal microtubule-based motile system displaying signatures of unconventional diffusion. The system consists of a single model cargo driven by an ensemble of N340K NCD motors along a single microtubule. Despite the absence of cytosolic or cytoskeleton complexity, the system shows complex behavior, characterized by sub-diffusive motion for short time lag scales and linear mean squared displacement dependence for longer time lags. The latter is also shown to have non-Gaussian character and cannot be ascribed to a canonical diffusion process. We use single particle tracking and analysis at varying temperatures and motor concentrations to identify the origin of these behaviors as enzymatic activity of mutant NCD. Our results show that signatures of non-Gaussian diffusivities can arise as a result of an active process and suggest that some immotility of cargos observed in cells may reflect the ensemble workings of mechanochemical enzymes and need not always reflect the properties of the cytoskeletal network or the cytosol.

Force sharing and force generation by two teams of elastically coupled molecular motors

Many active cellular processes such as long-distance cargo transport, spindle organization, as well as flagellar and ciliary beating are driven by molecular motors. These motor proteins act collectively and typically work in small teams. One particularly interesting example is two teams of antagonistic motors that pull a common cargo into opposite directions, thereby generating mutual interaction forces. Important issues regarding such multiple motor systems are whether or not motors from the same team share their load equally, and how the collectively generated forces depend on the single motor properties. Here we address these questions by introducing a stochastic model for cargo transport by an arbitrary number of elastically coupled molecular motors. We determine the state space of this motor system and show that this space has a rather complex and nested structure, consisting of multiple activity states and a large number of elastic substates, even for the relatively small system of two identical motors working against one antagonistic motor. We focus on this latter case because it represents the simplest tug-of-war that involves force sharing between motors from the same team. We show that the most likely motor configuration is characterized by equal force sharing between identical motors and that the most likely separation of these motors corresponds to a single motor step. These likelihoods apply to different types of motors and to different elastic force potentials acting between the motors. Furthermore, these features are observed both in the steady state and during the initial build-up of elastic strains. The latter build-up is non-monotonic and exhibits a maximum at intermediate times, a striking consequence of mutual unbinding of the elastically coupled motors. Mutual strain-induced unbinding also reduces the magnitude of the collectively generated forces. Our computational approach is quite general and can be extended to other motor systems such as motor teams working against an optical trap or mixed teams of motors with different single motor properties.

Non-Markovian intracellular transport with sub-diffusion and run-length dependent detachment rate

Intracellular transport of organelles is fundamental to cell function and health. The mounting evidence suggests that this transport is in fact anomalous. However, the reasons for the anomaly is still under debate. We examined experimental trajectories of organelles inside a living cell and propose a mathematical model that describes the previously reported transition from sub-diffusive to super-diffusive motion. In order to explain super-diffusive behaviour at long times, we introduce non-Markovian detachment kinetics of the cargo: the rate of detachment is inversely proportional to the time since the last attachment. Recently, we observed the non-Markovian detachment rate experimentally in eukaryotic cells. Here we further discuss different scenarios of how this effective non-Markovian detachment rate could arise. The non-Markovian model is successful in simultaneously describing the time averaged variance (the time averaged mean squared displacement corrected for directed motion), the mean first passage time of trajectories and the multiple peaks observed in the distributions of cargo velocities. We argue that non-Markovian kinetics could be biologically beneficial compared to the Markovian kinetics commonly used for modelling, by increasing the average distance the cargoes travel when a microtubule is blocked by other filaments. In turn, sub-diffusion allows cargoes to reach neighbouring filaments with higher probability, which promotes active motion along the microtubules.

Dissecting myosin-5B mechanosensitivity and calcium regulation at the single molecule level

Myosin-5B is one of three members of the myosin-5 family of actin-based molecular motors. Despite its fundamental role in recycling endosome trafficking and in collective actin network dynamics, the molecular mechanisms underlying its motility are inherently unknown. Here we combine single-molecule imaging and high-speed laser tweezers to dissect the mechanoenzymatic properties of myosin-5B. We show that a single myosin-5B moves processively in 36-nm steps, stalls at ~2 pN resistive forces, and reverses its directionality at forces >2 pN. Interestingly, myosin-5B mechanosensitivity differs from that of myosin-5A, while it is strikingly similar to kinesin-1. In particular, myosin-5B run length is markedly and asymmetrically sensitive to force, a property that might be central to motor ensemble coordination. Furthermore, we show that Ca2+ does not affect the enzymatic activity of the motor unit, but abolishes myosin-5B processivity through calmodulin dissociation, providing important insights into the regulation of postsynaptic cargoes trafficking in neuronal cells.

The Role of Supercoiling in the Motor Activity of RNA Polymerases

RNA polymerase (RNAP) is, in its elongation phase, an emblematic example of a molecular motor whose activity is highly sensitive to DNA supercoiling. After a review of DNA supercoiling basic features, we discuss how supercoiling controls polymerase velocity, while being itself modified by polymerase activity. This coupling is supported by single-molecule measurements. Physical modeling allows us to describe quantitatively how supercoiling and torsional constraints mediate a mechanical coupling between adjacent polymerases. On this basis, we obtain a description that may explain the existence and functioning of RNAP convoys.

Cargo Transport by Two Coupled Myosin Va Motors on Actin Filaments and Bundles

Myosin Va (myoVa) is a processive, actin-based molecular motor essential for intracellular cargo transport. When a cargo is transported by an ensemble of myoVa motors, each motor faces significant physical barriers and directional challenges created by the complex actin cytoskeleton, a network of actin filaments and actin bundles. The principles that govern the interaction of multiple motors attached to the same cargo are still poorly understood. To understand the mechanical interactions between multiple motors, we developed a simple in vitro model in which two individual myoVa motors labeled with different-colored Qdots are linked via a third Qdot that acts as a cargo. The velocity of this two-motor complex was reduced by 27% as compared to a single motor, whereas run length was increased by only 37%, much less than expected from multimotor transport models. Therefore, at low ATP, which allowed us to identify individual motor steps, we investigated the intermotor dynamics within the two-motor complex. The randomness of stepping leads to a buildup of tension in the linkage between motors-which in turn slows down the leading motor-and increases the frequency of backward steps and the detachment rate. We establish a direct relationship between the velocity reduction and the distribution of intermotor distances. The analysis of run lengths and dwell times for the two-motor complex, which has only one motor engaged with the actin track, reveals that half of the runs are terminated by almost simultaneous detachment of both motors. This finding challenges the assumptions of conventional multimotor models based on consecutive motor detachment. Similar, but even more drastic, results were observed with two-motor complexes on actin bundles, which showed a run length that was even shorter than that of a single motor.

Small teams of myosin Vc motors coordinate their stepping for efficient cargo transport on actin bundles

Myosin Vc (myoVc) is unique among vertebrate class V myosin isoforms in that it requires teams of motors to move continuously on single actin filaments. Single molecules of myoVc cannot take multiple hand-over-hand steps from one actin-binding site to the next without dissociating, in stark contrast to the well studied myosin Va (myoVa) isoform. At low salt, single myoVc motors can, however, move processively on actin bundles, and at physiologic ionic strength, even teams of myoVc motors require actin bundles to sustain continuous motion. Here, we linked defined numbers of myoVc or myoVa molecules to DNA nanostructures as synthetic cargos. Using total internal reflectance fluorescence microscopy, we compared the stepping behavior of myoVc versus myoVa ensembles and myoVc stepping patterns on single actin filaments versus actin bundles. Run lengths of both myoVc and myoVa teams increased with motor number, but only multiple myoVc motors showed a run-length enhancement on actin bundles compared with actin filaments. By resolving the stepping behavior of individual myoVc motors with a quantum dot bound to the motor domain, we found that coupling of two myoVc motors significantly decreased the futile back and side steps that were frequently observed for single myoVc motors. Changes in the inter-motor distance between two coupled myoVc motors affected stepping dynamics, suggesting that mechanical tension coordinates the stepping behavior of two myoVc motors for efficient directional motion. Our study provides a molecular basis to explain how teams of myoVc motors are suited to transport cargos such as zymogen granules on actin bundles.

Tug-of-war between two elastically coupled molecular motors: a case study on force generation and force balance

Intracellular transport is performed by molecular motors that pull cargos along cytoskeletal filaments. Many cellular cargos are observed to move bidirectionally, with fast transport in both directions. This behaviour can be understood as a stochastic tug-of-war between two teams of antagonistic motors. The first theoretical model for such a tug-of-war, the Müller-Klumpp-Lipowsky (MKL) model, was based on two simplifying assumptions: (i) both motor teams move with the same velocity in the direction of the stronger team, and (ii) this velocity matching and the associated force balance arise immediately after the rebinding of an unbound motor to the filament. In this study, we extend the MKL model by including an elastic coupling between the antagonistic motors, and by allowing the motors to perform discrete motor steps. Each motor step changes the elastic interaction forces experienced by the motors. In order to elucidate the basic concepts of force balance and force fluctuations, we focus on the simplest case of two antagonistic motors, one kinesin against one dynein. We calculate the probability distribution for the spatial separation of the motors and the dependence of this distribution on the motors' unbinding rate. We also compute the probability distribution for the elastic interaction forces experienced by the motors, which determines the average elastic force 〈F〉 and the standard deviation of the force fluctuations around this average value. The average force 〈F〉 is found to decrease monotonically with increasing unbinding rate ε₀. The behaviour of the MKL model is recovered in the limit of small ε₀. In the opposite limit of large ε₀, 〈F〉 is found to decay to zero as 1/ε₀. Finally, we study the limiting case with ε₀ = 0 for which we determine both the force statistics and the time needed to attain the steady state. Our theoretical predictions are accessible to experimental studies of in vitro systems consisting of two antagonistic motors attached to a synthetic scaffold or crosslinked via DNA hybridization.

Transport of organelles by elastically coupled motor proteins

Motor-driven intracellular transport is a complex phenomenon where multiple motor proteins simultaneously attached on to a cargo engage in pulling activity, often leading to tug-of-war, displaying bidirectional motion. However, most mathematical and computational models ignore the details of the motor-cargo interaction. A few studies have focused on more realistic models of cargo transport by including elastic motor-cargo coupling, but either restrict the number of motors and/or use purely phenomenological forms for force-dependent hopping rates. Here, we study a generic model in which N motors are elastically coupled to a cargo, which itself is subjected to thermal noise in the cytoplasm and to an additional external applied force. The motor-hopping rates are chosen to satisfy detailed balance with respect to the energy of elastic stretching. With these assumptions, an (N + 1) -variable master equation is constructed for dynamics of the motor-cargo complex. By expanding the hopping rates to linear order in fluctuations in motor positions, we obtain a linear Fokker-Planck equation. The deterministic equations governing the average quantities are separated out and explicit analytical expressions are obtained for the mean velocity and diffusion coefficient of the cargo. We also study the statistical features of the force experienced by an individual motor and quantitatively characterize the load-sharing among the cargo-bound motors. The mean cargo velocity and the effective diffusion coefficient are found to be decreasing functions of the stiffness. While the increase in the number of motors N does not increase the velocity substantially, it decreases the effective diffusion coefficient which falls as 1/N asymptotically. We further show that the cargo-bound motors share the force exerted on the cargo equally only in the limit of vanishing elastic stiffness; as stiffness is increased, deviations from equal load sharing are observed. Numerical simulations agree with our analytical results where expected. Interestingly, we find in simulations that the stall force of a cargo elastically coupled to motors is independent of the stiffness of the linkers.

Multimotor Driven Cargos: From Single Motor under Load to the Role of Motor-Motor Coupling

Motor proteins constitute an essential part of the cellular machinery. They have been the subject of intensive studies in the past two decades. Yet, when several motors simultaneously carry a single cargo, the effect of motor-motor coupling, such as mutual stalling and jamming, remains unclear. We commence by constructing a general model for single motor motion, which is a product of a derived load-dependent expression and a phenomenological motor specific function. Forming the latter according to recent single molecule measurements for a given load, the model correctly predicts the motor full step-size distribution for all other measured loads. We then use our proposed model to predict transport properties of multimotor complexes, with particular attention to 1-dimensional constructs with variable flexibility, motor density, and number of motors: (i) a chain of motors connected by springs, a recently studied construction of a pair, and (ii) an array of motors all connected by identical springs to a stiff rod, which is essentially a mirror image of standard gliding motility assays. In both systems, and for any number of carrying motors, we find that, while low flexibility results in a strongly damped velocity, increased flexibility renders an almost single motor velocity. Comparing our model based simulations to recent gliding assays we find remarkable qualitative agreement. We also demonstrate consistency with other multimotor motility assays. In all cases, the characteristic spring constant, that controls the crossover behavior between high and low velocity regimes, is found to be the stalling force divided by the mean step size. We conjecture that this characteristic spring constant can serve as a tool for engineering multimotor complexes.

Mechanism for Anaphase B: Evaluation of "Slide-and-Cluster" versus "Slide-and-Flux-or-Elongate" Models

Elongation of the mitotic spindle during anaphase B contributes to chromosome segregation in many cells. Here, we quantitatively test the ability of two models for spindle length control to describe the dynamics of anaphase B spindle elongation using experimental data from Drosophila embryos. In the slide-and-flux-or-elongate (SAFE) model, kinesin-5 motors persistently slide apart antiparallel interpolar microtubules (ipMTs). During pre-anaphase B, this outward sliding of ipMTs is balanced by depolymerization of their minus ends at the poles, producing poleward flux, while the spindle maintains a constant length. Following cyclin B degradation, ipMT depolymerization ceases so the sliding ipMTs can push the poles apart. The competing slide-and-cluster (SAC) model proposes that MTs nucleated at the equator are slid outward by the cooperative actions of the bipolar kinesin-5 and a minus-end-directed motor, which then pulls the sliding MTs inward and clusters them at the poles. In assessing both models, we assume that kinesin-5 preferentially cross-links and slides apart antiparallel MTs while the MT plus ends exhibit dynamic instability. However, in the SAC model, minus-end-directed motors bind the minus ends of MTs as cargo and transport them poleward along adjacent, parallel MT tracks, whereas in the SAFE model, all MT minus ends that reach the pole are depolymerized by kinesin-13. Remarkably, the results show that within a narrow range of MT dynamic instability parameters, both models can reproduce the steady-state length and dynamics of pre-anaphase B spindles and the rate of anaphase B spindle elongation. However, only the SAFE model reproduces the change in MT dynamics observed experimentally at anaphase B onset. Thus, although both models explain many features of anaphase B in this system, our quantitative evaluation of experimental data regarding several different aspects of spindle dynamics suggests that the SAFE model provides a better fit.

Collective dynamics of processive cytoskeletal motors

Major cellular processes are supported by various biomolecular motors that usually operate together as teams. We present an overview of the collective dynamics of processive cytokeletal motor proteins based on recent experimental and theoretical investigations. Experimental studies show that multiple motors function with different degrees of cooperativity, ranging from negative to positive. This effect depends on the mechanical properties of individual motors, the geometry of their connections, and the surrounding cellular environment. Theoretical models based on stochastic approaches underline the importance of intermolecular interactions, the properties of single motors, and couplings with cellular medium in predicting the collective dynamics. We discuss several features that specify the cooperativity in motor proteins. Based on this approach a general picture of collective dynamics of motor proteins is formulated, and the future directions and challenges are discussed.

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.

Quasi-steady-state analysis of coupled flashing ratchets

We perform a quasi-steady-state (QSS) reduction of a flashing ratchet to obtain a Brownian particle in an effective potential. The resulting system is analytically tractable and yet preserves essential dynamical features of the full model. We first use the QSS reduction to derive an explicit expression for the velocity of a simple two-state flashing ratchet. In particular, we determine the relationship between perturbations from detailed balance, which are encoded in the transitions rates of the flashing ratchet, and a tilted-periodic potential. We then perform a QSS analysis of a pair of elastically coupled flashing ratchets, which reduces to a Brownian particle moving in a two-dimensional vector field. We suggest that the fixed points of this vector field accurately approximate the metastable spatial locations of the coupled ratchets, which are, in general, impossible to identify from the full system.

Allosteric control of kinesin's motor domain by tubulin: a molecular dynamics study

Molecular motors such as kinesin are essential for many biological processes. These motors have two motor domains, which bind to tubulin filaments, hydrolyze ATP, and transduce the released chemical energy into directed movements. The general principles of this chemomechanical coupling are now well-established but the underlying molecular mechanisms remain elusive because small conformational changes within large proteins are difficult to detect experimentally. Here, we use atomistic molecular dynamics simulations to monitor such changes within a single motor domain of KIF1A, which belongs to the kinesin-3 motor family. The nucleotide binding pocket of this domain can be empty or occupied by ATP or ADP. For these three nucleotide states, we determine the mobility of the backbone of the protein, both in solution and attached to tubulin. Only one subdomain of the motor domain is found to exhibit a strongly increased mobility upon binding to tubulin: the neck linker that presumably acts as a mechanical transmitter to the other motor domain in dimeric kinesin-3 motors. Furthermore, upon binding to tubulin, the neck linker mobility becomes sensitive to the bound nucleotide and is highly increased after phosphate release, which implies undocking of this linker from the core of the motor domain. These simulation results are consistent with experimental data from EPR spectroscopy, FRET, and cryo-electron microscopy. A detailed analysis of our simulation data also reveals that the undocking of the neck linker in the ADP-kinesin-tubulin state arises from allosteric interactions between the nucleotide and tubulin and that the β-sheet core undergoes a twist both during phosphate release and ATP binding. The computational approach used here can be applied to other motor domains and mechanoenzymes in order to identify allosteric interactions between the subdomains of these proteins.

Emergent systems energy laws for predicting myosin ensemble processivity

In complex systems with stochastic components, systems laws often emerge that describe higher level behavior regardless of lower level component configurations. In this paper, emergent laws for describing mechanochemical systems are investigated for processive myosin-actin motility systems. On the basis of prior experimental evidence that longer processive lifetimes are enabled by larger myosin ensembles, it is hypothesized that emergent scaling laws could coincide with myosin-actin contact probability or system energy consumption. Because processivity is difficult to predict analytically and measure experimentally, agent-based computational techniques are developed to simulate processive myosin ensembles and produce novel processive lifetime measurements. It is demonstrated that only systems energy relationships hold regardless of isoform configurations or ensemble size, and a unified expression for predicting processive lifetime is revealed. The finding of such laws provides insight for how patterns emerge in stochastic mechanochemical systems, while also informing understanding and engineering of complex biological systems.

External forces influence the elastic coupling effects during cargo transport by molecular motors

Cellular transport is achieved by the cooperative action of molecular motors which are elastically linked to a common cargo. When the motors pull on the cargo at the same time, they experience fluctuating elastic strain forces induced by the stepping of the other motors. These elastic coupling forces can influence the motors' stepping and unbinding behavior and thereby the ability to transport cargos. Based on a generic single motor description, we introduce a framework that explains the response of two identical molecular motors to a constant external force. In particular, we relate the single motor parameters, the coupling strength and the external load force to the dynamics of the motor pair. We derive four distinct transport regimes and determine how the crossover lines between the regimes depend on the load force. Our description of the overall cargo dynamics takes into account relaxational displacements of the cargo caused by the unbinding of one motor. For large forces and weak elastic coupling these back-shifts dominate the displacements. To develop an intuitive understanding about motor cooperativity during cargo transport, we introduce a time scale for load sharing. This time scale allows us to predict how the regulation of single motor parameters influences the cooperativity. As an example, we show that up-regulating the single motor processivity enhances load sharing of the motor pair.

Synchronization of elastically coupled processive molecular motors and regulation of cargo transport

The collective work of motor proteins plays an important role in cellular transport processes. Since measuring intermotor coupling and hence a comparison to theoretical predictions is difficult, we introduce the synchronization as an alternative observable for motor cooperativity. This synchronization can be determined from the ratio of the mean times of motor resting and stepping. Results from a multistate Markov chain model and Brownian dynamics simulations, describing the elastically coupled motors, coincide well. Our model can explain the experimentally observed effect of strongly increased transport velocities and powers by the synchronization and coupling of myosin V and kinesin I.

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.

Influence of molecular motors on the motion of particles in viscoelastic media

We study theoretically and by numerical simulations the motion of particles driven by molecular motors in a viscoelastic medium representing the cell cytoplasm. For this, we consider a generalized Langevin equation coupled to a stochastic stepping dynamics for the motors that takes into account the action of each motor separately. In the absence of motors, the model produces subdiffusive motion of particles characterized by a power-law scaling of the mean square displacement versus the lag time as t^{α}, with 0<α<1, similar to that observed in cells. Our results show how the action of the motors can induce a transition to a superdiffusive regime at large lag times with the characteristics of those found in experiments reported in the literature. We also show that at small lag times, the motors can act as static crosslinkers that slow down the natural subdiffusive transport. An analysis of previously reported experimental data in the relevant time scales provides evidence of this phenomenon. Finally, we study the effect of a harmonic potential representing an optical trap, and we show a way to approach to a macroscopic description of the active transport in cells. This last point stresses the relevance of the molecular motors for generating not only directed motion to specific targets, but also fast diffusivelike random motion.

Nucleus-targeted drug delivery: theoretical optimization of nanoparticles decoration for enhanced intracellular active transport

A rational design for a nanoparticle is suggested, which will maximize its arrival efficiency from the plasma membrane to the nuclear surrounding. The design is based on grafting the particle surface with polymer spacers, each ending with a motor protein associating molecule, for example, nuclear localization signal peptide. It is theoretically shown that the spacer polymer molecular weight can be adjusted to significantly increase the effective particle processivity time. This should lead to appreciable enhancement of active transport of the nanocarrier, and consequently drug delivery, to the nucleus.

Coupling of two non-processive myosin 5c dimers enables processive stepping along actin filaments

Myosin 5c (Myo5c) is a low duty ratio, non-processive motor unable to move continuously along actin filaments though it is believed to participate in secretory vesicle trafficking in vertebrate cells. Here, we measured the ATPase kinetics of Myo5c dimers and tested the possibility that the coupling of two Myo5c molecules enables processive movement. Steady-state ATPase activity and ADP dissociation kinetics demonstrated that a dimer of Myo5c-HMM (double-headed heavy meromyosin 5c) has a 6-fold lower Km for actin filaments than Myo5c-S1 (single-headed myosin 5c subfragment-1), indicating that the two heads of Myo5c-HMM increase F-actin-binding affinity. Nanometer-precision tracking analyses showed that two Myo5c-HMM dimers linked with each other via a DNA scaffold and moved processively along actin filaments. Moreover, the distance between the Myo5c molecules on the DNA scaffold is an important factor for the processive movement. Individual Myo5c molecules in two-dimer complexes move stochastically in 30-36 nm steps. These results demonstrate that two dimers of Myo5c molecules on a DNA scaffold increased the probability of rebinding to F-actin and enabled processive steps along actin filaments, which could be used for collective cargo transport in cells.

Entrainment to periodic initiation and transition rates in a computational model for gene translation

Periodic oscillations play an important role in many biomedical systems. Proper functioning of biological systems that respond to periodic signals requires the ability to synchronize with the periodic excitation. For example, the sleep/wake cycle is a manifestation of an internal timing system that synchronizes to the solar day. In the terminology of systems theory, the biological system must entrain or phase-lock to the periodic excitation. Entrainment is also important in synthetic biology. For example, connecting several artificial biological systems that entrain to a common clock may lead to a well-functioning modular system. The cell-cycle is a periodic program that regulates DNA synthesis and cell division. Recent biological studies suggest that cell-cycle related genes entrain to this periodic program at the gene translation level, leading to periodically-varying protein levels of these genes. The ribosome flow model (RFM) is a deterministic model obtained via a mean-field approximation of a stochastic model from statistical physics that has been used to model numerous processes including ribosome flow along the mRNA. Here we analyze the RFM under the assumption that the initiation and/or transition rates vary periodically with a common period . We show that the ribosome distribution profile in the RFM entrains to this periodic excitation. In particular, the protein synthesis pattern converges to a unique periodic solution with period . To the best of our knowledge, this is the first proof of entrainment in a mathematical model for translation that encapsulates aspects such as initiation and termination rates, ribosomal movement and interactions, and non-homogeneous elongation speeds along the mRNA. Our results support the conjecture that periodic oscillations in tRNA levels and other factors related to the translation process can induce periodic oscillations in protein levels, and may suggest a new approach for re-engineering genetic systems to obtain a desired, periodic, protein synthesis rate.

Universality and nonuniversality of mobility in heterogeneous single-file systems and Rouse chains

We study analytically the tracer particle mobility in single-file systems with distributed friction constants. Our system serves as a prototype for nonequilibrium, heterogeneous, strongly interacting Brownian systems. The long time dynamics for such a single-file setup belongs to the same universality class as the Rouse model with dissimilar beads. The friction constants are drawn from a density ϱ(ξ), and we derive an asymptotically exact solution for the mobility distribution P[μ0(s)], where μ0(s) is the Laplace-space mobility. If ϱ is light tailed (first moment exists), we find a self-averaging behavior: P[μ0(s)]=δ[μ0(s)-μ(s)], with μ(s)∝s1/2. When ϱ(ξ) is heavy tailed, ϱ(ξ)≃ξ-1-α(0<α<1) for large ξ, we obtain moments 〈[μs(0)]n〉∝sβn, where β=1/(1+α) and there is no self-averaging. The results are corroborated by simulations.

Delineating cooperative responses of processive motors in living cells

Characterizing the collective functions of cytoskeletal motors is critical to understanding mechanisms that regulate the internal organization of eukaryotic cells as well as the roles various transport defects play in human diseases. Though in vitro assays using synthetic motor complexes have generated important insights, dissecting collective motor functions within living cells still remains challenging. Here, we show that the protein heterodimerization switches FKBP-rapalog-FRB can be harnessed in engineered COS-7 cells to compare the collective responses of kinesin-1 and myosinVa motors to changes in motor number and cargo size. The dependence of cargo velocities, travel distances, and position noise on these parameters suggests that multiple myosinVa motors can cooperate more productively than collections of kinesins in COS-7 cells. In contrast to observations with kinesin-1 motors, the velocities and run lengths of peroxisomes driven by multiple myosinVa motors are found to increase with increasing motor density, but are relatively insensitive to the higher loads associated with transporting large peroxisomes in the viscoelastic environment of the COS-7 cell cytoplasm. Moreover, these distinctions appear to be derived from the different sensitivities of kinesin-1 and myosinVa velocities and detachment rates to forces at the single-motor level. The collective behaviors of certain processive motors, like myosinVa, may therefore be more readily tunable and have more substantial roles in intracellular transport regulatory mechanisms compared with those of other cytoskeletal motors.

An exact approach for studying cargo transport by an ensemble of molecular motors

Background

Intracellular transport is crucial for many cellular processes where a large fraction of the cargo is transferred by motor-proteins over a network of microtubules. Malfunctions in the transport mechanism underlie a number of medical maladies.

Existing methods for studying how motor-proteins coordinate the transfer of a shared cargo over a microtubule are either analytical or are based on Monte-Carlo simulations. Approaches that yield analytical results, while providing unique insights into transport mechanism, make simplifying assumptions, where a detailed characterization of important transport modalities is difficult to reach. On the other hand, Monte-Carlo based simulations can incorporate detailed characteristics of the transport mechanism; however, the quality of the results depend on the number and quality of simulation runs used in arriving at results. Here, for example, it is difficult to simulate and study rare-events that can trigger abnormalities in transport.

Results

In this article, a semi-analytical methodology that determines the probability distribution function of motor-protein behavior in an exact manner is developed. The method utilizes a finite-dimensional projection of the underlying infinite-dimensional Markov model, which retains the Markov property, and enables the detailed and exact determination of motor configurations, from which meaningful inferences on transport characteristics of the original model can be derived.

Conclusions

Under this novel probabilistic approach new insights about the mechanisms of action of these proteins are found, suggesting hypothesis about their behavior and driving the design and realization of new experiments.

The advantages provided in accuracy and efficiency make it possible to detect rare events in the motor protein dynamics, that could otherwise pass undetected using standard simulation methods. In this respect, the model has allowed to provide a possible explanation for possible mechanisms under which motor proteins could coordinate their motion.

Ribosome flow model with positive feedback

Eukaryotic mRNAs usually form a circular structure; thus, ribosomes that terminate translation at the 3' end can diffuse with increased probability to the 5' end of the transcript, initiating another cycle of translation. This phenomenon describes ribosomal flow with positive feedback--an increase in the flow of ribosomes terminating translating the open reading frame increases the ribosomal initiation rate. The aim of this paper is to model and rigorously analyse translation with feedback. We suggest a modified version of the ribosome flow model, called the ribosome flow model with input and output. In this model, the input is the initiation rate and the output is the translation rate. We analyse this model after closing the loop with a positive linear feedback. We show that the closed-loop system admits a unique globally asymptotically stable equilibrium point. From a biophysical point of view, this means that there exists a unique steady state of ribosome distributions along the mRNA, and thus a unique steady-state translation rate. The solution from any initial distribution will converge to this steady state. The steady-state distribution demonstrates a decrease in ribosome density along the coding sequence. For the case of constant elongation rates, we obtain expressions relating the model parameters to the equilibrium point. These results may perhaps be used to re-engineer the biological system in order to obtain a desired translation rate.

Future challenges in single-molecule fluorescence and laser trap approaches to studies of molecular motors

Single-molecule analysis is a powerful modern form of biochemistry, in which individual kinetic steps of a catalytic cycle of an enzyme can be explored in exquisite detail. Both single-molecule fluorescence and single-molecule force techniques have been widely used to characterize a number of protein systems. We focus here on molecular motors as a paradigm. We describe two areas where we expect to see exciting developments in the near future: first, characterizing the coupling of force production to chemical and mechanical changes in motors, and second, understanding how multiple motors work together in the environment of the cell.

Complex degradation processes lead to non-exponential decay patterns and age-dependent decay rates of messenger RNA

Experimental studies on mRNA stability have established several, qualitatively distinct decay patterns for the amount of mRNA within the living cell. Furthermore, a variety of different and complex biochemical pathways for mRNA degradation have been identified. The central aim of this paper is to bring together both the experimental evidence about the decay patterns and the biochemical knowledge about the multi-step nature of mRNA degradation in a coherent mathematical theory. We first introduce a mathematical relationship between the mRNA decay pattern and the lifetime distribution of individual mRNA molecules. This relationship reveals that the mRNA decay patterns at steady state expression level must obey a general convexity condition, which applies to any degradation mechanism. Next, we develop a theory, formulated as a Markov chain model, that recapitulates some aspects of the multi-step nature of mRNA degradation. We apply our theory to experimental data for yeast and explicitly derive the lifetime distribution of the corresponding mRNAs. Thereby, we show how to extract single-molecule properties of an mRNA, such as the age-dependent decay rate and the residual lifetime. Finally, we analyze the decay patterns of the whole translatome of yeast cells and show that yeast mRNAs can be grouped into three broad classes that exhibit three distinct decay patterns. This paper provides both a method to accurately analyze non-exponential mRNA decay patterns and a tool to validate different models of degradation using decay data.

Metrics for characterizing collective transport by multiple dimeric kinesins

Kinesin is a processive molecular motor which transports various cellular cargos by converting chemical energy into mechanical movements. Although the motion of a single molecule has been characterized in several studies, the dynamics of collective transport remains controversial. Since the chemical reactions fueling molecular motors are stochastic processes, the movements of coupled motors are not perfectly synchronized. The goal of this study is to develop metrics to analyze the level of synchronization of coupled (stochastic) motors. The correlation among movements of coupled motors, the slackness, the cooperativity, and the power loss of kinesins are explored using the developed metrics. These metrics can be extended to characterize collective work done by other molecular motors also.