Properties of tug-of-war model for cargo transport by molecular motors

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.

Bifurcation of velocity distributions in cooperative transport of filaments by fast and slow motors

Several intracellular processes are governed by two different species of molecular motors, fast and slow ones, that both move in the same direction along the filaments but with different velocities. The transport of filaments arising from the cooperative action of these motors has been recently studied by three in vitro experiments, in which the filament velocity was measured for varying fraction of the fast motors adsorbed onto substrate surfaces in a gliding assay. As the fast motor fraction was increased, two experiments found a smooth change whereas the third one observed an abrupt increase of the filament velocity. Here, we show that all of these experimental results reflect the competition between fast and slow motors and can be understood in terms of an underlying saddle-node bifurcation. The comparison between theory and experiment leads to predictions for the detachment forces of the two motor species. Our theoretical study shows the existence of three different motility regimes: 1), fast transport with a single velocity; 2), slow transport with a single velocity; and 3), bistable transport, where the filament velocity stochastically switches between fast and slow transport. We determine the parameter regions for these regimes in terms of motility diagrams as a function of the surface fraction of fast motors and microscopic single-motor parameters. An abrupt increase of the filament velocity for an increasing fraction of fast motors is associated with the occurrence of bistable transport.

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.

Tug of war of molecular motors: the effects of uneven load sharing

We analyze theoretically the problem of cargo transport along microtubules by motors of two species with opposite polarities. We consider two different one-dimensional models previously developed in the literature: a quite widespread model which assumes equal force sharing, here referred to as the mean field model (MFM), and a stochastic model (SM) which considers individual motor-cargo links. We find that in generic situations, the MFM predicts larger cargo mean velocity, smaller mean run time and less frequent reversions than the SM. These phenomena are found to be the consequences of the load sharing assumptions and can be interpreted in terms of the probabilities of the different motility states. We also explore the influence of the viscosity in both models and the role of the stiffness of the motor-cargo links within the SM. Our results show that the mean cargo velocity is independent of the stiffness, while the mean run time decreases with such a parameter. We explore the case of symmetric forward and backward motors considering kinesin-1 parameters, and the problem of transport by kinesin-1 and cytoplasmic dyneins considering two different sets of parameters previously proposed for dyneins.

Cargo transport by several motors

In a cell, organelles and vesicles are usually transported by cooperation of several motor proteins, including plus-end-directed motor kinesin and minus-end-directed motor dynein. In recent years, many biophysical models have been constructed to understand the mechanism of this transport; however, so far, its basic principle has not been completely understood. In this paper, we will present a model that is based on recent experimental results and existing theoretical models. In this model, each motor is regarded as a head-spring system. The head can bind to or detach from the track stochastically, and step forward or backward with a fixed step size L and force-dependent transition rates. The spring connects the head to the cargo tightly. The position of the cargo is determined by the force-balance condition. An obvious characteristic of our model is that the motors interact with each other and do not share the external load equally. Results indicate that the basic properties of the cargo, including its mean velocity and stall force (definitions are given in the text), are greatly affected by the intermotor interaction. Stated simplistically, the mean velocity and (average) stall force decrease with the intermotor interaction. Therefore, in a sense, the intermotor interaction is negative to the cooperate motion of motors. However, if the cargo is pulled by motors from the same species, its (average) stall force is usually larger than that of the single motor; this means that under external load, the velocity of cargo transported by several motors might be higher than the single-motor cases. We can infer from this that the cooperation of motors is beneficial to draw big cargoes, while it is not as beneficial as one might expect to improve motion velocity when the external load is low.

Bidirectional transport by molecular motors: enhanced processivity and response to external forces

Intracellular transport along cytoskeletal filaments is often mediated by two teams of molecular motors that pull on the same cargo and move in opposite directions along the filaments. We have recently shown theoretically that this bidirectional transport can be understood as a stochastic tug-of-war between the two motor teams. Here, we further develop our theory to investigate the experimentally accessible dynamic behavior of cargos transported by strong motors such as kinesin-1 or cytoplasmic dynein. By studying the run and binding times of such a cargo, we show that the properties of biological motors, such as the large ratio of stall/detachment force and the small ratio of superstall backward/forward velocity, are favorable for bidirectional cargo transport, leading to fast motion and enhanced diffusion. In addition, cargo processivity is shown to be strongly enhanced by transport via several molecular motors even if these motors are engaged in a tug-of-war. Finally, we study the motility of a bidirectional cargo under force. Frictional forces arising, e.g., from the viscous cytoplasm, lead to peaks in the velocity distribution, while external forces as exerted, e.g., by an optical trap, lead to hysteresis effects. Our results, in particular our explicit expressions for the cargo binding time and the distance of the peaks in the velocity relation under friction, are directly accessible to in vitro as well as in vivo experiments.

Dynamics of the tug-of-war model for cellular transport

The transport of organelles and other cargoes in living cells has been described by a kinetic tug-of-war model advanced by Müller, Klumpp, and Lipowsky, in which, as a function of time, t, a team of n+ (t)=0,1,⋯,N+ molecular motors may attach a cargo to a filamentous track and pull it towards the plus end in competition with n- (t)=0,1,⋯,N- motors that pull towards the opposite end. In recent work [Y. Zhang, Phys. Rev. E 79, 061918 (2009)] this model was analyzed for N+,N->>1, establishing the existence, depending on the motor parameters and the ratio ν=N+/N-, of system states with either one, two, or three distinct stable stationary modes of motion. Here, adopting a theoretical perspective, we study the parametric and ν dependence of the transitions between these mono-, bi-, or tristable system states and examine their associated trajectories and domains of attraction in the flow space, (n+,n-), of the attached motor numbers. Various sequences of winning, losing, and "stalemate" or close-to-motionless modes are uncovered. When, as realistic, N+ and N- are of order 2 to 10, fluctuations will move the system from one of two or three modes of motion to another mode. An analysis of the associated probability fluxes demonstrates that the mean time between mode-to-mode transitions increases exponentially with N+ and N-. The overall stall force, i.e., the externally imposed load under which the mean cargo velocity vanishes, is similarly elucidated and shown to vary strongly but sublinearly with N+ and N-, as well as depending in a less than transparent manner on other model parameters beyond the stall forces of the individual + and - motors.

Biased transport of elastic cytoskeletal filaments with alternating polarities by molecular motors

We present a simple model for the bidirectional dynamics of actin bundles with alternating polarities in gliding assays with nonprocessive myosin motors. The bundle is represented as an elastic chain consisting of monomers with positive and negative polarities. The motion of the bundle is induced by the pulling forces of the underlying motors which stochastically attach to the monomers and, depending on their polarities, pull them in the right or left direction. We demonstrate that perfectly apolar chains consisting of equal numbers of monomers with positive and negative polarities may exhibit biased bidirectional motion with a nonzero drift. This effect is attributed to the elastic tension developed in the chain due to the action of the myosin motors. We also show that as a result of this tension, the attachment probability of the motors is greatly reduced and becomes strongly dependent on the length of the chain.