Cooperative molecular motors

Bidirectional bacterial gliding motility powered by the collective transport of cell surface proteins

The gliding motility of Flavobacterium johnsoniae is driven by moving surface adhesive proteins. Recently, these motility components were observed to travel along a closed loop on the cell surface. The mechanism by which such moving surface adhesins give rise to cell motion remains unknown. On the basis of the unique motility properties of F. johnsoniae, we present a generic model for bidirectional motion of rigidly coupled adhesins, which are propelled in opposite directions. Using analytical and numerical methods, we demonstrate that, for a sufficiently large adhesin speed, bidirectional motion arises from spontaneous symmetry breaking. The model also predicts that, close to the bifurcation point, a weak asymmetry in the binding dynamics is sufficient to facilitate directed motility, indicating that the direction of motion could be sensitively regulated internally in response to inhomogeneity of the environment.

Nonlinear cross-bridge elasticity and post-power-stroke events in fast skeletal muscle actomyosin

Generation of force and movement by actomyosin cross-bridges is the molecular basis of muscle contraction, but generally accepted ideas about cross-bridge properties have recently been questioned. Of the utmost significance, evidence for nonlinear cross-bridge elasticity has been presented. We here investigate how this and other newly discovered or postulated phenomena would modify cross-bridge operation, with focus on post-power-stroke events. First, as an experimental basis, we present evidence for a hyperbolic [MgATP]-velocity relationship of heavy-meromyosin-propelled actin filaments in the in vitro motility assay using fast rabbit skeletal muscle myosin (28-29°C). As the hyperbolic [MgATP]-velocity relationship was not consistent with interhead cooperativity, we developed a cross-bridge model with independent myosin heads and strain-dependent interstate transition rates. The model, implemented with inclusion of MgATP-independent detachment from the rigor state, as suggested by previous single-molecule mechanics experiments, accounts well for the [MgATP]-velocity relationship if nonlinear cross-bridge elasticity is assumed, but not if linear cross-bridge elasticity is assumed. In addition, a better fit is obtained with load-independent than with load-dependent MgATP-induced detachment rate. We discuss our results in relation to previous data showing a nonhyperbolic [MgATP]-velocity relationship when actin filaments are propelled by myosin subfragment 1 or full-length myosin. We also consider the implications of our results for characterization of the cross-bridge elasticity in the filament lattice of muscle.

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.

Guiding cell migration by tugging

The ability of cells to move directionally toward areas of stiffer extracellular matrix (ECM) via a process known as 'durotaxis' is thought to be critical for development and wound healing, but durotaxis can also drive cancer metastasis. Migration is driven by integrin-mediated focal adhesions (FAs), protein assemblies that couple contractile actomyosin bundles to the plasma membrane, transmit force generated by the cytoskeleton to the ECM, and convert the mechanical properties of the microenvironment into biochemical signals. To probe the stiffness of the ECM, motile fibroblasts modulate FA mechanics on the nanoscale and exert forces that are reminiscent of repeated tugging on the ECM. Within a single cell, all FAs tug autonomously and thus act as local rigidity sensors, allowing discernment of differences in the extracellular matrix rigidity at high spatial resolution. In this article, we review current advances that may shed light on the mechanism of traction force fluctuations within FAs. We also examine plausible downstream effectors of tugging forces which may regulate cytoskeletal and FA dynamics to guide cell migration in response to ECM stiffness gradients.

Muscle as a metamaterial operating near a critical point

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

Cargo transportation by two species of motor protein

The cargo motion in living cells transported by two species of motor protein with different intrinsic directionalities is discussed in this study. Similarly to single motor movement, the cargo steps forward and backward along a microtubule stochastically. Recent experiments found that cargo transportation by two motor species has a memory; it does not change its direction as frequently as expected, which means that its forward and backward step rates depend on its previous motion trajectory. By assuming that the cargo has only the least possible memory, i.e., its step direction depends only on the direction of its last step, two cases of cargo motion are analyzed in detail in this study: (I) cargo motion under constant external load, and (II) cargo motion in one fixed optical trap. Due to the existence of memory, in the first case, the cargo can keep moving in the same direction for a long distance. In the second case, the cargo will oscillate in the trap. The oscillation period decreases and the oscillation amplitude increases with increasing forward step rate of the motor, but both of them decrease with increasing trap stiffness. The most likely location of the cargo, where the probability of finding the oscillating cargo is maximum, may be the same as or may be different from the trap center, which depends on the step rates of the two motor species. Meanwhile, if the motors are robust, i.e., their forward to backward step rate ratios are high, there may be two such most likely locations, located one on each side of the trap center. The probability of finding the cargo in a given location, the probability of the cargo being in forward or backward motion, and various mean first passage times of the cargo to a given location or a given state are also analyzed.

Signalling through mechanical inputs: a coordinated process

There is growing awareness that mechanical forces - in parallel to electrical or chemical inputs - have a central role in driving development and influencing the outcome of many diseases. However, we still have an incomplete understanding of how such forces function in coordination with each other and with other signalling inputs in vivo. Mechanical forces, which are generated throughout the organism, can produce signals through force-sensitive processes. Here, we first explore the mechanisms through which forces can be generated and the cellular responses to forces by discussing several examples from animal development. We then go on to examine the mechanotransduction-induced signalling processes that have been identified in vivo. Finally, we discuss what is known about the specificity of the responses to different forces, the mechanisms that might stabilize cells in response to such forces, and the crosstalk between mechanical forces and chemical signalling. Where known, we mention kinetic parameters that characterize forces and their responses. The multi-layered regulatory control of force generation, force response and force adaptation should be viewed as a well-integrated aspect in the greater biological signalling systems.

Dimer motion on a periodic substrate: spontaneous symmetry breaking and absolute negative mobility

We consider two coupled particles moving along a periodic substrate potential with negligible inertia effects (overdamped limit). Even when the particles are identical and the substrate spatially symmetric, a sinusoidal external driving of appropriate amplitude and frequency may lead to spontaneous symmetry breaking in the form of a permanent directed motion of the dimer. Thermal noise restores ergodicity and thus zero net velocity, but entails arbitrarily fast diffusion of the dimer for sufficiently weak noise. Moreover, upon application of a static bias force, the dimer exhibits a motion opposite to that force (absolute negative mobility). The key requirement for all these effects is a nonconvex interaction potential of the two particles.

Effects of phase disorder on transport of globally coupled Brownian motors

The transport of N globally coupled Brownian motors driven by a periodic force with phase disorder is investigated. An approximate theoretical analysis of the model is presented. The effects of the phase disorder and the driving strength of the periodic force on the transport of the coupled Brownian motors are discussed both theoretically and numerically. It is found that the increase of the periodical driving force decreases the average velocity, while the coupled particles may benefit from the phase disorder to enhance collective transport.

Distinct transport regimes for two elastically coupled molecular motors

Cooperative cargo transport by two molecular motors involves an elastic motor-motor coupling, which can reduce the motors' velocity and/or enhance their unbinding from the filament. We show theoretically that these interference effects lead, in general, to four distinct transport regimes. In addition to a weak coupling regime, kinesin and dynein motors are found to exhibit a strong coupling and an enhanced unbinding regime, whereas myosin motors are predicted to attain a reduced velocity regime. All of these regimes, which we derive by explicit calculations and general time scale arguments, can be explored experimentally by varying the elastic coupling strength.

Walking motion of an overdamped active particle in a ratchet potential

An active particle can convert its internal energy into mechanical work. We study a generalized energy-depot model of an overdamped active particle in a ratchet potential. Using well-known biological parameters for kinesin-1 and modeling ATP influx as a pulsed energy supply, we apply our model to the molecular motor system. We find that our simple model can capture the essential properties of the kinesin motor such as forward stepping, stalling, backward stepping, dependence on ATP concentration, and stall force. Our model might be quite universal in the sense that it is able to describe dynamics of various types of motors as long as realistic parameters for each motor species are adopted.

ATP-independent cooperative binding of yeast Isw1a to bare and nucleosomal DNA

Among chromatin remodeling factors, the ISWI family displays a nucleosome-enhanced ATPase activity coupled to DNA translocation. While these enzymes are known to bind to DNA, their activity has not been fully characterized. Here we use TEM imaging and single molecule manipulation to investigate the interaction between DNA and yeast Isw1a. We show that Isw1a displays a highly cooperative ATP-independent binding to and bridging between DNA segments. Under appropriate tension, rare single nucleation events can sometimes be observed and loop DNA with a regular step. These nucleation events are often followed by binding of successive complexes bridging between nearby DNA segments in a zipper-like fashion, as confirmed by TEM observations. On nucleosomal substrates, we show that the specific ATP-dependent remodeling activity occurs in the context of cooperative Isw1a complexes bridging extranucleosomal DNA. Our results are interpreted in the context of the recently published partial structure of Isw1a and support its acting as a “protein ruler” (with possibly more than one tick).

Rectification of spatial disorder

We demonstrate that a large ensemble of noiseless globally coupled-pinned oscillators is capable of rectifying spatial disorder with spontaneous current activated through a dynamical phase transition mechanism, either of first or second order, depending on the profile of the pinning potential. In the presence of an external weak drive, the same collective mechanism can result in an absolute negative mobility, which, though not immediately related to symmetry breaking, is most prominent at the phase transition. Our results apply to a tug-of-war by competing molecular motors for bidirectional cargo transport.

Biomechanical regulation of contractility: spatial control and dynamics

Cells are active materials; they can change shape using internal energy to build contractile networks of actin filaments and myosin motors. Contractility of the actomyosin cortex is tightly regulated in space and time to orchestrate cell shape changes. Conserved biochemical pathways regulate actomyosin networks in subcellular domains which drive cell shape changes. Actomyosin networks display complex dynamics, such as flows and pulses, which participate in myosin distribution and provide a more realistic description of the spatial distribution and evolution of forces during morphogenesis. Such dynamics are influenced by the mechanical properties of actomyosin networks. Moreover, actomyosin can self-organize and respond to mechanical stimuli through multiple types of biomechanical feedback. In this review we propose a framework encapsulating spatiotemporal regulation of contractility from established pathways with the dynamics and mechanics of actomyosin networks. Through the comparison of cytokinesis, cell migration and epithelial morphogenesis, we delineate emergent properties of contractile activity, including self-organization, adaptability and robustness.

Growth and shortening of microtubules: a two-state model approach

In this study, a two-state mechanochemical model is presented to describe the dynamic instability of microtubules (MTs) in cells. The MT switches between two states, the assembly and disassembly states. In assembly state, the growth of MTs includes two processes: free GTP-tubulin binding to the tip of protofilament (PF) and conformation change of PF, during which the first tubulin unit that curls outwards is rearranged onto the MT surface, using the energy released from the hydrolysis of GTP in the penultimate tubulin unit. In the disassembly state, the shortening of MTs also includes two processes, the release of GDP-tubulin from the tip of PF and the curling of one new tubulin unit out of the MT surface. Switches between these two states, which are usually called rescue and catastrophe, happen stochastically with external force-dependent rates. Using this two-state model with parameters obtained by fitting the recent experimental data, detailed properties of MT growth are obtained. I find that MT is mainly in the assembly state, its mean growth velocity increases with both the external force and the GTP-tubulin concentration, and an MT will shorten on average without an external force. To know more about the external force and GTP-tubulin concentration-dependent properties of MT growth, and for future experimental verification of this two-state model, 11 critical forces are defined and discussed numerically.

Bidirectional motion of motor assemblies and the weak-noise escape problem

We present a detailed calculation that enables us to estimate the reversal time of a molecular motor assembly that displays bidirectional motion in the limit of weak noise. We derive a Fokker-Planck equation by taking a large volume expansion of a master equation, and we consider a simple choice of transition rates that enables us to reduce the number of variables to 2. We use the Wentzell-Freidlin theory to define an effective nonequilibrium potential and analytically estimate the reversal time. We also present the results of stochastic simulations that match very well our simulation results.

Conformational changes, diffusion and collective behavior in monomeric kinesin-based motility

Molecular motors convert chemical energy into mechanical motion and power the transport of material within living cells; the motion of a motor is thought to be influenced by stochastic chemical state transitions of the molecule as well as intramolecular diffusion of one motor head seeking the next binding site. Existing models for the motility of single-headed monomeric motors that map the system to a simplified two-state Brownian ratchet have some predictive power, but in general are unable to elucidate the contributions of different molecular level processes to the overall effective parameters. In this work, we build a detailed molecular level model of monomeric kinesin motility that naturally incorporates conformational changes (power strokes) and biased diffusion. Our results predict that mean velocity is most sensitive to the power stroke size, while run length distribution is sensitive primarily to the strength of the microtubule bias potential with a weak dependence on power stroke that can be tuned by the strength of an applied load. In addition, we demonstrate that motor pairs attached to the same cargo can cooperatively function to increase motility in both the plus- and minus-end directions. These findings illustrate the importance of a detailed mechanochemical model for dissecting the contributions of microscopic parameters to monomeric kinesin dynamics.

On the spontaneous collective motion of active matter

Spontaneous directed motion, a hallmark of cell biology, is unusual in classical statistical physics. Here we study, using both numerical and analytical methods, organized motion in models of the cytoskeleton in which constituents are driven by energy-consuming motors. Although systems driven by small-step motors are described by an effective temperature and are thus quiescent, at higher order in step size, both homogeneous and inhomogeneous, flowing and oscillating behavior emerges. Motors that respond with a negative susceptibility to imposed forces lead to an apparent negative-temperature system in which beautiful structures form resembling the asters seen in cell division.

Time-dependent motor properties of multipedal molecular spiders

Molecular spiders are synthetic biomolecular walkers that use the asymmetry resulting from cleavage of their tracks to bias the direction of their stepping motion. Using Monte Carlo simulations that implement the Gillespie algorithm, we investigate the dependence of the biased motion of molecular spiders, along with binding time and processivity, on tunable experimental parameters, such as number of legs, span between the legs, and unbinding rate of a leg from a substrate site. We find that an increase in the number of legs increases the spiders' processivity and binding time but not their mean velocity. However, we can increase the mean velocity of spiders with simultaneous tuning of the span and the unbinding rate of a spider leg from a substrate site. To study the efficiency of molecular spiders, we introduce a time-dependent expression for the thermodynamic efficiency of a molecular motor, allowing us to account for the behavior of spider populations as a function of time. Based on this definition, we find that spiders exhibit transient motor function over time scales of many hours and have a maximum efficiency on the order of 1%, weak compared to other types of molecular motors.

Motor-driven dynamics of cytoskeletal filaments in motility assays

We model analytically the dynamics of a cytoskeletal filament in a motility assay. The filament is described as rigid rod free to slide in two dimensions. The motor proteins consist of polymeric tails tethered to the plane and modeled as linear springs and motor heads that bind to the filament. As in related models of rigid and soft two-state motors, the binding and unbinding dynamics of the motor heads and the dependence of the transition rates on the load exerted by the motor tails play a crucial role in controlling the filament's dynamics. Our work shows that the filament effectively behaves as a self-propelled rod at long times, but with non-Markovian noise sources arising from the coupling to the motor binding and unbinding dynamics. The effective propulsion force of the filament and the active renormalization of the various friction and diffusion constants are calculated in terms of microscopic motor and filament parameters. These quantities could be probed by optical force microscopy.

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

We present a detailed analysis of the dynamical instabilities appearing in two kinetic theories for the collective behavior of molecular motors: the rigid two-state model and the two-state crossbridge (or power-stroke) model with continuous binding sites. We calculate force-velocity relations, discuss their stability, plot a diagram that summarizes the oscillation regimes, identify the location of the Hopf bifurcation with a memory effect, discuss the oscillation frequency and make a link with single-molecule experiments. We show that the instabilities present in these models naturally translate into non-linearities in force-displacement relations, and at linear order give forces that are similar to the delayed stretch activation observed in oscillating muscles. We also find that instabilities can appear for both apparent load-decelerated and load-accelerated detachment rates in a 3-state crossbridge model.

Particle diode: rectification of interacting Brownian ratchets

Transport of Brownian particles interacting with each other via the Morse potential is investigated in the presence of an ac driving force applied locally at one end of the chain. By using numerical simulations, we find that the system can behave as a particle diode for both overdamped and underdamped cases. For low frequencies, the transport from the free end to the ac acting end is prohibited, while the transport from the ac acting end to the free end is permitted. However, the polarity of the particle diode will reverse for medium frequencies. There exists an optimal value of the well depth of the interaction potential at which the average velocity takes its maximum. The average velocity υ decreases monotonically with the system size N by a power law υ ∝ N(-1).

Relation between cooperative molecular motors and active Brownian particles

Active Brownian particles (ABPs), obeying a nonlinear Langevin equation with speed-dependent drift and noise amplitude, are well-known models used to describe self-propelled motion in biology. In this paper we study a model describing the stochastic dynamics of a group of coupled molecular motors (CMMs). Using two independent numerical methods, one based on the stationary velocity distribution of the motors and the other one on the local increments (also known as the Kramers-Moyal coefficients) of the velocity, we establish a connection between the CMM and the ABP models. The parameters extracted for the ABP via the two methods show good agreement for both symmetric and asymmetric cases and are independent of N, the number of motors, provided that N is not too small. This indicates that one can indeed describe the CMM problem with a simpler ABP model. However, the power spectrum of velocity fluctuations in the CMM model reveals a peak at a finite frequency, a peak which is absent in the velocity spectrum of the ABP model. This implies richer dynamic features of the CMM model which cannot be captured by an ABP model.

Real-time monitoring of cell elasticity reveals oscillating myosin activity

The cytoskeleton is the physical and biochemical interface for a large variety of cellular processes. Its complex regulation machinery is involved upstream and downstream in various signaling pathways. The cytoskeleton determines the mechanical properties of a cell. Thus, cell elasticity could serve as a parameter reflecting the behavior of the system rather than reflecting the specific properties of isolated components. In this study, we used atomic force microscopy to perform real-time monitoring of cell elasticity unveiling cytoskeletal dynamics of living bronchial epithelial cells. In resting cells, we found a periodic activity of the cytoskeleton. Amplitude and frequency of this spontaneous oscillation were strongly affected by intracellular calcium. Experiments reveal that basal cell elasticity and superimposed elasticity oscillations are caused by the collective action of myosin motor proteins. We characterized the cell as a mechanically multilayered structure, and followed cytoskeletal dynamics in the different layers with high time resolution. In conclusion, the collective activities of the myosin motor proteins define overall mechanical cell dynamics, reflecting specific changes of the chemical and mechanical environment.

Motion reversal of molecular motor assemblies due to weak noise

Bidirectional motion is an example of collective behavior of molecular motors. It occurs at finite noise level in a nonequilibrium system. We consider this problem as a first exit problem. We identify the noise strength by doing an expansion of a master equation and apply the Wentzell-Freidlin theory to define an effective nonequilibrium potential and provide analytical estimates of the reversal time. Our results match very well with the results of stochastic simulations.

Spontaneous sarcomere dynamics

Sarcomeres are the basic force generating units of striated muscles and consist of an interdigitating arrangement of actin and myosin filaments. While muscle contraction is usually triggered by neural signals, which eventually set myosin motors into motion, isolated sarcomeres can oscillate spontaneously between a contracted and a relaxed state. We analyze a model for sarcomere dynamics, which is based on a force-dependent detachment rate of myosin from actin. Our numerical bifurcation analysis of the spontaneous sarcomere dynamics reveals notably Hopf bifurcations, canard explosions, and gluing bifurcations. We discuss possible implications for experiments.

Flow density of a reversible ratchet

By defining "flow density" in a reversible ratchet, a kind of Brownian motor that runs four processes in a two-dimensional parameter space to complete a cycle, we analytically investigate the behavior of the motor's integrated flow in terms of the magnitude and location of the density maximum. It turns out that flow density gives information that can serve as a guide for obtaining the most suitable integrated flow important in the design of reversible ratchets.

Collective behavior of antagonistically acting kinesin-1 motors

In many subcellular force-generating systems, groups of motor proteins act antagonistically. Here, we present an experimental study of the tug of war between superprocessive kinesin-1 motors acting on antiparallel microtubule doublets in vitro. We found distinct modes of slow and fast movements, as well as sharp transitions between these modes and regions of coexistence. We compare our experimental results to a quantitative theory based on the physical properties of individual motors. Our results show that mechanical interactions between motors can collectively generate coexisting transport regimes with distinct velocities.

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.

Microtubule organization by the antagonistic mitotic motors kinesin-5 and kinesin-14

During cell division, different molecular motors act synergistically to rearrange microtubules. Minus end-directed motors are thought to have a dual role: focusing microtubule ends to poles and establishing together with plus end-directed motors a balance of force between antiparallel microtubules in the spindle. We study here the competing action of Xenopus laevis kinesin-14 and -5 in vitro in situations in which these motors with opposite directionality cross-link and slide microtubules. We find that full-length kinesin-14 can form microtubule asters without additional factors, whereas kinesin-5 does not, likely reflecting an adaptation to mitotic function. A stable balance of force is not established between two antiparallel microtubules with these motors. Instead, directional instability is generated, promoting efficient motor and microtubule sorting. A nonmotor microtubule cross-linker can suppress directional instability but also impedes microtubule sorting, illustrating a conflict between stability and dynamicity of organization. These results establish the basic organizational properties of these antagonistic mitotic motors and a microtubule bundler.

Dynamic instabilities in assemblies of molecular motors with finite stiffness

We propose a two-state "soft-motor" model for the collective behavior of molecular motors which takes into account both the internal motor stiffness and the periodic interaction with the filament. As in the Prandtl-Tomlinson model of tribology, the important parameter of the model is the pinning parameter, which compares the stiffness of the motors to the stiffness of the potential. The model predicts dynamic instabilities in two disconnected regions of parameter space. These parameter ranges correspond to two existing theories of motor assemblies, the rigid two-state model and the crossbridge model. The model also predicts a discontinuity of the slope of the force-velocity relation at small velocities.

Sheared active fluids: thickening, thinning, and vanishing viscosity

We analyze the behavior of a suspension of active polar particles under shear. In the absence of external forces, orientationally ordered active particles are known to exhibit a transition to a state of nonuniform polarization and spontaneous flow. Such a transition results from the interplay between elastic stresses, due to the liquid crystallinity of the suspension, and internal active stresses. In the presence of an external shear, we find an extremely rich variety of phenomena, including an effective reduction (increase) in the apparent viscosity depending on the nature of the active stresses and the flow-alignment property of the particles, as well as more exotic behaviors such as a nonmonotonic stress-strain-rate relation and yield stress for large activities.

Molecular motors as an auto-oscillator

The organization of biomotile systems possesses structural and functional hierarchy, building up from single molecules via protein assemblies and cells further up to an organ. A typical example is the hierarchy of cardiac muscle, on the top of which is the heart. The heartbeat is supported by the rhythmic contraction of the muscle cells that is controlled by the Ca(2+) oscillation triggered by periodic electrical excitation of pacemaker cells. Thus, it is usually believed that the heartbeat is governed by the control system based on a sequential one-way chain with the electrical∕chemical information transfer from the upper to the lower level of hierarchy. On the other hand, it has been known for many years that the contractile system of muscle, i.e., skinned muscle fibers and myofibrils, itself possesses the auto-oscillatory properties even in the constant chemical environment. A recent paper [Plaçais, et al. (2009), Phys. Rev. Lett. 103, 158102] demonstrated the auto-oscillatory movement∕tension development in an in vitro motility assay composed of a single actin filament and randomly distributed myosin II molecules, suggesting that the auto-oscillatory properties are inherent to the contractile proteins. Here we discuss how the molecular motors may acquire the higher-ordered auto-oscillatory properties while stepping up the staircase of hierarchy.

Robust transport by multiple motors with nonlinear force-velocity relations and stochastic load sharing

Transport by processive molecular motors plays an important role in many cell biological phenomena. In many cases, motors work together to transport cargos in the cell, so it is important to understand the mechanics of the multiple motors. Based on earlier modeling efforts, here we study effects of nonlinear force-velocity relations and stochastic load sharing on multiple motor transport. We find that when two or three motors transport the cargo, then the nonlinear and stochastic effects compensate so that the mechanical properties of the transport are robust. Similarly, the transport is insensitive to compliance of the cargo-motor links. Furthermore, the rate of movement against moderate loads is not improved by increasing the small number of motors. When the motor number is greater than 4, correlations between the motors become negligible, and the earlier analytical mean-field theory of the multiple motor transport holds. We predict that the effective diffusion of the cargo driven by the multiple motors under load increases by an order of magnitude compared to that for the single motor. Finally, our simulations predict that the stochastic effects are responsible for a significant dispersion of velocities generated by the 'tug-of-war' of the multiple opposing motors.

Coordination and collective properties of molecular motors: theory

Many cellular processes require molecular motors to produce motion and forces. Single molecule experiments have led to a precise description of how a motor works. Under most physiological conditions, however, molecular motors operate in groups. Interactions between motors yield collective behaviors that cannot be explained only from single molecule properties. The aim of this paper is to review the various theoretical descriptions that explain the emergence of collective effects in molecular motor assemblies. These include bidirectional motion, hysteretic behavior, spontaneous oscillations, and self-organization into dynamical structures. We discuss motors acting on the cytoskeleton both in a prescribed geometry such as in muscles or flagella and in the cytoplasm.

Spindle checkpoint regulated by nonequilibrium collective spindle-chromosome interaction; relationship to single DNA molecule force-extension formula

The spindle checkpoint, which blocks segregation until all sister chromatid pairs have been stably connected to the two spindle poles, is perhaps the biggest mystery of the cell cycle. The main reason seems to be that the spatial correlations imposed by microtubules between stably attached kinetochores and the nonlinear dependence of the system on the increasing number of such kinetochores have been disregarded in earlier spindle checkpoint studies. From these missing parts a non-equilibrium collective spindle-chromosome interaction is obtained here for budding yeast (Saccharomyces cerevisiae) cells. The interaction, which is based on a non-equilibrium statistical mechanics, can sense and count the number of stably attached kinetochores and sense the threshold for segregation. It blocks segregation until all sister chromatids pairs have been bi-oriented and regulates tension such that segregation becomes synchronized, thus explaining how the cell might decide to segregate replicated chromosomes. The model also predicts kinetochore oscillations at a frequency which agrees well with observation. Finally, a relationship between this spindle-chromosome dynamics and the force-extension formula obtained in a single DNA molecule experiment is obtained.

Interacting molecular motors: efficiency and work fluctuations

We investigate the model of "reversible ratchet" with interacting particles, presented by us earlier [F. Slanina, EPL 84, 50009 (2008)]. We further clarify the effect of efficiency enhancement due to interaction and show that it is of energetic origin, rather than a consequence of reduced fluctuations. We also show complicated structures emerging in the interaction and density dependence of the current and response function. The fluctuation properties of the work and input energy indicate in detail the far-from-equilibrium nature of the dynamics.

Force-velocity relations for multiple-molecular-motor transport

A transition rate model of cargo transport by N molecular motors is proposed. Under the assumption of steady state, the force-velocity curve of multimotor system can be derived from the force-velocity curve of a single motor. Our work shows, in the case of low load, that the velocity of multimotor system can decrease or increase with increasing motor number, which is dependent on the single motor force-velocity curve; and most commonly, the velocity decreases. This gives a possible explanation to some recent experimental observations.

Cooperative molecular motors moving back and forth

We use a two-state ratchet model to study the cooperative bidirectional motion of molecular motors on cytoskeletal tracks with randomly alternating polarities. Our model is based on a previously proposed model [Badoual, Proc. Natl. Acad. Sci. U.S.A. 99, 6696 (2002)] for collective motor dynamics and, in addition, takes into account the cooperativity effect arising from the elastic tension that develops in the cytoskeletal track due to the joint action of the walking motors. We show, both computationally and analytically, that this additional cooperativity effect leads to a dramatic reduction in the characteristic reversal time of the bidirectional motion, especially in systems with a large number of motors. We also find that bidirectional motion takes place only on (almost) apolar tracks, while on even slightly polar tracks the cooperative motion is unidirectional. We argue that the origin of these observations is the sensitive dependence of the cooperative dynamics on the difference between the number of motors typically working in and against the instantaneous direction of motion.

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

Molecular motors are essential components for the biophysical functions of the cell. Current quantitative understanding of how multiple motors move along a single track is not complete, even though models and theories for a single motor mechanochemistry abound. Recently, Müller et al. have developed a tug-of-war model to describe the bidirectional movement of the cargo [Proc. Natl. Acad. Sci. U.S.A. 105, 4609 (2008)]. They found that the tug-of-war model exhibits several qualitative different motility regimes, which depend on the precise value of single motor parameters, and they suggested that the sensitivity can be used by a cell to regulate its cargo traffic. In the present paper, we will carry out a detailed theoretical analysis of a special case of tug-of-war model: in which the numbers of the two different motor species which bound to the cargo tend to infinite. Through the analysis, all the stable, i.e., biophysically observable, steady states and their stability domains can be obtained. Depending on values of the several parameters, the tug-of-war model exhibits uni-, bi-, or tristability. The steady-state movement of the cargo, which is transported by two different molecular motor species, is determined by the initial numbers of the motors which bound to the track.

The fusion of actin bundles driven by interacting motor proteins

The cooperative action of many molecular motors is essential for dynamic processes such as cell motility and mitosis. This action can be studied by using motility assays which track the motion of cytoskeletal filaments over a surface coated with motor proteins. Here, we propose to use a motility assay consisting of a-polar actin bundles subjected to the action of myosin II motors where no external loading is applied. In this work we focus on those bundles undergoing fusion with other nearby bundles. Specifically, we investigate the role of the bundles' dimension on the transition from bidirectional to directional motion and on the properties of their motion during fusion. Our experimental data reveal that only small bundles exhibit dynamic transition to directional motion, implying that the forces acting on them exceed the threshold value necessary to induce the transition. Moreover, these bundles accelerate along their trajectory, suggesting that the forces acting on them increase while approaching each other. We show that these forces do not originate from external loading but rather arise from the action of the motors on the bundles. These forces are transmitted through the medium over micron-scale distances without being cut off. Moreover, we show that the forces propagate to distances that are proportional to the size of the bundles, or equivalently, to the number of motors, which they interact with.

A kinetic model for the enzymatic action of cellulase

We develop a mechanochemical model for the dynamics of cellulase, a two-domain enzyme connected by a peptide linker, as it extracts and hydrolyzes a cellulose polymer from a crystalline substrate. We consider two random walkers, representing the catalytic domain (CD) and the carbohydrate binding module (CBM), whose rates for stepping are biased by the coupling through the linker and the energy required to lift the cellulose polymer from the crystalline surface. Our results show that the linker length and stiffness play a critical role in the cooperative action of the CD and CBM domains and that, for a given linker length, the steady-state hydrolysis shows a maximum at some intermediate linker stiffness. The maximum hydrolysis rate corresponds to a transition of the linker from a compressed to an extended conformation, where the system exhibits maximum fluctuation, as measured by the variance of the separation distance between the two domains and the dispersion around the mean hydrolysis speed. In the range of experimentally known values of the parameters of our model, improving the intrinsic hydrolytic activity of the CD leads to a proportional increase in the overall hydrolysis rate.

Self-organization and cooperativity of weakly coupled molecular motors under unequal loading

We study the collective dynamics of Brownian motors moving on a one-dimensional track when an external load is applied to the leading motor. Motors are driven by a two-state ratchet mechanism, which is appropriate to single-headed kinesins, and their relative motion is only constrained by their mutual interaction potential (weak coupling). We show that unequal loading enhances cooperativity, leading to the formation of clusters with velocities and efficiencies higher than those predicted by simple superposition. When a weak attraction between motors is present, we find nonmonotonic collective velocity-force curves, hysteretic phenomena, and a dynamic self-regulation mechanism that selects the cluster size for optimal performance.

Force amplification response of actin filaments under confined compression

Actin protein is a major component of the cell cytoskeleton, and its ability to respond to external forces and generate propulsive forces through the polymerization of filaments is central to many cellular processes. The mechanisms governing actin's abilities are still not fully understood because of the difficulty in observing these processes at a molecular level. Here, we describe a technique for studying actin-surface interactions by using a surface forces apparatus that is able to directly visualize and quantify the collective forces generated when layers of noninterconnected, end-tethered actin filaments are confined between 2 (mica) surfaces. We also identify a force-response mechanism in which filaments not only stiffen under compression, which increases the bending modulus, but more importantly generates opposing forces that are larger than the compressive force. This elastic stiffening mechanism appears to require the presence of confining surfaces, enabling actin filaments to both sense and respond to compressive forces without additional mediating proteins, providing insight into the potential role compressive forces play in many actin and other motor protein-based phenomena.