Monte Carlo modeling of single-molecule cytoplasmic dynein

Modeling of Chemomechanical Coupling of Cytoplasmic Dynein Motors

Cytoplasmic dynein homodimer is a motor protein that can step processively on microtubules (MTs) toward the minus end by hydrolyzing ATP molecules. Some dynein motors show a complicated stepping behavior with variable step sizes and having both hand-overhand and inchworm steps, while some mammalian dynein motors show simplistic stepping behavior with a constant step size and having only hand-overhand steps. Here, a model for the chemomechanical coupling of the dynein is presented, based on which an analytical theory is given on the dynamics of the motor. The theoretical results explain consistently and quantitatively the available experimental data on various aspects of the dynamics of dynein with complicated stepping behavior and the dynamics of dynein with simplistic stepping behavior. The very differences in the dynamic behavior between the two motors are due solely to different elastic coefficients of the linkage connecting the two dynein heads, with the dynein motors of the complicated and simplistic stepping behaviors having small and large coefficients, respectively. Moreover, it is analyzed that the ATPase rate of the dynein head with a docked linker being larger than that with an undocked linker is indispensable for the unidirectional motility of the motor, and the small free energy change for the linker docking in the strong MT-binding state facilitates the unidirectional motility.

Modelling intracellular transport in crowded environments: effects of motor association to cargos

In intracellular transports, motor proteins transport macromolecules as cargos to desired locations by moving on biopolymers such as microtubules. Recent experiments suggest that, while moving in crowded environments, cargos that can associate motor proteins during their translocation have larger run-length and association time compared to free motors. Here, we model the dynamics of a cargo that can associate at the most m free motors present on the microtubule track as obstacles to its motion. The proposed models display competing effects of association and crowding, leading to a peak in the run-length with the free-motor density. For m = 2 and 3, we show that this feature is governed by the largest eigenvalue of the transition matrix describing the cargo dynamics. In all the above cases, free motors are assumed to be present on the microtubule as stalled obstacles. We finally compare simulation results for the run-length for general scenarios where the free motors undergo processive motion in addition to binding and unbinding to or from the microtubule.

Coordination, cooperation, competition, crowding and congestion of molecular motors: Theoretical models and computer simulations

Cytoskeletal motor proteins are biological nanomachines that convert chemical energy into mechanical work to carry out various functions such as cell division, cell motility, cargo transport, muscle contraction, beating of cilia and flagella, and ciliogenesis. Most of these processes are driven by the collective operation of several motors in the crowded viscous intracellular environment. Imaging and manipulation of the motors with powerful experimental probes have been complemented by mathematical analysis and computer simulations of the corresponding theoretical models. In this article, we illustrate some of the key theoretical approaches used to understand how coordination, cooperation and competition of multiple motors in the crowded intra-cellular environment drive the processes that are essential for biological function of a cell. In spite of the focus on theory, experimentalists will also find this article as an useful summary of the progress made so far in understanding multiple motor systems.

Theoretical analysis of cargo transport by catch bonded motors in optical trapping assays

Dynein motors exhibit catch bonding, where the unbinding rate of the motors from microtubule filaments decreases with increasing opposing load. The implications of this catch bond on the transport properties of dynein-driven cargo are yet to be fully understood. In this context, optical trapping assays constitute an important means of accurately measuring the forces generated by molecular motor proteins. We investigate, using theory and stochastic simulations, the transport properties of cargo transported by catch bonded dynein molecular motors - both singly and in teams - in a harmonic potential, which mimics the variable force experienced by cargo in an optical trap. We estimate the biologically relevant measures of first passage time - the time during which the cargo remains bound to the microtubule and detachment force - the force at which the cargo unbinds from the microtubule, using both two-dimensional and one-dimensional force balance frameworks. Our results suggest that even for cargo transported by a single motor, catch bonding may play a role depending on the force scale which marks the onset of the catch bond. By comparing with experimental measurements on single dynein-driven transport, we estimate realistic bounds of this catch bond force scale. Generically, catch bonding results in increased persistent motion, and can also generate non-monotonic behaviour of first passage times. For cargo transported by multiple motors, emergent collective effects due to catch bonding can result in non-trivial re-entrant phenomena wherein average first passage times and detachment forces exhibit non-monotonic behaviour as a function of the stall force and the motor velocity.

Mechanics of microtubule organizing center clustering and spindle positioning in budding yeast Cryptococcus neoformans

The dynamic process of mitotic spindle assembly depends on multitudes of inter-dependent interactions involving kinetochores (KTs), microtubules (MTs), spindle pole bodies (SPBs), and molecular motors. Before forming the mitotic spindle, multiple visible microtubule organizing centers (MTOCs) coalesce into a single focus to serve as an SPB in the pathogenic budding yeast, Cryptococcus neoformans. To explain this unusual phenomenon in the fungal kingdom, we propose a "search and capture" model, in which cytoplasmic MTs (cMTs) nucleated by MTOCs grow and capture each other to promote MTOC clustering. Our quantitative modeling identifies multiple redundant mechanisms mediated by a combination of cMT-cell cortex interactions and inter-cMT coupling to facilitate MTOC clustering within the physiological time limit as determined by time-lapse live-cell microscopy. Besides, we screen various possible mechanisms by computational modeling and propose optimal conditions that favor proper spindle positioning-a critical determinant for timely chromosome segregation. These analyses also reveal that a combined effect of MT buckling, dynein pull, and cortical push maintains spatiotemporal spindle localization.

Dynein-Inspired Multilane Exclusion Process with Open Boundary Conditions

Motivated by the sidewise motions of dynein motors shown in experiments, we use a variant of the exclusion process to model the multistep dynamics of dyneins on a cylinder with open ends. Due to the varied step sizes of the particles in a quasi-two-dimensional topology, we observe the emergence of a novel phase diagram depending on the various load conditions. Under high-load conditions, our numerical findings yield results similar to the TASEP model with the presence of all three standard TASEP phases, namely the low-density (LD), high-density (HD), and maximal-current (MC) phases. However, for medium- to low-load conditions, for all chosen influx and outflux rates, we only observe the LD and HD phases, and the maximal-current phase disappears. Further, we also measure the dynamics for a single dynein particle which is logarithmically slower than a TASEP particle with a shorter waiting time. Our results also confirm experimental observations of the dwell time distribution: The dwell time distribution for dyneins is exponential in less crowded conditions, whereas a double exponential emerges under overcrowded conditions.

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.

How Cytoplasmic Dynein Couples ATP Hydrolysis Cycle to Diverse Stepping Motions: Kinetic Modeling

Cytoplasmic dynein is a two-headed molecular motor that moves to the minus end of a microtubule by ATP hydrolysis free energy. By employing its two heads (motor domains), cytoplasmic dynein exhibits various bipedal stepping motions: inchworm and hand-over-hand motions, as well as nonalternating steps of one head. However, the molecular basis to achieve such diverse stepping manners remains unclear because of the lack of an experimental method to observe stepping and the ATPase reaction of dynein simultaneously. Here, we propose a kinetic model for bipedal motions of cytoplasmic dynein and perform Gillespie Monte Carlo simulations that qualitatively reproduce most experimental data obtained to date. The model represents the status of each motor domain as five states according to conformation and nucleotide- and microtubule-binding conditions of the domain. In addition, the relative positions of the two domains were approximated by three discrete states. Accompanied by ATP hydrolysis cycles, the model dynein stochastically and processively moved forward in multiple steps via diverse pathways, including inchworm and hand-over-hand motions, similarly to experimental data. The model reproduced key experimental motility-related properties, including velocity and run length, as functions of the ATP concentration and external force, therefore providing a plausible explanation of how dynein achieves various stepping manners with explicit characterization of nucleotide states. Our model highlights the uniqueness of dynein in the coupling of ATPase with its movement during both inchworm and hand-over-hand stepping.

A model for the chemomechanical coupling of the mammalian cytoplasmic dynein molecular motor

Available single-molecule data have shown that some mammalian cytoplasmic dynein dimers move on microtubules with a constant step size of about 8.2 nm. Here, a model is presented for the chemomechanical coupling of these mammalian cytoplasmic dynein dimers. In contrast to the previous models, a peculiar feature of the current model is that the rate constants of ATPase activity are independent of the external force. Based on this model, analytical studies of the motor dynamics are presented. With only four adjustable parameters, the theoretical results reproduce quantitatively diverse available single-molecule data on the force dependence of stepping ratio, velocity, mean dwell time, and dwell-time distribution between two mechanical steps. Predicted results are also provided for the force dependence of the number of ATP molecules consumed per mechanical step, indicating that under no or low force the motors exhibit a tight chemomechanical coupling, and as the force increases the number of ATPs consumed per step increases greatly.

Allocating and Splitting Free Energy to Maximize Molecular Machine Flux

Biomolecular machines transduce between different forms of energy. These machines make directed progress and increase their speed by consuming free energy, typically in the form of nonequilibrium chemical concentrations. Machine dynamics are often modeled by transitions between a set of discrete metastable conformational states. In general, the free-energy change associated with each transition can increase the forward rate constant, decrease the reverse rate constant, or both. In contrast to previous optimizations, we find that in general flux is maximized neither by devoting all free-energy changes to increasing forward rate constants nor by solely decreasing reverse rate constants. Instead, the optimal free-energy splitting depends on the detailed dynamics. Extending our analysis to machines with vulnerable states (from which they can break down), in the strong driving corresponding to in vivo cellular conditions, processivity is maximized by reducing the occupation of the vulnerable state.

Viral highway to nucleus exposed by image correlation analyses

Parvoviral genome translocation from the plasma membrane into the nucleus is a coordinated multistep process mediated by capsid proteins. We used fast confocal microscopy line scan imaging combined with image correlation methods including auto-, pair- and cross-correlation, and number and brightness analysis, to study the parvovirus entry pathway at the single-particle level in living cells. Our results show that the endosome-associated movement of virus particles fluctuates from fast to slow. Fast transit of single cytoplasmic capsids to the nuclear envelope is followed by slow movement of capsids and fast diffusion of capsid fragments in the nucleoplasm. The unique combination of image analyses allowed us to follow the fate of intracellular single virus particles and their interactions with importin β revealing previously unknown dynamics of the entry pathway.

Heterogeneity in kinesin function

The kinesin family proteins are often studied as prototypical molecular motors; a deeper understanding of them can illuminate regulation of intracellular transport. It is typically assumed that they function identically. Here we find that this assumption of homogeneous function appears incorrect: variation among motors' velocities in vivo and in vitro is larger than the stochastic variation expected for an ensemble of "identical" motors. When moving on microtubules, slow and fast motors are persistently slow, and fast, respectively. We develop theory that provides quantitative criteria to determine whether the observed single-molecule variation is too large to be generated from an ensemble of identical molecules. To analyze such heterogeneity, we group traces into homogeneous sub-ensembles. Motility studies varying the temperature, pH and glycerol concentration suggest at least 2 distinct functional states that are independently affected by external conditions. We end by investigating the functional ramifications of such heterogeneity through Monte-Carlo multi-motor simulations.

Effect of fuel concentration and force on collective transport by a team of dynein motors

Motor proteins are essential components of intracellular transport inside eukaryotic cells. These protein molecules use chemical energy obtained from hydrolysis of ATP to produce mechanical forces required for transporting cargos inside cells, from one location to another, in a directed manner. Of these motors, cytoplasmic dynein is structurally more complex than other motor proteins involved in intracellular transport, as it shows force and fuel (ATP) concentration dependent step-size. Cytoplasmic dynein motors are known to work in a team during cargo transport and force generation. Here, we use a complete Monte-Carlo model of single dynein constrained by in vitro experiments, which includes the effect of both force and ATP on stepping as well as detachment of motors under force. We then use our complete Monte-Carlo model of single dynein motor to understand collective cargo transport by a team of dynein motors, such as dependence of cargo travel distance and velocity on applied force and fuel concentration. In our model, cargos pulled by a team of dynein motors do not detach rapidly under higher forces, confirming the experimental observation of longer persistence time of dynein team on microtubule under higher forces.

Effect of catch bonding on transport of cellular cargo by dynein motors

Recent experiments have demonstrated that dynein motors exhibit catch bonding behavior, in which the unbinding rate of a single dynein decreases with increasing force, for a certain range of force. Motivated by these experiments, we study the effect of catch bonding on unidirectional transport properties of cellular cargo carried by multiple dynein motors. We introduce a threshold force bond deformation (TFBD) model, consistent with the experiments, wherein catch bonding sets in beyond a critical applied load force. We find catch bonding can result in dramatic changes in the transport properties, which are in sharp contrast to kinesin-driven unidirectional transport, where catch bonding is absent. We predict that under certain conditions, the average velocity of the cellular cargo can actually increase as applied load is increased. We characterize the transport properties in terms of a velocity profile plot in the parameter space of the catch bond strength and the stall force of the motor. This plot yields predictions that may be experimentally accessed by suitable modifications of motor transport and binding properties.

Load-induced enhancement of Dynein force production by LIS1-NudE in vivo and in vitro

Most sub-cellular cargos are transported along microtubules by kinesin and dynein molecular motors, but how transport is regulated is not well understood. It is unknown whether local control is possible, for example, by changes in specific cargo-associated motor behaviour to react to impediments. Here we discover that microtubule-associated lipid droplets (LDs) in COS1 cells respond to an optical trap with a remarkable enhancement in sustained force production. This effect is observed only for microtubule minus-end-moving LDs. It is specifically blocked by RNAi for the cytoplasmic dynein regulators LIS1 and NudE/L (Nde1/Ndel1), but not for the dynactin p150^Glued subunit. It can be completely replicated using cell-free preparations of purified LDs, where duration of LD force production is more than doubled. These results identify a novel, intrinsic, cargo-associated mechanism for dynein-mediated force adaptation, which should markedly improve the ability of motor-driven cargoes to overcome subcellular obstacles.

Transport of large cargo through the cytoplasm can encounter physical impediments which should be overcome. Here the authors show that lipid droplets constrained by an optical trap respond with an increase in dynein-mediated force that is dependent on dynein regulators LIS1 and NudE/L, but not on p150^glued.

The winch model can explain both coordinated and uncoordinated stepping of cytoplasmic dynein

Cytoplasmic dynein moves processively along microtubules, but the mechanism of how its heads use the energy from ATP hydrolysis, coupled to a linker swing, to achieve directed motion, is still unclear. In this article, we present a theoretical model based on the winch mechanism in which the principal direction of the linker stroke is toward the microtubule-binding domain. When mechanically coupling two identical heads (each with postulated elastic properties and a minimal ATPase cycle), the model reproduces stepping with 8-nm steps (even though the motor itself is much larger), interhead coordination, and processivity, as reported for mammalian dyneins. Furthermore, when we loosen the elastic connection between the heads, the model still shows processive directional stepping, but it becomes uncoordinated and the stepping pattern shows a greater variability, which reproduces the properties of yeast dyneins. Their slower chemical kinetics allows processive motility and a high stall force without the need for coordination.

Biased Brownian motion as a mechanism to facilitate nanometer-scale exploration of the microtubule plus end by a kinesin-8

Kinesin-8s are plus-end-directed motors that negatively regulate microtubule (MT) length. Well-characterized members of this subfamily (Kip3, Kif18A) exhibit two important properties: (i) They are "ultraprocessive," a feature enabled by a second MT-binding site that tethers the motors to a MT track, and (ii) they dissociate infrequently from the plus end. Together, these characteristics combined with their plus-end motility cause Kip3 and Kif18A to enrich preferentially at the plus ends of long MTs, promoting MT catastrophes or pausing. Kif18B, an understudied human kinesin-8, also limits MT growth during mitosis. In contrast to Kif18A and Kip3, localization of Kif18B to plus ends relies on binding to the plus-end tracking protein EB1, making the relationship between its potential plus-end-directed motility and plus-end accumulation unclear. Using single-molecule assays, we show that Kif18B is only modestly processive and that the motor switches frequently between directed and diffusive modes of motility. Diffusion is promoted by the tail domain, which also contains a second MT-binding site that decreases the off rate of the motor from the MT lattice. In cells, Kif18B concentrates at the extreme tip of a subset of MTs, superseding EB1. Our data demonstrate that kinesin-8 motors use diverse design principles to target MT plus ends, which likely target them to the plus ends of distinct MT subpopulations in the mitotic spindle.

Model for bidirectional movement of cytoplasmic dynein

Cytoplasmic dynein exhibits a directional processive movement on microtubule filaments and is known to move in steps of varying length based on the number of ATP molecules bound to it and the load that it carries. It is experimentally observed that dynein takes occasional backward steps and the frequency of such backward steps increases as the load approaches the stall force. Using a stochastic process model, we investigate the bidirectional movement of single head of a dynein motor. The probability for backward step is implemented based on fluctuation theorem of non-equilibrium statistical mechanics. We find that the movement of dynein motor is characterized with negative velocity implying backward motion beyond stall force. We observe that the motor moves backward for super stall forces by hydrolyzing the ATP exactly the same way as it does while moving forward for sub-stall forces. Movement of dynein is also simulated using a kinetic Monte Carlo method and the simulated velocities are in good agreement with velocities obtained using a stochastic rate equation model.

Axonal transport: how high microtubule density can compensate for boundary effects in small-caliber axons

Long-distance intracellular axonal transport is predominantly microtubule-based, and its impairment is linked to neurodegeneration. In this study, we present theoretical arguments that suggest that near the axon boundaries (walls), the effective viscosity can become large enough to impede cargo transport in small (but not large) caliber axons. Our theoretical analysis suggests that this opposition to motion increases rapidly as the cargo approaches the wall. We find that having parallel microtubules close enough together to enable a cargo to simultaneously engage motors on more than one microtubule dramatically enhances motor activity, and thus minimizes the effects of any opposition to transport. Even if microtubules are randomly placed in axons, we find that the higher density of microtubules found in small-caliber axons increases the probability of having parallel microtubules close enough that they can be used simultaneously by motors on a cargo. The boundary effect is not a factor in transport in large-caliber axons where the microtubule density is lower.

Teamwork in microtubule motors

Diverse cellular processes are driven by the collective force from multiple motor proteins. Disease-causing mutations cause aberrant function of motors, but the impact is observed at a cellular level and beyond, therefore necessitating an understanding of cell mechanics at the level of motor molecules. One way to do this is by measuring the force generated by ensembles of motors in vivo at single-motor resolution. This has been possible for microtubule motor teams that transport intracellular organelles, revealing unexpected differences between collective and single-molecule function. Here we review how the biophysical properties of single motors, and differences therein, may translate into collective motor function during organelle transport and perhaps in other processes outside transport.

Cargo transport by cytoplasmic Dynein can center embryonic centrosomes

To complete meiosis II in animal cells, the male DNA material needs to meet the female DNA material contained in the female pronucleus at the egg center, but it is not known how the male pronucleus, deposited by the sperm at the periphery of the cell, finds the cell center in large eggs. Pronucleus centering is an active process that appears to involve microtubules and molecular motors. For small and medium-sized cells, the force required to move the centrosome can arise from either microtubule pushing on the cortex, or cortically-attached dynein pulling on microtubules. However, in large cells, such as the fertilized Xenopus laevis embryo, where microtubules are too long to support pushing forces or they do not reach all boundaries before centrosome centering begins, a different force generating mechanism must exist. Here, we present a centrosome positioning model in which the cytosolic drag experienced by cargoes hauled by cytoplasmic dynein on the sperm aster microtubules can move the centrosome towards the cell's center. We find that small, fast cargoes (diameter ∼100 nm, cargo velocity ∼2 µm/s) are sufficient to move the centrosome in the geometry of the Xenopus laevis embryo within the experimentally observed length and time scales.

Come together: group behavior of dynein motors

The relationship between single-molecule activity and emergent group function is just starting to be addressed. Reporting in a recent issue of Cell, Rai et al. (2013) show that groups of dynein motors function particularly well together and suggest specific features of the single-motors responsible for this cooperative function.

Molecular adaptations allow dynein to generate large collective forces inside cells

Many cellular processes require large forces that are generated collectively by multiple cytoskeletal motor proteins. Understanding how motors generate force as a team is therefore fundamentally important but is poorly understood. Here, we demonstrate optical trapping at single-molecule resolution inside cells to quantify force generation by motor teams driving single phagosomes. In remarkable paradox, strong kinesins fail to work collectively, whereas weak and detachment-prone dyneins team up to generate large forces that tune linearly in strength and persistence with dynein number. Based on experimental evidence, we propose that leading dyneins in a load-carrying team take short steps, whereas trailing dyneins take larger steps. Dyneins in such a team bunch close together and therefore share load better to overcome low/intermediate loads. Up against higher load, dyneins "catch bond" tenaciously to the microtubule, but kinesins detach rapidly. Dynein therefore appears uniquely adapted to work in large teams, which may explain how this motor executes bewilderingly diverse cellular processes.

Signaling endosomes and growth cone motility in axon regeneration

During development and regeneration, growth cones guide neurites to their targets by altering their motility in response to extracellular guidance cues. One class of cues critical to nervous system development is the neurotrophins. Neurotrophin binding to their cognate receptors stimulates their endocytosis into signaling endosomes. Current data indicate that the spatiotemporal localization of signaling endosomes can direct diverse processes regulating cell motility, including membrane trafficking, cytoskeletal remodeling, adhesion dynamics, and local translation. Recent experiments manipulating signaling endosome localization in neuronal growth cones support these views and place the neurotrophin signaling endosome in a central role regulating growth cone motility during axon growth and regeneration.

Nanoparticle-mediated signaling endosome localization regulates growth cone motility and neurite growth

Understanding neurite growth regulation remains a seminal problem in neurobiology. During development and regeneration, neurite growth is modulated by neurotrophin-activated signaling endosomes that transmit regulatory signals between soma and growth cones. After injury, delivering neurotrophic therapeutics to injured neurons is limited by our understanding of how signaling endosome localization in the growth cone affects neurite growth. Nanobiotechnology is providing new tools to answer previously inaccessible questions. Here, we show superparamagnetic nanoparticles (MNPs) functionalized with TrkB agonist antibodies are endocytosed into signaling endosomes by primary neurons that activate TrkB-dependent signaling, gene expression and promote neurite growth. These MNP signaling endosomes are trafficked into nascent and existing neurites and transported between somas and growth cones in vitro and in vivo. Manipulating MNP-signaling endosomes by a focal magnetic field alters growth cone motility and halts neurite growth in both peripheral and central nervous system neurons, demonstrating signaling endosome localization in the growth cone regulates motility and neurite growth. These data suggest functionalized MNPs may be used as a platform to study subcellular organelle localization and to deliver nanotherapeutics to treat injury or disease in the central nervous system.

Monte Carlo simulation from proton slip to "coupled" proton flow in ATP synthase based on the bi-site mechanism

ATP synthase couples proton flow to ATP synthesis, but is leaky to protons at very low nucleotide concentration. Based on the bi-site mechanism, we simulated the proton conduction from proton slip to "coupled" proton flow in ATP synthase using the Monte Carlo method. Good agreement is obtained between the simulated and available experimental results. Our model provides deeper insight into the nucleotide dependence of ATP catalysis, and the kinetic cooperativity in three catalysis subunits. The results of simulation support the bi-site mechanism in ATP synthesis.

LIS1 and NudE induce a persistent dynein force-producing state

Cytoplasmic dynein is responsible for many aspects of cellular and subcellular movement. LIS1, NudE, and NudEL are dynein interactors initially implicated in brain developmental disease but now known to be required in cell migration, nuclear, centrosomal, and microtubule transport, mitosis, and growth cone motility. Identification of a specific role for these proteins in cytoplasmic dynein motor regulation has remained elusive. We find that NudE stably recruits LIS1 to the dynein holoenzyme molecule, where LIS1 interacts with the motor domain during the prepowerstroke state of the dynein crossbridge cycle. NudE abrogates dynein force production, whereas LIS1 alone or with NudE induces a persistent-force dynein state that improves ensemble function of multiple dyneins for transport under high-load conditions. These results likely explain the requirement for LIS1 and NudE in the transport of nuclei, centrosomes, chromosomes, and the microtubule cytoskeleton as well as the particular sensitivity of migrating neurons to reduced LIS1 expression.

Traffic of cytoskeletal motors with disordered attachment rates

Motivated by experimental results on the interplay between molecular motors and tau proteins, we extend lattice-based models of intracellular transport to include a second species of particle which locally influences the motor-filament attachment rate. We consider various exactly solvable limits of a stochastic multiparticle model before focusing on the low-motor-density regime. Here, an approximate treatment based on the random-walk behavior of single motors gives good quantitative agreement with simulation results for the tau dependence of the motor current. Finally, we discuss the possible physiological implications of our results.

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.

Optimal cytoplasmic transport in viral infections

For many viruses, the ability to infect eukaryotic cells depends on their transport through the cytoplasm and across the nuclear membrane of the host cell. During this journey, viral contents are biochemically processed into complexes capable of both nuclear penetration and genomic integration. We develop a stochastic model of viral entry that incorporates all relevant aspects of transport, including convection along microtubules, biochemical conversion, degradation, and nuclear entry. Analysis of the nuclear infection probabilities in terms of the transport velocity, degradation, and biochemical conversion rates shows how certain values of key parameters can maximize the nuclear entry probability of the viral material. The existence of such "optimal" infection scenarios depends on the details of the biochemical conversion process and implies potentially counterintuitive effects in viral infection, suggesting new avenues for antiviral treatment. Such optimal parameter values provide a plausible transport-based explanation of the action of restriction factors and of experimentally observed optimal capsid stability. Finally, we propose a new interpretation of how genetic mutations unrelated to the mechanism of drug action may nonetheless confer novel types of overall drug resistance.

Multiscale approaches for studying energy transduction in dynein

Cytoplasmic dynein is an important motor that drives all minus-end directed movement along microtubules. Dynein is a complex motor whose processive motion is driven by ATP-hydrolysis. Dynein's run length has been measured to be several millimetres with typical velocities in the order of a few nanometres per second. Therefore, the average time between steps is a fraction of a second. When this time scale is compared with typical time scales for protein side chain and backbone movements (approximately 10(-9) s and approximately 10(-5) s, respectively), it becomes clear that a multi-timescale modelling approach is required to understand energy transduction in this protein. Here, we review recent efforts to use computational and mathematical modelling to understand various aspects of dynein's chemomechanical cycle. First, we describe a structural model of dynein's motor unit showing a heptameric organization of the motor subunits. Second, we describe our molecular dynamics simulations of the motor unit that are used to investigate the dynamics of the various motor domains. Third, we present a kinetic model of the coordination between the two dynein heads. Lastly, we investigate the various potential geometries of the dimer during its hydrolytic and stepping cycle.

Intracellular transport: kinesins working together

While most in vitro experiments with motor proteins focus on the behavior of individual motors, in cells most cargo are transported by multiple motors and even multiple classes of motor. How these motors cooperate and compete in transporting cargo is not clear. Recent experimental and theoretical work suggests that motors attached to a given cargo interact in both expected and unexpected ways.

Stepping, strain gating, and an unexpected force-velocity curve for multiple-motor-based transport

BACKGROUND

Intracellular transport via processive kinesin, dynein, and myosin molecular motors plays an important role in maintaining cell structure and function. In many cases, cargoes move distances longer than expected for single motors; there is significant evidence that this increased travel is in part due to multiple motors working together to move the cargoes. Although we understand single motors experimentally and theoretically, our understanding of multiple motors working together is less developed.

RESULTS

We theoretically investigate how multiple kinesin motors function. Our model includes stochastic fluctuations of each motor as it proceeds through its enzymatic cycle. Motors dynamically influence each other and function in the presence of thermal noise and viscosity. We test the theory via comparison with the experimentally observed distribution of step sizes for two motors moving a cargo, and by predicting slightly subadditive stalling force for two motors relative to one. In the presence of load, our predictions for travel distances and mean velocities are different from the steady-state model: with high motor-motor coupling, we predict a form of strain-gating, where-because of the underlying motor's dynamics-the motors share load unevenly, leading to increased mean travel distance of the multiple-motor system under load. Surprisingly, we predict that in the presence of small load, two-motor cargoes move slightly slower than do single-motor cargoes. Unpublished data from G.T. Shubeita, B.C. Carter, and S.P.G. confirm this prediction in vivo.

CONCLUSIONS

When only a few motors are active, fluctuations and unequal load sharing between motors can result in significant alterations of ensemble function.

Model for the unidirectional motion of a dynein molecule

Cytoplasmic dyneins transport cellular organelles by moving on a microtubule filament. It has been found recently that depending on the applied force and the concentration of the adenosine triphosphate molecules, dynein's step size varies. Based on these studies, we propose a simple model for dynein's unidirectional motion taking into account the variations in its step size. We study how the average velocity and the relative dispersion in the displacement vary with the applied load. The model is amenable to further extensions by inclusion of details associated with the structure and the processivity of the molecule.

Cargo Transport: Two Motors Are Sometimes Better Than One

Molecular motor proteins are crucial for the proper distribution of organelles and vesicles in cells. Much of our current understanding of how motors function stems from studies of single motors moving cargos in vitro. More recently, however, there has been mounting evidence that the cooperation of multiple motors in moving cargos and the regulation of motor-filament affinity could be key mechanisms that cells utilize to regulate cargo transport. Here, we review these recent advances and present a picture of how the different mechanisms of regulating the number of motors moving a cargo could facilitate cellular functions.

Mechanism of transport of IFT particles in C. elegans cilia by the concerted action of kinesin-II and OSM-3 motors

The assembly and function of cilia on Caenorhabditis elegans neurons depends on the action of two kinesin-2 motors, heterotrimeric kinesin-II and homodimeric OSM-3–kinesin, which cooperate to move the same intraflagellar transport (IFT) particles along microtubule (MT) doublets. Using competitive in vitro MT gliding assays, we show that purified kinesin-II and OSM-3 cooperate to generate movement similar to that seen along the cilium in the absence of any additional regulatory factors. Quantitative modeling suggests that this could reflect an alternating action mechanism, in which the motors take turns to move along MTs, or a mechanical competition, in which the motors function in a concerted fashion to move along MTs with the slow motor exerting drag on the fast motor and vice versa. In vivo transport assays performed in Bardet-Biedl syndrome (BBS) protein and IFT motor mutants favor a mechanical competition model for motor coordination in which the IFT motors exert a BBS protein–dependent tension on IFT particles, which controls the IFT pathway that builds the cilium foundation.

Model for unidirectional movement of axonemal and cytoplasmic dynein molecules

A model for the unidirectional movement of dynein is presented based on the structural observations and biochemical experimental results available. In this model, the binding affinity of dynein for microtubule (MT) is independent of its nucleotide state and the change between strong and weak MT-binding is determined naturally by the variation of relative orientation between the stalk and MT, as the stalk rotates following nucleotide-state transition. Thus the enigmatic communication from the adenosine triphosphate (ATP)-binding site in the globular domain to the far MT-binding site in the tip of the stalk, which is a prerequisite in conventional models, is not required. Using the present model, the previous experimental results such as the effect of ATP and adenosine diphosphate (ADP) bindings on dissociation of dynein from MT, the movement of single-headed axonemal dyneins at saturating ATP concentration, the load dependence of step-size for the movement of two-headed cytoplasmic dyneins and the dependence of stall force on ATP concentration can be well explained.

Flexible light-chain and helical structure of F-actin explain the movement and step size of myosin-VI

Myosin-VI is a dimeric isoform of unconventional myosins. Single molecule experiments indicate that myosin-VI and myosin-V are processive molecular motors, but travel toward opposite ends of filamentous actin. Structural studies show several differences between myosin-V and VI, including a significant difference in the light-chain domain connecting the motor domains. Combining the measured kinetics of myosin-VI with the elasticity of the light chains, and the helical structure of F-actin, we compare and contrast the motility of myosin-VI with myosin-V. We show that the elastic properties of the light-chain domain control the stepping behavior of these motors. Simple models incorporating the motor elastic energy can quantitatively capture most of the observed data. Implications of our result for other processive motors are discussed.

Building complexity: an in vitro study of cytoplasmic dynein with in vivo implications

BACKGROUND

Cytoplasmic dynein is the molecular motor responsible for most retrograde microtubule-based vesicular transport. In vitro single-molecule experiments suggest that dynein function is not as robust as that of kinesin-1 or myosin-V because dynein moves only a limited distance (approximately 800 nm) before detaching and can exert a modest (approximately 1 pN) force. However, dynein-driven cargos in vivo move robustly over many microns and exert forces of multiple pN. To determine how to go from limited single-molecule function to robust in vivo transport, we began to build complexity in a controlled manner by using in vitro experiments.

RESULTS

We show that a single cytoplasmic dynein motor frequently transitions into an off-pathway unproductive state that impairs net transport. Addition of a second (and/or third) dynein motor, so that cargos are moved by two (or three) motors rather than one, is sufficient to recover several properties of in vivo motion; such properties include long cargo travels, robust motion, and increased forces. Part of this improvement appears to arise from selective suppression of the unproductive state of dynein rather than from a fundamental change in dynein's mechanochemical cycle.

CONCLUSIONS

Multiple dyneins working together suppress shortcomings of a single motor and generate robust motion under in vitro conditions. There appears to be no need for additional cofactors (e.g., dynactin) for this improvement. Because cargos are often driven by multiple dyneins in vivo, our results show that changing the number of dynein motors could allow modulation of dynein function from the mediocre single-dynein limit to robust in vivo-like dynein-driven motion.

Model of chromosome motility in Drosophila embryos: adaptation of a general mechanism for rapid mitosis

During mitosis, ensembles of dynamic MTs and motors exert forces that coordinate chromosome segregation. Typically, chromosomes align at the metaphase spindle equator where they oscillate along the pole-pole axis before disjoining and moving poleward during anaphase A, but spindles in different cell types display differences in MT dynamicity, in the amplitude of chromosome oscillations and in rates of chromatid-to-pole motion. Drosophila embryonic mitotic spindles, for example, display remarkably dynamic MTs, barely detectable metaphase chromosome oscillations, and a rapid rate of "flux-pacman-dependent" anaphase chromatid-to-pole motility. Here we develop a force-balance model that describes Drosophila embryo chromosome motility in terms of a balance of forces acting on kinetochores and kMTs that is generated by multiple polymer ratchets and mitotic motors coupled to tension-dependent kMT dynamics. The model shows that i), multiple MTs displaying high dynamic instability can drive steady and rapid chromosome motion; ii), chromosome motility during metaphase and anaphase A can be described by a single mechanism; iii), high kinetochore dynein activity is deployed to dampen metaphase oscillations, to augment the basic flux-pacman mechanism, and to drive rapid anaphase A; iv), modulation of the MT rescue frequency by the kinetochore-associated kinesin-13 depolymerase promotes metaphase chromosome oscillations; and v), this basic mechanism can be adapted to a broad range of spindles.

Modeling mitosis

The mitotic spindle is a fascinating protein machine that uses bipolar arrays of dynamic microtubules and many mitotic motors to coordinate the accurate segregation of sister chromatids. Here we discuss recent mathematical models and computer simulations that, in concert with experimental studies, help explain the molecular mechanisms by which the spindle machinery performs its crucial functions. We review current models of spindle assembly, positioning, maintenance and elongation; of chromosome capture and congression; and of the spindle assembly checkpoint. We discuss some limitations of the application of modeling to other aspects of mitosis and the feasibility of building more comprehensive system-level models.