Coordination and collective properties of molecular motors: theory

Migration and Separation of Polymers in Nonuniform Active Baths

Polymerlike structures are ubiquitous in nature and synthetic materials. Their configurational and migration properties are often affected by crowded environments leading to nonthermal fluctuations. Here, we study an ideal Rouse chain in contact with a nonhomogeneous active bath, characterized by the presence of active self-propelled agents which exert time-correlated forces on the chain. By means of a coarse-graining procedure, we derive an effective evolution for the center of mass of the chain and show its tendency to migrate toward and preferentially localize in regions of high and low bath activity depending on the model parameters. In particular, we demonstrate that an active bath with nonuniform activity can be used to separate efficiently polymeric species with different lengths and/or connectivity.

KnotResolver: tracking self-intersecting filaments in microscopy using directed graphs

MOTIVATION

Quantification of microscopy time series of in vitro reconstituted motor-driven microtubule transport in "gliding assays" is typically performed using computational object tracking tools. However, these are limited to non-intersecting and rod-like filaments.

RESULTS

Here, we describe a novel computational image-analysis pipeline, KnotResolver, to track image time series of highly curved self-intersecting looped filaments (knots) by resolving cross-overs. The code integrates filament segmentation and cross-over or "knot" identification based on directed graph representation, where nodes represent cross-overs and edges represent the path connecting them. The graphs are mapped back to contours and the distance to a reference minimized. The accuracy of contour detection is sub-pixel with a robustness to noise. We demonstrate the utility of KnotResolver by automatically quantifying "flagella-like" curvature dynamics and wave-like oscillations of clamped microtubules in a "gliding assay."

AVAILABILITY AND IMPLEMENTATION

The MATLAB-based source code is released as OpenSource and is available at https://github.com/CyCelsLab/MTKnotResolver.

Non-specific cargo-filament interactions slow down motor-driven transport

Active, motor-based cargo transport is important for many cellular functions and cellular development. However, the cell interior is complex and crowded and could have many weak, non-specific interactions with the cargo being transported. To understand how cargo-environment interactions will affect single motor cargo transport and multi-motor cargo transport, we use an artificial quantum dot cargo bound with few (~ 1) to many (~ 5-10) motors allowed to move in a dense microtubule network. We find that kinesin-driven quantum dot cargo is slower than single kinesin-1 motors. Excitingly, there is some recovery of the speed when multiple motors are attached to the cargo. To determine the possible mechanisms of both the slow down and recovery of speed, we have developed a computational model that explicitly incorporates multi-motor cargos interacting non-specifically with nearby microtubules, including, and predominantly with the microtubule on which the cargo is being transported. Our model has recovered the experimentally measured average cargo speed distribution for cargo-motor configurations with few and many motors, implying that numerous, weak, non-specific interactions can slow down cargo transport and multiple motors can reduce these interactions thereby increasing velocity.

A mutation in switch I alters the load-dependent kinetics of myosin Va

Myosin Va is the molecular motor that drives intracellular vesicular transport, powered by the transduction of chemical energy from ATP into mechanical work. The coupling of the powerstroke and phosphate (Pi) release is key to understanding the transduction process, and crucial details of this process remain unclear. Therefore, we determined the effect of elevated Pi on the force-generating capacity of a mini-ensemble of myosin Va S1 (WT) in a laser trap assay. By increasing the stiffness of the laser trap we determined the effect of increasing resistive loads on the rate of Pi-induced detachment from actin, and quantified this effect using the Bell approximation. We observed that WT myosin generated higher forces and larger displacements at the higher laser trap stiffnesses in the presence of 30 mM Pi, but binding event lifetimes decreased dramatically, which is most consistent with the powerstroke preceding the release of Pi from the active site. Repeating these experiments using a construct with a mutation in switch I of the active site (S217A) caused a seven-fold increase in the load-dependence of the Pi-induced detachment rate, suggesting that the S217A region of switch I may help mediate the load-dependence of Pi-rebinding.

Flagella-like beating of actin bundles driven by self-organized myosin waves

Wave-like beating of eukaryotic cilia and flagella-threadlike protrusions found in many cells and microorganisms-is a classic example of spontaneous mechanical oscillations in biology. This type of self-organized active matter raises the question of the coordination mechanism between molecular motor activity and cytoskeletal filament bending. Here we show that in the presence of myosin motors, polymerizing actin filaments self-assemble into polar bundles that exhibit wave-like beating. Importantly, filament beating is associated with myosin density waves initiated at twice the frequency of the actin-bending waves. A theoretical description based on curvature control of motor binding to the filaments and of motor activity explains our observations in a regime of high internal friction. Overall, our results indicate that the binding of myosin to actin depends on the actin bundle shape, providing a feedback mechanism between the myosin activity and filament deformations for the self-organization of large motor filament assemblies.

Myosin II Adjusts Motility Properties and Regulates Force Production Based on Motor Environment

Introduction

Myosin II has been investigated with optical trapping, but single motor-filament assay arrangements are not reflective of the complex cellular environment. To understand how myosin interactions propagate up in scale to accomplish system force generation, we devised a novel actomyosin ensemble optical trapping assay that reflects the hierarchy and compliancy of a physiological environment and is modular for interrogating force effectors.

Methods

Hierarchical actomyosin bundles were formed in vitro. Fluorescent template and cargo actin filaments (AF) were assembled in a flow cell and bundled by myosin. Beads were added in the presence of ATP to bind the cargo AF and activate myosin force generation to be measured by optical tweezers.

Results

Three force profiles resulted across a range of myosin concentrations: high force with a ramp-plateau, moderate force with sawtooth movement, and baseline. The three force profiles, as well as high force output, were recovered even at low solution concentration, suggesting that myosins self-optimize within AFs. Individual myosin steps were detected in the ensemble traces, indicating motors are taking one step at a time while others remain engaged in order to sustain productive force generation.

Conclusions

Motor communication and system compliancy are significant contributors to force output. Environmental conditions, motors taking individual steps to sustain force, the ability to backslip, and non-linear concentration dependence of force indicate that the actomyosin system contains a force-feedback mechanism that senses the local cytoskeletal environment and communicates to the individual motors whether to be in a high or low duty ratio mode.

Supplementary Information

The online version contains supplementary material available at 10.1007/s12195-022-00731-1.

Nano-Particles Carried by Multiple Dynein Motors Self-Regulate Their Number of Actively Participating Motors

Intra-cellular active transport by native cargos is ubiquitous. We investigate the motion of spherical nano-particles (NPs) grafted with flexible polymers that end with a nuclear localization signal peptide. This peptide allows the recruitment of several mammalian dynein motors from cytoplasmic extracts. To determine how motor–motor interactions influenced motility on the single microtubule level, we conducted bead-motility assays incorporating surface adsorbed microtubules and combined them with model simulations that were based on the properties of a single dynein. The experimental and simulation results revealed long time trajectories: when the number of NP-ligated motors N_m increased, run-times and run-lengths were enhanced and mean velocities were somewhat decreased. Moreover, the dependence of the velocity on run-time followed a universal curve, regardless of the system composition. Model simulations also demonstrated left- and right-handed helical motion and revealed self-regulation of the number of microtubule-bound, actively transporting dynein motors. This number was stochastic along trajectories and was distributed mainly between one, two, and three motors, regardless of N_m. We propose that this self-regulation allows our synthetic NPs to achieve persistent motion that is associated with major helicity. Such a helical motion might affect obstacle bypassing, which can influence active transport efficiency when facing the crowded environment of the cell.

Novel mechanism for oscillations in catchbonded motor-filament complexes

Generation of mechanical oscillations is ubiquitous to a wide variety of intracellular processes ranging from activity of muscle fibres to oscillations of the mitotic spindle. The activity of motors plays a vital role in maintaining the integrity of the mitotic spindle structure and in generating spontaneous oscillations. While the structural features and properties of the individual motors are well characterized, their implications on the functional behaviour of motor-filament complexes is more involved. We show that force-induced allosteric deformations in dynein, which results in catchbonding behaviour, provide a generic mechanism to generate spontaneous oscillations in motor-cytoskeletal filament complexes. The resultant phase diagram of such motor-filament systems - characterized by force-induced allosteric deformations - exhibits bistability and sustained limit cycle oscillations in biologically relevant regimes, such as for catchbonded dynein. The results reported here elucidate the central role of this mechanism in fashioning a distinctive stability behaviour and oscillations in motor-filament complexes, such as mitotic spindles.

Navigating the crowd: visualizing coordination between genome dynamics, structure, and transcription

The eukaryotic genome is hierarchically structured yet highly dynamic. Regulating transcription in this environment demands a high level of coordination to permit many proteins to interact with chromatin fiber at appropriate sites in a timely manner. We describe how recent advances in quantitative imaging techniques overcome caveats of sequencing-based methods (Hi-C and related) by enabling direct visualization of transcription factors and chromatin at high resolution, from single genes to the whole nucleus. We discuss the contribution of fluorescence imaging to deciphering the principles underlying this coordination within the crowded nuclear space in living cells and discuss challenges ahead.

Different motilities of microtubules driven by kinesin-1 and kinesin-14 motors patterned on nanopillars

Kinesin is a motor protein that plays important roles in a variety of cellular functions. In vivo, multiple kinesin molecules are bound to cargo and work as a team to produce larger forces or higher speeds than a single kinesin. However, the coordination of kinesins remains poorly understood because of the experimental difficulty in controlling the number and arrangement of kinesins, which are considered to affect their coordination. Here, we report that both the number and spacing significantly influence the velocity of microtubules driven by nonprocessive kinesin-14 (Ncd), whereas neither the number nor the spacing changes the velocity in the case of highly processive kinesin-1. This result was realized by the optimum nanopatterning method of kinesins that enables immobilization of a single kinesin on a nanopillar. Our proposed method enables us to study the individual effects of the number and spacing of motors on the collective dynamics of multiple motors.

Active cargo positioning in antiparallel transport networks

Cytoskeletal filaments assemble into dense parallel, antiparallel, or disordered networks, providing a complex environment for active cargo transport and positioning by molecular motors. The interplay between the network architecture and intrinsic motor properties clearly affects transport properties but remains poorly understood. Here, by using surface micropatterns of actin polymerization, we investigate stochastic transport properties of colloidal beads in antiparallel networks of overlapping actin filaments. We found that 200-nm beads coated with myosin Va motors displayed directed movements toward positions where the net polarity of the actin network vanished, accumulating there. The bead distribution was dictated by the spatial profiles of local bead velocity and diffusion coefficient, indicating that a diffusion-drift process was at work. Remarkably, beads coated with heavy-mero-myosin II motors showed a similar behavior. However, although velocity gradients were steeper with myosin II, the much larger bead diffusion observed with this motor resulted in less precise positioning. Our observations are well described by a 3-state model, in which active beads locally sense the net polarity of the network by frequently detaching from and reattaching to the filaments. A stochastic sequence of processive runs and diffusive searches results in a biased random walk. The precision of bead positioning is set by the gradient of net actin polarity in the network and by the run length of the cargo in an attached state. Our results unveiled physical rules for cargo transport and positioning in networks of mixed polarity.

Signatures of motor susceptibility to forces in the dynamics of a tracer particle in an active gel

We study a model for the motion of a tracer particle inside an active gel, exposing the properties of the van Hove distribution of the particle displacements. Active events of a typical force magnitude can give rise to non-Gaussian distributions having exponential tails or side peaks. The side peaks are predicted to appear when the local bulk elasticity of the gel is large enough and few active sources are dominant. We explain the regimes of the different distributions and study the structure of the side peaks for active sources that are susceptible to the elastic stress that they cause inside the gel. We show how the van Hove distribution is altered by both the duty cycle of the active sources and their susceptibility, and suggest it as a sensitive probe to analyze microrheology data in active systems with restoring elastic forces.

Kinetic theory of pattern formation in mixtures of microtubules and molecular motors

In this study we formulate a theoretical approach, based on a Boltzmann-like kinetic equation, to describe pattern formation in two-dimensional mixtures of microtubular filaments and molecular motors. Following the previous work by Aranson and Tsimring [Phys. Rev. E 74, 031915 (2006)PLEEE81539-375510.1103/PhysRevE.74.031915] we model the motor-induced reorientation of microtubules as collision rules, and devise a semianalytical method to calculate the corresponding interaction integrals. This procedure yields an infinite hierarchy of kinetic equations that we terminate by employing a well-established closure strategy, developed in the pattern-formation community and based on a power-counting argument. We thus arrive at a closed set of coupled equations for slowly varying local density and orientation of the microtubules, and study its behavior by performing a linear stability analysis and direct numerical simulations. By comparing our method with the work of Aranson and Tsimring, we assess the validity of the assumptions required to derive their and our theories. We demonstrate that our approximation-free evaluation of the interaction integrals and our choice of a systematic closure strategy result in a rather different dynamical behavior than was previously reported. Based on our theory, we discuss the ensuing phase diagram and the patterns observed.

Construction of artificial cilia from microtubules and kinesins through a well-designed bottom-up approach

Self-organized structures of biomolecular motor systems, such as cilia and flagella, play key roles in the dynamic processes of living organisms, like locomotion or the transportation of materials. Although fabrication of such self-organized structures from reconstructed biomolecular motor systems has attracted much attention in recent years, a systematic construction methodology is still lacking. In this work, through a bottom-up approach, we fabricated artificial cilia from a reconstructed biomolecular motor system, microtubule/kinesin. The artificial cilia exhibited a beating motion upon the consumption, by the kinesins, of the chemical energy obtained from the hydrolysis of adenosine triphosphate (ATP). Several design parameters, such as the length of the microtubules, the density of the kinesins along the microtubules, the depletion force among the microtubules, etc., have been identified, which permit tuning of the beating frequency of the artificial cilia. The beating frequency of the artificial cilia increases upon increasing the length of the microtubules, but declines for the much longer microtubules. A high density of the kinesins along the microtubules is favorable for the beating motion of the cilia. The depletion force induced bundling of the microtubules accelerated the beating motion of the artificial cilia and increased the beating frequency. This work helps understand the role of self-assembled structures of the biomolecular motor systems in the dynamics of living organisms and is expected to expedite the development of artificial nanomachines, in which the biomolecular motors may serve as actuators.

Bistability, Oscillations, and Bidirectional Motion of Ensemble of Hydrodynamically Coupled Molecular Motors

We analyze the collective behavior of hydrodynamically coupled molecular motors. We show that the local fluxes induced by motor displacement can induce the experimentally observed bidirectional motion of cargoes and vesicles. By means of a mean-field approach we show that sustained oscillations as well as bistable collective motor motion arise even for very large collection of motors, when thermal noise is irrelevant. The analysis clarifies the physical mechanisms responsible for such dynamics by identifying the relevant coupling parameter and its dependence on the geometry of the hydrodynamic coupling as well as on system size. We quantify the phase diagram for the different phases that characterize the collective motion of hydrodynamically coupled motors and show that sustained oscillations can be reached for biologically relevant parameters, hence, demonstrating the relevance of hydrodynamic interactions in intracellular transport.

An Emergence Framework of Carcinogenesis

Experimental paradigms provide the framework for the understanding of cancer, and drive research and treatment, but are rarely considered by clinicians. The somatic mutation theory (SMT), in which cancer is considered a genetic disease, has been the predominant traditional model of cancer for over 50 years. More recently, alternative theories have been proposed, such as tissue organization field theory (TOFT), evolutionary models, and inflammatory models. Key concepts within the various models have led to them being difficult to reconcile. Progressively, it has been recognized that biological systems cannot be fully explained by the physicochemical properties of their constituent parts. There is an increasing call for a 'systems' approach. Incorporating the concepts of 'emergence', 'systems', 'thermodynamics', and 'chaos', a single integrated framework for carcinogenesis has been developed, enabling existing theories to become compatible as alternative mechanisms, facilitating the integration of bioinformatics and providing a structure in which translational research can flow from both 'benchtop to bedside' and 'bedside to benchtop'. In this review, a basic understanding of the key concepts of 'emergence', 'systems', 'system levels', 'complexity', 'thermodynamics', 'entropy', 'chaos', and 'fractals' is provided. Non-linear mathematical equations are included where possible to demonstrate compatibility with bioinformatics. Twelve principles that define the 'emergence framework of carcinogenesis' are developed, with principles 1-10 encapsulating the key concepts upon which the framework is built and their application to carcinogenesis. Principle 11 relates the framework to cancer progression. Principle 12 relates to the application of the framework to translational research. The 'emergence framework of carcinogenesis' collates current paradigms, concepts, and evidence around carcinogenesis into a single framework that incorporates previously incompatible viewpoints and ideas. Any researcher, scientist, or clinician involved in research, treatment, or prevention of cancer can employ this framework.

Coordinated force generation of skeletal myosins in myofilaments through motor coupling

In contrast to processive molecular motors, skeletal myosins form a large motor ensemble for contraction of muscles against high loads. Despite numerous information on the molecular properties of skeletal myosin, its ensemble effects on collective force generation have not been rigorously clarified. Here we show 4 nm stepwise actin displacements generated by synthetic myofilaments beyond a load of 30 pN, implying that steps cannot be driven exclusively by single myosins, but potentially by coordinated force generations among multiple myosins. The simulation model shows that stepwise actin displacements are primarily caused by coordinated force generation among myosin molecules. Moreover, the probability of coordinated force generation can be enhanced against high loads by utilizing three factors: strain-dependent kinetics between force-generating states; multiple power stroke steps; and high ATP concentrations. Compared with other molecular motors, our findings reveal how the properties of skeletal myosin are tuned to perform cooperative force generation for efficient muscle contraction.

Skeletal muscle myosin forms large ensembles to generate force against high loads. Using optical tweezers and simulation Kaya et al. provide experimental evidence for cooperative force generation, and describe how the molecular properties of skeletal myosins are tuned for coordinated power strokes.

Collective colloid diffusion under soft two-dimensional confinement

This work presents a numerical and theoretical investigation of the collective dynamics of colloids in an unbounded solution but trapped in a harmonic potential. Under strict two-dimensional confinement (infinitely stiff trap) the collective colloidal diffusion is enhanced and diverges at zero wave number (like k^{-1}), due to the hydrodynamic propagation of the confining force across the layer. The analytic solution for the collective diffusion of colloids under a Gaussian trap of width δ still shows enhanced diffusion for large wavelengths kδ<1, while a gradual transition to normal diffusion for kδ>1. At intermediate and short wavelengths, we illustrate to what extent the hydrodynamic enhancement of diffusion is masked by the conservative forces between colloids. At very large wavelengths, the collective diffusion becomes faster than the solvent momentum transport and a transition from Stokesian dynamics to inertial dynamics takes place. Using our inertial coupling method code (resolving fluid inertia), we study this transition by performing simulations at small Schmidt number. Simulations confirm theoretical predictions for the k→0 limit [Phys. Rev. E 90, 062314 (2014)PLEEE81539-375510.1103/PhysRevE.90.062314] showing negative density-density time correlations. However, at finite k simulations show deviations from the theory.

Engineering Circular Gliding of Actin Filaments Along Myosin-Patterned DNA Nanotube Rings To Study Long-Term Actin-Myosin Behaviors

Nature has evolved molecular motors that are critical in cellular processes occurring over broad time scales, ranging from seconds to years. Despite the importance of the long-term behavior of molecular machines, topics such as enzymatic lifetime are underexplored due to the lack of a suitable approach for monitoring motor activity over long time periods. Here, we developed an "O"-shaped Myosin Empowered Gliding Assay (OMEGA) that utilizes engineered micron-scale DNA nanotube rings with precise arrangements of myosin VI to trap gliding actin filaments. This circular gliding assay platform allows the same individual actin filament to glide over the same myosin ensemble (50-1000 motors per ring) multiple times. First, we systematically characterized the formation of DNA nanotubes rings with 4, 6, 8, and 10 helix circumferences. Individual actin filaments glide along the nanotube rings with high processivity for up to 12.8 revolutions or 11 min in run time. We then show actin gliding speed is robust to variation in motor number and independent of ring curvature within our sample space (ring diameter of 0.5-4 μm). As a model application of OMEGA, we then analyze motor-based mechanical influence on "stop-and-go" gliding behavior of actin filaments, revealing that the stop-to-go transition probability is dependent on motor flexibility. Our circular gliding assay may provide a closed-loop platform for monitoring long-term behavior of broad classes of molecular motors and enable characterization of motor robustness and long time scale nanomechanical processes.

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.

Ratchet rectification effect on the translocation of a flexible polyelectrolyte chain

We report a three dimensional Langevin dynamics simulation of a uniformly charged flexible polyelectrolyte chain, translocating through an asymmetric narrow channel with periodically varying cross sections under the influence of a periodic external electric field. When reflection symmetry of the channel is broken, a rectification effect is observed with a favored direction for the chain translocation. For a given volume of the channel unit and polymer length, the rectification occurs below a threshold frequency of the external periodic driving force. We have also observed that the extent of the rectification varies non-monotonically with increasing molecular weight and the strength of geometric asymmetry of the channel. Observed non-monotonicity of the rectification performance has been interpreted in terms of a competition between two effects arising from the channel asymmetry and change in conformational entropy. An analytical model is presented with predictions consistent with the simulation results.

Elasticity of a semiflexible filament with a discontinuous tension due to a cross-link or a molecular motor

We analyze the stretching elasticity of a wormlike chain with a tension discontinuity resulting from a Hookean spring connecting its backbone to a fixed point. The elasticity of isolated semiflexible filaments has been the subject in a significant body of literature, primarily because of its relevance to the mechanics of biological matter. In real systems, however, these filaments are usually part of supramolecular structures involving cross-linkers or molecular motors, which cause tension discontinuities. Our model is intended as a minimal structural element incorporating such a discontinuity. We obtain analytical results in the weakly bending limit of the filament, concerning its force-extension relation and the response of the two parts in which the filament is divided by the spring. For a small tension discontinuity, the linear response of the filament extension to this discontinuity strongly depends on the external tension. For large external tension f, the spring force contributes a subdominant correction ∼1/f^{3/2} to the well-known ∼1/sqrt[f]-dependence of the end-to-end extension.

Multiscale modeling of virus replication and spread

Replication and spread of human viruses is based on the simultaneous exploitation of many different host functions, bridging multiple scales in space and time. Mathematical modeling is essential to obtain a systems-level understanding of how human viruses manage to proceed through their life cycles. Here, we review corresponding advances for viral systems of large medical relevance, such as human immunodeficiency virus-1 (HIV-1) and hepatitis C virus (HCV). We will outline how the combination of mathematical models and experimental data has advanced our quantitative knowledge about various processes of these pathogens, and how novel quantitative approaches promise to fill remaining gaps.

How molecular motors extract order from chaos (a key issues review)

Molecular motors are the workhorses of living cells. Seemingly by 'magic', these molecules are able to complete purposeful tasks while being immersed in a sea of thermal chaos. Here, we review the current understanding of how these machines work, present simple models based on thermal ratchets, discuss implications for statistical physics, and provide an overview of ongoing research in this important and fascinating field of study.

Forced desorption of semiflexible polymers, adsorbed and driven by molecular motors

We formulate and characterize a model to describe the dynamics of semiflexible polymers in the presence of activity due to motor proteins attached irreversibly to a substrate, and a transverse pulling force acting on one end of the filament. The stochastic binding-unbinding of the motor proteins and their ability to move along the polymer generate active forces. As the pulling force reaches a threshold value, the polymer eventually desorbs from the substrate. Performing underdamped Langevin dynamics simulation of the polymer, and with stochastic motor activity, we obtain desorption phase diagrams. The correlation time for fluctuations in the desorbed fraction increases as one approaches complete desorption, captured quantitatively by a power law spectral density. We present theoretical analysis of the phase diagram using mean field approximations in the weakly bending limit of the polymer and performing linear stability analysis. This predicts an increase in the desorption force with the polymer bending rigidity, active velocity and processivity of the motor proteins to capture the main features of the simulation results.

Mechanical coordination in motor ensembles revealed using engineered artificial myosin filaments

The sarcomere of muscle is composed of tens of thousands of myosin motors that self-assemble into thick filaments and interact with surrounding actin-based thin filaments in a dense, near-crystalline hexagonal lattice. Together, these actin-myosin interactions enable large-scale movement and force generation, two primary attributes of muscle. Research on isolated fibres has provided considerable insight into the collective properties of muscle, but how actin-myosin interactions are coordinated in an ensemble remains poorly understood. Here, we show that artificial myosin filaments, engineered using a DNA nanotube scaffold, provide precise control over motor number, type and spacing. Using both dimeric myosin V- and myosin VI-labelled nanotubes, we find that neither myosin density nor spacing has a significant effect on the gliding speed of actin filaments. This observation supports a simple model of myosin ensembles as energy reservoirs that buffer individual stochastic events to bring about smooth, continuous motion. Furthermore, gliding speed increases with cross-bridge compliance, but is limited by Brownian effects. As a first step to reconstituting muscle motility, we demonstrate human β-cardiac myosin-driven gliding of actin filaments on DNA nanotubes.

Motor function in interpolar microtubules during metaphase

We analyze experimental motility assays of microtubules undergoing small fluctuations about a "balance point" when mixed in solution of two different kinesin motor proteins, KLP61F and Ncd. It has been proposed that the microtubule movement is due to stochastic variations in the densities of the two species of motor proteins. We test this hypothesis here by showing how it maps onto a one-dimensional random walk in a random environment. Our estimate of the amplitude of the fluctuations agrees with experimental observations. We point out that there is an initial transient in the position of the microtubule where it will typically move of order its own length. We compare the physics of this gliding assay to a recent theory of the role of antagonistic motors on restricting interpolar microtubule sliding of a cell's mitotic spindle during prometaphase. It is concluded that randomly positioned antagonistic motors can restrict relative movement of microtubules, however they do so imperfectly. A variation in motor concentrations is also analyzed and shown to lead to greater control of spindle length.

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

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

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

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

Cooperative protofilament switching emerges from inter-motor interference in multiple-motor transport

Within living cells, the transport of cargo is accomplished by groups of molecular motors. Such collective transport could utilize mechanisms which emerge from inter-motor interactions in ways that are yet to be fully understood. Here we combined experimental measurements of two-kinesin transport with a theoretical framework to investigate the functional ramifications of inter-motor interactions on individual motor function and collective cargo transport. In contrast to kinesin's low sidestepping frequency when present as a single motor, with exactly two kinesins per cargo, we observed substantial motion perpendicular to the microtubule. Our model captures a surface-associated mode of kinesin, which is only accessible via inter-motor interference in groups, in which kinesin diffuses along the microtubule surface and rapidly “hops” between protofilaments without dissociating from the microtubule. Critically, each kinesin transitions dynamically between the active stepping mode and this weak surface-associated mode enhancing local exploration of the microtubule surface, possibly enabling cellular cargos to overcome macromolecular crowding and to navigate obstacles along microtubule tracks without sacrificing overall travel distance.

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

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

Efficiency at maximum power of motor traffic on networks

We study motor traffic on Bethe networks subject to hard-core exclusion for both tightly coupled one-state machines and loosely coupled two-state machines that perform work against a constant load. In both cases we find an interaction-induced enhancement of the efficiency at maximum power (EMP) as compared to noninteracting motors. The EMP enhancement occurs for a wide range of network and single-motor parameters and is due to a change in the characteristic load-velocity relation caused by phase transitions in the system. Using a quantitative measure of the trade-off between the EMP enhancement and the corresponding loss in the maximum output power we identify parameter regimes where motor traffic systems operate efficiently at maximum power without a significant decrease in the maximum power output due to jamming effects.

The role of self-organization in developmental evolution

In developmental and evolutionary biology, particular emphasis has been given to the relationship between transcription factors and the cognate cis-regulatory elements of their target genes. These constitute the gene regulatory networks that control expression and are assumed to causally determine the formation of structures and body plans. Comparative analysis has, however, established a broad sequence homology among species that nonetheless display quite different anatomies. Transgenic experiments have also confirmed that many developmentally important elements are, in fact, functionally interchangeable. Although dependent upon the appropriate degree of gene expression, the actual construction of specific structures appears not directly linked to the functions of gene products alone. Instead, the self-formation of complex patterns, due in large part to epigenetic and non-genetic determinants, remains a persisting theme in the study of ontogeny and regenerative medicine. Recent evidence indeed points to the existence of a self-organizing process, operating through a set of intrinsic rules and forces, which imposes coordination and a holistic order upon cells and tissue. This has been repeatedly demonstrated in experiments on regeneration as well as in the autonomous formation of structures in vitro. The process cannot be wholly attributed to the functional outcome of protein-protein interactions or to concentration gradients of diffusible chemicals. This phenomenon is examined here along with some of the methodological and theoretical approaches that are now used in understanding the causal basis for self-organization in development and its evolution.

The kinetics of mechanically coupled myosins exhibit group size-dependent regimes

Naturally occurring groups of muscle myosin behave differently from individual myosins or small groups commonly assayed in vitro. Here, we investigate the emergence of myosin group behavior with increasing myosin group size. Assuming the number of myosin binding sites (N) is proportional to actin length (L) (N = L/35.5 nm), we resolve in vitro motility of actin propelled by skeletal muscle myosin for L = 0.2-3 μm. Three distinct regimes were found: L < 0.3 μm, sliding arrest; 0.3 μm ≤ L ≤ 1 μm, alternation between arrest and continuous sliding; L > 1 μm, continuous sliding. We theoretically investigated the myosin group kinetics with mechanical coupling via actin. We find rapid actin sliding steps driven by power-stroke cascades supported by postpower-stroke myosins, and phases without actin sliding caused by prepower-stroke myosin buildup. The three regimes are explained: N = 8, rare cascades; N = 15, cascade bursts; N = 35, continuous cascading. Two saddle-node bifurcations occur for increasing N (mono → bi → mono-stability), with steady states corresponding to arrest and continuous cascading. The experimentally measured dependence of actin sliding statistics on L and myosin concentration is correctly predicted.

Kinesin-8 is a low-force motor protein with a weakly bound slip state

During the cell cycle, kinesin-8s control the length of microtubules by interacting with their plus ends. To reach these ends, the motors have to be able to take many steps without dissociating. However, the underlying mechanism for this high processivity and how stepping is affected by force are unclear. Here, we tracked the motion of yeast (Kip3) and human (Kif18A) kinesin-8s with high precision under varying loads using optical tweezers. Surprisingly, both kinesin-8 motors were much weaker compared with other kinesins. Furthermore, we discovered a force-induced stick-slip motion: the motor frequently slipped, recovered from this state, and then resumed normal stepping motility without detaching from the microtubule. The low forces are consistent with kinesin-8s being regulators of microtubule dynamics rather than cargo transporters. The weakly bound slip state, reminiscent of a molecular safety leash, may be an adaptation for high processivity.

Power-stroke-driven actomyosin contractility

In ratchet-based models describing actomyosin contraction the activity is usually associated with actin binding potential while the power-stroke mechanism, residing inside myosin heads, is viewed as passive. To show that contraction can be propelled directly through a conformational change, we propose an alternative model where the power stroke is the only active mechanism. The asymmetry, ensuring directional motion, resides in steric interaction between the externally driven power-stroke element and the passive nonpolar actin filament. The proposed model can reproduce all four discrete states of the minimal actomyosin catalytic cycle even though it is formulated in terms of continuous Langevin dynamics. We build a conceptual bridge between processive and nonprocessive molecular motors by demonstrating that not only the former but also the latter can use structural transformation as the main driving force.

Micron-scale coherence in interphase chromatin dynamics

Chromatin structure and dynamics control all aspects of DNA biology yet are poorly understood, especially at large length scales. We developed an approach, displacement correlation spectroscopy based on time-resolved image correlation analysis, to map chromatin dynamics simultaneously across the whole nucleus in cultured human cells. This method revealed that chromatin movement was coherent across large regions (4-5 µm) for several seconds. Regions of coherent motion extended beyond the boundaries of single-chromosome territories, suggesting elastic coupling of motion over length scales much larger than those of genes. These large-scale, coupled motions were ATP dependent and unidirectional for several seconds, perhaps accounting for ATP-dependent directed movement of single genes. Perturbation of major nuclear ATPases such as DNA polymerase, RNA polymerase II, and topoisomerase II eliminated micron-scale coherence, while causing rapid, local movement to increase; i.e., local motions accelerated but became uncoupled from their neighbors. We observe similar trends in chromatin dynamics upon inducing a direct DNA damage; thus we hypothesize that this may be due to DNA damage responses that physically relax chromatin and block long-distance communication of forces.

Maximum power operation of interacting molecular motors

We study the mechanical and thermodynamic properties of different traffic models for kinesin which are relevant in biological and experimental contexts. We find that motor-motor interactions play a fundamental role by enhancing the thermodynamic efficiency at maximum power of the motors, as compared to the noninteracting system, in a wide range of biologically compatible scenarios. We furthermore consider the case where the motor-motor interaction directly affects the internal chemical cycle and investigate the effect on the system dynamics and thermodynamics.

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.

Master equation approach for a cross-bridge power-stroke model with a finite number of motors

The cross-bridge power-stroke model has been widely used to describe the motion of large motor assemblies connected to a common rigid filament. In this paper, we go beyond the original velocity-ensemble approach and propose a master equation approach to account for the cooperative motion of a finite number of motors based on the cross-bridge model. By studying the force-velocity relationship for motors with strain-independent detachment rate, we show the convergence of our approach to the velocity-ensemble approach in the limit of large motor numbers. In the case that the detachment rate of motors is strain dependent, based on two assumptions for the strain distribution among motors, we show the occurrence of the bimodal distribution of the number of motors bound to the filament. This provides a new perspective to look at the instability of a multimotor system, which is essential for all the experimentally observed complex motions displayed by a group of motors, such as hysteresis, bidirectional motion, and spontaneous oscillation. By comparing the velocities calculated using the two assumptions with the stochastic simulation, it suggests that the coupling between motors via the common connection to the filament might facilitate the fast movement of filaments at small loading forces.

Efficiency at maximum power of interacting molecular machines

We investigate the efficiency of systems of molecular motors operating at maximum power. We consider two models of kinesin motors on a microtubule: for both the simplified and the detailed model, we find that the many-body exclusion effect enhances the efficiency at maximum power of the many-motor system, with respect to the single motor case. Remarkably, we find that this effect occurs in a limited region of the system parameters, compatible with the biologically relevant range.

Transport properties of melanosomes along microtubules interpreted by a tug-of-war model with loose mechanical coupling

In this work, we explored theoretically the transport of organelles driven along microtubules by molecular motors of opposed polarities using a stochastic model that considers a Langevin dynamics for the cargo, independent cargo-motor linkers and stepping motion for the motors. It has been recently proposed that the stiffness of the motor plays an important role when multiple motors collectively transport a cargo. Therefore, we considered in our model the recently reported values for the stiffness of the cargo-motor linker determined in living cells (∼0.01 pN/nm, [1]) which is significantly lower than the motor stiffness obtained in in vitro assays and used in previous studies. Our model could reproduce the multimodal velocity distributions and typical trajectory characteristics including the properties of the reversions in the overall direction of motion observed during melanosome transport along microtubules in Xenopus laevis melanophores. Moreover, we explored the contribution of the different motility states of the cargo-motor system to the different modes of the velocity distributions and could identify the microscopic mechanisms of transport leading to trajectories compatible with those observed in living cells. Finally, by changing the attachment and detachment rates, the model could reproduce the different velocity distributions observed during melanosome transport along microtubules in Xenopus laevis melanophores stimulated for aggregation and dispersion. Our analysis suggests that active tug-of-war processes with loose mechanical coupling can account for several aspects of cargo transport along microtubules in living cells.

Active compaction of crosslinked driven filament networks

The contractile ability of active materials relies on the interplay of force-exerting and force-bearing structures. However, the complexity of interactions and limited parameter control of many model systems are major obstacles in advancing our understanding of the underlying fundamental principles. To shed light on these principles we introduce and analyse a minimal reconstituted system, consisting of highly concentrated actin filaments that are crosslinked by α-actinin and actively transported in the two-dimensional geometry of a motility assay. This minimal system actively compacts and evolves into highly compact fibres that exceed the length of the individual filaments by two orders of magnitude. We identify the interplay between active transport and crosslinking to be responsible for the observed active compaction. This enables us to control the structure and the length scale of active compaction.

United we stand: integrating the actin cytoskeleton and cell-matrix adhesions in cellular mechanotransduction

Many essential cellular functions in health and disease are closely linked to the ability of cells to respond to mechanical forces. In the context of cell adhesion to the extracellular matrix, the forces that are generated within the actin cytoskeleton and transmitted through integrin-based focal adhesions are essential for the cellular response to environmental clues, such as the spatial distribution of adhesive ligands or matrix stiffness. Whereas substantial progress has been made in identifying mechanosensitive molecules that can transduce mechanical force into biochemical signals, much less is known about the nature of cytoskeletal force generation and transmission that regulates the magnitude, duration and spatial distribution of forces imposed on these mechanosensitive complexes. By focusing on cell-matrix adhesion to flat elastic substrates, on which traction forces can be measured with high temporal and spatial resolution, we discuss our current understanding of the physical mechanisms that integrate a large range of molecular mechanotransduction events on cellular scales. Physical limits of stability emerge as one important element of the cellular response that complements the structural changes affected by regulatory systems in response to mechanical processes.

Stochastic force generation by small ensembles of myosin II motors

Forces in the actin cytoskeleton are generated by small groups of nonprocessive myosin II motors for which stochastic effects are highly relevant. Using a cross-bridge model with the assumptions of fast power-stroke kinetics and equal load sharing between equivalent states, we derive a one-step master equation for the activity of a finite-sized ensemble of mechanically coupled myosin II motors. For constant external load, this approach yields analytical results for duty ratio and force-velocity relation as a function of ensemble size. We find that stochastic effects cannot be neglected for ensemble sizes below 15. The one-step master equation can be used also for efficient computer simulations with linear elastic external load and reveals the sequence of buildup of force and ensemble rupture that is characteristic for reconstituted actomyosin contractility.

Mechano-electrical vibrations of microtubules--link to subcellular morphology

Spontaneous mechanical oscillations were predicted and experimentally proven on almost every level of cellular structure. Besides morphogenetic potential of oscillatory mechanical force, oscillations may drive vibrations of electrically polar structures or these structures themselves may oscillate on their own natural frequencies. Vibrations of electric charge will generate oscillating electric field, role of which in morphogenesis is discussed in this paper. This idea is demonstrated in silico on the conformation of two growing microtubules.