Force and motion generation of myosin motors: muscle contraction

Brownian ratchet theory refers to the phenomenon that non-equilibrium fluctuations in an isothermal medium and anisotropic system can induce mechanical force and motion. This concept of noise-induced transport has motivated an abundance of theoretical and applied research. One of the exciting applications of the ratchet theory lies in the possible explanation of the operating mode of biological molecular motors. Biomolecular motors are proteins able of converting chemical reactions into mechanical motion and force. Operating at energy levels only a few times greater than the energy levels of thermal baths, their operating mode has to be stochastic in nature. Here, we review the theoretical concepts of the Brownian ratchet theory and its possible link to the operation of the myosin II motors involved in muscle contraction.     

A modified active Brownian dynamics model using asymmetric energy conversion and its application to the molecular motor system

We consider a modified energy depot model in the overdamped limit using an asymmetric energy conversion rate, which consists of linear and quadratic terms in an active particle’s velocity. In order to analyze our model, we adopt a system of molecular motors on a microtubule and employ a flashing ratchet potential synchronized to a stochastic energy supply. By performing an active Brownian dynamics simulation, we investigate effects of the active force, thermal noise, external load, and energy-supply rate. Our model yields the stepping and stalling behaviors of the conventional molecular motor. The active force is found to facilitate the forwardly processive stepping motion, while the thermal noise reduces the stall force by enhancing relatively the backward stepping motion under external loads. The stall force in our model decreases as the energy-supply rate is decreased. Hence, assuming the Michaelis–Menten relation between the energy-supply rate and the an ATP concentration, our model describes ATP-dependent stall force in contrast to kinesin-1.     

Mechanical transduction mechanisms of RecA-like molecular motors

A majority of ATP-dependent molecular motors are RecA-like proteins, performing diverse functions in biology. These RecA-like molecular motors consist of a highly conserved core containing the ATP-binding site. Here I examined how ATP binding within this core is coupled to the conformational changes of different RecA-like molecular motors. Conserved hydrogen bond networks and conformational changes revealed two major mechanical transduction mechanisms: (1) intra-domain conformational changes and (2) inter-domain conformational changes. The intra-domain mechanism has a significant hydrogen bond rearrangement within the domain containing the P-loop, causing relative motion between two parts of the protein. The inter-domain mechanism exhibits little conformational change in the P-loop domain. Instead, the major conformational change is observed between the P-loop domain and an adjacent domain or subunit containing the arginine finger. These differences in the mechanical transduction mechanisms may link to the underlying energy surface governing a Brownian ratchet or a power stroke.     

Considerations on muscle contraction

The independent force generator and the power-stroke cross-bridge model have dominated the thinking on mechanisms of muscular contraction for nearly the past five decades. Here, we review the evolution of the cross-bridge theory from its origins as a two-state model to the current thinking of a multi-state mechanical model that is tightly coupled with the hydrolysis of ATP. Finally, we emphasize the role of skeletal muscle myosin II as a molecular motor whose actions are greatly influenced by Brownian motion. We briefly consider the conceptual idea of myosin II working as a ratchet rather than a power stroke model, an idea that is explored in detail in the companion paper.     

Direct observation of stepped proteolipid ring rotation in E. coli F₀F₁-ATP synthase

Although single‐molecule experiments have provided mechanistic insight for several molecular motors, these approaches have proved difficult for membrane bound molecular motors like the F_oF₁‐ATP synthase, in which proton transport across a membrane is used to synthesize ATP. Resolution of smaller steps in F_o has been particularly hampered by signal‐to‐noise and time resolution. Here, we show the presence of a transient dwell between F_o subunits a and c by improving the time resolution to 10 μs at unprecedented S/N, and by using Escherichia coli F_oF₁ embedded in lipid bilayer nanodiscs. The transient dwell interaction requires 163 μs to form and 175 μs to dissociate, is independent of proton transport residues aR210 and cD61, and behaves as a leash that allows rotary motion of the c‐ring to a limit of ～36° while engaged. This leash behaviour satisfies a requirement of a Brownian ratchet mechanism for the F_o motor where c‐ring rotational diffusion is limited to 36°.     

Fluctuation driven transport and models of molecular motors and pumps

Non-equilibrium fluctuations can drive vectorial transport along an anisotropic structure in an isothermal medium by biasing the effect of thermal noise (k _B T). Mechanisms based on this principle are often called Brownian ratchets and have been invoked as a possible explanation for the operation of biomolecular motors and pumps. We discuss the thermodynamics and kinetics for the operation of microscopic ratchet motors under conditions relevant to biology, showing how energy provided by external fluctuations or a non-equilibrium chemical reaction can cause unidirectional motion or uphill pumping of a substance. Our analysis suggests that molecular pumps such as Na,K-ATPase and molecular motors such as kinesin and myosin may share a common underlying mechanism. Received: 18 February 1998 / Revised version: 5 May 1998 / Accepted: 14 May 1998     

Active Nucleosome Displacement: A Theoretical Approach

Three-quarters of eukaryotic DNA are wrapped around protein cylinders forming so-called nucleosomes that block the access to the genetic information. Nucleosomes need therefore to be repositioned, either passively (by thermal fluctuations) or actively (by molecular motors). Here we introduce a theoretical model that allows us to study the interplay between a motor protein that moves along DNA (e.g., an RNA polymerase) and a nucleosome that it encounters on its way. We aim at describing the displacement mechanisms of the nucleosome and the motor protein on a microscopic level to understand better the intricate interplay between the active step of the motor and the nucleosome-repositioning step. Different motor types (Brownian ratchet versus power-stroke mechanism) that perform very similarly under a constant load are shown to have very different nucleosome repositioning capacities.     

一种生物分子马达的冲激力模型

基于动力冲程模型并结合布朗棘轮模型的扩散机制,提出了分子马达的冲激力模型。该模型基于一系列具有时间或空间周期性的啄函数来模拟分子马达的做功冲击,得到了关于几率流（速度）的解析结果。计算结果与实验数据相符合。为了产生非零的几率流,新模型并不要求分子马达与轨道间的势能必须是不对称的,因而相对于布朗棘轮模型来说更加稳健。     

Bacterial Translocation Ratchets: Shared Physical Principles with Different Molecular Implementations

Secretion systems enable bacteria to import and secrete large macromolecules including DNA and proteins. While most components of these systems have been identified, the molecular mechanisms of macromolecular transport remain poorly understood. Recent findings suggest that various bacterial secretion systems make use of the translocation ratchet mechanism for transporting polymers across the cell envelope. Translocation ratchets are powered by chemical potential differences generated by concentration gradients of ions or molecules that are specific to the respective secretion systems. Bacteria employ these potential differences for biasing Brownian motion of the macromolecules within the conduits of the secretion systems. Candidates for this mechanism include DNA import by the type II secretion/type IV pilus system, DNA export by the type IV secretion system, and protein export by the type I secretion system. Here, we propose that these three secretion systems employ different molecular implementations of the translocation ratchet mechanism.     

Understanding a molecular motor walking along a microtubule: an asymmetric Brownian motor driven by bubble formation with a focus on binding affinity

In recent years, many studies on a molecular motor have been conducted in the fields of biorheology and nanoengineering. The molecular motor is a molecule that converts the chemical energy obtained by ATP hydrolysis into mechanical energy. Explaining this mechanism is important for nanoengineering. A kinesin, which is a type of molecular motor, has the characteristics to move on a microtubule with hand-over-hand steps. The kinesin walking behaviour is explained by the ‘asymmetric Brownian ratchet model’. Previously, we had suggested that the walking mechanism was achieved by the bubble formation in a nanosized channel surrounded by hydrophobic atoms with the transition between the two states – bubble state and liquid state. However, the walking behaviour of the model motor was different from that of a single molecule measurement of a kinesin. In this study, we constructed a new motor system focused on the asymmetric binding affinity of a motor protein and performed a model simulation using the dissipative particle dynamics method. As a result, it was observed that hand-over-hand walking depends on the transition position ratio and the transition frequency coefficient. Moreover, the efficiency of the new motor system is higher than that of the previous motor systems. The new motor model can provide a simulation guide for the design of biomimetic nanomachines.     

RNA polymerase as a molecular machine

Structure and function of DNA-dependent RNA polymerase is considered in terms of a conveying molecular machine. The use of mechanical energy and mechanical devices, such as "power-stroke motor", is supposed unlikely in the conveying function of RNA polymerase, as well as other molecular machines. Brownian motion and thermal mobility of macromolecules and their parts are postulated as the only motive impulse at the molecular level. Binding of substrates and subsequent chemical reaction as the energy input may provide successive selection and fixation of alternative conformational states of the enzyme complex thus providing the directionality of the conveyance ("Brownian ratchet mechanism"). The following sequence of events "substrate binding--fixation of a certain conformational state--chemical reaction--fixation of an alternative conformational state--translocation (dissociation and downstream reassociation) of product-template duplex" is proposed as the principal scheme of the forward movement of RNA polymerase along DNA template.     

A chemically reversible Brownian motor: application to kinesin and Ncd.

Kinesin and nonclaret disjunctional protein (ncd) are two microtubule-based molecular motors that use energy from ATP hydrolysis to drive motion in opposite directions. They are structurally very similar and bind with similar orientations on microtubule. What is the origin of the different directionality? Is it some subtle feature of the structure of the motor domains, not apparent in x-ray diffraction studies, or possibly some difference near the neck regions far from the microtubule binding site? Perhaps because the motors function as dimers, the explanation involves differences in the strength of the interaction between the two motor monomers themselves. Here we present another possibility, based on a Brownian ratchet, in which the direction of motion of the motor is controlled by the chemical mechanism of ATP hydrolysis and is an inherent property of a single head. In contrast to conventional power stroke models, dissociation of the individual heads is not obligatory in the chemomechanical cycle, and the steps during which motion and force generation occurs are best described as one-dimensional thermally activated transitions that take place while both heads are attached to the microtubule. We show that our model is consistent with experiments on kinesin in which the velocity is measured as a function of external force and with the observed stiochiometry of one ATP/8-nm step at low load. Further, the model provides a way of understanding recent experiments on the ATP dependence of the variance (randomness) of the distance moved in a given time.     

Single molecule processes on the stepwise movement of ATP-driven molecular motors

Movement is a fundamental characteristic of all living things. This biogenic function that is attributed to the molecular motors such as kinesin, dynein and myosin. Molecular motors generate forces by using chemical energy derived from the hydrolysis reaction of ATP molecules. Despite a large number of studies on this topic, the chemomechanical energy transduction mechanism is still unsolved. In this study, we have investigated the chemomechanical coupling of the ATPase cycle to the mechanical events of the molecular motor kinesin using single molecule detection (SMD) techniques. The SMD techniques allowed to detection of the movement of single kinesin molecules along a microtubule and showed that kinesin steps mainly in the forward direction, but occasionally in the backward. The stepping direction is determined by a certain load-dependent process, on which the stochastic behavior is well characterized by Feynman's thermal ratchet model. The driving force of the stepwise movement is essentially Brownian motion, but it is biased in the forward direction by using the free energy released from the hydrolysis of ATP.     

Pushing, pulling and trapping--modes of motor protein supported protein translocation

Protein translocation across the cellular membranes is an ubiquitous and crucial activity of cells. This process is mediated by translocases that consist of a protein conducting channel and an associated motor protein. Motor proteins interact with protein substrates and utilize the free energy of ATP binding and hydrolysis for protein unfolding, translocation and unbinding. Since motor proteins are found either at the cis- or trans-side of the membrane, different mechanisms for translocation have been proposed. In the Power stroke model, cis-acting motors are thought to push, while trans-motors pull on the substrate protein during translocation. In the Brownian ratchet model, translocation occurs by diffusion of the unfolded polypeptide through the translocation pore while directionality is achieved by trapping and refolding. Recent insights in the structure and function of the molecular motors suggest that different mechanisms can be employed simultaneously.     

The protein import motor of mitochondria: a targeted molecular ratchet driving unfolding and translocation

Unfolding and import of preproteins into mitochondria are facilitated by a molecular motor in which heat shock protein 70 (Hsp70) in the matrix plays an essential role. Here we present two different experimental approaches to analyze mechanisms underlying this function of Hsp70. First, preproteins containing stretches of glutamic acid (polyE) or glycine (polyG) repeats in front of folded domains were imported into mitochondria. This occurred although Hsp70 cannot pull on these stretches to unfold the folded domains, since it does not bind to polyE and polyG. Secondly, preproteins containing titin immunoglobulin (Ig)-like domains were imported into mitochondria, despite the fact that forces of >200 pN are required to mechanically unfold these domains. Since molecular motors generate forces of approximately 5 pN, Hsp70 could not promote unfolding of the Ig-like domains by mechanical pulling. Our observations suggest that Hsp70 acts as an element of a Brownian ratchet, which mediates unfolding and translocation of preproteins across the mitochondrial membranes.     

Viscoelasticity of living materials: mechanics and chemistry of muscle as an active macromolecular system

At the molecular and cellular level, mechanics and chemistry are two aspects of the same macromolecular system. We present a bottom-up approach to such systems based on Kramers' diffusion theory of chemical reactions, the theory of polymer dynamics, and the recently developed models for molecular motors. Using muscle as an example, we develop a viscoelastic theory of muscle in terms of an simple equation for single motor protein movement. Both A.V. Hill's contractile component and A.F. Huxley's equation of sliding-filament motion are shown to be special cases of the general viscoelastic theory of the active material. Some disparity between the mechanical and the chemical views of cross-bridges and motor proteins are noted, and a duality between force and energy in discrete states and transitions of macromolecular systems is discussed.     

Mobility of Molecular Motors Regulates Contractile Behaviors of Actin Networks

Cells generate mechanical forces primarily from interactions between F-actin, cross-linking proteins, myosin motors, and other actin-binding proteins in the cytoskeleton. To understand how molecular interactions between the cytoskeletal elements generate forces, a number of in vitro experiments have been performed but are limited in their ability to accurately reproduce the diversity of motor mobility. In myosin motility assays, myosin heads are fixed on a surface and glide F-actin. By contrast, in reconstituted gels, the motion of both myosin and F-actin is unrestricted. Because only these two extreme conditions have been used, the importance of mobility of motors for network behaviors has remained unclear. In this study, to illuminate the impacts of motor mobility on the contractile behaviors of the actin cytoskeleton, we employed an agent-based computational model based on Brownian dynamics. We find that if motors can bind to only one F-actin like myosin I, networks are most contractile at intermediate mobility. In this case, less motor mobility helps motors stably pull F-actins to generate tensile forces, whereas higher motor mobility allows F-actins to aggregate into larger clustering structures. The optimal intermediate motor mobility depends on the stall force and affinity of motors that are regulated by mechanochemical rates. In addition, we find that the role of motor mobility can vary drastically if motors can bind to a pair of F-actins. A network can exhibit large contraction with high motor mobility because motors bound to antiparallel pairs of F-actins can exert similar forces regardless of their mobility. Results from this study imply that the mobility of molecular motors may critically regulate contractile behaviors of actin networks in cells.     

Molecular motors: thermodynamics and the random walk

The biochemical cycle of a molecular motor provides the essential link between its thermodynamics and kinetics. The thermodynamics of the cycle determine the motor's ability to perform mechanical work, whilst the kinetics of the cycle govern its stochastic behaviour. We concentrate here on tightly coupled, processive molecular motors, such as kinesin and myosin V, which hydrolyse one molecule of ATP per forward step. Thermodynamics require that, when such a motor pulls against a constant load f, the ratio of the forward and backward products of the rate constants for its cycle is exp [-(DeltaG + u(0)f)/kT], where -DeltaG is the free energy available from ATP hydrolysis and u(0) is the motor's step size. A hypothetical one-state motor can therefore act as a chemically driven ratchet executing a biased random walk. Treating this random walk as a diffusion problem, we calculate the forward velocity v and the diffusion coefficient D and we find that its randomness parameter r is determined solely by thermodynamics. However, real molecular motors pass through several states at each attachment site. They satisfy a modified diffusion equation that follows directly from the rate equations for the biochemical cycle and their effective diffusion coefficient is reduced to D-v(2)tau, where tau is the time-constant for the motor to reach the steady state. Hence, the randomness of multistate motors is reduced compared with the one-state case and can be used for determining tau. Our analysis therefore demonstrates the intimate relationship between the biochemical cycle, the force-velocity relation and the random motion of molecular motors.