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Myosin II plays a pivotal role in muscle contraction by generating force through the cooperative action of multiple motors on actin filaments. In this study, we integrate the nonlinear elasticity of the neck linker in individual myosin II and comprehensively investigate the evolution of cooperativity and dynamics at {\it microstate} and {\it mesostate} levels using a combined model of single and multiple motors. We find that a substantial proportion of actin-bound motors reside in the {\it mid-} and {\it post-power stroke} states, and our nonlinear model reveals their increased capacity for load sharing. Additionally, we systematically explore the impact of mechanical load and ATP concentration on myosin II motors. Notably, we observe that the average net distance of actin undergoes a transition from a weak load-sensitive regime at low ATP concentrations to a load-sensitive regime at higher ATP concentrations. Furthermore, increasing the load or raising the ATP concentration to saturation can enhance the efficiency and output power of myosin filament. Moreover, the efficiency of the myosin filament increases with the power stroke strength, reaching a maximum at a specific range, and subsequently declining beyond that threshold. Finally, we explore the mean run time/length and mean existence probability of myosin filament, shedding light on its overall behavior., Comment: 10 pages, 7 figures

Chaperone-assisted translocation through a nanopore embedded in membrane holds a prominent role in the transport of biopolymers. Inspired by classical Brownian ratchet, we develop a theoretical framework characterizing such translocation process through a master equation approach. In this framework, the polymer chain, provided with reversible binding of chaperones, undergoes forward/backward diffusion, which is rectified by chaperones. We drop the assumption of timescale separation and keep the length of a polymer chain finite, both of which happen to be the key points in most of the previous studies. Our framework makes it accessible to derive analytical expressions for mean translocation velocity and effective diffusion constant in stationary state, which is the basis of a comprehensive understanding towards the dynamics of such process. Generally, the translocation of polymer chain across membrane consists of three subprocesses: initiation, termination, and translocation of the main body part of a polymer chain, where the translocation of the main body part depends on the binding/unbinding kinetics of chaperones. That is the main concern of this study. Our results show that the increase of forward/backward diffusion rate of a polymer chain and the binding/unbinding ratio of chaperones both raise the mean translocation velocity of a polymer chain, and roughly speaking, the dependence of effective diffusion constant on these two factors achieves similar behavior.

Transcription is the first step of gene expression, in which a particular segment of DNA is copied to RNA by the enzyme RNA polymerase (RNAP). Despite many details of the complex interactions between DNA and RNA synthesis disclosed experimentally, much of physical behavior of transcription remains largely unknown. Understanding torsional mechanics of DNA and RNAP together with its nascent RNA and RNA-bound proteins in transcription maybe the first step towards deciphering the mechanism of gene expression. In this study, based on the balance between viscous drag on RNA synthesis and torque resulted from untranscribed supercoiled DNA template, a simple model is presented to describe mechanical properties of transcription. With this model, the rotation and supercoiling density of the untranscribed DNA template are discussed in detail. Two particular cases of transcription are considered, transcription with constant velocity and transcription with torque dependent velocity. Our results show that, during the initial stage of transcription, rotation originated from the transcribed part of DNA template is mainly released by the rotation of RNAP synthesis. During the intermediate stage, the rotation is usually released by both the supercoiling of the untranscribed part of DNA template and the rotation of RNAP synthesis, with proportion depending on the friction coefficient in environment and the length of nascent RNA. However, with the approaching to the upper limit of twisting of the untranscribed DNA template, the rotation resulted from transcription will then be mainly released by the rotation of RNAP synthesis.

In gene expression, various kinds of proteins need to bind to specific locus of DNA. It is still not clear how these proteins find their target locus. In this study, the mean first-passage time (FPT) of protein binding to its target locus on DNA chain is discussed by a chain-space coupled model. Our results show that the 1-dimensional diffusion constant has a critical value, with which the mean time spent by a protein to find its target locus is almost independent of the binding rate of protein to DNA chain and the detachment rate from DNA chain. Which implies that, the frequency of protein binding to DNA and the sliding time on DNA chain have little influence on the search efficiency, and therefore whether or not the 1-dimensional sliding on DNA chain increases the search efficiency depends on the 1-dimensional diffusion constant of the protein on DNA chain. This study also finds that only protein bindings to DNA loci which are close to the target locus help to increase the search efficiency, while bindings to those loci which are far from the target locus might delay the target binding process. As expected, the mean FPT increases with the distance between the initial position of protein in cell space and its target locus on DNA chain. The direct binding probability, which can be regarded as one index to describe if the 1-dimensional sliding along DNA chain is helpful to increase the search efficiency is calculated. Our results show that the influence of 1-dimensional sliding along DNA chain on the search process depends on both diffusion constants of protein in cell space and on the 1-dimensional DNA chain.

In cells, most of cargos are transported by motor proteins along microtubule. Biophysically, unidirectional motion of large number of motor proteins along a single track can be described by totally asymmetric simple exclusion process (TASEP). From which many meaningful properties, such as the appearance of domain wall (defined as the borderline of high density and low density of motor protein along motion track) and boundary layers, can be obtained. However, it is biologically obvious that a single track may be occupied by different motor species. So previous studies based on TASEP of one particle species are not reasonable enough to find more detailed properties of the motion of motors along a single track. To address this problem, TASEP with two particle species is discussed in this study. Theoretical methods to get densities of each particle species are provided. Using these methods, phase transition related properties of particle densities are obtained. Our analysis show that domain wall and boundary layer of single species densities always appear simultaneously with those of the total particle density. The height of domain wall of total particle density is equal to the summation of those of single species. Phase diagrams for typical model parameters are also presented. The methods presented in this study can be generalized to analyze TASEP with more particle species.

Experimental evidences show that in gene transcription, RNA polymerase has the possibility to be stalled at certain position of the transcription template. This may be due to the template damage, or protein barriers. Once stalled, polymerase may backtrack along the template to the previous nucleotide to wait for the repair of the damaged site, or simply bypass the barrier or damaged site and consequently synthesize an incorrect messenger RNA, or degrade and detach from the template. Thus, the {\it effective} transcription rate (the rate to synthesize correct product mRNA) and the transcription {\it effectiveness} (the ratio of the {\it effective} transcription rate to the {\it effective} transcription initiation rate) are both influenced by polymerase stalling events. This study shows that, Without backtracking, detachment of stalled polymerase can also help to increase the {\it effective} transcription rate and transcription {\it effectiveness}. Generally, the increase of bypass rate of the stalled polymerase will lead to the decrease of the {\it effective} transcription rate and transcription {\it effectiveness}. But when both detachment rate and backtracking rate of the stalled polymerase vanish, the {\it effective} transcription rate may also be increased by the bypass mechanism.

Translation is an important process for prokaryotic and eukaryotic cells to produce necessary proteins for cell growth. Numerious experiments have been performed to explore the translational properties. Diverse models have also been developed to determine the biochemical mechanism of translation. However, to simplify the majority of the existing models, the frameshifting of ribosome along the mRNA transcript is neglected, which actually occurs in real cells and has been extensively experimentally studied. The frameshifting of ribosome evidently influences the efficiency and speed of translation, considering that the peptide chains synthesized by shifted ribosomes will not fold into functional proteins and will degrade rapidly. In this study, a theoretical model is presented to describe the translational process based on the model for totally asymmetric simple exclusion process. In this model, the frameshifting of the ribosome along the mRNA transcript and the attachment/detachment of the ribosome to/from the main body of mRNA codons during translation elongation process, are explicitly included. The results show that, with ribosome frameshifing, the speed of correctly synthesized peptide chains may increase first and then decrease with both the translation initiation rate $\alpha$ and the ribosome detachment rate $\omega_d$. This results indicates that regulating the translation process to reach maximal synthesized speed of proteins is theoretically feasible. Traffic-related problems of ribosome motion along the mRNA transcript are also addressed theoretically. Depending on parameter values, shock wave (or domain wall) may exist for ribosome probabilities along the mRNA.

Bacterial flagellar motor (BFM) is one of the ion-driven molecular machines, which drives the rotation of flagellar filaments and enable bacteria to swim in viscous solutions. Understanding its mechanism is one challenge in biophysics. Based on previous models and inspired by the idea used in description of motor proteins, in this study one two-state model is provided. Meanwhile, according to corresponding experimental data, mathematical relationship between BFM membrane voltage and pH value of the environment, and relationship between internal and external sodium concentrations are given. Therefore, with model parameter values obtained by fitting theoretical results of torque-speed relation to recent experimental data, many biophysical properties of bacterial flagellar motor can be obtained for any pH values and any external sodium concentrations. Including the rotation speed, stall torque (i.e. the torque generated by BFM), rotation dispersion, and rotation randomness. In this study, the single-stator BFM will be firstly analyzed, and then properties of multiple-stator BFM are addressed briefly.

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

In living cells, motor proteins, such as kinesin and dynein can move processively along microtubule (MT), and also detach from or attach to MT stochastically. Experiments have found that, the traffic of motor might be jammed, and various theoretical models have been designed to understand this traffic jam phenomenon. But previous studies mainly focus on motor attachment/detachment rate dependent properties. Recent experiment of Leduc {\it et al.} found that the traffic jam formation of motor protein kinesin depends also on the length of MT [Proc. Natl. Acad. Sci. U.S.A. {\bf 109}, 6100-6105 (2012)]. In this study, the MT length dependent properties of motor traffic will be analyzed. We found that MT length has one {\it critical value} $N_c$, traffic jam occurs only when MT length $N>N_c$. The jammed length of MT increases with total MT length, while the non-jammed MT length might not change monotonically with the total MT length. The critical value $N_c$ increases with motor detachment rate from MT, but decreases with motor attachment rate to MT.

In this study, through phenomenological comparison of the velocity-force data of processive motor proteins, including conventional kinesin, cytoplasmic dynein and myosin V, we found that, the ratio between motor velocities of two different ATP concentrations is almost invariant for any substall, superstall or negative external loads. Therefore, the velocity of motor can be well approximated by a Michaelis-Menten like formula $V=\atp k(F)L/(\atp +K_M)$, with $L$ the step size, and $k(F)$ the external load $F$ dependent rate of one mechanochemical cycle of motor motion in saturated ATP solution. The difference of Michaelis-Menten constant $K_M$ for substall, superstall and negative external load indicates, the ATP molecule affinity of motor head for these three cases are different, though the expression of $k(F)$ as a function of $F$ might be unchanged for any external load $F$. Verifications of this Michaelis-Menten like formula has also been done by fitting to the recent experimental data.

Recently, a three-state model is presented to describe the intracellular traffic of unconventional (single-headed) kinesin KIF1A [Phys. Rev. Lett. {\bf 95}, 118101 (2005)], in which each motor can bind strongly or weakly to its microtubule track, and each binding site of the track might be empty or occupied by one motor. As the usual two-state model, i.e. the totally asymmetric simple exclusion process (TASEP) with motor detachment and attachment, in steady state of the system, this three-state model also exhibits shock (or domain wall separating the high-density and low density phases) and boundary layers. In this study, using mean-field analysis, the conditions of existence of shock and boundary layers are obtained theoretically. Combined with numerical calculations, the properties of shock are also studied. This study will be helpful to understand the biophysical properties of the collective transport of kinesin KIF1A.

In living cells, molecular motors convert chemical energy into mechanical work. Its thermodynamic energy efficiency, i.e. the ratio of output mechanical work to input chemical energy, is usually high. However, using two-state models, we found the motion of molecular motors is loosely coupled to the chemical cycle. Only part of the input energy can be converted into mechanical work. Others is dissipated into environment during substeps without contributions to the macro scale unidirectional movement.

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

Molecular motors are single macromolecules that generate forces at the piconewton range and nanometer scale. They convert chemical energy into mechanical work by moving along filamentous structures. In this paper, we study the velocity of two-head molecular motors in the framework of a mechanochemical network theory. The network model, a generalization of the recently work of Liepelt and Lipowsky (PRL 98, 258102 (2007)), is based on the discrete mechanochemical states of a molecular motor with multiple cycles. By generalizing the mathematical method developed by Fisher and Kolomeisky for single cycle motor (PNAS(2001) 98(14) P7748-7753), we are able to obtain an explicit formula for the velocity of a molecular motor.

Molecular motors are essential components for the biophysical functions of the cell. Our current quantitative understanding of how multiple motors move along a single track is not complete; even though models and theories for single motor chemomechanics abound. Recently, M.J.I. M$\ddot{\rm u}$ller {\em et al.} have developed a tug-of-war model to describe the bidirectional movement of the cargo (PNAS(2008) 105(12) P4609-4614). Through Monte Carlo simulations, they discovered that the tug-of-war model exhibits several qualitative different motility regimes, which depend on the precise value of single motor parameters, and they suggested the sensitivity can be used by a cell to regulate its cargo traffic. In the present paper, we carry out a thorough analysis of the tug-of-war model. All the stable, i.e., biophysically observable, steady states are obtained. Depending on several parameters, the system exhibits either uni-, bi- or tristability. Based on the separating boundary of the different stable states and the initial numbers of the different motor species that are bound to the track, the steady state of the cargo movement can be predicted, and consequently the steady state velocity can be obtained. It is found that, the velocity, even the direction, of the cargo movement change with the initial numbers of the motors which are bound to the track and several other parameters.

Molecular motors convert chemical energy into mechanical work while operating in an environment dominated by Brownian motion. The aim of this paper is to explore the flow of energy between the molecular motors and its surroundings, in particular, its efficiency. Based on the Fokker-Planck equation with either $N$ or infinite chemical states, we find that the energy efficiency of the molecular motors, whether the Stokes efficiency or the usual thermodynamic efficiency, is strictly bounded by 1, because of the dissipation of the energy in both the overdamped surroundings and in the process of the chemical reaction., Comment: 14 pages