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The optimal transport and Wasserstein barycenter of Gaussian distributions have been solved. In literature, the closed form formulas of the Monge map, the Wasserstein distance and the Wasserstein barycenter have been given. Moreover, when Gaussian distributions extend more generally to elliptical contoured distributions, similar results also hold true. In this case, Gaussian distributions are regarded as elliptical contoured distribution with generator function $e^{-x/2}$. However, there are few results about optimal transport for elliptical contoured distributions with different generator functions. In this paper, we degenerate elliptical contoured distributions to radial contoured distributions and study their optimal transport and prove their Wasserstein barycenter is still radial contoured. For general elliptical contoured distributions, we give two numerical counterexamples to show that the Wasserstein barycenter of elliptical contoured distributions does not have to be elliptical contoured.

Cellular migration is crucial for biological processes including embryonic development, immune response, and wound healing. The myosin-clutch model is a framework that describes how cells control migration through the interactions between myosin, the clutch mechanism, and the substrate. This model is related to how cells regulate adhesion, generate traction forces, and move on compliant substrates. In this study, we present a five-dimensional nonlinear autonomous system to investigate the influences of myosin, clutches, substrate, and external load on the system's stability. Moreover, we analyze the effects of various parameters on fixed points and explore the frequency and amplitude of the limit cycle associated with oscillations. We discovered that the system demonstrates oscillatory behavior when the velocity of the myosin motor is relatively low, or when the ratio of the motor attachment rate to motor detachment rate is relatively high. The external load shares a fraction of the force exerted by myosin motors, thereby diminishing the force endured by the clutches. Within a specific range, an increase in external load not only diminishes and eventually eliminates the region lacking fixed points but also decelerates clutch detachment, enhancing clutch protein adherence., Comment: 16 pages, 14 figures

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

Due to the wide applications of X-ray computed tomography (CT) in medical imaging activities, radiation exposure has become a major concern for public health. Sparse-view CT is a promising approach to reduce the radiation dose by down-sampling the total number of acquired projections. However, the CT images reconstructed by this sparse-view imaging approach suffer from severe streaking artifacts and structural information loss. In this work, an end-to-end dual-domain attention-based deep network (DDANet) is proposed to solve such an ill-posed CT image reconstruction problem. The image-domain CT image and the projection-domain sinogram are put into the two parallel sub-networks of the DDANet to independently extract the distinct high-level feature maps. In addition, a specified attention module is introduced to fuse the aforementioned dual-domain feature maps to allow complementary optimizations of removing the streaking artifacts and mitigating the loss of structure. Numerical simulations, anthropomorphic thorax phantom and in vivo pre-clinical experiments are conducted to verify the sparse-view CT imaging performance of the DDANet. Results demonstrate that this newly developed approach is able to robustly remove the streaking artifacts while maintaining the fine structures. As a result, the DDANet provides a promising solution in achieving high quality sparse-view CT imaging.

Kinesin-1 is an ATP-driven, two-headed motor protein that transports intracellular cargoes along microtubule. Based on recent experimental observations, we formulate a mechanochemical model for it, in which forward/backward/futile cycle of kinesin-1 can be realized through multiple biochemical pathways. Our results show that both forward and futile cycles consist of two biochemical pathways, while backward cycle may be realized through six possible pathways. Backward motion of kinesin-1 is mainly through backward sliding along microtubule when it is in semi-detach state, and usually coupled with ATP hydrolysis. Under low external load, about 80\% of ATP is wasted by kinesin-1, and this proportion of waste is almost independent of ATP concentration. At high ATP concentration or under high external load, both heads of kinesin are always in ATP or ADP$\cdot$Pi binding state and bind to microtubule tightly. But at low ATP concentration and low load, kinesin mainly stays at the one head bound state. In contrast to mean run length, the mean run time of kinesin-1 along microtubule decreases with ATP concentration. Unless external load is near the stall force, the motion of kinesin-1 is almost deterministic, not as predicted by usual Browian ratchet models.

With nontrivial entropy production, first passage process is one of the most common nonequilibrium process in stochastic thermodynamics. Using one dimensional birth and death precess as a model framework, approximated expressions of mean first passage time (FPT), mean total number of jumps (TNJ), and their coefficients of variation (CV), are obtained for the case far from equilibrium. Consequently, uncertainty relations for FPT and TNJ are presented. Generally, mean FPT decreases exponentially with entropy production, while mean TNJ decreases exponentially first and then tends to a starting site dependent limit. For forward biased process, the CV of TNJ decreases exponentially with entropy production, while that of FPT decreases exponentially first and then tends to a starting site dependent limit. For backward biased process, both CVs of FPT and TNJ tend to one for large absolute values of entropy production. Related properties about the case of equilibrium are also addressed briefly for comparison.

We consider a randomly flashing ratchet, where the potential acting can be switched to another at random. Using coupled Fokker-Planck equations, we formulate the expressions of quantities measuring dynamics and thermodynamics. Extended numerical calculations present how the potential landscapes and the transitions affect the motility and energetics. Load-dependent velocity and energetic efficiency of motor proteins, kinesin and dynein, further exemplify the randomly flashing ratchet model. We also discuss the system with two shifted sawtooth potentials.

In recent letter [Phys.~Rev.~Lett {\bf 123}, 110602 (2019)], Y.~Hasegawa and T.~V.~Vu derived a thermodynamic uncertainty relation. But the bound of their relation is loose. In this comment, through minor changes, an improved bound is obtained. This improved bound is the same as the one obtained in [Phys.~Rev.~Lett {\bf 123}, 090604 (2019)] by A.~M.~Timpanaro {\it et. al.}, but the derivation here is straightforward.

We study the existence and uniqueness of solution of a evolutionary partial differential equation originating from the continuum limit of a coupled process of totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the TASEP-LK coupled process has been extensively studied by Monte Carlo simulations, numerical computations, and detailed experiments. However, no rigorous mathematical analysis so far has been given for the corresponding differential equations, especially the existence and uniqueness of their solutions. In this paper, we prove the existence of the $C^\infty[0,1]$ steady-state solution by the method of upper and lower solution, and the uniqueness in both $W^{1,2}(0,1)$ and $L^\infty(0,1)$ by a generalized maximum principle. We further prove the global existence and uniqueness of the time-dependent solution in $C([0,1]\times [0,+\infty))\cap C^{2,1}([0,1]\times (0,+\infty))$, which, for any continuous initial value, converges to the steady-state solution in $C[0,1]$ (global attractivity). Our results support the numerical calculations and Monte Carlo simulations, and provide theoretical foundations for the TASEP-LK coupled process, especially the most important phase diagram of particle density along the travel track under different model parameters, which is difficult because the boundary layers (at one or both boundaries) and domain wall (separating high and low particle densities) may appear as the length of the travel track tends to infinity. The methods used in this paper may be instructive for studies of the more general cases of the TASEP-LK process, such as the one with multiple travel tracks and/or multiple particle species., Comment: 38 pages, 5 figures

In this study, the minimum amount of work needed to drive a thermodynamic system from one initial distribution to another in a given time duration is discussed. Equivalently, for given amount of work, the minimum time duration required to complete such a transition is obtained. Results show that the minimum amount of work is used to achieve the following three objectives, to increase the internal energy of the system, to decrease the system entropy, to change the mean position of the system, and with other nonzero part dissipated into environment. To illustrate the results, an example with explicit solutions is presented.

The study of stochastic thermodynamic machines is one of the main topics in nonequilibrium thermodynamics. In this study, within the framework of Fokker-Planck equation, and using the method of characteristics of partial differential equation as well as the variational method, performance of stochastic thermodynamic machines is optimized according to the external potential, with the irreversible work $W_{irr}$, or the total entropy production $\Delta S_{\rm tot}$ equivalently, reaching its lower bound. Properties of the optimal thermodynamic machines are discussed, with explicit expressions of upper bounds of work output $W$, power $P$, and energy efficiency $\eta$ are presented. To illustrate the results obtained, typical examples with optimal protocols (external potentials) are also presented.

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.

In recent letter [Phys. Rev. Lett {\bf 121}, 070601 (2018), arXiv:1802.06554], the speed limit for classical stochastic Markov processes is considered, and a trade-off inequality between the speed of the state transformation and the entropy production is given. In this comment, a more accurate inequality will be presented.

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.

A specific class of stochastic heat engines driven cyclically by time-dependent potential, which is defined in the half-line ($0

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 this paper, the existence and uniqueness of solution of a specific differential equation is studied. This equation originates from the description of a coupled process by totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the properties of the TASEP-LK coupled process have been extensively studied by Monte Carlo simulations and numerical calculations, as well as detailed experiments. However, so far, no rigorous mathematical analysis has been given to the corresponding differential equations, especially their existence and uniqueness of solution. In this paper, using the upper and lower solution method, the existence of solution of the steady state equation is obtained. Then using a generalized maximum principle, we show that the solution constructed from the upper and lower solution method is actually the unique solution in C∞ space. Moreover, the existence and uniqueness of solution of the time dependent differential equation are also obtained in one specific space X\b{eta}. Our results imply that the previous results obtained by numerical calculations and Monte Carlo simulations are theoretically correct, especially the most important phase diagram of particle density along the travel track under different model parameters. The study in this paper provides theoretical foundations for the analysis of TASEP-LK coupled process. At the same time, the methods used in this paper may be instructive for studies about the more general cases of the TASEP-LK process, such as the one with multiple travel tracks or the one with multiple particle species., Comment: This paper has a newer version. Because of my mistake, I submit it twice with the new identifier as arXiv:1905.12235

In this short article, we establish a rigidity theorem for pairs of hyperquadrics in a weaker sense, i.e., we impose a condition that minimal rational curves are preserved, which is stronger than inheriting a sub-VMRT structure, a notion raised by Mok & Zhang (2014). This problem has its source in a theorem of Tsai (1993), and the main result of this article can be applied back to give a more intrinsic proof of Tsai's theorem., Comment: 9 pages

Transcription is one of the essential processes for cells to read genetic information encoded in genes, which is initiated by the binding of RNA polymerase to related promoter. Experiments have found that the nucleotide sequence of promoter has great influence on gene expression strength, or promoter activity. In synthetic biology, one interesting question is how we can synthesize a promoter with given activity, and which positions of promoter sequence are important for determining its activity. In this study, based on recent experimental data, correlations between promoter activity and its sequence positions are analyzed by various methods. Our results show that, except nucleotides in the two highly conserved regions, $-35$ box and $-10$ box, influences of nucleotides in other positions are also not neglectable. For example, modifications of nucleotides around position $-19$ in spacing region may change promoter activity in a large scale. The results of this study might be helpful to our understanding of biophysical mechanism of gene transcription, and may also be helpful to the design of synthetic cell factory.

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.

We study a pair (\mathcal{S}_0,\mathcal{S}) of irreducible Hermitian Symmetric Spaces of compact type (cHSS) in this paper, with the first aim being classifying all the admissible pairs (\mathcal{S}_0,\mathcal{S})). This notion is a natural generalization of the pairs of sub-diagram type originated by Jaehyun Hong and Ngaiming Mok ([HoM 10]). Based on this classification, we partially solve the rigidity problem for the admissible pairs (\mathcal{S}_0,\mathcal{S}) which was raised by Mok and Zhang (2014) ([MoZ 14]), culminating in determining a sufficient condition for the pairs being non-rigid and proving that special pairs, which show up in the classification procedure, are algebraic, as a weaker result than being rigid. However, whether special pairs are rigid or not remains unknown and needs further investigation in the framework of VMRT theory., Comment: 29 pages

In this study, through various tests one theoretical model is presented to describe the relationship between promoter strength and its nucleotide sequence. Our analysis shows that, promoter strength is greatly influenced by nucleotide groups with three adjacent nucleotides in its sequence. Meanwhile, nucleotides in different regions of promoter sequence have different effects on promoter strength. Based on experimental data for {\it E. coli} promoters, our calculations indicate, nucleotides in -10 region, -35 region, and the discriminator region of promoter sequence are more important than those in spacing region for determining promoter strength. With model parameter values obtained by fitting to experimental data, four promoter libraries are theoretically built for the corresponding experimental environments under which data for promoter strength in gene expression has been measured previously.

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.

Wind is one of the important reasonable resources. However, wind turbine towers are sure to be threatened by hurricanes. In this paper, method to estimate the number of wind turbine towers that would be buckled by hurricanes is discussed. Monte Carlo simulations show that our method is much better than the previous one. Since in our method, the probability density function of the buckling probability of a single turbine tower in a single hurricane is obtained accurately but not from one approximated expression. The result in this paper may be useful to the design and maintenance of wind farms.

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.

As a mathematical theory for the stochasstic, nonlinear dynamics of individuals within a population, Delbr\"{u}ck-Gillespie process (DGP) $n(t)\in\mathbb{Z}^N$, is a birth-death system with state-dependent rates which contain the system size $V$ as a natural parameter. For large $V$, it is intimately related to an autonomous, nonlinear ordinary differential equation as well as a diffusion process. For nonlinear dynamical systems with multiple attractors, the quasi-stationary and stationary behavior of such a birth-death process can be underestood in terms of a separation of time scales by a $T^*\sim e^{\alpha V}$ $(\alpha>0)$: a relatively fast, intra-basin diffusion for $t\ll T^*$ and a much slower inter-basin Markov jump process for $t\gg T^*$. In the present paper for one-dimensional systems, we study both stationary behavior ($t=\infty$) in terms of invariant distribution $p_n^{ss}(V)$, and finite time dynamics in terms of the mean first passsage time (MFPT) $T_{n_1\rightarrow n_2}(V)$. We obtain an asymptotic expression of MFPT in terms of the "stochastic potential" $\Phi(x,V)=-(1/V)\ln p^{ss}_{xV}(V)$. We show in general no continuous diffusion process can provide asymptotically accurate representations for both the MFPT and the $p_n^{ss}(V)$ for a DGP. When $n_1$ and $n_2$ belong to two different basins of attraction, the MFPT yields the $T^*(V)$ in terms of $\Phi(x,V)\approx \phi_0(x)+(1/V)\phi_1(x)$. For systems with a saddle-node bifurcation and catastrophe, discontinuous "phase transition" emerges, which can be characterized by $\Phi(x,V)$ in the limit of $V\rightarrow\infty$. In terms of time scale separation, the relation between deterministic, local nonlinear bifurcations and stochastic global phase transition is discussed. The one-dimensional theory is a pedagogic first step toward a general theory of DGP., Comment: 32 pages, 3 figures

Comment on "Efficiency of Isothermal Molecular Machines at Maximum Power" (PRL 108, 210602 (2012), arXiv:1201.6396), Comment: So far, this manuscript has not been accepted for publication, and I do not want to work on it now. So I want to withdraw it.

In cells, helicase translocation along nucleic acid is essential for many biological processes. However, so far, the mechanism of this translocation is not fully understood. Recent studies show that helicase might translocate through two processes, active process and passive process, with different translocation rate. In this study, a model including such two processes is presented. In which, each of these two processes consists of two sub-processes, chemical sub-process in which needed translocation factors are attached, and mechanochemical sub-process in which helicase makes a forward translocation step. Helicase can switch stochastically between these two processes with external force dependent rates. By this model, ribosome translocation along message RNA is detailed discussed. We found that, with the increase of external force, the mean translocation rate of ribosome increases from one lower limit to one upper limit, and both of these two limits increase with concentrations of the translocation factors. Under high external force, ribosome translocates mainly through active process. At saturating concentrations of translocation factors, the translocation is governed by mechanochemical sub-processes., Comment: an error has been found

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.

Due to inter-subunit communication, multisubunit enzymes usually hydrolyze ATP in a concerted fashion. However, so far the principle of this process remains poorly understood. In this study, from the viewpoint of statistical thermodynamics, a simple model is presented. In this model, we assume that the binding of ATP will change the potential of the corresponding enzyme subunit, and the degree of this change depends on the state of its adjacent subunits. The probability of enzyme in a given state satisfies the Boltzmann's distribution. Although it looks much simple, this model can fit the recent experimental data of chaperonin TRiC/CCT well. From this model, the dominant state of TRiC/CCT can be obtained. This study provided a new way to understand biophysical processes by statistical thermodynamics analysis.

In the recent paper of Valleriani {\it et al} [Phys. Rev. E {\bf 83}, 042903 (2011)], a simple model for describing the translation of messenger RNA (mRNA) by ribosomes is presented, and an expression of the translational ratio $r$, defined as the ratio of translation rate $\omega_{\rm tl}$ of protein from mRNA to degradation rate $\omega_p$ of protein, is obtained. The key point to get this ratio $r$ is to get the translation rate $\omega_{\rm tl}$. In the study of Valleriani {\it et al}, $\omega_{\rm tl}$ is assumed to be the mean value of measured translation rate, i.e. the mean value of ratio of the translation number of protein to the lifetime of mRNA. However, in experiments different methods might be used to get $\omega_{\rm tl}$. Therefore, for the sake of future application of their model to more experimental data analysis, in this comment three methods to get the translation rate $\omega_{\rm tl}$, and consequently the translational ratio $r$, are provided. Based on one of the methods which might be employed in most of the experiments, we find that the translational ratio $r$ decays exponentially with the length of mRNA in prokaryotic cells, and decays reciprocally with the length of mRNA in eukaryotic cells. This result is slight different from that obtained in Valleriani's study.

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.