The stepping motor protein as a feedback control ratchet

It is explained how going from a one headed motor protein to a stepping two headed motor protein is equivalent to going from a stochastically flashing ratchet to a feedback control ratchet. Both these ratchets have been well studied in the literature and their speeds and efficiencies are briefly reviewed. Next it is shown how a feedback control ratchet mechanism model can account for very accurate recent data obtained on kinesin. Finally, the role of internal friction in the operation of stepping motor proteins is discussed.     

驱动蛋白定向运动的动力学分析

将布朗粒子的定向运动,看作是系统受到外部非平衡涨落作用的结果,并建立相应的扩散模型。通过蒙特卡罗模拟方法,得到布郎粒子定向梯跳运动曲线。结果表明：非对称锯齿势,外部含时闪烁力及加性色噪声协调与联合作用,可使布朗粒子做梯跳或锯齿运动。作为一种可能的解释,将驱动蛋白Ｋｉｎｅｓｉｎｓ沿微管定向梯跳运动看作是上述协调作用的结果。     

Transport characteristics of molecular motors

Properties of transport of molecular motors are investigated. A simplified model based on the concept of Brownian ratchets is applied. We analyze a stochastic equation of motion by means of numerical methods. The transport is systematically studied with respect to its energetic efficiency and quality expressed by an effective diffusion coefficient. We demonstrate the role of friction and non-equilibrium driving on the transport quantifiers and identify regions of a parameter space where motors are optimally transported.     

Direction reversal of fluctuation-induced biased Brownian motion on distorted ratchets

The biased movement of Brownian particles on a fluctuating two-state periodic potential made of identical distorted ratchets is studied. The purpose is to investigate how the direction of the particle movement is related to the asymmetry of the potential. In general, distorting one of the two linear arms of a regular symmetric ratchet (with equal arm lengths) can create a driving force for the Brownian particle to execute biased movement. The direction of the induced biased movement depends on the type of the distortion. It has been found that if one linear arm is kinked into two linear sub-arms, the direction of the movement can be either positive or negative depending on the frequency of the fluctuation and the location and the degree of the kink. In contrast, if one arm of the symmetric ratchet is replaced by a continuous nonlinear sinusoidal function, the movement is always unidirectional. Thus, for the latter case to generate the direction reversal phenomenon, the ratchets have to have an additional asymmetry. We also have found that two potentials with different distorted ratchets can generate identical fluxes if the distortions are polar symmetric about the mid-point of the arm(s) of the basic linear two-arm ratchet. The results are useful for designing experimental apparatuses for the separation of protein particles based on their sizes and charges and the viscosity of the medium.     

Mesoscopic Model of Actin-Based Propulsion

Two theoretical models dominate current understanding of actin-based propulsion: microscopic polymerization ratchet model predicts that growing and writhing actin filaments generate forces and movements, while macroscopic elastic propulsion model suggests that deformation and stress of growing actin gel are responsible for the propulsion. We examine both experimentally and computationally the 2D movement of ellipsoidal beads propelled by actin tails and show that neither of the two models can explain the observed bistability of the orientation of the beads. To explain the data, we develop a 2D hybrid mesoscopic model by reconciling these two models such that individual actin filaments undergoing nucleation, elongation, attachment, detachment and capping are embedded into the boundary of a node-spring viscoelastic network representing the macroscopic actin gel. Stochastic simulations of this ‘in silico’ actin network show that the combined effects of the macroscopic elastic deformation and microscopic ratchets can explain the observed bistable orientation of the actin-propelled ellipsoidal beads. To test the theory further, we analyze observed distribution of the curvatures of the trajectories and show that the hybrid model''s predictions fit the data. Finally, we demonstrate that the model can explain both concave-up and concave-down force-velocity relations for growing actin networks depending on the characteristic time scale and network recoil. To summarize, we propose that both microscopic polymerization ratchets and macroscopic stresses of the deformable actin network are responsible for the force and movement generation.     

Controlling the ratchet effect through the symmetries of the systems: application to molecular motors

We discuss a novel generic mechanism for controlling the ratchet effect through the breaking of relevant symmetries. We review previous works on ratchets where directed transport is induced by the breaking of standard temporal symmetries f(t)=-f(t+T/2) and f(t)=f(-t) (or f(t)=-f(-t)). We find that in seemingly unrelated systems the average velocity (or the current) of particles (or solitons) exhibits common features. We show that, as a consequence of Curie's symmetry principle, the average velocity (or the current) is related to the breaking of the symmetries of the system. This relationship allows us to control the transport in a systematic way. The qualitative agreement between the present analytical predictions and previous experimental, numerical, and theoretical results leads us to suggest that for the given breaking of the temporal symmetries there is an optimal wave form for a given time-periodic force. Also, we comment on how this mechanism can be applied to the case where a ratchet effect is induced by breaking of spatial symmetries. Finally, we conjecture that the ratchet potential underlying biological motor proteins might be optimized according to the breaking of the relevant symmetries.     

RNA polymerase pushing

Molecular motors can exhibit Brownian ratchet or power stroke mechanisms. These mechanistic categories are related to transition state position: An early transition state suggests that chemical energy is stored and then released during the step (stroke) while a late transition state suggests that the release of chemical energy rectifies thermally activated motion that has already occurred (ratchet). Cellular RNA polymerases are thought to be ratchets that can push each other forward to reduce pausing during elongation. Here, by constructing a two-dimensional energy landscape from the individual landscapes of active and backtracked enzymes, we identify a new pushing mechanism which is the result of a saddle trajectory that arises in the two-dimensional energy landscape of interacting enzymes. We show that this mechanism is more effective with an early transition state suggesting that interacting RNAPs might translocate via a power stroke.     

The Fokker-Planck approach for the cooperative molecular motor model with finite number of motors

We provide the methodology for the analysis of the cooperative molecular motor model with finite number of motors, which are linearly and rigidly coupled, based on the Fokker-Planck approach. The probability density functions for the position of motors are solved numerically from the stationary Fokker-Planck equations. By using these probability density functions, we provide the analytical expressions, such as the velocity, the rate of the ATP consumption, the energetic efficiency, and the dissipation energy rates. Furthermore, we investigate three specific examples, such as single motor model, 2-motor model, and infinitely coupled motor model. Numerical algorithm to solve the Fokker-Planck equations is also provided.     

Are histones, tubulin, and actin derived from a common ancestral protein?

Histones and the cytoskeletal components tubulin and actin all act as thermal ratchets, using the energy present in Brownian motion to do work. All three also bind to nucleotides. Here we suggest that histones, tubulin, and actin derive from a common ancestral protein. There is some sequence similarity between histone 2A and the bacterial tubulin homologue FtsZ. Histones and actin also share some sequence similarity in the nucleotides and at phosphate-binding sites. Thus, actin and tubulin may also be related, although this is not obvious from sequence analysis. Indeed, actin and tubulin are closely functionally related and cooperate in many cellular processes. Interestingly, recent advances in nanotechnology suggest that thermal ratchets may be able to impart lifelike properties; thus, the evolution of the ancestral histone, tubulin, and actin thermal ratchet may have been crucial in the development of complexity in living organisms.     

Chlamydomonas reinhardtii as a model system for pro-active herbicide resistance evolution research

Modern herbicides greatly contribute to world agricultural production but their sustainability is threatened by the widespread evolution of herbicide resistant weedy plant populations. Despite the commercial and scientific importance of resistance, there has not been an experimental model system for pro-actively evaluating the potential for herbicide resistance evolution. Here, utilizing the rapidly growing, unicellular photosynthetic microalgae Chlamydomona s reinhardtii (Dangeard), a ratchet protocol has been developed that solves the problem of maintaining both large populations and strong herbicide selection. The ratchet protocol is a progressive set of cycles, each cycle commencing with a population of approximately one million individuals apportioned amongst three herbicide doses for 14 days. Whenever the evolving population demonstrates growth across the three herbicide selection intensities, then the population ratchets to the next cycle of higher herbicide dose. Therefore, by always maintaining large populations under selection pressure, this system offers the opportunity for beneficial mutations to arise and be enriched. Using the well-characterized atrazine herbicide, the ratchet protocol resulted in rapid evolution of populations with different levels of resistance. This robust laboratory based Chlamydomonas system is proposed for application in establishing the respective propensity for resistance evolution to herbicides or other selecting agents.  © 2007 The Linnean Society of London, Biological Journal of the Linnean Society , 2007, 91 , 257–266.     

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.     

Analytical aspects of the Brownian motor effect in randomly flashing ratchets

The muscle contraction, operation of ATP synthase, maintaining the shape of a cell are believed to be secured by motor proteins, which can be modelled using the Brownian ratchet mechanism. We consider the randomly flashing ratchet model of a Brownian motor, where the particles can be in two states, only one of which is sensitive the applied spatially periodic potential (the mathematical setting is a pair of weakly coupled reaction-diffusion and Fokker–Planck equations). We prove that this mechanism indeed generates unidirectional transport by showing that the amount of mass in the wells of the potential decreases/increases from left to right. The direction of transport is unambiguously determined by the location of each minimum of the potential with respect to the so-called diffusive mean of its adjacent maxima. The transport can be generated not only by an asymmetric potential, but also by a symmetric potential and asymmetric transition rates, and as a consequence of the general result we derive explicit conditions when the latter happens. When the transitions are localized on narrow active sites in the protein conformation space, we find a more explicit characterization of the bulk transport direction, and infer that some common preconditions of the motor effect are redundant.     

Proteolytic degradation routes for turkey beta 1-adrenoceptor probed with antipeptide antibodies against the N-terminal sequence of the receptor

Anti-peptide antibodies, raised against the N-terminal sequence (amino acids 2-10) of the turkey beta 1-adrenoceptor [Yarden et al., Proc. Natl. Acad. Sci. USA (1986) 83, 6795-6799] recognized the 50 kDa- but not the 40 kDa-form of the receptor, thus confirming the previous assumption that the N-terminus of the 50 kDa form is lost during its conversion to the 40 kDa-form [Jür beta, R., Hekman, M. & Helmreich, E.J.M. (1985) Biochemistry 24, 3349-3354]. By in situ proteolysis small amounts of receptor fragments were formed, which could be recognized by the N-terminus specific antibody. Therefore, although the production of the stable 40 kDa receptor species by proteolytic removal of a portion of the N-terminal appears to be the predominant route, there exists an additional pathway of degradation which must involve the initial cleavage of the carboxyl terminal.     

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.     

Kick-starting the ratchet: the fate of mutators in an asexual population

Muller's ratchet operates in asexual populations without intergenomic recombination. In this case, deleterious mutations will accumulate and population fitness will decline over time, possibly endangering the survival of the species. Mutator mutations, i.e., mutations that lead to an increased mutation rate, will play a special role for the behavior of the ratchet. First, they are part of the ratchet and can come to dominance through accumulation in the ratchet. Second, the fitness-loss rate of the ratchet is very sensitive to changes in the mutation rate and even a modest increase can easily set the ratchet in motion. In this article we simulate the interplay between fitness loss from Muller's ratchet and the evolution of the mutation rate from the fixation of mutator mutations. As long as the mutation rate is increased in sufficiently small steps, an accelerating ratchet and eventual extinction are inevitable. If this can be countered by antimutators, i.e., mutations that reduce the mutation rate, an equilibrium can be established for the mutation rate at some level that may allow survival. However, the presence of the ratchet amplifies fluctuations in the mutation rate and, even at equilibrium, these fluctuations can lead to dangerous bursts in the ratchet. We investigate the timescales of these processes and discuss the results with reference to the genome degradation of the aphid endosymbiont Buchnera aphidicola.     

A stochastic model for a single click of Muller's ratchet

This work presents a new approach to Muller's ratchet, where Haigh's model is approximately mapped into a simpler model that describes the behaviour of a population after a click of the ratchet, i.e., after loss of what was the fittest class. This new model predicts the distribution of times to the next click of the ratchet and is equivalent to a Wright-Fisher model for a population of haploid asexual individuals with one locus and two alleles. Within this model, the fittest members of a population correspond to carriers of one allele, while all other individuals have suboptimal fitness and are represented as carriers of the other allele. In this way, all suboptimal fitness individuals are amalgamated into a single “mutant” class.The approach presented here has some limitations and the potential for improvement. However, it does lead to results for the rate of the ratchet that, over a wide range of parameters, are accurate within one order of magnitude of simulation results. This contrasts with existing approaches, which are designed for only one or other of the two different parameter regimes known for the ratchet and are more accurate only in the parameter regime they were designed for.Numerical results are presented for the mean time between clicks of the ratchet for (i) the Wright-Fisher model, (ii) a diffusion approximation of this model and (iii) individually based simulations of a full model. The diffusion approximation is validated over a wide range of parameters by its close agreement with the Wright-Fisher model.The present work predicts that: (a) the time between clicks of the ratchet is insensitive to the value of the selection coefficient when the genomic mutation rate is large compared with the selection coefficient against a deleterious mutation, (b) the time interval between clicks of the ratchet has, approximately, an exponential distribution (or its discrete analogue). It is thus possible to determine the variance in times between clicks, given the expected time between clicks. Evidence for both (a) and (b) is seen in simulations.     

Distribution of the Fittest Individuals and the Rate of Muller's Ratchet in a Model with Overlapping Generations

Muller''s ratchet is a paradigmatic model for the accumulation of deleterious mutations in a population of finite size. A click of the ratchet occurs when all individuals with the least number of deleterious mutations are lost irreversibly due to a stochastic fluctuation. In spite of the simplicity of the model, a quantitative understanding of the process remains an open challenge. In contrast to previous works, we here study a Moran model of the ratchet with overlapping generations. Employing an approximation which describes the fittest individuals as one class and the rest as a second class, we obtain closed analytical expressions of the ratchet rate in the rare clicking regime. As a click in this regime is caused by a rare, large fluctuation from a metastable state, we do not resort to a diffusion approximation but apply an approximation scheme which is especially well suited to describe extinction events from metastable states. This method also allows for a derivation of expressions for the quasi-stationary distribution of the fittest class. Additionally, we confirm numerically that the formulation with overlapping generations leads to the same results as the diffusion approximation and the corresponding Wright-Fisher model with non-overlapping generations.     

Analysis of the mitotic exit control system using locked levels of stable mitotic cyclin

Cyclin‐dependent kinase (Cdk) both promotes mitotic entry (spindle assembly and anaphase) and inhibits mitotic exit (spindle disassembly and cytokinesis), leading to an elegant quantitative hypothesis that a single cyclin oscillation can function as a ratchet to order these events. This ratchet is at the core of a published ODE model for the yeast cell cycle. However, the ratchet model requires appropriate cyclin dose–response thresholds. Here, we test the inhibition of mitotic exit in budding yeast using graded levels of stable mitotic cyclin (Clb2). In opposition to the ratchet model, stable levels of Clb2 introduced dose‐dependent delays, rather than hard thresholds, that varied by mitotic exit event. The ensuing cell cycle was highly abnormal, suggesting a novel reason for cyclin degradation. Cdc14 phosphatase antagonizes Clb2–Cdk, and Cdc14 is released from inhibitory nucleolar sequestration independently of stable Clb2. Thus, Cdc14/Clb2 balance may be the appropriate variable for mitotic regulation. Although our results are inconsistent with the aforementioned ODE model, revision of the model to allow Cdc14/Clb2 balance to control mitotic exit corrects these discrepancies, providing theoretical support for our conclusions.     

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     

The polymerization ratchet model explains the force-velocity relation for growing microtubules

The polymerization of filamentous proteins generates mechanical forces which drive many cellular processes. Dogterom and Yurke measured the force-velocity relation generated by a single microtubule. They found that the force is generally in the range predicted by the “polymerization ratchet” mechanism, but the force-velocity relationship decreased faster than that theory predicted. Here we generalize the polymerization ratchet model to take into account the “subsidy effect” that arises because a microtubule consists of 13 protofilaments. With this generalization the model fits the experimental data well. The biological implications of the polymerization ratchet model are discussed. Received: 18 May 1998 / Revised version: 4 November 1998 / Accepted: 30 November 1998