The entropy and efficiency of a molecular motor model (original) (raw)
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Thermodynamics and Kinetics of Molecular Motors
Biophysical Journal, 2010
Molecular motors are first and foremost molecules, governed by the laws of chemistry rather than of mechanics. The dynamical behavior of motors based on chemical principles can be described as a random walk on a network of states. A key insight is that any molecular motor in solution explores all possible motions and configurations at thermodynamic equilibrium. By using input energy and chemical design to prevent motion that is not wanted, what is left behind is the motion that is desired. This review is focused on two-headed motors such as kinesin and Myosin V that move on a polymeric track. By use of microscopic reversibility, it is shown that the ratio between the number of forward steps and the number of backward steps in any sufficiently long time period does not directly depend on the mechanical properties of the linker between the two heads. Instead, this ratio is governed by the relative chemical specificity of the heads in the front-versus-rear position for the fuel, adenosine triphosphate and its products, adenosine diphosphate and inorganic phosphate. These insights have been key factors in the design of biologically inspired synthetic molecular walkers constructed out of DNA or out of small organic molecules.
Generalized Efficiency and its Application to Microscopic Engines
Physical Review Letters, 1999
We generalize the definition of the energy output of an engine as the minimum energy input required to accomplish the same task as the engine. Applying this new concept to molecular motors makes it possible to measure their efficiency even without attaching any external load to them. This way we can compare and characterize the operation of molecular motors in various situations. We also investigate how the thermally driven motors differ from other motors. PACS numbers: 05.40.Jc, 02.50.Ey, 05.70.Ce, 87.10. + e Recently, micron-scale devices have been built [1] to drive the forward motion of microscopic particles not with a net macroscopic field but with small imposed fluctuations of an anisotropic periodic "ratchet" potential [2]. Motor proteins have been experimentally studied on the level of individual proteins and appear to work by the same principles. It seems only a matter of time before the first man-made molecular motors will be assembled . Such small engines are subject to a physics that is fundamentally different from the physics of our macroscopic world. First of all, when the length scales go down the Reynolds number goes down also, and we approach the overdamped limit in which inertia no longer plays a role and where the velocity of a particle is directly proportional to the force acting on that particle at that moment. Second, there is Brownian motion. Particles are being randomly kicked around by molecules of the surrounding medium and any deterministic motion comes on top of a thermal noise term. The motion of such an overdamped Brownian particle is described by the Langevin equation
Energy transfer in a molecular motor in the Kramers regime
Physical Review E, 2013
We present a theoretical treatment of energy transfer in a molecular motor described in terms of overdamped Brownian motion on a multidimensional tilted periodic potential. The tilt acts as a thermodynamic force driving the system out of equilibrium and, for non-separable potentials, energy transfer occurs between degrees of freedom. For deep potential wells, the continuous theory transforms to a discrete master equation that is tractable analytically. We use this master equation to derive formal expressions for the hopping rates, drift, diffusion, efficiency and rate of energy transfer in terms of the thermodynamic force. These results span both strong and weak coupling between degrees of freedom, describe the near and far from equilibrium regimes, and are consistent with generalized detailed balance and the Onsager relations. We thereby derive a number of diverse results for molecular motors within a single theoretical framework.
Biosystems, 2007
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.
2004
In recent literature there has been a lot of interest in the phenomena of noise induced transport in the absence of an average bias occurring in spatially periodic systems far from equilibrium. One of the main motivations in this area is to understand the mechanism behind the operation of biological motors at molecular scale. These molecular motors convert chemical energy available during the hydrolysis of ATP into mechanical motion to transport cargo and vesicles in living cells with very high reliability, adaptability and efficiency in a very noisy environment. The basic principle behind such a motion, namely the Brownian ratchet principle, has applications in nanotechnology as novel nanoparticle separation devices. Also, the mechanism of ratchet operation finds applications in game theory. Here, we briefly focus on the physical concepts underlying the constructive role of noise in assisting transport at a molecular level. The nature of particle currents, the energetic efficiency of these motors, the entropy production in these systems and the phenomenon of resonance/coherence are discussed.
Nonequilibrium Fluctuations and Mechanochemical Couplings of a Molecular Motor
Physical Review Letters, 2007
We investigate theoretically the violations of Einstein and Onsager relations, and the efficiency for a single processive motor operating far from equilibrium using an extension of the two-state model introduced by Kafri et al. [Biophys. J. 86, 3373 (2004)]. With the aid of the Fluctuation Theorem, we analyze the general features of these violations and this efficiency and link them to mechanochemical couplings of motors. In particular, an analysis of the experimental data of kinesin using our framework leads to interesting predictions that may serve as a guide for future experiments. 87.16.Nn, 05.70.Ln Motor proteins are nano-machines that convert chemical energy into mechanical work and motion . Important examples include kinesin, myosin, and RNA polymerase. Despite a number of theoretical models , understanding the mechanochemical transduction mechanisms behind these motors remains a significant challenge . Recent advances in experimental techniques to probe the fluctuations of single motors provide ways to gain insight into their kinetic pathways [10]. However, a general description for fluctuations of systems driven out of equilibrium, and in particular of motors, is still lacking. Recently, the Fluctuation Theorem (FT) has emerged as a promising framework to characterize fluctuations in far-from-equilibrium regimes where Einstein and Onsager relations no longer hold . In a nutshell, FT states that the probability distribution for the entropy production rate obeys a symmetry relation, and it has been verified in a number of beautiful experiments on biopolymers and colloidal systems . In this Letter, we demonstrate that FT provides a natural framework in which thermodynamic constraints can be imposed on the operation of nanomachines far from equilibrium.
Transport characteristics of molecular motors
Biosystems, 2008
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.
Efficiency of molecular machines with continuous phase space
EPL (Europhysics Letters), 2012
We consider a molecular machine described as a Brownian particle diffusing in a tilted periodic potential. We evaluate the absorbed and released power of the machine as a function of the applied molecular and chemical forces, by using the fact that the times for completing a cycle in the forward and the backward direction have the same distribution, and that the ratio of the corresponding splitting probabilities can be simply expressed as a function of the applied force. We explicitly evaluate the efficiency at maximum power for a simple sawtooth potential. We also obtain the efficiency at maximum power for a broad class of 2-D models of a Brownian machine and find that loosely coupled machines operate with a smaller efficiency at maximum power than their strongly coupled counterparts.
Traffic by multiple species of molecular motors
Physical Review E, 2009
We study the traffic of two types of molecular motors using the two-species asymmetric simple exclusion process (ASEP) with periodic boundary conditions and with attachment and detachment of particles. We determine characteristic properties such as motor densities and currents by simulations and analytical calculations. For motors with different unbinding probabilities, mean field theory gives the correct bound density and total current of the motors, as shown by numerical simulations. For motors differing in their stepping probabilities, the particle-hole symmetry of the current-density relationship is broken and mean field theory fails drastically. The total motor current exhibits exponential finite-size scaling, which we use to extrapolate the total current to the thermodynamic limit. Finally, we also study the motion of a single motor in the background of many non-moving motors.