Molecular motors drive mechanical motions utilizing the free energy liberated from chemical reactions such as ATP hydrolysis. Although it is essential to know the efficiency of this free energy transduction, it has been a challenge due to the system's microscopic scale. Here, we evaluate the single-molecule energetics of a rotary molecular motor, F1-ATPase, by applying a recently derived nonequilibrium equality together with an electrorotation method. We show that the sum of the heat flow through the probe's rotational degree of freedom and the work against external load is almost equal to the free energy change per a single ATP hydrolysis under various conditions. This implies that F1-ATPase works at an efficiency of nearly 100% in a thermally fluctuating environment.
F1-ATPase (or F1), the highly-efficient and reversible biochemical engine, has motivated physicists as well as biologists to imagine the design principles governing machines in the fluctuating world. Recent experiments have clarified yet another interesting property of F1; the dissipative heat inside the motor is very small, irrespective of the velocity of rotation and energy transport. Conceptual interest is devoted to the fact that the amount of internal dissipation is not simply determined by the sequence of equilibrium pictures, but also relies on the rotational-angular dependence of nucle…
F$_\mathrm{o}$F$_1$-ATP synthase is a factory for synthesizing ATP in virtually all cells. Its core machinery is the subcomplex F$_1$-motor (F$_1$-ATPase) and performs the reversible mechanochemical coupling. Isolated F$_1$-motor hydrolyzes ATP, which is accompanied by unidirectional rotation of its central $γ$-shaft. When a strong opposing torque is imposed, the $γ$-shaft rotates in the opposite direction and drives the F$_1$-motor to synthesize ATP. This mechanical-to-chemical free-energy transduction is the final and central step of the multistep cellular ATP-synthetic pathway. Here, we det…
We have developed a novel method to evaluate the potential profile of a molecular motor at each chemical state from only the probe's trajectory and applied it to a rotary molecular motor F$_1$-ATPase. By using this method, we could also obtain the information regarding the mechanochemical coupling and energetics. We demonstrate that the position-dependent transition of the chemical states is the key feature for the highly efficient free-energy transduction by F$_1$-ATPase.
F$_{1}$-ATPase is a rotary molecular motor that \emph{in vivo} is subject to strong nonequilibrium driving forces. There is great interest in understanding the operational principles governing its high efficiency of free-energy transduction. Here we use a near-equilibrium framework to design a non-trivial control protocol to minimize dissipation in rotating F$_{1}$ to synthesize ATP. We find that the designed protocol requires much less work than a naive (constant-velocity) protocol across a wide range of protocol durations. Our analysis points to a possible mechanism for energetically efficie…
A model of coupled molecular oscillators is proposed to study nonequilibrium thermodynamics of synchronization. We find that synchronization of nonequilibrium oscillators costs energy even when the oscillator-oscillator coupling is conservative. By solving the steady state of the many-body system analytically, we show that the system goes through a nonequilibrium phase transition driven by energy dissipation, and the critical energy dissipation depends on both the frequency and strength of the exchange reactions. Moreover, our study reveals the optimal design for achieving maximum synchronizat…
In the absence of external driving, a system exposed to thermal fluctuations will relax to equilibrium. However, the constant input of work makes it possible to counteract this relaxation, and maintain the system in a nonequilibrium steady state. In this Article, we use the stochastic thermodynamics of Markov jump processes to compute the minimum rate at which energy must be supplied and dissipated to maintain an arbitrary nonequilibrium distribution in a given energy landscape. This lower bound depends on two factors: the undriven probability current in the equilibrium state, and the distance…
We give a decomposition expression for dissipative heat using the instantaneous diffusion coefficient in a nonequilibrium steady state. The dissipative heat can be expressed using three diffusion coefficients: instantaneous, equilibrium, and drift. An experimental application of the decomposition expression permits us to evaluate the heat dissipation rate from single-trajectory data only. We also numerically demonstrate this method.
Stochastic thermodynamics has largely succeeded in characterizing both equilibrium and far-from-equilibrium phenomena. Yet many opportunities remain for application to mesoscopic complex systems -- especially biological ones -- whose effective dynamics often violate detailed balance and whose microscopic degrees of freedom are often unknown or intractable. After reviewing excess and housekeeping energetics -- the adaptive and homeostatic components of a system's dissipation -- we extend stochastic thermodynamics with a trajectory class fluctuation theorem for nonequilibrium steady-state, nonde…
Nonequilibrium energetics of sensing and actuation by a smart active particle
University of Cambridge（剑桥大学）；Max Planck Institute for Dynamics and Self-Organization (MPIDS)（马克斯·普朗克动力学与自组织研究所）；University of Rome “Tor Vergata”（罗马托维尔加塔大学）
A Stirling engine made of a colloidal particle in contact with a nonequilibrium bath is considered and analyzed with the tools of stochastic energetics. We model the bath by non Gaussian persistent noise acting on the colloidal particle. Depending on the chosen definition of an isothermal transformation in this nonequilibrium setting, we find that either the energetics of the engine parallels that of its equilibrium counterpart or, in the simplest case, that it ends up being less efficient. Persistence, more than non Gaussian effects, are responsible for this result.
A novel equality relating the rate of energy dissipation to a degree of violation of the fluctuation-response relation (FRR) in nonequilibrium Langevin systems is described. The FRR is a relation between the correlation function of the fluctuations and the response function of macroscopic variables. Although it has been established that the FRR holds in equilibrium, physical significance of violation of the FRR in nonequilibrium systems has been under debate. Recently, the authors have found that an extent of the FRR violation is related in a simple equality to the rate of energy dissipation i…
There are many functional contexts where it is desirable to maintain a mesoscopic system in a nonequilibrium state. However, such control requires an inherent energy dissipation. In this article, we unify and extend a number of works on the minimum energetic cost to maintain a mesoscopic system in a prescribed nonequilibrium distribution using ancillary control. For a variety of control mechanisms, we find that the minimum amount of energy dissipation necessary can be cast as an information-theoretic measure of distinguishability between the target nonequilibrium state and the underlying equil…
Living organisms are inherently out-of-equilibrium systems. We employ new developments in stochastic energetics and rely on a minimal microscopic model to predict the amount of mechanical energy dissipated by such dynamics. Our model includes complex rheological effects and nonequilibrium stochastic forces. By performing active microrheology and tracking micron-sized vesicles in the cytoplasm of living oocytes, we provide unprecedented measurements of the spectrum of dissipated energy. We show that our model is fully consistent with the experimental data, and we use it to offer predictions for…
An expression for the energetic efficiency of a molecular motor is presented in terms of an effective temperature, which was defined based on the ratio of the correlation function to the susceptibility of its velocity. We also present a numerical result regarding this temperature for a thermal ratchet model. Furthermore, our expression of efficiency using known values for kinesin can adequately explain the experimental results. The framework presented here has a closed form phenomenologically, and is independent of any detailed models.
We investigate the nonequilibrium energetics near a critical point of a non-linear driven oscillator immersed in an active bacterial bath. At the critical point, we reveal a scaling exponent of the average power $\langle\dot{W}\rangle\sim (D_{\rm a}/τ)^{1/4}$ where $D_{\rm a}$ is the effective diffusivity and $τ$ the correlation time of the bacterial bath described by a Gaussian colored noise. Other features that we investigate are the average stationary power and the variance of the work both below and above the saddle-node bifurcation. Above the bifurcation, the average power attains an opti…
Microscopic machines utilize free energy to create and maintain out-of-equilibrium organization in virtually all living things. Often this takes the form of converting the free energy stored in nonequilibrium chemical potential differences into useful work, via a series of reactions involving the binding, chemical catalysis, and unbinding of small molecules. Such chemical reactions occur on timescales much faster than the protein conformational rearrangements they induce. Here, we derive the energetic cost for driving a system out of equilibrium via a series of such effectively instantaneous (…
Using a one-dimensional macromolecule in aqueous solution as an illustration, we demonstrate that the relative entropy from information theory, $\sum_k p_k\ln(p_k/p_k^*)$, has a natural role in the energetics of equilibrium and nonequilibrium conformational fluctuations of the single molecule. It is identified as the free energy difference associated with a fluctuating density in equilibrium, and is associated with the distribution deviate from the equilibrium in nonequilibrium relaxation. This result can be generalized to any other isothermal macromolecular systems using the mathematical theo…
We study the stochastic energetic exchanges in quantum heat engines. Due to microreversibility, these obey a fluctuation relation, called the heat engine fluctuation relation, which implies the Carnot bound: no machine can have an efficiency larger than Carnot's efficiency. The stochastic thermodynamics of a quantum heat engine (including the joint statistics of heat and work and the statistics of efficiency) is illustrated by means of an optimal two-qubit heat engine, where each qubit is coupled to a thermal bath and a two-qubit gate determines energy exchanges between the two qubits. We disc…
We study 1-D diffusion of $N$ hard-core interacting Brownian particles driven by the space- and time-dependent external force. We give the exact solution of the $N$-particle Smoluchowski diffusion equation. In particular, we investigate the nonequilibrium energetics of two interacting particles under the time-periodic driving. The hard-core interaction induces entropic repulsion which differentiates the energetics of the two particles. We present exact time-asymptotic results which describe the mean energy, the accepted work and heat, and the entropy production of interacting particles and we …
Single-molecule experiments have found near-perfect thermodynamic efficiency in the rotary motor F1-ATP synthase. To help elucidate the principles underlying nonequilibrium energetic efficiency in such stochastic machines, we investigate driving protocols that minimize dissipation near equilibrium in a simple model rotary mechanochemical motor, as determined by a generalized friction coefficient. Our simple model has a periodic friction coefficient that peaks near system energy barriers. This implies a minimum-dissipation protocol that proceeds rapidly when the system is overwhelmingly in a si…
Accurate and robust spatial orders are ubiquitous in living systems. In 1952, Alan Turing proposed an elegant mechanism for pattern formation based on spontaneous breaking of the spatial translational symmetry in the underlying reaction-diffusion system. Much is understood about dynamics and structure of Turing patterns. However, little is known about the energetic cost of Turing pattern. Here, we study nonequilibrium thermodynamics of a small spatially extended biochemical reaction-diffusion system by using analytical and numerical methods. We find that the onset of Turing pattern requires a …
The hallmark of active matter is the autonomous directed motion of its microscopic constituents driven by consumption of energy resources. This leads to the emergence of large scale dynamics and structures without any equilibrium equivalent. Though active field theories offer a useful hydrodynamic description, it is unclear how to properly quantify the energetic cost of the dynamics from such a coarse-grained description. We provide a thermodynamically consistent framework to identify the energy exchanges between active systems and their surrounding thermostat at the hydrodynamic level. Based …
