Efficiency of molecular motors at maximum power

Molecular motors transduce chemical energy obtained from hydrolizing ATP into mechanical work exerted against an external force. We calculate their efficiency at maximum power output for two simple generic models and show that the qualitative behaviour depends crucially on the position of the transition state. Specifically, we find a transition state near the initial state (sometimes characterized as a "power stroke") to be most favorable with respect to both high power output and high efficiency at maximum power. In this regime, driving the motor further out of equilibrium by applying higher chemical potential differences can even, counter-intuitively, increase the efficiency.Comment: published in EPL: http://www.iop.org/EJ/abstract/0295-5075/83/3/3000

Optimal finite-time processes in stochastic thermodynamics

For a small system like a colloidal particle or a single biomolecule embedded in a heat bath, the optimal protocol of an external control parameter minimizes the mean work required to drive the system from one given equilibrium state to another in a finite time. In general, this optimal protocol obeys an integro-differential equation. Explicite solutions both for a moving laser trap and a time-dependent strength of such a trap show finite jumps of the optimal protocol to be typical both at the beginning and the end of the process.Comment: 4 pages, 1 figure, accepted for publication in Phys. Rev. Let

Can one identify non-equilibrium in a three-state system by analyzing two-state trajectories?

For a three-state Markov system in a stationary state, we discuss whether, on the basis of data obtained from effective two-state (or on-off) trajectories, it is possible to discriminate between an equilibrium state and a non-equilibrium steady state. By calculating the full phase diagram we identify a large region where such data will be consistent only with non-equilibrium conditions. This regime is considerably larger than the region with oscillatory relaxation, which has previously been identified as a sufficient criterion for non-equilibrium.Comment: 4 pages, 2 figures, J. Chem. Phys. (2010) (in press

Stochastic thermodynamics of chemical reaction networks

For chemical reaction networks described by a master equation, we define energy and entropy on a stochastic trajectory and develop a consistent nonequilibrium thermodynamic description along a single stochastic trajectory of reaction events. A first-law like energy balance relates internal energy, applied (chemical) work and dissipated heat for every single reaction. Entropy production along a single trajectory involves a sum over changes in the entropy of the network itself and the entropy of the medium. The latter is given by the exchanged heat identified through the first law. Total entropy production is constrained by an integral fluctuation theorem for networks arbitrarily driven by time-dependent rates and a detailed fluctuation theorem for networks in the steady state. Further exact relations like a generalized Jarzynski relation and a generalized Clausius inequality are discussed. We illustrate these results for a three-species cyclic reaction network which exhibits nonequilibrium steady states as well as transitions between different steady states.Comment: 14 pages, 2 figures, accepted for publication in J. Chem. Phy

Optimal protocols for Hamiltonian and Schr\"odinger dynamics

For systems in an externally controllable time-dependent potential, the optimal protocol minimizes the mean work spent in a finite-time transition between given initial and final values of a control parameter. For an initially thermalized ensemble, we consider both Hamiltonian evolution for classical systems and Schr\"odinger evolution for quantum systems. In both cases, we show that for harmonic potentials, the optimal work is given by the adiabatic work even in the limit of short transition times. This result is counter-intuitive because the adiabatic work is substantially smaller than the work for an instantaneous jump. We also perform numerical calculations of the optimal protocol for Hamiltonian dynamics in an anharmonic quartic potential. For a two-level spin system, we give examples where the adiabatic work can be reached in either a finite or an arbitrarily short transition time depending on the allowed parameter space.Comment: submitted to J. Stat. Mech.: Theor. Exp

Entropy production for mechanically or chemically driven biomolecules

Entropy production along a single stochastic trajectory of a biomolecule is discussed for two different sources of non-equilibrium. For a molecule manipulated mechanically by an AFM or an optical tweezer, entropy production (or annihilation) occurs in the molecular conformation proper or in the surrounding medium. Within a Langevin dynamics, a unique identification of these two contributions is possible. The total entropy change obeys an integral fluctuation theorem and a class of further exact relations, which we prove for arbitrarily coupled slow degrees of freedom including hydrodynamic interactions. These theoretical results can therefore also be applied to driven colloidal systems. For transitions between different internal conformations of a biomolecule involving unbalanced chemical reactions, we provide a thermodynamically consistent formulation and identify again the two sources of entropy production, which obey similar exact relations. We clarify the particular role degenerate states have in such a description

Optimale Prozesse in stochastischer Thermodynamik

The concept of Stochastic Thermodynamics deals with the question how to define thermodynamic quantities for nonequilibrium mesoscopic systems. Here, thermal fluctuations must be considered. The main objective of this thesis is the analysis of optimization problems in the context of Stochastic Thermodynamcis. A quite natural optimization principle for nonequilibrium processes is the requirement that a defined result should be achieved with the smallest possible amount of dissipation. For a transition between two given equilibrium states in a given finite time, this is directly linked to a process schedule which leads to a minimal (mean) work. For systems in an externally controllable time-dependent potential, the optimal protocol minimizes the mean work spent in a finite-time transition between two given equilibrium states. Surprisingly, the optimal protocol involves jumps for overdamped Langevin dynamics and even delta-type singularities for underdamped Langevin dynamics. For purely Hamiltonian and Schrödinger dynamics in harmonic potentials, we show that the optimal protocol is highly degenerate and that even in the limit of short transition times, the optimal work is given by the adiabatic work which is substantially smaller than the work for an instantaneous jump. These optimal protocols significantly improve free energy calculations via the Jarzynski equality. Most processes in the biological cell, however, cannot be described by a nonequilibrium transition between equilibrium states. Rather, these systems are permanently driven out of equilibrium, e.g. by chemical potential differences. An important model class of such dynamics are Brownian motors which transfer either chemical or thermal energy into mechanical work leading to directed transport against a load force. It is meaningful to characterize such thermodynamic machines by their performance at maximum power output rather than at maximum efficiency. The efficiency at this maximum power then is a relevant quantity. We consider a Carnot engine on the mesoscale which can be constructed by using a Brownian particle instead of the working gas and a time-dependent trapping potential instead of the confining vessel. The efficiency at maximum power output can be calculated analytically. Surprisingly, it is given by a quite universal expression which does only depend on the viscosity (or more generally on the mobility matrices) at the two temperatures. This result is independent of the shape of the potential used to trap the particle. In contrast to heat engines, molecular motors in the biological cell are mostly driven by chemical potential differences. For two simple motor models, the efficiency of the molecular motor at maximum power shows two unexpected features: (i) Both the power output and the efficiency increase when the transition state position is moved closer to the initial motor position and (ii) for appropriate parameters, the efficiency increases when the system is driven further out of equilibrium by a higher chemical potential difference. Beyond their relevance for directed transport within the cell, molecular motors are also important for the synthesis of proteins. We study the protein production rate at a given error rate for the second stage of gene expression (translation). We find that for a given error rate equivalent to the experimentally observed value, the protein production rate is not at its theoretical maximum. We therefore conjecture that other evolutionary goals or structural reasons are responsible for the observed rate constants.Die Stochastische Thermodynamic beschäftigt sich mit der Fragestellung, wie thermodynamische Größen auch für mesoskopische Systeme im Nichtgleichgewicht definiert werden können. Dabei müssen thermische Fluktuationen berücksichtigt werden. Das Hauptziel der vorliegenden Arbeit ist die Analyse von Optimierungs-Problemen im Umfeld der Stochastischen Thermodynamik. Ein natürliches Optimierungsprinzip für Nichtgleichgewichtsprozesse ist die Forderung, dass ein definiertes Ziel mit einem Minimum an benötigter Dissipation erreicht wird. Für einen Übergang zwischen zwei Gleichgewichtszuständen is dies eng verknüpft mit der Frage nach einer Prozessführung, die zu einer minimalen mittleren Arbeit führt. Das optimale Protokoll minimiert die mittlere Arbeit für einen Übergang zwischen zwei Gleichgewichtszuständen in einer gegebenen Zeit. Für überdämpfte Langevin-Dynamik zeigt das optimal Protokoll Sprünge am Anfang und Ende, für unterdämpfte Langevin-Dynamik ergeben sich sogar Delta-Peaks am Anfang und Ende. Für Hamilton'sche und Schrödinger-Dynamik in harmonischen Potentialen kann die adiabatische Arbeit schon in beliebig kurzer Übergangszeit erreicht werden. Optimale Protokolle verbessern im allgemeinen die Berechnung von freien Energiedifferenzen mit Hilfe der Jarzynski-Relation. Die meisten Prozesse in der biologischen Zelle können nicht durch einfache Nichtgleichgewichts-Übergänge zwischen Gleichgewichtszuständen beschrieben werden. Viele Systeme werden z. B. durch chemische Potentialdifferenzen permanent aus dem Gleichgewicht getrieben. Eine wichtige Klasse solcher Prozesse sind Brown'sche Motoren, die entweder durch chemische oder thermische Energie getrieben werden und diese in mechanisch Arbeit umwandeln. Es ist sinnvoll, statt des Wirkungsgrades die Leistung einer solchen Maschine zu maximieren. Eine wichtige Kenngröße ist dann der Wirkungsgrad bei dieser maximalen Leistung. Nun wird das Analogon zu einer Carnot-Maschine auf einer mesoskopischen Skala betrachten. Dabei kann ein Brown'sches Teilchen als Arbeitsgas und ein zeitabhängiges Potential anstatt eines Zylinders verwendet werden. Der Wirkungsgrad am Arbeitspunkt mit maximaler Leistung kann analytisch berechnet werden. Überraschenderweise ergibt sich ein sehr universeller Ausdruck, der nur von den Viskositäten (oder allgemein in mehreren Dimensionen von den Mobilitätsmatrizen) der umgebenden Flüssigkeit bei den beiden Temperaturen abhängt. Insbesondere ist dieses Ergebnis unabhängig von der Form des verwendeten zeitabhängigen Potentials. Molekulare Motoren in der Zelle werden meist nicht durch Temperaturdifferenzen sondern durch chemische Potentialdifferenzen angetrieben. Für zwei verschiedene einfache Motormodelle zeigt sich, dass der Wirkungsgrad des molekularen Motors bei maximaler Leistung dann folgendes überraschendes Verhalten zeigt: (i) Der Wirkungsgrad und die Leistung nehmen gleichzeitig zu, wenn die Position des Übergangszustands näher an den ursprünglichen Zustand rückt. (ii) Bei geeigneter Parameterwahl wächst der Wirkungsgrad, wenn das System durch eine wachsende chemische Potentialdifferenz weiter aus dem Gleichgewicht getrieben wird. Molekularen Motoren sorgen nicht nur für gerichteten Transport in der Zelle sondern sind auch an der Synthese von Proteinen beteiligt. Wir untersuchen die Produktionsrate für Proteine bei einer gegebenen Fehlerrate im zweiten Schritt der Proteinsynthese (tranlation). Dabei zeigt sich, dass die Protein-Produktionsrate bei einer festgehaltenen (durch den experimentellen Wert gegebenen) Fehlerrate nicht maximal ist. Deshalb kann vermutet werden, dass andere evolutionäre Ziele oder strukturelle Gründe für die beobachten Ratenkonstanten verantwortlich sind