Thermodynamic uncertainty relation for biomolecular processes

A thermodynamically consistent model of finite-state machines

Finite-state machines (FSMs) are a theoretically and practically important model of computation. We propose a general, thermodynamically consistent model of FSMs and characterize the resource requirements of these machines. We model FSMs as time-inhomogeneous Markov chains. The computation is driven by instantaneous manipulations of the energy levels of the states. We calculate the entropy production of the machine, its error probability, and the time required to complete one update step. We find that a sequence of generalized bit-setting operations is sufficient to implement any FSM.

Nanothermometry Reveals Calcium-Induced Remodeling of Myosin

Ions greatly influence protein structure-function and are critical to health and disease. A 10, 000-fold higher calcium in the sarcoplasmic reticulum (SR) of muscle suggests elevated calcium levels near active calcium channels at the SR membrane and the impact of localized high calcium on the structure-function of the motor protein myosin. In the current study, combined quantum dot (QD)-based nanothermometry and circular dichroism (CD) spectroscopy enabled detection of previously unknown enthalpy changes and associated structural remodeling of myosin, impacting its function following exposure to elevated calcium. Cadmium telluride QDs adhere to myosin, function as thermal sensors, and reveal that exposure of myosin to calcium is exothermic, resulting in lowering of enthalpy, a decrease in alpha helical content measured using CD spectroscopy, and the consequent increase in motor efficiency. Isolated muscle fibers subjected to elevated levels of calcium further demonstrate fiber lengthening and decreased motility of actin filaments on myosin-functionalized substrates. Our results, in addition to providing new insights into our understanding of muscle structure-function, establish a novel approach to understand the enthalpy of protein-ion interactions and the accompanying structural changes that may occur within the protein molecule.

Hidden entropy production and work fluctuations in an ideal active gas

Collections of self-propelled particles that move persistently by continuously consuming free energy are a paradigmatic example of active matter. In these systems, unlike Brownian "hot colloids," the breakdown of detailed balance yields a continuous production of entropy at steady state, even for an ideal active gas. We quantify the irreversibility for a noninteracting active particle in two dimensions by treating both conjugated and time-reversed dynamics. By starting with underdamped dynamics, we identify a hidden rate of entropy production required to maintain persistence and prevent the rapidly relaxing momenta from thermalizing, even in the limit of very large friction. Additionally, comparing two popular models of self-propulsion with identical dissipation on average, we find that the fluctuations and large deviations in work done are markedly different, providing thermodynamic insight into the varying extents to which macroscopically similar active matter systems may depart from equilibrium.

Unified Thermodynamic Uncertainty Relations in Linear Response

Thermodynamic uncertainty relations (TURs) are recently established relations between the relative uncertainty of time-integrated currents and entropy production in nonequilibrium systems. For small perturbations away from equilibrium, linear response (LR) theory provides the natural framework to study generic nonequilibrium processes. Here, we use LR to derive TURs in a straightforward and unified way. Our approach allows us to generalize TURs to systems without local time-reversal symmetry, including, e.g., ballistic transport and periodically driven classical and quantum systems. We find that, for broken time reversal, the bounds on the relative uncertainty are controlled both by dissipation and by a parameter encoding the asymmetry of the Onsager matrix. We illustrate our results with an example from mesoscopic physics. We also extend our approach beyond linear response: for Markovian dynamics, it reveals a connection between the TUR and current fluctuation theorems.

Force-dependent diffusion coefficient of molecular Brownian ratchets

We study the mean velocity and diffusion constant in three related models of molecular Brownian ratchets. Brownian ratchets can be used to describe translocation of biopolymers like DNA through nanopores in cells in the presence of chaperones on the trans side of the pore. Chaperones can bind to the polymer and prevent it from sliding back through the pore. First, we study a simple model that describes the translocation in terms of an asymmetric random walk. It serves as an introductory example but already captures the main features of a Brownian ratchet. We then provide an analytical expression for the diffusion constant in the classical model of a translocation ratchet that was first proposed by Peskin et al. [C. S. Peskin, G. M. Odell, and G. F. Oster, Cellular motions and thermal fluctuations: The Brownian ratchet, Biophys. J. 65, 316 (1993)BIOJAU0006-349510.1016/S0006-3495(93)81035-X]. This model is based on the assumption that the binding and unbinding of the chaperones are much faster than the diffusion of the DNA strand. To remedy this shortcoming, we propose a modified model that is also applicable if the (un)binding rates are finite. We calculate the force-dependent mean velocity and diffusivity for this model and compare the results to the original one. Our analysis shows that for large pulling forces the predictions of both models can differ strongly even if the (un)binding rates are large in comparison to the diffusion timescale but still finite. Furthermore, implications of the thermodynamic uncertainty relation on the efficiency of Brownian ratchets are discussed.

Topological localization in out-of-equilibrium dissipative systems

In this paper, we report that notions of topological protection can be applied to stationary configurations that are driven far from equilibrium by active, dissipative processes. We consider two physically disparate systems: stochastic networks governed by microscopic single-particle dynamics, and collections of driven interacting particles described by coarse-grained hydrodynamic theory. We derive our results by mapping to well-known electronic models and exploiting the resulting correspondence between a bulk topological number and the spectrum of dissipative modes localized at the boundary. For the Markov networks, we report a general procedure to uncover the topological properties in terms of the transition rates. For the active fluid on a substrate, we introduce a topological interpretation of fluid dissipative modes at the edge. In both cases, the presence of dissipative couplings to the environment that break time-reversal symmetry are crucial to ensuring topological protection. These examples constitute proof of principle that notions of topological protection do indeed extend to dissipative processes operating out of equilibrium. Such topologically robust boundary modes have implications for both biological and synthetic systems.

Cycling Tames Power Fluctuations near Optimum Efficiency

According to the laws of thermodynamics, no heat engine can beat the efficiency of a Carnot cycle. This efficiency traditionally comes with vanishing power output and practical designs, optimized for power, generally achieve far less. Recently, various strategies to obtain Carnot's efficiency at large power were proposed. However, a thermodynamic uncertainty relation implies that steady-state heat engines can operate in this regime only at the cost of large fluctuations that render them immensely unreliable. Here, we demonstrate that this unfortunate trade-off can be overcome by designs operating cyclically under quasistatic conditions. The experimentally relevant yet exactly solvable model of an overdamped Brownian heat engine is used to illustrate the formal result. Our study highlights that work in cyclic heat engines and that in quasistatic ones are different stochastic processes.

Arcsine Laws in Stochastic Thermodynamics

We show that the fraction of time that a thermodynamic current spends above its average value follows the arcsine law, a prominent result obtained by Lévy for Brownian motion. Stochastic currents with long streaks above or below their average are much more likely than those that spend similar fractions of time above and below their average. Our result is confirmed with experimental data from a Brownian Carnot engine. We also conjecture that two other random times associated with currents obey the arcsine law: the time a current reaches its maximum value and the last time a current crosses its average value. These results apply to, inter alia, molecular motors, quantum dots, and colloidal systems.

Phase transition in thermodynamically consistent biochemical oscillators

Biochemical oscillations are ubiquitous in living organisms. In an autonomous system, not influenced by an external signal, they can only occur out of equilibrium. We show that they emerge through a generic nonequilibrium phase transition, with a characteristic qualitative behavior at criticality. The control parameter is the thermodynamic force which must be above a certain threshold for the onset of biochemical oscillations. This critical behavior is characterized by the thermodynamic flux associated with the thermodynamic force, its diffusion coefficient, and the stationary distribution of the oscillating chemical species. We discuss metrics for the precision of biochemical oscillations by comparing two observables, the Fano factor associated with the thermodynamic flux and the number of coherent oscillations. Since the Fano factor can be small even when there are no biochemical oscillations, we argue that the number of coherent oscillations is more appropriate to quantify the precision of biochemical oscillations. Our results are obtained with three thermodynamically consistent versions of known models: the Brusselator, the activator-inhibitor model, and a model for KaiC oscillations.

Stochastic Thermodynamic Interpretation of Information Geometry

In recent years, the unified theory of information and thermodynamics has been intensively discussed in the context of stochastic thermodynamics. The unified theory reveals that information theory would be useful to understand nonstationary dynamics of systems far from equilibrium. In this Letter, we have found a new link between stochastic thermodynamics and information theory well-known as information geometry. By applying this link, an information geometric inequality can be interpreted as a thermodynamic uncertainty relationship between speed and thermodynamic cost. We have numerically applied an information geometric inequality to a thermodynamic model of a biochemical enzyme reaction.

Broken detailed balance and non-equilibrium dynamics in living systems: a review

Living systems operate far from thermodynamic equilibrium. Enzymatic activity can induce broken detailed balance at the molecular scale. This molecular scale breaking of detailed balance is crucial to achieve biological functions such as high-fidelity transcription and translation, sensing, adaptation, biochemical patterning, and force generation. While biological systems such as motor enzymes violate detailed balance at the molecular scale, it remains unclear how non-equilibrium dynamics manifests at the mesoscale in systems that are driven through the collective activity of many motors. Indeed, in several cellular systems the presence of non-equilibrium dynamics is not always evident at large scales. For example, in the cytoskeleton or in chromosomes one can observe stationary stochastic processes that appear at first glance thermally driven. This raises the question how non-equilibrium fluctuations can be discerned from thermal noise. We discuss approaches that have recently been developed to address this question, including methods based on measuring the extent to which the system violates the fluctuation-dissipation theorem. We also review applications of this approach to reconstituted cytoskeletal networks, the cytoplasm of living cells, and cell membranes. Furthermore, we discuss a more recent approach to detect actively driven dynamics, which is based on inferring broken detailed balance. This constitutes a non-invasive method that uses time-lapse microscopy data, and can be applied to a broad range of systems in cells and tissue. We discuss the ideas underlying this method and its application to several examples including flagella, primary cilia, and cytoskeletal networks. Finally, we briefly discuss recent developments in stochastic thermodynamics and non-equilibrium statistical mechanics, which offer new perspectives to understand the physics of living systems.

Entropic bounds on currents in Langevin systems

We derive a bound on generalized currents for Langevin systems in terms of the total entropy production in the system and its environment. For overdamped dynamics, any generalized current is bounded by the total rate of entropy production. We show that this entropic bound on the magnitude of generalized currents imposes power-efficiency tradeoff relations for ratchets in contact with a heat bath: Maximum efficiency-Carnot efficiency for a Smoluchowski-Feynman ratchet and unity for a flashing or rocking ratchet-can only be reached at vanishing power output. For underdamped dynamics, while there may be reversible currents that are not bounded by the entropy production rate, we show that the output power and heat absorption rate are irreversible currents and thus obey the same bound. As a consequence, a power-efficiency tradeoff relation holds not only for underdamped ratchets but also for periodically driven heat engines. For weak driving, the bound results in additional constraints on the Onsager matrix beyond those imposed by the second law. Finally, we discuss the connection between heat and entropy in a nonthermal situation where the friction and noise intensity are state dependent.

Exact Fluctuations of Nonequilibrium Steady States from Approximate Auxiliary Dynamics

We describe a framework to reduce the computational effort to evaluate large deviation functions of time integrated observables within nonequilibrium steady states. We do this by incorporating an auxiliary dynamics into trajectory based Monte Carlo calculations, through a transformation of the system's propagator using an approximate guiding function. This procedure importance samples the trajectories that most contribute to the large deviation function, mitigating the exponential complexity of such calculations. We illustrate the method by studying driven diffusion and interacting lattice models in one and two spatial dimensions. Our work offers an avenue to calculate large deviation functions for high dimensional systems driven far from equilibrium.

Mapping of uncertainty relations between continuous and discrete time

Lower bounds on fluctuations of thermodynamic currents depend on the nature of time, discrete or continuous. To understand the physical reason, we compare current fluctuations in discrete-time Markov chains and continuous-time master equations. We prove that current fluctuations in the master equations are always more likely, due to random timings of transitions. This comparison leads to a mapping of the moments of a current between discrete and continuous time. We exploit this mapping to obtain uncertainty bounds. Our results reduce the quests for uncertainty bounds in discrete and continuous time to a single problem.

Quantifying fluctuations in reversible enzymatic cycles and clocks

Biochemical reactions are fundamentally noisy at a molecular scale. This limits the precision of reaction networks, but it also allows fluctuation measurements that may reveal the structure and dynamics of the underlying biochemical network. Here, we study nonequilibrium reaction cycles, such as the mechanochemical cycle of molecular motors, the phosphorylation cycle of circadian clock proteins, or the transition state cycle of enzymes. Fluctuations in such cycles may be measured using either of two classical definitions of the randomness parameter, which we show to be equivalent in general microscopically reversible cycles. We define a stochastic period for reversible cycles and present analytical solutions for its moments. Furthermore, we associate the two forms of the randomness parameter with the thermodynamic uncertainty relation, which sets limits on the timing precision of the cycle in terms of thermodynamic quantities. Our results should prove useful also for the study of temporal fluctuations in more general networks.

Universal Trade-Off between Power, Efficiency, and Constancy in Steady-State Heat Engines

Heat engines should ideally have large power output, operate close to Carnot efficiency and show constancy, i.e., exhibit only small fluctuations in this output. For steady-state heat engines, driven by a constant temperature difference between the two heat baths, we prove that out of these three requirements only two are compatible. Constancy enters quantitatively the conventional trade-off between power and efficiency. Thus, we rationalize and unify recent suggestions for overcoming this simple trade-off. Our universal bound is illustrated for a paradigmatic model of a quantum dot solar cell and for a Brownian gyrator delivering mechanical work against an external force.

Phase Transition in Protocols Minimizing Work Fluctuations

For two canonical examples of driven mesoscopic systems-a harmonically trapped Brownian particle and a quantum dot-we numerically determine the finite-time protocols that optimize the compromise between the standard deviation and the mean of the dissipated work. In the case of the oscillator, we observe a collection of protocols that smoothly trade off between average work and its fluctuations. However, for the quantum dot, we find that as we shift the weight of our optimization objective from average work to work standard deviation, there is an analog of a first-order phase transition in protocol space: two distinct protocols exchange global optimality with mixed protocols akin to phase coexistence. As a result, the two types of protocols possess qualitatively different properties and remain distinct even in the infinite duration limit: optimal-work-fluctuation protocols never coalesce with the minimal-work protocols, which therefore never become quasistatic.

Motion of kinesin in a viscoelastic medium

Kinesin is a molecular motor that transports cargo along microtubules. The results of many in vitro experiments on kinesin-1 are described by kinetic models in which one transition corresponds to the forward motion and subsequent binding of the tethered motor head. We argue that in a viscoelastic medium like the cytosol of a cell this step is not Markov and has to be described by a nonexponential waiting time distribution. We introduce a semi-Markov kinetic model for kinesin that takes this effect into account. We calculate, for arbitrary waiting time distributions, the moment generating function of the number of steps made, and determine from this the average velocity and the diffusion constant of the motor. We illustrate our results for the case of a waiting time distribution that is Weibull. We find that for realistic parameter values, viscoelasticity decreases the velocity and the diffusion constant, but increases the randomness (or Fano factor).

Current fluctuations in quantum absorption refrigerators

Absorption refrigerators transfer thermal energy from a cold bath to a hot bath without input power by utilizing heat from an additional "work" reservoir. Particularly interesting is a three-level design for a quantum absorption refrigerator, which can be optimized to reach the maximal (Carnot) cooling efficiency. Previous studies of three-level chillers focused on the behavior of the averaged cooling current. Here, we go beyond that and study the full counting statistics of heat exchange in a three-level chiller model. We explain how to obtain the complete cumulant generating function of the refrigerator in a steady state, then derive a partial cumulant generating function, which yields closed-form expressions for both the averaged cooling current and its noise. Our analytical results and simulations are beneficial for the design of nanoscale engines and cooling systems far from equilibrium, with their performance optimized according to different criteria, efficiency, power, fluctuations, and dissipation.

Design principles for enhancing phase sensitivity and suppressing phase fluctuations simultaneously in biochemical oscillatory systems

Biological systems need to function accurately in the presence of strong noise and at the same time respond sensitively to subtle external cues. Here we study design principles in biochemical oscillatory circuits to achieve these two seemingly incompatible goals. We show that energy dissipation can enhance phase sensitivity linearly by driving the phase-amplitude coupling and increase timing accuracy by suppressing phase diffusion. Two general design principles in the key underlying reaction loop formed by two antiparallel pathways are found to optimize oscillation performance with a given energy budget: balancing the forward-to-backward flux ratio between the two pathways to reduce phase diffusion and maximizing the net flux of the phase-advancing pathway relative to that of the phase-retreating pathway to enhance phase sensitivity. Experimental evidences consistent with these design principles are found in the circadian clock of cyanobacteria. Future experiments to test the predicted dependence of phase sensitivity on energy dissipation are proposed.

Large deviation function for a driven underdamped particle in a periodic potential

Employing large deviation theory, we explore current fluctuations of underdamped Brownian motion for the paradigmatic example of a single particle in a one-dimensional periodic potential. Two different approaches to the large deviation function of the particle current are presented. First, we derive an explicit expression for the large deviation functional of the empirical phase space density, which replaces the level 2.5 functional used for overdamped dynamics. Using this approach, we obtain several bounds on the large deviation function of the particle current. We compare these to bounds for overdamped dynamics that have recently been derived, motivated by the thermodynamic uncertainty relation. Second, we provide a method to calculate the large deviation function via the cumulant generating function. We use this method to assess the tightness of the bounds in a numerical case study for a cosine potential.

Thermodynamic Bounds on Precision in Ballistic Multiterminal Transport

For classical ballistic transport in a multiterminal geometry, we derive a universal trade-off relation between total dissipation and the precision, at which particles are extracted from individual reservoirs. Remarkably, this bound becomes significantly weaker in the presence of a magnetic field breaking time-reversal symmetry. By working out an explicit model for chiral transport enforced by a strong magnetic field, we show that our bounds are tight. Beyond the classical regime, we find that, in quantum systems far from equilibrium, the correlated exchange of particles makes it possible to exponentially reduce the thermodynamic cost of precision.

Derivation of Markov processes that violate detailed balance

Time-reversal symmetry of the microscopic laws dictates that the equilibrium distribution of a stochastic process must obey the condition of detailed balance. However, cyclic Markov processes that do not admit equilibrium distributions with detailed balance are often used to model systems driven out of equilibrium by external agents. I show that for a Markov model without detailed balance, an extended Markov model can be constructed, which explicitly includes the degrees of freedom for the driving agent and satisfies the detailed balance condition. The original cyclic Markov model for the driven system is then recovered as an approximation at early times by summing over the degrees of freedom for the driving agent. I also show that the widely accepted expression for the entropy production in a cyclic Markov model is actually a time derivative of an entropy component in the extended model. Further, I present an analytic expression for the entropy component that is hidden in the cyclic Markov model.

First-passage times in renewal and nonrenewal systems

Fluctuations in stochastic systems are usually characterized by full counting statistics, which analyzes the distribution of the number of events taking place in the fixed time interval. In an alternative approach, the distribution of the first-passage times, i.e., the time delays after which the counting variable reaches a certain threshold value, is studied. This paper presents the approach to calculate the first-passage time distribution in systems in which the analyzed current is associated with an arbitrary set of transitions within the Markovian network. Using this approach, it is shown that when the subsequent first-passage times are uncorrelated, there exist strict relations between the cumulants of the full counting statistics and the first-passage time distribution. On the other hand, when the correlations of the first-passage times are present, their distribution may provide additional information about the internal dynamics of the system in comparison to the full counting statistics; for example, it may reveal the switching between different dynamical states of the system. Additionally, I show that breaking of the fluctuation theorem for first-passage times may reveal the multicyclic nature of the Markovian network.

Energetic Costs, Precision, and Transport Efficiency of Molecular Motors

An efficient molecular motor would deliver cargo to the target site at a high speed and in a punctual manner while consuming a minimal amount of energy. According to a recently formulated thermodynamic principle, referred to as the thermodynamic uncertainty relation, the travel distance of a motor and its variance are, however, constrained by the free energy being consumed. Here we use the principle underlying the uncertainty relation to quantify the transport efficiency of molecular motors for varying ATP concentration ([ATP]) and applied load (f). Our analyses of experimental data find that transport efficiencies of the motors studied here are semioptimized under the cellular condition. The efficiency is significantly deteriorated for a kinesin-1 mutant that has a longer neck-linker, which underscores the importance of molecular structure. It is remarkable to recognize that, among many possible directions for optimization, biological motors have evolved to optimize the transport efficiency in particular.

The importance of thermodynamics for molecular systems, and the importance of molecular systems for thermodynamics

Improved understanding of molecular systems has only emphasised the sophistication of networks within the cell. Simultaneously, the advance of nucleic acid nanotechnology, a platform within which reactions can be exquisitely controlled, has made the development of artificial architectures and devices possible. Vital to this progress has been a solid foundation in the thermodynamics of molecular systems. In this pedagogical review and perspective, we discuss how thermodynamics determines both the overall potential of molecular networks, and the minute details of design. We then argue that, in turn, the need to understand molecular systems is helping to drive the development of theories of thermodynamics at the microscopic scale.

Limits of predictions in thermodynamic systems: a review

The past twenty years have seen a resurgence of interest in nonequilibrium thermodynamics, thanks to advances in the theory of stochastic processes and in their thermodynamic interpretation. Fluctuation theorems provide fundamental constraints on the dynamics of systems arbitrarily far from thermal equilibrium. Thermodynamic uncertainty relations bound the dissipative cost of precision in a wide variety of processes. Concepts of excess work and excess heat provide the basis for a complete thermodynamics of nonequilibrium steady states, including generalized Clausius relations and thermodynamic potentials. But these general results carry their own limitations: fluctuation theorems involve exponential averages that can depend sensitively on unobservably rare trajectories; steady-state thermodynamics makes use of a dual dynamics that lacks any direct physical interpretation. This review aims to present these central results of contemporary nonequilibrium thermodynamics in such a way that the power of each claim for making physical predictions can be clearly assessed, using examples from current topics in soft matter and biophysics.

Liquid-Gas Transitions in Steady Heat Conduction

We study liquid-gas transitions of heat conduction systems in contact with two heat baths under constant pressure in the linear response regime. On the basis of local equilibrium thermodynamics, we propose an equality with a global temperature, which determines the volume near the equilibrium liquid-gas transition. We find that the formation of the liquid-gas interface is accompanied by a discontinuous change in the volume when increasing the mean temperature of the baths. A supercooled gas near the interface is observed as a stable steady state.

Stochastic thermodynamics of periodically driven systems: Fluctuation theorem for currents and unification of two classes

Periodic driving is used to operate machines that go from standard macroscopic engines to small nonequilibrium microsized systems. Two classes of such systems are small heat engines driven by periodic temperature variations, and molecular pumps driven by external stimuli. Well-known results that are valid for nonequilibrium steady states of systems driven by fixed thermodynamic forces, instead of an external periodic driving, have been generalized to periodically driven heat engines only recently. These results include a general expression for entropy production in terms of currents and affinities, and symmetry relations for the Onsager coefficients from linear-response theory. For nonequilibrium steady states, the Onsager reciprocity relations can be obtained from the more general fluctuation theorem for the currents. We prove a fluctuation theorem for the currents for periodically driven systems. We show that this fluctuation theorem implies a fluctuation dissipation relation, symmetry relations for Onsager coefficients, and further relations for nonlinear response coefficients. The setup in this paper is more general than previous studies, i.e., our results are valid for both heat engines and molecular pumps. The external protocol is assumed to be stochastic in our framework, which leads to a particularly convenient way to treat periodically driven systems.

Fundamental Bounds on First Passage Time Fluctuations for Currents

Current is a characteristic feature of nonequilibrium systems. In stochastic systems, these currents exhibit fluctuations constrained by the rate of dissipation in accordance with the recently discovered thermodynamic uncertainty relation. Here, we derive a conjugate uncertainty relationship for the first passage time to accumulate a fixed net current. More generally, we use the tools of large-deviation theory to simply connect current fluctuations and first passage time fluctuations in the limit of long times and large currents. With this connection, previously discovered symmetries and bounds on the large-deviation function for currents are readily transferred to first passage times.

Frenetic Bounds on the Entropy Production

We give a systematic derivation of positive lower bounds for the expected entropy production (EP) rate in classical statistical mechanical systems obeying a dynamical large deviation principle. The logic is the same for the return to thermodynamic equilibrium as it is for steady nonequilibria working under the condition of local detailed balance. We recover there recently studied "uncertainty" relations for the EP, appearing in studies about the effectiveness of mesoscopic machines. In general our refinement of the positivity of the expected EP rate is obtained in terms of a positive and even function of the expected current(s) which measures the dynamical activity in the system, a time-symmetric estimate of the changes in the system's configuration. Also underdamped diffusions can be included in the analysis.

Generic Properties of Stochastic Entropy Production

We derive an Itô stochastic differential equation for entropy production in nonequilibrium Langevin processes. Introducing a random-time transformation, entropy production obeys a one-dimensional drift-diffusion equation, independent of the underlying physical model. This transformation allows us to identify generic properties of entropy production. It also leads to an exact uncertainty equality relating the Fano factor of entropy production and the Fano factor of the random time, which we also generalize to non-steady-state conditions.

Allocating dissipation across a molecular machine cycle to maximize flux

Biomolecular machines consume free energy to break symmetry and make directed progress. Nonequilibrium ATP concentrations are the typical free energy source, with one cycle of a molecular machine consuming a certain number of ATP, providing a fixed free energy budget. Since evolution is expected to favor rapid-turnover machines that operate efficiently, we investigate how this free energy budget can be allocated to maximize flux. Unconstrained optimization eliminates intermediate metastable states, indicating that flux is enhanced in molecular machines with fewer states. When maintaining a set number of states, we show that-in contrast to previous findings-the flux-maximizing allocation of dissipation is not even. This result is consistent with the coexistence of both "irreversible" and reversible transitions in molecular machine models that successfully describe experimental data, which suggests that, in evolved machines, different transitions differ significantly in their dissipation.

Proof of the finite-time thermodynamic uncertainty relation for steady-state currents

The thermodynamic uncertainty relation offers a universal energetic constraint on the relative magnitude of current fluctuations in nonequilibrium steady states. However, it has only been derived for long observation times. Here, we prove a recently conjectured finite-time thermodynamic uncertainty relation for steady-state current fluctuations. Our proof is based on a quadratic bound to the large deviation rate function for currents in the limit of a large ensemble of many copies.

Information-Theoretic Bound on the Entropy Production to Maintain a Classical Nonequilibrium Distribution Using Ancillary Control

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 equilibrium distribution. This work offers quantitative insight into the intuitive idea that more energy is needed to maintain a system farther from equilibrium.

Physical insight into the thermodynamic uncertainty relation using Brownian motion in tilted periodic potentials

Using Brownian motion in periodic potentials V(x) tilted by a force f, we provide physical insight into the thermodynamic uncertainty relation, a recently conjectured principle for statistical errors and irreversible heat dissipation in nonequilibrium steady states. According to the relation, nonequilibrium output generated from dissipative processes necessarily incurs an energetic cost or heat dissipation q, and in order to limit the output fluctuation within a relative uncertainty ε, at least 2k_{B}T/ε^{2} of heat must be dissipated. Our model shows that this bound is attained not only at near-equilibrium [f≪V^{'}(x)] but also at far-from-equilibrium [f≫V^{'}(x)], more generally when the dissipated heat is normally distributed. Furthermore, the energetic cost is maximized near the critical force when the barrier separating the potential wells is about to vanish and the fluctuation of Brownian particles is maximized. These findings indicate that the deviation of heat distribution from Gaussianity gives rise to the inequality of the uncertainty relation, further clarifying the meaning of the uncertainty relation. Our derivation of the uncertainty relation also recognizes a bound of nonequilibrium fluctuations that the variance of dissipated heat (σ_{q}^{2}) increases with its mean (μ_{q}), and it cannot be smaller than 2k_{B}Tμ_{q}.

Finite-time generalization of the thermodynamic uncertainty relation

For fluctuating currents in nonequilibrium steady states, the recently discovered thermodynamic uncertainty relation expresses a fundamental relation between their variance and the overall entropic cost associated with the driving. We show that this relation holds not only for the long-time limit of fluctuations, as described by large deviation theory, but also for fluctuations on arbitrary finite time scales. This generalization facilitates applying the thermodynamic uncertainty relation to single molecule experiments, for which infinite time scales are not accessible. Importantly, often this finite-time variant of the relation allows inferring a bound on the entropy production that is even stronger than the one obtained from the long-time limit. We illustrate the relation for the fluctuating work that is performed by a stochastically switching laser tweezer on a trapped colloidal particle.

Coherence of biochemical oscillations is bounded by driving force and network topology

Biochemical oscillations are prevalent in living organisms. Systems with a small number of constituents cannot sustain coherent oscillations for an indefinite time because of fluctuations in the period of oscillation. We show that the number of coherent oscillations that quantifies the precision of the oscillator is universally bounded by the thermodynamic force that drives the system out of equilibrium and by the topology of the underlying biochemical network of states. Our results are valid for arbitrary Markov processes, which are commonly used to model biochemical reactions. We apply our results to a model for a single KaiC protein and to an activator-inhibitor model that consists of several molecules. From a mathematical perspective, based on strong numerical evidence, we conjecture a universal constraint relating the imaginary and real parts of the first nontrivial eigenvalue of a stochastic matrix.

Thermodynamic Paradigm for Solution Demixing Inspired by Nuclear Transport in Living Cells

Living cells display a remarkable capacity to compartmentalize their functional biochemistry. A particularly fascinating example is the cell nucleus. Exchange of macromolecules between the nucleus and the surrounding cytoplasm does not involve traversing a lipid bilayer membrane. Instead, large protein channels known as nuclear pores cross the nuclear envelope and regulate the passage of other proteins and RNA molecules. Beyond simply gating diffusion, the system of nuclear pores and associated transport receptors is able to generate substantial concentration gradients, at the energetic expense of guanosine triphosphate hydrolysis. In contrast to conventional approaches to demixing such as reverse osmosis and dialysis, the biological system operates continuously, without application of cyclic changes in pressure or solvent exchange. Abstracting the biological paradigm, we examine this transport system as a thermodynamic machine of solution demixing. Building on the construct of free energy transduction and biochemical kinetics, we find conditions for the stable operation and optimization of the concentration gradients as a function of dissipation in the form of entropy production.

Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables

Recent large deviation results have provided general lower bounds for the fluctuations of time-integrated currents in the steady state of stochastic systems. A corollary are so-called thermodynamic uncertainty relations connecting precision of estimation to average dissipation. Here we consider this problem but for counting observables, i.e., trajectory observables which, in contrast to currents, are non-negative and nondecreasing in time (and possibly symmetric under time reversal). In the steady state, their fluctuations to all orders are bound from below by a Conway-Maxwell-Poisson distribution dependent only on the averages of the observable and of the dynamical activity. We show how to obtain the corresponding bounds for first-passage times (times when a certain value of the counting variable is first reached) and their uncertainty relations. Just like entropy production does for currents, dynamical activity controls the bounds on fluctuations of counting observables.

Mapping current fluctuations of stochastic pumps to nonequilibrium steady states

We show that current fluctuations in a stochastic pump can be robustly mapped to fluctuations in a corresponding time-independent nonequilibrium steady state. We thus refine a recently proposed mapping so that it ensures equivalence of not only the averages, but also optimal representation of fluctuations in currents and density. Our mapping leads to a natural decomposition of the entropy production in stochastic pumps similar to the "housekeeping" heat. As a consequence of the decomposition of entropy production, the current fluctuations in weakly perturbed stochastic pumps are shown to satisfy a universal bound determined by the steady state entropy production.

First Passage under Restart

First passage under restart has recently emerged as a conceptual framework suitable for the description of a wide range of phenomena, but the endless variety of ways in which restart mechanisms and first passage processes mix and match hindered the identification of unifying principles and general truths. Hope that these exist came from a recently discovered universality displayed by processes under optimal, constant rate, restart-but extensions and generalizations proved challenging as they marry arbitrarily complex processes and restart mechanisms. To address this challenge, we develop a generic approach to first passage under restart. Key features of diffusion under restart-the ultimate poster boy for this wide and diverse class of problems-are then shown to be completely universal.

Stochastic Thermodynamics of Learning

Virtually every organism gathers information about its noisy environment and builds models from those data, mostly using neural networks. Here, we use stochastic thermodynamics to analyze the learning of a classification rule by a neural network. We show that the information acquired by the network is bounded by the thermodynamic cost of learning and introduce a learning efficiency η≤1. We discuss the conditions for optimal learning and analyze Hebbian learning in the thermodynamic limit.

Nonequilibrium Markov state modeling of the globule-stretch transition

We describe a systematic approach to construct coarse-grained Markov state models from molecular dynamics data of systems driven into a nonequilibrium steady state. We apply this method to study the globule-stretch transition of a single tethered model polymer in shear flow. The folding and unfolding rates of the coarse-grained model agree with the original detailed model. We demonstrate that the folding and unfolding proceeds through the same narrow region of configuration space but along different cycles.

Work fluctuation and total entropy production in nonequilibrium processes

Work fluctuation and total entropy production play crucial roles in small thermodynamic systems subject to large thermal fluctuations. We investigate a trade-off relation between them in a nonequilibrium situation in which a system starts from an arbitrary nonequilibrium state. We apply a variational method to study this problem and find a stationary solution against variations over protocols that describe the time dependence of the Hamiltonian of the system. Using the stationary solution, we find the minimum of the total entropy production for a given amount of work fluctuation. An explicit protocol that achieves this is constructed from an adiabatic process followed by a quasistatic process. The obtained results suggest how one can control the nonequilibrium dynamics of the system while suppressing its work fluctuation and total entropy production.

Design principles for nonequilibrium self-assembly

We consider an important class of self-assembly problems, and using the formalism of stochastic thermodynamics, we derive a set of design principles for growing controlled assemblies far from equilibrium. The design principles constrain the set of configurations that can be obtained under nonequilibrium conditions. Our central result provides intuition for how equilibrium self-assembly landscapes are modified under finite nonequilibrium drive.

Optimal inference strategies and their implications for the linear noise approximation

We study the information loss of a class of inference strategies that is solely based on time averaging. For an array of independent binary sensors (e.g., receptors, single electron transistors) measuring a weak random signal (e.g., ligand concentration, gate voltage) this information loss is up to 0.5  bit per measurement irrespective of the number of sensors. We derive a condition related to the local detailed balance relation that determines whether or not such a loss of information occurs. Specifically, if the free-energy difference arising from the signal is symmetrically distributed among the forward and backward rates, time integration mechanisms will capture the full information about the signal. As an implication, for the linear noise approximation, we can identify the same loss of information, arising from its inherent simplification of the dynamics.