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The steady-state work fluctuations for an overdamped Brownian particle driven by an external stochastic force are analyzed by using the Langevin approach. The first two moments of the work distribution are given analytically, both are proportional to the time duration of the work injection. The first two moments determine a unique Gaussian function, and the deviations of the probability density functions (PDFs) of work from the Gaussian functions are measured. The tails of the PDFs are shown to approximate the Gaussian functions. In the limit of weak external force and large time interval, the PDF can be approximated by the Gaussian form and the steady-state fluctuation theorem is held. [ABSTRACT FROM AUTHOR]

The analysis of cellular signaling cascades based on information thermodynamics has recently developed considerably. A signaling cascade may be considered a binary code system consisting of two types of signaling molecules that carry biological information, phosphorylated active, and non-phosphorylated inactive forms. This study aims to evaluate the signal transduction step in cascades from the viewpoint of changes in mixing entropy. An increase in active forms may induce biological signal transduction through a mixing entropy change, which induces a chemical potential current in the signaling cascade. We applied the fluctuation theorem to calculate the chemical potential current and found that the average entropy production current is independent of the step in the whole cascade. As a result, the entropy current carrying signal transduction is defined by the entropy current mobility. [ABSTRACT FROM AUTHOR]

It is now well established that most if not all enzymatic proteins display a slow stochastic dynamics of transitions between a variety of conformational substates composing their native state. A hypothesis is stated that the protein conformational transition networks, as just as higher-level biological networks, the protein interaction network, and the metabolic network, have evolved in the process of self-organized criticality. Here, the criticality means that all the three classes of networks are scale-free and, moreover, display a transition from the fractal organization on a small length-scale to the small-world organization on the large length-scale. Good mathematical models of such networks are stochastic critical branching trees extended by long-range shortcuts. Biological molecular machines are proteins that operate under isothermal conditions and hence are referred to as free energy transducers. They can be formally considered as enzymes that simultaneously catalyze two chemical reactions: the free energy-donating (input) reaction and the free energy-accepting (output) one. The far-from-equilibrium degree of coupling between the output and the input reaction fluxes have been studied both theoretically and by means of the Monte Carlo simulations on model networks. For single input and output gates the degree of coupling cannot exceed unity. Study simulations of random walks on model networks involving more extended gates indicate that the case of the degree of coupling value higher than one is realized on the mentioned above critical branching trees extended by long-range shortcuts.

Some critical trends in information theory, its role in living systems and utilization in fluctuation theory are discussed. The mutual information of thermodynamic coupling is incorporated into the generalized fluctuation theorem by using information theory and nonequilibrium thermodynamics. Thermodynamically coupled dissipative structures in living systems are capable of degrading more energy, and processing complex information through developmental and environmental constraints. The generalized fluctuation theorem can quantify the hysteresis observed in the amount of the irreversible work in nonequilibrium regimes in the presence of information and thermodynamic coupling.

Mutual information and its causal variant, directed information, have been widely used to quantitatively characterize the performance of biological sensing and information transduction. However, once coupled with selection in response to decision-making, the sensing signal could have more or less evolutionary value than its mutual or directed information. In this work, we show that an individually sensed signal always has a better fitness value, on average, than its mutual or directed information. The fitness gain, which satisfies fluctuation relations (FRs), is attributed to the selection of organisms in a population that obtain a better sensing signal by chance. A new quantity, similar to the coarse-grained entropy production in information thermodynamics, is introduced to quantify the total fitness gain from individual sensing, which also satisfies FRs. Using this quantity, the optimizing fitness gain of individual sensing is shown to be related to fidelity allocations for individual environmental histories. Our results are supplemented by numerical verifications of FRs, and a discussion on how this problem is linked to information encoding and decoding.

Fluctuation theorems are a class of equalities that express universal properties of the probability distribution of a fluctuating path functional such as heat, work or entropy production over an ensemble of trajectories during a non-equilibrium process with a well-defined initial distribution. Jinwoo and Tanaka (Jinwoo, L.; Tanaka, H. Sci. Rep. 2015, 5, 7832) have shown that work fluctuation theorems hold even within an ensemble of paths to each state, making it clear that entropy and free energy of each microstate encode heat and work, respectively, within the conditioned set. Here we show that information that is characterized by the point-wise mutual information for each correlated state between two subsystems in a heat bath encodes the entropy production of the subsystems and heat bath during a coupling process. To this end, we extend the fluctuation theorem of information exchange (Sagawa, T.; Ueda, M. Phys. Rev. Lett. 2012, 109, 180602) by showing that the fluctuation theorem holds even within an ensemble of paths that reach a correlated state during dynamic co-evolution of two subsystems.

Sagawa and Ueda established a fluctuation theorem of information exchange by revealing the role of correlations in stochastic thermodynamics and unified the non-equilibrium thermodynamics of measurement and feedback control. They considered a process where a non-equilibrium system exchanges information with other degrees of freedom such as an observer or a feedback controller. They proved the fluctuation theorem of information exchange under the assumption that the state of the other degrees of freedom that exchange information with the system does not change over time while the states of the system evolve in time. Here we relax this constraint and prove that the same form of the fluctuation theorem holds even if both subsystems co-evolve during information exchange processes. This result may extend the applicability of the fluctuation theorem of information exchange to a broader class of non-equilibrium processes, such as a dynamic coupling in biological systems, where subsystems that exchange information interact with each other.

We formulate a box model of coupled ocean-atmosphere to examine the differential fields interactive with the thermohaline circulation (THC) and their response to global warming. We discern a robust convective bound on the atmospheric heat transport, which would divide the climate regime into warm and cold branches; but unlike the saline mode of previous box models, the cold state, if allowed, has the same-signed-though weaker-density contrast and THC as the present climate, which may explain its emergence from coupled general circulation models. We underscore the nondeterminacy of the THC due to random eddy shedding and apply the fluctuation theorem to constrain the shedding rate, thus closing the problem. The derivation reveals an ocean propelled toward the maximum entropy production (MEP) on millennial timescale (termed 'MEP-adjustment'), the long timescale arising from the compounding effect of microscopic fluctuations in the shedding rate and their slight probability bias. Global warming may induce hysteresis between the two branches, like that seen in GCMs, but the cold transition is far more sensitive to the moistening than the heating effects as the latter would be countered by the hydrological feedback. The uni- or bi-modality of the current state-hence whether the THC may recover after the cold transition-depends on the global-mean convective flux and may not be easily assessed due to its observed uncertainty. [ABSTRACT FROM AUTHOR]

The adaptability to ever-changing environment is one of the fundamental characteristics that differentiates living matters from non-living ones. The combination of passive adaptation by Darwinian evolution and active adaptation by sensing and decision-making shapes the mechanisms of adaptability. In this work, we outline that the biological adaptation shares a very similar structure with the stochastic and information-thermodynamics. Fitness, the macroscopic quantity to characterize evolutionary success, plays the similar role as the free energy in this structure. The fluctuation relation of fitness and information derived from the structure can be regarded as an extension of the concept of the evolutionary stable strategy. [ABSTRACT FROM AUTHOR]

Close to equilibrium, the exchange of particles and heat between macroscopic systems at different temperatures and different chemical potentials is known to be governed by a matrix of transport coefficients which are positive and symmetric. We investigate the amounts of heat and particles that are exchanged between two small quantum systems within a given time, and find them characterized by a transport matrix which neither needs to be symmetric nor positive. At larger times even spontaneous transport can be observed in the total absence of temperature and chemical potential differences provided that the two systems are different in size. All these deviations from standard transport behavior can be attributed to the fact that work is done on the system in the processes contacting and separating those parts of the system that initially possess different temperatures and chemical potentials. The standard transport properties are recovered for vanishing work and also in the limit of large systems and sufficiently large contact times. The general results are illustrated by an example. [ABSTRACT FROM AUTHOR]

Stochastic motion of macrospins is similar to driven diffusion of Brownian particles on the surface of a sphere. One crucial difference is in how the micro-states transform under time reversal. This dictates the form of stochastic entropy production (sEP). An excess sEP in the reservoir, in addition to a Clausius term, may appear depending on the interpretation of stochastic trajectory, thereby, precluding such analysis without a detailed knowledge of the governing dynamics. To show this, we derive expressions of sEP using Fokker-Planck equation, and the ratio of probability distributions of time-forward and time-reversed trajectories. We calculate probability distributions of sEP using numerical simulations, and obtain good agreement with the detailed fluctuation theorem. Within adiabatic approximation, analytic form for the distribution function is also derived. [ABSTRACT FROM AUTHOR]

Based on the fluctuation theorem of a small engine, we present a description of the stochastic dynamics of a mesoscopic system. We apply it to the problem of the escape of a particle from a metastable state subjected to a fluctuating nonconservative force, which is beyond the description of the fluctuation-dissipation theorem. As an example, we consider a reversal of a small magnetic moment by fluctuating spin-torque. The joint probability distribution of input heat and output work is derived from the full-counting statistics under adiabatic pumping. We predict a threshold voltage above which the spin-torque shot noise induces probabilistic switching events, and below which such events are blocked. [ABSTRACT FROM AUTHOR]

We consider a simple model of the Markovian stochastic dynamics on complex networks to examine the statistical properties of the probability fluxes. The additional transition, called hereafter a gate, powered by the external constant force breaks a detailed balance in the network. We argue, using a theoretical approach and numerical simulations, that the stationary distributions of the probability fluxes emergent under such conditions converge to the Gaussian distribution. By virtue of the stationary fluctuation theorem, its standard deviation depends directly on the square root of the mean flux. In turn, the nonlinear relation between the mean flux and the external force, which provides the key result of the present study, allows us to calculate the two parameters that entirely characterize the Gaussian distribution of the probability fluxes both close to as well as far from the equilibrium state. Also, the other effects that modify these parameters, such as the addition of shortcuts to the tree-like network, the extension and configuration of the gate and a change in the network size studied by means of computer simulations are widely discussed in terms of the rigorous theoretical predictions. [ABSTRACT FROM AUTHOR]

Several stochastic processes modeling molecular motors on a linear track are given by random walks (not necessarily Markovian) on quasi 1d lattices and share a common regenerative structure. Analyzing this abstract common structure, we derive information on the large fluctuations of the stochastic process by proving large deviation principles for the first-passage times and for the position.We focus our attention on the Gallavotti-Cohen-type symmetry of the position rate function (fluctuation theorem), showing its equivalence with the independence of suitable random variables. In the special case of Markov random walks, we show that this symmetry is universal only inside a suitable class of quasi 1d lattices. [ABSTRACT FROM AUTHOR]

We survey new extensions of continuum mechanics incorporating spontaneous violations of the Second Law (SL), which involve the viscous flow and heat conduction. First, following an account of the Fluctuation Theorem (FT) of statistical mechanics that generalizes the SL, the irreversible entropy is shown to evolve as a submartingale. Next, a stochastic thermomechanics is formulated consistent with the FT, which, according to a revision of classical axioms of continuum mechanics, must be set up on random fields. This development leads to a reformulation of thermoviscous fluids and inelastic solids. These two unconventional constitutive behaviors may jointly occur in nano-poromechanics. [ABSTRACT FROM AUTHOR]

As shown by Jarzynski, free energy differences between equilibrium states can be expressed in terms of the statistics of work carried out on a system during non-equilibrium transformations. This exact result, as well as the related Crooks fluctuation theorem, provide the basis for the computation of free energy differences from fast switching molecular dynamics simulations, in which an external parameter is changed at a finite rate, driving the system away from equilibrium. In this article, we first briefly review the Jarzynski identity and the Crooks fluctuation theorem and then survey various algorithms building on these relations. We pay particular attention to the statistical efficiency of these methods and discuss practical issues arising in their implementation and the analysis of the results.

Under the infrastructure of three gradually deepening layers consisting of System, Service and Software, the information entropy of the Trustworthy Workflow Management System (TWfMS) will evolve from being more precise to more undetermined, due to a series of exception event X occurring on certain components (ExCs), along with the life cycle of TWfMS, experienced in its phased original, as-is, to-be, and agile-consistent stages, and recover, more precisely again, by turning back to the original state from the agile-consistent stage, due to its self-autonomous improvement. With a special emphasis on the system layer, to assure the trustworthiness of WfMS, this paper firstly introduces the preliminary knowledge of the hierarchical information entropy model with correlation theories. After illustrating the fundamental principle, the transformation rule is deduced, step by step, followed by a case study, which is conducive to generating discussions and conclusions in the different research areas of TWfMS. Overall, in this paper, we argue that the trustworthiness maintenance of WfMS could be analyzed and computational, through the viewpoint that all the various states of TWfMS can be considered as the transformation between WfMS and its trustworthiness compensate components, whose information entropy fluctuate repeatedly and comply with the law of the dissipative structure system.

A model of signal transduction from the perspective of informational thermodynamics has been reported in recent studies, and several important achievements have been obtained. The first achievement is that signal transduction can be modelled as a binary code system, in which two forms of signalling molecules are utilised in individual steps. The second is that the average entropy production rate is consistent during the signal transduction cascade when the signal event number is maximised in the model. The third is that a Szilard engine can be a single-step model in the signal transduction. This article reviews these achievements and further introduces a new chain of Szilard engines as a biological reaction cascade (BRC) model. In conclusion, the presented model provides a way of computing the channel capacity of a BRC.

We present a general method to identify an arbitrary number of fluctuating quantities which satisfy a detailed fluctuation theorem for all times within the framework of time-inhomogeneous Markovian jump processes. In doing so, we provide a unified perspective on many fluctuation theorems derived in the literature. By complementing the stochastic dynamics with a thermodynamic structure (i.e., using stochastic thermodynamics), we also express these fluctuating quantities in terms of physical observables.

Kullback–Leibler divergence (KLD) is a type of extended mutual entropy, which is used as a measure of information gain when transferring from a prior distribution to a posterior distribution. In this study, KLD is applied to the thermodynamic analysis of cell signal transduction cascade and serves an alternative to mutual entropy. When KLD is minimized, the divergence is given by the ratio of the prior selection probability of the signaling molecule to the posterior selection probability. Moreover, the information gain during the entire channel is shown to be adequately described by average KLD production rate. Thus, this approach provides a framework for the quantitative analysis of signal transduction. Moreover, the proposed approach can identify an effective cascade for a signaling network.

Cell signal transduction is a non-equilibrium process characterized by the reaction cascade. This study aims to quantify and compare signal transduction cascades using a model of signal transduction. The signal duration was found to be linked to step-by-step transition probability, which was determined using information theory. By applying the fluctuation theorem for reversible signal steps, the transition probability was described using the average entropy production rate. Specifically, when the signal event number during the cascade was maximized, the average entropy production rate was found to be conserved during the entire cascade. This approach provides a quantitative means of analyzing signal transduction and identifies an effective cascade for a signaling network.

A cell signaling system is in a non-equilibrium state, and it includes multistep biochemical signaling cascades (BSCs), which involve phosphorylation of signaling molecules, such as mitogen-activated protein kinase (MAPK) pathways. In this study, the author considered signal transduction description using information thermodynamic theory. The ideal BSCs can be considered one type of the Szilard engine, and the presumed feedback controller, Maxwell’s demon, can extract the work during signal transduction. In this model, the mutual entropy and chemical potential of the signal molecules can be redefined by the extracted chemical work in a mechanicochemical model, Szilard engine, of BSC. In conclusion, signal transduction is computable using the information thermodynamic method.

We establish quantum thermodynamics for open quantum systems weakly coupled to their reservoirs when the system exhibits degeneracies. The first and second law of thermodynamics are derived, as well as a finite-time fluctuation theorem for mechanical work and energy and matter currents. Using a double quantum dot junction model, local eigenbasis coherences are shown to play a crucial role on thermodynamics and on the electron counting statistics. [ABSTRACT FROM AUTHOR]

We consider a model of N interacting two-colour Friedman urns. The interaction model considered is such that the reinforcement of each urn depends on the fraction of balls of a particular colour in that urn as well as the overall fraction of balls of that colour in all the urns combined together. We show that the urns synchronize almost surely and that the fraction of balls of each colour converges to the deterministic limit of one-half, which matches with the limit known for a single Friedman urn. Furthermore, we use the notion of stable convergence to obtain limit theorems for fluctuations around the synchronization limit. [ABSTRACT FROM AUTHOR]

In this note, we consider the fluctuation theorem for $$X_{f_n}^{(n)}:=\sum _{i=1}^n f(\lambda _i)I(\lambda _i\ge \theta _n)$$ , where $$\lambda _i, i=1,\ldots ,n$$ are eigenvalues from a Wigner matrix and $$\theta _n\rightarrow 2^-$$ . We prove that in the edge case $$X_{f_n}^{(n)}$$ behaves like the counting function of Wigner matrix. Our results can be viewed as a complement of Bao et al. (J Stat Phys 150(1):88-129, 2013). [ABSTRACT FROM AUTHOR]

Spontaneous, random violations of the second law become relevant as the length and/or timescales become very small. Modern statistical physics tells us that the second law then needs to be replaced by the fluctuation theorem and, mathematically, the irreversible entropy evolves as a submartingale. This is illustrated on the Couette flow of a molecular fluid. On the continuum level, such phenomena lead to a framework of thermomechanics relying on stochastic (rather than deterministic) functionals of energy and entropy, which are applied in heat diffusion and thermoelasticity settings. Counterintuitive thermomechanical behaviors are also discussed in (1) the evolution of acceleration wavefronts of nanoscale thickness and (2) the fluid mechanics which has to enter a permeability model of a poroelastic medium with nanoscale pores. [ABSTRACT FROM PUBLISHER]

The mechanisms of the strong inward rectification in inward rectifier K + (Kir) channels are controversial because the drop in electrical potential due to the movement of the blocker and coupling ions is insufficient to explain the steep voltage-dependent block near the equilibrium potential. Here, we study the “driving force”-dependent block in Kir channels with a novel approach incorporating concepts from the non-equilibrium thermodynamics of small systems, and computer kinetic simulations based on the experimental data of internal Ba 2 + block on Kir2.1 channels. The steep exponential increase in the apparent binding rate near the equilibrium potential is explained, when the encounter frequency is construed as the likelihood of transfer events down or against the electrochemical potential gradient. The exponent of flux ratio, n f = 2.62, implies that the blockage of the internal blocker may be coupled with the outward transport of 2 to 3 K + ions. The flux-coupled block in the single-file multi-ion pore can be demonstrated by the concentration gradient alone, as well as when the driving force is the electrochemical potential difference across the membrane. [ABSTRACT FROM AUTHOR]

We consider two different methods of calculating the relevant average for the non-equilibrium partition identity (NPI), i.e., which result in two different values. At best only one of these will accurately correspond to what is observed. In order to better understand the two outcomes we carry out a detailed error analysis. This analysis is difficult due to the importance of extremely rare events in forming the average, resulting in the necessity to go beyond linear approximations for the error estimates. We begin by analysing the error in the fluctuation relation, and build upon this to estimate the errors in the NPI average. At short durations the full ensemble average always gives the observed average (i.e. the NPI holds). However, at very long durations, given a fixed amount of sampling, the observed average is predicted by treating the probability distribution as a Dirac-delta function. At intermediate times, neither corresponds to the observed average. This has profound implications for non-equilibrium work relations, as first introduced by Jarzynski. [ABSTRACT FROM AUTHOR]

The violations of the second law are relevant as the length and/or time scales become very small. The second law then needs to be replaced by the fluctuation theorem and mathematically, the irreversible entropy is a submartingale. First, we discuss the consequences of these results for the axioms of continuum mechanics, arguing in favor of a framework relying on stochastic functionals of energy and entropy. We next determine a Lyapunov function for diffusion-type problems governed by stochastic rather than deterministic functionals of internal energy and entropy, where the random field coefficients of diffusion are not required to satisfy the positive definiteness everywhere. Next, a formulation of micropolar fluid mechanics is developed, accounting for the lack of symmetry of stress tensor on molecular scales. This framework is then applied to employed to show that spontaneous random fluctuations of the microrotation field will arise in Couette-and Poiseuille-type flows in the absence of random (turbulence-like) fluctuations of the classical velocity field. Finally, while the permeability is classically modeled by the Darcy law or its modifications, besides considering the violations of the second law, one also needs to account for the spatial randomness of the channel network, implying a modification of the hierarchy of scale-dependent bounds on the macroscopic property of the network. [ABSTRACT FROM AUTHOR]

We consider a system of N two-colors urns in which the reinforcement of each urn depends also on the content of all the other urns. This interaction is of mean-field type and it is tuned by a parameter α ∈ [ 0 , 1 ] ; in particular, for α = 0 the N urns behave as N independent Pólya’s urns. For α > 0 urns synchronize, in the sense that the fraction of balls of a given color converges a.s. to the same (random) limit in all urns. In this paper we study fluctuations around this synchronized regime. The scaling of these fluctuations depends on the parameter α . In particular the standard scaling t − 1 / 2 appears only for α > 1 / 2 . For α ≥ 1 / 2 we also determine the limit distribution of the rescaled fluctuations. We use the notion of stable convergence, which is stronger than convergence in distribution. [ABSTRACT FROM AUTHOR]

The first part of the paper is devoted to the foundations, that is the mathematical and physical justification, of equilibrium statistical mechanics. It is a pedagogical attempt, mostly based on Khinchin's presentation, which purpose is to clarify some aspects of the development of statistical mechanics. In the second part, we discuss some recent developments that appeared out of equilibrium, such as fluctuation theorem and Jarzynski equality.

Starting from the most general formulation of stochastic thermodynamics—i.e. a thermodynamically consistent nonautonomous stochastic dynamics describing systems in contact with several reservoirs—we define a procedure to identify the conservative and the minimal set of nonconservative contributions in the entropy production. The former is expressed as the difference between changes caused by time-dependent drivings and a generalized potential difference. The latter is a sum over the minimal set of flux-force contributions controlling the dissipative flows across the system. When the system is initially prepared at equilibrium (e.g. by turning off drivings and forces), a finite-time detailed fluctuation theorem holds for the different contributions. Our approach relies on identifying the complete set of conserved quantities and can be viewed as the extension of the theory of generalized Gibbs ensembles to nonequilibrium situations.

We reformulate stochastic thermodynamics in terms of noise realizations for Langevin systems in contact with multiple reservoirs and investigate the structure of the second laws of thermodynamics. We derive a hierarchy of fluctuation theorems when one degree of freedom of the system is affected by multiple reservoirs simultaneously, that is, when noise-mixing occurs. These theorems and the associated second laws of thermodynamics put stricter bounds on the thermodynamics of Langevin systems. We apply our results to a stochastic machine in noise-mixing environments and demonstrate that our new bounds play a crucial role in determining the potential function and performance of the machine.

Growing economic inequalities are observed in several countries throughout the world. Following Pareto, the power-law structure of these inequalities has been the subject of much theoretical and empirical work. But their nonequilibrium dynamics , e.g. after a policy change, remains incompletely understood. Here we introduce a thermodynamical theory of inequalities based on the analogy between economic stratification and statistical entropy. Within this framework we identify the combination of upward mobility with precariousness as a fundamental driver of inequality. We formalize this statement by a “second-law” inequality displaying upward mobility and precariousness as thermodynamic conjugate variables. We estimate the time scale for the “relaxation” of the wealth distribution after a sudden change of the after-tax return on capital. Our method can be generalized to gain insight into the dynamics of inequalities in any Markovian model of socioeconomic interactions. [ABSTRACT FROM AUTHOR]

Cytoplasmic dynein exhibits a directional processive movement on microtubule filaments and is known to move in steps of varying length based on the number of ATP molecules bound to it and the load that it carries. It is experimentally observed that dynein takes occasional backward steps and the frequency of such backward steps increases as the load approaches the stall force. Using a stochastic process model, we investigate the bidirectional movement of single head of a dynein motor. The probability for backward step is implemented based on fluctuation theorem of non-equilibrium statistical mechanics. We find that the movement of dynein motor is characterized with negative velocity implying backward motion beyond stall force. We observe that the motor moves backward for super stall forces by hydrolyzing the ATP exactly the same way as it does while moving forward for sub-stall forces. Movement of dynein is also simulated using a kinetic Monte Carlo method and the simulated velocities are in good agreement with velocities obtained using a stochastic rate equation model. [ABSTRACT FROM AUTHOR]

Directly observing protein folding in real time using atomic force microscopy (AFM) is challenging. Here the use of AFM to directly monitor the folding of an α/β protein, NuG2, by using low-drift AFM cantilevers is demonstrated. At slow pulling speeds (<50 nm s−1), the refolding of NuG2 can be clearly observed. Lowering the pulling speed reduces the difference between the unfolding and refolding forces, bringing the non-equilibrium unfolding-refolding reactions towards equilibrium. At very low pulling speeds (ca. 2 nm s−1), unfolding and refolding were observed to occur in near equilibrium. Based on the Crooks fluctuation theorem, we then measured the equilibrium free energy change between folded and unfolded states of NuG2. The improved long-term stability of AFM achieved using gold-free cantilevers allows folding-unfolding reactions of α/β proteins to be directly monitored near equilibrium, opening the avenue towards probing the folding reactions of other mechanically important α/β and all-β elastomeric proteins. [ABSTRACT FROM AUTHOR]