Gavin E. Crooks

Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences

There are only a very few known relations in statistical dynamics that are valid for systems driven arbitrarily far-from-equilibrium. One of these is the fluctuation theorem, which places conditions on the entropy production probability distribution of nonequilibrium systems. Another recently discovered far from equilibrium expression relates nonequilibrium measurements of the work done on a system to equilibrium free energy differences. In this paper, we derive a generalized version of the fluctuation theorem for stochastic, microscopically reversible dynamics. Invoking this generalized theorem provides a succinct proof of the nonequilibrium work relation.

Path-ensemble averages in systems driven far from equilibrium

The Kawasaki nonlinear response relation, the transient fluctuation theorem, and the Jarzynski nonequilibrium work relation are all expressions that describe the behavior of a system that has been driven from equilibrium by an external perturbation. In contrast to linear response theory, these expressions are exact no matter the strength of the perturbation, or how far the system has been driven away from equilibrium. In this paper, I show that these three relations (and several other closely related results) can all be considered special cases of a single theorem. This expression is explicitly derived for discrete time and space Markovian dynamics, with the additional assumptions that the unperturbed dynamics preserve the appropriate equilibrium ensemble, and that the energy of the system remains finite.

Self-Organization of the Escherichia Coli Chemotaxis Network Imaged with Super-Resolution Light Microscopy

Photoactivated localization microscopy analysis of chemotaxis receptors in bacteria suggests that the non-random organization of these proteins results from random self-assembly of clusters without direct cytoskeletal involvement or active transport.

Measuring thermodynamic length.

Thermodynamic length is a metric distance between equilibrium thermodynamic states. Among other interesting properties, this metric asymptotically bounds the dissipation induced by a finite time transformation of a thermodynamic system. It is also connected to the Jensen-Shannon divergence, Fisher information, and Raos entropy differential metric. Therefore, thermodynamic length is of central interest in understanding matter out of equilibrium. In this Letter, we will consider how to define thermodynamic length for a small system described by equilibrium statistical mechanics and how to measure thermodynamic length within a computer simulation. Surprisingly, Bennetts classic acceptance ratio method for measuring free energy differences also measures thermodynamic length.

Protein secondary structure: entropy, correlations and prediction

MOTIVATION Is protein secondary structure primarily determined by local interactions between residues closely spaced along the amino acid backbone or by non-local tertiary interactions? To answer this question, we measure the entropy densities of primary and secondary structure sequences, and the local inter-sequence mutual information density. RESULTS We find that the important inter-sequence interactions are short ranged, that correlations between neighboring amino acids are essentially uninformative and that only one-fourth of the total information needed to determine the secondary structure is available from local inter-sequence correlations. These observations support the view that the majority of most proteins fold via a cooperative process where secondary and tertiary structure form concurrently. Moreover, existing single-sequence secondary structure prediction algorithms are almost optimal, and we should not expect a dramatic improvement in prediction accuracy. AVAILABILITY Both the data sets and analysis code are freely available from our Web site at http://compbio.berkeley.edu/Motivation: RNA structure motifs contained in mRNAs have been found to play important roles in regulating gene expression. However, identification of novel RNA regulatory motifs using computational methods has not been widely explored. Effective tools for predicting novel RNA regulatory motifs based on genomic sequences are needed. Results: We present a new method for predicting common RNA secondary structure motifs in a set of functionally or evolutionarily related RNA sequences. This method is based on comparison of stems (palindromic helices) between sequences and is implemented by applying graph-theoretical approaches. It first finds all possible stable stems in each sequence and compares stems pairwise between sequences by some defined features to find stems conserved across any two sequences. Then by applying a maximum clique finding algorithm, it finds all significant stems conserved across at least k sequences. Finally, it assembles in topological order all possible compatible conserved stems shared by at least k sequences and reports a number of the best assembled stem sets as the best candidate common structure motifs. This method does not require prior structural alignment of the sequences and is able to detect pseudoknot structures. We have tested this approach on some RNA sequences with known secondary structures, in which it is capable of detecting the real structures completely or partially correctly and outperforms other existing programs for similar purposes. Availability: The algorithm has been implemented in C++ in a program called comRNA, which is available at http:// ural.wustl.edu/softwares.html Contact: stormo@genetics.wustl.edu

Scaling laws governing stochastic growth and division of single bacterial cells

Significance Growth and division of individual cells are the fundamental events underlying many biological processes, including the development of organisms, the growth of tumors, and pathogen–host interactions. Quantitative studies of bacteria can provide insights into single-cell growth and division but are challenging owing to the intrinsic noise in these processes. Now, by using a unique combination of measurement and analysis technologies, together with mathematical modeling, we discover quantitative features that are conserved across physiological conditions. These universal behaviors reflect the physical principle that a single timescale governs noisy bacterial growth and division despite the complexity of underlying molecular mechanisms. Uncovering the quantitative laws that govern the growth and division of single cells remains a major challenge. Using a unique combination of technologies that yields unprecedented statistical precision, we find that the sizes of individual Caulobacter crescentus cells increase exponentially in time. We also establish that they divide upon reaching a critical multiple (≈1.8) of their initial sizes, rather than an absolute size. We show that when the temperature is varied, the growth and division timescales scale proportionally with each other over the physiological temperature range. Strikingly, the cell-size and division-time distributions can both be rescaled by their mean values such that the condition-specific distributions collapse to universal curves. We account for these observations with a minimal stochastic model that is based on an autocatalytic cycle. It predicts the scalings, as well as specific functional forms for the universal curves. Our experimental and theoretical analysis reveals a simple physical principle governing these complex biological processes: a single temperature-dependent scale of cellular time governs the stochastic dynamics of growth and division in balanced growth conditions.

Thermodynamics of Prediction

A system responding to a stochastic driving signal can be interpreted as computing, by means of its dynamics, an implicit model of the environmental variables. The systems state retains information about past environmental fluctuations, and a fraction of this information is predictive of future ones. The remaining nonpredictive information reflects model complexity that does not improve predictive power, and thus represents the ineffectiveness of the model. We expose the fundamental equivalence between this model inefficiency and thermodynamic inefficiency, measured by dissipation. Our results hold arbitrarily far from thermodynamic equilibrium and are applicable to a wide range of systems, including biomolecular machines. They highlight a profound connection between the effective use of information and efficient thermodynamic operation: any system constructed to keep memory about its environment and to operate with maximal energetic efficiency has to be predictive.

Thermodynamic Metrics and Optimal Paths

A fundamental problem in modern thermodynamics is how a molecular-scale machine performs useful work, while operating away from thermal equilibrium without excessive dissipation. To this end, we derive a friction tensor that induces a Riemannian manifold on the space of thermodynamic states. Within the linear-response regime, this metric structure controls the dissipation of finite-time transformations, and bestows optimal protocols with many useful properties. We discuss the connection to the existing thermodynamic length formalism, and demonstrate the utility of this metric by solving for optimal control parameter protocols in a simple nonequilibrium model.

Length of time's arrow.

An unresolved problem in physics is how the thermodynamic arrow of time arises from an underlying time reversible dynamics. We contribute to this issue by developing a measure of time-symmetry breaking, and by using the work fluctuation relations, we determine the time asymmetry of recent single molecule RNA unfolding experiments. We define time asymmetry as the Jensen-Shannon divergence between trajectory probability distributions of an experiment and its time-reversed conjugate. Among other interesting properties, the length of times arrow bounds the average dissipation and determines the difficulty of accurately estimating free energy differences in nonequilibrium experiments.

Beyond Boltzmann-Gibbs statistics: Maximum entropy hyperensembles out of equilibrium

What is the best description that we can construct of a thermodynamic system that is not in equilibrium, given only one, or a few, extra parameters over and above those needed for a description of the same system at equilibrium? Here, we argue the most appropriate additional parameter is the nonequilibrium entropy of the system. Moreover, we should not attempt to estimate the probability distribution of the system directly, but rather the metaprobability (or hyperensemble) that the system is described by a particular probability distribution. The result is an entropic distribution with two parameters, one a nonequilibrium temperature, and the other a measure of distance from equilibrium. This dispersion parameter smoothly interpolates between certainty of a canonical distribution at equilibrium and great uncertainty as to the probability distribution as we move away from equilibrium. We deduce that, in general, large, rare fluctuations become far more common as we move away from equilibrium.