Publications

The Jarzynski equality and the fluctuation theorem relate equilibrium free energy differences to nonequilibrium measurements of the work. These relations extend to single-molecule experiments that have probed the finite-time thermodynamics of proteins and nucleic acids. The effects of experimental error and instrument noise have not been considered previously. Here, we present a Bayesian formalism for estimating free energy changes from nonequilibrium work measurements that compensates for instrument noise and combines data from multiple driving protocols. We reanalyze a recent set of experiments in which a single RNA hairpin is unfolded and refolded using optical tweezers at three different rates. Interestingly, the fastest and farthest-from-equilibrium measurements contain the least instrumental noise and, therefore, provide a more accurate estimate of the free energies than a few slow, more noisy, near-equilibrium measurements. The methods we propose here will extend the scope of single-molecule experiments; they can be used in the analysis of data from measurements with atomic force microscopy, optical, and magnetic tweezers.

The dynamics of an open quantum system can be described by a quantum operation, a linear, complete positive map of operators. Here, I exhibit a compact expression for the time reversal of a quantum operation, which is closely analogous to the time reversal of a classical Markov transition matrix. Since open quantum dynamics are stochastic, and not, in general, deterministic, the time reversal is not, in general, an inversion of the dynamics. Rather, the system relaxes towards equilibrium in both the forward and reverse time directions. The probability of a quantum trajectory and the conjugate, time reversed trajectory are related by the heat exchanged with the environment.

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 Rao's entropy differential metric. Therefore, thermodynamic length is of central interest in understanding matter out-of-equilibrium. In this paper, 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, Bennett's classic acceptance ratio method for measuring free energy differences also measures thermodynamic length.

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.

We consider a simple, physically motivated model of a dilute classical gas of interacting particles, initially equilibrated with a heat bath, undergoing adiabatic and quasistatic compression or expansion. This provides an example of a thermodynamic process for which non-Gaussian work fluctuations can be computed exactly from microscopic principles. We find that the work performed during this process is described statistically by a gamma distribution, and we use this result to show that the model satisfies the nonequilibrium work and fluctuation theorems, but not a prediction based on linear response theory.

Background: Benchmarking algorithms in structural bioinformatics often involves the construction of datasets of proteins with given sequence and structural properties. The SCOP database is a manually curated structural classification which groups together proteins on the basis of structural similarity. The ASTRAL compendium provides non redundant subsets of SCOP domains on the basis of sequence similarity such that no two domains in a given subset share more than a defined degree of sequence similarity. Taken together these two resources provide a 'ground truth' for assessing structural bioinformatics algorithms. We present a small and easy to use API written in python to enable construction of datasets from these resources. Results: We have designed a set of python modules to provide an abstraction of the SCOP and ASTRAL databases. The modules are designed to work as part of the Biopython distribution. Python users can now manipulate and use the SCOP hierarchy from within python programs, and use ASTRAL to return sequences of domains in SCOP, as well as clustered representations of SCOP from ASTRAL. Conclusion: The modules make the analysis and generation of datasets for use in structural genomics easier and more principled.

Motivation: Protein sequence comparison methods are routinely used to infer the intricate network of evolutionary relationships found within the rapidly growing library of protein sequences, and thereby to predict the structure and function of uncharacterized proteins. Here, we detail an improved statistical benchmark of pairwise protein sequence comparison algorithms. We use bootstrap resampling techniques to determine standard statistical errors, and to estimate the confidence of our conclusions. We show that the underlying structure within benchmark databases causes Efron's standard, nonparametric bootstrap to be biased. Consequently, the standard bootstrap under-predicts average performance when used in the context of evaluating sequence comparison methods. As an alternative, we have developed an unbiased statistical evaluation based upon the Bayesian bootstrap, a resampling method operationally similar to the standard bootstrap.
Results: We apply our analysis to the comparative study of amino acid substitution matrix families and find that using modern matrices results in a small, but statistically significant improvement in remote homology detection compared to the classic PAM and BLOSUM matrices.

Motivation: Standard algorithms for pairwise protein sequence alignment make the simplifying assumption that amino acid substitutions at neighboring sites are uncorrelated. This assumption allows implementation of fast algorithms for pairwise sequence alignment, but it ignores information that could conceivably increase the power of remote homolog detection. We examine the validity of this assumption by constructing extended substitution matrixes that encapsulate the observed correlations between neighboring sites, by developing an efficient and rigorous algorithm for pairwise protein sequence alignment that incorporates these local substitution correlations, and by assessing the ability of this algorithm to detect remote homologies.
Results: Our analysis demonstrates that local correlations between substitutions are not strong on the average. Consequentially, incorporating local substitution correlations into pairwise alignment does not lead to a statistically significant improvement in remote homology detection. Therefore, the standard assumption (that individual residues within protein sequences evolve independent of neighboring positions) is an efficient and appropriate approximation.

Sequence alignment underpins common tasks in molecular biology, including genome annotation, molecular phylogenetics, and homology modelling. Fundamental to sequence alignment is the placement of gaps, which represent character insertions or deletions. We assessed the ability of a generalized affine gap cost model to reliably detect remote protein homology and to produce high-quality alignments. Generalized affine gap alignment with optimal gap parameters performed as well as the traditional affine gap model in remote homology detection. Evaluation of alignment quality showed that the generalized affine model aligns fewer residue pairs than the traditional affine model but achieves significantly higher per-residue accuracy. We conclude that generalized affine gap costs should be used when alignment accuracy carries more importance than aligned sequence length.

We propose an alternative model of amino acid replacement during protein evolution. The observed correlations between pairs of homologous protein sequences are typically explained in terms of a Markovian dynamic of amino acid substitution. This model assumes that every location on the protein sequence has the same background distribution of amino acids, an assumption that is incompatible with the observed heterogeneity of protein amino acid profiles and with the success of profile multiple sequence alignment. On investigation, we find that the variation of the amino acid background distribution from one residue to the next is sufficient to explain the observed sequence correlations of homologs. The resulting dynamical model of independent replacements drawn from heterogeneous backgrounds is simple and consistent, and provides a unified homology match score for sequence-sequence, sequence-profile and profile-profile alignment.

Most natural processes occur far from equilibrium and cannot be treated within the framework of classical thermodynamics. In 1998, Oono and Paniconi [Oono, Y. & Paniconi, M. (1998) Prog. Theor. Phys. Suppl. 130, 29-44] proposed a general phenomenological framework, steady-state thermodynamics, encompassing nonequilibrium steady states and transitions between such states. In 2001, Hatano and Sasa [Hatano, T. & Sasa, S. (2001) Phys. Rev. Lett. 86, 3463-3466] derived a testable prediction of this theory. Specifically, they were able to show that the exponential average of Y, a quantity similar to a dissipated work, should be equal to zero for arbitrary transitions between nonequilibrium steady states, -lne-Y = 0. We have tested this strong prediction by measuring the dissipation and fluctuations of microspheres optically driven through water. We have found that -lne-Y 0 for three different nonequilibrium systems, supporting Hatano and Sasa's proposed extension of thermodynamics to arbitrary steady states and irreversible transitions.

Correlations between protein structures and amino acid sequences are widely used for protein structure prediction. For example, secondary structure predictors generally use correlations between a secondary structure sequence and corresponding primary structure sequence, whereas threading algorithms, and similar tertiary structure predictors, typically incorporate interresidue contact potentials. To investigate the relative importance of these interactions we measured the mutual information between the primary structure, secondary structure and side-chain surface exposure, both for adjacent residues along the amino acid sequence, and for tertiary structure contacts between residues distantly separated along the backbone. We find that local interactions along the amino acid chain are far more important than non-local contacts, and that correlations between proximate amino acids are essentially uninformative. This suggests that knowledge-based contact potentials may be less important for structure predication than is generally believed.

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 have measured the entropy densities of primary structure and secondary structure sequences, and the local inter-sequence mutual information density. We find that the important inter-sequence interactions are short ranged, that correlations between neighboring amino acids are essentially uninformative, and that only 1/4 of the total information needed to determine the secondary structure is available from local inter-sequence correlations. Since the remaining information must come from non-local interactions, this observation supports the view that the majority of most proteins fold via a cooperative process where secondary and tertiary structure form concurrently. To provide a more direct comparison to existing secondary structure prediction methods, we construct a simple hidden Markov model (HMM) of the sequences. This HMM achieves a prediction accuracy comparable to other single sequence secondary structure prediction algorithms, and can extract almost all of the inter-sequence mutual information. This suggests that these algorithms are almost optimal, and that we should not expect a dramatic improvement in prediction accuracy. However, local correlations between secondary and primary structure are probably of under-appreciated importance in many tertiary structure prediction methods, such as threading.

WebLogo (http://weblogo.berkeley.edu/) generates sequence logos, graphical representations of the patterns within a multiple sequence alignment. Sequence logos provide a richer and more precise description of sequence similarity than consensus sequences and can rapidly reveal significant features of the alignment otherwise difficult to perceive. Each logo consists of stacks of letters, one stack for each position in the sequence. The overall height of each stack indicates the sequence conservation at that position (measured in bits), while the height of symbols within the stack reflects the relative frequency of the corresponding amino or nucleic acid at that position. WebLogo has recently been enhanced with additional features and options, to provide a convenient and highly configurable sequence logo generator. A command line interface and the complete, open, WebLogo source code are available for local installation and customization.

The transition path sampling methodology is adapted to the efficient sampling of large fluctuations in nonequilibrium systems evolving according to Langevin's equations of motion. This technique is used to simulate the behavior of the bistable Maier-Stein system at noise intensities much lower than those previously possible.

Cellular automata can be designed that allow the simulation of a large variety of polymer problems including isolated polymers in dilute solution, polymers in high density melts and polymers embedded in media. The two-space algorithm is a particularly efficient algorithm for polymer simulation that is easy to implement and generalize on both conventional serial hardward and Cellular Automaton (CA) Machines. We describe the implementation of this algorithm and two applications: two dimensions (2-D) melts and polymer collapse. Simulations of high density melts in 2-D show that contrary to expectations polymers do not segregate at high density, there is significant interpenetration as there is in 3-D. Polymer collapse is studied in the regime far from equilibrium. Collapse is found to be dominated by migration of the chain ends. The kinetic process of collapse can systematically and reproducibly restrict the possible conformations that are explored during protein folding. This suggests that the kinetics of collapse may help lead to the desired folded conformation of proteins.

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 single time step dynamics preserve the appropriate equilibrium ensemble, and that the energy of the system remains finite.

We investigate the internal structure of a polymer during collapse from an expanded coil to a compact globule. Collapse is more probable in local regions of high curvature, so a smoothing of the fractal polymer structure occurs that proceeds systematically from the shortest to the longest length scales. A proposed universal scaling relationship is tested by comparison with Monte Carlo simulations. We speculate that the universal form applies to various fractal systems with local processes that promote smoothness over time. The results complement earlier work showing that on the macroscale polymer collapse proceeds by driven diffusion of the polymer ends.

There are only a very few know 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.

An equality has recently been shown relating the free energy difference between two equilibrium ensembles of a system, and an ensemble average of the work required to switch between these two configurations. In the current paper it is shown that this result can be derived under the assumption that the system's dynamics are Markovian and microscopically reversible.

Errata: The final line of this paper should read "This work was supported by NSF grant number CHE-9508336."

By computer simulation of the hard sphere fluid, we have determined the probabilities of observing N molecular centers within molecular sized volumes of the fluid. These probability distributions are found to be almost exactly Gaussian at medium densities. A maximum entropy prediction constructed from knowledge of the first two moments of the distribution and a prior distribution consistent with ideal gas behavior quantitatively predicts the occupation distribution for low and medium densities.

Errata: The lines in Fig. 3 are mislabeled. The dashed line is the fit to the data using an uninformative prior, while the solid line is a fit using the ideal gas prior. The final line of this paper should read "This work was supported by NSF grant number CHE-9508336."

Lipolase and Lipozyme are produced in large quantities (as a result of genetic engineering and overexpression) for the detergents market and provide a cheap source of highly active biocatalysts. Humicola lanuginosa lipase (HIL) and Rhizomucor miehei lipase (RmL) have been isolated in partially purified form from commercial preparations of Lipolase and Lipozyme, respectively. These lipases were solubilized in Aerosol-OT (AOT)-stabilized water-in-oil (w/o) microemulsions in n-heptane. HIL and RmL activity in these microemulsions was assayed by spectrophotometric measurement of the initial rate of p-nitrophenyl butyrate hydrolysis, and by chromatographic determination of the initial rate of octyl decanoate synthesis from l-octanol and decanoic acid. The hydrolytic activity of RmL in microemulsions measured as a function of buffer pH prior to dispersal, followed a sigmoidal profile with the highest activities observed at alkaline pi-is. This broadly matches the pH-activity profile for tributyrin hydrolysis by Lipolase in an aqueous emulsion assay. The hydrolytic activity of RmL in the same microemulsions, measured as a function of pH, gave a bell-shaped profile with a maximum activity at pH 7.5. Again, the observed pH-activity profile was similar to that reported for a purified RmL in a tributyrin-based aqueous emulsion assay. In contrast, the esterification activity exhibited by both HIL and RmL in AOT microemulsions over the available range pH 6.1 to 10.4, decreases as the pH increases, most likely reflecting the effect of substrate ionization. The dependence of the hydrolytic and condensation activity of HIL on R, the mole ratio of water to surfactant, were similar with both profiles exhibiting a maximum at R = 5. The hydrolytic and esterification activities of RmL followed similar R-dependent profiles, but the profiles in this case exhibited a maximum at R = 10. The water activities at these R values were directly measured as 0.78 and 0.9, respectively. Measured water activities were unperturbed by the presence of lipase at the concentrations used in these studies.

There are only a very few known relations in statistical dynamics that are valid for systems driven arbitrarily far away from equilibrium by an external perturbation. 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 contrast to linear response theory, these expressions are exact no matter the strength of the perturbation, or how far the system has been driven from equilibrium. In this work I show that these 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.

These theoretical results indicate that the most interesting nonequilibrium phenomena will be observed during rare excursions of the system away from the stable states. However, direct simulation of the dynamics is inherently inefficient, since the majority of the computation time is taken watching small, uninteresting fluctuations about the stable states. Transition path sampling has been developed as a Monte Carlo algorithm to efficiently sample rare transitions between stable or metastable states in equilibrium systems. Here, the transition path sampling methodology is adapted to the efficient sampling of large fluctuations in nonequilibrium systems evolving according to Langevin's equations of motion. Simulations are then used to study the behavior of the Maier-Stein system, an important model for a large class of nonequilibrium systems. Path sampling is also implemented for the kinetic Ising model, which is then employed to study surface induced evaporation.

This thesis describes studies of the catalytic activity of Humicola lanuginosa Lipase (HlL) and Rhizomucor miehei Lipase (RmL) in AOT stabilised Water in Oil microemulsions. The activities of the enzymes were assayed by monitoring the rate of hydrolysis of p-nitrophenylbutyrate (pNPC4), using UV spectrophotometry, and by the synthesis of octyldecanoate from 1-octanol and decanoic acid, monitored by HPLC.

The hydrolytic activity of HlL in microemulsions exhibited a classical sigmodal dependence of activity on the pH of the aqueous phase (measured before dispersal in the microemulsion) indicating that the activity of the enzyme is affected by a single ionisation, with an apparent pKa of 7. Maximum activity occurred at alkaline pHs (pH>=9.3). In a comparable study of the effect of pH on the rate of synthesis of octyldecanoate under similar conditions also showed a maximum at alkali pHs. The dependence of HlL activity on the R-value of the microemulsion exhibited a maximum at R 5 for both pNPC4 hydrolysis and octyldecanoate synthesis. The stability of the enzyme over 1 month was good for all pHs (6.1, 7.2, 9.3) and R-values studied (R=5, 7.5, 10, 20) except when high pHs and low R values were combined. (pH=9.3, R=5 & 7.5)

The dependence of pNPC4 hydrolysis by RmL on the pH of the water phase showed a peak at pH 7.5, indicating that the activity of the enzyme is affected by two ionisations, with apparent pKas of 6 & 9. In contrast the pH dependence of RmL assayed by octyldecanoate synthesis showed no loss of activity at high pHs. The reason for the difference in pH dependencies is not known at present. The R dependence of RmL activity showed a peak in activity at R 10 for both octyldecanoate synthesis and pNPC4 hydrolysis. The stability of the enzyme in various microemulsions was monitored over 1 month. With respect to the R-value the enzyme was most stable at R=10. The stability decreased with increasing pH.

The activities of the enzyme preparations were calibrated against a standard tributyrin emulsion assay. Comparison of the maximum observed activities of HlL and RmL in microemulsions, compared to the tributyrin assay, indicate that RmL is more active towards both pNPC4 hydrolysis (~*20) and octyldecanoate synthesis (~*35) than HlL.

Jeffreys' J-divergence and the Jensen-Shannon divergence are shown to be related by an inequality that involves a transcendental function of the Jeffreys divergence.

The Amoroso distribution is the natural unification of the generalized gamma and generalized extreme value families of distributions. At least 40 distinct, named distributions (and as many synonyms) occur as special cases or limiting forms. Consequentially, this single simple functional form encapsulates and systematizes an extensive menagerie of interesting and common probability distributions.