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PubMed2005

sFFT: a faster accurate computation of the p-value of the entropy score.

We present sFFT, an algorithm for efficiently computing the p-value of the information content, or the entropy score of an alignment of DNA sequences. Applying the FFT algorithm to an exponentially shifted probability mass function allows us perform fast convolutions that do not suffer from the otherwise overwhelming effect of accumulated numerical roundoff errors. Through a rigorous analysis of the propagation of numerical errors across the various steps of sFFT, we provide a theoretical bound on the overall error of our computed p-value. The accuracy of the computed p-value, as well as the utility of the error bound, are empirically demonstrated. Although there are faster algorithms that would compute this p-value, they can err significantly; sFFT is the fastest reliable algorithm. Finally, we note that the basic algorithm is likely to be applicable in a wider context than the one considered here.

Algorithms↗
PubMed2006

Genome-wide tagging SNPs with entropy-based Monte Carlo method.

The number of common single nucleotide polymorphisms (SNPs) in the human genome is estimated to be around 3-6 million. It is highly anticipated that the study of SNPs will help provide a means for elucidating the genetic component of complex diseases and variable drug responses. High-throughput technologies such as oligonucleotide arrays have produced enormous amount of SNP data, which creates great challenges in genome-wide disease linkage and association studies. In this paper, we present an adaptation of the cross entropy (CE) method and propose an iterative CE Monte Carlo (CEMC) algorithm for tagging SNP selection. This differs from most of SNP selection algorithms in the literature in that our method is independent of the notion of haplotype block. Thus, the method is applicable to whole genome SNP selection without prior knowledge of block boundaries. We applied this block-free algorithm to three large datasets (two simulated and one real) that are in the order of thousands of SNPs. The successful applications to these large scale datasets demonstrate that CEMC is computationally feasible for whole genome SNP selection. Furthermore, the results show that CEMC is significantly better than random selection, and it also outperformed another block-free selection algorithm for the dataset considered.

Algorithms↗
PubMed2004

Learning kernels from biological networks by maximizing entropy.

MOTIVATION: The diffusion kernel is a general method for computing pairwise distances among all nodes in a graph, based on the sum of weighted paths between each pair of nodes. This technique has been used successfully, in conjunction with kernel-based learning methods, to draw inferences from several types of biological networks. RESULTS: We show that computing the diffusion kernel is equivalent to maximizing the von Neumann entropy, subject to a global constraint on the sum of the Euclidean distances between nodes. This global constraint allows for high variance in the pairwise distances. Accordingly, we propose an alternative, locally constrained diffusion kernel, and we demonstrate that the resulting kernel allows for more accurate support vector machine prediction of protein functional classifications from metabolic and protein-protein interaction networks. AVAILABILITY: Supplementary results and data are available at noble.gs.washington.edu/proj/maxent

Algorithms↗
PubMed1982

The changes in heat capacity and entropy of troponin C induced by calcium binding.

We have used the method of enthalpy titration to analyze the structural changes of troponin C caused by calcium binding. Successive additions of calcium to metal-free troponin C in a microcalorimeter cell result in at least three distinguishable transitions. Analysis of the results has shown that troponin C has at least three classes of calcium binding sites; one site of highest affinity (binding constant, 10(8)-10(10) M-1), one site of next highest affinity (binding constant, 10(6)-10(7) M-1) and two low affinity sites (binding constant, 10(5)-10(6) M-1). Titrations of troponin C with calcium at various temperatures have shown that calcium binding causes large changes in the heat capacity of troponin C. Following the method of Sturtevant (1977), the magnitudes of the hydrophobic and intramolecular vibrational contributions to the heat capacity and entropy changes of troponin C on calcium binding have been estimated. In Mg-free solutions, calcium binding to the 1st site of highest affinity gives rise to a strong hydrophobic effect and to a tightening of the molecular structure. In contrast, calcium binding to the 2nd site of next highest affinity gives rise to a strong hydrophobic effect in the reverse direction and to a "softening" of the structure. Calcium binding to the two low affinity sites has a moderately strong hydrophobic effect and also causes a moderate tightening of the structure. These results are in many respects similar to those obtained with proton magnetic resonance spectroscopy by Levine et al. (1977). These studies are mutually complementary. When 1 mM magnesium is present the changes caused by calcium binding to the two high affinity sites are greatly altered, whereas those involving the two low affinity sites are not much affected. The moderate tightening of the structure which is caused by calcium binding to the two low affinity sites, and which is seen both in the absence and presence of magnesium, is most likely to be involved in the regulation of contraction.

Animals↗
PubMed1999

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.

Journal Article↗
PubMed2000

Fluctuation theorems for entropy production in open systems

We derive a fluctuation theorem to describe entropy fluctuations in steady states of systems with density gradients due to open boundaries. The fluctuations are related to the growth rate of the phase-space density, instead of the phase-space contraction rate. Explicit derivations are presented for a multibaker map, but the arguments are rather general, and should hold for a much wider class of dynamical systems. A comparison with recent results for stochastic systems is also given.

Journal Article↗
PubMed2000

Cluster diversity and entropy on the percolation model: the lattice animal identification algorithm

We present an algorithm to identify and count different lattice animals (LA's) in the site-percolation model. This algorithm allows a definition of clusters based on the distinction of cluster shapes, in contrast with the well-known Hoshen-Kopelman algorithm, in which the clusters are differentiated by their sizes. It consists in coding each unit cell of a cluster according to the nearest neighbors (NN) and ordering the codes in a proper sequence. In this manner, a LA is represented by a specific code sequence. In addition, with some modification the algorithm is capable of differentiating between fixed and free LA's. The enhanced Hoshen-Kopelman algorithm [J. Hoshen, M. W. Berry, and K. S. Minser, Phys. Rev. E 56, 1455 (1997)] is used to compose the set of NN code sequences of each cluster. Using Monte Carlo simulations on planar square lattices up to 2000x2000, we apply this algorithm to the percolation model. We calculate the cluster diversity and cluster entropy of the system, which leads to the determination of probabilities associated with the maximum of these functions. We show that these critical probabilities are associated with the percolation transition and with the complexity of the system.

Journal Article↗
PubMed2000

Tsallis maximum entropy principle and the law of large numbers

Tsallis has suggested a nonextensive generalization of the Boltzmann-Gibbs entropy, the maximization of which gives a generalized canonical distribution under special constraints. In this Brief Report, we show that the generalized canonical distribution so obtained may differ from that predicted by the law of large numbers when empirical samples are held to the same constraint. This conclusion is based on a result regarding the large deviation property of conditional measures and is confirmed by numerical evidence.

Journal Article↗
PubMed2000

Entropy, dynamics, and instantaneous normal modes in a random energy model

It is shown that the fraction f(u) of imaginary-frequency instantaneous normal modes (INM) may be defined and calculated in a random energy model (REM) of liquids. The configurational entropy S(c) and the averaged hopping rate among the states, R, are also obtained and related to f(u) with the results R approximately f(u) and S(c)=a+b ln(f(u)). The proportionality between R and f(u) is the basis of existing INM theories of diffusion, so the REM further confirms their validity. A link to S(c) opens new avenues for introducing INM into dynamical theories. Liquid states are usually defined by assigning a configuration to the minimum to which it will drain, but the REM naturally treats saddle barriers on the same footing as minima, which may be a better mapping of the continuum of configurations to discrete states. Requirements for a detailed REM description of liquids are discussed.

Journal Article↗
PubMed2000

Steady adiabatic state: its thermodynamics, entropy production, energy dissipation, and violation of onsager relations

A class of statistical systems is considered where different degrees of freedom have well-separated characteristic times, and are described by different temperatures. The stationary state is a nonequilibrium state with a heat flow. A generalized statistical thermodynamics is constructed and a universal variational principle is proposed. Entropy production and energy dissipation occur at a constant rate. To leading order in the small ratio of the characteristic times, there exists a universal relation between them. Onsager relations in the context of heat transfer are also considered. They are always broken, except close to equilibrium.

Journal Article↗
PubMed2001

Entropy fluctuations for directed polymers in 2+1 dimensions.

We find numerically that the sample to sample fluctuation of the entropy DeltaS is a more sensitive tool in distinguishing low from high temperature behaviors than the common corresponding fluctuation in the free energy. In 1+1 dimensions we find a single phase for all temperatures, since (DeltaS)(2) is always extensive. In 2+1 dimensions we find a behavior that at first sight might appear to be a transition from a low temperature phase where (DeltaS)(2) is extensive to a high temperature phase where it is subextensive. This is observed in spite of the relatively large system we use. The observed behavior is explained not as a phase transition but as a strong crossover behavior. We use an analytical argument to obtain (DeltaS)(2) for high temperature, and find that while it is always extensive it is also extremely small, and that the leading extensive part decays very quickly to zero as the temperature is increased.

Journal Article↗
PubMed2001

Relative entropy: free energy associated with equilibrium fluctuations and nonequilibrium deviations.

Using a one-dimensional macromolecule in aqueous solution as an illustration, we demonstrate that the relative entropy from information theory, Sigma(k)p(k) ln(p(k)/p(*)(k)), has a natural role in the energetics of equilibrium and nonequilibrium conformational fluctuations of the single molecule. It is identified as the free energy difference associated with a fluctuating density in equilibrium, and is associated with the distribution deviate from the equilibrium in nonequilibrium relaxation. This result can be generalized to any other isothermal macromolecular system using the mathematical theories of large deviations and Markov processes, and at the same time provides the well-known mathematical results with interesting physical interpretations.

Journal Article↗
PubMed2001

Sticky spheres, entropy barriers, and nonequilibrium phase transitions.

A sticky spheres model to describe slow dynamics of a nonequilibrium system is proposed. The dynamical slowing down is due to the presence of entropy barriers. An exact steady state analysis of the representative mean field equations, in the case when the clusters are chosen with the same a priori probability, demonstrates a nonequilibrium phase transition from an exponential cluster size distribution to a power law.

Journal Article↗
PubMed2001

Noise robust estimates of correlation dimension and K2 entropy.

Using Gaussian kernels to define the correlation sum we derive simple formulas that correct the noise bias in estimates of the correlation dimension and K2 entropy of chaotic time series. The corrections are only based on the difference of correlation dimensions for adjacent embedding dimensions and hence preserve the full functional dependencies on both the scale parameter and embedding dimension. It is shown theoretically that the estimates, which are derived for additive white Gaussian noise, are also robust for moderately colored noise. Simulations underline the usefulness of the proposed correction schemes. It is demonstrated that the method gives satisfactory results also for non-Gaussian and dynamical noise.

Journal Article↗
PubMed2001

Noise properties of stochastic processes and entropy production.

Based on a Fokker-Planck description of external Ornstein-Uhlenbeck noise and cross-correlated noise processes driving a dynamical system we examine the interplay of the properties of noise processes and the dissipative characteristic of the dynamical system in the steady state entropy production and flux. Our analysis is illustrated with appropriate examples.

Journal Article↗
PubMed2001

Decoherence in a classically chaotic quantum system: entropy production and quantum-classical correspondence.

We study the decoherence process for an open quantum system that is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators). We carefully analyze the time dependence of the rate of entropy production showing that it has two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system-environment coupling strength); for longer times (but before equilibration) it is fixed by dynamical properties of the system (and is related to the Lyapunov exponent). The nature of the transition time between both regimes is investigated and the issue of quantum to classical correspondence is addressed. Finally, the impact of the interaction with the environment on coherent tunneling is analyzed.

Journal Article↗
PubMed2002

Upper bound for the time derivative of entropy for nonequilibrium stochastic processes.

We have shown how the intrinsic properties of a noise process can set an upper bound for the time derivative of entropy in a nonequilibrium system. The interplay of dissipation and the properties of noise processes driving the dynamical systems in presence and absence of external forces, reveals some interesting extremal nature of the upper bound.

Journal Article↗
PubMed2002

Temperature-entropy formulation of thermoelectric thermodynamic cycles.

A temperature-entropy formulation is derived for thermoelectric devices. Thermoelectric chiller and generator cycles can then be cast in the same irreversible thermodynamics framework commonly applied to conventional large-scale cooling and power generation equipment, including a transparent identification of the principal energy flows and performance bottlenecks (dissipation). Distinct differences in chiller versus generator mode are highlighted and illustrated with data from commercial thermoelectric units.

Journal Article↗