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PubMed2006

Variance estimators for the Lempel-Ziv entropy rate estimator.

Lempel and Ziv's 1976 algorithm provides an easy-to-compute way to automatically estimate the entropy rate for symbolic time series, requiring no free parameters. Here we derive an analytical variance estimate for the Lempel-Ziv entropy rate estimator that is easily computable from observations with negligible extra effort beyond the entropy rate itself, and compare to another procedure, a time-series-based bootstrap method. These provide a justified "error bar" quantifying the size of expected fluctuations on the estimate itself, given by the single time series of symbols.

Journal Article↗
PubMed2006

Excess-entropy-based anomalies for a waterlike fluid.

Many thermodynamic and dynamic properties of water display unusual behavior at low enough temperatures. In a recent study, Yan et al. [Phys. Rev. Lett. 95, 130604 (2005)] identified a spherically symmetric two-scale potential that displays many of the same anomalous properties as water. More specifically, for select parametrizations of the potential, one finds that the regions where isothermal compression anomalously (i) decreases the fluid's structural order, (ii) increases its translational self-diffusivity, and (iii) increases its entropy form nested domes in the temperature-density plane. These property relationships are similar to those found for more realistic models of water. In this work, the authors provide evidence that suggests that the anomalous regions specified above can all be linked through knowledge of the excess entropy. Specifically, the authors show how entropy scaling relationships developed by Rosenfeld [Phys. Rev. A 15, 2545 (1977)] can be used to describe the region of diffusivity anomalies and to predict the state conditions for which anomalous viscosity and thermal conductivity behavior might be found.

Journal Article↗
PubMed2005

Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive.

Phase space can be constructed for N equal and distinguishable subsystems that could be probabilistically either weakly correlated or strongly correlated. If they are locally correlated, we expect the Boltzmann-Gibbs entropy S(BG) identical with -k Sigma(i) p(i) ln p(i) to be extensive, i.e., S(BG)(N) proportional, variant N for N --> infinity. In particular, if they are independent, S(BG) is strictly additive, i.e., S(BG)(N) = NS(BG)(1), for allN. However, if the subsystems are globally correlated, we expect, for a vast class of systems, the entropy S(q) identical with k[1 - Sigma(i) p(q)(i)]/(q - 1) (with S(1) = S(BG)) for some special value of q not equal 1 to be the one which is extensive [i.e., S(q)(N) proportional, variant N for N --> infinity]. Another concept which is relevant is strict or asymptotic scale-freedom (or scale-invariance), defined as the situation for which all marginal probabilities of the N-system coincide or asymptotically approach (for N --> infinity) the joint probabilities of the (N - 1)-system. If each subsystem is a binary one, scale-freedom is guaranteed by what we hereafter refer to as the Leibnitz rule, i.e., the sum of two successive joint probabilities of the N-system coincides or asymptotically approaches the corresponding joint probability of the (N - 1)-system. The kinds of interplay of these various concepts are illustrated in several examples. One of them justifies the title of this paper. We conjecture that these mechanisms are deeply related to the very frequent emergence, in natural and artificial complex systems, of scale-free structures and to their connections with nonextensive statistical mechanics. Summarizing, we have shown that, for asymptotically scale-invariant systems, it is S(q) with q not equal 1, and not S(BG), the entropy which matches standard, clausius-like, prescriptions of classical thermodynamics.

Journal Article↗
PubMed2003

Extending the definition of entropy to nonequilibrium steady states.

We study the nonequilibrium statistical mechanics of a finite classical system subjected to nongradient forces xi and maintained at fixed kinetic energy (Hoover-Evans isokinetic thermostat). We assume that the microscopic dynamics is sufficiently chaotic (Gallavotti-Cohen chaotic hypothesis) and that there is a natural nonequilibrium steady-state rho(xi). When xi is replaced by xi + deltaxi, one can compute the change deltarho of rho(xi) (linear response) and define an entropy change deltaS based on energy considerations. When xi is varied around a loop, the total change of S need not vanish: Outside of equilibrium the entropy has curvature. However, at equilibrium (i.e., if xi is a gradient) we show that the curvature is zero, and that the entropy S(xi + deltaxi) near equilibrium is well defined to second order in deltaxi.

Journal Article↗
PubMed1989

Role of coupling entropy in establishing the nature and magnitude of allosteric response.

The coupling free energy between an allosteric ligand and a substrate, delta Gax, is an explicit measure of the nature as well as the magnitude of impact that an allosteric ligand has on the binding of the substrate ligand to the enzyme, with positive values indicating inhibition and negative values indicating activation. By measuring the variation with temperature of the coupling free energy between the allosteric ligand and the substrate, it is possible to determine the enthalpic and entropic components that give rise to the coupling free energy. We have performed this analysis on two different K-type allosteric systems: the allosteric inhibition of rat liver phosphofructokinase by MgATP, and the allosteric activation of beef heart NAD+-dependent isocitrate dehydrogenase by ADP. In both cases the coupling free energy arises as the net result of opposing enthalpic and entropic components, with the coupling enthalpy (delta Hax) favoring activation and the coupling entropy (delta Sax) favoring inhibition. For phosphofructokinase at 25 degrees C, the absolute value of T delta Sax is greater than the absolute value of delta Hax, and net inhibition of rat liver phosphofructokinase by MgATP is realized. For isocitrate dehydrogenase, delta Hax dominates; however, the net activation is substantially mitigated by the magnitude of T delta Sax. Hence, the coupling entropy plays an important role in establishing both the nature and magnitude of the allosteric response. We hypothesize that the negative coupling entropy arises from the particular constraint placed upon the internal dynamical properties of the enzyme by the simultaneous binding of both allosteric and substrate ligands.

Allosteric Regulation↗
PubMed1992

Side-chain entropy opposes alpha-helix formation but rationalizes experimentally determined helix-forming propensities.

In recent host-guest studies, the helix-forming tendencies of amino acid residues have been quantified by three groups, each obtaining similar results [Padmanabhan, S., Marqusee, S., Ridgeway, T., Laue, T. M. & Baldwin, R. L. (1990) Nature (London) 344, 268-270; O'Neil, K. T. & DeGrado, W. F. (1990) Science 250, 646-651; Lyu, P. C., Liff, M. I., Marky, L. A. & Kallenbach, N. R. (1990) Science 250, 669-673]. Here, we explore the hypothesis that these measured helix-forming propensities are due primarily to conformational restrictions imposed upon residue side chains by the helix itself. This proposition is tested by calculating the extent to which the bulky helix backbone "freezes out" available degrees of freedom in helix side chains. Specifically, for a series of apolar residues, the difference in configurational entropy, delta S, between each side chain in the unfolded state and in the alpha-helical state is obtained from a simple Monte Carlo calculation. These computed entropy differences are then compared with the experimentally determined values. Measured and calculated values are found to be in close agreement for naturally occurring amino acids and in total disagreement for non-natural amino acids. In the calculation, delta S(Ala) = 0. The rank order of entropy loss for the series of natural apolar side chains under consideration is Ala less than Leu less than Trp less than Met less than Phe less than Ile less than Tyr less than Val. Among these, none favor helix formation; Ala is neutral, and all remaining residues are unfavorable to varying degrees. Thus, applied to side chains, the term "helix preference" is a misnomer. While side chain-side chain interactions may modulate stability in some instances, our results indicate that the drive to form helices must originate in the backbone, consistent with Pauling's view of four decades ago [Pauling, L., Corey, R. B. & Branson, H. R. (1951) Proc. Natl. Acad. Sci. USA 37, 205-210].

Amino Acids↗
PubMed1998

An entropy criterion to detect minimally frustrated intermediates in native proteins.

The analysis of the information flow in a feed-forward neural network suggests that the output of the network can be used to compute a structural entropy for the sequence-to-secondary structure mapping. On this basis, we formulate a minimum entropy criterion for the identification of minimally frustrated traits with helical conformation that correspond to initiation sites of protein folding. The entropy of protein segments can be viewed as a nucleation propensity that is useful to characterize putative regions where folding is likely to be initiated with the formation of stretches of alpha-helices under the predominant influence of local interactions. Our procedure is successfully tested in the search for initiation sites of protein folding for which independent experimental and computational evidence exists. Our results lend support to the view that folding is a hierarchical event in which, in harmony with the minimal frustration principle, the final conformation preserves structural modules formed in the early stages of the process.

Proteins↗
PubMed2002

Vector entropy imaging theory with application to computerized tomography.

Medical imaging theory for x-ray CT and PET is based on image reconstruction from projections. In this paper a novel vector entropy imaging theory under the framework of multiple criteria decision making is presented. We also study the most frequently used image reconstruction methods, namely, least square, maximum entropy, and filtered back-projection methods under the framework, of the single performance criterion optimization. Finally, we introduce some of the results obtained by various reconstruction algorithms using computer-generated noisy projection data from the Hoffman phantom and real CT scanner data. Comparison of the reconstructed images indicates that the vector entropy method gives the best in error (difference between the original phantom data and reconstruction), smoothness (suppression of noise), grey value resolution and is free of ghost images.

Algorithms↗
PubMed2004

Maximum entropy modeling of short sequence motifs with applications to RNA splicing signals.

We propose a framework for modeling sequence motifs based on the maximum entropy principle (MEP). We recommend approximating short sequence motif distributions with the maximum entropy distribution (MED) consistent with low-order marginal constraints estimated from available data, which may include dependencies between nonadjacent as well as adjacent positions. Many maximum entropy models (MEMs) are specified by simply changing the set of constraints. Such models can be utilized to discriminate between signals and decoys. Classification performance using different MEMs gives insight into the relative importance of dependencies between different positions. We apply our framework to large datasets of RNA splicing signals. Our best models out-perform previous probabilistic models in the discrimination of human 5' (donor) and 3' (acceptor) splice sites from decoys. Finally, we discuss mechanistically motivated ways of comparing models.

Base Sequence↗
PubMed2001

Onset of propofol-induced burst suppression may be correctly detected as deepening of anaesthesia by approximate entropy but not by bispectral index.

The bispectral index (BIS) is a complex EEG variable that combines several disparate descriptors of the EEG into a single value. Approximate entropy is a novel EEG measure that quantifies the regularity of a data time series such as EEG. We report two patients in which the EEG effect of propofol was quantified very similarly by BIS and approximate entropy. However, at the beginning of burst suppression of the EEG, BIS did not indicate an increased anaesthetic drug effect, while approximate entropy did.

Adult↗
PubMed1993

On the stability of nucleic acid structures in solution: enthalpy-entropy compensations, internal rotations and reversibility.

The thermodynamics of self-association (stacking) of free bases and nucleotides, intramolecular stacking in dinucleotides, nearest-neighbour base pair stacking interactions in duplex DNA and RNA, and the formation of hairpin loops illustrate enthalpy/entropy compensations. Large stacking exothermicities are associated with large negative entropy changes that ensure that delta G is small, permitting readily reversible associations in solution. We rationalise enthalpy/entropy compensations with reference to residual motions and torsional vibrations which make a larger entropic contribution to binding when - delta H approximately kT (thermal energy at room temperature), than when - delta H >> kT. We present a factorisation of experimental free energies for helix formation in terms of approximate contributions from the restriction of rotations, hydrophobic interactions, electrostatic interactions due to base stacking, and contributions from hydrogen bonding, and estimate the adverse free energy cost per rotor (mainly entropy) of ordering the phosphate backbone as between 1.9 and 5.4 kJ mol-1 [averaged over 12 rotors per base pair for A-U on A-U stacking (lower limit), and G-C on C-G stacking (upper limit)]. The largest cost is associated with the most exothermic stacking interactions, while the range of values is consistent with earlier conclusions from data on the fusion of hydrocarbon chains (lower value), and with entropy changes in covalent isomerisations of small molecules involving severe restrictions (upper value).

Base Composition↗
PubMed1981

Enthalpy and entropy of transfer of amino acids and diglycine from water to aqueous polyol solutions.

The enthalpies of transfer of several amino acids and diglycine from water to aqueous polyols (glycerol, xylitol, sorbitol, and inositol) have been calorimetrically determined in order to clarify the mechanism of polyol-induced stabilization of proteins. The obtained enthalpy data were combined with the data on free energy of transfer previously determined to calculate the corresponding entropy of transfer. The enthalpies of transfer of nonpolar side chains of amino acids and peptide group were positive and negative, respectively, depending on the polyol concentration and hydrophobicity of the side chains. This indicates that the non-spontaneous transfer of nonpolar side chains to aqueous polyols is due to an enthalpy effect while that of peptide group is due to an entropy effect. These thermodynamic data are discussed in terms of changes in water structure or solvent ordering around the solute molecules. It was found that the polyol-induced stabilization of proteins is a result of a complicated enthalpy-entropy compensation phenomenon closely related to the solvent ordering around the solute molecules; consequently, the polyol effects on nonpolar side chains do not necessarily dominate those on peptide groups in an enthalpic sense.

Amino Acids↗
PubMed2002

Associating genes with gene ontology codes using a maximum entropy analysis of biomedical literature.

Functional characterizations of thousands of gene products from many species are described in the published literature. These discussions are extremely valuable for characterizing the functions not only of these gene products, but also of their homologs in other organisms. The Gene Ontology (GO) is an effort to create a controlled terminology for labeling gene functions in a more precise, reliable, computer-readable manner. Currently, the best annotations of gene function with the GO are performed by highly trained biologists who read the literature and select appropriate codes. In this study, we explored the possibility that statistical natural language processing techniques can be used to assign GO codes. We compared three document classification methods (maximum entropy modeling, naïve Bayes classification, and nearest-neighbor classification) to the problem of associating a set of GO codes (for biological process) to literature abstracts and thus to the genes associated with the abstracts. We showed that maximum entropy modeling outperforms the other methods and achieves an accuracy of 72% when ascertaining the function discussed within an abstract. The maximum entropy method provides confidence measures that correlate well with performance. We conclude that statistical methods may be used to assign GO codes and may be useful for the difficult task of reassignment as terminology standards evolve over time.

Algorithms↗
PubMed1999

Entropy of classical histories.

We consider a number of proposals for the entropy of sets of classical coarse-grained histories based on the procedures of Jaynes, and we prove a series of inequalities relating these measures. We then examine these as a function of the coarse-graining for various classical systems, and show explicitly that the entropy is minimized by the finest-grained description of a set of histories. We propose an extension of the second law of thermodynamics to the entropy of histories. We briefly discuss the implications for decoherent or consistent history formulations of quantum mechanics.

Journal Article↗
PubMed2001

Entropy-driven phase transition in binary mixtures.

Based on the principle of entropy maximum, a transparent method to study the phase separation is proposed. The excluded volume effects of binary mixtures of hard spheres with two different diameters are analyzed and the role of entropy is emphasized. As a result of the entropy variation caused by the packing of large spheres, there is a critical volume fraction to denote the phase boundary. It is shown that the variation of free volume fraction is influenced by the ratio alpha=d(L)/d(S) of large to small sphere diameters and the ratio x=eta(L)/(eta(L)+eta(S)) of large-sphere volume fraction to the total volume fraction of large- and small-spheres. We introduce a modification factor beta to describe the overlap degree of two large spheres excluded volumes when they pack together. The critical volume fractions for large-sphere packing with different values of alpha and x are calculated, and the corresponding phase boundaries are determined. Our results are in quite good agreement with previous experimental measurements.

Journal Article↗
PubMed2001

Entropy production and wave packet dynamics in the Fock space of closed chaotic many-body systems.

Highly excited many-particle states in quantum systems such as nuclei, atoms, quantum dots, spin systems, quantum computers, etc., can be considered as "chaotic" superpositions of mean-field basis states (Slater determinants, products of spin or qubit states). This is due to a very high level density of many-body states that are easily mixed by a residual interaction between particles (quasiparticles). For such systems, we have derived simple analytical expressions for the time dependence of the energy width of wave packets, as well as for the entropy, number of principal basis components, and inverse participation ratio, and tested them in numerical experiments. It is shown that the energy width Delta(t) increases linearly and very quickly saturates. The entropy of a system increases quadratically, S(t) approximately t(2), at small times, and afterward can grow linearly, S(t) approximately t, before saturation. Correspondingly, the number of principal components determined by the entropy N(pc) approximately exp[S(t)] or by the inverse participation ratio increases exponentially fast before saturation. These results are explained in terms of a cascade model which describes the flow of excitation in the Fock space of basis components. Finally, the striking phenomenon of damped oscillations in the Fock space at the transition to equilibrium is discussed.

Journal Article↗
PubMed2001

Mesoscopic nonequilibrium thermodynamics of single macromolecules and dynamic entropy-energy compensation.

We introduce axiomatically a complete thermodynamic formalism for a single macromolecule, either with or without detailed balance, in an isothermal ambient fluid based on its stochastic dynamics. With detailed balance, the theory yields mesoscopic, nonequilibrium for entropy (Upsilon(t)) and free energy (Psi(t)) of the macromolecule. Upsilon(t) and Psi(t) fluctuate. Expectation (d/dt)E[Psi(t)]< or =0, "="holds if and only if the macromolecule is at thermal equilibrium, in which we show that Upsilon(t) still fluctuates but Psi(t) is a constant. The entropy fluctuation of Landau, E[(DeltaUpsilon(t))(2)], precisely matches the fluctuation in the internal energy, which in turn equals the fluctuation in heat dissipation. As a generalization of Clausius' classic result, the dynamic fluctuations in the entropy and energy of the macromolecule are exactly compensated at thermal equilibrium. For systems with detailed balance, Helmholtz free energy is shown to be the potential of Onsager's thermodynamic force.

Journal Article↗