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Critical evaluation of methods to incorporate entropy loss upon binding in high-throughput docking.

Proper accounting of the positional/orientational/conformational entropy loss associated with protein-ligand binding is important to obtain reliable predictions of binding affinity. Herein, we critically examine two simplified statistical mechanics-based approaches, namely a constant penalty per rotor method, and a more rigorous method, referred to here as the partition function-based scoring (PFS) method, to account for such entropy losses in high-throughput docking calculations. Our results on the estrogen receptor beta and dihydrofolate reductase proteins demonstrate that, while the constant penalty method over-penalizes molecules for their conformational flexibility, the PFS method behaves in a more "DeltaG-like" manner by penalizing different rotors differently depending on their residual entropy in the bound state. Furthermore, in contrast to no entropic penalty or the constant penalty approximation, the PFS method does not exhibit any bias towards either rigid or flexible molecules in the hit list. Preliminary enrichment studies using a lead-like random molecular database suggest that an accurate representation of the "true" energy landscape of the protein-ligand complex is critical for reliable predictions of relative binding affinities by the PFS method.

Algorithms↗

Contributions to protein entropy and heat capacity from bond vector motions measured by NMR spin relaxation.

The backbone dynamics of both folded and unfolded states of staphylococcal nuclease (SNase) and the N-terminal SH3 domain from drk (drkN SH3) are studied at two different temperatures. A simple method for obtaining order parameters, describing the amplitudes of motion of bond vectors, from NMR relaxation measurements of both folded and unfolded proteins is presented and the data obtained for 15N-NH bond vectors in both the SNase and drkN SH3 systems analyzed with this approach. Using a recently developed theory relating the amplitude of bond vector motions to conformational entropy, the entropy change between the folded and unfolded forms of SNase is calculated on a per residue basis. It is noteworthy that the region of the molecule with the smallest entropy change includes those residues showing native-like structure in the unfolded form of the molecule, as established by NOE-based experiments. Order parameters of backbone 15N-NH bond vectors show significantly larger changes with temperature in the unfolded states of both proteins relative to the corresponding folded forms. The differential temperature dependence is interpreted in terms of differences in the heat capacities of folded and unfolded polypeptide chains. The contribution to the heat capacity of the unfolded chain from rapid 15N-NH bond vector motions is calculated and compared with estimates of the heat capacity of the backbone unit, -CHCONH-, obtained from calorimetric data. Methyl dynamics measured at 14 and 30 degrees C establish that the amplitudes of side-chain motions in the folded SH3 domain are more sensitive to changes in temperature than the backbone dynamics, suggesting that over this temperature range side-chain ps to ns time-scale motions contribute more to the heat capacity than backbone motions for this protein.

Drosophila Proteins↗

Using Shannon entropy on measuring the individual variability in the Rufous-bellied thrush Turdus rufiventris vocal communication.

We applied the information theory concepts to notes repertoire characteristics combined with temporal parameters of the Rufous-bellied thrush Turdus rufiventris song, using this particular case to test a new method of analysing quantitatively complex animal communication systems. Like most Turdus thrushes, Rufous-bellied thrushes are remarkable for their long, varied and melodious songs. For the analysis of the species repertoire, we used recordings of 44 individuals from 24 localities covering its full geographical range. We measured the repertoire size, note duration and rhythm (frequency of note utterance), and combined these parameters with the Shannon entropy values calculated for each individual. Although individuals maintain species-specific recognition capacity, we find a large variation between their song parameters and show that the information theory can be useful to analyse large and varied animal vocal repertoires. We are introducing two new parameters, temporal average entropy (E(t)) and utterance frequency average entropy (E(f)), for measuring such communication systems.

Animals↗

Analysis of HIV by entropy evolution rate.

We analyze the variation of HIV after infection by means of an information measure, called the entropy evolution rate. In our analysis, we use a part of the external glycoprotein gp120 including the V3 region observed from six patients. Then we could make the following two aspects clear; (1) the relation between the change of the entropy evolution rate and the appearance of symptoms of disease, and (2) the relation between the change of the entropy evolution rate and that of the CD4 count of the patients.

Acquired Immunodeficiency Syndrome↗

Analysis of postural sway using entropy measures of signal complexity.

A stochastic complexity analysis is applied to centre-of-pressure (COP) time series, by using different complexity features, namely the spectral entropy, the approximate entropy, and the singular value decomposition spectrum entropy. A principal component analysis allows an estimate of the overall signal complexity in terms of the ensemble complexity score; the difference in values between open-eyes (OE) and closed-eyes (CE) trials is used for clustering purposes. In experiments on healthy young adults, the complexity of the mediolateral component is shown not to depend on the manipulation of vision. Conversely, the increase of the anteroposterior complexity in OE conditions can be statistically significant, leading to a functional division of the subjects into two groups: the Romberg ratios (RRs), namely the ratios of the CE measure to the OE measure, are: RR = 1.19 +/- 0.15 (group 1 subjects), and RR = 1.05 +/- 0.14 (group 2 subjects). Multivariate statistical techniques are applied to the complexity features and the parameters of a postural sway model recently proposed; the results suggest that the complexity change is the sign of information-generating behaviours of postural fluctuations, in the presence of a control strategy which aims at loosening long-range correlation and decreasing stochastic activity when visual feedback is allowed.

Adult↗

Complexity analysis of stride interval time series by threshold dependent symbolic entropy.

The stride interval of human gait fluctuates in complex fashion. It reflects the rhythm of the locomotor system. The temporal fluctuations in the stride interval provide us a non-invasive technique to evaluate the effects of neurological impairments on gait and its changes with age and disease. In this paper, we have used threshold dependent symbolic entropy, which is based on symbolic nonlinear time series analysis to study complexity of gait of control and neurodegenerative disease subjects. Symbolic entropy characterizes quantitatively the complexity even in time series having relatively few data points. We have calculated normalized corrected Shannon entropy (NCSE) of symbolic sequences extracted from stride interval time series. This measure of complexity showed significant difference between control and neurodegenerative disease subjects for a certain range of thresholds. We have also investigated complexity of physiological signal and randomized noisy data. In the study, we have found that the complexity of physiological signal was higher than that of random signals at short threshold values.

Biological Clocks↗

Wavelet entropy in event-related potentials: a new method shows ordering of EEG oscillations.

In this work we show the application of a measure of entropy defined from the wavelet transform, namely the wavelet entropy (WS), to the study of event-related potentials (ERPs). WS was computed for ERPs recorded from nine healthy subjects with three different types of stimuli, among them target stimuli in a cognitive task. A significant decrease of entropy was correlated with the responses to target stimuli (P300), thus showing that these responses correspond to a more "ordered" state than the spontaneous EEG. Furthermore, we propose the WS as a quantitative measure for such transitions between EEG ("disordered state") and ERP ("ordered state").

Adult↗

Measuring regularity by means of a corrected conditional entropy in sympathetic outflow.

A new method for measuring the regularity of a process over short data sequences is reported. This method is based on the definition of a new function (the corrected conditional entropy) and on the extraction of its minimum. This value is taken as an index in the information domain quantifying the regularity of the process. The corrected conditional entropy is designed to decrease in relation to the regularity of the process (like other estimates of the entropy rate), but it is able to increase when no robust statistic can be performed as a result of a limited amount of available samples. As a consequence of the minimisation procedure, the proposed index is obtained without an a-priori definition of the pattern length (i.e. of the embedding dimension of the reconstructed phase space). The method is validated on simulations and applied to beat-to-beat sequences of the sympathetic discharge obtained from decerebrate artificially ventilated cats. At control, regular, both quasiperiodic and periodic (locked to ventilation) dynamics are observed. During the sympathetic activation induced by inferior vena cava occlusion, the presence of phase-locked patterns and the increase in regularity of the sympathetic discharge evidence an augmented coupling between the sympathetic discharge and ventilation. The reduction of complexity of the neural control obtained by spinalization decreases the regularity in the sympathetic outflow, thus pointing to a weaker coupling between the sympathetic discharge and ventilation.

Animals↗

Biophysics of aging and therapeutic interventions by entropy-variation systems.

The cell is thermodynamically an open system and aging is characterized by an increasingly higher structural disorder (increase of entropy) and functional loss. If a variation of negative entropy is introduced by an external source, an anti-clockwise effect leading to regenerative processes and/or increase of the functional reserve supporting regenerative tissue changes is theoretically expected. The achievement of a negative variation of entropy is the main principle of a new technology which implies an exogenous delivery of energy with higher performance than the physiological production. Promising clinical experiences in liver cirrhosis and in long-standing scarring lesions seem to confirm the clinical applicability of the theoretical model.

Aging↗

The distribution of the local entropy in ultrasound images.

In this article, a model for the amplitude statistics of the backscattered ultrasonic signal is presented. We propose to view a tissue as being composed of a large number of small units, each having slightly different characteristics. This variability within the tissue is expressed by fluctuations in the parameters of the local probability distribution function (PDF). Based on analogous expressions derived for radio propagation and optical scintillations, the local PDF is modulated by a lognormal distribution of the local standard deviation. Integrating the local contributions yields the amplitude PDF for the entire tissue. We show that the local entropy is a normal variable, since for four different local PDFs it is linearly related to the logarithm of the local standard deviation. When the local entropy histogram exhibits distinct and multiple peaks, the local entropy distribution can be used for region segmentation. This fact is demonstrated for ultrasonic images of ovarian cysts.

Entropy↗

Dynamical entropy is conserved during cocaine-induced changes in fetal rat motor patterns.

Our previous studies demonstrated that the intra-cisternal (IC) administration of cocaine to fetal rats increased motor activity and decreased responsiveness to perioral stimulation. One explanation for these observations comes from the behavioral pharmacology of stimulant drugs: increased motor activity is often associated with a decrease in its variety. Previous power spectral transformation of this data suggests an alternative explanation: cocaine-induced hyperactivity fixates a new behavioral pattern with complexity equal to that of saline controls. We explore these possibilities using statistical techniques derived from studies of nonlinear dynamical systems, examining patterns of the total motor activity of the individual fetus as counts per 5 s interval on either gestational day E20 or E21 for 20 min following IC injections of saline, 2.5 or 10 mg/kg of cocaine. The results are consistent with a state in which increased spontaneous activity is associated with the emergence of a new dynamical pattern which conserves entropy and provides experimental support for a fundamental conservation-variational relation, hT approximately equal to lambda 1 x DR, that has been proven for abstract models of chaotic dynamical systems. A multivariate analysis of variance (MANOVA) followed by appropriate analyses of variance (ANOVAs) and pairwise comparisons revealed that, whereas cocaine induced increases in the total amount of motor activity, the rate of increase in the variety of new sequences in activity counts over time did not change with treatment and age conditions. This invariant is quantified by an absence of change in topological entropy, delta hT = 0. The analyses also showed that, in order to maintain hT values, compensatory changes took place in the leading Lyapounov characteristic exponent, lambda 1 (the distance between sequential values 'stretched' along the increasing amplitudes of the variations) such that delta lambda 1 > 0, and the correlation dimension, DR (the hierarchical range of possible values, 'complicated clustering') was reduced, so that delta DR < 0. Our findings are consistent with the idea that the association between cocaine-induced increases in activity and decreases in adaptive response are not due to the dynamical entropy loss of decreased behavioral variety, but are rather the result of competitive interference by a drug-induced, equally complex, new pattern of spontaneous behavior.

Animals↗

Entropies for detection of epilepsy in EEG.

The electroencephalogram (EEG) is a representative signal containing information about the condition of the brain. The shape of the wave may contain useful information about the state of the brain. However, the human observer cannot directly monitor these subtle details. Besides, since bio-signals are highly subjective, the symptoms may appear at random in the time scale. Therefore, the EEG signal parameters, extracted and analyzed using computers, are highly useful in diagnostics. The aim of this work is to compare the different entropy estimators when applied to EEG data from normal and epileptic subjects. The results obtained indicate that entropy estimators can distinguish normal and epileptic EEG data with more than 95% confidence (using t-test). The classification ability of the entropy measures is tested using ANFIS classifier. The results are promising and a classification accuracy of about 90% is achieved.

Algorithms↗

The 'weighted sum of relative entropy': a new index for synonymous codon usage bias.

Shannon entropy from information theory has been applied to estimate the degree of deviation from equal usage of synonymous codons; however, previous attempts have failed to take into account all three aspects of amino acid usage, i.e. (i) the number of distinct amino acids, (ii) their relative frequencies, and (iii) their degree of codon degeneracy. A new index taking into account all of these aspects is proposed. The index, designated as the 'weighted sum of relative entropy' (E(w)), is defined as the sum of the relative entropy of each amino acid weighted by its relative frequency in the sequence. In this paper, we demonstrate that E(w) allows us to avoid some amino acid usage biases and can yield results contradictory to those obtained by previous methods.

Algorithms↗

Competitive spiking and indirect entropy minimization of rate code: efficient search for hidden components.

Our motivation, which originates from the psychological and physiological evidences of component-based representations in the brain, is to find neural methods that can efficiently search for structures. Here, an architecture made of coupled parallel working reconstruction subnetworks is presented. Each subnetwork utilizes non-negativity constraint on the generative weights and on the internal representation. 'Spikes' are generated within subnetworks via winner take all mechanism. Memory components are modified in order to directly minimize the reconstruction error and to indirectly minimize the entropy of the spike rate distribution, via a combination of a stochastic gradient search and a novel tuning method. This tuning dynamically changes the learning rate: the higher the entropy of the spike rate, the higher the learning rate of the gradient search in the subnetworks. This method effectively reduces the search space and increases the escape probability from high entropy local minima. We demonstrate that one subnetwork can develop localized and oriented components. Coupled networks can discover and sort components into the subnetworks; a problem subject to combinatorial explosion. Synergy between spike code and rate code is discussed.

Action Potentials↗

Efficiency, power, and entropy in event-related fMRI with multiple trial types. Part II: design of experiments.

The performance of an experimental design for functional magnetic resonance imaging (fMRI) can be characterized by its estimation efficiency, which is the ability to make an estimate of the hemodynamic response, its detection power, which is the ability to detect an activation, and its conditional entropy, which is a measure of the randomness of the design. In Liu and Frank [Neuroimage 21 (2004) 387-400], it is shown that there is a fundamental theoretical trade-off between estimation efficiency and detection power for experiments with multiple trial types and that there is an empirical relation between estimation efficiency and conditional entropy. This paper provides an intuitive interpretation of the theoretical results and examines the practical implications of these results for the optimal design of fMRI experiments with multiple trial types. The properties of block designs, permuted block designs, m-sequence designs, clustered m-sequence designs, and mixed designs are explored. It is shown that these designs nearly achieve the theoretically predicted performance and can be used in practice to obtain advantageous trade-offs among efficiency, power, and entropy.

Artifacts↗

Flexibility and conformational entropy in protein-protein binding.

To better understand the interplay between protein-protein binding and protein dynamics, we analyzed molecular dynamics simulations of 17 protein-protein complexes and their unbound components. Complex formation does not restrict the conformational freedom of the partner proteins as a whole, but, rather, it leads to a redistribution of dynamics. We calculate the change in conformational entropy for seven complexes with quasiharmonic analysis. We see significant loss, but also increased or unchanged conformational entropy. Where comparison is possible, the results are consistent with experimental data. However, stringent error estimates based on multiple independent simulations reveal large uncertainties that are usually overlooked. We observe substantial gains of pseudo entropy in individual partner proteins, and we observe that all complexes retain residual stabilizing intermolecular motions. Consequently, protein flexibility has an important influence on the thermodynamics of binding and may disfavor as well as favor association. These results support a recently proposed unified model for flexible protein-protein association.

Animals↗

Towards a unifying approach to diversity measures: bridging the gap between the Shannon entropy and Rao's quadratic index.

The diversity of a species assemblage has been studied extensively for many decades in relation to its possible connection with ecosystem functioning and organization. In this view most diversity measures, such as Shannon's entropy, rely upon information theory as a basis for the quantification of diversity. Also, traditional diversity measures are computed using species relative abundances and cannot account for the ecological differences between species. Rao first proposed a diversity index, termed quadratic diversity (Q) that incorporates both species relative abundances and pairwise distances between species. Quadratic diversity is traditionally defined as the expected distance between two randomly selected individuals. In this paper, we show that quadratic diversity can be interpreted as the expected conflict among the species of a given assemblage. From this unusual interpretation, it naturally follows that Rao's Q can be related to the Shannon entropy through a generalized version of the Tsallis parametric entropy.

Animals↗

The fractional contributions of elementary modes to the metabolism of Escherichia coli and their estimation from reaction entropies.

The metabolism of a cell can be viewed as a weighted sum of elementary modes. Due to the multiplicity of modes the identification of the individual weights represents a non-trivial problem. To enable the determination of weighting factors we have identified and implemented two gene deletions in combination with defined growth conditions that limit the metabolism from 4374 original elementary modes to 24 elementary modes for a non-PHB synthesizing control and 40 modes for a PHB synthesizing strain. These remaining modes can be further grouped into five families that have the same overall stoichiometry. Thus, the complexity of the problem is significantly reduced, and weighting factors for each family of modes could be determined from the measurement of accumulation rates of metabolites. Moreover, it is shown that individual weights are inversely correlated with the entropy generated by the operation of the used pathways defined in elementary modes. This suggests that evolution developed cellular regulatory patterns that permit diversity of pathways while favoring efficient pathways with low entropy generation. Furthermore, such correlation provides a rational way of estimating metabolic fluxes based on the thermodynamic properties of elementary modes. This is demonstrated with an example in which experimentally determined, intracellular fluxes are shown to be highly correlated with fluxes computed based on elementary modes and reaction entropies. The analysis suggests that the set of elementary modes can be interpreted analogous to a metabolic ensemble of quantum states of a macroscopic system.

Anaerobiosis↗