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PubMed2000

Correlation between changes in nuclear magnetic resonance order parameters and conformational entropy: molecular dynamics simulations of native and denatured staphylococcal nuclease.

Recent work has suggested that changes in NMR order parameters may quantitatively reflect changes in the conformational entropy of a protein ensemble. The extent of the mathematical relationship between local entropy changes as seen by NMR order parameters and the full protein entropy change is a complex issue. As a step towards a fuller understanding of this problem, molecular dynamics calculations of both native and denatured staphylococcal nuclease were performed. The N-H bond vector motion, in both explicit and implicit solvent, was analyzed to estimate local and global entropy changes. The calculated N-H bond vector order parameters from simulation agreed on average with experimental values for both native and denatured structures. However, the inverted-U profile of order parameters versus residue number observed experimentally for denatured nuclease was only partially reproduced by simulation of compact denatured structures. Comparisons made across the full set of simulations revealed a correlation between the N-H order parameter-based conformational entropy change and the total quasiharmonic-based conformational entropy change between the native and denatured structures. The calculations showed that about 25% of the total entropy change was reflected by changes in simulated S2 values. This result suggests that NMR-derived order parameters may be used to provide a reasonable estimate of the total conformational entropy change on protein folding.

Magnetic Resonance Spectroscopy↗

PubMed1993

Contribution of hydration to protein folding thermodynamics. II. The entropy and Gibbs energy of hydration.

The entropy of hydration of non-polar and polar groups upon protein unfolding has been estimated for four globular proteins in the temperature range 5 to 125 degrees C, using structural information on the groups of these proteins exposed to water in the native and unfolded states and the entropies of transfer of various model compounds from the gaseous phase to water. The latter was calculated using the molar scale. It is shown that the entropies of hydration of non-polar and polar groups are both negative, but change in different directions with increasing temperature: the entropy of hydration of polar groups increases in absolute magnitude, while the entropy of hydration of non-polar groups decreases and becomes zero at 122 degrees C for aliphatic groups and at 104 degrees C for aromatic groups, with a change in sign at higher temperature. The configurational entropy was estimated by comparing the entropy of overall hydration with the total entropy of protein unfolding. Using previously determined enthalpies of hydration of polar and non-polar groups and disruption of the internal bonds, the Gibbs energies of hydration of protein groups and breaking of the hydrogen bonds and van der Waals interactions have been estimated. It was shown that hydration of the polar groups and aromatic non-polar groups destabilizes the native protein structure, while hydration of aliphatic non-polar groups (hydrophobic hydration) stabilizes it; however, the main stabilization effect comes from the internal van der Waals interactions and hydrogen bonding. Analysis of the factors contributing to the stability of the folded protein conformation shows that it is the hydration of polar groups that is mainly responsible for the cold denaturation of proteins.

Cytochrome c Group↗

PubMed1997

Estimating the entropy of DNA sequences.

The Shannon entropy is a standard measure for the order state of symbol sequences, such as, for example, DNA sequences. In order to incorporate correlations between symbols, the entropy of n-mers (consecutive strands of n symbols) has to be determined. Here, an assay is presented to estimate such higher order entropies (block entropies) for DNA sequences when the actual number of observations is small compared with the number of possible outcomes. The n-mer probability distribution underlying the dynamical process is reconstructed using elementary statistical principles: The theorem of asymptotic equi-distribution and the Maximum Entropy Principle. Constraints are set to force the constructed distributions to adopt features which are characteristic for the real probability distribution. From the many solutions compatible with these constraints the one with the highest entropy is the most likely one according to the Maximum Entropy Principle. An algorithm performing this procedure is expounded. It is tested by applying it to various DNA model sequences whose exact entropies are known. Finally, results for a real DNA sequence, the complete genome of the Epstein Barr virus, are presented and compared with those of other information carriers (texts, computer source code, music). It seems as if DNA sequences possess much more freedom in the combination of the symbols of their alphabet than written language or computer source codes.

Algorithms↗

PubMed1977

Public nuisance and entropy increase.

Public nuisance if the inevitable physical conclusion of the second law of thermodynamics, the increase of entropy, under which humans make efforts to build up order, that is, to make states of lower entropy, scattering much entropy into the environment. In other words, humans live on negentropy, negative entropy. Saving of negentropy is the first means of conserving the environment. Soon after C. E. Shanon had introduced the idea of entropy in information theory, N. Wiener showed the relation between entropy of thermodynamics and entropy of information. Wiener showed that the paradox of Maxwell is not a contradiction if we consider the entropy of demon's information. This idea is very suggestive. Moral principles, tax systems, and price mechanisms are effective for the conservation of the environment. Good information systems can conserve good environments. This is the second means. The third way is the use of biological systems. For example, in sewage treatment plants the active sludge purifies polluted water. The reason why microorganisms can select some substances in polluted water is that they have received the information by heredity, the information carried by the double helical structure of DNA.

Environment↗

PubMed2006

Increase in high frequency EEG activity explains the poor performance of EEG spectral entropy monitor during S-ketamine anesthesia.

OBJECTIVE: To study the effects of S-ketamine on the EEG and to investigate whether spectral entropy of the EEG can be used to assess the depth of hypnosis during S-ketamine anesthesia. METHODS: The effects of sub-anesthetic (159 (21); mean (SD) ng/ml) and anesthetic (1,959 (442) ng/ml) serum concentrations of S-ketamine on state entropy (SE), response entropy (RE) and classical EEG spectral power variables (recorded using the Entropy Module, GE Healthcare, Helsinki, Finland) were studied in 8 healthy males. These EEG data were compared with EEG recordings from 6 matching subjects anesthetized with propofol. RESULTS: The entropy values decreased from the baseline SE 85 (3) and RE 96 (3) to SE 55 (18) and RE 72 (17) during S-ketamine anesthesia but both inter- and intra-individual variation of entropy indices was wide and their specificity to indicate unconsciousness was poor. Propofol induced more pronounced increase in delta power (P<0.02) than S-ketamine, whereas anesthetic S-ketamine induced more high frequency EEG activity in the gamma band (P<0.001). Relative power of 20-70 Hz EEG activity was associated with high SE (P=0.02) and RE (P=0.03) values during S-ketamine anesthesia. CONCLUSIONS: These differences in low and high frequency EEG power bands probably explain why entropy monitor, while adequate for propofol, is not suitable for assessing the depth of S-ketamine anesthesia. SIGNIFICANCE: The entropy monitor is not adequate for monitoring S-ketamine-induced hypnosis.

Adult↗

PubMed1999

Does microbial life always feed on negative entropy? Thermodynamic analysis of microbial growth.

Schrödinger stated in his landmark book, What is Life?, that life feeds on negative entropy. In this contribution, the validity of this statement is discussed through a careful thermodynamic analysis of microbial growth processes. In principle, both feeding on negative entropy, i.e. yielding products of higher entropy than the substrates, and generating heat can be used by microorganisms to rid themselves of internal entropy production resulting from maintenance and growth processes. Literature data are reviewed in order to compare these two mechanisms. It is shown that entropy-neutral, entropy-driven, and entropy-retarded growth exist. The analysis of some particularly interesting microorganisms shows that enthalpy-retarded microbial growth may also exist, which would signify a net uptake of heat during growth. However, the existence of endothermic life has never been demonstrated in a calorimeter. The internal entropy production in live cells also reflects itself in the Gibbs energy dissipation accompanying growth, which is related quantitatively to the biomass yield. An empirical correlation of the Gibbs energy dissipation in terms of the physico-chemical nature of the growth substrate has been proposed in the literature and can be used to predict the biomass yield approximately. The ratio of enthalpy change and Gibbs energy change can also be predicted since it is shown to be approximately equal to the same ratio of the relevant catabolic process alone.

Aerobiosis↗

PubMed1987

Heat capacity and entropy changes of the two major isotypes of bullfrog (Rana catesbeiana) parvalbumins induced by calcium binding.

The possible structural changes of the two major isotypes (PA1 and PA2) of parvalbumins from bullfrog (Rana catesbeiana) skeletal muscle caused by Ca2+ binding have been analyzed by microcalorimetric titrations. Titrations of the parvalbumins with Ca2+ have been made in both the absence and presence of Mg2+ at pH 7.0 and at 5, 15, and 25 degrees C. The reactions of the parvalbumins with Ca2+ are exothermic in both the presence and absence of Mg2+ and at every temperature. But the contributions of enthalpy and entropy changes are variable; Mg2+-Ca2+ exchange on PA1 at 25 degrees C is driven almost entirely by a favorable enthalpy change, whereas Ca2+ binding to PA2 at 5 degrees C is driven for the most part by a favorable entropy change. The magnitudes of the hydrophobic and internal vibrational contributions to the heat capacity and entropy changes of the parvalbumins on Ca2+ binding and Mg2+-Ca2+ exchange have been estimated by the empirical method of Sturtevant [Sturtevant, J. M. (1977) Proc. Natl. Acad. Sci. U.S.A. 74, 2236-2240]. Although PA1 (beta) and PA2 (alpha) belong to genetically different lineages, the parvalbumins indicate very similar conformational changes to each other on both Ca2+ binding and Mg2+-Ca2+ exchange. On Mg2+-Ca2+ exchange, the vibrational as well as hydrophobic entropy is slightly increased in a parallel manner. In contrast, on Ca2+ binding, the hydrophobic entropy increases but the vibrational entropy decreases. The increase in the hydrophobic entropy indicates the sequestering of nonpolar groups from the surface to the interior of molecules, while the changes in the vibrational entropy suggest that the overall structures are tightened on Ca2+ binding but loosened on Mg2+-Ca2+ exchange.

Animals↗

PubMed2005

Absolute and relative entropies from computer simulation with applications to ligand binding.

A comparison between two related methods, Schlitter's formula and quasiharmonic analysis, for calculating absolute entropies from the covariance matrix of atomic fluctuations using molecular dynamics (MD) simulations is presented. Calculations for a set of organic compounds in the gas phase are compared to the corresponding statistical thermodynamics results for translational and rotational entropies and to experimental data for vibrational entropies. Encouraging agreement is obtained for translational entropies, but for the rotational contribution, both methods fail to reproduce the theoretically calculated values. Absolute and relative vibrational entropies are found to be better reproduced using quasiharmonic analysis compared to Schlitter's formula. For rotational entropies, we propose a method based on the variances in Euler angles, which gives good agreement with theory. Alternative methods for estimating translational entropies based on principal root mean-square (rms) fluctuations of the center of mass are also presented, and these reproduce theoretically calculated values well. These methodologies are applied to the binding of benzene to T4-lysozyme, where close agreement with the literature is obtained for translational and rotational entropies.

Bacteriophage T4↗

PubMed2006

Confusing cause and effect: energy-entropy compensation in the preferential solvation of a nonpolar solute in dimethyl sulfoxide/water mixtures.

We performed molecular simulations to analyze the thermodynamics of methane solvation in dimethyl sulfoxide (DMSO)/water mixtures (298 K, 1 atm). Two contributions to the interaction thermodynamics are studied separately: (i) the introduction of solute-solvent interactions (primary contribution) and (ii) the solute-induced disruption of cohesive solvent-solvent interactions (secondary contribution). The energy and entropy changes of the secondary contribution always exactly cancel in the free energy (energy-entropy compensation), hence only the primary contribution is important for understanding changes of the free energy. We analyze the physical significance of the solute-solvent energy and solute-solvent entropy associated with the primary contribution and discuss how to obtain these quantities from experiments combining solvation thermodynamic and solvent equation of state data. We show that the secondary contribution dominates changes in the methane solvation entropy and enthalpy: below 30 mol % DMSO in the mixture, methane, because of more favorable dispersion interactions with DMSO molecules, preferentially attracts DMSO molecules, which, in response, release water molecules into the bulk, causing an increase in the entropy. This large energy-entropy compensating process easily causes a confusion in the cause for and the effect of preferred methane-DMSO interactions. Methane-DMSO dispersion interactions are the cause, and the entropy change is the effect. Procedures that infer thermodynamic driving forces from analyses of the solvation entropies and enthalpies should therefore be used with caution.

Journal Article↗

PubMed1993

Approximate entropy of heart rate as a correlate of postoperative ventricular dysfunction.

BACKGROUND: Instantaneous changes in the heart rate, i.e., heart rate variation, traditionally have been quantified by the standard deviation of a series of intervals between successive heart beats. Approximate entropy provides another measure of variability by calculating the logarithmic likelihood that patterns that are similar remain similar on the next incremental comparisons. Approximate entropy is a nonnegative number that will distinguish data sets by their amount of regularity, with larger numbers indicating more randomness. We hypothesized that a decrease in the approximate entropy of heart rate would be associated with postoperative ventricular dysfunction (e.g., myocardial infarction, unstable angina, congestive heart failure, prolonged inotropic support). METHODS: Twenty-three high-risk noncardiac patients were continuously monitored by ambulatory electrocardiographic recorders from the evening before surgery up to 80 h during the postoperative period: 9 demonstrated postoperative ventricular dysfunction, and 14 had an uncomplicated postoperative course. Hourly approximate entropy average values were calculated. RESULTS: Approximate entropy was high (> 0.7) in all but two patients preoperatively. Postoperative approximate entropy <0.55 had a sensitivity of 88% and a specificity of 71% for being associated with postoperative ventricular dysfunction; preoperative approximate entropy values were not significantly different between the two groups. CONCLUSIONS: These results suggest that changes in approximate entropy can distinguish between patients who sustained poor outcome and those who had an uncomplicated course.

Aged↗

PubMed1999

Topological and spatial aspects of the hydration of solutes of extreme solvation entropy.

The hydration of charged Lennard-Jones spheres by simple point charge water is considered. Molecular dynamics and expanded ensemble simulations were used to compare the hydration structures surrounding solutes with extreme solvation entropy. The variations in the solvation entropy were analyzed in terms of changes in the spatial and topological structure of the hydration shells. The solvation entropy was found to be maximal for solutes that can replace water molecules in the hydrogen-bond network. Further, using a Kirkwood-type factorization, the solvation entropy was expanded as a sum over the partial n-body distribution functions. The two-body solute-water contribution to the solvation entropy was found to exceed the full solvation entropy for solutes with low charge, whereas the converse is true for the other solutes. This is consistent with the idea that water-water correlations are enhanced by solvation of, for example, noble gases, whereas they are disrupted by solvation of ions. Further, the orientational and radial parts of the two-body solute-water entropy were calculated as functions of the charge of the solute. The orientational part has a single maximum, whereas the radial part maintains the bimodal form of the full solvation entropy.

Journal Article↗

PubMed2001

Trajectory versus probability density entropy.

We show that the widely accepted conviction that a connection can be established between the probability density entropy and the Kolmogorov-Sinai (KS) entropy is questionable. We adopt the definition of density entropy as a functional of a distribution density whose time evolution is determined by a transport equation, conceived as the only prescription to use for the calculation. Although the transport equation is built up for the purpose of affording a picture equivalent to that stemming from trajectory dynamics, no direct use of trajectory time evolution is allowed, once the transport equation is defined. With this definition in mind we prove that the detection of a time regime of increase of the density entropy with a rate identical to the KS entropy is possible only in a limited number of cases. The proposals made by some authors to establish a connection between the two entropies in general, violate our definition of density entropy and imply the concept of trajectory, which is foreign to that of density entropy.

Journal Article↗

PubMed2002

Entropy of electromagnetic polarization.

The entropy of electromagnetic polarization is considered in this paper. It is shown that unless the non-field entropy, and not the total entropy, is used as the independent variable in the expression for the internal energy, the first law is violated and the meaning of heat flow, as given by the second law, is contradicted. The total entropy and its field and non-field components are shown to be state functions. The field entropy comprises contributions from the field generated by the contents of the system and stored within as well as outside its boundaries. The contribution of the field stored outside the system boundaries is derived and demonstrated for the case of a uniformly polarized sphere. Finally, expressions are derived for field entropies and entropy densities, in composite systems, using the concept of interaction entropy. The results are shown to be fundamentally different compared to those used in the current literature.

Journal Article↗

PubMed1993

The effect of noise on entropy.

A complementary relationship between the entropy (S) and the variance (sigma(2)) of an electron-density map is derived by approximating the logarithmic term in the entropy expression by a series expansion around the average map density. The resulting expression is S approximately ln N - 1/2sigma(2), where N is the number of grid points and sigma is the r.m.s. deviation from the mean in a map normalized to unit mean. The algebraic expression is of interest because it is consistent with and allows numerical evaluation of the surprising argument that noise decreases the entropy of a map. The argument is that a noise contribution by itself generates a certain variance that is independent of the atomic structure and that adds to the variance due to the structure. Increased variance corresponds to decreased entropy. This property of noise provides an intuitively reasonable justification for maximizing the entropy of an electron-density map in the quest for more readily interpretable maps of macromolecules. The entropy-variance relationship also extends the range of applicability of the entropy concept to maps with a limited amount of negative density. The approximation which leads to the entropy-variance relationship is most applicable where it is most likely to be useful - in experimental maps of relatively low structure definition.

Journal Article↗

PubMed2004

Loss of translational entropy in molecular associations.

Molecular associations in solution are opposed by the loss of entropy (DeltaS) that results from the restriction of motion of each component in the complex. Theoretical estimates of DeltaS are essential for rationalizing binding affinities, as well as for calculating entropic contribution to enzyme catalysis. Recently a statistical-mechanical framework has been proposed for estimating efficiently the translational entropy loss (DeltaS(trsl)), while taking explicitly into account the complex intermolecular interactions between the solute and the solvent. This framework relates the translational entropy of a solute in solution to its "free volume," defined as the volume accessible to the center of mass of the solute in the presence of the solvent and calculated by using an extension of the cell model (CM) for condensed phases. The translational entropy of pure water, estimated with the CM algorithm, shows good agreement with the experimental information. The free volume of various solutes in water, calculated within the CM by using molecular dynamics simulations with explicit solvent, displays a strong correlation with the solutes' polar and total surface areas. This correlation is used to propose a parameterization that can be used to calculate routinely the translational entropy of a solute in water. We also applied the CM formalism to calculate the free volume and translational entropy loss (DeltaS(trsl)) on binding of benzene to a cavity in a mutant T4-lysozyme. Our results agree with previously published estimates of the binding of benzene to this mutant T4-lysozyme. These and other considerations suggest that the cell model is a simple yet efficient theoretical framework to evaluate the translational entropy loss on molecular association in solution.

Algorithms↗

PubMed1997

Maximum entropy, analysis of kinetic processes involving chemical and folding-unfolding changes in proteins.

We show that numerical inversion of the Laplace transform by using the maximum entropy method can be successfully applied to the analysis of complex kinetic processes involving chemical and folding-unfolding changes in proteins. First, we present analyses of simulated data which support that: (i) the maximum entropy calculation of rate distributions, combined with Monte Carlo analyses of the associated uncertainties, yields results consistent with the information actually supplied by the data, thus preventing their over-interpretation; (ii) maximum entropy analysis may be used to extract discrete rates (corresponding to individual exponential contributions), as well as broad rate distributions (provided, of course, that the adequate information is supplied by the data). We further illustrate the applicability of the maximum entropy analysis with experimental data corresponding to two nontrivial model processes: (a) the kinetics of chemical modification of sulfhydryl groups in glycogen synthase by reaction with Ellman's reagent; (b) the kinetics of folding of ribonuclease a under strongly folding conditions, as monitored by fluorescence and optical absorption. Finally, we discuss that the maximum entropy approach should be particularly useful in studies on protein folding kinetics, which generally involve the comparison between several complex kinetic profiles obtained by using different physical probes. Thus, protein folding kinetics is usually interpreted in terms of kinetic mechanisms involving a comparatively small number of kinetic steps between well-defined protein states. According to this picture, rate distributions derived from experimental kinetic profiles by maximum entropy analysis are expected to show a small number of comparatively narrow peaks, from which we can determine, without a priori assumptions, the number of exponential contributions required to describe each experimental kinetic profile (the number of peaks), together with their amplitudes (from the peak areas), time constant values (from the peak positions), and associated Monte Carlo uncertainties. On the other hand, recent theoretical studies describe protein folding kinetics in terms of the protein energy landscape (the multidimensional surface of energy versus conformational degrees of freedom), emphasize the difficulty in defining a single reaction coordinate for folding, and point out that individual chains may fold by multiple pathways. This indicates that, in some cases at least, folding kinetics might have to be described in terms of broad rate distributions (rather than in terms of a small number of discrete exponential contributions related to kinetic steps between well-defined protein states). We suggest that the maximum entropy procedures described in this work may provide a method to detect this situation and to derive such broad rate distributions from experimental data.

Entropy↗

PubMed1996

Contributions to conformational entropy arising from bond vector fluctuations measured from NMR-derived order parameters: application to protein folding.

The relation between order parameters derived from NMR spin relaxation experiments and the contribution to conformational entropy from ns-ps timescale bond vector dynamics is investigated by considering a number of simple models describing bond vector motion. In a few cases both classical and quantum mechanical derivations are included to establish the validity of obtaining order parameter-entropy relations using classical mechanics only. For these cases it is found that classical and quantum mechanical derivations give very similar results so long as the square of the order parameter of the bond vector is less than approximately 0.95. For a given change in order parameter, the change in conformational entropy is sensitive to the model employed, with the absolute value of the entropy change increasing with the number of degrees of freedom in the model. The entropy-order parameter profile calculated from a 1.12 ns molecular dynamics trajectory of fully hydrated Escherichia coli ribonuclease HI is well fit using a simple expression based on a model assuming bond vector diffusion in a cone, suggesting that it may well be possible to extract meaningful entropy changes reflecting changes in ps-ns time scale motions from changes in NMR-derived order parameters. Contributions to the conformational entropy change associated with a folding-unfolding transition of an SH3 domain and calculated from changes in rapid N-HN backbone dynamics are presented.

Entropy↗

PubMed1997

Entropy in protein folding and in protein-protein interactions.

The reduction of conformational entropy is a major barrier that has to be overcome in protein folding and binding. Changes in solvent entropy are also a major factor. Recent advances include clarification of the fundamental issues concerning the separation of entropy into components, the treatment of association entropy in binding, and the role of size and shape effects in solvation entropy. Advances in the application of entropy calculations include an emerging consensus for estimates of backbone and sidechain entropy loss in protein folding via use of numerically intensive methods for sampling, and use of the expanding protein-structure database.

Algorithms↗