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PubMed2006

Computational assessment of the entropy of solvation of small-sized hydrophobic entities.

A high level polarizable force field is used to study the temperature dependence of hydrophobic hydration of small-sized molecules from computer simulations. Molecular dynamics (MD) simulations of liquid water at various temperatures form the basis of free energy perturbation calculations that consider the onset and growth of a repulsive sphere. This repulsive sphere acts as a model construct for the hydrophobic species. In the present study, an extension is pursued for seven independent target temperatures, ranging from close to the freezing point almost up to the boiling point of liquid water under standard conditions. Care is taken to maintain proper physico-chemical model descriptions by cross-checking with experimental water densities at the selected target temperatures. The polarizable force field description of molecular water turns out to be suitable throughout the entire temperature domain considered. Derivatives of the computed free energies of hydrophobic hydration with respect to the temperature give access to the changes in entropy. In practice the entropy differential is determined from the negative of the slope of tangential lines formed at a certain target temperature in the free energy profile. The obtained changes in entropy are negative for small-sized cavities, and hence reconfirm the basic ideas of the Lum-Chandler-Weeks theory on hydrophobic hydration of small-sized solutes.

Computer Simulation↗

PubMed2004

Maximum entropy formulation of the Kirkwood superposition approximation.

Using a variational formulation, we derive the Kirkwood superposition approximation for systems at equilibrium in the thermodynamic limit. We define the entropy of the triplet correlation function and show that the Kirkwood closure brings the entropy to its maximal value. This approach leads to a different interpretation for the Kirkwood closure relation, usually explained by probabilistic considerations of dependence and independence of particles. The Kirkwood closure is generalized to finite volume systems at equilibrium by computing the pair correlation function in finite domains. Closure relations for high order correlation functions are also found using a variational approach. In particular, maximizing the entropy of quadruplets leads to the high order closure g(1234)=g(123)g(124)g(134)g(234)/[g(12)g(13)g(14)g(23)g(24)g(34)] used in the Born-Green-Yvon 2 equations which are a pair of integral equations for the triplet and pair correlation functions.

Entropy↗

PubMed2005

Calculation of the entropy and free energy by the hypothetical scanning Monte Carlo method: application to peptides.

A new approach, the hypothetical scanning Monte Carlo (HSMC), for calculating the absolute entropy, S, and free energy, F, has been introduced recently and applied first to fluids (argon and water) and later to peptides. In this paper the method is further developed for peptide chains in vacuum. S is calculated from a given MC sample by reconstructing each sample conformation i step-by-step, i.e., calculating transition probabilities (TPs) for the dihedral and bond angles and fixing the related atoms at their positions. At step k of the process the chain's coordinates that have already been determined are kept fixed (the "frozen past") and TP(k) is obtained from a MC simulation of the "future" part of the chain whose TPs as yet have not been determined; when the process is completed the contribution of conformation i to the entropy is, S(i) approximately -ln Pi(k) TP(k). In a recent paper we studied polyglycine chains, modeled by the AMBER force field with constant bond lengths and bond angles (the rigid model). Decaglycine [(Gly)(10)] was studied in the helical, extended, and hairpin microstates, while (Gly)(16) was treated only in the first two microstates. In this paper the samples are increased and restudied, (Gly)(16) is also investigated in the hairpin microstate, and for (Gly)(10) approximations are tested where only part of the future is considered for calculating the TPs. We calculate upper and lower bounds for F and demonstrate that like for fluids, F can be obtained from multiple reconstructions of a single conformation. We also test a more realistic model of (Gly)(10) where the bond angles are allowed to move (the flexible model). Very accurate results for S and F are obtained which are compared to results obtained by the quasiharmonic approximation and the local states method. Thus, differences in entropy and free energy between the three microstates are obtained within errors of 0.1-0.3 kcal/mol. The HSMC method can be applied to a macromolecule with any degree of flexibility, ranging from local fluctuations to a random coil. The present results demonstrate that the difference in stability, DeltaF(mn)=F(m)-F(n) between significantly different microstates m and n, can be obtained from two simulations only without the need to resort to thermodynamic integration. Our long-term goal is to extend this method to any peptide and apply it to a peptide immersed in a box with explicit water.

Computer Simulation↗

PubMed2006

Pair-correlation entropy of hydrophobic hydration: decomposition into translational and orientational contributions and analysis of solute-size effects.

We develop an efficient method to evaluate the translational and orientational contributions to the solute-water pair-correlation entropy that is a major component of the hydration entropy. A water molecule is modeled as a hard sphere of diameter dS=0.28 nm in which a point dipole and a point quadrupole of tetrahedral symmetry are embedded. A hard sphere of diameter dM, a hydrophobic solute, is immersed at infinite dilution in the model water. The pair-correlation entropy is decomposed into the translational and orientational contributions in an analytical manner using the angle-dependent Ornstein-Zernike integral equation theory. The two contributions are calculated for solutes with a variety of sizes (0.6<or=dM/dS<or=30). The effects of the solute-water attractive interaction are also studied. As dM becomes larger, the percentage of the orientational contribution first increases, takes a maximum value at dM=DM (DM/dS depends on the strength of the solute-water attractive interaction and is in the range of 1.4-2), and then decreases toward a limiting value. The percentage of the orientational contribution reduces progressively as the solute-water attractive interaction becomes stronger. The physical origin of the maximum orientational restriction at dM=DM is discussed in detail.

Chemistry, Physical↗

PubMed2006

Validation of Markov state models using Shannon's entropy.

Markov state models are kinetic models built from the dynamics of molecular simulation trajectories by grouping similar configurations into states and examining the transition probabilities between states. Here we present a procedure for validating the underlying Markov assumption in Markov state models based on information theory using Shannon's entropy. This entropy method is applied to a simple system and is compared with the previous eigenvalue method. The entropy method also provides a way to identify states that are least Markovian, which can then be divided into finer states to improve the model.

Bayes Theorem↗

PubMed2006

Calculation of the entropy and free energy of peptides by molecular dynamics simulations using the hypothetical scanning molecular dynamics method.

Hypothetical scanning (HS) is a method for calculating the absolute entropy S and free energy F from a sample generated by any simulation technique. With this approach each sample configuration is reconstructed with the help of transition probabilities (TPs) and their product leads to the configuration's probability, hence to the entropy. Recently a new way for calculating the TPs by Monte Carlo (MC) simulations has been suggested, where all system interactions are taken into account. Therefore, this method--called HSMC--is in principle exact where the only approximation is due to insufficient sampling. HSMC has been applied very successfully to liquid argon, TIP3P water, self-avoiding walks on a lattice, and peptides. Because molecular dynamics (MD) is considered to be significantly more efficient than MC for a compact polymer chain, in this paper HSMC is extended to MD simulations as applied to peptides. Like before, we study decaglycine in vacuum but for the first time also a peptide with side chains, (Val)(2)(Gly)(6)(Val)(2). The transition from MC to MD requires implementing essential changes in the reconstruction process of HSMD. Results are calculated for three microstates, helix, extended, and hairpin. HSMD leads to very stable differences in entropy TDeltaS between these microstates with small errors of 0.1-0.2 kcal/mol (T=100 K) for a wide range of calculation parameters with extremely high efficiency. Various aspects of HSMD and plans for future work are discussed.

Chemistry, Physical↗

PubMed2007

Estimation of absolute solvent and solvation shell entropies via permutation reduction.

Despite its prominent contribution to the free energy of solvated macromolecules such as proteins or DNA, and although principally contained within molecular dynamics simulations, the entropy of the solvation shell is inaccessible to straightforward application of established entropy estimation methods. The complication is twofold. First, the configurational space density of such systems is too complex for a sufficiently accurate fit. Second, and in contrast to the internal macromolecular dynamics, the configurational space volume explored by the diffusive motion of the solvent molecules is too large to be exhaustively sampled by current simulation techniques. Here, we develop a method to overcome the second problem and to significantly alleviate the first one. We propose to exploit the permutation symmetry of the solvent by transforming the trajectory in a way that renders established estimation methods applicable, such as the quasiharmonic approximation or principal component analysis. Our permutation-reduced approach involves a combinatorial problem, which is solved through its equivalence with the linear assignment problem, for which O(N3) methods exist. From test simulations of dense Lennard-Jones gases, enhanced convergence and improved entropy estimates are obtained. Moreover, our approach renders diffusive systems accessible to improved fit functions.

Algorithms↗

PubMed2002

Entropy of protein sequences: an integral approach.

Several classifications of protein spatial structures and their structural elements are known. This makes revealing of the relation between these structural elements and sequence fragments rather topical. The most important move in this direction would be the determination of positional sensitivity levels and ranges between the residues in protein sequences. In this work the Shannon-Weaver informational entropy was used as a disorder criterion for solving this problem. This entropy was computed as function of the distance between the amino acid residues in different sets of unhomological protein sequences. Similarity of this function for different sets of protein sequences was shown. Analysis of informational entropy allows detecting a long-range positional correlation (> or =30) between the amino acid residues and oscillations with periods of 3.6 and 2.9. These oscillation periods correspond to periodicity of alpha- and 3(10)-helices.

Computational Biology↗

PubMed2000

Predicting the standard enthalpy (deltaH0f) and entropy (S0) of alkanes by artificial neural networks.

Artificial Neural Networks (ANNs) with Extended Delta-Bar-Delta (EDBD) back propagation learning algorithm have been developed to predict the standard enthalpy and entropy of 87 acyclic alkanes. Molecular weight, boiling point and density of the compounds were used as input parameters. The network's architecture and parameters were optimized to give maximum performances. The best network was a 3-6-2 ANN, and the optimum learning epoch was about 1320. The results show that the maximum relative errors of enthalpy and entropy are less than 3%. They reveal that the performances of ANNs for predicting the enthalpy and entropy of alkanes are satisfying.

Algorithms↗

PubMed2004

Spectral entropy measurement of patient responsiveness during propofol and remifentanil. A comparison with the bispectral index.

BACKGROUND: We compared two spectral entropies, state entropy (SE) and response entropy (RE), based on the irregularity of the EEG, to measure loss of response to verbal command (LOR(verbal)) and noxious stimulus (LOR(noxious)) with the bispectral index (BIS) during propofol infusion with and without remifentanil. METHODS: Three groups of 20 patients received an effect-site controlled propofol infusion (Ce(PROP)) starting at 1 microg ml(-1) and increased in steps of 0.5 microg ml(-1) at 4 min intervals. In addition, a remifentanil infusion was maintained at a group-dependent, fixed effect-site target concentration (Ce(REMI)) (0, 2 or 4 ng ml(-1)). The ability of BIS, SE or RE to predict LOR(verbal) and LOR(noxious) were compared with the changes in BIS, SE and RE using logistic regression, prediction probability (P(K)), and sensitivity/specificity. RESULTS: In all groups, BIS, SE and RE decreased with increasing Ce(PROP). However, BIS decreased more smoothly than SE and RE at deeper levels of sedation. At LOR(verbal), BIS(50), SE(50) and RE(50) increased with increasing Ce(REMI). BIS, SE and RE all detected LOR(verbal) accurately but BIS performed better at 100% sensitivity. Sensitivity/specificity for detection of LOR(verbal) decreased for all methods with increasing Ce(REMI). LOR(noxious) was poorly described by all measures. CONCLUSION: LOR(verbal) was detected accurately by BIS, SE and RE except for 100% sensitivity, where BIS performed better. Though BIS, SE and RE were influenced by remifentanil during propofol administration, their ability to detect LOR(verbal) remained accurate. None of the measures predicted LOR(noxious).

Acoustic Stimulation↗

PubMed2004

Comparative effects of ketamine on Bispectral Index and spectral entropy of the electroencephalogram under sevoflurane anaesthesia.

BACKGROUND: The Bispectral Index (BIS) and spectral entropy of the electroencephalogram can be used to assess the depth of hypnosis. Ketamine is known to increase BIS in anaesthetized patients and may confound that index as a guide to steer administration of hypnotics. We compared the effects of ketamine on BIS, response entropy (RE) and state entropy (SE) during surgery under sevoflurane anaesthesia. METHODS: Twenty-two women undergoing gynaecological surgery were enrolled in this double-blind, randomized study. Anaesthesia was induced i.v. and maintained with sevoflurane. Under stable surgical and anaesthetic conditions, patients were assigned to receive either a bolus of ketamine 0.5 mg kg(-1) or the same volume of saline. Blood pressure, heart rate, BIS, RE and SE were measured every 2.5 min from 10 min before (baseline) until 15 min after ketamine or saline administration. The maximum relative increase in BIS, RE and SE compared with baseline was calculated for each patient. Values are mean (sd). RESULTS: Baseline values were BIS 33 (4), RE 31 (5), SE 30 (5) for the ketamine patients and BIS 35 (3), RE 33 (5) and SE 32 (6) for the patients receiving saline. BIS, RE and SE increased significantly from 5 min (BIS) and 2.5 min (RE and SE) after ketamine administration, peaking at 46 (8) (BIS), 52 (12) (RE) and 50 (12) (SE) respectively. The maximum relative increase in RE [42.2 (10.4%)] and SE [41.6 (10.9)%] was higher than that of BIS [29.4 (10.4%)]. Blood pressure, heart rate and RE-SE gradient did not change in either group. CONCLUSIONS: Ketamine administered under sevoflurane anaesthesia causes a significant increase in BIS, RE and SE without modification of the RE-SE gradient. This increase is paradoxical in that it is associated with a deepening level of hypnosis.

Adult↗

PubMed1999

Comparison of the entropy technique with two other techniques for detecting disease clustering using data from children with high blood lead levels.

The entropy technique was compared with two other case-control techniques for detecting disease clustering using data on blood lead levels of children who were patients at the King/Drew Medical Center in South-Central Los Angeles in 1991 to 1994. The other two methods are the nearest neighbor technique (NNT) and Moran's IPOP technique, a variation of Moran's I test, in which rates are adjusted for population size. Four different blood lead levels (15 microg/dl, 20 microg/dl, 30 microg/dl, 35 microg/dl) were used as cutoff levels to designate cases. Persons with blood lead levels greater than or equal to the cutoff level were designated as cases. The authors found significant clustering for all four cutoff levels using the entropy method, and for the first three cutoff levels using the NNT. They found significant clustering with Moran's IPOP for some scales for two of the cutoff levels. While performance of the entropy technique and the NNT were independent of scale, that of Moran's IPOP was highly scale-dependent.

Adolescent↗

PubMed2006

Electroencephalographic approximate entropy changes in healthy volunteers during remifentanil infusion.

BACKGROUND: The aim of this study was to investigate the independent effect of remifentanil on the approximate entropy (ApEn) in frontoparietal montages. The authors investigated which montages were relevant to assess the remifentanil effect on the electroencephalogram. Spectral edge frequency and the canonical univariate parameter were used as comparators. METHODS: Twenty-eight healthy volunteers were enrolled. With recording of the electroencephalogram at the F3, F4, Cz, P3, and P4 montages, remifentanil was infused at the rate of 1-8 mug . kg . min for 15-20 min. The relation between remifentanil concentration and the electroencephalographic parameters were tested by Spearman correlation. Signal-to-noise ratio, artifact robustness, coefficient of variation of the median baseline and maximal electroencephalographic effects, and ratio of average maximal electroencephalographic effect to interindividual baseline variability were measured. The performance of ApEn as an index of remifentanil effect site concentrations was tested by prediction probability. RESULTS: Approximate entropy showed significant correlation (R = -0.6465, P < 0.0001) with remifentanil concentration. It provided comparable signal-to-noise ratio, artifact robustness, and ratio of average maximal electroencephalographic effect to interindividual baseline variability to 95% spectral edge frequency. The coefficients of variation of the median baseline and maximal electroencephalo graphic effects were smallest in ApEn. Parietal montages showed higher ratios of average maximal electroencephalographic effect to interindividual baseline variability for all electroencephalographic parameters and lower coefficients of variation of the baseline values for ApEn and 95% spectral edge frequency than frontal montages. The prediction probability of ApEn was 0.7730. CONCLUSIONS: Approximate entropy derived from a parietal montage is appropriate for the assessment of the remifentanil effect on the electroencephalogram.

Adult↗

PubMed1998

Approximate entropy of respiratory rate and tidal volume during weaning from mechanical ventilation.

OBJECTIVE: To determine the effects of respiratory failure on respiratory rate pattern and tidal volume pattern. DESIGN: Prospective, clinical study. SETTING: Cardiovascular intensive care unit. PATIENTS: Ten patients within 12 hrs of cardiac surgery, and 21 patients who required prolonged (>7 days) mechanical ventilation. INTERVENTIONS: Patients were placed on spontaneous ventilation for weaning trials. MEASUREMENTS AND MAIN RESULTS: During spontaneous ventilation, each breath's instantaneous respiratory rate and tidal volume were recorded for later analysis. Approximate entropy (ApEn) was calculated for respiratory rate and tidal volume series of the terminal 1000 breaths on each spontaneous ventilation trial in series of 100, 300, and 1000 breaths. Ten patients (controls) were studied and extubated within 12 hrs of cardiac surgery. The other 21 patients were studied during attempts to wean them from mechanical ventilation. These patients passed (Group V-Pass) 59 and failed (Group V-Fail) 14 weaning trials. Mean tidal volume did not vary between groups, but respiratory rate increased progressively from the control group to Group V-Pass to Group V-Fail (p < .017). Conversely, approximate entropy of respiratory rate (ApEn-RR) did not vary among the three groups at any time series length, but approximate entropy of tidal volume (ApEn-VT) increased from the control group to Group V-P (p< .017) to Group V-F (p< .017) at all time series lengths. ApEn-VT was very specific but only moderately sensitive at identifying respiratory failure. CONCLUSION: Respiratory failure causes tidal volume patterns to become increasingly irregular, but increasing respiratory rate has no effect on respiratory rate pattern.

Aged↗

PubMed2005

Response entropy increases during painful stimulation.

Frontal electromyography (FEMG) may increase during painful stimulation and indicate patient arousal. The Datex-Ohmeda Entropy Module calculates state entropy (SE) of the electroencephalogram (EEG; 0.8-32 Hz) and response entropy (RE) of EEG and FEMG (0.8-47 Hz). We determined whether RE increases above SE (RE--SE), an indication of FEMG, increase during painful stimuli and if this is related to paralysis or level of anesthesia. With the unanesthetized baseline measurement, SE was 89 +/- 2 and RE was 98 +/- 2. During paralysis and anesthesia with either 0.8% (n = 10) or 1.4% (n = 10) isoflurane, SE decreased to 63 +/- 7 and 34 +/- 14, respectively, and the RE--SE difference decreased 90%. Before recovery from paralysis, arterial catheter or head pin placement increased RE--SE above unanesthetized levels in eight patients (five treated with 0.8% and three with 1.4% isoflurane), consistent with an increase in FEMG. The elevated RE--SE difference was related to a significant increase in SE, blood pressure, and heart rate. After recovery from paralysis, tetanic stimulation of the ulnar nerve increased the RE--SE difference above unanesthetized levels in 8 of 20 patients (6 treated with 0.8% and 2 with 1.4% isoflurane). In these patients, SE increased significantly. The remaining 12 patients did not show an increase in RE--SE during tetanic stimulation and SE did not increase. We conclude that increased RE during painful stimulation was not dependent on recovery from paralysis but was seen more often in patients anesthetized with 0.8% compared with 1.4% isoflurane. This suggests that RE reflects FEMG and may be useful to identify inadequate anesthesia and patient arousal during painful stimuli.

Aged↗

PubMed2001

Toward a theory of the general-anesthetic-induced phase transition of the cerebral cortex. II. Numerical simulations, spectral entropy, and correlation times.

In our two recent papers [M.L. Steyn-Ross et al., Phys. Rev. E 60, 7299 (1999); 64, 011917 (2001)] we presented clinical evidence for a general anesthetic-induced phase change in the cerebral cortex, and showed how the significant features of the cortical phase change (biphasic power surge, spectral energy redistribution, "heat capacity" divergence), could be explained using a stochastic single-macrocolumn model of the cortex. The model predictions were based on rather strong "adiabatic" assumptions which assert that the mean-field excitatory and inhibitory macrocolumn voltages are "slow" variables whose equilibration times are much longer than those of the input "currents" that drive the macrocolumn. In the present paper we test the adiabatic assumption by running numerical simulations of the stochastic differential equations. These simulations confirm the number and nature of the steady-state solutions, the growth of fluctuation power at transition, and the redistribution of spectral energy towards lower frequencies. We use spectral entropy to quantify these changes in the power spectral density, and to show that the spectral entropy should decrease markedly at the point of transition. This prediction agrees with recent clinical findings by Viertiö-Oja and colleagues [J. Clinical Monitoring Computing 16, 60 (2000)]. Our modeling work shows that there is an inverse relationship between spectral entropy H and correlation time T of the soma-voltage fluctuations: H inversely proportional to (ln T). In a theoretical analysis we prove that this proportionality becomes exact for an ideal Lorentzian process. These findings suggest that by monitoring the changes in EEG correlation time, it should be possible to track changes in the state of patient consciousness.

Anesthesia, General↗

PubMed2002

Comment on "Kullback-Leibler and renormalized entropies: applications to electroencephalograms of epilepsy patients".

In a recent paper Quian Quiroga et al. [R. Quian Quiroga et al., Phys. Rev. E 62, 8380 (2000)] found renormalized entropy, formerly introduced as a complexity measure for the different regimes of a dynamical system, to be closely related to the standard Kullback-Leibler entropy. They assure this finding by reanalyzing electroencephalographic data of epilepsy patients, previously examined by exclusive use of renormalized entropy [K. Kopitzki et al., Phys. Rev. E 58, 4859 (1998)]. We argue that the general considerations undertaken by the authors and the experimental results do not justify this conclusion.

Data Interpretation, Statistical↗

PubMed2004

Observability of Rényi's entropy.

Despite recent claims we argue that Rényi's entropy is an observable quantity. It is shown that, contrary to popular belief, the reported domain of instability for Rényi entropies has zero measure (Bhattacharyya measure). In addition, we show that the instabilities can be easily emended by introducing a coarse graining into an actual measurement. We also clear up any doubts regarding the observability of Rényi's entropy in (multi)fractal systems and in systems with absolutely continuous probability density functions.

Entropy↗