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PubMed2001

Entropy-enthalpy compensation: fact or artifact?

The phenomenon of entropy-enthalpy (S-H) compensation is widely invoked as an explanatory principle in thermodynamic analyses of proteins, ligands, and nucleic acids. It has been suggested that this compensation is an intrinsic property of either complex, fluctuating, or aqueous systems. The questions examined here are whether the observed compensation is extra-thermodynamic (i.e., reflects anything more than the well-known laws of statistical thermodynamics) and if so, what does it reveal about the system? Compensation is rather variably defined in the literature and different usages are discussed. The most precise and interesting one, which is considered here, is a linear relationship between DeltaH and DeltaS for some series of perturbations or changes in experimental variable. Some recent thermodynamic data on proteins purporting to show compensation is analyzed and shown to be better explained by other causes. A general statistical mechanical model of a complex system is analyzed to explore whether and under what conditions extra-thermodynamic compensation can occur and what it reveals about the system. This model shows that the most likely behavior to be seen is linear S-H compensation over a rather limited range of perturbations with a compensation temperature Tc = dDeltaH/dDeltaS within 20% of the experimental temperature. This behavior is insensitive to the details of the model, thus revealing little extra-thermodynamic or causal information about the system. In addition, it will likely be difficult to distinguish this from more trivial forms of compensation in real experimental systems.

Artifacts↗

PubMedDate not supplied

The development of an intuitive understanding of entropy.

Previous studies of the concept of entropy have indicated that before 7 to 9 years of age, children have difficulty understanding the effects of randomizing forces and often predict highly improbable events, such as a return to the original ordered state. In the present research, 306 children between 3 and 11 years of age were asked directly whether undifferentiated forces, such as the wind or objects being thrown into the air, could create order or disorder in a set of objects. The results showed that even 4-year-olds are sensitive to the asymmetrical effects of such events. Older children apply this principle more consistently and are able to use it in explaining their answers.

Age Factors↗

PubMed2005

Approximate entropy of fetal heart rate variability as a predictor of fetal distress in women at term pregnancy.

BACKGROUND: The aim of this study is to investigate the relationship between approximate entropy (ApEn) of fetal heart rate (FHR) with umbilical blood gas parameters and the power spectrum of FHR variability in Chinese and to test whether ApEn of FHR variability could be used as a predictor of fetal distress in women at term pregnancy. METHODS: Sixty-seven pregnant women with singleton, term fetus were recruited for the recording of FHR variability and the data were used for the estimate of ApEn. Blood gases after birth were measured through umbilical artery. RESULTS: In all 67 neonates, there was some amount of interinfant variability in the ApEn values with a mean of 1.139 +/- 0.169. The ApEn values were significantly (P < 0.05) correlated with pO(2), SO(2), pCO(2), pH, HCO(3), or base excess (BE). The 15 fetuses with low ApEn (ApEn <1.0) had higher risk of metabolic acidosis (BE less than -12 mmol/l) than those with high ApEn (ApEn > or =1) (likelihood ratio = 12.301, P < 0.001). The powers of FHR variability in all frequency ranges (0-0.256 Hz) were lower in the low-ApEn group than those in the high-ApEn group. CONCLUSIONS: The ApEn of FHR variability significantly decreased during fetal asphyxia, including hypoxia, hypercapnia, and both respiratory and metabolic acidosis; low ApEn was linked to decreased power spectrum density in all frequency domains. The ApEn values may be used as a predictor of fetal distress in women at term pregnancy.

Acidosis↗

PubMed1999

Multiple comparison of entropies with application to dinosaur biodiversity.

Did the biodiversity of dinosaurs decline, or did it remain more or less constant before their mass extinction 65 million years ago? Sheehan et al. (1991, Science, 835-839) reported that the biodiversity of families of dinosaur species remained more or less constant preceding their extinction, suggesting extinction due to a cataclysmic event such as an asteroid strike. But that claim was based on the incorrect interpretation that a large p value associated with a test of null hypothesis of equality supports that null hypothesis. To assess whether there is a basis for such a claim, we formulate the problem as one of practical equivalence, in analalogy to bioequivalence. We then develop reliable practical equivalence confidence intervals for differences of entropies by applying the bootstrap-t technique to a nearly pivotal quantity. Confidence intervals for changes in the biodiversity of dinosaurs are then computed, allowing the reader to assess whether there is evidence of near constancy of dinosaur biodiversity before extinction.

Animals↗

PubMed2006

Approximate entropy of self-reported mood prior to episodes in bipolar disorder.

BACKGROUND: Approximate entropy (ApEn) measures regularity in time series data, while traditional linear statistics measure variability. Using self-reported mood data from patients with bipolar disorder, this preliminary study addressed whether ApEn could distinguish (i) the 60 days prior to the start of a manic or depressed episode from the 60 days prior to a month of euthymia, and (ii) the 60 days prior to a manic episode from the 60 days prior to a depressed episode. METHODS: Self-reported mood data from 49 outpatients with bipolar disorder receiving standard treatment were analysed. The data contained 27 episodes (12 manic and 15 depressed), and 43 periods of 1 month of euthymia. For the 60 days prior to episode or euthymia, the ApEn, linear statistics and the correlation between linear and non-linear measures were calculated. RESULTS: ApEn was significantly greater in the 60 days prior to a manic or depressive episode than the 60 days prior to a month of euthymia. The onset of an episode was associated with greater irregularity in mood. Variability was also significantly larger and correlated with ApEn. ApEn was significantly greater in the 60 days prior to a manic episode than in the 60 days prior to a depressed episode, whereas measures of variability were not significantly different. Mood in the 60 days prior to mania was more irregular than prior to depression. CONCLUSIONS: Non-linear measures may complement traditional linear measures in the analysis of longitudinal data in bipolar disorder. A larger study is indicated.

Adult↗

PubMed2006

Approximate entropy as a measure of irregularity for psychiatric serial metrics.

OBJECTIVES: The quantification of subtle patterns in sequential data, and their changes, has considerable potential utility throughout psychiatry, including the analyses of mood ratings, heart rate, respiratory, and electroencephalographic recordings. METHODS: Approximate entropy (ApEn), a relatively recently developed statistic quantifying serial irregularity, has been applied in numerous studies throughout mathematics and other fields of study, especially biology. RESULTS: We discussed applications of ApEn, both extant and potential, of most relevance to psychiatrists. We provided a mechanistic interpretation of lowered ApEn values, and discusses the relationship between ApEn and other (both classical and complexity) measures of serial dynamics. We also briefly discussed cross-ApEn, a thematically similar quantification of two-variable asynchrony that can aid in uncovering subtle disruptions in complicated network dynamics. CONCLUSIONS: ApEn and cross-ApEn have significant potential to consequentially enhance present statistical methodologies of analysis of psychiatric data, in both clinical and in research settings.

Adult↗

PubMed2005

Characteristics of heart rate variability entropy and blood pressure during hemodialysis in patients with end-stage renal disease.

Entropy (ENT) is a newly developed measure of the complexity of heart rate variability (HRV). The aim of this study was to characterize the complexity of HRV in patients with end-stage renal disease (ESRD) and to find a possible clinical utility. Healthy subjects and patients with ESRD undergoing hemodialysis (HD) were recruited. The HD population consisted of patients with and without diabetes mellitus (DM). An electrocardiogram was recorded before HD, and blood pressure was measured during HD. The coefficients of variation of R-R intervals, high- and low-frequency components, and ratio of the low- to high-frequency components were measured as variables of HRV. The ENT was used to describe the complexity of HRV. Forty-six healthy subjects and 27 HD patients participated in this study. The ENT negatively correlated with the duration of DM (p = 0.001), systolic blood pressure (p = 0.003), and mean blood pressure (p = 0.004) before a HD session. ENT in HD patients was lower than that in healthy subjects (p < 0.01). ENT in HD patients with DM was lower than that in HD patients without DM (p < 0.01). The change in systolic blood pressure (DeltaSBP) during a HD session showed high correlations to ENT and ultrafiltration rate (UFR) of the dialyzer. The following equation was obtained: DeltaSBP = 2.25 x ENT - 2.28 x UFR - 21.27 (R2 = 0.805; p < 0.0001). ENT decreased with uremic and diabetic status. ENT also represents a possible prediction of hypotension during a HD session.

Adult↗

PubMed2005

Assessing phylogenetic relationships of Lycium samples using RAPD and entropy theory.

AIM: To evaluate the phylogenetic relationships among related species of Lycium samples. METHODS: Random amplified polymorphic DNA (RAPD) fingerprinting and lab-on-a-chip electrophoresis techniques were used to analyze the characteristics of Lycium species. Seven species and 3 varieties of Lycium were studied. Based on RAPD fingerprint data obtained from 11 primers, we proposed a new index, called dispersivity, using entropy theory and projection methods to depict the diversity of the DNA fingerprints. RESULTS: Using the proposed dispersivity, primers were sorted and the dendrograms of the 7 species and 3 varieties of Lycium were constructed synthetically by merging primer information. CONCLUSION: Phylogenetic relationships among Lycium samples were constructed synthetically based on RAPD fingerprint data generated from 11 primers.

DNA Fingerprinting↗

PubMed2000

Approximate entropy studies of hormone pulsatility from plasma concentration time series: influence of the kinetics assessed by simulation.

Approximate entropy (ApEn) is a method developed in the early nineties to quantify the "regularity" of a time series. In recent years, it has been vigorously employed to study the oscillatory/pulsatile secretory behavior of many hormones and found capable of successfully identifying pathological or prepathological states characterized by an enhanced secretion irregularity. Since hormone secretion rate is nonaccessible to direct measurement, ApEn is usually calculated from the time series of the hormone concentrations in plasma. However, the plasma concentration time course also reflects the whole-body kinetics of the hormone and can thus only provide a distorted portrait of the secretion rate at the gland level. In this paper, we investigate by simulation whether and how this distortion can influence the study of the regularity of hormone pulsatility by ApEn. Pulsatile secretion time series with different degrees of irregularity are simulated by varying the statistics of the random parameters which describe the secretory pulses. Then, plasma concentration time series are obtained by convolution with the hormone impulse response. Different degrees of impulse response smoothness are also considered in order to vary the amount of the distortion introduced. Results show that ApEn computed from secretion time series consistently discriminated better than ApEn calculated from plasma concentration time series among processes with different degrees of regularity. In addition, smoother impulse responses decreased the ApEn differences between plasma concentration time series corresponding to different degrees of secretion regularity. Therefore, the power of the ApEn index in the study of hormone pulsatility can potentially be enhanced by applying it to the hormone secretion time series.

Algorithms↗

PubMedDate not supplied

Ratio of entropy to enthalpy in thermal transitions in biological tissues.

Thermal transitions in biological tissues that have been reported in the literature are summarized in terms of the apparent molar entropy (DeltaS) and molar enthalpy (DeltaH) involved in the transition. A plot of DeltaS versus DeltaH for all the data yields a straight line, consistent with the definition of free energy, DeltaG=DeltaH+TDeltaS. Various bonds may be involved in cooperative bond breakage during thermal transitions; however, for the sake of description, the equivalent number of cooperative hydrogen bonds can be cited. Most of the tissue data behave as if 10 to 20 hydrogen bonds are cooperatively broken during coagulation, with one transition, the expression of heat shock protein, involving 90 cooperative hydrogen bonds. The data are consistent with DeltaS=a+bDeltaH, where a=-327.5 J(mol K) and b=31.47 x 10(-4) K(-1). If each additional hydrogen bond adds 19 x 10(3) Jmol to DeltaH, then each additional bond adds 59.8 J(mol K) to DeltaS. Hence, the dynamics of irreversible thermal transitions can be described in terms of one free parameter, the apparent number of cooperative hydrogen bonds broken during the transition.

Animals↗

PubMed2006

Geometric partition functions of cellular systems: explicit calculation of the entropy in two and three dimensions.

A method is proposed for the characterisation of the entropy of cellular structures, based on the compactivity concept for granular packings. Hamiltonian-like volume functions are constructed both in two and in three dimensions, enabling the identification of a phase space and making it possible to take account of geometrical correlations systematically. Case studies are presented for which explicit calculations of the mean vertex density and porosity fluctuations are given as functions of compactivity. The formalism applies equally well to two- and three-dimensional granular assemblies.

Aerosols↗

PubMed2004

Complexity quantification of dense array EEG using sample entropy analysis.

In this paper, a time series complexity analysis of dense array electroencephalogram signals is carried out using the recently introduced Sample Entropy (SampEn) measure. This statistic quantifies the regularity in signals recorded from systems that can vary from the purely deterministic to purely stochastic realm. The present analysis is conducted with an objective of gaining insight into complexity variations related to changing brain dynamics for EEG recorded from the three cases of passive, eyes closed condition, a mental arithmetic task and the same mental task carried out after a physical exertion task. It is observed that the statistic is a robust quantifier of complexity suited for short physiological signals such as the EEG and it points to the specific brain regions that exhibit lowered complexity during the mental task state as compared to a passive, relaxed state. In the case of mental tasks carried out before and after the performance of a physical exercise, the statistic can detect the variations brought in by the intermediate fatigue inducing exercise period. This enhances its utility in detecting subtle changes in the brain state that can find wider scope for applications in EEG based brain studies.

Adult↗

PubMed1997

Effect of convective stretching and folding on aerosol mixing deep in the lung, assessed by approximate entropy.

There is a surprisingly substantial amount of aerosol mixing and deposition deep in the lung, which cannot be explained by classic transport mechanisms such as streamline crossing, inertial impaction, or gravitational sedimentation with reversible acinar flow. Mixing associated with "stretch and fold" convective flow patterns can, however, be a potent source of transport. We show such patterns in experimental preparations using rat lungs and in the theoretical Baker Transform. In both cases, mixing is associated with the temporal evolution of two length scales. The first is the slowly increasing diffusive length scale. The second is the rapidly decreasing lateral length scale, due to "stretching and folding," over which diffusion must take place. This interaction leads to aerosol mixing in much shorter times than previously appreciated. Finally, we propose a new method by which to quantify the state of mixing, using an approximation to the entropy of the aerosol concentration distribution. The results of the analysis suggest that stretching and folding may be a key feature underlying peripheral aerosol transport.

Aerosols↗

PubMed2002

Entropy as a measure for linkage disequilibrium over multilocus haplotype blocks.

OBJECTIVE: The presence of linkage disequilibrium (LD) forms the basis for a range of uses, including the fine-mapping of diseases and studies on human genealogy. Recent findings indicate that single nucleotide polymorphisms (SNP) can occur in blocks of limited haplotypic diversity with high degrees of LD. Commonly used measures for LD, such as r(2) and D', consider only two loci and might miss information to appropriately describe LD in larger haplotypic structures. METHODS: We introduce the Normalized Entropy Difference, epsilon, as a new multilocus measure for LD. A related quantity, deltaS, provides an approximate chi(2) test for the significance of LD. The ability of the measure to detect haplotype blocks is investigated using simulated data sets as well as a real data set previously analyzed by Daly et al. (2001). RESULTS: epsilon allows for arbitrary numbers of loci, describes LD with regard to the loci sequence, and can be interpreted as a multilocus extension of r(2). The application of epsilon to the data sets demonstrated the measure's ability to appropriately describe simultaneous multilocus LD and to detect haplotype blocks. CONCLUSIONS: epsilon is a reasonable multilocus LD measure and might be of potential use in the construction of the human haplotype map.

Chromosome Mapping↗

PubMed2005

Edgeworth approximation of multivariate differential entropy.

We develop the general, multivariate case of the Edgeworth approximation of differential entropy and show that it can be more accurate than the nearest-neighbor method in the multivariate case and that it scales better with sample size. Furthermore, we introduce mutual information estimation as an application.

Algorithms↗

PubMed2006

Enhancing density-based data reduction using entropy.

Data reduction algorithms determine a small data subset from a given large data set. In this article, new types of data reduction criteria, based on the concept of entropy, are first presented. These criteria can evaluate the data reduction performance in a sophisticated and comprehensive way. As a result, new data reduction procedures are developed. Using the newly introduced criteria, the proposed data reduction scheme is shown to be efficient and effective. In addition, an outlier-filtering strategy, which is computationally insignificant, is developed. In some instances, this strategy can substantially improve the performance of supervised data analysis. The proposed procedures are compared with related techniques in two types of application: density estimation and classification. Extensive comparative results are included to corroborate the contributions of the proposed algorithms.

Algorithms↗

PubMed2005

The estimation of distributions and the minimum relative entropy principle.

Estimation of Distribution Algorithms (EDA) have been proposed as an extension of genetic algorithms. In this paper we explain the relationship of EDA to algorithms developed in statistics, artificial intelligence, and statistical physics. The major design issues are discussed within a general interdisciplinary framework. It is shown that maximum entropy approximations play a crucial role. All proposed algorithms try to minimize the Kullback-Leibler divergence KLD between the unknown distribution p(x) and a class q(x) of approximations. However, the Kullback-Leibler divergence is not symmetric. Approximations which suppose that the function to be optimized is additively decomposed (ADF) minimize KLD(q||p), the methods which learn the approximate model from data minimize KLD(p||q). This minimization is identical to maximizing the log-likelihood. In the paper three classes of algorithms are discussed. FDA uses the ADF to compute an approximate factorization of the unknown distribution. The factors are marginal distributions, whose values are computed from samples. The second class is represented by the Bethe-Kikuchi approach which has recently been rediscovered in statistical physics. Here the values of the marginals are computed from a difficult constrained minimization problem. The third class learns the factorization from the data. We analyze our learning algorithm LFDA in detail. It is shown that learning is faced with two problems: first, to detect the important dependencies between the variables, and second, to create an acyclic Bayesian network of bounded clique size.

Algorithms↗

PubMed2001

An entropy based method to evaluate hazardous inorganic substance balances of waste treatment systems.

At present, methods to assess the environmental performance of waste treatment systems are based mainly on emission standards. Since waste treatment yields also products used for recycling and/or land filling, new additional instruments are needed to analyse and evaluate the partitioning of substances during waste treatment. It is shown that in order to fulfil the goals of European Waste Management, waste treatment systems have to concentrate hazardous inorganic substances. Based on substance flow analysis and the statistical entropy function, a method is presented to classify waste treatment systems in view of their power to concentrate or dilute substances. The method allows for the determination of the Substance Concentrating Efficiency (SCE) of waste treatment systems quantitatively. The SCE approach is applied in a case study comparing waste incineration and mechanical-biological waste processing. The results show that this new tool is useful to support decisions regarding the selection of waste treatment systems.

Biodegradation, Environmental↗