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Entropy effects during sorption of alkanes in zeolites.

Recent developments in Configurational-Bias Monte Carlo (CBMC) techniques allow the accurate calculation of the sorption isotherms for alkanes, and their mixtures, in various zeolites. The CBMC simulations give new insights into subtle entropy effects affecting mixture adsorption. Three types of entropy effects can be distinguished. (1) Size entropy effects favour the component with the smaller number of C atoms because the smaller molecule finds it easier to fill in the 'gaps' within the zeolite matrix at high molecular loadings. (2) Configurational entropy effects come into play for mixtures of alkanes that differ in the degree of branching. For a mixture of linear and branched alkanes with the same number of C atoms, configurational entropy effects favour the linear isomer because such molecules 'pack' more efficiently within, say, the intersecting channel topology of MFI zeolite. (3) Length entropy effects comes into force for sorption of linear and branched alkanes within the cylindrical channels of say AFI and MOR zeolites; here the double branched alkane has the shortest length and can be packed more efficiently within the channels. We demonstrate that CBMC simulations allow the efficient screening of zeolite structures for a given separation duty and aid the development of novel separation processes exploiting entropy effects.

Journal Article↗

Estimating entropies from molecular dynamics simulations.

While the determination of free-energy differences by MD simulation has become a standard procedure for which many techniques have been developed, total entropies and entropy differences are still hardly ever computed. An overview of techniques to determine entropy differences is given, and the accuracy and convergence behavior of five methods based on thermodynamic integration and perturbation techniques was evaluated using liquid water as a test system. Reasonably accurate entropy differences are obtained through thermodynamic integration in which many copies of a solute are desolvated. When only one solute molecule is involved, only two methods seem to yield useful results, the calculation of solute-solvent entropy through thermodynamic integration, and the calculation of solvation entropy through the temperature derivative of the corresponding free-energy difference. One-step perturbation methods seem unsuitable to obtain entropy estimates.

Journal Article↗

Voronoi cell volume distribution and configurational entropy of hard-spheres.

The Voronoi cell volume distributions for hard-disk and hard-sphere fluids have been studied. The distribution of the Voronoi free volume vf, which is the difference between the actual cell volume and the minimal cell volume at close packing, is well described by a two-parameter (2gamma) or a three-parameter (3gamma) gamma distribution. The free parameter m in both the 2gamma and 3gamma models is identified as the "regularity factor." The regularity factor is the ratio of the square of the mean and the variance of the free volume distribution, and it increases as the cell volume distribution becomes narrower. For the thermodynamic structures, the regularity factor increases with increasing density and it increases sharply across the freezing transition, in response to the onset of order. The regularity factor also distinguishes between the dense thermodynamic structures and the dense random or quenched structures. The maximum information entropy (max-ent) formalism, when applied to the gamma distributions, shows that structures of maximum information entropy have an exponential distribution of vf. Simulations carried out using a swelling algorithm indicate that the dense random-packed states approach the distribution predicted by the max-ent formalism, though the limiting case could not be realized in simulations due to the structural inhomogeneities introduced by the dense random-packing algorithm. Using the gamma representations of the cell volume distribution, we check the numerical validity of the Cohen-Grest expression [M. H. Cohen and G. S. Grest, Phys. Rev. B 20, 1077 (1979)] for the cellular (free volume) entropy, which is a part of the configurational entropy. The expression is exact for the hard-rod system, and a correction factor equal to the dimension of the system, D, is found necessary for the hard-disk and hard-sphere systems. Thus, for the hard-disk and hard-sphere systems, the present analysis establishes a relationship between the precisely defined Voronoi free volume (information) entropy and the thermodynamic entropy. This analysis also shows that the max-ent formalism, when applied to the free volume entropy, predicts an exponential distribution which is approached by disordered states generated by a swelling algorithm in the dense random-packing limit.

Journal Article↗

Entropy, diffusivity, and structural order in liquids with waterlike anomalies.

The excess entropy, defined as the difference between the entropies of the liquid and the ideal gas under identical density and temperature conditions, is studied as a function of density and temperature for liquid silica and a two-scale ramp potential, both of which are known to possess waterlike liquid state anomalies. The excess entropy for both systems is evaluated using a fairly accurate pair correlation approximation. The connection between the excess entropy and the density and diffusional anomalies is demonstrated. Using the pair correlation approximation to the excess entropy, it can be shown that if the energetically favorable local geometries in the low and high density limits have different symmetries, then a structurally anomalous regime can be defined in terms of orientational and translational order parameters, as in the case of silica and the two-scale ramp system but not for the one-scale ramp liquid. Within the category of liquids with waterlike anomalies, we show that the relationship between the macroscopic entropy and internal energy is sufficient to distinguish between those with local anisotropy and consequent open packings at low densities and those with isotropic interactions but multiple length scales. Since it is straightforward to evaluate the pair correlation entropy and internal energy from simulations or experimental data, such plots should provide a convenient means to diagnose the existence as well as type of anomalous behavior in a range of liquids, including ionic and intermetallic melts and complex fluids with ultrasoft repulsions.

Journal Article↗

Enthalpy-entropy compensations in drug-DNA binding studies.

We present a comparative study of calorimetrically derived thermodynamic profiles for the binding of a series of drugs with selected DNA host duplexes. We use these data to demonstrate that comparisons between complete thermodynamic profiles (delta G zero, delta H zero, delta S zero, delta Cp) are required before drug binding can be used as a probe of DNA conformation, since enthalpy-entropy compensations can cause two drug-DNA binding events to exhibit similar binding free energies (delta G zero) despite being driven by entirely different thermodynamic forces (delta H zero, delta S zero). In this work, we employ a combination of spectroscopic and calorimetric techniques to characterize thermodynamically the DNA binding of netropsin and distamycin (two minor groove-directed ligands), ethidium (an intercalator), and daunomycin (a combined intercalator/groove binder). Our free energy data (delta G zero) show that each drug exhibits similar binding affinities at 25 degrees C for the alternating copolymer duplex poly[d(A-T)].poly[d(A-T)] and for the homopolymer duplex poly(dA).poly(dT). However, our calorimetric measurements reveal that the nature of the thermodynamic forces (delta H zero, delta S zero) that drive drug binding to these two host duplexes at 25 degrees C are entirely different, despite similar binding free energies (delta G zero) and similar salt dependencies (lnK/ln[Na+]). Specifically, the 25 degrees C binding of all four drugs to the alternating copolymer poly[d(A-T)].poly[d(A-T)] is overwhelmingly enthalpy driven, whereas the corresponding binding of each drug to the homopolymer duplex poly(dA).poly(dT) is overwhelmingly entropy driven. Thus, the similar binding free energies (delta G zero) we measure for complexation of each drug with poly[d(A-T)].poly[d(A-T)] and poly(dA).poly(dT) result from compensating changes in the enthalpy and entropy terms. Comparison with the thermodynamic profiles for the complexation of these drug molecules to other DNA host duplexes at 25 degrees C reveals that the binding of each is strongly enthalpy driven, except when the poly(dA).poly(dT) homopolymer serves as the host duplex. This comparison allows us to conclude that poly[d(A-T)].poly[d(A-T)] behaves thermodynamically as the more "normal" host duplex toward drug binding, whereas the entropy-driven binding to the poly(dA).poly(dT) duplex represents "aberrant" behavior. Furthermore, since each of the four drugs exhibits different modes of DNA binding, we conclude that the observed entropy-driven behavior for binding to poly(dA).poly(dT) reflects an intrinsic property of the homopolymer duplex that is perturbed in a common manner upon ligation rather than a common property of all four binding ligands. To rationalize the large positive entropy changes that drive drug complexation with poly(dA).poly(dT) duplex, we propose a model that emphasizes binding-induced perturbations of the more highly hydrated, altered B conformation of the homopolymer. Our results suggest that an aberrant thermodynamic binding profile may reflect an unusual DNA conformation in the host duplex. However, before such a conclusion can be reached, complete thermodynamic binding profiles must be examined, since enthalpy-entropy compensations can cause two binding events to exhibit similar binding constants even when they are driven by very different thermodynamic forces.

Calorimetry↗

Two-parameter deformations of logarithm, exponential, and entropy: a consistent framework for generalized statistical mechanics.

A consistent generalization of statistical mechanics is obtained by applying the maximum entropy principle to a trace-form entropy and by requiring that physically motivated mathematical properties are preserved. The emerging differential-functional equation yields a two-parameter class of generalized logarithms, from which entropies and power-law distributions follow: these distributions could be relevant in many anomalous systems. Within the specified range of parameters, these entropies possess positivity, continuity, symmetry, expansibility, decisivity, maximality, concavity, and are Lesche stable. The Boltzmann-Shannon entropy and some one-parameter generalized entropies already known belong to this class. These entropies and their distribution functions are compared, and the corresponding deformed algebras are discussed.

Journal Article↗

Entropy on charge density: making the quantum mechanical connection.

Entropy maximization has proven effective in treating certain aspects of the phase problem of X-ray diffraction. Much of its development has been expressed in probabilistic language, although image enhancement has been somewhat more physical or geometric in description. Here phasing and entropy maximization are embedded in the quantum mechanical problem of reconstructing an electronic one-matrix under experimental constraints. Entropy on an N-representable one-particle density matrix is well defined. The entropy is the expected form, and it is a simple function of the one-matrix eigenvalues which all must be non-negative. Certain other properties are pertinent to phasing which is implicit in one-matrix reconstruction governed by entropy maximization. Throughout this work reference is made to informational entropy, not the entropy of thermodynamics.

Journal Article↗

Time-dependent entropy estimation of EEG rhythm changes following brain ischemia.

Our approach is motivated by the need to generate a rigorous measure of the degree of disorder (or complexity) of the EEG signal in brain injury. Entropy is a method to quantify the order/disorder of a time series. It is the first time that a time-dependent entropy (TDE) is used in the quantification of brain injury level. The TDE was sensitive enough to monitor the significant changes in the subject's brain rhythms during recovery from global ischemic brain injury. Among the different entropy measures, we used Tsallis entropy. This entropy is parametrized and is able to match with the particular properties of EEG, like long-range rhythms, spikes, and bursts. The method was tested in a signal composed of segments of synthetic signals (Gaussian and uniform distributions) and segments of real signals. The real signal segments were extracted from normal EEG, EEG recordings from early recovery, and normal EEG corrupted by simulated spikes and bursts. Adult Wistar rats were subjected to asphyxia-cardiac arrest for 3 and 5 min. The TDE detected the pattern of ischemia-induced EEG alterations and was able to discriminate the different injury levels. Two parameters seem to be good descriptors of the recovery process; the mean entropy and the variance of the estimate followed opposite trends, with the mean entropy decreasing and its variance increasing with injury.

Algorithms↗

Research on multi-trait genome association study method based on Shannon information entropy.

BACKGROUND: Genetic analysis of complex traits is crucial for elucidating disease mechanisms and biological inheritance processes. However, traditional Genome-wide Association Study (GWAS) for single trait often fail to capture the synergistic effects of genetic loci on multiple traits. METHODS: This study proposes a method for analyzing the association between multiple traits and gene regions based on Shannon information entropy. Innovatively, Shannon information entropy is introduced to integrate gene region information as genetic entropy, thereby constructing an Inverse Shannon Entropy-Multi-Trait Association Analysis of Gene Region genetic model (InvSE-MTAGR). Furthermore, a partial regression test is applied to the model to establish the Inverse Partial Shannon Entropy-Multi-Trait Association Analysis of Gene Region method (InvPSE-MTAGR). When performing multi-trait analysis with InvSE-MTAGR, the method achieved statistical significance by accumulating minor effects, thereby enhancing the ability to identify pleiotropic gene regions. RESULTS: The simulation results showed that the proposed multi-trait gene region association analysis method performed well in terms of both Type I error rate control and statistical power. Leveraging tomato and sorghum datasets for validation, the proposed multi-trait gene region association analysis method based on Shannon information entropy accurately pinpointed most of the gene regions harboring candidate genes. CONCLUSION: The study reveals the advantage of multi-trait method in integrating weak-effect pleiotropic signals and capturing the correlation among traits, which provides an efficient theoretical tool for dynamic analysis of complex multi-trait genetic networks and multi-target collaborative breeding of crops.

Genome-Wide Association Study↗

Minimal entropy probability paths between genome families.

We develop a metric for probability distributions with applications to biological sequence analysis. Our distance metric is obtained by minimizing a functional defined on the class of paths over probability measures on N categories. The underlying mathematical theory is connected to a constrained problem in the calculus of variations. The solution presented is a numerical solution, which approximates the true solution in a set of cases called rich paths where none of the components of the path is zero. The functional to be minimized is motivated by entropy considerations, reflecting the idea that nature might efficiently carry out mutations of genome sequences in such a way that the increase in entropy involved in transformation is as small as possible. We characterize sequences by frequency profiles or probability vectors, in the case of DNA where N is 4 and the components of the probability vector are the frequency of occurrence of each of the bases A, C, G and T. Given two probability vectors a and b, we define a distance function based as the infimum of path integrals of the entropy function H( p) over all admissible paths p(t), 0 < or = t< or =1, with p(t) a probability vector such that p(0)=a and p(1)=b. If the probability paths p(t) are parameterized as y(s) in terms of arc length s and the optimal path is smooth with arc length L, then smooth and "rich" optimal probability paths may be numerically estimated by a hybrid method of iterating Newton's method on solutions of a two point boundary value problem, with unknown distance L between the abscissas, for the Euler-Lagrange equations resulting from a multiplier rule for the constrained optimization problem together with linear regression to improve the arc length estimate L. Matlab code for these numerical methods is provided which works only for "rich" optimal probability vectors. These methods motivate a definition of an elementary distance function which is easier and faster to calculate, works on non-rich vectors, does not involve variational theory and does not involve differential equations, but is a better approximation of the minimal entropy path distance than the distance //b-a//(2). We compute minimal entropy distance matrices for examples of DNA myostatin genes and amino-acid sequences across several species. Output tree dendograms for our minimal entropy metric are compared with dendograms based on BLAST and BLAST identity scores.

Algorithms↗

Rényi continuous entropy of DNA sequences.

Entropy measures of DNA sequences estimate their randomness or, inversely, their repeatability. L-block Shannon discrete entropy accounts for the empirical distribution of all length-L words and has convergence problems for finite sequences. A new entropy measure that extends Shannon's formalism is proposed. Renyi's quadratic entropy calculated with Parzen window density estimation method applied to CGR/USM continuous maps of DNA sequences constitute a novel technique to evaluate sequence global randomness without some of the former method drawbacks. The asymptotic behaviour of this new measure was analytically deduced and the calculation of entropies for several synthetic and experimental biological sequences was performed. The results obtained were compared with the distributions of the null model of randomness obtained by simulation. The biological sequences have shown a different p-value according to the kernel resolution of Parzen's method, which might indicate an unknown level of organization of their patterns. This new technique can be very useful in the study of DNA sequence complexity and provide additional tools for DNA entropy estimation. The main MATLAB applications developed and additional material are available at the webpage . Specialized functions can be obtained from the authors.

Animals↗

Association entropy in adsorption processes.

The association of two species to form a bound complex, e.g., the binding of a ligand to a protein or the adsorption of a peptide on a lipid membrane, involves an entropy loss, reflecting the conversion of free translational and rotational degrees of freedom into bound motions. Previous theoretical estimates of the standard entropy change in bimolecular binding processes, DeltaS(o), have been derived from the root-mean-square fluctuations in protein crystals, suggesting DeltaS(o) approximately -50 e.u., i.e., TDeltaS degrees approximately -25 kT = -15 kcal/mol. In this work we focus on adsorption, rather than binding processes. We first present a simple statistical-thermodynamic scheme for calculating the adsorption entropy, including its resolution into translational and rotational contributions, using the known distance-orientation dependent binding (adsorption) potential. We then utilize this scheme to calculate the free energy of interaction and entropy of pentalysine adsorption onto a lipid membrane, obtaining TDeltaS(o) approximately -1.7 kT approximately -1.3 kcal/mol. Most of this entropy change is due to the conversion of one free translation into a bound motion, the rest arising from the confinement of two rotational degrees of freedom. The smaller entropy loss in adsorption compared to binding processes arises partly because a smaller number of degrees of freedom become restricted, but mainly due to the fact that the binding potential is much "softer."

Adsorption↗

The Shannon information entropy of protein sequences.

A comprehensive data base is analyzed to determine the Shannon information content of a protein sequence. This information entropy is estimated by three methods: a k-tuplet analysis, a generalized Zipf analysis, and a "Chou-Fasman gambler." The k-tuplet analysis is a "letter" analysis, based on conditional sequence probabilities. The generalized Zipf analysis demonstrates the statistical linguistic qualities of protein sequences and uses the "word" frequency to determine the Shannon entropy. The Zipf analysis and k-tuplet analysis give Shannon entropies of approximately 2.5 bits/amino acid. This entropy is much smaller than the value of 4.18 bits/amino acid obtained from the nonuniform composition of amino acids in proteins. The "Chou-Fasman" gambler is an algorithm based on the Chou-Fasman rules for protein structure. It uses both sequence and secondary structure information to guess at the number of possible amino acids that could appropriately substitute into a sequence. As in the case for the English language, the gambler algorithm gives significantly lower entropies than the k-tuplet analysis. Using these entropies, the number of most probable protein sequences can be calculated. The number of most probable protein sequences is much less than the number of possible sequences but is still much larger than the number of sequences thought to have existed throughout evolution. Implications of these results for mutagenesis experiments are discussed.

Algorithms↗

A study on the enthalpy-entropy compensation in protein unfolding.

A large number of thermodynamic data including the free energy, enthalpy, entropy, and heat capacity changes were collected for the denaturation of various proteins. Regression indicated that remarkable enthalpy-entropy compensation occurred in protein unfolding, which meant that the change in enthalpy was almost compensated by a corresponding change in entropy resulting in a smaller net free energy change. This behavior was proposed to result from the water molecule reorganization, which contributed significantly to the enthalpy and entropy changes but little to the free energy change in protein unfolding. It turned out that the enthalpy-entropy compensation could provide novel insights into the problem of enthalpy and entropy convergence in protein unfolding.

Calorimetry, Differential Scanning↗

Hydration entropy change from the hard sphere model.

The gas to liquid transfer entropy change for a pure non-polar liquid can be calculated quite accurately using a hard sphere model that obeys the Carnahan-Starling equation of state. The same procedure fails to produce a reasonable value for hydrogen bonding liquids such as water, methanol and ethanol. However, the size of the molecules increases when the hydrogen bonds are turned off to produce the hard sphere system and the volume packing density rises. We show here that the hard sphere system that has this increased packing density reproduces the experimental transfer entropy values rather well. The gas to water transfer entropy values for small non-polar hydrocarbons is also not reproduced by a hard sphere model, whether one uses the normal (2.8 A diameter) or the increased (3.2 A) size for water. At least part of the reason that the hard sphere model with 2.8 A size water produces too small entropy change is that the size of water is too small for a system without hydrogen bonds. The reason that the 3.2 A model also produces too small entropy values is that this is an overly crowded system and that the free volume introduced in the system by the addition of a solute molecule produces too much of a relief to this crowding. A hard sphere model, in which the free volume increase is limited by requiring that the average surface-to-surface distance between the solute and water molecules is the same as that between the increased-size water molecules, does approximately reproduce the experimental hydration entropy values.

Entropy↗

The entropy generation in visual-pigment system by the absorption of light.

We first showed a general theory that reception of information from the outside in a receptor system is accompanied both by an inflow of entropy and by a generation of entropy depending on the reliability of the actual reception mechanism. Then, considering a case for the absorption of light by the visual-pigment system of visual cell, we calculated the time(t)-dependent change in the number N2(t) of excited visual-pigments to obtain the entropy increase. We thus arrived at the following four conclusions: (1) One or two photons can be detected with a reliability of at least 54%; (2) In compensation for this detection, entropy > 1.12 x log2 is generated; (3) An incident photon of frequency v from a ligh source of temperature Ts yields an entropy of hv/Ts; and (4) Depending on the characteristics of the visual-pigment system, another entropy being different from (2) is generated in proportion to N2(t).

Absorption↗

Configurational entropies of lipids in pure and mixed bilayers from atomic-level and coarse-grained molecular dynamics simulations.

Single-chain and single-fragment configurational entropies of lipid tails in hydrated lipid bilayers are evaluated from molecular dynamics simulations using the quasi-harmonic approximation. The entropy distribution along individual acyl tails is obtained and compared to that of corresponding hydrocarbon chains in the liquid phase. We consider pure dipalmitoylphosphatidylcholine and mixed dioleoylphosphatidylcholine/dioleoylphosphatidylethanolamine bilayers. The systems are modeled at different levels of spatial resolution: In an atomic-level (AL) model all (heavy) atoms are explicitly simulated; in a coarse-grained (CG) model particles (beads) representing groups of covalently bound atoms are used, which map approximately four non-hydrogen atoms to one interaction site. Single-chain and single-fragment entropies and correlations between the motions of (single) acyl chains are compared. A good correspondence is found between the flexibility of the AL and CG models. The loss in configurational entropy due to the reduction in the number of degrees of freedom upon coarse-graining of the model is estimated. The CG model shows about 4 times faster convergence of the chain entropies than the more detailed AL model. Corrections to the quasi-harmonic entropy estimates were found to be small for the CG model. For the AL model, the correction due to mode anharmonicities is small, but the correction due to pairwise (supralinear) mode correlations is sizable.

Computer Simulation↗

Calculations of solute and solvent entropies from molecular dynamics simulations.

The translational, rotational and conformational (vibrational) entropy contributions to ligand-receptor binding free energies are analyzed within the standard formulation of statistical thermodynamics. It is shown that the partitioning of the binding entropy into different components is to some extent arbitrary, but an appropriate method to calculate both translational and rotational entropy contributions to noncovalent association is by estimating the configurational volumes of the ligand in the bound and free states. Different approaches to calculating solute entropies using free energy perturbation calculations, configurational volumes based on root-mean-square fluctuations and covariance matrix based quasiharmonic analysis are illustrated for some simple molecular systems. Numerical examples for the different contributions demonstrate that theoretically derived results are well reproduced by the approximations. Calculation of solvent entropies, either using total potential energy averages or van't Hoff plots, are carried out for the case of ion solvation in water. Although convergence problems will persist for large and complex simulation systems, good agreement with experiment is obtained here for relative and absolute ion hydration entropies. We also outline how solvent and solute entropic contributions are taken into account in empirical binding free energy calculations using the linear interaction energy method. In particular it is shown that empirical scaling of the nonpolar intermolecular ligand interaction energy effectively takes into account size dependent contributions to the binding free energy.

Chymotrypsin↗