Statistical Inference for Ergodic Algorithmic Model (EAM), Applied to Hydrophobic Hydration Processes

Abstract: The thermodynamic properties of hydrophobic hydration processes can be represented in probability space by a Dual‐Structure Partition Function {DS‐PF} = {M‐PF} ∙ {T‐PF}, which is the product of a Motive Partition Function {M‐PF} multiplied by a Thermal Partition Function {T‐PF}. By development of {DS‐PF}, parabolic binding potential functions α) RlnKdual = (–ΔG°dual/T) ={f(1/T)*g(T)} and β) RTlnKdual = (–ΔG°dual) = {f(T)*g(lnT)} have been calculated. The resulting binding functions are “convoluted” functions dependent on the reciprocal interactions between the primary function f(1/T) or f(T) with the secondary function g(T) or g(lnT), respectively. The binding potential functions carry the essential thermodynamic information elements of each system. The analysis of the binding potential functions experimentally determined at different temperatures by means of the Thermal Equivalent Dilution (TED) principle has made possible the evaluation, for each compound, of the pseudo‐stoichiometric coefficient ξw, from the curvature of the binding potential functions. The positive value indicates convex binding functions (Class A), whereas the negative value indicates concave binding function (Class B). All the information elements concern sets of compounds that are very different from one set to another, in molecular dimension, in chemical function, and in aggregation state. Notwithstanding the differences between, surprising equal unitary values of niche (cavity) formation in Class A <hfor>A= –22.7 kJmol−1 ξw−1 sets with standard deviation σ= 3.1% and <sfor>A = –445JK−1mol−1ξw−1JK−1mol−1ξw−1 with standard deviation σ= 0.7%. Other surprising similarities have been found, demonstrating that all the data analyzed belong to the same normal statistical population. The Ergodic Algorithmic Model (EAM) has been applied to the analysis of important classes of reactions, such as thermal and chemical denaturation, denaturation of proteins, iceberg formation or reduction, hydrophobic bonding, and null thermal free energy. The statistical analysis of errors has shown that EAM has a general validity, well beyond the limits of our experiments. Specifically, the properties of hydrophobic hydration processes as biphasic systems generating convoluted binding potential functions, with water as the implicit solvent, hold for all biochemical and biological solutions, on the ground that they also are necessarily diluted solutions, statistically validated

Ergodic Algorithmic Model (EAM), with Water as Implicit Solvent, in Chemical, Biochemical, and Biological Processes

For many years, we have devoted our research to the study of the thermodynamic properties of hydrophobic hydration processes in water, and we have proposed the Ergodic Algorithmic Model (EAM) for maintaining the thermodynamic properties of any hydrophobic hydration reaction at a constant pressure from the experimental determination of an equilibrium constant (or other potential functions) as a function of temperature. The model has been successfully validated by the statistical analysis of the information elements provided by the EAM model for about fifty compounds. The binding functions are convoluted functions, RlnKeq = {f(1/T)* g(T)} and RTlnKeq = {f(T)* g(lnT)}, where the primary linear functions f(1/T) and f(T) are modified and transformed into parabolic curves by the secondary functions g(T) and g(lnT), respectively. Convoluted functions are consistent with biphasic dual-structure partition function, {DS-PF} = {M-PF} · {T-PF} · {ζw}, composed by ({M-PF} (Density Entropy), {T-PF}) (Intensity Entropy), and {ζw} (implicit solvent). In the present paper, after recalling the essential aspects of the model, we outline the importance of considering the solvent as “implicit” in chemical and biochemical reactions. Moreover, we compare the information obtained by computer simulations using the models till now proposed with “explicit” solvent, showing the mess of information lost without considering the experimental approach of the EAM model

Molecular thermodynamic model for equilibria in solution. I. Reacting and non-reacting ensembles

The partition functions of solution thermodynamics are mathematical representations of the properties of molecular ensembles statistically distributed according to specifc characteristics. The ensembles are classi®ed as non-reacting or reacting. The non-reacting ensembles are characterized by one mean enthalpy level with dispersion around the mean. The reacting ensembles are characterized by two or more distinctly separated enthalpy levels over which the different species are variably distributed, depending on concentration and/or temperature. The non-reacting ensembles can be distinguished into microcanonical, thermal, osmotic, thermo-osmotic ensembles, depending on the type of exchange with the surroundings which is connected to the ¯uctuations of the ensemble variables. The reacting ensembles can be distinguished into thermal, osmotic, thermo-osmotic, electrochemical, electro-osmotic, electrothermal, electro-thermo-osmotic ensembles, depending on the type of reaction and of exchange with the surroundings

Ergodic Algorithmic Model (EAM), with Water as Implicit Solvent, in Chemical, Biochemical, and Biological Processes

For many years, we have devoted our research to the study of the thermodynamic properties of hydrophobic hydration processes in water, and we have proposed the Ergodic Algorithmic Model (EAM) for maintaining the thermodynamic properties of any hydrophobic hydration reaction at a constant pressure from the experimental determination of an equilibrium constant (or other potential functions) as a function of temperature. The model has been successfully validated by the statistical analysis of the information elements provided by the EAM model for about fifty compounds. The binding functions are convoluted functions, RlnKeq = {f(1/T)* g(T)} and RTlnKeq = {f(T)* g(lnT)}, where the primary linear functions f(1/T) and f(T) are modified and transformed into parabolic curves by the secondary functions g(T) and g(lnT), respectively. Convoluted functions are consistent with biphasic dual-structure partition function, {DS-PF} = {M-PF} ∙ {T-PF} ∙ {ζw}, composed by ({M-PF} (Density Entropy), {T-PF}) (Intensity Entropy), and {ζw} (implicit solvent). In the present paper, after recalling the essential aspects of the model, we outline the importance of considering the solvent as “implicit” in chemical and biochemical reactions. Moreover, we compare the information obtained by computer simulations using the models till now proposed with “explicit” solvent, showing the mess of information lost without considering the experimental approach of the EAM model

Hydrophobic Hydration Processes: Intensity Entropy and Null Thermal Free Energy and Density Entropy and Motive Free Energy

ABSTRACT: The processes at the molecule level, which are the source of the ergodic properties of thermodynamic systems, are analyzed with special reference to entropy. The entropy change produced by increasing the temperature T depends on the increase of velocity of the particles with a decrease of the squared mean sojourn time (τm 2) and gradual loss of instant energy intensity. The diminution, which is due to dilution, of the number of terms in the summation of cumulative sojourn time (τi 2)Σ produces loss of energy density, thus generating a gradual increase of density entropy, dSDens. The ergodic property of thermodynamic systems consists of the equivalence of density entropy (dependent on dilution) with intensity entropy (dependent on temperature). This equivalence has been experimentally verified in every hydrophobic hydration process as thermal equivalent dilution. An ergodic dual-structure partition function {DS-PF} represents the state probability of every hydrophobic hydration process, corresponding to the biphasic composition of these systems. The dualstructure partition function {DS-PF} (Kmot·ζth) is the product of a motive partition function {M-PF} (Kmot) multiplied by a thermal partition function {T-PF} (ζth = 1). {M-PF} gives rise to changes of density entropy, whereas {T-PF} gives rise to changes of intensity entropy. {M-PF} is referred to a reacting mole ensemble (reacting solute) composed of few elements (moles), ruled by binomial distribution, whereas {T-PF} is referred to a nonreacting molecule ensemble (NoremE) (nonreacting solvent), which is composed of a very large population of elements (molecules), ruled by Boltzmann statistics. Statistical thermodynamic methods cannot be applied to {M-PF} that can be calculated by numerical methods from the experimental titration data. By development of the dual-structure partition function {DS-PF}, parabolic convoluted binding functions are obtained. The tangents to the binding functions represent the dual enthalpy, −ΔHdual = (−ΔHmot − ΔHth), and the dual entropy, ΔSdual = (ΔSmot + ΔSth). The connections between canonical and grand-canonical partition functions of statistical thermodynamics with thermal and motive partition functions of chemical thermodynamics, respectively, are discussed. Special attention has been devoted to the equality ΔHth/T + ΔSth = 0, typical of NoremEs, as an entropy−enthalpy compensation with ΔGth/T = 0. The thermodynamic potential change Δμ, as proposed by potential distribution theorem (PDT) for iceberg formation from {T-PF} of the solvent, is nonexistent because the excess solvent is at a constant potential (Δμsolv = 0). The information level offered by the ergodic algorithmic model (EAM) is more complete and correct than that offered by the potential distribution theorem (PDT): the stoichiometry of the water reaction in hydrophobic hydration processes is determined by the EAM as the function of the number ±ξw. Quasi-chemical approximation, renamed the chemical molecule/mole scaling function (Che. m/M. sF), is a fundamental breakthrough in the application of statistical thermodynamics to chemical reactions. Boltzmann statistical molecule distribution of the thermal partition function {T-PF} is scaled with binomial mole distribution of the motive partition function {M-PF}. For computer-assisted drug design, the alternative calculation procedure of Talhout, based on the previous experimental determination of binding functions, is recommended. The ergodic algorithmic model (EAM), applied to the experimental convoluted binding functions, can recover the distinct terms of intensity entropy (ΔHmot/T) and density entropy (ΔSmot), together with other essential information elements, lost by computer simulations

Protonation equilibria and hydration in disubstituted benzoic acids

The protonation constants log kapp of a series of disubstituted benzoic acids in aqueous solution at different temperatures between 5” and 55°C have been determined potentiometrically. The data of log k,, have been analyzed under the light of a statistical thermodynamic model. The curvature of the function log k,, = f( l/ T) is related to the number n, of water molecules involved in the protonation and hydration reaction. The upward concavity of the curves of dinitro compounds are steeper that those for monosubstituted acids and imply higher number of water molecules. The curves of polyalkyl-substituted benzoic acids as determined by other authors show opposite (downward concavity) curvatures corresponding to negative numbers n, of water molecules. The values of log kapp at 25°C of disubstituted and polyalkyl-substituted benzoic acids plotted against the Hammett substituent constants o&,, deviate significantly from the line of the Hammett model.