Hydrophobic hydration processes. General thermodynamic model by thermal equivalent dilution determinations

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

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) RlnK_dual = (−ΔG°_dual/T) ={f(1/T)*g(T)} and β) RTlnK_dual = (−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 ±ξ_w, 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 <Δh_for>_A = −22.7 ± 0.7 kJ·mol⁻¹·ξ_w⁻¹ sets with standard deviation σ = ±3.1% and <Δs_for>_A = −445 ± 3J·K⁻¹·mol⁻¹·ξ_w⁻¹J·K⁻¹·mol⁻¹·ξ_w⁻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.

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

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 (τ_m 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 (τ_i ²)_Σ produces loss of energy density, thus generating a gradual increase of density entropy, dS _Dens. 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 dual-structure partition function {DS-PF} (K _mot·ζ_th) is the product of a motive partition function {M-PF} (K _mot) multiplied by a thermal partition function {T-PF} (ζ_th = 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, -ΔH _dual = (-ΔH _mot - ΔH _th), and the dual entropy, ΔS _dual = (ΔS _mot + ΔS _th). 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 ΔH _th/T + ΔS _th = 0, typical of NoremEs, as an entropy-enthalpy compensation with ΔG _th/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 (Δμ_solv = 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 ±ξ_w. 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 (ΔH _mot/T) and density entropy (ΔS _mot), together with other essential information elements, lost by computer simulations.

Self-association of cyclodextrins and cyclodextrin complexes in aqueous solutions

Cyclodextrins (CDs) are oligosaccharides that self-assemble in aqueous solutions to form transient clusters, nanoparticles and small microparticles. The critical aggregation concentration (cac) of the natural αCD, βCD and γCD in pure aqueous solutions was estimated to be 25, 8 and 9 mg/ml, respectively. The cac of 2-hydroxypropyl-β-cyclodextrin (HPβCD), that consists of mixture of isomers, was estimated to be significantly higher or 118 mg/ml. Addition of chaotropic agents (i.e. that disrupts non-covalent bonds such as hydrogen bonds) to the aqueous media increases the cac. Formation of drug/CD complexes can increase or decrease the cac. Due to the transient nature of the CD clusters and nanoparticles they can be difficult to detect and their presence is frequently ignored. However, they have profound effect on the physiochemical properties of CDs and their pharmaceutical applications. For example, the values of stability constants of drug/CD complexes can be both concentration dependent and method dependent. Like in the case of micelles water-soluble polymers can enhance the solubilizing effect of CDs. Also, formation of drug/CD complex nanoparticles appears to increase the ability of CDs to enhance drug delivery through some mucosal membranes.

Hydrophobic Hydration Processes. I: Dual-Structure Partition Function for Biphasic Aqueous Systems

The thermodynamic properties of hydrophobic hydration processes have been analyzed and assessed. The thermodynamic binding functions result to be related to each other by the mathematical relationships of an ergodic algorithmic model (EAM). The active dilution d _A of species A in solution is expressed as d _A = 1/(Φ·x _A) with thermal factor Φ = T ^{-(C _p,A/R)} and (1/x _A) = d _id(A), where d _id(A) = ideal dilution. Entropy function is set as S = f(d _id(A),T). Thermal change of entropy (i.e., entropy intensity change) is represented by the equation (dS) _d = C _p dln T. Configuration change of entropy (i.e., entropy density change) is represented by the equation (dS)_T = (-R dln x _A)_T = (R dln d _id(A)) _T . Because every logarithmic function in thermodynamic space corresponds to an exponential function in probability space, the sum functions ΔH _dual = (ΔH _mot + ΔH _th) and ΔS _dual = (ΔS _mot + ΔS _th) of the thermodynamic space give birth, in exponential probability space, to a dual-structure partition function { DS-PF }: exp(-ΔG _dual/RT) = K _dual = (K _mot·ζ_th) = {(exp(-ΔH _mot/RT))(exp(ΔS _mot/R))}·{exp(-ΔH _th/RT) exp(ΔS _th/R)}. Every hydrophobic hydration process can be represented by { DS-PF } = { M-PF }·{ T-PF }, indicating biphasic systems. { M-PF } = f(T,d _id(A)), concerning the solute, is monocentric and produces changes of entropy density, contributing to free energy -ΔG _mot, whereas { T-PF } = g(T), concerning the solvent, produces changes of entropy intensity, not contributing to free energy. Entropy density and entropy intensity are equivalent and summed with each other (i.e., they are ergodic). From the dual-structure partition function { DS-PF }, the ergodic algorithmic model (EAM) can be developed. The model EAM consists of a set of mathematical relationships, generating parabolic convoluted binding functions R ln K _dual = -ΔG _dual/T = {f(1/T)*g(T)} and RT ln K _dual = -ΔG _dual = {f(T)*g(ln T)}. The first function in each convoluted couple f(1/T) or f(T) is generated by { M-PF }, whereas the second function, g(T) or g(ln T), respectively, is generated by { T-PF }. The mathematical properties of the thermodynamic functions of hydrophobic hydration processes, experimentally determined, correspond to the geometrical properties of parabolas, with constant curvature amplitude C _ampl = 0.7071/ΔC _p,hydr. The dual structure of the partition function conforms to the biphasic composition of every hydrophobic hydration solution, consisting of a diluted solution, with solvent in excess at constant potential.

Successful amphiphiles as the key to crystallization of membrane proteins: Bridging theory and practice

BACKGROUND

Membrane proteins constitute a major group of proteins and are of great significance as pharmaceutical targets, but underrepresented in the Protein Data Bank. Particular reasons are their low expression yields and the constant need for cautious and diligent handling in a sufficiently stable hydrophobic environment substituting for the native membrane. When it comes to protein crystallization, such an environment is often established by detergents.

SCOPE OF REVIEW

In this review, 475 unique membrane protein X-ray structures from the online data bank "Membrane proteins of known 3D structure" are presented with a focus on the detergents essential for protein crystallization. By systematic analysis of the most successful compounds, including current trends in amphiphile development, we provide general insights for selection and design of detergents for membrane protein crystallization.

MAJOR CONCLUSIONS

The most successful detergents share common features, giving rise to favorable protein interactions. The hydrophile-lipophile balance concept of well-balanced hydrophilic and hydrophobic detergent portions is still the key to successful protein crystallization. Although a single detergent compound is sufficient in most cases, sometimes a suitable mixture of detergents has to be found to alter the resulting protein-detergent complex. Protein crystals with a high diffraction limit involve a tight crystal packing generally favored by detergents with shorter alkyl chains.

GENERAL SIGNIFICANCE

The formation of well-diffracting membrane protein crystals strongly depends on suitable surfactants, usually screened in numerous crystallization trials. The here-presented findings provide basic criteria for the assessment of surfactants within the vast space of potential crystallization conditions for membrane proteins.

Hydrophobic hydration processes thermal and chemical denaturation of proteins

The hydrophobic hydration processes have been analysed under the light of a mixture model of water that is assumed to be composed by clusters (W(5))(I), clusters (W(4))(II) and free water molecules W(III). The hydrophobic hydration processes can be subdivided into two Classes A and B. In the processes of Class A, the transformation A(-ξ(w)W(I)→ξ(w)W(II)+ξ(w)W(III)+cavity) takes place, with expulsion from the bulk of ξ(w) water molecules W(III), whereas in the processes of Class B the opposite transformation B(-ξ(w)W(III)-ξ(w)W(II)→ξ(w)W(I)-cavity) takes place, with condensation into the bulk of ξ(w) water molecules W(III). The thermal equivalent dilution (TED) principle is exploited to determine the number ξ(w). The denaturation (unfolding) process belongs to Class A whereas folding (or renaturation) belongs to Class B. The enthalpy ΔH(den) and entropy ΔS(den) functions can be disaggregated in thermal and motive components, ΔH(den)=ΔH(therm)+ΔH(mot), and ΔS(den)=ΔS(therm)+ΔS(mot), respectively. The terms ΔH(therm) and ΔS(therm) are related to phase change of water molecules W(III), and give no contribution to free energy (ΔG(therm)=0). The motive functions refer to the process of cavity formation (Class A) or cavity reduction (Class B), respectively and are the only contributors to free energy ΔG(mot). The folded native protein is thermodynamically favoured (ΔG(fold)≡ΔG(mot)<0) because of the outstanding contribution of the positive entropy term for cavity reduction, ΔS(red)≫0. The native protein can be brought to a stable denatured state (ΔG(den)≡ΔG(mot)<0) by coupled reactions. Processes of protonation coupled to denaturation have been identified. In thermal denaturation by calorimetry, however, is the heat gradually supplied to the system that yields a change of phase of water W(III), with creation of cavity and negative entropy production, ΔS(for)≪0. The negative entropy change reduces and at last neutralises the positive entropy of folding. In molecular terms, this means the gradual disruption by cavity formation of the entropy-driven hydrophobic bonds that had been keeping the chains folded in the native protein. The action of the chemical denaturants is similar to that of heat, by modulating the equilibrium between W(I), W(II), and W(III) toward cavity formation and negative entropy production. The salting-in effect produced by denaturants has been recognised as a hydrophobic hydration process belonging to Class A with cavity formation, whereas the salting-out effect produced by stabilisers belongs to Class B with cavity reduction. Some algorithms of denaturation thermodynamics are presented in the Appendices.