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.

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

The "hydrophobic hydration processes" can be satisfactorily interpreted on the basis of a common molecular model for water, consisting of two types of clusters, namely W(I) and W(II) accompanied by free molecules W(III). The principle of thermal equivalent dilution (TED) is the potent tool (Ergodic Hypothesis) employed to monitor the water equilibrium and to determine the number xi(w) of water molecules W(III) involved in each process. The hydrophobic hydration processes can be subdivided into two Classes: Class A includes those processes for which the transformation A(-xi(w)W(I)-->xi(w)W(II)+xi(w)W(III)+cavity) takes place with the formation of a cavity, by expulsion of xi(w) water molecules W(III) whereas Class B includes those processes for which the opposite transformation B(-xi(w)W(II)-xi(w)W(III)-->xi(w)W(I)-cavity) takes place with reduction of the cavity, by condensation of xi(w) water molecules W(III). The number xi(w) depends on the size of the reactants and measures the extent of the change in volume of the cavity. Disaggregating the thermodynamic functions DeltaH(app) and DeltaS(app) as the functions of T (or lnT) and xi(w) has enabled the separation of the thermodynamic functions into work and thermal components. The work functions DeltaG(Work), DeltaH(Work) and DeltaS(Work) only refer specifically to the hydrophobic effects of cavity formation or cavity reduction, respectively. The constant self-consistent unitary (xi(w)=1) work functions obtained from both large and small molecules indicate that the same unitary reaction is taking place, independent from the reactant size. The thermal functions DeltaH(Th) and DeltaS(Th) refer exclusively to the passage of state of water W(III). Essential mathematical algorithms are presented in the appendices.

Thermodynamics of micelle formation in water, hydrophobic processes and surfactant self-assemblies.

The critical micelle concentration (c.m.c.) for four cationic surfactants, alkyl-trimethyl-ammonium bromides, was determined as a function of temperature by conductivity measurements. The values of the standard free energy of micellisation DeltaG degrees(mic) at different temperatures were calculated by using a pseudo-phase transition model. Then, from the diagram (-DeltaG degrees(mic)/T)=f(1/T), the thermodynamic functions DeltaH(app) and DeltaS(app) were calculated. From the plots DeltaH(app)=f(T) and DeltaS(app) = f(ln T) the slopes DeltaC(p) = n(w(H))C(p,w) and DeltaC(p)=n(w(S))C(p,w) were calculated, with the numbers n(w(H)) and n(w(S)) negative and equal and therefore defined simply as n(w). The number n(w)<0, indicating condensed water molecules, depends on the reduction of cavity that takes place as a consequence of the coalescence of the cavities previously surrounding the separate aliphatic or aromatic moieties. The analysis, based on a molecular model consisting of three forms of water, namely W(I), W(II), and W(III), respectively, was extended to several other types of surfactants for which c.m.c. data had been published by other authors. The results of this analysis form a coherent scheme consistent with the proposed molecular model. The enthalpy for all the types of surfactant is described by DeltaH(app)= -3.6 + 23.1xi(w)-xi(w)C(p,w)T and the entropy by DeltaS(app)= +10.2+428xi(w)-xi(w)C(p,w) ln T where xi(w)= |n(w)| represents the number of molecules W(III) involved in the reaction. The term Deltah(w)= +23.1 kJ mol(-1) xi(w)(-1) indicates an unfavourable endothermic contribution to enthalpy for reduction of the cavity whereas the term Deltas(w)= +428 J K(-1) mol(-1) xi(w)(-1) represents a positive entropy contribution for reduction of the cavity, what is the driving force of hydrophobic association. The processes of non polar gas dissolution in water and of micelle formation were found to be strictly related: they are, however, exactly the opposite of one another. In micelle formation no intermolecular electronic short bond is formed. We propose, therefore, to substitute the term "hydrophobic bond" with that of "hydrophobic association".

Envirometrics. Part I: Modeling of water salinity and air quality data.

Envirometrics utilises advanced mathematical, statistical and information tools to extract information. Two typical environmental data sets are analysed using MVATOB (Multi Variate Analysis TOol Box). The first data set corresponds to the variable river salinity. Least median squares (LMS) detected the outliers whereas linear least squares (LLS) could not detect and remove the outliers. The second data set consists of daily readings of air quality values. Outliers are detected by LMS and unbiased regression coefficients are estimated by multi-linear regression (MLR). As explanatory variables are not independent, principal component regression (PCR) and partial least squares regression (PLSR) are used. Both examples demonstrate the superiority of LMS over LLS.

A software for the estimation of binding parameters of biochemical equilibria based on statistical probability model.

An algorithm is proposed for the estimation of binding parameters for the interaction of biologically important macromolecules with smaller ones from electrometric titration data. The mathematical model is based on the representation of equilibria in terms of probability concepts of statistical molecular thermodynamics. The refinement of equilibrium concentrations of the components and estimation of binding parameters (log site constant and cooperativity factor) is performed using singular value decomposition, a chemometric technique which overcomes the general obstacles due to near singularity. The present software is validated with a number of biochemical systems of varying number of sites and cooperativity factors. The effect of random errors of realistic magnitude in experimental data is studied using the simulated primary data for some typical systems. The safe area within which approximate binding parameters ensure convergence has been reported for the non-self starting optimization algorithms.

Data-, knowledge- and method-bases in chemical sciences I. BAQOR-database for equilibrium constants in aquo-organic media.

BAQOR is a computer readable database for equilibrium constants in presence of different percentages of water miscible cosolvents. The present version with user friendly software in dBase III+ contains 740 records and runs on any IBM compatible PC. The physico-chemical properties of binary and ternary water-cosolvent mixtures, the equilibrium constants of proton- and metal-ligand complexes are retrievable through pop-up menus. Specific searches by metal-, ligand-, solvent-, and stoichiometry-wise and their combinations is possible. Several display modes-monitor, file and hard copy-are available for the numerical fields as well as for literature citation.

Calculation of site affinity constants and cooperativity coefficients for binding of ligands and/or protons to macromolecules. I. Generation of partition functions and mass balance equations.

The thermodynamics of binding of a ligand A and/or proton H to a macromolecule M is treated by the partition function method. In complex systems, the representation of the equilibria by means of cumulative constants beta PQR used as coefficients in partition functions ZM, ZA, and ZH is ill-suited to least-squares refinement procedures because the cumulative constants are interrelated by common cooperativity functions gamma j(i) and common site affinity constants kappa j. There is therefore the need to express ZM, ZA, ZH as functions of site constants kappa j and cooperativity coefficients bj. This is done by developing an algebra of partition functions based on the following concepts: (i) factorability of partition functions; (ii) binary generating function Jj = (1 + kappa j[Y])i tau for each class j of sites, represented by column (Jj) and row (Jj) vectors; (iii) cooperativity between sites of one class described by functions gamma j(i), represented by diagonal matrices gamma j; (iv) probability of finding microspecies represented by elements of tensor product matrix Ll = (J1)[J2]; (v) statistical factors mij obtained from Newton polynomials, Jj; (vi) power operators Oi', O(i-l)', and O(i tau-l)', transforming vectors Jj; and (vii) operators Oi or O(i-l) indicating tensor products of i or (i-l) vectors Jj. Vectors Jj combined in tensors Ll give rise to both an affinity/cooperativity space and a parallel index space. The partition functions ZM, ZA, and ZH and the total amounts TM, TA, and TH can be obtained as an appropriate sum of elements of matrices Ll, each of which is represented in an index space by a combination p1, p2,...q1, q2,...r1, r2,... of indices ij. From these indices the contribution of that element to partition function ZM, ZA, or ZH and to total amount TM, TA, or TH is calculated in the affinity/cooperativity space as product of factors: [i tau !/i !(i tau-i)!]kappa ij(exp[bj (i-1)i])[X]i, i being any index p, q, r and X any component M, A, or H. Future applications of this algorithm to practical problems of macromolecule-ligand-proton equilibria are outlined.

Calculation of site affinity constants and cooperativity coefficients for binding of ligands and/or protons to macromolecules. II. Relationships between chemical model and partition function algorithm.

The relationships between the chemical properties of a system and the partition function algorithm as applied to the description of multiple equilibria in solution are explained. The partition functions ZM, ZA, and ZH are obtained from powers of the binary generating functions Jj = (1 + kappa j gamma j,i[Y])i tau j, where i tau j = p tau j, q tau j, or r tau j represent the maximum number of sites in sites in class j, for Y = M, A, or H, respectively. Each term of the generating function can be considered an element (ij) of a vector Jj and each power of the cooperativity factor gamma ij,i can be considered an element of a diagonal cooperativity matrix gamma j. The vectors Jj are combined in tensor product matrices L tau = (J1) [J2]...[Jj]..., thus representing different receptor-ligand combinations. The partition functions are obtained by summing elements of the tensor matrices. The relationship of the partition functions with the total chemical amounts TM, TA, and TH has been found. The aim is to describe the total chemical amounts TM, TA, and TH as functions of the site affinity constants kappa j and cooperativity coefficients bj. The total amounts are calculated from the sum of elements of tensor matrices Ll. Each set of indices (pj..., qj..., rj...) represents one element of a tensor matrix L tau and defines each term of the summation. Each term corresponds to the concentration of a chemical microspecies. The distinction between microspecies MpjAqjHrj with ligands bound on specific sites and macrospecies MpAqHR corresponding to a chemical stoichiometric composition is shown. The translation of the properties of chemical model schemes into the algorithms for the generation of partition functions is illustrated with reference to a series of examples of gradually increasing complexity. The equilibria examined concern: (1) a unique class of sites; (2) the protonation of a base with two classes of sites; (3) the simultaneous binding of ligand A and proton H to a macromolecule or receptor M with four classes of sites; and (4) the binding to a macromolecule M of ligand A which is in turn a receptor for proton H. With reference to a specific example, it is shown how a computer program for least-squares refinement of variables kappa j and bj can be organized. The chemical model from the free components M, A, and H to the saturated macrospecies MpAQHR, with possible complex macrospecies MpAq and AHR, is defined first.(ABSTRACT TRUNCATED AT 250 WORDS)

Potentiometric titrations in methanol/water medium: Intertitration variability.

Homogeneous sets of data from strong acid-strong base potentiometric titrations in a mixed solvent medium (9:1 v/v methanol/water), performed by means of the glass electrode, at various constant ionic strengths and with different reference electrodes, have been analysed by statistical criteria. A standardized procedure has been established to obtain reliable potentiometric data in mixed solvents. It has been demonstrated how, with the aid of a proper linearized model, analysis of variance (ANOVA) applied to the standardization titrations can be used to test the reliability of a potentiometric chain in a medium other than pure water. The error expected in the stability constants thus evaluated is related to the intertitration error of the operational pK*(')(w), for the same medium and the same chain. The results obtained by applying ANOVA to the mixed solvent data substantially confirm the trend found for aqueous media, the intertitration error being larger than the intratitration error for all the parameters investigated (E(')(0), pK*(')(w), mean equivalence volume). The stochastic error thus obtained depends on the ionic medium used and on the kind of reference electrode employed in the electrochemical chain.

Analysis of variance in determinations of equivalence volume and of the ionic product of water in potentiometric titrations.

Homogeneous sets of data from strong acid-strong base potentiometric titrations in aqueous solution at various constant ionic strengths have been analysed by statistical criteria. The aim is to see whether the error distribution matches that for the equilibrium constants determined by competitive potentiometric methods using the glass electrode. The titration curve can be defined when the estimated equivalence volume VEM, with standard deviation (s.d.) sigma (VEM), the standard potential E(0), with s.d. sigma(E(0)), and the operational ionic product of water K(*)(w) (or E(*)(w) in mV), with s.d. sigma(K(*)(w)) [or sigma(E(*)(w))] are known. A special computer program, BEATRIX, has been written which optimizes the values of VEM, E(0) and K(*)(w) by linearization of the titration curve as a Gran plot. Analysis of variance applied to a set of 11 titrations in 1.0M sodium chloride medium at 298 K has demonstrated that the values of VEM belong to a normal population of points corresponding to individual potential/volume data-pairs (E(i); v(i)) of any titration, whereas the values of pK(*)(w) (or of E(*)(w)) belong to a normal population with members corresponding to individual titrations, which is also the case for the equilibrium constants. The intertitration variation is attributable to the electrochemical component of the system and appears as signal noise distributed over the titrations. The correction for junction-potentials, introduced in a further stage of the program by optimization in a Nernst equation, increases the noise, i.e., sigma(pK(*)(w)). This correction should therefore be avoided whenever it causes an increase of sigma(pK(*)(w)). The influence of the ionic medium has been examined by processing data from acid-base titrations in 0.1M potassium chloride and 0.5M potassium nitrate media. The titrations in potassium chloride medium showed the same behaviour as those in sodium chloride medium, but with an s.d. for pK(*)(w) that was smaller and close to the expected instrumental noise, whereas the titrations in nitrate medium had a high noise level and even the determination of VEM was less certain. Procedures are also proposed for obtaining reference sets of data and checking the conformity of the solutions and apparatus to the chosen reference.

Analysis of variance applied to determinations of equilibrium constants.

The statistical analysis of variance has been applied to the values of the equilibrium constants of the glycinate-proton and glycinate-nickel systems, determined in different laboratories by pH-titration in aqueous solution. The analysis shows how the main part of the error derives from the variability from one titration to another even in the same laboratory. Therefore the data for a single titration (k) must be processed separately, thus yielding a mean value for the equilibrium constant logbeta (pqr)(k) of the species M(p)H(q)L(r); from these mean values for different titrations in each laboratory l, a within-laboratory grand average, logbeta (pqr)(l), can be calculated; the variance of this grand average measures the experimental error. A further analysis of the data from the different participating laboratories shows that there were no significant differences between laboratories for the constants reported. From these results it can be inferred that all the values of the mean constants logbeta (pqr)(k) for one species, as determined separately for each titration in four laboratories, belong to the same population. A chi(2) analysis of these populations demonstrates that the stability constants of the species HL, H(2)L(+), NiL(+), NiL(2) (with L(-) = glycinate) are normally distributed, but not that for NiL(-)(3). Therefore, general mean values of the first four constants can be calculated and proposed as reliable standard values at 25 degrees and I = 1.0M Na(Cl): protonation of glycinate, log beta(011) = 9.651(12), log beta(021) = 12.071(26); nickel-glycinate complexes, log beta(101) = 5.615(35), log beta(102) = 10.363(62). These values indicate that the standard deviations are rather higher than those often reported in the literature.