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) 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.

1. Introduction

The thermodynamic properties of hydrophobic hydration processes in water have been analyzed by us in a set of articles [1,2,3,4]. The thermodynamic properties of these systems are described by setting a Dual-Structure Partition Function {DS-PF} = {M-PF} · {T-PF}, as product of a Motive Partition Function {M-PF} multiplied by a Thermal Partition Function {T-PF}. This means that every hydrophobic hydration system in water is biphasic, constituted by diluted solution and by water, as implicit solvent at constant potential μ_s. By development of {DS-PF}, the 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 proved to hold for every hydrophobic process. The binding potential functions are convoluted functions, indicating as convolution the reciprocal interactions of the primary function f(1/T) or f(T) with the secondary function g(T) or g(lnT), respectively. The extrapolatio ns of the thermodynamic functions f(1/T) or f(T) for each compound ΔH_dual to T = 0 and ΔS_dual to lnT = 0, respectively, are allowed because the coefficient ΔC_p,hydr has been demonstrated to be constant and independent from the temperature for both mathematical and chemical constraints [5]. The extrapolations are correct because, for experimental data taken above 273 K, the inferior integration limits to T = 0 K and to lnT = 0, respectively, are equivalent because the interval (0 → 273 or more) is practically equal to the interval (1 → 273 or more).

The binding potential functions have been determined for many different systems and processes, for example for the solubility in water of noble gases and for protein denaturation, for protonation of carboxylic acids and for micelle formation [1,2 and reference therein]. Notwithstanding the evident structural, functional, and dimensional differences, every binding function has been found to be dependent on the same coefficients. At the end of the calculations, we have found that beyond significant information, extracted from the terms extrapolated to ξ_w = 0, the unitary (unitary means for ξ_w = 1) binding functions presented important similarities and analogies. Every function depends on the heat capacity of the system ΔC_p,hydr and on a stoichiometric coefficient ξ_w. The latter coefficient measures the number of water molecules W_III involved. Then, we have chosen to pass to an analysis of the general statistical validity of the coefficient found in different compounds and in different classes of reactions.

2. Results

Statistical Inference: User-Friendly Functions

The extrapolation of the thermodynamic functions for each compound ΔH_dual to T = 0 and ΔS_dual to lnT = 0, respectively has made possible the disaggregation of the thermodynamic functions ΔH_dual and ΔS_dual in two parts, motive (or work) and thermal (or compensative), as proposed by Lumry [6] since 1980. The disaggregation in motive and thermal functions has been done successfully for all the experimental data in the literature concerning hydrophobic hydration process. The analysis of the thermal components by means of Thermal Equivalent Dilution (TED) [7] has made possible the evaluation of the pseudo-stoichiometric coefficient ±ξ_w in each compound, from the curvature of the convoluted binding potential functions [1,2,3,4].

The separate motive functions, instead, can be analyzed and disaggregated by considering groups of compounds, i.e., for all the compounds in one family of reactions, by taking advantage of the previously determined coefficient ξ_w of each compound. For instance, we have considered together, either obtained by us either by other researchers all the values of ΔH_mot and ΔS_mot for all non-polar gases, then for all liquids, then for all carboxylic acids, then for all micelles having analogous structures with increasing length of the chain, then for all proteins belonging to a homogeneous group such as the family of lysozymes, etc. For each family, we have plotted ΔH_mot and ΔS_mot of each component of the family against the respective number ξ_w, previously determined by applying TED [5]. The disaggregation of enthalpy and entropy motive functions for non–polar gases are reported in

Table 1. Disaggregation of motive functions for non-polar gases (ξ_w = |n_w|). Data from E. Wilhelm, R. Battino, R.J. Wilcock, Chem. Rev., 1977, 77, 219–262.

Motive Function	Equation	R2 Factor	At Null Iceberg (ξw = 0)	St. dev.
ΔHmot =	+13.124 − 31687 nw	0.983	ΔH0(ξw = 0) = −31.7 kJ·mol−1	2.1
ΔSmot =	−82.681 − 445.44 nw	0.993	ΔS0(ξw = 0) = −82.7 J·K− 1·mol−1	5.4

.

We want to stress the point that the difference between one another group of compounds in structure, in molecular size, in aggregation state is remarkable. We call the attention, for example, on the differences existing between the determination of the solubility in water of non-polar gases and the denaturation of a protein: the molecular size of a gas is extremely different from that of a macromolecule, and the aggregation state as well. Moreover, radically different experimental methods have been employed in the various processes. Nonetheless, the motive functions of all the groups, when plotted as the functions of ξ_w, have given in any case significant linear diagrams with the same unitary slope [1,2].

For the groups where the heat capacity ΔC_p,hydr was positive (Class A) the analysis has yielded the following expressions—for enthalpy:

ΔH_mot = ΔH₀^{(ξw = 0)} + ξ_wΔh_for

(1)

and for entropy:

ΔS_mot = ΔS₀^{(ξw = 0)} + ξ_w Δs_for

(2)

In the families having negative heat capacity ΔC_p,hydr < 0 (Class B), the expressions were found to be for enthalpy:

ΔH_mot = ΔH₀^{(ξw = 0)} + ξ_w Δh_red

(3)

and for entropy:

ΔS_mot = ΔS₀^{(ξw = 0)} + ξ_w Δs_red

(4)

The unitary functions calculated by disaggregation of the motive thermodynamic functions of every hydrophobic hydration process can be represented by a paradigmatic scheme (

Table 2. General paradigm for hydrophobic hydration processes.

Transformations
Class A	A(ξwWI → ξwWII(iceberg) + ξwWIII)		iceberg formation
Class B	B(−ξwWIII − ξwWII (iceberg) → ξwWI)		iceberg reduction
Thermodynamic Functions
Class A
Apparent	Motive	Iceberg	Thermal
ΔHdual = ΔH0(ξw = 0) + ΔHfor + ΔHth	ΔHmot = ΔH0(ξw = 0) + ΔHfor	ΔHfor = ξw·Δhfor < 0	ΔHth = +ξw Cp,w T
ΔSdual = ΔS0(ξw = 0) + ΔSfor + ΔSth	Δsmot = ΔS0(ξw = 0) + ΔSfor	ΔSfor = ξw·Δsfor < 0	ΔSth = +ξw Cp,w lnT
Class B
Apparent	Motive	Iceberg	Thermal
ΔHdual = ΔH0(ξw = 0) + ΔHred + ΔHth	ΔHmot = ΔH0(ξw = 0) + ΔHred	ΔHred = ξw·Δhred > 0	ΔHth = −ξw Cp,w T
ΔSdual = ΔS0(ξw = 0) + ΔSred + ΔSth	ΔSmot = ΔS0(ξw = 0) + ΔSred	ΔSred = ξw·Δsred > 0	ΔSth = −ξw Cp,w lnT

) composed by three terms, whereby each term can be associated to a specific reaction step. The single terms in the paradigm of Table 2 can be calculated.

The motive enthalpies in each family of homologous compounds and the unitary functions calculated by disaggregation of the motive entropies in each family are reported for Class A and for Class B in

Table 3. Unitary functions by ergodic algorithmic model.

Class A. Unitary Enthalpy Function: Δhfor
Process	Δhfor	Unit	ξw range
Gas dissolut.	−21.6	kJ·mol−1·ξw−1	2–6
Liquid dissol.	−23.3	kJ·mol−1·ξw−1	2.7–5.4
Protein denat.	−22.1	kJ·mol−1·ξw−1	80–140
Carbox Proton.(*)	−21.8	kJ·mol−1·ξw−1	1.8–2.3
Class A. Unitary Entropy Function: Δsfor
Process	Δsfor	Unit	ξw range
Gas dissolut.	−445.4	J·K−1mol−1·ξw−1	2–6
Gas (Privalov)	−450	J·K−1mol−1·ξw−1	2–6
Liquid dissol.	−447	J·K−1mol−1·ξw−1	2.7–5.4
Protein denat.	−428.5	J·K−1mol−1·ξw−1	80–140
Carbox Proton.(*)	−442.6	J·K−1mol−1·ξw−1	1.8–2.3
Class B. Unitary Enthalpy Function: Δhred
Process	Δhred	Unit	ξw range
Micelle	+23.2	kJ·mol−1·ξw−1	4–19
Bio-complx	+24.3	kJ·mol−1·ξw−1	19–189
Benz.Cl-tryps.	+23.41	kJ·mol−1·ξw−1	5.3–11.3
Class B. Unitary Entropy Function: Δsred
Process	Δhred	Unit	ξw range
Micelle	−428±33	J·K−1mol−1·ξw−1	4–19
Bio-complx	−436.2	J·K−1mol−1·ξw−1	19–189
Benz.Cl-tryps.	−434.4	J·K−1mol−1·ξw−1	5.3–11.3

(*)Carboxylic acids.

.

The unitary functions Δh_for and Δs_for in Class A and Δh_red and Δs_red in Class B, respectively, show important similarities and analogies. The mean values of the unitary functions in the two Classes A and B are reported in

Table 4. Validation of the model. Analysis of unitary thermodynamic functions (*) [3].

Analysis within Classes
Class A: iceberg formation	Unit	Relative error
<Δhfor>A = −22.7 ± 0.7	kJ·mol−1 ·ξw−1	±3.1%
<Δsfor>A = −445 ± 3	J·K−1·mol−1·ξw−1	±0.7%
Class B: iceberg reduction	Unit	Relative error
<Δhred>B = +23.7 ± 0.6	kJ·mol−1 ·ξw−1	±2.51%
<Δsred> B = +432 ± 4	J·K−1·mol−1·ξw−1	±0.9%
Comparison among Classes
Enthalpy	Entropy
<Δhfor>A = −22.7 ± 0.7 kJ·mol−1·ξw−1	<Δsfor> A = −445 ± 3 J·K−1·mol−1·ξw−1
<Δhred>B = +23.7 ± 0.6 kJ·mol−1·ξw−1	<Δsred> B = +432 ± 4 J·K−1·mol−1·ξw−1
mean abs.value <Δh>A,B = 22.95 ± 0.75	mean abs.value <Δs> A,B = 438.5 ± 6.5
mean sd: (0.72 + 0.62) 1/2 = 0.92	mean sd: (32 + 42) 1/2 = 5
Student’s ratio: 0.75/0.92 = 0.815	Student’s ratio: 6.5/5 = 1.3
Hypothesis: absolute values in Class A and B are equal: hypothesis accepted (Mean in Class A = mean in Class B with sign reversed)

(*) Mean values obtained from more than eighty different sets, with about 600 data points. Note the small variability ±σ, indicating that all the points belong to a unique homogeneous statistical population.

. The values of each unitary function present a small variability around the mean in both Classes. The same homogeneity of results was found in both Classes. The residual error is within the limits of a possible experimental error, thus validating statistically the model. Moreover, we have found that the unitary values of Class B are equal to the corresponding values of Class A, with sign reversed (Table 4).

The statistical self-consistency of unitary data obtained by extremely different systems represents a decisive validation of the Ergodic Algorithmic Model (EAM).

From a thorough analysis of the data in Table 4, a one can remark the large value of the negative unitary entropy change in Class A, <Δs_for>_A = −445 ± 3 J·K⁻¹·mol⁻¹·ξ_w⁻¹ and the corresponding unitary positive entropy change <Δs_red>_B = +432 ± 4 J·K⁻¹·mol⁻¹·ξ_w⁻¹ in Class B.

The fact that the values of the unitary thermodynamic functions in Class B are the same as the corresponding values in Class A with opposite sign confirms the hypothesis that in Class A and in Class B we are dealing with the same unitary processes but in opposite directions in the two Classes: reaction A(ξ_wW_I → ξ_wW_II (iceberg + ξ_wW_III) with phase transition in Class A and reaction B(−ξ_wW_III– ξ_wW_II (iceberg → ξ_wW_I), in Class B with opposite phase transition. The large value of the negative unitary entropy change in Class A can be assumed to be a result of iceberg formation in connection with the dissociation [5] of +ξ_w water molecules W_III. By the name of “iceberg” we intend the complex between the hydrophobic molecule and the water W_II, surrounding it, as a whole. In contrast, the large value of the positive unitary entropy change in Class B is in accordance with the association [5] of –ξ_w water molecules W_III with iceberg reduction.

The iceberg formation in Class A is equivalent to reducing the volume of the solvent (ΔV_solvent < 0 = −V_cav, with V_cav = +ξ_wV_WI) and to increasing the concentration of the solute with configuration entropy loss, whereas in Class B the positive value of the unitary entropy change indicates that there is a process of iceberg reduction. Iceberg reduction means an increase of solvent volume (ΔV_solvent > 0 = −V_cav, with V_cav = −ξ_wV_WI) with dilution of the solute and consequent configuration entropy gain. The enthalpy-change for iceberg formation (Class A) is negative (<Δh_for>_A = −22.7 ± 0.7 kJ·mol⁻¹·ξ_w⁻¹) thus indicating that clustering interaction with hydrophobic molecules and water (W_II) is favored with respect to the clustering interaction H₂O (W_I) -- H₂O (W_I). We note that W_II is part of the solute and the preferred reaction is of concern of the Motive Partition Function {M-PF}.

The favorable enthalpy change is contrasted by the large negative entropy change due to concentration of the molecule, again an element of {M-PF}. The low solubility of gas re-establishes the free energy balance. On the other hand, in the process of Class B for iceberg reduction the enthalpy change is unfavorable but the entropy change is largely positive (<Δs_red>_B = +432 ± 4 J·K⁻¹·mol⁻¹·ξ_w⁻¹). We observe that in processes of Class B, as for example in micelle formation, the need of a balance by low solubility has decayed and the micelles are soluble.

The preference by a system for a process of Class B (hydrophobic hydration) rather than that of Class A (hydrophilic hydration) is determined by the favorable large entropic unitary functions.

The finding that it is the entropy change to lead the reactions disproves the theory, as described by Ben Naim [8], that hydration might be determined by solute-solvent energetic interactions that should be either stronger (attractive) or weaker (repulsive) than the corresponding solvent–solvent interaction in the system, giving rise to hydrophobic or hydrophilic hydration, respectively.

We can calculate the concentration change, corresponding to the unitary entropy change. We recall (see Table 3) that the mean value for Classes A and B is <Δs_r>_A,B = 438.5 ± 6.5 J·K⁻¹·mol⁻¹. From ΔS_X/R = −ln[X] (438.5/8.3145 = 52.74236) we calculate the unitary number of molecules N_Un = 0.802304·10²³. Hence, by dividing by the Avogadro number N_Avo = 6.02214·10²³, we can calculate the molecular (and molar) fraction x = 0.802304/6.02214 = 0.13322575 (13.32%).

It is important to point out that the cavity as calculated by statistical thermodynamic methods (e.g., Potential Distribution Theorem (PDT) [9]) is inconsistent with iceberg formation or reduction considered in this article. The former statistical cavity concerns exclusively the non-reacting ensemble of the solvent (NoremE), involving thermal functions without any effect on free energy and on thermodynamic potential μ_s. The Potential Distribution of the Theorem (PDT) is not-existent because the solvent (as Implicit Solvent) in hydrophobic hydration processes and in biochemical reactions is at constant potential μ_s: the Thermal Partition Function {T-PF} cannot give any contribution to free energy and to potential μ_s. In contrast, the iceberg formation (or iceberg reduction) considered in this article concerns the motive functions with formation (or reduction) of iceberg by transforming water W_I into W_II and W_III, or vice-versa. This process of iceberg formation, in Class A, reduces the volume of the solvent W_I. In such a way, it modifies the thermodynamic state of the solute (by changing dilution and, therefore, configuration density entropy), which is of concern for motive functions (ΔG_mot ≠ 0). Water W_II, in fact, becomes part of the molecule of the solute (as iceberg sheath) and enters as component of the equilibrium constant in the solute motive partition function {M-PF}. In Class B, the opposite process takes place, with expansion of the solvent W_I thus modifying the thermodynamic state of the solute. The transformations in the thermodynamic state of the solute, including W_II, are, again, of concern for motive partition functions {M-PF} and, consequently, for motive free energy.

The Ergodic Algorithmic Model (EAM) offers a complete picture of every hydrophobic hydration process, considering the steps of:

(i)

Subdividing the apparent partition function into thermal {T-PF} and motive {M-PF} partition functions;

(ii)

Determining iceberg formation or iceberg reduction in a niche within the field of motive partition function;

(iii)

Considering two types of water clusters W_I and W_II, together with free molecules W_III;

(iv)

Determining the number ±ξ_w of water clusters W_I (phase change) from curvatures of the binding potential functions α) RlnK_dual = {f(1/T)*g(T)} and β) RTlnK_dual = {f(T)*g(lnT)}, calculated from sets of equilibrium constants measured at different temperatures and treated by thermal equivalent dilution principle;

(v)

assigning to W_I the role of solvent (implicit solvent);

(vi)

attributing any change of configuration entropy of W_II and W_III to the motive partition function {M-PF} (solute) and not to the thermal partition function {T-PF} of implicit solvent.

Altogether these elements concur to describe in detail the behavior of many biochemical reactions, so important for the description of biological processes. In this regard, we cannot forget the criticism by Lumry [6], who considered that the thermodynamic functions applied to biochemical equilibria were not user-friendly. He thought that the causes of the unreliability of these thermodynamic functions were that nobody used to subdivide the thermodynamic functions into thermal and work components, and nobody had considered the dual nature of hydrophobic solutions. These statements by Lumry represent a splendid insight into the problem of hydrophobic hydration. By applying the two conditions foreseen by Lumry, we have been able to calculate reliable unitary functions <Δh_for>_A, <Δs_for>_A, <Δh_red>_B, and <Δs_red>_B. The statistical analysis of the unitary data in Table 4 not only demonstrates that ΔC_p,hydr, is constant but also that we can identify in the unitary functions the user-friendly functions hoped for by Lumry [10]. The statistical analysis has been extended to a significant number of different compounds. By employing the unitary functions of Table 4, if we can define previously, by applying TED, to sets of equilibrium constants measured at different temperature, the number ±ξ_w of molecules W_III involved, we can calculate the motive thermodynamic functions for iceberg formation or reduction in any new hydrophobic hydration process.

3. Discussion

3.1. Water in Thermal and Chemical Denaturation

The statistical analysis has validated the unitary values found in different compounds. In the following paragraphs we want to analyze the specific reactions that were experimentally found to solve many questions so far not yet explained in the literature. We start by analyzing the displacement of the equilibrium between the different forms of water, to explain any transformation of the macromolecules, particularly in denaturation processes. A reasonable explanation of the mechanism of thermal denaturation is that the folded native protein had been formed through a process of hydrophobic association analogous to that of micelle formation, with an outstanding positive entropic contribution. We recall that the hydrophobic bonding is driven by the positive entropy change, ΔS_red > 0 produced as the consequence of the condensation of ξ_w water molecules (W_II + W_III) into water W_I, with iceberg reduction B(−ξ_wW_III − ξ_wW_II (iceberg) → ξ_wW_I. To the folded native protein can be assigned unitary values of the thermodynamic stepwise functions equal to those of the denaturation steps, with sign reversed. In the opposite reaction A(ξ_wW_I → ξ_wW_II (iceberg) + ξ_wW_III), taking place at denaturation, W_I is that part of the system that is giving rise to a change of phase, from structured to fluid state. When the heat supply starts, the heat moves a cluster from the solvent W_I thus generating one cluster W_II and a molecule of water W_III, thus displacing the equilibrium toward the fluid state and formation of iceberg. The whole process takes place through three steps (Figure 1):

(1)

Start: The heat supplied to the system generates melting of some clusters of water W_I to give W_II +W_III, creating the niche wherein iceberg is formed. In fact, the creation of the niche with iceberg reduces the solvent (W_I) volume and produces negative entropy (dS_for < 0), thus beginning to cancel the positive entropic contribution of protein folding (ΔS_red + dS_for).

(2)

Scanning: The process of heat supply continues until the integral entropy:

_T2
ΔS_for = ∫ dS_for
^T1

(5)

cancels completely the positive entropy of folding and causes disruption of the hydrophobic bonds, at least those around the active site, that had been keeping the native protein folded.

(3)

Final: At this stage (ΔS_red + ΔS_for = 0), the whole positive entropy contribution produced by folding is cancelled: the disruption of every hydrophobic bond is completed, and the denatured state has become the stable one. The denaturation process consists in the disruption, through iceberg creation and negative entropy production, of the hydrophobic bonds that had been keeping the chains folded in the native protein.

A mechanism analogous to that of thermal denaturation (Figure 2a) can explain the chemical denaturation of proteins (Figure 2b). The added denaturant tends to combine with water W_II thus displacing the equilibrium in water toward hydrophobic hydration with iceberg formation. The negative entropy produced by iceberg formation neutralises the positive entropy of folding. In other words, we can say that heat displaces the equilibrium of water forms toward hydrophobic hydration by acting on W_III whereas the denaturant displaces the equilibrium by acting on W_II. The negative entropy ΔS_for << 0 that is produced has the same effect as in heat denaturation, destroying the hydrophobic bonds of the native protein. With reference to the equilibrium in water, we can explain [4] the action of the so-called stabilizers, substances that favor protein folding. We can consider (Figure 2c) that the stabilizers act as templates for the tetrahedral structure of water W_I, thus displacing the equilibrium in the opposite direction, toward the reduction of the iceberg and hydrophilic hydration.

3.2. Motive Free Energy and Iceberg Formation/Reduction

As already explained, the hydrophobic hydration processes are based on the formation of a niche filled with an iceberg in Class A and on iceberg reduction in Class B. The iceberg is formed in Class A, by a phase transition in the bulk of the solvent (W_I) of +ξ_w water clusters W_I which then transform into icebergs W_II. In contrast, the opposite process takes place in Class B whereby an iceberg is reduced in Class B by condensation to solvent (W_I) of −ξ_w water clusters W_I, reconstituted by molecules W_III combined with W_II set free by iceberg reduction. The number ξ_w depends on the size of the reacting molecule or moiety of macromolecule. As shown in the previous sections, we have accepted the idea that ξ_w water clusters W_I give origin to a change of phase forming (W_II + W_III) in such a way that the values of enthalpy divided by T, ΔH_th/T and entropy, ΔS_th, produced by this change of phase, consist of the only contribution by the hydrophobic isobaric heat capacity ΔC_p,hydr in every hydrophobic hydration process. The existence of thermal entropy and thermal enthalpy as distinct from motive entropy and motive enthalpy, respectively, follows from the Dual-Structure Partition Function {DS-PF} [5]. If we apply this principle, by subtracting ΔH_th from ΔH_dual and ΔS_th from ΔS_dual, thus obtaining ΔH_mot and ΔS_mot, respectively, we can calculate a motive free energy,

ΔG_mot = ΔH _mot − TΔS_mot

(6)

that in protein denaturation, and in general in Class A, is positive (ΔG_mot > 0), exclusively due to the prominent negative configuration entropy contribution ΔS_for for niche formation. The processes of Class A can be classified as entropy opposed. We remind that the values of ΔH_mot and ΔS_mot are simply

Obtained from the experimental data of ΔH_dual and ΔS_dual extrapolated to T = 0 and to lnT = 0, respectively. This finding is a correction of the opinion of Lumry [6], who retained that motive and thermal parts of enthalpy and entropy were usually not experimentally determinable. The numerical results of the disaggregation of the motive functions for different types of hydrophobic hydration processes are reported in

Table 5. Motive function disaggregation (*).

Process	Motive Function	Disaggregation Equation	ΔGmot(298)	Unit	ξw Range	<ξw> Mean
Class A
Gas Dissolut.	ΔHmot	−17.7 − 21.6 ξw		kJ·mol−1
	ΔSmot	−86.4(!) − 445.4ξw		J·K−1mol−1
	ΔGmot(298)	8.04(**) + 111.133ξw	+452.4	kJ·mol−1	2–6	4
Liq. Dissolut.	ΔHmot	+4.6 − 23.3 ξw		kJ·mol−1
	ΔSmot	−0.5(!!) − 447 ξw		J·K−1mol−1
	ΔGmot(298)	4.74 (**) + 109.9 ξw	+437.9	kJ·mol−1	2.7–5.4	4
Protein denat.	ΔHmot	+211.82 − 22.5 ξw		kJ·mol−1
	ΔSmot	+415 − 428.5 ξw		J·K−1mol−1
	ΔGmot(298)	−88.15(**) + 105.04ξw	+8319	kJ·mol−1	80–140	120
Protonation	ΔHmot	+0.1 − 21.8 ξw		kJ·mol−1
	ΔSmot	+104 − 442.6 ξw		J K−1·mol−1
	ΔGmot(298)	−30.9(**) + 104.6ξw	+1783	kJ·mol−1	1.8–2.3	2.1
Class B
Protein fold.	ΔHmot	−211.82 + 22.5ξw		kJ·mol−1
	ΔSmot	−415.8 + 424.2ξw		J K−1·mol−1
	ΔGmot(298)	+88.15 − 103.9ξw	−8281	kJ·mol−1	80–140	120
Micelle form.	ΔHmot	−3.97 + 23.13 ξw		kJ·mol−1
	ΔSmot	+10.2 + 428 ξw		J K−1·mol−1
	ΔGmot(298)	–7.01(**) − 104.4ξw	–1569	kJ·mol−1	4–19	15
Deprotonat.	ΔHmot	–0.1 − 21.8ξw		kJ·mol−1
	ΔSmot	−104 + 442.6ξw		J K−1·mol−1
	ΔGmot(298)	+30.9(**) − 104.6ξw	−1783	kJ·mol−1	1.8–2.3	2.1

(*)ΔH₀^{(ξw = 0)} and ΔS₀^{(ξw = 0)} are the motive functions extrapolated to null iceberg (ξ_w = 0); (**)ΔG₀^{(ξw = 0)} (T) = ΔH₀^{(ξw = 0)} − T ΔS₀^{(ξw = 0)}; (!) configuration entropy loss at gas condensation; (!!) in liquids: no condensation = no entropy loss.

. Each motive function is composed by two terms. For example, the first term of enthalpy in Class A is ΔH₀^{(ξw = 0)} and represents the motive enthalpy extrapolated to ξ_w = 0, i.e., at null iceberg. The second term –Δh_for·ξ_w represents the contribution to enthalpy by the process of iceberg formation (or iceberg reduction in Class B). Analogous distinctions hold for motive entropy. The reliability of the motive functions, with the terms corresponding to the process of iceberg formation/ reduction, can be checked with reference to the process of protonation of carboxylate anions. This process belongs to Class A with iceberg formation whereas the process of deprotonation of the corresponding acid belong to Class B with iceberg reduction. By considering a reaction of this kind involving ξ_w = 2.1 water molecules W_III, we can calculate the free energy for iceberg formation as ΔG_for = −ξ_w 22.95 –T (−ξ_w 0.4385) = −48.195 + 0.921 T. This equation is represented as vector composition in Figure 3a. On the contrary, the free energy for iceberg reduction is represented by the equation ΔG_red = 48.195–0.921 T and is represented in Figure 3b. In both diagrams, we can verify how the prominent contribution, either positive or negative, respectively, derives from the entropic component. The process in Class A (ΔG_for > 0) is unstable whereas the process in Class B is thermodynamically favored (ΔG_red < 0). The result is that the dissociated state is the stable one. This corresponds to the experimental finding that carboxylic acids are dissociated in aqueous solution in their stable state.

The strength of each acid is mainly determined by the positive entropic contribution for iceberg reduction. The dissociation of carboxylic acids is entropy driven. We have verified [3] that by subtracting the free energy for iceberg formation (ΔG°_for/T = RlnK_for) from the free energy for protonation (ΔG°_prot/T = RlnK_prot) at different temperatures we obtain a residual (ΔG°_x/T = RlnK_x). The values of lnK_x for every acid plotted against 1/T present perfectly linear van’t Hoff plots (i.e., linear binding function). This confirms that was the iceberg reaction to produce the constant curvature of the binding functions.

The analysis of the functions extrapolated to null iceberg ΔH₀^{(ξw = 0)}, ΔS₀^{(ξw = 0)} and ΔG₀^{(ξw = 0)}(298) offer important pieces of information. The null iceberg functions are referred to the initial step of a reaction: water-gas, water-liquid, lateral chain-lateral chain, etc. In every case, either in Class A or in Class B, the null-iceberg functions, are a small portion of the whole respective motive function. This confirms again that the process of iceberg formation or reduction is ruling the whole reaction.

The values of free energy ΔG₀^{(ξw = 0)}(298) is positive in Class A and negative in Class B, thus showing which processes are thermodynamically favored with ΔG₀^{(ξw = 0)}(298) < 0. As for enthalpy, the value of ΔH₀^{(ξw = 0)} = +211.6 kJ·mol⁻¹ shows that at uncoiling, the detaching from one another of parts of the coiled external chains of a protein, requires expenditure of energy.

As for enthalpy, important pieces of information can be extracted from the analysis of ΔS₀^{(ξw = 0)}, at null iceberg. In gas dissolution, the value ΔS₀^{(ξw = 0)} = −86.4 J K⁻¹·mol⁻¹ is coincident, with opposite sign, with the entropy change ΔS_evap = +86.9 ± 1.4 J·K⁻¹·mol⁻¹ (thermal entropy change) given by the Trouton constant referring to the passage in general from liquid to vapour. If we recall that evaporation is the passage from condensed liquid to gaseous vapour we can explain how in the dissolution in aqueous solution, the gas molecule, when trapped in the solvent water, is losing an equivalent amount of configuration entropy. Indirectly, this finding is confirmed by analyzing the process of dissolution in water of liquid substances. In the case of liquids, the entropy change at null iceberg is ΔS₀^{(ξw = 0)} = −0.5 J·K⁻¹·mol⁻¹, i.e., almost zero, because the liquid is already condensed before dissolution in water and no condensation process takes place.

The coincidence of ΔS₀^{(ξw = 0)} = −86.4 J·K⁻¹·mol⁻¹, (configuration entropy change) calculated in non-polar gases [3] with the entropy change ΔS_condens = −86.9 ± 1.4 J·K⁻¹·mol⁻¹ (thermal entropy change) given by the Trouton constant referring to the passage in general from vapour to liquid (equivalence between thermal and configuration entropy) is a highly significant validation of the model.

The motive functions ΔH_mot, ΔS_mot, and ΔG_mot can be calculated also for micelle formation and for any element of Class B, where the function ΔG_mot results to be negative (ΔG_mot < 0). The process of micelle formation by hydrophobic bonding, notwithstanding the positive enthalpy contribution, is thermodynamically favored (ΔG_mot < 0), because of the overwhelming effect of the favorable entropy contribution (−TΔS_red << 0) for iceberg reduction. The processes of Class B can be classified as entropy driven [5].

The analysis of the null iceberg function ΔS₀^{(ξw = 0)} explains the mistake taken by Chandler [11,12] by introducing the concept of a length effect and of a cross-over point for the passage from volume hydrophobic effect to surface hydrophobic effect. The length scale effect supposed by Chandler is referred to the specific initial state ΔS₀^{(ξw = 0)} of the reactants, analogous to the initial passage of whole molecules in the dissolution of gases in water (ΔS₀^{(ξw = 0)} = −86.4 J K⁻¹·mol⁻¹) or to the dissolution of whole liquid molecules in water (ΔS₀^{(ξw = 0)} = −0.5 J·K⁻¹·mol⁻¹) or to the passage from molecular solution to macromolecular solution. In macromolecular solution, the initial state is a free unit with unfolded moieties and the resulting state is that with folded moieties: the macromolecular behavior of folding starts at large macromolecular size. In contrast with Chandler theory, there is no length scale effect for the specific reaction of iceberg formation or iceberg reduction: the kind of iceberg reaction is independent from the size of the solute molecule, rather the affinity of the iceberg reaction is strictly proportional to the size of the entering molecule or moiety, as shown by the constant values of the unitary functions referred to ξ_w = −1 in Table 4, <Δh_red> = +23.7 ± 0.6 kJ·mol⁻¹ ·ξ_w⁻¹ and <Δs_red>_B = +432 ± 4 J·K⁻¹·mol⁻¹·ξ_w⁻¹ calculated from any kind of molecule, from noble gases to macromolecules.

It is worth-noting that the prominent entropic effects, negative in Class A and positive in Class B, respectively, are not consistent with some theories (Ben Naim [8]) attributing hydrophobic or hydrophilic hydration to solute-solvent energetic interactions, that should be stronger or weaker than the solvent–solvent energetic interactions.

Even the process of cold denaturation of protein can be explained by the EAM model [13] by referring to the motive free energy. The change of sign of the motive free energy of folding ΔG_mot (here named ΔG_fold) indicates which is the stable state: either folded or denatured. Above T_fold the folded state is stable being ΔG_fold < 0, whereas below T_fold the protein denatures for ΔG_fold > 0.

3.3. Null Thermal Free Energy

Lee and Graziano [14] expressed the opinion that in biochemical processes there are some side reactions where enthalpy and entropy compensate for each other and do not influence the free energy. The same hypothesis had been launched by Benzinger [15]. The Ergodic Algorithmic Model (EAM) confirms how thermal enthalpy, ΔH_th and thermal entropy, ΔS_th satisfy the conditions foreseen by these authors. A necessary consequence of this property is that thermal free energy is zero (ΔG_th = 0). Regrettably, thermal free energy is considered different from zero in too many text-books and articles, with specific reference to protein unfolding [16,17,18,19,20] and to micelle formation [21]. Moreover, no mention of the motive functions is reported in these texts [13].

It is worth mentioning that the equation,

ΔG(T) = 0 − (ΔC_p,hydr/T) {(T_d –T) + T ln(T/T_d)}

(7)

with ΔG(T) ≠ 0, represents a heresy for general thermodynamic theory because thermal intensity entropy cannot produce any chemical work in a non-reacting system (NoremE ensemble), the solvent, wherein no concentration change is possible and, consequently no free energy can be produced. Mechanical work, however, is possible in these non-reacting systems (cf. Carnot cycle). On the other hand, the mathematical expression of Equation (7), if correctly developed, results to be equal to zero, as required by thermal partition functions. Therefore, we obtain, at variance with the erroneous Equation (7):

ΔG(T)/T = ΔH(T)/T–ΔS(T) = 0

(8)

[5] in accordance to the invariable property of null free energy (ΔG_th/T = 0) of thermal functions.

We would like to underline that the identification of the solvent as a non-reacting system (implicit solvent) has been possible because of the introduction of distinct partition functions for Implicit Solvent, with thermal probability factor {T-PF}, and for solution with motive probability factor {M-PF}. The null free energy is a constitutional invariable property of every non-reacting molecule ensemble (NoremE) [5].

3.4. Water W_I, W_II, W_III, and Hydrophobic Bond

Characterization of hydrophobic bonding is another point of the thermodynamics of hydrophobic hydration processes where the Ergodic Algorithmic Model (EAM) can offer a positive contribution to modify erroneous assumptions, unfortunately accepted by the literature. While examining the applicability of the second law of thermodynamics to the living organisms, Edsall and Gutfreund [22] have considered the assembly of a virus molecule from its subunits, which, according to these authors, apparently involves an increase of order in the system. If the virus is considered an isolated system, this process—according to them—would be in defiance of the Second Law. However, a virus molecule—they arbitrarily assume—interacts directly with its environment. The assembly of a virus molecule was assumed by Edsall and Gutfreund (Figure 4) to increase the entropy of the whole system, due to the supposed liberation of solvation water from the components and the resulting increase in rotational and translational entropy of solvent molecules, when detached from the interface between subunits. The Ergodic Algorithmic Model rejects the generally accepted interpretation (cf. Wikipedia) of “the assembling of a virus molecule from the components with expulsion of solvent molecules from the intermolecular interface, with increase in rotational and translational entropy of the solvent molecules expelled”.

The hydrophobic association processes present invariably negative ΔC_p.hydr (ΔC_p.hydr < 0). According to the Ergodic Algorithmic Model (EAM), negative heat capacity means negative n_w and condensation of water molecules to form W_I. (We recall that ΔC_p.hydr = n_wC_p.wr and ξ_w = |n_w|). Therefore, association by hydrophobic bonding means that the reaction B(−ξ_wW_III– ξ_wW_II(iceberg) → ξ_wW_I has taken place, with condensation of water molecules W_II+W_III to W_I and consequent iceberg reduction. If we would accept the suggestion of Edsall and Gutfreund of the liberation of water of solvation from the interface, we should have ΔC_p.hydr > 0 contrary to the experimental evidence. In any case, we have shown above that the thermal components of the thermodynamic functions cannot give any contribution to free energy (−ΔG_th/T = 0). In contrast, the Ergodic Algorithmic Model (EAM) (Figure 5) suggests that the initially separated units (supposed to be four) stick together with coalescing of the icebergs. The reduction of iceberg by condensation of water molecules W_II and W_III to W_I, results in an increase (emeraldine) of the solvent volume.

The increase of the solvent volume (ΔV_solvent > 0) is combined with the reduction of the number of independent molecular units of the solute from 4 to 1, so that the solute is diluted. Dilution of solute means increase of density entropy. Moreover, water molecules W_III disappear from the solution for condensation, thus becoming more diluted as ligand. Altogether, these combined processes make the dilution of the solute to increase.

Correspondingly, Tanford also considers the formation of hydrophobic bonds. According to this author, when two or more hydrophobic molecular units present in the solution associate one another the extension of the solute–solvent interface should be reduced.

Consequently, the number of hydrogen bonds rearranged should be reduced, thus producing, according to Tanford [23], positive entropy gaining in the solvent water. The last process has some resemblance with the reaction B(−ξ_wW_III–ξ_wW_II → ξ_wW_I). The process of iceberg reduction is completely ignored and the entropy increase erroneously attributed to the solvent but not to the solute. We confirm again that only the implicit solvent is consistent with the convoluted binding potential functions.

According to the Ergodic Algorithmic Model (EAM), the increase of configuration density entropy due to dilution is the driving force that moves the reaction toward the association of the units by hydrophobic bonding. In other words, the entropy change can be attributed exclusively to changes in the thermodynamic state of the solute, {M-PF}. and not of the solvent {T-PF}.

Alternatively to the scheme of Edsall and Gutfreund, the dissolution of molecules has been interpreted by Tanford [23] as a rearrangement of water-to-water hydrogen bonds. This rearrangement should be caused by the introduction into the solvent water of molecules of a hydrophobic compound, e.g., an aliphatic hydrocarbon. This process should involve entropy-consuming reactions in the solvent. The rearrangements of hydrogen bonds take place at the solute-water interface. This process resembles the reaction A(ξ_wW_I → ξ_wW_II + ξ_wW_III) without iceberg formation. The step of iceberg formation is again completely ignored by Tanford: the entropy changes is still erroneously attributed to the solvent and not to the solute. The difference between the Edsall and Gutfreund [22] scheme and that proposed by the Ergodic Algorithmic Model (EAM) (Figure 5) is evident. In the Edsall and Gutfreund scheme (see Figure 4), and in that of Tanford as well, the entropy producing processes take place in the structure of the solvent, with changes in the thermodynamic thermal parameters of the solvent itself.

In contrast, in the Ergodic Algorithmic Model (EAM), the entropy consuming process of iceberg formation or the entropy producing process of iceberg reduction taking place in the solvent yield changes in the thermodynamic configuration state of the solute {M-PF}. In fact, iceberg formation means diminution of dilution of the solute and, therefore, density entropy diminution, whereas iceberg reduction with extension of the solvent volume is equivalent to increasing the dilution and hence increasing the density entropy of the solute. These entropy changes of the solute are more pertinent to the problem at hand: we are studying, in fact, the thermodynamic properties of the solute.

This new view of hydrophobic bonding represents a complete change of perspective with respect to the Edsall and Gufreund scheme. It is important from a theoretical point of view, to underline this point. The assignment of a change of configuration density entropy to the solvent W_I is in principle contradictory. A chemical reaction consists of changes of the concentrations of the reactants, corresponding to changes in configuration density entropy: it is impossible to have a change of concentration (or dilution) of an excess component, the solvent (NoremE ensemble, as Implicit Solvent), that has, by definition, no concentration change. The solvent in a diluted solution has the same role of vacuum in a gas. The vacuum can only change its volume but not its concentration and is at constant potential. Therefore, the solvent cannot produce per se any configuration density entropy. In fact, in the Ergodic Algorithmic Model (EAM) that part of water giving origin to increase, or diminution of configuration entropy is water W_II, or W_III as factors of equilibrium constant, whereas the function of solvent (as the implicit solvent) is reserved for water W_I. Therefore, the idea of considering that the favorable entropy causing hydrophobic bonds is generated within the solvent water is unacceptable.

The free energy change for hydrophobic bonding can be evaluated from the mean values for iceberg reduction, <Δh_red>_B = +23.7 ± 0.6 kJ·mol⁻¹·ξ_w⁻¹ and <Δs_red> _B = +432 ± 4 J·K⁻¹·mol⁻¹·ξ_w⁻¹ reported in Table 4. At 298 K, the free energy for hydrophobic bond results to be ΔG₂₉₈ = −105.04 kJ·mol⁻¹·ξ_w⁻¹ for each water molecule W_III involved. The composition of this free energy is represented in Figure 6 where we can appreciate the overwhelming effect of the entropy term. The hydrophobic bond is confirmed to be entropy driven. We repeat, that we are dealing with positive entropy change of the solute, due to iceberg reduction, with consequent expansion of the solvent volume with dilution of the solute.

The separation of the thermal functions ΔH_th and ΔS_th from the motive functions ΔH_mot and ΔS_mot, respectively, raises the question whether the temperature is conditioning, or not, hydrophobic bonding. The separate determination of the thermal functions ΔH_th/T and ΔS_th which are equal to each other leads to conclude that both represent the same entropy change (both are measured in J·K⁻¹·mol⁻¹). The thermal portions ΔH_th and ΔS_th of the observed thermodynamic functions ΔH_dual and ΔS_dual, respectively, concern the transformation (phase transition) of water W_I.

The thermal functions, concerning the solvent partition function, do not contribute to free energy of iceberg reduction or iceberg formation which are the basic steps for formation or disruption, respectively, of the hydrophobic bond. Therefore, this shows that it is vain to search for a direct effect of the temperature on the hydrophobic processes of iceberg formation or reduction. Indirectly, it is the supply of heat at denaturation that promotes [4] melting of some water clusters W_I and causes iceberg formation.

3.5. Water W_I: Implicit Solvent

The question of the changes in the volume of the solvent is of some concern for General bio-thermodynamics because it is connected to the type of solution model adopted. In the theory of ideal solution, in fact, the solvent is like a vacuum. The solute molecules move in this surrounding as if they were gaseous. The only transformation undergone by the solvent concerns its volume. According to the Ergodic Algorithmic Model (EAM), the solvent water, in its component W_I, as Implicit Solvent, keeps the properties of the bulk solvent. Changes in the volume of the solvent produce changes in the thermodynamic properties of the solute. The only changes that are relevant to the thermodynamic properties of the solvent are “iceberg formation” (ΔV_solvent < 0), increasing the concentration of the solute, or “iceberg reduction” (ΔV_solvent > 0) increasing the dilution of the solute. Iceberg formation is entropy consuming (<Δs_for>_A = −445 ± 3 J·K⁻¹·mol⁻¹·ξ_w⁻¹), whereas iceberg reduction is entropy producing (<Δs_red>_B = +432 ± 4 J·K⁻¹·mol⁻¹ ·ξ_w⁻¹). We recall that the large entropy production:

(i)

is developing in the solute motive partition function {M-PF}, and

(ii)

is the driving force forming hydrophobic bonds between solute units.

The picture of a gain of entropy by the solute, as due to dilution of the solute itself, is coherent with the so-called molecule-frame (MF) approach. According to Henchman et al. [24] the MF approach is associated with theories such as “continuum solvent”. This approach ignores the explicit nature of the solvent using vacuum in the ideal gas as reference model. The MF approach is valid under the condition of diluted solutions, condition that hydrophobic hydration processes satisfy. Alternatively, Henchman proposes to refer to the system frame (SF) whereby the molecules, either of solute and solvent, are referred to a unique common reference system. The MF frame seems more adequate for the ergodic algorithmic model, whereby water W_I represents the Implicit Solvent with thermal partition function {T-PF}. In fact, the cluster W_II surrounding the solute forms a unique molecular unit with a solute moiety and it follows the thermodynamic state of that solute moiety, in the realm of {M-PF}. The (solute +W_II) molecular unit is dissolved in the bulk solvent W_I, again conform to the MF scheme. At the same time, the water molecules W_III are free to move in the full volume of the solvent W_I, as in a vacuum, and this part of the process is again conformed to the MF picture.

The processes of iceberg reduction and iceberg formation imply enthalpy changes also. The enthalpy for iceberg formation indicates an exothermic reaction A(ξ_wW_I → ξ_wW_II(iceberg) + ξ_wW_III (<Δh_for>_A = −22.2 ± 0.7 kJ·mol⁻¹·ξ_w⁻¹). This is the unitary enthalpy change for the transformation from clusters (W_x)_I to ((W_x-1)_II + W_III) with iceberg formation and chemical combination with solute to form the solvation sheath. This reaction producing the ligands of the solute (W_x-1)_II and W_III with solute solvation takes place in the domain of motive partition function and is of concern, therefore, for the motive thermodynamic functions. The transformation from clusters (W_x)_I to clusters (W_x-1)_II implies changes in the strength of water-water hydrogen bonds that are stronger in W_II than in W_I. The component W_II is credited, in fact, of higher density than W_I. The reaction step of iceberg formation with solvation of the solute is, therefore, exothermic. On the other hand, the process of iceberg reduction B(−ξ_wW_III − ξ_wW_II(iceberg) → ξ_wW_I), that involves a back reaction from ((W_x-1)_II+W_III) to (W_x)_I, is endothermic as shown by the unitary enthalpy <Δh_red>_B = +23.7 ± 0.6 kJ·mol⁻¹·ξ_w⁻¹. The enthalpy changes are again coherent with the MF approach because the enthalpy effects concern only water clusters W_II and molecules W_III solvating the solute (i.e., all solute components), whereas clusters (W_x)_I, composing the bulk solvent, do not change their concentration but simply expand or reduce their total volume by addition or subtraction, respectively, of some clusters.

3.6. From Ergodic Algorithmic Model to Computer Chemistry

In Reference [13], we have already discussed the connections between Ergodic Algorithmic Model (EAM) and computo-chemistry, particularly between EAM and Potential Distribution Theorem (PDT). The comparison has revealed the relevant weak points of PDT. Many calculations aiming at obtaining potential functions μ_s for iceberg formation in the solvent, based on the partition function of the solvent itself, are not appropriate because, as shown by the Ergodic Algorithmic Model (EAM), the thermal partition function {T-PF} of the solvent cannot give origin to any free energy change and, of course, to any change of potential μ_s. In contrast, another point of PDT has been properly developed, in conformity with the dual structure of the hydrophobic hydration systems. The introduction in PD by Pratt and La Violette of the quasi-chemical approximation [25,26], which could be more appropriately renamed as chemical molecule/mole scaling function, keeps the point that the solute of any hydrophobic hydration system constitutes a REME ensemble, not ruled by Boltzmann statistics, rather by binomial distribution of chemical reactions. The partition function of the solute is the Motive Partition Function {M-PF} and the binding function RTlnK_mot = f(T) can be calculated by applying the quasi-chemical approximation, proposed by Pratt and La Violette [25,26] either to the dissociation constant of water

K_diss = (a_A)·(a_B)⁻¹·(a_WII)^ξw

(9)

or to the association constant of water

K_assoc = (a_A)·(a_B)⁻¹·(a_WII)^–ξw

(10)

chosen by reference to the sign of ±ξ_w, experimentally determined. The ergodic activity a_A is calculated as product of thermal activity factor Φ times molar fraction x_A

a_A = Φ·x_A

(11)

where Φ = T ^{–(Cp,A /R)} is preserving the ergodic property of the solution. By taking advantage of the previous calculation of the thermal functions ΔH_th = ΔC_p,hydrT and ΔS_th = ΔC_p,hydr lnT, the calculated binding potential functions α) lnK_calc = (−ΔG_calc/T) = {f(1/T)_*g(T)}, and β) RTlnK_calc = (−ΔG_calc) = {f(T)_*g(lnT)} can be obtained and numerically compared with the observed binding potential function RlnK_dual = (−ΔG_dual/T) = {f(1/T)_*g(T)} and RTlnK_dual = (−ΔG_dual) = {f(T)_*g(lnT)}.

Unfortunately, too many computer simulations are not recognizing the dual structure of the partition function {DS-PF} of the hydrophobic hydration processes, and do not combine the binomial distribution of a Mole ensemble {M-PF} with the Boltzmann distribution of a molecule ensemble {T-PF}. The computer calculations should be amended, by following the procedure indicated by Talhout, et al. [27], who have determined experimentally the curved binding functions at different temperatures for a series of hydrophobically modified benzamidinium chloride inhibitors to trypsin, and then have checked the results of simulations with the experimental findings.

We recall the point that the statistical validation of the whole set of experimental data of a significant large population of experimental points relative to hydrophobic hydration processes of any kind, presented in this article, qualifies these data as representative of any type of hydrophobic hydration processes. These results realize the user-friendly functions hoped for by Lumry. Therefore, in every example of computer-assisted drug design, we must assume that binding potential functions α) RTlnK_dual = (−ΔG_dual/RT) = {f(T)_*g(lnT)} and and β) RlnK_dual = (−ΔG_dual) = f(1/T)_*(T) necessarily exist. These functions can be experimentally determined in advance of computer simulation. Then, computer simulations will be assessed in comparison with experimental equilibrium constants.

4. Conclusions

The hydrophobic hydration processes are characterized, from a thermodynamic point of view by curvilinear shapes of the binding potential functions α) RlnK_dual = (−ΔG_dual/RT) = {f(1/T)*g(T)} and β) RTlnK_dual = (−ΔG_dual) = {f(T)*g(lnT)}. The Class A processes present each function (convex), with a minimum whereas the Class B processes present each function (concave), with a maximum. The type of curvature depends on the type of reaction in water. In Class A, the reaction of water with phase transition (solvent → iceberg) is

A{ξ_wW_I(solvent) → ξ_wW_II(iceberg) + ξ_wW_III}

whereas in Class B, the reaction of water with opposite phase transition (iceberg → solvent) is

B{– ξ_wW_III– ξ_wW_II(iceberg)→ ξ_wW_I(solvent)}

The curvatures of the binding functions depend on the non-zero value of the hydrophobic heat capacity ΔC_p.hydr (ΔC_p.hydr ≠ 0). The hydrophobic heat capacity ΔC_p.hydr (ΔC_p.hydr = ±ξ_wC_p.w) is constant and independent from the temperature because it depends on the number ±ξ_w of water molecules W_III involved in each specific hydrophobic hydration process. Being ±ξ_w a pseudo-stoichiometric coefficient, it must remain necessarily constant if the reaction remains the same at different temperatures. The determination, therefore, of the curvatures of the binding functions α) RlnK_app = (−ΔG_dual/T) = {f(1/T)_*g(T)} and β) RTlnK_dual = (−ΔG_dual) = {f(T)_*g(lnT)}, respectively, represents a new reliable and efficient method, based on TED, for measuring the number ±ξ_w of water clusters W_I and hence of water icebergs W_II, and water molecules W_III.

The observed dual enthalpy ΔH_dual and the observed dual entropy ΔS_dual are composed each by two terms, thermal and motive: ΔH_dual = ΔH_mot + ΔH_th and ΔS_dual = ΔS_mot + ΔS_th, respectively. A typical property of ΔC_p.hydr is that it contributes exclusively, and is the only contribution, to the thermal components of the thermodynamic functions, ΔH_th and ΔS_th. These thermal functions concerning the solvent W_I only, represent the thermal entropy acquired (ΔH_th/T = +ξ_wC_p.w) in Class A by those water molecules W_I that, by a phase change, become water W_II with W_III. In Class B, the thermal functions represent thermal entropy lost (ΔH_th/T = −ξ_wC_p.w) by those water molecules W_II with W_III that go back to water W_I. The thermal components give a null contribution to free energy (−ΔG_th/T = 0), although they significantly affect the observed enthalpy and entropy values. The compensative properties follow from the thermal probability factor {T-PF}, referred to the solvent.

The motive functions ΔH_mot and ΔS_mot are obtained by subtracting the contributions of the thermal functions from the observed enthalpy, ΔH_dual and entropy, ΔS_dual, respectively. The motive functions, which derive from the motive probability factor {M-PF}, are independent from T but depend on the stoichiometry ±ξ_w of water clusters W_I (solvent) to water W_II (iceberg)). The motive functions of each compound in a homogeneous series, disaggregated by plotting them as the function of the respective number ξ_w, give self-consistent unitary values of enthalpy and entropy, in Class A

<Δh_for>_A = −22.7 ± 0.7; kJ·mol⁻¹ ·ξ_w⁻¹; <Δs_for>_A = −445 ± 3; J·K⁻¹·mol⁻¹·ξ_w⁻¹

and in Class B

<Δh_red>_B = +23.7 ± 0.6; kJ·mol⁻¹ ·ξ_w⁻¹; <Δs_red> _B = +432 ± 4; J·K⁻¹·mol⁻¹·ξ_w⁻¹

These unitary values present low variability, in the limits of experimental error, notwithstanding they were obtained from data concerning molecules of different size, in different aggregation states and measured by different experimental methods. This means that the about 600 experimental data from about 80 different compounds give origin to a normal population of experimental errors. The statistical inference confirms that ΔC_p.hydr is constant and that the unitary functions calculated are user-friendly to calculate the motive functions in every biochemical equilibrium.

The motive functions concern the partition function {M-PF} of the solute. The solute includes water molecules W_III, as free ligand, and clusters W_II, as iceberg sheaths joined to other solute units. The motive functions, combined in a Gibbs equation, give the free energy change ΔG_mot concerning the solute in every hydrophobic hydration process. Recognition of the peculiarities of thermal and motive functions is essential for a correct analysis of the thermodynamics of many biochemical equilibria. It is worth note the essential role played by the reaction steps of iceberg formation from water W_I (Class A) or iceberg reduction to water W_I (Class B) in regard of the motive configuration density entropy of the solute (and not of the solvent) in every hydrophobic hydration process. The motive configuration density entropy change of the solute for iceberg formation is negative in Class A by reducing the volume of the solvent water W_I and positive for iceberg reduction in Class B by expanding the volume of the solvent water W_I. In both Classes, the changes of iceberg have effect on the motive partition function {M-PF} of the solute.

The processes of iceberg formation or iceberg reduction are ubiquitous in bio-fluids. The knowledge of the user-friendly unitary functions reported above, coupled to Thermal Equivalent Dilution (TED) method for the determination of the number ξ_w, will be of fundamental help for anybody interested in the studies of the biochemical equilibria. By employing the user-friendly unitary functions, in the future, anybody can calculate the motive functions (enthalpy, entropy, and free energy) for iceberg reaction in any compound, if one has previously determined, by applying TED, the coefficient ξ_w for each compound. Even computer simulations can take advantage of the information provided by Ergodic Algorithmic Model (EAM) to check the reliability of the numbers obtained by statistical mechanics calculations. The quasi-chemical approximation [25,26] can be employed to feel the gap between NoremE and REME ensembles.

The validity of the iceberg model in the literature is controversial. Indeed, the results obtained from experiments and calculations carried out to prove the usual iceberg model are conflicting: the first time-resolved observations concluded that some water molecules are immobilized by hydrophobic groups [28], in strong contrast to previous NMR conclusions [29]. Molecular dynamics simulations of aqueous solutions of various hydrophobic solutes, for a wide range of concentrations, show that the rate of water reorientation in the vicinity of the hydrophobic solutes is decreased only moderately [30]. Our model, even if assumes the iceberg formation, has as a focal point the existence of implicit solvent and a completely different approach to the hydrophobic process. In fact, the positive density entropy (configuration) gain ΔS_red >> 0 for iceberg reduction to water W_I with consequent expansion of solvent volume, is the driving force that causes the formation of the hydrophobic bonds. This reappraisal of the hydrophobic bond represents a complete change of perspective with respect to the mechanism proposed in the literature. The role of water is completely reversed: reduction of iceberg W_II associated to W_III with condensation as W_I, with consequent density entropy gain by the more diluted solute. No more dissociation of W_II erroneously is considered as density entropy increase of the solvent.