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Copyright © 2005 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 89, Issue 4, 2258-2276, 1 October 2005

doi:10.1529/biophysj.104.057331

Biophysical Theory and Modeling

A Contribution to the Theory of Preferential Interaction Coefficients

J. Michael Schurr*, Go To Corresponding Author,  , David P. Rangel* and Sergio R. Aragon†

* Department of Chemistry, University of Washington, Seattle, Washington 98195-1700
† Department of Chemistry/Biochemistry, San Francisco State University, San Francisco, California 94132

Address reprint requests to J. Michael Schurr, Tel.: 206-543-6681; Fax: 206-685-8665.

Abstract

A simple and complete derivation of the relation between concentration-based preferential interaction coefficients and integrals over the relevant pair correlation functions is presented for the first time. Certain omissions from the original treatment of pair correlation functions in multicomponent thermodynamics are also addressed. Connections between these concentration-based quantities and the more common molality-based preferential interaction coefficients are also derived. The pair correlation functions and preferential interaction coefficients of both solvent (water) and cosolvent (osmolyte) in the neighborhood of a macromolecule contain contributions from short-range repulsions and generic long-range attractions originating from the macromolecule, as well as from osmolyte-solvent exchange reactions beyond the macromolecular surface. These contributions are evaluated via a heuristic analysis that leads to simple insightful expressions for the preferential interaction coefficients in terms of the volumes excluded to the centers of the water and osmolyte molecules and a sum over the contributions of exchanging sites in the surrounding solution. The preferential interaction coefficients are predicted to exhibit the experimentally observed dependence on osmolyte concentration. Molality-based preferential interaction coefficients that were reported for seven different osmolytes interacting with bovine serum albumin are analyzed using the this formulation together with geometrical parameters reckoned from the crystal structure of human serum albumin. In all cases, the excluded volume contribution, which is the volume excluded to osmolyte centers minus that excluded to water centers in units of exceeds in magnitude the contribution of the exchange reactions. Under the assumption that the exchange contribution is dominated by sites in the first surface-contiguous layer, the ratio of the average exchange constant to its neutral random value is determined for each osmolyte. These ratios all lie in the range 1.0±0.15, which indicates rather slight deviations from random occupation near the macromolecular surface. Finally, a mechanism is proposed whereby the chemical identity of an osmolyte might be concealed from partially ordered multilayers of water in clefts, grooves, and pits, and its consequences are noted.

Introduction

The effects of weakly interacting osmolytes on the conformational equilibria and ligand binding reactions of biological macromolecules have been studied intensively over the past two decades 1,2,3,4. A major objective in many cases was to ascertain the difference between the number of water molecules “associated” with the products of a particular reaction on one hand and the corresponding number “associated” with its reactants on the other. The precise meaning, or interpretation, of the numbers of “associated” waters and the differences therein remains a subject of discussion and some debate 5,6,7,8,9. This general approach to studying changes in “associated” waters has come to be known as the osmotic stress method. In the case of a solution, consisting of water (solvent, component 1), dilute macromolecules (components 2J, J=1, … M), and neutral osmolyte (cosolvent, component 3), the osmotic stress method yields the slope where K is the equilibrium constant for the reaction when written so as to take no account of either water or osmolyte, a1 is the activity of the water, and denotes the concentrations of each kind of macromolecule. This slope is extrapolated to the limit of infinite dilution, The difference in “associated” waters between products and reactants is sometimes taken to be the aforementioned slope,

(1)
where the index p or r denotes macromolecular products or reactants, respectively, and denote the respective standard state chemical potentials, and denote the respective stoichiometric coefficients of the reaction under consideration, and
(2)
and the symmetrically defined are concentration-based “preferential interaction coefficients”, which characterize the variation of that part of that does not depend upon with either respectively.

Alternative preferential interaction coefficients are defined in connection with equilibrium dialysis experiments and are usually molality based. The molalities of species 1, 2, and 3 are denoted by, respectively, and Two common molality-based preferential interaction coefficients are: and where the index J denoting the macromolecular conformation has been suppressed. Although relations between these and other molality-based preferential interaction coefficients have been intensively investigated, the connections between molality-based and concentration-based preferential interaction coefficients, like have received less attention. Clever and intuitive thermodynamic approaches indicate that for any given macromolecular species 2,

(3)
where N12 and N32 denote the total number of water and osmolyte molecules, respectively, in a domain of sufficient size surrounding a single isolated macromolecule, and c1 and c3 denote the respective bulk concentrations in an exterior domain, no part of which is near any macromolecule 1,2,3,4,5,6,7,8,9. can be regarded as the excess number of osmolyte molecules in the vicinity of the macromolecule above the quantity that would be expected from the number of water molecules in that region and the bulk concentration ratio,

Although the analysis below indicates that Eq. (3) is correct, the rigor of the thermodynamic approaches used to derive it is debatable. For example, the neglect of the osmotic pressure due to the macromolecule within its local domain is justifiable only for a domain of very great size, yet in many cases that domain was assumed to extend no more than one or two hydration layers beyond the macromolecule. The likely resolution of this paradoxical circumstance is noted briefly below.

Recently several articles appeared in which or the equivalent was expressed in terms of the so-called Kirkwood-Buff integrals 10, and where g12(r) and g32(r) are the pair correlation functions, which are described in greater detail below 11,12,13,14. The derivation of the main relation followed an unusually circuitous, piecewise, and technically demanding route that took place over three different articles and a book that collectively spanned 26 years 11,15,16,17. Chitra and Smith combined two relations that appeared earlier in Ben-Naim’s book 17, namely his Eq. 6.7.49 for and Eq. 6.17.16 for to obtain the final expression for The Eq. 6.7.49 was explicitly derived in Ben-Naim’s book, but the derivation of the much more difficult Eq. 6.17.16 was simply described as quite lengthy and omitted entirely. In fact, the first stage of that proof was presented in his 1975 article 15, and the second stage was presented in his 1988 article 16. Unfortunately, neither Chitra and Smith 11 nor Ben-Naim 17 referenced directly those earlier articles, from which the entire proof could be assembled. Chitra and Smith 11 demonstrated the approximate validity of their expression for by molecular dynamics simulations of both the pair correlation functions and the free energies of insertion of different small species 2 into aqueous solutions over a wide range of concentrations of various cosolvents. Shimizu 13 suggested a way to obtain the separate G12 and G32 from the measured and where is the partial molecular volume. He employed a relation between and G12 and G32 that was also first presented in Ben-Naim’s book 17 (Eq. 6.17.22), but the derivation, described as quite lengthy, was also omitted entirely. Again, a two-stage proof of the relevant relation can be found in the same two earlier articles 15,16. Shimizu 12 also extended his idea to determine the changes, ΔG32 and ΔG12, accompanying a reaction of species 2 from the measured and which was assumed to be the entire ΔV associated with the reaction. Shimizu and Smith 14 examined the differences between the effects of crowders, such as polyethylene glycol, and small osmolytes, such as glycerol, that stabilize native protein structures, on the separate G12 and G23. Schellman 18 undertook a related analysis in terms of the cross-second virial coefficients ().

The initial objective of this study is to provide a complete and much simpler derivation of the relevant expression for directly from the results of Kirkwood and Buff 10, as well as some important details that are missing from their original treatment of multicomponent thermodynamics. Such details include the choice of origin of the coordinate frame in a highly deformable macromolecule, its manifestation in the pair correlation functions, the invariance of the integrals of to that choice, a complete definition of the pair correlation function in the classical grand ensemble, and a derivation of the partial molecular volume. This derivation of follows a considerably more direct line than the Ben-Naim-Chitra-Smith development, and is technically much simpler. All of the results of Kirkwood and Buff that are needed to derive were rederived and found to be correct. In addition, a short proof of Ben-Naim’s expression for is provided in Appendix D .

Connections between this concentration-based and the molality-based and are derived via thermodynamic arguments that make use of certain expressions of Anderson et al. 19,20, which were also verified by rederivation.

The main objective of this study is to clarify the meaning(s) of the and and especially to relate them to more familiar quantities such as excluded volumes and equilibrium constants for osmolyte-solvent exchange in the region surrounding the macromolecule 21,22,23,24,25,26. Although this development is more heuristic than rigorous, useful predictions and significant insights emerge. As an example, the experimental data for seven different osmolytes interacting with bovine serum albumin (BSA) are analyzed using this formulation in conjunction with geometrical parameters reckoned from the crystal structure of human serum albumin (HSA). The separate excluded volume and exchange contributions are evaluated. Under the assumption that only the surface-contiguous layer of osmolyte sites is important, the ratio of the average exchange constant to its neutral random value is obtained in each case.

Finally, a mechanism is proposed whereby the chemical identity of the osmolyte may be concealed from partially ordered hydration multilayers in clefts, grooves, and pits, and its consequences are briefly noted.


A Derivation of Γ1(2)

Let us consider a system comprising ν different molecular species, α, β, … η, at constant T, V. In this case, when each species j undergoes a change of

(4)
where
(5)

Thus, the column vector containing the ν different dμk is related to the ν different dNj by the matrix relation dμ=M dN, where the elements of M are given by Eq. (5). Inversion of this matrix relation gives dN=M−1dμ, or

(6)
where
(7)

Kirkwood and Buff 10 established that the in Eq. (7) are directly related to integrals of the relevant pair correlation functions,

(8)
where is the αβ-pair correlation function, or radial distribution function, and r=|r1-r2| is the distance between the arbitrarily chosen central atom of an α-molecule at r1 and that of a β-molecule at r2, as indicated in Appendix A . A complete definition of gαβ(r12) in the grand ensemble 27 is given in Eq. (A1) in Appendix A . It must be emphasized that gαβ(r12) pertains to no atoms other than the arbitrarily chosen central atom of each molecule, and will in general depend upon that choice. Because the relations presented here derive ultimately from fluctuations in the numbers of molecules in a volume V that is large enough to contain on average a great many molecules of each kind, those relations must be independent of the choice of central atom. It may be concluded from Eq. (8) that integrals of the gαβ(r)-1 over the volume V, or at least from 0 out to a distance where gαβ(r) has converged to 1.0, are independent of the choice of central atom. The grand ensemble used to derive Eq. (8) can itself be derived by considering that the volume V is a tiny fraction of an enormously larger supersystem with a fixed number of molecules 27.

The pair correlation function has the following physical meaning. If the chosen central atom of a molecule of kind α is located at r1, then cβgαβ(r) is the probability per unit volume of finding the chosen central atom of a molecule of kind β at r2, such that r=|r1-r2|. A completely random disposition of β-molecules in the vicinity of α corresponds to gαβ(r)=1.0. In general, gαβ(r) is the factor by which the purely random probability per unit volume (i.e., cβ) must be multiplied to reckon the actual probability per unit volume of finding a β-molecule at distance r from an α-molecule. The pair correlation functions are by definition symmetric, so gαβ(r)=gβα (r), and also Bαβ=Bβα. We shall later regard cβgαβ(r) as the rotationally averaged mean density of centers of β-molecules at a distance r from the center of an α-molecule.

The matrix relation in Eq. (8) can be written as B=(kT/V)M−1, which can be inverted to give M=(kT/V)B−1, and

(9)
where |B|αβ is the cofactor of Bαβ (i.e., (−1)α+β times the determinant of the matrix obtained by striking out the αth row and the βth column) and |B| denotes the determinant of B10.

For the particular case of a three-component system held at constant T and V, the chemical potential μ2(T,c1,c2,c3) depends on all three concentrations, so

(10)
The constant T subscript is suppressed in Eqs. (10) below. When is held constant, then and it follows from Eq. (10) that
(11)

An equation analogous to Eq. (10) holds for dμ1, from which it follows that

(12)
The change in c1(T,P,c2,c3) at constant T,P,c2 is
(13)

It is shown in Appendix B that where denotes the partial molecular volume (m3/molecule). Then Eq. (13) yields

(14)

After substituting Eq. (14) into Eq. (12) and rearranging one finds

(15)

After substituting Eqs. (14) into Eq. (11), and Eq. (11) into Eq. (2), there results

(16)
Equation (9) was used to obtain the second line of Eq. (16) from the first, and the superscript ∞ on the |B|αβ indicates that they are to be evaluated in the limit The must be evaluated in the same limit.

The right-hand side of Eq. (16) is partially evaluated by leaving the factors in place, but expanding the in terms of elements of the three-component B-matrix, where

(17)

Every term in both the numerator and denominator of the right-hand side of Eq. (16) contains at least one factor of c2, which can be divided out. Any remaining terms that still contain a factor of c2 will vanish in the limit and are therefore omitted. After effecting some factorization and cancellation, the result can be expressed as

(18)

It remains to evaluate the factor in parentheses on the right-hand side of Eq. (18). An expression for was presented by Kirkwood and Buff 10 without explicit derivation. That derivation is sketched briefly in Appendix C and the result is given in Eq. (C6). Note that the denominator of Eq. (C6) is independent of α, and cancels out of the ratio, An important simplification is that applies in the limit which leaves just a two-component (2×2), rather than a three-component (3×3) B-matrix, so the indicated cofactors become just elements of the two-component B-matrix. In fact, Eq. (C6) gives the simple expressions, and where D is the denominator, which cancels out of After performing straightforward algebra, invoking the symmetry, Bαβ=Bβα, and omitting any canceling terms, the entire factor in parentheses reduces to c1/c3, and Eq. (18) becomes simply

(19)


Interpretation and discussion

From the definition of G12 in Eq. (17), it is clear that c1G12 is the excess number of 1-molecules in the vicinity of a 2-molecule beyond what would be expected from a random disposition of 1-molecules. An analogous meaning holds for c3G32. Although the c1G12 and c3G32 in Eq. (19) are explicitly excess numbers, rather than the total numbers of molecules in a domain surrounding the 2-molecule, Eq. (19) for can be written in a form that is completely analogous to Eq. (3) for as will be seen.

The pair correlation functions g12(r) and g32(r) must converge to the value 1.0 at large distances. Typically, for small osmolytes in a solution of dilute macromolecules, this occurs within, at most, a few nanometers beyond the maximum extension of the macromolecule (species 2). Thus, the upper limit of the integral in G12 or G32 can be reduced from ∞ to R, where R is any value sufficiently great that both g12(r) and g32(r) have converged to 1.0. Then Eq. (19) can be written as

(20)
where is the number of α-molecules within a sphere of radius R around the 2-molecule. The relevant criterion for the minimum size, Rmin, of the domain surrounding the macromolecule is clearly the convergence of the relevant pair correlation functions to 1.0 at all Because standard osmolytes are typically at least a few times larger than water, species 3 is typically excluded by the macromolecule from a larger volume than is species 1. Consequently, g32(r) cannot possibly converge to 1.0 within the volume defined by the centers of 1-molecules in the first hydration shell, and the minimum domain size generally must involve more water molecules than those in the first hydration shell in order for Eq. (20) to be valid.

Equation (20) is rigorously valid for a finite domain size of radius even though no account was taken of the osmotic pressure due to the macromolecule. This is likely a consequence of allowing the domain boundary to move with the macromolecule, so that it can never be contacted by the macromolecule and never experience its contribution to the osmotic pressure inside the macromolecular domain.

The preferential interaction coefficient can also be written in the simple form

(21)
which is most useful for our analysis. Corresponding expressions for can also be obtained simply by replacing the index 1 by 3 and vice versa in Eqs. (2), which is permitted by the evident symmetry of the theory in regard to 1↔3 interchange. It follows from Eqs. (19) that
(22)

The right-hand side of Eq. (22) is just which matches the right-hand side of Eq. (3). Furthermore, it is shown by thermodynamic arguments in Appendix E that in the limit,

(23)
where (c.f. Eqs. (E16) and (E19)). The relations in Eq. (23) were obtained by assuming that and remain constant, independent of and This should be a rather good approximation, when and which correspond to prevailing conditions in many studies. is obtained via vapor pressure osmometry, and is measured by equilibrium dialysis. At typically low osmolyte volume fractions is quite close to but is rather different 5,19,20. In any case, most experimental work has reported or or both. Equation (23) thus provides the principal connections between these theoretical expressions for (or ) in terms of pair correlation functions and the experimentally measured quantities.

We note that this cannot be simply expressed as because there is no Maxwell relation equating to Moreover, is also not equivalent to because direct evaluation of the latter in terms of and 9 yielded a result that is not equivalent to the right-hand side of Eq. (22).

Radial distribution functions of multicomponent systems have not yet been treated rigorously and analytically, and no suitable approximate formulation in terms of basic quantities, such as excluded volumes and exchange constants for specific sites, was presented previously. Heuristic approximate evaluations of various contributions to c1g12(r), c3g32(r), and are presented in the following section.


Heuristic evaluation of Γ1(2) and Γ3(2)

In general, both repulsive exclusion forces and attractive binding forces contribute simultaneously to and These contributions are evaluated approximately below. Comparisons with the models adopted by other workers will be discussed after this model is developed.

Repulsive exclusion forces

To simplify the discussion, let us first consider the effects of repulsive hard-core exclusion forces between the water (species 1) and the macromolecule (species 2). The superscript “ex” is used to indicate a contribution arising from such forces. A substantial void region, where is expected around r=0, as illustrated in Fig. 1. If both species 1 and 2 were perfectly spherical, then this void region would be followed at larger r by the region of the first coordination shell, where 11,17,28. This is true even in the case of hard spheres with no attractive interactions whatsoever. The first coordination shell would then be followed by a dip of g12(r) below 1.0, which in turn would be followed by a weaker second coordination shell, a second shallower dip, and so on, finally leveling off to g12(r)=1.0. In the case of a nonspherical macromolecule, the dips and peaks associated with the void volumes and coordination shells arising from different parts of the surface are superposed with a distribution of relative “phases”, so that likely exhibits simply a more or less smooth rise to a plateau at 1.0, as indicated in Fig. 1. Because typical neutral osmolytes (species 3) are larger than water, the void regions of would extend outward somewhat farther than in the case of as indicated also in Fig. 1. The volumes excluded to the centers of species 3 and 1 can be expressed as and respectively. The difference between the volumes accessible to the centers of species 1 and 3 within the macromolecular domain is defined by, which is also the difference between the volumes excluded to species 3 and 1.

Display large version of this figure
Figure 1
Schematic illustration of and versus the distance r between the central atoms of species 2 and either 1 or 3, respectively. The and are those parts of the pair correlation functions that arise solely from repulsive exclusion forces between species 2 and either 1 or 3, respectively.

The contribution to can be understood heuristically in terms of the osmotic pressure-volume work required to introduce a 2-molecule into the solution. The 2-molecule must effectively extrude the centers of the osmolytes (species 3) from a region occupied by the centers of the waters (species 1), which requires the input of work equal to where π is the osmotic pressure of species 1 in the bulk solution. This work appears as a term in which is the increase in solution free energy upon adding a 2-molecule to the solution. The variation of the osmotic pressure of species 1 with its activity is given by Thus, the osmotic work contribution to is when is sufficiently dilute that This simple analysis breaks down, when becomes comparable to

In the void regions, where and vanish, and are practically independent of either or The contribution of repulsive exclusion forces to is obtained from Eq. (21) as

(24)

Any variation of with c1 or c3 should be rather slight, due to the constancy of the void volumes, so should remain nearly constant, so long as doesn’t change much from the value, which will be the case, provided that Due to the generally larger void volume of g32(r) in comparison to g12(r), both and should be generally positive. In view of Eqs. (22), it is also expected that

(25)
where denotes the contribution of purely repulsive exclusion forces. Hence, is expected to be proportional to c3 and negative.


Generic long-range attractive forces

Let us now consider generic attractive forces, long-range van der Waals forces in particular, that may affect the densities of (centers of) species 1 and 3 in the region immediately beyond the void volume. Such mean densities are denoted by and where the superscript “ga” denotes generic attractions. For simplicity it will be assumed here that such generic attractions do not discriminate significantly between species 1 or 3, so that the ratio of their densities at any r beyond the void volume matches that of the bulk solution, that is which implies that even though both may differ significantly from 1.0. In that case, the net contributions to and reckoned from Eqs. (21), respectively, are Thus, generic, but nondiscriminating, attractions may alter the local densities of species 1 and 3, but make no net contribution to the preferential interaction coefficients. Nonvanishing contributions of attractive interactions presumably arise from discriminatory exchange reactions, as indicated in the following section.


Osmolyte-water exchange reactions

Schellman 21,22,23,24,25,26 introduced the notion that the relevant reactions in solution were exchange reactions at sites or regions near the surface of the macromolecule (species 2). The objective here is to incorporate such exchanges within this formulation of the preferential interaction coefficients in terms of integrals over particular pair correlation functions.

Let us consider first the jth site, which may contain either a single osmolyte (species 3) or water molecules (species 1). For osmolytes that do not bear charged groups, it is expected that but that assumption need not be invoked at this point. The exchange reaction for this site is written as

(26)
where denotes a complex with bound waters on average in the jth site and M·L denotes a complex with a single bound osmolyte at the jth site. It is not required that be an integer. When the macromolecule M (species 2) is sufficiently dilute, the equilibrium constant for Eq. (26) is
(27)
where a1=aw is the water activity for the mol fraction 1.0 standard state and is the osmolyte activity for its hypothetical Henry’s Law mol fraction 1.0 standard state, wherein each osmolyte experiences only the environment of its infinitely dilute solution (in water). The fraction of occupied (by osmolyte) j-sites is
(28)

The instantaneous density of the central atom of a 3-molecule in the jth site for any fixed configuration of the 2-molecule is a three-dimensional δ-function, δ(r-rj), where rj is the variable position of the central atom of the 3-molecule in the jth site in a coordinate frame originating on the central atom of the 2-molecule. When this density is averaged (with appropriate statistical weights) over the rj for all allowed positions and configurations of the 3-molecule in the site and over all configurations of the 2-molecule, and that result is in turn rotationally averaged about the chosen central atom of the 2-molecule, there results a distributed or smeared density function, which depends only on the distance r from that central atom and should be peaked near the average distance The preceding averages are taken only over those configurations, wherein rj lies within the somewhat arbitrarily defined boundaries of the jth exchange site for each configuration of species 2. This density function is still normalized, so

The density function for those 1-molecules that occupy the jth site, when the 3-molecule is absent, is defined in the following way. First the center of a 3-molecule with a fixed configuration is placed at rj in the jth site of a 2-molecule with a fixed configuration. The surrounding solution is assumed to consist entirely of 1-molecules. The density of all the η1 1-molecules in the solution, is then averaged over all positions and configurations of those same 1-molecules. The resulting mean density of 1-molecules will practically vanish over an excluded volume, that depends upon the particular rj and fixed configurations ξ and ζ of the 2- and 3-molecules, respectively. The quantities ξ and ζ should be regarded as generalized vectors, or lists, of the coordinates of all the atoms in the 2-molecule and 3-molecule, respectively. Now, the 3-molecule is removed, but the configuration of the 2-molecule is held fixed at ξ. The 1-molecules are allowed to equilibrate with the 2-molecule in that same configuration ξ. The mean density of those 1-molecules, whose centers lie within the particular excluded volume, is defined by where the sum runs only over the (variable) 1-molecules in each configuration, whose centers at lie within V(rj,ξ,ζ), and the average is taken over all configurations of 1-molecules. This mean density of 1-molecules in V(rj,ξ,ζ) is further averaged over the rj (within the jth site), ξ, and ζ by repeating this initial averaging process for various rj, ξ, and ζ, and then averaging the results over rj, ξ, and ζ. One obtains where the subscripts denote the final averages over rj, ξ, and ζ. By definition, the average value of for the jth site is When species 3 has no charged groups, so electrostriction effects are negligible, it is expected that the average number of 1-molecules that occupy an empty exchange site is Finally, rotational averaging of around the central atom of the 2-molecule yields which depends only upon the scalar distance r from the central atom of the 2-molecule. The normalization integral remains unchanged, so It is expected that the final smeared density, will normally be peaked near and exhibit a slightly greater width than because the centers of multiple 1-molecules are involved.

In light of the preceding remarks, the contribution of the jth site to the mean density of 1-molecules in the vicinity of the 2-molecule is

(29)
and to the mean density of 3-molecules is
(30)
The fraction of occupied sites, fj, from Eq. (28) appears in Eq. (30) and 1-fj appears in Eq. (29). The contributions of the exchange reaction at the jth site to the preferential interaction coefficients follow from Eqs. (21) and the respective normalizations of and :
(31)
(32)

The total contributions of exchange reactions at all such sites are and where the sums run over all sites (j), which lie beyond the macromolecular void volume.


A model for preferential interactions

Let us now consider a model system that exhibits simultaneously all of the aforementioned repulsive exclusion forces, generic attractions, and discriminatory interactions that are responsible for exchange. For simplicity, we shall assume that the contributions of the various interactions to the total mean densities, c1g12(r) and c3g32(r), are additive. This important assumption is not generally valid and merits some discussion. For any given fixed configuration of species 2, the repulsive hard-core exclusion forces between 2 and either 1 or 3 affect the densities of species 1 and 3 in one region of space, whereas attractions or repulsions of longer range act on 1 and 3 in a different region (outside the hard core, but still inside the macromolecular domain of radius R). Hence, the effects of the short-range and longer-range interactions are largely spatially complementary, and would be expected to be nearly additive, even after configurational and rotational averaging of species 2. Nondiscriminatory generic attractions make no net contribution to or and are not considered further here. In regard to exchange reactions, some interaction between exchanging sites is generally expected. The neglect of such interactions renders this discussion oversimplified in an important regard, whenever is not small compared to Nevertheless, useful insights may emerge, and quantitatively useful accuracy may be obtained whenever

Under this additivity assumption

(33)
(34)

Equations (24) give the in terms of c1, c3 and and Eqs. (31) express the in terms of νj, Kj, a3, and a1.

To examine the regime of small c3 in more detail, additional approximations are invoked. First, it is assumed that and are independent of c3 (which has units of molecules/m3) up to a molar concentration of 1.0. To lowest order in c3, that gives and where γ3 is the activity coefficient of species 3. With these approximations, and the exact relation, Eqs. (33) become,

(35)
(36)


Generalization of the exchange model

We imagine that a lattice of exchanging sites (or cells) with initial volume fills the entire osmolyte-accessible region of the macromolecular domain of radius R An osmolyte is regarded as bound to a particular site, when its central atom lies within that cell. The initial cell volume is taken as so the cell size matches the partial molecular volume of the osmolyte. Thus, if all of the initial sites were filled, species 3 would just fill the entire volume. The average number of 1-molecules that occupy a cell, when the osmolyte is absent, is assumed to be which is exact far from the macromolecular surface, and is almost certainly a fairly good approximation even near the macromolecular surface, except when electrostriction effects are large. Thus, the species 1 would just fill the lattice volume in the absence of species 3. While holding the overall lattice volume constant, one could now choose a smaller uniform cell size for the lattice of exchange sites, namely where is an integer, provided that the contributions of each site to are reduced by the same factor, and that the j-sums in Eqs. (35) are extended from the original L sites of volume to the mL smaller sites of volume For some sufficiently large value of m, when should become entirely independent of m or A lattice cell size in that range is adopted here. The center of the jth cell is taken at position q(j), and its exchange constant, Kq(j), may vary from one cell to the next in a limited way, so as to create a gradient of the Kq(j) along any reasonably smooth path in the discrete q(j) space. Both and the exchange constant, Kq(j), for each smaller cell of volume, are taken as the values typical of a site with the initial volume, whose center lies within that smaller cell, with the understanding that m-1 adjacent sites are closed, whenever an osmolyte binds to the smaller cell in question. In this way, any region of volume will bind one and only one osmolyte in approximately the same way as a function of a3 or a1, regardless of the number of cells into which is it subdivided, and the maximum densities of species 1 and 3 will remain unchanged. The smaller lattice cell volumes are employed simply to represent the spatial variation of the exchange constants at higher resolution than is afforded by cells of volume By suitable adjustment of the Kj associated with the various sites in the lattice, it is possible to create any conceivable mean densities c1g12(r) and c3g32(r) at a level of resolution set by the lattice cell size, subject to the implicit volume conservation rule invoked here (i.e., ). The approximate validity of this model is limited to the regime of small volume fraction of species 3, so that events in any one region of volume do not affect events in neighboring regions of the same size. The large anticooperativity associated with the closure of m-1 binding sites surrounding a given site, when it becomes occupied, generally has a very strong influence on the system, except when the volume fraction of species 3 is small. In that special case, for a cell volume Eqs. (35) become

(37)
(38)


Neutral binding

When the standard free energy change for an exchange reaction vanishes, Kj=1.0. First, let us consider the limit of small c3, where, γ3→1.0 and →1.0, so the numerator of each term in the j-sums of Eqs. (37) becomes For typical small neutral osmolytes, excluding molecules the size of trehalose and sucrose, one expects that Note that, if as was assumed in early treatments of exchange by Schellman 21,22,23,24,25,26, then and the entire exchange contribution of the jth site would vanish. Although the condition, Kj=1.0, is the point of neutrality in terms of vanishing standard state free energy change, it is not generally the point of neutrality in regard to purely random binding in the neighborhood of a 2-molecule, because ν 1-molecules are released for every 3-molecule bound. The point of neutrality in regard to random binding of 1- and 3-molecules at the jth site, when γ3=1.0 and is clearly Kj=ν.

In general, sites that lie out in the bulk solution sufficiently far from the surface of the 2-molecule can make no net contribution to or so for such sites it is absolutely required that which can be taken as the general condition for neutrality of any site in regard to random binding of 1’s and 3’s. Smaller values of Kj yield a positive contribution of the jth site to

We consider next the limit, wherein so the second terms in the denominators of Eqs. (37) can be ignored. The product, is unitless, and has the same value in any units, so one can take c3 in mol/L and in L/mol. For small neutral osmolytes, one typically has and up to c3=1.0M. Thus, for the inequality, will be satisfied, when Hence, Kj could be as large as 10, and still satisfy this inequality for c3=1.0M. In other words, K could be up to 2–3 times greater than the neutral random binding value, and still the second terms in the denominators of Eqs.