Originally published as Biophys J. BioFAST on July 29, 2005.
doi:10.1529/biophysj.104.057331
Biophysical Journal 89:2258-2276 (2005)
© 2005 The Biophysical Society
A Contribution to the Theory of Preferential Interaction Coefficients
J. Michael Schurr *,
David P. Rangel * and
Sergio R. Aragon 
* Department of Chemistry, University of Washington, Seattle, Washington 98195-1700; and Department of Chemistry/Biochemistry, San Francisco State University, San Francisco, California 94132
Correspondence: Address reprint requests to J. Michael Schurr, Tel.: 206-543-6681; Fax: 206-685-8665; E-mail: schurr{at}chem.washington.edu.
 |
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
–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 (5–9). 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 (1–9). 
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 (11–14). 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 (11,15–17). 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, 
G32 and G12, accompanying a reaction of species 2 from the measured 
and 
which was assumed to be the entire V 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 (21–26). 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  1(2)
|
|---|
Let us consider a system comprising 
different molecular species, 
, ß, ... 
, at constant T, V. In this case, when each species j undergoes a change of 
 | (4) |
where
 | (5) |
Thus, the column vector containing the different dµk is related to the 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 ß-pair correlation function, or radial distribution function, and r = |r1 – r2| is the distance between the arbitrarily chosen central atom of an -molecule at r1 and that of a ß-molecule at r2, as indicated in Appendix A. A complete definition of gß(r12) in the grand ensemble (27) is given in Eq. A1 in Appendix A. It must be emphasized that gß(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 gß(r) – 1 over the volume V, or at least from 0 out to a distance where gß(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 is located at r1, then cßgß(r) is the probability per unit volume of finding the chosen central atom of a molecule of kind ß at r2, such that r = |r1 – r2|. A completely random disposition of ß-molecules in the vicinity of corresponds to gß(r) = 1.0. In general, gß(r) is the factor by which the purely random probability per unit volume (i.e., cß) must be multiplied to reckon the actual probability per unit volume of finding a ß-molecule at distance r from an -molecule. The pair correlation functions are by definition symmetric, so gß(r) = gß (r), and also Bß = Bß. We shall later regard cßgß(r) as the rotationally averaged mean density of centers of ß-molecules at a distance r from the center of an -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|ß is the cofactor of Bß (i.e., (–1)+ß times the determinant of the matrix obtained by striking out the th row and the ßth column) and |B| denotes the determinant of B (10).
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. 10–16 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 and 15 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|ß 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 , and cancels out of the ratio, 
An important simplification is that applies in the limit which leaves just a two-component (2 x 2), rather than a three-component (3 x 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, Bß = Bß, 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 -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 and 3 and 19–21, which is permitted by the evident symmetry of the theory in regard to 1 
3 interchange. It follows from Eqs. 19–21 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 1(2) AND 3(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.
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 
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 and 24, 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 and 22, 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 (21–26) 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 
-function, (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 1 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 
and 
of the 2- and 3-molecules, respectively. The quantities and 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 . The 1-molecules are allowed to equilibrate with the 2-molecule in that same configuration . 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,,), and the average is taken over all configurations of 1-molecules. This mean density of 1-molecules in V(rj,,) is further averaged over the rj (within the jth site), , and by repeating this initial averaging process for various rj, , and , and then averaging the results over rj, , and . One obtains 
where the subscripts denote the final averages over rj, , and . 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 22 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 and 25 give the 
in terms of c1, c3 and 
and Eqs. 31 and 32 express the 
in terms of j, 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 
3 is the activity coefficient of species 3. With these approximations, and the exact relation, 
Eqs. 33 and 34 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 and 36 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 and 36 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, 3 
1.0 and 
1.0, so the numerator of each term in the j-sums of Eqs. 37 and 38 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 (21–26), 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 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 3 = 1.0 and 
is clearly Kj = .
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 and 38 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 = 1.0 M. Thus, for 
the inequality, 
will be satisfied, when 
Hence, Kj could be as large as 10, and still satisfy this inequality for c3 = 1.0 M. 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. 37 and 38 would be negligibly small for all c3 up to 1.0 M. In this limit, Eqs. 37 and 38 can be written as
 | (39) |
 | (40) |
where 
is the difference between the volumes accessible to 1 and 3 in units of 
As noted above, 
is generally positive, because the osmolyte generally exceeds the water in size, so X should also be generally positive. The j-sum (of binding terms) can in principle take either sign, depending upon the magnitude of Kj.
Variation of 1(2) and 3(2) with c3
Equations 39 and 40 predict that should be nearly constant independent of c3, and that should vary nearly in proportion to 
with constant slope, up to c3 = 1.0 M. In fact, for seven different osmolytes interacting with BSA, it was found that 
hence also varied in proportion to m3 with a constant negative slope up to m3 = 1.0 molal (5,29,30; J. G. Cannon, personal communication, 2005). The negative slope implies that the total j-sum is either positive or not so negative that it overwhelms the positive value of X. The constant slope indicates that the second terms in the denominators of Eqs. 37 and 38 are negligible up to m3 = 1.0 m, which in turn implies that (in the case of BSA) most of the contribution from the j-sum must arise from sites with Kj-values that do not exceed by more than approximately threefold the random binding value, 
Because X derives from a shell volume with a thickness equal to the difference in radius between the osmolyte and water, it is expected to vary nearly in proportion to the area in the case of macromolecules with homologous surfaces. Likewise, the j-sum concerns primarily just the contact layer and a few additional layers of osmolyte or water, so that it too is expected to vary nearly in proportion to the area in the case of macromolecules with homologous surfaces. Courtenay et al. (6) noted that numerous globular proteins exhibit similar values of the ratio, 
where 
is the water accessible area.
Analysis of 1(2)-values for BSA
Experimental values of for different osmolytes interacting with BSA are obtained from the corresponding determined by the Record group (5,29,30; and J. G. Cannon, personal communication, 2005) via the relations Eqs. 22, 23, and 40, which are combined to give
 | (41) |
where 
and 
By combining the measured value of with an estimate of X obtained from the protein structure, it is possible to obtain an experimental estimate of S.
Although no crystal structure has been reported for BSA, it is assumed to be satisfactorily modeled by human serum albumin. BSA has 607 amino acid residues and HSA has 609, which are 76% homologous with the BSA sequence. Only 578 of the 609 residues of HSA are resolved in the crystal structure (31). Here the molecular volume reckoned for the crystal structure is simply scaled by 609/578 = 1.054 to estimate the corresponding volume for the full HSA (or BSA). However, the reckoned for the crystal structure corresponds to the volume of a relatively thin shell of a given thickness about the macromolecule, so it is scaled by the factor 
The osmolyte and water accessible areas are also scaled by 1.035 to estimate the corresponding areas for HSA (or BSA). The various volumes and areas are reckoned using the program MSROLL (32). The crystal structure contains an HSA dimer and seven water molecules. The nonhydrogen atoms of both the water molecules and the second dimer and their coordinates are deleted from the list of atomic coordinates, leaving just the atoms and coordinates of the 578 resolved residues of the first monomer. The program assigns a van der Waals radius to each atom or group of HSA. An effective radius, 
is assigned to water (i = 1) and to each osmolyte (i = 3).
The molecular displacement volume (
) of HSA in water is determined by rolling a water-size sphere of radius 
Å around its exterior van der Waals surface in each of a series of closely spaced parallel planes. The program reckons the volume inside the continuous surface formed by the contact surface(s) of the sphere with the van der Waals surface of the protein plus the so-called reentrant surface(s) that bridge the gaps in the contact surface by following the interior surface of the bridging sphere. We obtain 
for HSA, which is then scaled by 1.054 to estimate 
for BSA. This molecular displacement volume cannot be occupied by any part of the 1.48 Å sphere, and for numerous globular proteins is found to lie within 1–2% of the partial molecular volume (S. Aragon, unpublished data). Courtenay et al. (5) report 
for BSA, which differs by 
1% from the calculated above. This agreement provides an important check on the structure and computational protocols used, but does not pertain directly to the preferential interaction coefficients.
Next we obtain the volumes excluded by HSA to the centers of water-size or osmolyte-size spheres. This is the volume inside a surface that is traced out by the center of the osmolyte sphere or water sphere, as it rolls over the surface of the protein, and represents the void volume in or 
Although MSROLL does not calculate the volume inside the excluded-center surface directly, that volume can be reckoned by first inflating the atomic van der Waals radii by 
or 
and using a probe sphere of zero radius. The resulting contact plus (vanishing) reentrant surface is the same surface traced out by the center of a sphere of radius or 
as it rolls over the uninflated van der Waals surface, and its interior volume is calculated by the program. The difference between the volume excluded to an osmolyte and that excluded to a water center is just for that osmolyte, as illustrated in Fig. 2. After scaling by 1.035, it is divided by the molecular volume of water, 
to obtain 
Then the exchange contribution, 
is finally evaluated. The results for X and S are presented for seven different osmolytes interacting with BSA in Table 1. In every case, X exceeds the magnitude of S. Thus, in the case of BSA, the largest contribution to is simply a geometrical consequence of the fact that the osmolytes are substantially larger than water and therefore have a larger effective sphere radius. Four of the osmolytes, urea, glycerol, proline, and trehalose, exhibit a negative value of S, which indicates that 
is on average greater than its neutral value, 
and implies a greater than random preference of the osmolyte for exchanging sites within the macromolecular domain. The remaining three osmolytes, trimethylamine N-oxide (TMAO), K+glutamate, and betaine glycine, exhibit a positive value of S, which indicates that is on average less than its neutral value, and implies a lower than random preference of these osmolytes for the exchanging sites.
View larger version (73K):
[in this window]
[in a new window]
|
FIGURE 2 Schematic illustration (not to scale) of the difference between the volume excluded to osmolyte (large sphere) and to water (small sphere) by human serum albumin. The desired volume is that of the shaded shell between the surfaces traced out by the center of an osmolyte-size sphere and that traced out by a water-size sphere, as those spheres are rolled over the surface of the protein. The sizes of the water and osmolyte relative to that of BSA are exaggerated for illustrative purposes.
|
|
In previous work in this field, it was commonly assumed that (or ) is determined primarily by exchange sites within the first surface-contact layer. We now also assume that the exchanging sites are confined to the first surface-contact layer. We take the number (
) of surface-contiguous sites for species 3 to be the accessible area traced out by the center of a sphere of radius rolled over the van der Waals surface of HSA, scaled by the factor 1.035, and divided by the area, 
of a single site of volume Values of for the different osmolytes are also included in Table 1. For these sites, the cell volume is 
so m = 1.0, and mL = L = 
In this case, 
where 
is the average exchange constant for the surface-contiguous sites of species 3. From the values of S, 
and we estimate 
which is the ratio of the average exchange constant to its neutral (or random) value. These values are listed in the final column of Table 1. These 
ratios are all remarkably close to 1.0, with a maximum deviation of <0.15. By this criterion the average interactions of these osmolytes with the BSA surface are all surprisingly similar.
Comparisons with prior work
An advantage of this formulation for (or ) in terms of the pair correlation functions (Eqs. 21 and 22) in comparison to the thermodynamic formulation in terms of numbers of molecules in the macromolecular domain (Eq. 3) is that the excluded volume contribution is unambiguously given by 
which is the volume of the shell in Fig. 2.
Shimizu (13) and Shimizu and Smith (14) employed a single "excluded" volume, 
that is independent of the water or osmolyte, and is essentially the macromolecular displacement volume, 
that is excluded to any part of a water or osmolyte molecule. Those authors approximated the excluded volume contribution to the numbers of water and osmolyte molecules in the macromolecular domain by 
and 
respectively. Their use of the same value, for both 
and
TOP
ABSTRACT
INTRODUCTION