Salt Dependence of Oligoion-Polyion Binding - American Chemical

The generally large power dependence of Kob on the concentration of salt, C3, is quantified .... oligoion-polyion binding equilibria, and preferential...
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J . Phys. Chem. 1993,97, 71 16-7126

7116

Salt Dependence of Oligoion-Polyion Binding: A Thermodynamic Description Based on Preferential Interaction Coefficients Charles F. Anderson..? and M. Thomas Record, Jr.'J** Departments of Chemistry and Biochemistry, University of Wisconsin-Madison,

Madison, Wisconsin 53706

Received: March 3, 1993

The binding of a ligand (such as an oligolysine) to a polyion (such as DNA) in aqueous solution is characterized by KO, CB/C&L, the quotient of molar concentrations of the participants in the complexation equilibrium, LZL FZFF! B Z ~ .The generally large power dependence of Kob on the concentration of salt, C3, is quantified experimentally by evaluating SK,, (d In K o ~ / Ind C3)~,pat constant temperature and pressure. Introducing approximations that can be justified under typical experimental conditions (C3 >> CB,CF,CL),we show that SK,, = A(lZJI - 2T2~3),a stoichiometrically weighted combination of terms pertaining to the participants in the binding reaction (J = F, B, or L). Here T2j3 = (a In U Z J / ~In u3)~,p,,,,~, a3 is the activity of uniunivalent salt, and (I2J (mu) is the activity (molality) of an electroneutral component, 25, that consists of an ionic species J, bearing Z J charges, and an equivalent number [ Z Jof~ oppositely charged univalent ions. We establish the conditions under which -T2~3 = = (dc3/dc2J)T,a3,the form of the preferential interaction coefficient that can be evaluated from a series of grand canonical Monte Carlo (GCMC) simulations at constant T and u3. We also demonstrate the conditions under which -T2j3 = I ' 3 2 ~ (dc3/dczJ)T,r,,r3, the preferential interaction coefficient that can be determined by membrane dialysis. For systems that exhibit a linear dependence of In Kok on In C3 over a wide range of C3, any salt dependences of the individual T2j3 must effectively cancel so that A(~ZJI- 2T2~3)is constant. Various salt-dependent corrections to this expression must be considered when C3 approaches the molar level. The rigorous thermodynamic relationship between SK,, and the coefficients I'zF will enable the use of G C M C simulations to investigate the molecular determinants of the effect of salt on oligoion-polyion binding and on other equilibria involving polyions, including aggregation and conformational transitions.

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I'Ey

origins can be traced ultimately to changes in the long-range Coulombic interactions of mobile salt ions with structural charges Equilibrium and kinetic parameters characterizing the nonon a cylindrical polyion that has a sufficiently high axial charge covalent binding interactions of nucleic acids with positively density."JJs-'7 When this charge density is reduced by ligand charged ligands typically vary as a power of the (uniunivalent) binding (or denaturation), the concomitant reduction in longsalt concentration.1" These power-law salt dependences have range interactions of small ions with a polyion generally causes been observed in studies of the binding interactionsof intercalating, the extent of ligand binding (or denaturation) to exhibit a strong groove-binding,and phosphate-binding ligands with nucleic acids. dependence on C3, the concentration of excess uniunivalent salt For these systems under typical experimental conditions,changes in the s o l u t i ~ n . ~ J ~ ~ J ~ J ~ in salt concentration affect the extent and kinetics of complexThe fundamental polyelectrolyte basis of the dependences on ation/dissociation much more profoundly than do comparable C3 of equilibrium' and kinetic) parameters that characterize the changes (on a percentage basis) in reactant concentrations or binding of oligocations and proteins to nucleic acids was changes in other physical variables (T,P, pH). These effects of established by Record, Lohman, and deHaseth. By analyzing salt concentration on equilibria and rate processes involving nucleic data on the binding of oligolysines to double-stranded polynuacids are typically much larger than those observed for processes cleotides,*O Record et a1.l demonstrated the large power dependinvolving only low molecular weight solutes or polyampholytic ences on C, of the equilibrium binding quotients: Kob [C#K.b., proteins. Moreover, salt effects on processes involving a cylindrical where SKo, is an intrinsically negative quantity whose magnitude polyelectrolyte of high axial charge density exhibit a distinctive exceeds unity. Values of this exponent were interpreted in terms functional form that is characteristic of a cation-exchange process, of thermodynamic stoichiometries of ion exchange, which for rather than a Hofmeister salt effect or a general ionic strength dependence attributable to DebyeHuckel ~ c r e e n i n g . l * ~ - ~ . ~ - I ~ these systems is dominated by cation exchange from the nucleic acid.' Upon binding, the cationic ligand in effect displaces a The thermodynamic consequences of binding a cationic ligand definite number of thermodynamically bound salt cations from to a highly charged polyanion (which reduces the exposure of the nucleic acid. On the basis of Manning's original (limitingpolyion charges to the aqueous salt solution) are so distinctive as law) version of condensation theory,I4 Record et a1.l showed that to warrant the designation "polyelectrolyte effect", by analogy the extent of thermodynamic binding of salt cations to a highly with the "hydrophobic effect" that accompanies the binding of charged cylindrical polyanion is (1-(2[)-l) per monomer charge a nonpolar ligand to a nonpolar macromolecule (which reduces and hence that the exposure of nonpolar surface to solvent ~ a t e r ) . ~ . ~ JSince ~J* the polyelectrolyte effect is observed in systems where the salt SK,, = -ZL(l - (2t)-') (1) ions do not bind to sites on the ligand or the nucleic acid, its Here, Z, is the number of charges on the oligocationic ligand, * Correspondence should be.addressed to either author at the Department and the parameter [ is determined by the mean axial charge of Chemistry, University of Wisconsin, 1101 University Ave., Madison, WI density of the polyanion, the valence of the salt cation, and the 53706. product of temperature and the dielectric constant of the solvent. Department of Chemistry. This expression is in excellent agreement with the oligolysine t Department of Biochemistry. 1. Introduction

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0022-365419312097-71 16$04.00/0

0 1993 American Chemical Society

Salt Dependence of Oligoion-Polyion Binding polynucleotide binding data20 and with more recent data for the binding interactions of a wide variety of oligocations including oligopeptides,2*-22polyamines,23and inert inorganic cations24 with double- or single-stranded nucleic acids over a wide range of C3. A comprehensive description of the effects of salt concentration on conformational and ligand (protein) binding equilibria involving nucleic acids has been presented.2 In this exposition, SKob was analyzed as a sum of contributions from the polyelectrolyte effect (expressed analytically as in eq l), from the competitive site binding of salt ions to the ligand (protein), and from changes in hydration of the ligand and/or nucleic acid. In subsequent papers,15J6 SKob was expressed (approximately) as -ZL( 1 + 2r3,),where the coefficient I'3u reflects "preferential interactions" between the (uncomplexed) DNA polyelectrolyte ("u" on a nucleotide basis) and electrolyte ("3") components. Estimates of I'su for DNA have been obtained by alternative theoretical approaches, including the Poisson-Boltzmann (PB) cell modell6 and grand canonical Monte Carlo (GCMC) simulations.23-27 The primary objective of this paper is to derive a generally applicable thermodynamic relationship between SKob,, the common experimental measure of the effect of salt concentration on oligoion-polyion binding equilibria, and preferential interaction coefficients (I';') in the particular form that can be evaluated by GCMC simulations. Although the development presented here is in most respects purely thermodynamic, and hence not predicated on any particular model, the course of the derivation is determined in important ways by the anticipation that GCMC simulations will be the theoretical means whereby the preferential interaction coefficients are evaluated. The approximations introduced can be justified, in general, by consideration of prevalent experimental conditions ("excess salt") or of certain properties that can be established empirically (such as the insensitivity of partial molar volumes to composition changes). The theoretical expression relating SKob, to the appropriate I'K' is presented in its simplest and most frequently applicable form, but sufficient details are given to permit the introduction of correction terms that could be required at elevated salt concentrations. We also show all approximations required to establish an analogous relationship between SKoband preferential interaction coefficients in the form that can be evaluated experimentally (for example) by membrane dialysis. Section 1I.A reviews the type of experimental information that is available to quantify the dependence on salt concentration of oligoion-polyion binding equilibria. The calculation of preferential interaction coefficients (I';') from GCMC simulations is explained in section 1I.B. In section I K A , we introduce the notation used here to describe the thermodynamic properties of interest. The derivation presented in section 1II.B shows how SKob can be related to derivatives of the type T23, each of which reflects the dependence on salt activity of the activity of a participant in the oligoion-polyion binding equilibrium. In section 1II.C we derive the relationship between T23 and,':'l which establishes the role that can be taken by GCMC simulations in a quantitative analysis of the polyelectrolyte effect on the binding of a charged ligand to a polyion. In section IV.A, we review the basic justifications for, and the cumulative implications of, the assumptions invoked at various stages in our derivation. Comparisons with previous work and future applications areconsidered in 1V.B. An Appendix summarizes the derivation of relationships between "23 and Donnan coefficients that can (in favorable cases) be measured experimentally by membrane dialysis. 11. Experimental and Theoretical Background

A. Experimental Characterization of Ligand-Nucleic Acid Binding. Nonspecific binding, due primarily to Coulombic attraction, localizes an oligocationic ligand on or close to the polyanion surface, but not at any particular chemically distinct site. Studies of nonspecific ligand binding to nucleic acids,

The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7117 typically at concentrations in the range pM-mM, have employed equilibrium dialysis and various spectroscopic methods.28~29In noncooperative "specific" binding, the ligand is localized at a definite (isolated) site on the polyion surface, as a result of various short-range attractive interactions whose contributions to the binding free energy greatly predominate over the contribution of Coulombic interactions. The specific binding of proteins having a locally cationic binding surface (though their overall charge may be negative) to nucleic acids, at concentrations typically in the nM-pM range, has been studied in a few favorable cases by fluorescence and more often by separation procedures (e.g., filter binding, gel electrophoresis with radioisotopic detection of bound and/or free DNA).30 Both specific and nonspecific binding equilibria exhibit a dependence on the concentration of salt that can be analyzed with the theoretical expressions derived in this paper. For specific binding, and for nonspecific binding under conditions where the spatial separation between nearest bound ligands is large enough, each binding interaction is independent of all the others. In many systems that have been investigated, the complexation reaction can be represented as the binding of a single ligand, L z ~ with , a single segment of nucleic acid, FZF, to form one thermodynamically distinct product, B ~ Baccording , to the stoichiometry

In eq 2, Fz~representsa consecutive sequence of lZF1 univalent charged monomers on a polymer, such as phosphate groups on a nucleic acid, and BZe represents a segment of the same total length, where a centrally situated sequence of ZLconsecutive charges has been eliminated by the binding of a ligand (according to the simplest possible model for the ligand-polyion complex). Formally, 12~1 could be chosen equal to 12~1so , that Z B = ZF+ ZL= 0. However, the Monte Carlo method of evaluating the preferential interaction coefficient for the complex B requires consideration not only of the polyion charges actually "occupied" by the ligand but of all charges on the polyion where the effective field (potential of mean force) is influenced by the presence of the ligand.31 There are 1zB(/2such charges extended along the polyion axis in both directions away from the bound ligand. The number IzB1/2 always can be integral because of the flexibility in the specification of 12~1,which only must exceed the lower limit dictated by the range of influence of the bound ligand on the field experienced by (unbound) polyion charges. The effective range of this influence does not appear to be a strong function of thesalt concentration, under conditions that have been examined thus far.3l The equilibrium distribution of the participants in the complexation reaction 2 conventionally is quantified as an "observed equilibrium quotient":

(3) Here CJ is the molar concentration (mol/dm3) of J = B, F, or L. In practice, Kobis evaluated by nonlinear least-squares fitting of data obtained from a titration of DNA with ligand (protein) at constant T,P, and a fixed concentration, C3, of excess salt. The functional form of the isotherm used to fit the titration curve depends on the type of binding. For specific binding, eq 3 is directly applicable. For nonspecific binding, the McGhee-von H i ~ p e isotherm 1 ~ ~ (or a similar functional form) is fitted to the experimental titration curve in order to correct for the reduced availability of potential binding sites as the number of bound ligands increases.' Since Kob is not a quotient of thermodynamic activities, it depends in general on the concentrations of all components in the system (as well as on T and P ) . In most experimental determinations of Kob, C3is at least an order of magnitude larger than the concentration of any participant, J, in the binding reaction. (Usually the concentrations CJare in the submillimolar range, and C3 1 10 mM.) Under these typical invitroconditions,

7118 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 changing the total amount of ligand while C3 is fixed causes a shift in the equilibrium distribution of reactants and products but produces no detectable change in the magnitude of K,b. Thus, the dependence of Kob on the composition of the solution can be attributed entirely to C3. Changes in C3 generally produce profound changes in K0b,even when salt is not a direct participant in the binding equilibrium (when neither the cation (u+) nor the anion (u-) of the salt is site bound to any of the ionic species J). When equilibrium dialysis measurements20923 are used to quantify the complexation of ligands with a nucleic acid, the pressure on the solution is not fixed, and in some cases K0b may depend upon composition variables other than C3. These conditions will not be explicitly considered in the present paper. Experimental information about the dependence of Kob on C, generally is represented by the derivative: SK,,

= (a In Koala In C3)EQ

(4)

Here and elsewhere in this paper, the subscript EQ on a derivative implies that it is evaluated under the conditions of material equilibrium with respect to the complexation reaction 2, at constant T and P. Equation 4 implies that SKobcan be evaluated as the slope of a line through (or tangent to) data on a plot of In Kob vs In C3. With few exceptions, such plots are linear for systems containing only one type of ligand, complex and excess salt. More precisely, deviations from linearity cannot be discerned outside the scatter of the data; the typical uncertainty in experimental determinations of SK,b is f 10-20%. Possible deviations from linearity in plots of In Kob vs In C3 have been noted at C3 > 0.5 M in a study of the effects of uniunivalent salts on lac-repressor lac operator binding.33 The theoretical development presented here does not presuppose the constancy of SKob. Experimental studies on the binding of oligocations to nucleic acids haveestablished that themagnitudeof S K h , an intrinsically negative quantity, is correlated with the mean axial charge density onthenucleicacidand withthevalenceontheligand. Specifically, S K e b is directly proportional to ZL,with a constant of proportionality equal to -0.9 for double-stranded nucleic acids1.23 or -0.74 for single-stranded nucleicacids.21922 Theseobservations are in quantitative agreement with eq 1. No general theory is capable of predicting the magnitude of Kob for a specific binding interaction of LZL with FZF, Even for systems where nonspecific binding may be attributed entirely to Coulombic interactions, various practical considerationspreclude the use of GCMC simulations to calculate the activity coefficients required for evaluation of Kob (as explained in the following subsection). Although values of Kobcannot be calculated a priori, the dependence of Kobon C3 is determined primarily by Coulombic interactions, and therefore SKob can be predicted by a variety of theoretical approaches. The following subsection reviews the evaluation of preferential interaction coefficientsfrom the output of GCMC simulations. Then, in section I11 we show how these coefficients are related to the measurable quantity, SKob,. B. Use of Monte Carlo Simulations To Characterize Nonideality in a Three-Component System. Unlike all of the standard analytic approaches to the calculation of thermodynamic quantities, Monte Carlo (MC) simulations are free of the various approximations that are introduced, often implicitly, into the evaluation of statistical mechanical integrals in order to avoid accounting for interactions among intractable numbers of particles.34 Consequently, the predictive capability of MC simulations should in principle be limited by only two interrelated considerations: (1) the amount of computer time available to achieve acceptably small uncertainties in the computed quantities of interest and (2) the amount, and accuracy, of molecular detail built into the model assumptions. In recent years, MC simulations based on a particular type of grand canonical ensemble have been used as a rigorous theoretical route to the quantitative prediction of thermodynamic coefficients that characterize the nonideality of charged solutes in aqueous s0lution.~5~2 During such “grand

Anderson and Record canonical” Monte Carlo (GCMC) simulations, the model system is equilibrated through a sequence of changes in the configuration of the model system. (In the present context, “configuration” refers exclusively to the spatial coordinates of the salt ions.) A detailed specification of the types of configurational changes and their associated transition probabilities has been given by Torrie and Valleau.4 Any thermodynamic coefficient determined by GCMC simulations is a function of the parameters needed to specify the model system (such as particle dimensions, dielectric constants, etc.) and of the thermodynamic variables that are characteristic of the grand canonical ensemble: temperature, volume, and the activity of the electrolyte component (as). Representing electrolyte solutions by the primitive model, Valleau and c o - w ~ r k e r s ~demonstrated ~-~~ that GCMC simulations can be used to predict mean ionic activity coefficients (on the molar concentration scale) whose values are in reasonable agreement with experiment. The adequacy of the primitive model for this application may be attributed to the predominance of long-range Coulombic interactions as determinants of thermodynamic nonideality. Only at high salt concentrations (C3 i: 1 mol/dm3) do accurate predictions of thermodynamic activity coefficients begin to require improvementson the primitive model (introduction of more realistic potentials for short-range repulsions between charged solutes and explicit consideration of interactions involving solvent molecules). The concentration of the polyelectrolyte component cannot in practice be equilibrated by the usual GCMC method (as introduced by Valleau and co-workers3’) because random insertions and deletions of units of the polyelectrolyte component would require including in the simulation such a large number of particles (and hence pairwise interactions) that practical limits on (most) current computational capacities would be exceeded. Addition or subtraction of a single unit of the polyelectrolyte component would constitutea drastic change in the configurational energy due to solute interactions within the MC cell, unless its volume were much larger than that which suffices when MC simulations are used to evaluate activity coefficients of simple electrolytes. Although GCMC simulations of the type introduced by Valleau and co-workers do not (at present) provide a practical route to the evaluation of the activity coefficient of a polyelectrolyte component, they do offer a means of quantifying the interdependence of the nonideality of the polyelectrolyte and electrolyte components. Mills et have used a seriesof GCMC simulations to evaluate the following derivative:

The model system for which I’Ec is evaluated contains only three electroneutral components: (1) solvent; (2) ( U * ) ~ J [J z ~ where , the sign of the charge on the univalent ion is opposite that of the charge on Jz~;and (3) uniunivalent salt, u+u-. Within VMC,there are, at equilibrium, N3 units of the simple electrolyte component and one unit of the ‘‘second’’ component (which may be either a polyele~trolyte~~ or an 0ligoelectrolyte2~~~2). The practical limitations outlined above dictate that Nz = 1, because explicit consideration of interactions among large charged molecules in an MC simulation generally is not feasible with current computational capacities. As a way of neglecting interpolyion interactions, the conventional “electroneutral cell model” has been widely adopted for theoretical calculations of thermodynamic properties of polyelectrolyte solutions by various methods including those based on the PB equati0n4~and on more accurate analytic description^^^-^ (that are still, in some respects, approximate). For a particular specification of T,a3 and N2 = 1, a series of simulations is carried out to determine the equilibrated values of N3 that correspond to values of VMCchosen over a suitable range. At sufficiently large VMC,N3/ VMCis found to be a linear function of 1/VMC+ According to eq 5, the slope of a plot of N3/VMc vs ~ / V MisC I’Ec. The intercept, extrapolated over the linear

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The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7119

regime to 1/ VMC= 0, must equal @/ VMc,the density of salt ions determined in the absence of the multiply charged solute (and its complement of counterions) by an independent M C simulation carried out at the same fixed values of a3 and T as in the threecomponent system. In principle, the definition given by eq 5 pertains even when plots of N3/ VMCvs 1/ VMCare nonlinear, a t relatively small VMC.Under theseconditions, however, the neglect of interpolyion interactions (an approximation inherent to the electroneutral cell model) degrades the reliability of this method of evaluating,‘!”I which would in any case be complicated by the problem of determining accurately the tangent line to a set of data points for which the functional form of the curvature is not known. The condition of high dilution (linearity of N3/ VMCvs 1/ VMC) can be attained at computationally manageable numbers of salt ions under most of the conditions that have been examined thus far.ZSZ7 For cationic ligands (or oligomeric nucleic acids) with ZL< 10, N3/ VMCmay be a linear function of 1/ VMConly a t such large values of VMCthat GCMC simulations for the electroneutral cell model are not practical (because of the huge number of pairwise interactions that would have to be reckoned).31 Various theoretical alternatives have been explored to evaluate I’32 for the interactions of salt with short cylindrical ~ligoelectrolytes.~~ At the opposite extreme, the number of particles from the M C cell must be large enough so that statistical fluctuations are reduced (after the simulations have been equilibrated) to the point where particle densities can be accurately evaluated (J. Bond, private communication). Provided that this condition on the particle densities has been met, N ~ / V M Cis accurately proportional to n3/ V= C3, themolarity ofsalt in thecorresponding macroscopic system, and 1/VMC Cz. Hence, eq 5 can be expressed in the equivalent thermodynamic form:

-

rKC= W 3 / W , C

(6)

In the remainder of this paper, the subscript M C will be used to denote the Monte Carlo constraints specified in eq 5 euen when the derivative so labeled is not evaluated directly by MC simulations. Experimental studies of ligand-binding equilibria that can be used to evaluate SK,b almost always are subject to the constraint of either constant pressure or constant chemical potential of the solvent as well as constant chemical potential of the salt (usually). Neither of these constraints is fulfilled in a MC simulation based on the conventional form of the grand canonical ensemble. Whereas constant pressure ensembles have been investigated by M C simulations, such simulations require the sampling of volume fluctuations, which can be described realistically only by considering explicitly the coordinates of all types of particles, including the solvent. To control the solvent activity, fluctuations in the number of solvent particles within the M C cell must be equilibrated during the simulation. Under typical experimental conditions for the systems of interest here, M C simulations on systems containing realistic numbers of explicit solvent particles are not feasible. Such computations would require the capacity to account for interactions among a large number of water molecules and accurate functional forms for the potentials describing their interactions with each other and with each type of charged solute. To avoid the huge commitment of computational time entailed by explicit treatment of water molecules, their dielectric properties typically are accounted for by introducing one or more parameters into the functional form of the intersolute interaction potentials that are assumed for the model system in the M C simulation^.^^*^^ According to the standard primitive model, the entire volume of the solution enclosed by the M C cell is a dielectric continuum having the polarization characteristic of pure water in an electric field. Other features of the standard primitive model and its applicability for theoretical calculations of thermodynamic properties of polyelectrolyte solutions have been considered in a recent review.’’ The magnitude of thermodynamic coefficients

TABLE I

I’gc

such as is determined by averaging the energy due to interparticle interactions over all configurations of the system. For the purpose of evaluating such broadly averaged properties by M C simulations, the standard primitive model may yield an adequate description of the system, especially in solutions that are sufficiently dilute so that the thermodynamic coefficients of interest are determined chiefly by long-range Coulombic interparticle interactions. In summary, current computational limitations effectively prohibit explicit incorporation of the typical experimental constraint of constant pressure (or constant solvent chemical potential) into M C simulations on systems represented by the standard primitive model. Thus, the consequences of variations in P must be considered in order to relate theoretical calculations of to experimental quantities, such as S&s, that are determined at constant P. This problem is addressed in section 1II.C.

I’gC

111. Results and Discussion

A. General Thermodynamic Description of the System. In addition to the multiply charged participants in the complexation reaction (eq 2), the system to be considered here also contains univalent cations, univalent anions, and solvent water. The five distinct ionic species in the solution constitute four electroneutral solute components. In keeping with the usual convention, the indices “1” and “3” hereare assigned to water and salt, respectively. Each of the multiply charged solutes (J = FZF, LZL, or BZs), together with an equivalent number of univalent salt ions, comprises an electroneutral component denoted 25. This label implies that each of these components in turn can be viewed as the “second” in a solution where the only other components are solvent and salt. (The rationale for this point of view is given in section 1II.B). Each of the solute components (2B, 2F, 2L, 3) is assumed to be a strong electrolyte, fully dissociated into its constituent ions. The following development does not address situations where u+ or u- bind directly to any of the multiply charged species, or where changes in the hydration of the ligand or polyion upon complexation affect any of the thermodynamic properties of interest. Changes in the extent of site binding of small ions and changes in hydration may affect the magnitude of Kobs for some ligand-binding equilibria, especially when the concentration of salt is relatively high.2,33.45 However, neither short-range interactions involving solvent molecules (explicitly) nor the nonCoulombic interactions that may give rise to specific ion site binding have yet been incorporated into models that have been investigated by M C simulations of the type described in section 1I.B. For the purpose of analyzing the isobaric salt dependence of Kob, the composition of the system can be specified either in terms of the molar concentrations (mol/dm3) of all of the components, including the solvent (C1, CZB,C ~ FC, ~ LC3), , or in terms of the molalities (mol/kg) of the electroneutral solute components (mZB,m2G, m2L, m3). For each solute component k , the two concentration scales are interrelated by ck = mk/vs, where Vsis the volume of solution containing 1 kg of the solvent. For J = B, F, or L, the molarity (or molality) of component 25 equals that of the corresponding multiply charged ionic species: C ~= J CJ (and mJ = m2J). The thermodynamic activities of the four electroneutral solute components are defined as products of the single ion activities of the five ionic species according to the entries in Table I. For each ionic species i (u+,u-, F, L, B), the single ion activity coefficient is defined, with respect to the molal scale, as y i ai/mi. From these coefficients the molal scale activity coefficients of the four electroneutral solute components are constructed by analogy to

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The Journal of Physical Chemistry, Vol. 97, No. 27, 1993

the corresponding activities, as shown in Table I. For reasons given below, activity coefficients defined on the molar concentration scale are not used in this development. In most experimental studies of salt effects on ligand binding equilibria, the multiply charged species J all are at such high dilutions that the numerical difference between each mJ and the corresponding CJ is less than the uncertainties with which these quantities can be measured. The samevirtual equality often, but not always, pertains to m3 and C3. Thus, for the purpose of specifying the composition of the system, the choice of concentration scale is effectively arbitrary. Moreover, when the numerical difference between CJ and MJ for a given ion is negligible, so also is the numerical difference between the activity coefficients defined with respect to these twoconcentration scales. However, analyzing the thermodynamic interdependence of solute activities requires consideration of the functional characteristics of each particular type of composition variable (or activity coefficient), rather than its numerical value. For example, exact expressions for the difference between (8m3/8m2~) T , P , ~ ,and (8c3/8c2J)T,p,a3,or the difference between (8 In aula In a3)T,p,mu and (8 In In a3)T,p,~U, can be derived by application of appropriate thermodynamic transformations. These differences are not necessarily negligible unless all solute components are a t high dilution. Thus, molarity and molality are not (in general) directly interchangeable as functions appearing in partial derivatives that are useful for a thermodynamic analysis of the salt dependence of oligoion-polyion binding equilibria. In view of the functional nonequivalence of molality and molarity, the choice of concentration scale is significant in formulating a general thermodynamic description of salt effects on oligoion-polyion binding equilibria. Extant experimental studies of these salt effects specify solution compositions in terms of solute molarities. (In practice, molalities generally are more difficult to quantify accurately.) Moreover, for each solute species, molarity is the macroscopic equivalent of the particle density that can be calculated by a theoretical method (such as M C simulations) on the basis of a suitably detailed molecular model of the system. Thus, the molar concentration scale is used both in the description of the experimental data from which SK,k is determined (section 1I.A) and in the theoretical evaluation of the I'Ec that will be used to predict SKobs (section 1I.B). However, at a crucial intermediate stage in the derivation presented in section III.B, a partial derivative expressing the interdependence of In QJ and In a3 is transformed to a partial derivative expressing the interdependence of the concentrations of the two solute components. This transformation is greatly facilitated by the use of molalities rather than molarities, because the latter option would impose an additional constraint (on the molarity of the solvent). Nevertheless, the molar scale ultimately must be reintroduced because I'Ec is evaluated from particle densities. The remainder of this section summarizes the standard thermodynamic relationships required for the interconversion of concentration scales. When a thermodynamic derivative expressed in terms of molarities is represented alternatively as a derivative in terms of molalities, thevariation in V,must be taken into account (because ck = I?Ik/vs).A general isothermal change in v, can be represented by the total differential

be expressed as a derivative of V, with respect to its molality:

The isothermal compressibility of the solution, K , in eq 7 is defined conventionally as the positive coefficient:

(9) GCMC simulations of the type outlined in section 1I.B can provide information about the thermodynamic consequences of intersolute interactions in a solution containing polyelectrolyte and electrolyte solutes. However, when this system is represented by the standard primitive model, no theoretical approach can be expected to yield accurate predictions of any properties, such as K and the vi, that are determined to a significant extent by interactions involving the solvent. These properties, which must be considered in the following derivation of the relationship between SK,k and the pertinent I'Ec, can be evaluated (estimated) from independent measurements on comparable systems. B. Thermodynamic Analysis of S L . In this subsection, we present all the thermodynamic transformations, and the simplifying approximations, whereby SKok can be expressed as a stoichiometrically weighted linear combination of partial derivatives of the type (8 In azJ/a In a 3 ) ~ , p , ~In~ .the following subsection, each of these derivatives is related to the corresponding preferential interaction coefficients: I's, which can be evaluated as described in section II.B, and I'321, which can be evaluated by membrane dialysis measurements (for F and L). The rather lengthy route between SKok and either of these types of preferential interaction coefficients would be unnecessary if the direct determination of the activity coefficients of each of the components 25 were currently feasible either by a sufficiently accurate theoretical calculation (GCMC simulations) or by any experimental means. As noted in the preceding subsection, one of the thermodynamic transformations that will be used to establish the link between SKob, and the relevant is facilitated by expressing the composition of the system in terms of solute molalities. Converting the molarities in eq 4 into molalities requires consideration of the dependence of V, on m3. Since a change in m3 at constant Tand P causes a shift in the complexation equilibrium (eq 2), eq 7 implies that the concomitant change in V, depends in general upon the partial molar volumes of all the solute constituents. However, we have already stipulated the neglect of any changes in the hydration of the equilibrating ionic solutes B, F, and L. Accordingly, their partial molar volumes may be related by the equality

I'Ec

On the basis of the assumption specified by eq 10 and the typical

in vitro condition of excess salt (m3 >> M2J for all J), the change in V, with respect to a change in m3 can be well approximated as

(The subscript EQ has the same significance as it does in eq 4.) By use of eq 11, the conversion of the molar concentrations specified in eq 4 into the corresponding molalities yields a relatively simple expression:

(7) Here, the partial molar volume of each solute component (25) is ~ Z J VJ [ZJI~* (J = B, F,L); the sign of the subscript on (the partial molar volume of a univalent ion, ui) is opposite that of ZJ. The partial molar volume of each solute species can

vi

+

The latter equality in eq 12 will be taken as exact in order to simplify the expressions presented in the following development. In section IV.A, further consideration will be given to conditions under which the neglect of C3p3 in eq 12 is appropriate.

The Journal of Physical Chemistry, Vol. 97,No.27, 1993 7121

Salt Dependence of Oligoion-Polyion Binding Because the complexation reaction 2 is at equilibrium, the quotient of single ion activities UF/UBUL is constant at a given T and P. Therefore, the derivative of the “mass action” quotient of molalities that appears in eq 12 equals the derivative of the corresponding (inverse) quotient of single ion activity coefficients defined on the molal scale, and SKob has the equivalent form:

= (a ln[yFyL/yBl/a In m3)EQ (13) Equation 13 indicates that the salt dependence of Kob can be attributed entirely to the activity coefficients YJ. Their salt dependences in general may be due not only to long-range Coulombic interactions and short-range repulsions but also to the binding of salt ions to sites on F, L, or B and to changes in solvation of these species. The theoretical evaluation of SK,b could be readily accomplished if each of the In YJ were known explicitly as a function of salt concentration. The analytic limiting law dependence of In yFon In C3 given by CC theory14was assumed in thederivation ofeq 1. l Additional contributions toSKobarising from the site binding of salt ions and from changes in solvation also have been expressed analytically, on the basis of certain model assumptions.2 At typical experimental levels of salt concentration, GCMC simulations offer a rigorous way of computing the thermodynamic properties of the standard primitive model representation of a polyelectrolyte-electrolyte solution. Although this approach cannot predict single ion activity coefficients, the quotient y ~ c y ~ / YB can be readily recast in terms of the activity coefficient of the four electroneutral components. If computational capacities were sufficient to permit taking into account all interactions among each of the five multiply charged species under typical experimental conditions, then the activity coefficient of each of the four electroneutral components could be evaluated from a single grand canonical MC simulation. The approach would be precisely analogous to that developed by Valleau and co-workers for evaluation of the activity coefficients of mixed simple electrolytes in aqueous s o l ~ t i o n ,but ~ ~the , ~ ~requisite computation would be of a dimensionality far grander than anything currently feasible. Since the “direct” approach to evaluating the salt dependence of SKob by GCMC simulations is effectively blocked, a succession of thermodynamic transformations, simplified by approximations that can be validated under typical experimental conditions, will be used to convert each derivative of the type (a In yJ/d In m3)EQ in eq 13 to a corresponding preferential interaction coefficient that can be evaluated either theoretically (by GCMC simulations) or experimentally (by membrane dialysis). To initiate the analysis of the derivatives appearing in eq 13, a general change in In YJ with the composition of the system a t constant T and P is represented as the total differential sKobs

(a In a3/a In m3)T,P,(mzK)(16) The first equality in eq 16 follows from the definition YJ aJ/mJ and the fact that when mu is held constant mJ (but not m+ and m-) also is constant. Then In a3 is introduced by application of the chain rule, because in the following subsection the resulting derivative of In U J with respect to In a3 will be transformed into a derivative of m3 with respect to m2~.The remaining derivative in eq 16 can be expressed more explicitly by taking into account the definition 7 3 a3/(m3 + Z ~ m ~ ) ( m + sIZFlmF + (ZBlmB),so that (a In a3/a In ~ ~ ) T P , I , ,=, ~m3/(m3 K] + ZLmd + m3/(m3 + IZFlmF + IZBlmB) + (a In Y S / ~In ms)~,~,(m~,+ More compactly: Neglect of e3 can be justified when m3 >> m2K for all K and the salt dependence of In 7 3 is small enough. The use of independent thermodynamic information to estimate €3 is considered in section 1V.A. Combining eqs 14-17 (setting e3 = 0) and introducing the resulting expression for each of the In YJ into the derivative specified in eq 13 yield sKo,

= 2(a In

[aflL/aBl/a

= 2(a In

[a2f12L/aZBl/a

(a In YJ/a

m2K)T,P,m3,(mzK+2K,) =O

(1 5)

Physical implications Of the approximation introduced in eq 15 and conditions under which it is warranted are considered in section 1V.A. Theremainingderivativeineq 14can be transformedas follows:

In a3)T,P,(m2K) - 2zL

(18)

In the second equality, the partial derivative of each of the UJ is replaced with the corresponding derivative of a2J by introducing the single ion activities for u+ and u-, according to the definitions given in Table I. For the purpose of evaluating thermodynamic coefficients, both experimental measurements and GCMC simulations (in practice) can be performed only on systems where electroneutrality is maintained. An important consequence of this restriction is that only electroneutral combinations of single ion activities, or variables that are thermodynamically equivalent to them, can be compared directly with predictions of GCMC simulations (or with experimental data). Since by the assumption introduced in eq 15 YJ does not depend on any of the { m 4 , UJ depends only on mJ and m3. Moreover, the activities of the salt ions (a+, a-) are determined entirely by m3 (according to an assumption already introduced in eq 17). Consequently, the constraints on the partial derivative of In a2J with respect to In a3 can be reduced to three (T, P, m2~),so that

(a In u2J/d In u3)T,P,(mzKJ= (a In “2Jld In u3)T,P.m,

E T2J3

(19) In terms of the coefficients defined in eq 19, eq 18can be expressed more compactly:

SK,,, = 2T2,, (Here K = F, L, B). No approximation is implied by using the molalities of the four electroneutral solute components, rather than the molalities of the five ionic species to specify the composition-dependence of In y J. The partial derivatives appearing in eq 14 are evaluated at mechanical and thermal equilibrium (constant P and T ) but not under the constraint of material equilibrium with respect to the complexation reaction 2. Equation 14 is simplified considerably by assuming that each In YJ is independent of variations in all of the m 2 ~ :

In a3)T,P,(mzK)

A(lzJl

+ 2T2,,

- 2T2,,

- 2T2J3)

- 22,

(20) Here, the symbol A signifies a stoichiometrically weighted linear combination of terms pertaining to each participant in the complexation reaction 2. (The negative sign on 2T2~3appears because the stoichiometric coefficient of the product is conventionally positive.) The derivatives rzJ3in eq 20 are not evaluated under the constraint of chemical equilibrium. The distinction between r 2 J 3 and (a In azJ/aIn a 3 ) ~may o be clarified by contrasting eq 20 with the equation A(lZjl-’2(a In azJ/a In a3)EQ) = 0, which can be derived simply by differentiating In KEQwith respect to In a3. For the complexation reaction 2, the thermodynamic equilibrium constant K~~can be expressed either as the quotient of single ion activities a ~ / a or~ inuterms ~ of the activities of the corresponding electroneutral solute components as a2B(a3)ZL/a2Fa2L. The E

7122 The Journal of Physical Chemistry, Vol. 97, No. 27, 19'93

constraints on T 2 j 3 imply that it must be evaluated at constant mzJ (and T and P ) in a solution where 25 and 3 are the only interacting solute components. Under these constraints, a direct determination of TzJ3 (particularly for J = B) is not feasible, either theoretically or experimentally, but an alternative route to this objective can be taken by appropriate thermodynamic transformations, as shown in the following subsection. C. Relationshipof SI&,, to Preferential Interaction Coefficients Obtainable by GCMC Simulations and/or Equilibrium Dialysis. In general, concentrations of solute components can be measured (or calculated theoretically) more accurately and conveniently than can the corresponding chemical activities. Consequently, the evaluation of SK,b is facilitated by transforming each of the T2,3 in eq 20 into a derivative expressing the interdependence of solute concentrations. Elementary mathematical relationships among partial derivatives yield the following well-known transformation:46 '23

= (a

a2/am3)T,P,m2(dm3/a

In

'3)T,P,m2

= (a In

a3/am2)T,P,m3(am3/a

In

'3)T,P,m2

T,P,o,

(21)

Anderson and Record determinations of I;':

Here, v3' is the partial molar volume of the electrolyte component in the reference state. If the chemical potential of the salt, rather than its activity, were fixed in the GCMC simulations, the coefficient of d P in eq 22 would be V3 instead of the (negligible) difference - P 3 O . Introducing eq 23 into eq 22 and using one of the transformations in eq 21 yield

v3

Here, as in eq 6, the subscript M C implies that Tand a3, but not P, are held constant. Since solvent molecules are not explicitly considered in GCMC simulations on the standard primitive model, solute molalities cannot be calculated. With appropriate empirical input, however, "23 can be expressed on the molar concentration scale, and then related to.'?:I In converting the molal solute concentrations appearing in eq 24 to corresponding molarities, a second potential effect of pressure variability on the relationship between T23 and I'zCmust be examined, because V, depends on P via the compressibility term given in eq 7. However, if changes in the partial molar volumes of all three components in the system are negligible over the concentration range covered in evaluating ,'"I: so also is the compressibility term in eq 7:

Here and wherever possible in the rest of this paper, thecomponent 25 will be designated "2" for the sake of simpler notation. Simultaneous constraints on P and a3 (or p3) cannot be strictly achieved either in a series of (conventional) GCMC simulations or by any experimental means. However, by introducing appropriate transformations and approximations (am3/dm2)~ , p , ~ , can be related,to ';'I as defined in eq 6, or to coefficients that This equation is an analog of the isothermal Gibbs-Duhem can be evaluated experimentally. In this subsection, we first relationship for differential variations of the partial molar volumes, establish the relationship between eq 21 and the form of the rather than the partial molar free energies ( H i ) . An alternative preferential interaction coefficient that can be calculated from way of expressing the assumption made in eq 25 is to stipulate GCMC simulations. that In yi for each of the ionic species (and the solvent) does not vary with P over the concentration range covered in evaluating In relating SKobsra quantity originally defined in terms of molarities, to coefficients of the type I;', which must be .':I' (If (a In _ri/dP)T,fir= 6 = 0, then & never differs evaluated in terms of molarities (particle densities), the reason from its value V,O in the reference state.) for introducing the molal concentration scale in eq 21 may not The dependence of V, on m2 can be determined by imposing be obvious. If the molar concentration scale were retained for the MC constraints on eq 7 and incorporating eqs 24 and 25: all components, a derivation parallel to that presented in section (avs/am2)MC = v2 - "2JV3 (26) 1II.B would yield an equation relating SKobto a stoichiometrically weighted linear combination (as in eq 20) of derivatives of the Differentiating C3 and C2 with respect to m2 under the MC type (a In a z J / aIn a3)T,p,CU,CI. Each of thesederivatives, by steps constraints and introducing eqs 24 and 26 yield analogous to those shown in eq 21, could then be transformed to -(aC3/aC2J)T,p,~3,cl. Although this derivative is mathematically well-defined, its relationship to coefficients that can be calculated by GCMC simulations is complicated substantially by the constraints fixing a3, P , and C1 (which cannot be achieved in any practical experiment, either). Relating ( ~ 3 m ~ / a m to ~)~,~,~, I;' requires only an examination of the possible consequences After the quotient of eqs 27 and 28 is rearranged, the following of variations in pressure over a series of GCMC simulations relationship is obtained between "23 :and ':"I conducted at constant T and a3. Consequently, the present analysis has been simplified by introducing molalities, for the purpose of the transformation given in eq 21, even though molarities must be reintroduced because I'E' is calculated from particle densities. The approximations invoked in assuming that the second equality in eq 29 is exact are consistent with typical experimental Fixing the value of a3 while the composition of a real solution conditions; they will be considered further in the following section. is altered would (in principle) require compensating adjustments in the pressure on the system. Since such adjustments cannot be In eq 29, the term C3v2 has a purely thermodynamic origin made in a series of GCMC simulations, the potential consequences inasmuch as it appears when the molal solute concentrations are of pressure variability must be examined. Under the constraints converted to the corresponding molarities. For many polyelecof constant T and a3,the thermodynamically linked differential trolytes, including double-stranded nucleic acids, Vz is large enough so that a t moderate salt concentrations the salt-dependent changes in m3, m2, and P must conform to the following equation: term c3v2 may not be negligible in comparison to = (a In a3/am3)T,P,m2 dm3 + (a u3/am2)T,P,m3 dm2 + However, when the form of eq 29 appropriate for each of the participants in the complexation equilibrium is introduced into (a In y 3 / d P ) T , m , , m , dP (22) eq 20, the equality among the partial molar volumes of the ionic Neglect of the third term in eq 22 is justified if the pressure species assumed in eq 10brings about substantial (but not perfect) dependence of In y 3 is negligible a t all concentrations covered in cancellation among terms of the type C3vu. The remaining term

I'gc.

Salt Dependence of Oligoion-Polyion Binding in C3V3 can be neglected in accordance with previous simplifications of eq 12 and eq 29, so eq 20 can be transformed to

In principle, each of the preferential interaction coefficients specified in eq 30 can be evaluated from the output of GCMC simulations. An equation equivalent in form to eq 30 can be derived in order to relate SK,b to preferential interaction coefficients that can (in favorable cases) be determined by experimental methods. In particular, membrane dialysis experiments can be performed to evaluate the Donnan coefficient at fixed chemical potentials of salt (p3) and solvent (rl). Defined on the molal scale as (am3/ dm2)T,,,,,r3 or, more commonly, on the molar scale as ( X 3 / aC2)~,,,~,,,,, these coefficients are evaluated at high dilution in the secondcomponent. Derivatives of the type ( a m 3 / a m 2 ) , , ,which , are not immediately accessible to experimental determination, often are equated to ( a ~ ~ / a m ~ ) ~ , ,Since , ~ , ,the , , difference .~~~~ between these derivatives may not be negligible under all conditions where the analysis developed here is applicable, the details of this derivation are reviewed in the Appendix, which also reviews the conversion of a Donnan coefficient expressed on the molal scale to one expressed on the molar scale. The derivations summarized in the Appendix are nor applicable to the transformation of ( d m 3 / a m 2 ) ~ pto,'"I: because neither P nor pl (nor al) is controlled when I$ is evaluated from a series of GCMC simulations on a system represented by the standard primitive model.

IV. Concluding Discussion A. Justification of Approximations and Scope of Applicability. The simplest and most frequently useful form of the relationship between SK,a and the set of r g c i s given by eq 30. Applicability of this expression is determined by two categories of approximations: those that have been introduced in various purely thermodynamic expressions beginning with eq 10 and those that must be incorporated into the molecular model assumed for the calculation of the I'gCby the GCMC method. The amount of detail that can be built into this model is limited both by the current state of knowledge about molecular structure and interaction potentials and by the computer time available to achieve accurate GCMC simulations. In this subsection, we consider how the simplifying approximations made at various stages in the derivation leading to eq 30 can be justified and how some of these are closely related to assumptions specified in the standard model system for which the '":I can be calculated. (Assumptions that comprise the standard primitive model have been considered in some detail.") In the derivation of eq 30, the approximations introduced to simplify various exact thermodynamicexpressions arevalid under most conditions that have been investigated. In vitro studies of oligoion-polyion binding in biological systems typically are conducted on solutions containing excess salt (C3 >> CF,CB,CL), where each of the CJis at high dilution. Under these conditions, the terms omitted from eqs 12,17, and 29 are patently negligible, and the use of eq 15 to simplify eq 14 can be justified by the following considerations. At the molecular level, eq 15 implies that interactions among the multiply charged species (F, L, and B) make negligible contributions to the composition dependence of each of the YJ, which is determined entirely by interactions of the ion J with small salt ions (u+ and u-). This assumption is realistic provided that the concentrations of all multiply charged species are sufficiently dilute in comparison to the concentration of added salt. When relatively few multiply charged species are present in the solution, and especially when their Coulombic interactions with each other are effectively screened by an excess of salt ions, each of the YJ is insensitive to changes in any of the {m~}. The approximation specified in eq 15 also justifies the use

The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7123

of the electroneutral cell model in GCMC simulations to calculate ':'I for separate three-component systems, rather than for a single five-component system. At salt concentrations approaching the molar level, neglecting C3 V3 in eqs 12 and 29 and e3 in eq 17 may no longer be justified. These additional terms can be estimated from independent measurements. For the common inorganic uniunivalent salts, such as NaCl, v 3 is on the order of 0.03 dm3/mol, and so when C3 I 1 mol/dm3 the error incurred by neglecting C3V3in eqs 12 and 29 is less than 10%. When C3 and/or V3 is so large that C3V3 cannot be neglected in these equations, the true (experimental) value of SK,b is underestimated by eq 12. This underestimate is compounded by the neglect of e3 in eq 17 because e3 is always positive when m3 R 0.4 mol/dm3. Under the condition of excess salt, where Z L ~ L JZF(mF, , and ~Z&B all are negligible compared to m3, e3 can be approximated as (a In y+/a In m3)Tg. This derivative can be estimated from measurements of the mean ionic activity coefficient (y+) of salt in a solution where it is the only solute component. Alternatively, e3 can be estimated analytically by differentiating the Davies equation,50 a "semiempirical" expression devised to fit the dependence of In yt on m3 for a wide variety of common inorganic salts. By this approach, c3 is predicted to be large enough (-0.20) at C3 = 1 mol/dm3 to exceed the (typical) experimental uncertainty in SKoa. A more general form of eq 30 could readily be derived by restoring the terms in e3 and C3V3 that were neglected in eqs 12, 18, and 29. The resulting expression would account in principle for all effects arising from purely thermodynamic origins. A less tractable potential consequence of elevated salt concentrations is that the standard primitive model may become inadequate as a basis for calculating rEC.For example, this model does not include short-range attractive potentials of the kind that would produce localization of a salt ion at a specific site on LZLor F ~ F . At high enough C3, the salt dependence of the extent of any site binding may make a nonnegligible contribution to one or more of the preferential interaction coefficients in eq 30. An explicit expression for the contribution to SK,b arising from changes in theextent of anion binding upon complexation has been presented? The standard primitive model also takes no explicit account of any interactions involving solvent. In deriving eq 30, various thermodynamicmanifestationsof solvent-solute interactions have been either ignored or idealized. For example, a significant difference between VBand ( VF+ VL)most probably would result from changes in solvation when B z is ~ formed by the complexation of LZLand FZp. If the equality of partial molar volumes assumed in eq 10 is not valid, eq 30 must be supplemented with a term C3AV22Jwhose magnitude grows monotonically with increasing 9. (As in eq 20, the *A'' here denotes a stoichiometrically weighted linear combination of terms pertaining to each participant in the complexation reaction 2.) If this reaction is accompanied by a net stoichiometricrelease (or uptake) of solvent, then the equilibrium distribution of FZP, BZB, and LZL would become a function of al, the activity of the solvent. At high enough salt concentrations, a change in C3 could produce a significant change in a1 through the Gibbs-Duhem linkage between solvent and solute activities. Therefore, the dependence on C3 of the preferential interaction coefficients may be due not only to changes in the thermodynamic effects of long-range Coulombic interactions (which are included in the standard primitive model) but also to changes in solvation that accompany shifts in the binding equilibrium (which are nor included in the standard primitive model). An analytic description of the contribution to SKob attributable to changes in the amount of bound water has been presented.2 When GCMC simulations are applied to predict the thermodynamic properties of solutions containing a polyelectrolyte component, practical limitations on the time available for computations generally impose two major restrictions on the types of interactions considered in evaluating the configurational energy

7124

The Journal of Physical Chemistry, Vol. 97, No. 27, 1993

of the system. Neglect of interpolyion interactions effectively precludes the direct use of GCMC simulations (as applied, for example, by Valleau and co-~orkers)~'for the purpose of evaluating the activity coefficients yZFand 7 2 ~ However, . explicit consideration of interpolyion interactions is avoided by use of the electroneutral cell model, under conditions where eq 15 is valid, in order to evaluate r32~and I'32B. Neglect of interactions involving the solvent effectively precludes fixing the solvent activity ai, when preferential interaction coefficients are evaluated by the GCMC method. However, even if a l could be fixed in a series of GCMC simulations, to relate the resulting preferential interaction coefficients ((dC3/dC2J)03,0,,T)to the corresponding derivatives T2j3 in eq 20 would still require consideration of the effects of pressure variability on the activity coefficients of all species in the system (including solvent). Since the standard primitive model does not recognize the existence of any interactions involving discrete solvent molecules, calculations based on this model cannot be expected to yield accurate predictions of the mechanical pressure, or of any property that is a sensitive function of pressure. However, in a real solution where the nonideality of the solute components is determined primarily by Coulombic interactions, significant changes in partial molar volumes with composition occur only at extreme concentrations. On these grounds, the approximations specified in eqs 23 and 25 are generally applicable. The underlying premise here is that any information required to evaluate thedifference between T23, as transformed in eq 21, and ,'?:I as calculated from eq 5, must be sought empirically. Only properties determined primarily by long-range Coulombic interactions, such as ':'I itself, may be predicted reliably by GCMC simulations on the standard primitive model. Binding equilibria in more complicated systems containing multiple ligands and mixed simple electrolytes could be analyzed along the lines of the present development, but a presentation of the analysis would entail notation that is cumbersome (or highly implicit). Moreover, the degree of complexity of the system considered here is consistent with current restrictions on the feasibility of GCMC simulations, which are the most rigorous theoretical means of evaluating the coefficients specified in eq 30. As derived here, this equation pertains to oligoion-polyion binding in solutions for which Kobhas been determined at constant Tand Punder experimental conditions such that this equilibrium quotient does not depend on the concentrations of any of the multiply charged participants in the complexation reaction. B. Relation to Previous Work and Potential for Future Applications. In this paper, we have presented a general thermodynamic analysis of the effect of salt concentration on oligoion-polyion binding equilibria. The resulting equation, eq 20, for SKob can be expressed in terms of preferential interaction coefficients that may be either calculated from GCMC simulations (section 1II.C) or measured by membrane dialysis or other experimental methods (Appendix). Subject to theapproximations discussed in the preceding subsection (IV.A), eq 20 is generally applicable in the analysis and interpretation of effects of salt concentration on any process involving charged molecules, including polyelectrolyte, Hofmeister, and ionic strength salt effects. (The distinctions among these effects have been discussed.2) Numerous thermodynamic analyses of effects of solutes (both charged and uncharged) on equilibria involving proteins51 have been based on the work of Wyman,46 who showed that for nonelectrolytes (in the present notation) SKob = A(-T23). This expression can be regarded as a special case of eq 20, which differs in two significant respects. The factor of 2 originates in eq 17 from the dissociation of the uniunivalent electrolyte component; the term IZJI for each participant in the binding equilibrium originates in eq 18 from the specification that all solute components are electroneutral. (In some previous review articles,6~i0J5J6 expressions pertaining to an undissociated poly-

Anderson and Record electrolyte were given instead of those pertaining to a fully dissociated polyelectrolyte, which were intended. Specifically, eqs 5 and 8 of ref 15 are valid only if component "2" is an undissociated polyelectrolyte or nonelectrolyte. In all of these articles, however, the explicit expressions for the measurable quantities SKob, and dT,/d In [M+] are correct as given.) Whether participants in the equilibrium of interest, and the solute affecting it, are charged or uncharged, SKob can be interpreted as a measure of differences in the thermodynamic extent of binding of solute component (3) with product and reactant components (25). However, the explicit expression for the extent of thermodynamic binding does depend on whether the interacting species are charged. The extent of thermodynamic binding of an uncharged solute (3) to an uncharged polymer (2) is given simply by r32. In the absence of any interactions, the ideal value of I'32 is zero. The extent of thermodynamic binding of a uniunivalent electrolyte with a polyelectrolyte component comprising lZJl univalent counterions is lZJl - 2T325 GZ IZJl(1 + 2r3,,), or (1 + 2I'3,) per polyion monomer charge (u). The ideal value of r3,, is -0.5, so in the absence of any interactions the thermodynamic extent of binding is again zero. The preferential interactions of a fully dissociated electrolyte component can only be measured (or computed by GCMC simulations) for electroneutral combinations of ions. Nevertheless, the phenomenon of preferential interaction (or thermodynamic binding) in general is attributable at the microscopic level to individual contributions from each of the small ions. Thus, in describing the preferential interactions of dissociated salt ions with a macromolecule, referring to the binding of a "unit" of salt may convey the erroneous impression that the contributions of the individual ions are equivalent. The utility of single ion preferential interaction coefficients (I'+, I?-) in analyses of preferential interactions of electrolytes with charged polymers will be developed in a subsequent paper (Record and Anderson, in preparation). Since nonideality due to long-range Coulombic interactions can be of paramount importance in solutions containing a highly charged polyion, such as DNA, all of the derivatives appearing in eq 14 must, in general, be considered in describing salt effects on an oligoion-polyion binding equilibrium. Thus, the relatively compact form for SKob given in eq 20 is critically dependent upon the simplification of eq 14. Under typical in vitro conditions, the approximation specified in eq 15 can be justified on the grounds indicated in the preceding section. Neglecting eachof these partial derivatives of In YJ is the most obvious, but not the only, way to simplify eq 14. If the extent ofthe association (or thedissociation) reaction is not large, then each of the (dmz,/a In m 3 ) may ~ ~ be sufficiently small to warrant neglect of all derivatives other than that which expresses the dependence of In YJ on 1723. (Details of this analysis will be presented elsewhere.) Neither eq 20 nor eq 30 has been published previously. Equation 20 is more fundamental than the expression for SKob derived by Record et al.2using binding polynomials and CC theory, which was applied to analyze binding, screening, and water activity effects of salt concentration on various categories of equilibria involving biopolymers. The fact that the pressure cannot be controlled over a series of GCMC simulations used to calculate ':I' introduces complications that (naturally) were not considered in previous formulation^.^^^ Although the applicability of eq 30 is governed by the approximations discussed in the preceding subsection, this expression for SKob is more general than those that have been previously published for the specific purpose of predicting the effects of salt concentration on oligoion-polyion binding equilibria. The original derivation1 of eq 1 was based on an analytic expression for the excess chemical potential of the polyelectrolyte component (obtained from CC theory14)and hence follows most directly from eq 13 of the present paper. The simplest (but not the only) way to reconcile eq 1 with eq 30 of this paper is to set r 3 2 L = -2J2 (the ideal value) and to assume that the preferential interaction coefficients for the species FZF and BZe are equal on a monomer basis: I ' ~ ~ F=/ IIZ' F~~~/ ~~ZWith ~ ~ .these

Salt Dependence of Oligoion-Polyion Binding assumptions, SKob = -&(1 + 2r32F/IZF()* Incorporating the CC limiting law expression for a highly charged cylindrical polyion, r 3 2 F = -(lzFl/4[), yields eq 1. At sufficiently high dilutions, the analytic expression I ’ 3 2 ~ / (ZFI= -(4{)-l, which has been obtained16from the PB cell model for a highly charged cylindrical polyion, is in accord with the corresponding C C “limiting law” expressionl4 and hence with the description of the cation-exchange process that was presented by Record et a1.l However, even at relatively low salt concentrations, the PB cell model predicts significant deviations from the highdilution value of r32F/IZFlr which have been corroborated by GCMC simulation^.^^-^^ For the purpose of analyzing the effects of salt concentration on oligoion-polyion binding equilibria, the applicability of the approach developed by Record et al.192 does not necessarily depend upon the validity of C C theory over the entire experimental range of salt concentrations. Various additional assumptions could tend to compensate for the inaccuracy of C C limiting laws at ordinary salt concentrations that is implied by calculations based on more rigorous theoretical descriptions of the same model. As potential sources of such compensation, we have considered explicitly each assumption that enters into the derivation of the theoretical expression for SKob in terms of preferential interaction coefficients. We have shown that only at salt concentrations C3 5 1 mol/dm3 are there salt-dependent terms whose magnitudes may not always be negligible compared to the values of the l’zy in eq 30 (or to the corresponding Donnan coefficients). Although the salt dependence of r32~for an uncomplexed nucleic acid ( F z ~ )is predicted to be s u b ~ t a n t i a lthis , ~ ~prediction is not necessarily at variance with the (typical) experimental finding that SKob is constant. As we have shown in this paper, the general thermodynamic form of SKob also depends on the preferential interaction coefficients of the free ligand (LZL) and the complex (BZB). GCMC simulations are in progress to determine whether the predicted salt dependence of r 3 2 may ~ be in effect canceled by the predicted salt dependences of r 3 2 B and I’32~, in order to account for the common experimental observations that SKob is saltinvariant. If this cancellation is not predicted for a system where all prerequisites discussed in the preceding subsection for the applicability of eq 30 have been met, then the standard primitive model must be judged deficient in some respect for this application. Of the various improvements to this model that are at present computationally feasible, a more detailed description of shortrange interactions within a few angstroms of the polyion surface seems most worthy of investigation at salt concentrations approaching the molar 1evel.l’ When GCMC simulations are used to calculate r 3 2 ~for a complex formed by nonspecific binding, the model system can be substantially simpler under conditions such that each bound ligand is effectively isolated. Otherwise, configurations with varying interligand separations would have to be included in the M C cell, and fluctuations among these configurations would have to be taken into account during the equilibration. The highest binding density at which thiscomplication can beavoided may beinferred from an examination of the axial profile of the local counterion charge density at the polyion surface.3’ At locations along the polyion surface that are relatively close to a site occupied by a ligand, the diminished local field causes a diminution of the potential of mean force and hence of the local density of unbound counterions. This effect decreases with increasing distance from the bound ligand along the polyion axis in either direction. The axial distance from a bound ligand a t which its influence on the local counterion concentration can no longer be detected is a reasonable measure of the minimum (average) distance required between bound ligands so that only one thermodynamically distinct type of complex between FZF and LZL need be included in VMC for simulations used to evaluate I’E:. In a subsequent paper (Olmsted et al., in preparation), M C calculations of the local

The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 7125 surface counterion concentration as a function of axial position will be reported for cylindrical oligoions of variable length. We have established the conditions under which GCMC simulations, for solutions represented by the standard primitive model, generally are applicable to the analysis of SKob. The suitability of the model assumed for the purpose of calculating M C predictions of any thermodynamic property can, of course, be established only by comparisons with experimental data. Such comparisons will be undertaken in subsequent papers. With input from GCMC simulations, eq 30 can be used directly for the purpose of analyzing experimentally observed effects of salt concentration on equilibria involving nucleic acids. Specifically, work is in progress to analyze data on the salt dependence of the transition temperature characteristic of nucleic acid denaturation32 and data on the salt dependence of the binding to nucleic acids of model ligands whose structure and charge distribution are known.2”24

Acknowledgment. This work was stimulated in part by some GCMC simulations of effects of salt concentration and chain length on the interactions of oligocations with DNA by Dr. Martha Olmsted in this laboratory. We acknowledge with thanks some consultations with Dr. Olmsted, Dr. Jeff Bond, and Harry Guttman. The assistance of Sheila Aiello in preparation of the manuscript and financial support of N I H GM3435 1 are gratefully recognized. Appendix: Relationships of (am,/am) Tp,,3 with (am/am)T,,,lo and (aC3/aC2)T , , , ~in~ a~ Three-Component System We here provide concise derivations of thermodynamic expressions useful in relating SKob, to coefficients that may be measured by membrane dialysis. Alternative, generally more elaborate, derivations of expressions that are in most respects equivalent may be found a t various places in standard references.48,49 For a general isothermal change of the chemical potential of the salt

Differentiating this equation with respect to m2 under the “Donnan” constraints of constant T, 1.11, and p3 gives

+

= (ap3/am2)T,P,m, + (a~3/am3)T,,m2(am3/am2)T,pl,p,

v3‘3(ap/am2)T,pI,p3 (A2) By introducing the definition of e3 given in eq 17, the dependence of p3 on m3 can be represented (ap3/am3)T,P,m2

2(1

+ c3)RT/m3

(A31

By the transformation given in eq 21, eq A2 can be rearranged to give (am3/am2)TJ,p3 = (am3/am2)T,pl,p, + [m3v3/2(1

+ e3)RT](aP/dm2)T,p,#3 (A4)

The derivative (aP/am2)T,,,,,, reflects the dependence of the “Donnan” osmotic pressure on the molality of the “second” component, consisting of a multiply charged ion (with charge 2,) and an equivalent number of univalent counterions. Under the Donnan constraints, the Gibbs-Duhem relationship for a threecomponent system can be expressed simply as:

V, d P = mz dp2

645) The explicit dependence of ~2 on composition variables is introduced with the standard definition of the chemical potential, in terms of the activity coefficient 7 2 , defined on the molal scale:

7126 The Journal of Physical Chemistry, Vol. 97,No. 27, 1993

m2 dp2 = m2v2’ d P

+ m2RT d In y2 +

RT[l + Z2m2/(m3 + z2m2)1 dm, (A6) At high dilution of m2 (Z2m2