Solvent Effect on Binding Thermodynamics of Biopolymers A. BEN-NAIM,* K. L. TING, and R. L. JERNIGAN Laboratory of Mathematical Biology, National Cancer Institute, National Institutes of Health, Bethesda, M a r v l a n d 20892

SYNOPSIS

The indirect solvent-induced effect on the free energy of binding of biopolymers is examined within the framework of classical statistical mechanics. We focus specifically on the role of the solute-solvent hydrogen bonding. In particular, we have estimated the first order solvent effect on the indirect interaction between two biopolymers. We find that the solvent-induced interactions between two hydrophilic groups through water-bridged hydrogen bonds could significantly enhance the binding free energy. Some preliminary estimates indicate that this effect is significant and perhaps could be crucial in molecular recognition processes. Furthermore, we have calculated, from crystal structure data, the distance distribution between all the oxygens and nitrogens on the surface of some proteins that do not belong to the binding domain. In most cases we found an enhanced peak in the range of 4-5 A, which is where we expect to find strong solvent-induced interactions.

INTRODU CTI 0N Association of two or more molecules to form a noncovalent aggregate is a ubiquitous phenomenon in biological systems. Such processes encompass an extremely broad range of functional biological subsystems. Highly specific binding processes are common with a ligand (L) binding preferentially to a specific site on the biopolymer (P). The best known case of such specific binding (molecular recognition) processes is the binding of substrates to enzymes. Recently, the proteins binding to DNA also have been studied from the point of view of their specificity.’The common question relevant to all these processes is how the ligand “recognizes” the binding site on P. In the case of the substrate-enzyme complexes, it has been traditional to invoke the “lock and key” me~hanism,~ which may include the geometric fit but also the complementarity of

’ IYYO .ionn wiley &L Sons, h e . CCC O( )06-:~5‘25/90/050901 I9 $04.00 Biopolymers, Vol. 29, 901-919 (1990) *Permanent address: Department of Physical Chemistry, Hehrew University, Jerusalem, Israel. ~

interactions such as hydrogen bonds and electrostatics. This metaphoric model requires that the two binding partners be tightly fit in the bound state (Figure la). It is clear, however, that such a geometrical fit alone is not a necessary and sufficient condition for the recognition or identification of a binding site. If the recognition phenomenon is to be studied within the realm of equilibrium thermodynamics, then the essential thermodynamic requirement is that the free energy of binding AGB(A) to the specific site A has the lowest value compared with all possible binding sites, i.e.,

AGB(A) = minAGB( I i)

The requirement of Eq. (1) can be used as a definition of the recognition of the site A. Clearly, this condition may be satisfied by mechanisms other than the geometric fit model. For instance, L might provide a specific pattern of functional groups (FG) that can form hydrogen bonds (HB) with complementary FGs on P. In such a case, the site A will be recognized (i.e., preferen901

902

BEN-NAIM, TING, AND JEltNIGAN

a

b

Figure 1. (a) Key and lock fit of a ligand L to a biopolymer P. (b) Recognition of a site through a specific pattern of HBs between L and P (indicated by dotted lines).

tially selected by L) for its specific pattern of FGs (Figure lb). This compatibility of the FGs does not necessarily conform strictly to the geometric compatibility. In usual examples treated in the literature, the direct energy of interaction rather than the free energy of the binding has been considered. The relation between the two is

AGB( i)

=

AU,( z )

+ SG( z )

(2)

where AU,(i) is the direct or the vacuum interaction between L and P at the site i, and SG( i) is the indirect or solvent-induced contribution to AG B( z). Although the solvent contribution has been frequently mentioned in conjunction with the binding process, no effort has been expended to clarify how the solvent affects the binding thermodynamics and to what extent it might contribute to the specificity of the binding process. In this paper, we address ourselves exclusively to the study of the solvent-induced contribution SG. In contrast to AUB, which involves only two interacting bodies, the quantity SG involves averages over the configurations of all solvent molecules. Because of this feature of 6G, i t is extremely difficult to arrive a t a quantitative conclusion regarding the magnitude of SG. Nevertheless, some qualitative conclusions, as well as some approximate estimates, will be discussed in subsequent sections. In the next section we introduce the general definition of the binding process and the corresponding thermodynamic quantities. In the third section we examine the first-order term in the power expansion of SG in the solvent density. This semiquantitative analysis reveals the essential types of solvent-induced contributions to SG. A more general formalism to identify the various

contributions to SG is discussed in the fourth section. In the fifth section we examine some aspect of the higher order terms in the density expansion of SG, and some concluding remarks are presented in the last section.

DEFINITION A N D THERMODYNAMICS We begin by defining the binding process of an L to a P as the process of bringing the ligand from a fixed position and orientation, a t infinite separation from P, to form a complex PL a t some fixed position and orientation. During this process, the temperature, the pressure, and the composition of the system are kept fixed. For simplicity, we assume that both L and P have a rigid conformation that is unchanged in the course of binding. In the more general case where either L or P (or both) are flexible, the thermodynamic formalism would be the same except for an additional averaging process over all possible conformations of L and P. (For a detailed treatment of such averaging processes, see Refs. 6 to 9.) Such illduction of conformational change is anticipated to plaj a larger role for intrinsically flexible parts of macromolecules, e.g., the side chains on the surfaces of globular proteins. Once the binding process has been defined, we can define the free energy, entropy, enthalpy, etc., of this process. I t should be realized, however, that these thermodynamic quantities differ from the conventional standard thermodynamic quantities of dimerization. For example, the standard free energy of dimerization, based on the molar concentration scale, is the free energy change associated with a unit reaction process P+L@PL

(3)

SOLVENT EFFECT ON BINDING THERMODYNAMICS

when each of the reactants and products are a t the hypothetical ideal dilute solution of 1M concentration. The relationship between the standard free energy of this reaction AG: and the binding free energy, as defined above, is

where the term SGk, includes all the internal partition functions (translational, rotational, vibrational, etc.) of all the species involved. Typically, this term has the form

SG,,

=

kT In( q p 'q, 'qpL)

,

~

AG,

'

=

AUB -t 6G

=

AU,

solvent. (Here, for notational simplicity we assume that the solvent is a one-component liquid. However, Eq. (6) also applies to the more general case where the solvent might include any number of components and any concentration of the solute species P , L, or PL.) The symbol ( )o stands for a TPN ensemble average over all possible configurations of the solvent molecules. The quantity SG will be of central concern in the present paper. Although it is not a directly measurable quantity, it can be expressed as the difference in the standard free energies of the binding process in the solvent and in an ideal gas, i.e.,

(5)

Since we shall be primarily interested in the solvent-induced contribution to the binding thermodynamics, we need not be concerned with this term. Therefore, from now on, we shall focus only on the binding free energy AG in Eq. (4). The (classical) statistical mechanical expression for the binding free energy is given by8 [ P = ( k T ) with k the Boltzmann constant]

where AU, may be referred to as the direct (or the vacuum) binding energy between P and L. This would have been the binding free energy if the binding process had been carried out in vacuum. The solvent-induced contribution will be referred to as the indirect part of AGB. The latter may be expressed in terms of isothermal, isobaric and constant composition (TPN) ensemble averages as shown on the right-hand side (rhs) of Eq. (6). For the species a , B, is the total interaction energy between cy and all the solvent molecules a t some specific configuration, i.e.,

(7) r=l

where V,, is the solute-solvent pair potential a t the specified configuration Xa,X,. B, may also be referred to as the binding energy of a to the

903

SG

=

AGO,'- AGF

(8)

Note that Eq. (8) may serve as a general definition of SG in terms of two measurable quantities. The more explicit form of SG in Eq. (6) also includes the assumption that the internal degrees of freedom as well as the direct binding energy AU, are not affected by the presence of the solvent. Now, suppose that the polymer P has several binding sites for L. In principle these binding sites do not have to be discrete, but for simplicity we assume that there are n discrete binding sites, so that n different complexes may be formed between P and L. Let the polymer P be a t some fixed configuration, and a ligand L approaches it from infinite separation. L may "land" on different sites of P. The probability of landing a t site i is given by

P( i )

1 =

-

N"

exp[ -DAUB( i ) - PSG( i ) ] (9)

where No is a normalization constant, and AU,(i) and S G ( i ) are the direct and the indirect parts of the binding free energy to the specific site i. If there is one site for which P ( i ) is larger than all other P( j ) ( j # i ) , then we say that the ligand will preferentially recognize this site. Thus, the extent of recognition is intimately related to the value of the binding free energy.* *It should be noted, however, that here we assume that the system is in equilibrium; hence, the recognition is defined by the thermodynamics of the binding process. In nonequilibrium systems, it is possible that recognition could be determined by kinetic effects. For instance, there might be a relatively quick path that leads to binding t o site i, such that the steady state concentration of the complex PtL is larger than all P,L ( j f 2 ) . In such cases, the binding thermodynamics alone may not be the only cause for the recognition of that site.

904

BEN-NAIM, TING, AND JERNIGAN

The recognition of a particular binding site is traditionally discussed in terms of the "key and lock" me~hanisrn.~ In its strict sense, this model implies a tight fit between the binding surfaces of L and P. However, this geometric fit is certainly not necessary to maximize the total (direct) interaction between L and P (Figure 1). The quantity AU,( i) might include van der Waals forces, charge-charge interactions, or multipole-multipole interactions, as well as HBs between FGs on L and P. The "fit," in the general sense, means that the binding site provides the maximum binding interaction between L and P a t the particular site i. In the following sections we shall focus on the solvent-induced contribution 6G. Although some discussions of recognition have included the term " hydrophobic interaction," it seems that this concept has been added as an additional parameter to AU, or effectively identified with van der Waals interaction between two nonpolar side chains on L and P, respectively.

FIRST-ORDER EFFECT O F 6G In this section we examine the first-order term in the expansion of 6G in a power series in the solvent density p,. This study strictly applies to the limit of very low solvent density. However, as we shall see in the fifth section, the study of this particular low limit behavior might also be useful in the estimation of the effect of higher solvent densities. It is well known that 6G, or the related function y, can be expanded in power series in the solvent density pW,lo.l1 y

=

The explicit expression for the linear coefficient a in Eq. (10) is12

where f , and f L w are the so-called Mayer f functions defined, for any pair of species i, j , in terms of the intermolecular potential Uij, as

For the present study, we are interested only in one configuration of P and L, i.e., when L is bound to P a t some specific site. Thus, in Eq. (11) we assume that the configurations X , and X, are fixed (at the binding state or the dimer state), and the integration is carried over all possible locations and orientations of a solvent molecule including the excluded volume. X , stands for R,, a, which comprises the three coordinates for the location and three coordinates for the orientation of a solvent molecule. More explicitly, the six-dimensional integral in Eq. (11) is 1

--JdX, 8a2

1

=

-8Jn 2

v

dR,

where +,f?,+, are the three angles that describe the orientation of the solvent molecule. In the following, we shall not need the explicit form of the integral in Eq. (13), but rather, we will use the shorthand notation d X , = d R , d0,. We now split the total volume V into four regions as depicted in Figure 2, namely,

exp[ - P 6 G ]

dR,

=

/dR, 1

where the coefficients a, b, c . . . may be written as integrals over the solute-solvent and solventsolvent pair potential functions. At present, there exists no way of evaluating the function y a t liquid densities. However, for our purposes, the study of even the first-order term is useful. As we shall see below, this term does reveal some of the most characteristic features of the solvent-induced contribution to the binding free energy.

+

/ dR, I1

A more general classification into regions is discussed in appendix A. These regions are defined as follows: We assume that the solute-solvent pair potentials U , and U , may be written as

u,,= UL", + UL",+ UL","

SOLVENT EFFECT ON BINDING THERMODYNAMICS

IV I.\

905

or

....,,

111. The region for which both f P w and f L w are positive. IV. The region for which either f P w or f L w is zero.

Four regions of integrations defined in the third section following Eq. (14).

Figure 2.

where U H is basically the hard-core repulsive part. The region of configurations (XLXw) for which U H is practically infinite (or for which f = exp[ - pUH] - 1 is practically - 1) will be referred to as the excluded volume of P (or L) toward a water molecule. The boundaries of these regions are delineated by dashed lines in Figure 2. If w is very far from P (or from L), then the interaction energy U , is practically zero or equivalently f p w = 0. In an intermediate range, U,, and U , get significant contributions from van der Waals interactions ( U s )or from hydrogen bonding (UHB).In these regions, f P w or fLw will be positive.’ The boundaries of these regions are indicated by the dotted lines in Figure 2. With the above rough classification of the regions around P and L, we can define the four regions of integration in Eq. (14) according to the values of the f functions as follows:

I . The intersection of the excluded volume of L and P. In this region both f L w and f , are practically -1; hence, the integrand is unity. 11. The region for which w is in the excluded volume of either P or L but in the attractive region of L or P; i.e., in this region we have either fLw

=

-1,

fPw

The various regions are depicted in Figure 2. Clearly, these encompass the entire space of integration over R, in Eq. (13). Since the integrand in Eq. (11) is a product of the two functions f L w and f, it is clear that the integrand will be zero unless both f L w and f P w are nonzero. Thus, the integration over region IV is zero, and the nonzero contributions to the integral arise from regions I to I11 only. In region I we have the immediate result

=

- -1J l d R w J d Q w ( -1)( -1) 871

where V F i is the overlapping volume of the two excluded volumes of L and P with respect to w. In region 11, we assume that w interacts with either P or L through a uniform van der Waals interaction and some discrete HBs. The former extends through the entire region V(II), whereas the latter is restricted to relatively small regions, which we shall refer to as the “hydrogen-bonding regions.” In an attempt a t a qualitative assessment, we also assume that the soft interaction is of the order of kT whereas the I-IB energy is about - 6 kcal/mol, or 10kT. Thus a t room temperature:

-pu,”, = -putw= 1

’O

(van der Waals)

~

We shall assume for simplicity that, except for the excluded volume, f L w and f p , are always positive. In principle, it is possible that repulsive forces, such as charge-charge repulsion, would produce negative values of the f function. We shall not treat these cases in this paper, however.

-

pu,”,” = -pu,“,”

= 10

(hydrogen bond)

BEN-NAIM, TING, AND JERNIGAN

906

Thus, the integral over region I1 can be split into two parts:

suming that the hydrogen-bonding region is very small compared with the volume of region 111, which we denote by V(III), we write

+ NgB[exp(lO) - 11 1 +

8,rr2

Q

+ NLEB[exp(10)

- 1)

-

[exp( 1) - 11V( 11)

Here, 17 = a3QHB/8a2is the region (element of volume u 3 and range of angles ),a, of integration for which a water molecule can form a HB with a FG on either P or L. The first term on the rhs of Eq. (21) is due to the van der Waals interaction between w and either P or L in region 11. V(I1) is the volume of region 11. Strictly, we should have subtracted the region of hydrogen bonding from V(II), which is (NEB + NEB)q. However, since the latter is much smaller than V(II), we have taken V(I1) in the first term on the rhs of Eq. (21). The second term is due to hydrogen bonding between all the FGs on both L and P and the water molecule. Each HB contributes an energy factor of [exp(ll) - 11 (i.e., van der Waals and HB energy). The total number of such FGs are denoted by NEB + NFJ. Finally, in region 111, a water molecule interacts with both L and P in such a way that f p w > 0 and f L w > 0. The region of integration can be further split into four subregions as follows-

-

-

where in III,, f L w fPw [exp(l) - 11; in 111,, fLw [exp(ll) - 11, and f p , [exp(l) - 11; in fLw [exp(l) 11 and f p , - [exp(ll) - 11; III,, and in III,, fLw fPw [exp(ll) - 13. Again, as-

-

-

-

11[exp( 1) - 1117

- I} dXw

x {exp[ -pU& =

-

-

Here, N E B is the number of FGs in region I11 that form HBs between P and w, with a similar meaning for NLCB.The quantity NF2w is the number of HBs that w can form simultaneously with FGs on both L and P. The corresponding region of integration is denoted by 17’ = U ‘ ~ Q ~ ; I ~ /which ~T~, is smaller than the region defined for one hydrogen bond, 17. We now distinguish between four cases: (1)when no HBs are involved in either region I1 or 111, (2) when HBs are involved in I1 but not 111, (3) when HBs are involved in I11 but not in 11, and (4) when HBs are involved in both I1 and 111. In case 1, we have for a in Eq. (11)

a,

=

Vfg - [exp( 1) - 11V(II)

- VLEpx - 1.72V(II) + 2.95V(III)

(24)

The first term on the rhs of Eq. (24) is the overlapping volume of the two excluded volumes of L and P. This term is always positive. When y is translated into SG, this term always enhances the binding between L and P. This phenomenon exists for any solvent, even for hard spheres, and is not characteristic of liquid water. The second and third terms on the rhs of Eq. (24) arise from the van der Waals interactions between L and P and a water molecule. One is due to the “loss” of solvation of region 11 and the second is due to the “gain” of the simultaneous interaction between a water molecule and L and P in region 111. Again, these terms are not special for liquid water, and we expect that such terms, with

907

SOLVENT EFFECT ON BINDING THERMODYNAMICS

similar magnitudes, would appear for any real solvent, such as water, alcohol, hexane, etc. We also feel that once HBs are operative these last two terms will be relatively small compared with the contribution due to HBs between water and L and P. In the following, we shall discuss cases 2 and 4 where hydrogen bonding is involved. We note that the term a , in Eq. (24) will be common to all the cases 2-4. However, we shall now focus on specific effects due to hydrogen bonding. In case 2 we have the following first-order coefficient in Eq. (ll),which we denote by a,:

a,

=

a,

-

[NEB + NEB] [exp(ll) - 11q (25)

We see that, in this case, turning on the hydrogen bonding produces a negative contribution to a in Eq. (11).The physical reason is quite evident. This term includes exactly the contribution of the HBs that are lost when P and L form a dimer. (Note that we are now examining only the solvent effect. Clearly, there is always a possibility that P and L will form direct HBs, but these will affect the direct interaction energy AU, and not 6G.) This contribution will be proportional to the total number of FGs on L and P (in region 11). In addition, we have two factors that are characteristic of the HB; one involves the HB energy and the other involves the range of the HB interaction. A rough estimate of the product of these two factors may be obtained from the experimental data on the second virial coefficients for water.l3,l 4 T h e second virial coefficient for water as a function o f temperature has been presented by Keyes, quoted from Eisenberg and Kauzmann14 as -1

= 2.062

I

-

(2.9017 X 103/T)

~exp[1.7095x 105/T2] cm3/g

(26)

As we have done for the solute-solvent pair potential, we also assume here that the waterwater interaction may be split into three partsthe hard, the soft, and a HB part. We also assume that the range of configurations for which a HB is formed is quite small and is outside the range of the hard interaction. In this case, we write Eq. (26)

as

where uww = 2.8 A is chosen as the effective hardcore diameter of a water molecule. The factor 8 in the second term on the rhs of Eq. (27) accounts for the fact that there are 8 identical ranges for which the two water molecules can form HBs. For each of these ranges, we approximate the integral by taking its maximum value [exp(ll) - 11 times the range of configurations IJ for which this value is attainable. From Eqs. (26) and (27), we may estimate 17 a t 298 K to be - B,

‘

+ 27.69

=

4[exp(ll) - 11

=

5.37

x 10-3cm3 mol-’

(28)

Introducing this approximation in Eq. (25), we obtain u 2 (in cm3 mol-l) a,

=

a,

-

(Np“,“+ NEB) x 322

(29)

Thus, the larger the number of FGs in this region, the more negative the contribution to a [ in Eq. (11)]and, hence, the larger the tendency for dissociation of P and L. Had we taken higher values of the HB energy as used by Dahl and Andersen15 we would have obtained lower estimates of TJ. Thus for the “strong” HB, cHB = -8.6 kcal/mol and “weak” HB, c H B = -6.9, we obtain rl tstrong) = 7.05 x lor5 and IJ (weak) = 1.2 x c r 3 mol-’ respectively. In case 3 we have only HBs in region 111; hence, from Eq. (23), we have a3 = a,

+ ( N E B+ NiEB)[exp( 10)

-

11

x [exp(l) - 1177

+ NF:w[exp(lO)

- 1]2771

(30)

Here, N E B and NLZB are the number of FGs that can form HBs with the solvent in region 111. Besides a,,which is the same as in Eq. (24), we have a

908

BEN-NAIM, TING, AND JERNIGAN

term here that results from a water molecule forming a HB with one solute and interacting via van der Waals interaction with the other. The last term includes the simultaneous HBs between a water molecule and two FGs on L and P in region 111. The number of such pairs of FGs is NEBw. Although the range of configurations for which a water molecular can form two HBs (with FGs on both L and P) will be somewhat smaller than the range for a single HB, i.e. (see appendix B), 17' < 17

(31)

the factor [exp(lO) - 11' will be large enough to more than compensate for the reduction in the range of configurations. We believe this term might contribute significantly to a , and hence, to the binding free energy of P and L. Case 4 is essentially a combination of cases 2 and 3. It is probably the most common case in actual examples of protein-protein and proteinDNA interactions. Of course, for any quantitative estimate of the solvent-induced contribution to the binding free energy, we must know the number of FGs and their distribution on the surfaces of the biopolymers. To summarize, there are three distinctly different ways that the solvent may affect the binding between two biopolymers. First, the overlapping of the excluded volumes is always positive and tends to enhance binding. Qualitatively, its effect is similar to the key and lock model; i.e., the tighter the binding region, the larger is the overlapping between the excluded volumes of L and P, and hence the greater the contribution to SG. The second is due to the loss of solvation in region 11. Qualitatively, this effect is similar to the effect of direct hydrogen bonding between L and P; i.e., it will depend on the number of FGs in region 11. Here, the larger the solvation of the FGs, the greater the effect on the dissociation of L and P. The third is due to the simultaneous hydrogen bonding between water and P and L in region 111. We believe, because of the factor [exp(lO) - 112,the last effect is probably an important one in enhancing association and can provide a means for obtaining strong specificity of the binding. To conclude, we make a rough estimate of the contribution due to the third effect, assuming that this is the only significant term contributing to SG. In Eq. (28) we had an estimate for 17 = 5.37 X cm3 mol-l. This is the range of configurations for the formation of one HB between two water molecules. In Eq. (30) we need an estimate of q' for

the range of configurations available to one water molecule simultaneously forming two HBs with two water molecules or two FGs. We consider one specific case when the two FGs are a t a distance of 4.5 A and correctly oriented to form two HBs with one water molecule. If we place a water molecule in such a way that it forms two HBs with these two FGs, we estimate that the range of locations and orientations will be similar to the one in Eq. (28) except for one rotational angle; i.e., in the case of one HB, the water molecule can rotate about the 0-0 axis without affecting the strength of the HB. However, when this molecule is required to form a second HB, the total range of orientation of 360" is reduced to either 13" or 28", according to whether we take the weak or strong HB in the Dahl and Ander~en'~ model. Thus, to convert from 71 to q', we have to multiply by the factor of about 20/360 (for details, see appendix B), i.e.,

We can now make a rough estimate of the contribution of one such pair of FGs in region TI1 to SG (assuming that this is the only contribution to SG).

SG

=

- k T In y

=

- k T ln(1

+ up,)

(33)

Taking the solvent density of water a t 1 atm pressure as p, = 4.1 x loA5mol omp3, we obtain

SG

=

- k T In y

=

-1.164 kcal mol-'

(34)

This is quite significant for such a first-order term. Note that a t this solvent density the first-order correction to the pressure is P/kT

=

1

+ B 2 p w = 1 - 5.2 X lo-'

which is quite a small correction to the ideal gas behavior. Perhaps a somewhat more realistic case would be to take the density of water of 4.1 x mol ~ m - for ~ . which P/kT

=

1

+ B2pw = 1

-

5.2

X

lo-'

S G = - k T l n y = -2.46kcalmol-'

(35)

SOLVENT EFFECT ON BINDING THERMODYNAMICS

This is quite significant. In a recent estimate made for SG a t liquid densities it was found16 that SG is about - 3 kcal/mol. This value has been also found by recent Monte Car10 calculations.”

DIVISION OF 6G INTO THE HYDROPHOBIC AND HYDROPHILIC INTERACTIONS In the preceding section we assumed that the solute-solvent pair potential may be divided into three components-the hard (H), the soft (S), and the hydrogen bond (HB) parts. Corresponding to this division of the pair potentials in Eqs. (15) and (16), we can define the three components of the binding energies, i.e., N

BP

=

C UPw(XP,Xi) 1=

=

909

where U, is the total interaction energy among the N solvent molecules, and the symbol ( ), stands for an average over all configurations of the solvent molecules, whereas the symbol ( )HS is a conditional average, i.e., the average over the configurations of the solvent molecules given that the hard and soft parts of the solute P have been “turned on” a t some specific configuration Xp. The latter average is taken with the probability density

If we perform the same factorization as in Eq. (37) on each of the solutes P, L, and PL, we can rewrite the solvent-induced contribution to AG as

1

~ p +” B;

+B

~ H= ~B~~ P + B~~ P

(36)

and similarly, we can write expressions for B , and for BpL. The summation in Eq. (36) is over all the N solvent molecules a t some particular configuration of the entire system Xp, X, . . .X,. In the last form on the rhs of Eq. (36), we combined the first two contributions into BY. Each of the average quantities appearing in Eq. (6) are supposed to be averages in the TPN ensemble. However, for notational simplicity, we shall use the TVN ensemble in the following. The final division of SG is independent of which ensemble we use for the averaging process. Thus, we write

where SGHS contains only the H , S interactions between the solute and the solvent, i.e.,

1

’I

and SG HB/HS contains the corresponding conditional averages

(exp[ +BPI),

The significance of the division of SG into two contributions is the following. The first term SGHS may be referred to as the conventional hydrophobic interaction (HI) between P and L. It is conventional in the sense that the term HI traditionally has been applied to describe interactions between two nonpolar solutes in water. When we “strip off” the HB interactions of P, L, and PL, the solutes behave as if nonpolar. We are not treating charged

910

BEN-NAIM, TING, AND JERNIGAN

groups on the surfaces of the solutes; however, these can also be separated in the same manner as in Eq. (39). Therefore, any solvent-induced attraction between P and L a t this stage can be referred to as HI. However, the concept of HI, in its more general sense, may include any solvent-induced driving forces that help bring P and L together. Specifically, in SGHBIHS,which may be referred to as hydrophilic interactions, we also add all the HBs that are formed between the solvent and P and L; these may affect the total solvent-induced part of AGB. A simple case to demonstrate this effect has been discussed in the preceding section. Finally, it is worthwhile noting that, for computational purposes, one might want to split SGHS into two terms as well, i.e.,

SGHS = SGH + SGSIH

(42)

where SGH may be referred to as the (conventional) HI between the hard-core part of the solute-solvent interaction and SGSIH is the conditional HI of the soft part given the hard part is turned on.

HIGHER ORDER TERMS IN THE DENSITY EXPANSION OF 6G In the third section we examined the first-order term in the solvent density expansion of SG (or equivalently, of the function y = exp[ -PSG]). Here, we point out some general features of the higher order terms that might be of particular importance in aqueous solutions of biopolymers. In the general theory of simple fluids, the density expansion of the function y(1,2) is written, symbolically, as follows:

2

+p”j{2 2

b;J + 4

+M+ + 3!q p f ) + & + . . - } xd(3) 4 4 ) 4 5 )

+ ...

(43)

where the integrands are represented by graphs.

The open circles are the “base points.” These are particles 1 and 2 (which in our case are the two solutes P and L), the configuration of which is presumed to be fixed. The full circles are the “field points.” These represent solvent (here water) molecules on the configurations of which the integrations are carried out. Each line in these graphs represents an f function connecting the two points a t its edges. As is well known, the coefficients for the higher order terms become more and more complicated, and even for the simplest liquids, only a few of these coefficients can be calculated analytically. In the case of liquid water, an f function includes hard, soft, and HB interactions; all these make even the first-order coefficient quite complicated. A detailed analysis of the cluster expansion for liquid water has been carried out by Dahl and Andersen.15 In the case of biopolymers in water, clearly the situation is more complex. However, we noticed in the preceding section that SG may be split into two parts [see Eq. (39)], one due to the hard and soft interactions, and the second due to hydrogen bonding between the solute and the solvent. We also noticed that SGHBlHShas the same formal form as SG [see Eq. (37)], except for the conditional character of the average quantities. Therefore, we expect an expansion similar to Eq. (43) to be applicable to SGHB/HS.Here, we can exploit the fact that HBs are very restricted in the configurational range, a fact that leads us to examine only a certain group of graphs that we believe give the most important contributions to 6GHB/HS. For simplicity, we now examine only FGs that can form a bridge of HBs through one water molecule (i.e., we focus on region J as defined in appendix A). We assume that we have NFCw pairs of FGs, one on P and the other on L. We further assume that each pair of FGs is a t the distance of 4.5 A and is oriented in such a way as to allow the formation of a bridge through a water molecule (see Figure 3). The pairs are also assumed to be far apart so that they act independently. In this case, since the configurations of P and L are presumed to be fixed, it is highly improbable that chains of more than one water molecule will bridge between the two FGs of each pair. Furthermore, because of the highly restricted range of configurations for hydrogen bonding, we can ignore all graphs in which a water molecule forms more than two HBs a t a time. These considerations drastically reduce the number of graphs that are likely to contribute to SGHBIHS

SOLVENT EFFECT ON BINDING THERMODYNAMICS

911

Similarly, for the third order term, we have the contribution d(4) d ( 5 )

4,&d(3) 3!

\fL Figure 3. Two h ~ d r o x groups ~l in the J regon of p and L at a distance of 4.5 A, and oriented in such a way that they can form HBs with one water molecule.

We recall the first-order term in the density expansion of y (see the third section), which in our

Note that in all of the integrals (44)-(46) the f functions involve only HB interactions, and the integration is restricted by the conditions imposed by the conditional character of the average quantities in SGHB/HS. Thus, if we sum over all graphs of the type discussed above, we Obtain

SGHBIHS= - k T In 1 + pwa + -a2 Pw 2

i

2 3 Pw +-a3+

...

3! = - kTpwa =

PwJfh(l,3)fLw(2r3)4 3 )

- kTp, exp( 20) q’NFcw (47)

9’ PwNZBweXP(20)

(44)

Since we have taken a relatively strong HB energy pUHB= lo), we shall estimate 7’ from the parameters of Dahl and Ander~en,’~ for the so-called strong hydrogen bond. From appendix B, we have estimated 9‘ for the weak and strong HBs based on estimations by Dahl and Ander~en.’~ The value of 7’ for the weak bond gives an unreasonably large value of SG HB/HS, probably overestimating the range of configurations allowable for hydrogen bonding. However, for the strong HB, we have (appendix B) (-

Thus, we ignore all other contributions to a except for the contribution due to the configuration in which both P and L form HBs with a water molecule. We now look a t the following graph that contributes to the second-order term in the expansion (43):

9’ =

1.87 x 10-7cm3mol-’

(48)

Using p, = 5.5 x lop2 mol cmP3,we can estimate SGHBlHSper pair of FGs as

SGHBIHS= - kTp, exp( 20) 9’

=

2.95 kcal mol-’ (49)

We see that in this case the integrations over the configurations of the field points (3) and (4) can be separated, and the integral under the squared brackets is the same as the one in Eq. (44).

This figure is quite close to the one obtained by theoretical calculations16 and from Monte Carlo calculations.’7 It also indicates that probably many terms in the full cluster expansion (43) cancel out for 8GHBIHS. We have estimated that in hemoglobin there are about 70 pairs of FGs at a distance between 4 and 5 A. Of course, not all FGs will be a t a distance of exactly 4.5 A and correctly oriented as required above. But even if only 10

912

BEN-NAIM, TING, AND JERNIGAN

pairs of FGs have the correct orientations, we would obtain about 29.5 kcal mol-’ due to the solventinduced effect on the binding free energy. Perhaps the most important aspect of 8GHBIHS is its extreme sensitivity to the way we define the HB range and energy. This is a result of the product of very large and very small numbers in Eq. (47). We stress, however, that these figures correspond to the ideal configuration of the two FGs (Figure 3) where the HB energy attains its maximum value. Clearly, in any real cases we might have pairs of FGs a t configurations that deviate from these ideal configurations; hence, the energy parameter ( -/3UHB = 20) used in Eq. (47) will have to be reduced. At present, there is no way we can tell how many pairs of FGs have the ideal, or nearly ideal, c&figuration to form two HBs with a single water molecule. What we can find is only the distances (and not the orientations) between such FGs.

In Figure 4 we present some histograms showing the number of pairs of Ns and 0 s (i.e., FGs that potentially can form HBs) in the J regions of P and L of some complexes of biopolymers. All the figures except GBNA are proteins. GBNA is a complex of d(CGCGAATTCGCG) with netropsin, 4HHB is human deoxyhemoglobin, 2CTS is the citrate synthase-CoA-citrate complex, 4CPA is the carboxypeptidase complex with potato inhibitor, lTGS is the trypsinogen complex with porcine pancreatic secretory trypsin inhibitor, 2PTC is the P-trypsin complex with pancreatic trypsin inhibitor, 2TPI is the trypsinogen complex with pancreatic trypsin inhibitor and Ile-Val, 2TGP is the trypsinogen complex with pancreatic trypsin inhibitor, and 4TPI2 is the trypsinogen complex with the Arg-15 analogue of pancreatic trypsin inhibitor and Val-Val. I t is clear that in each case shown in Figure 4 we have a t least 4 to 5 pairs of FGs that are a t a distance between 4 and 5 A, and therefore are potential candidates for formation of HB bridge

7

RIA

‘1

RIA

NL d ’

RIA

4CPA

T

dN:L 4TP12

RIA

RIA

Figure 4. Number of pairs of oxygens and nitrogens that belong to different biopolymers and that are not interacting directly, at different distances R . Data are for a variety of reported crystal structures. See the text for details.

SOLVENT EFFECT ON BINDING THERMODYNAMICS

Figure 5. Two hydroxyl groups in the J region of P and L at a distance, and orientation such that they can form HBs with a chain of two water molecules.

through one water molecule. This is a limited sample of interacting molecules, and it indicates just how limited such reported cases are. These examples were all taken from the Brookhaven Protein Data Bank.ls However, the enhanced numbers in between 4 and 5 A would indicate the likeliness of this process playing a role in favoring binding. Also in this section we assumed from the outset that pairs of FGs a t a distance of R = 4.5 are available, and therefore we have summed a specific series of graphs in Eq. (47). Clearly, for other distances and orientations, e.g., the one depicted in Figure 5, other graphs in the expansion of y should be considered. The first-order term will be

A

To this, one should add all graphs that can be reduced to powers of the integral as in Eq. (50). The main point we wanted to stress in this section is that, when a pair of FGs is in the right configuration to form HBs with a single water molecule, the indirect solvent-induced effects are likely to contribute quite significantly to the binding free energy.

DISCUSSION AND CONCLUSIONS In ths paper the traditional view that molecular recognition is essentially due to direct interactions has been expanded to include the indirect solventinduced effect as well. The characteristic feature of the direct recognition is that it can be described in terms of pointto-point interactions. These might include van der Waals, charge-charge, or HB interactions between

913

FGs on P and L. All these can be described in terms of a generalized key and lock model. The solvent-induced effect, in general, cannot be attributed to point-to-point interactions between P and L. Specifically for water as a solvent, the concept of hydrophobic interactions has sometimes been invoked to explain the solvent effect on the binding.4 Usually, the hydrophobic interactions are assigned to pairs of nonpolar groups that interact directly in the PL complex. Therefore, this assignment effectively modifies the van der Waals interactions between the two nonpolar groups. Such an assignment clearly is not suggested by the exact analysis of the content of the quantity SG. We deem it more appropriate to directly and explicitly account for such a solvent effect. In our examination of the various contributions to SG, we found essentially three different types of effects. (These were present both in the first-order term examined in the third section and in the more general treatment discussed in appendix A). The first effect is due to the overlapping of the excluded volumes of P and L with respect to a solvent molecule. The larger this volume, the greater will be the contribution to the binding free energy. Clearly, this effect is common to any solvent. This effect is not unique to liquid water. The second effect arises from the loss of solvation of FGs that were originally exposed to the solvent but are shielded from interaction with the solvent in the complex PL. This effect depends on the type of FGs that are in the binding region (denoted by I in appendix A). For instance, a nonpolar group in this region weakly interacts with the solvent before the binding; hence, the loss of solvation free energy will be minimal. On the other hand, a strongly interacting FG (say hydroxyl or a charged group) will cause a larger loss in solvation free energy upon binding. We see that this effect, like the direct interaction AUB, does depend on the FGs in the region I. But unlike the direct interaction, it depends on the extent of solvation of the FGs, i.e., the larger the solvation free energy of a particular FG, the greater will be the tendency toward dissociation of P and L. An approximate method of estimating the solvation free energy of such a FG was published re~ent1y.l~ The third effect arises from FGs on both P and L that can simultaneously interact through chains of water molecules. (These groups belong to region J according to the notation in appendix A). We found that, under favorable conditions (i.e., correct distances and orientations of the FGs), bridges through single water molecules could contribute

914

BEN-NAIM, TING, AND JERNIGAN

P provides an apparently featureless surface, and no site can be directly recognized by the ligand L. However, some specific pattern of FGs on both P and L can create recognizable sites (site A in the figure), which depends only on the HB bridges through solvent molecules. A t present, it is impossible to verify our conclusions experimentally. The difficulty lies in the fact that one needs very accurate information on both the locations and the orientations of the FGs in the I and J regions. The experimental distinction between regions I and J (see appendix A) is not easy. There have been several studies aimed a t identifying the binding regions between proteins and DNA. This can be done either by mutation or by chemical alteration. In each case the effect of a modification in the base pair (bp) on the free energy of the binding is noted. A change in a bp that leads to a significant change in the binding free energy is presumed to occur in the “binding region.” Here the binding region includes all the bp (or groups) that, when changed, affect the binding free energy. The latter includes both the direct and the indirect (solvent-induced) effect. A more stringent test, which in principle could be used to make such a distinction, is to modify the

significantly to 6G. This effect certainly cannot be “assigned” t o FGs in the binding region (region I in the notation of appendix A). All of the three effects discussed above should contribute to the binding free energy and therefore to “recognition thermodynamics.” In Figure 6, we show two examples where the solvent-induced effect might be important. In Figure 6a, the key and lock mechanism is operative. We have two identical sites A and B with identical direct interaction energies; i.e., AU,(A) = AU,(B). Furthermore, because of the identity of the two sites, the first two effects discussed above (the volume and the loss of solvation in region I) also will be identical. Therefore, the only effect that might lead to preferential binding, i.e., recognition, would be due to bridges through water molecules. In Figure 6a, we see that a carbonyl group on the surface of P (in region J), which is not directly “seen” by L, can form a chain of HBs with a carbonyl group on L. This can lead to preferential recognition of the site A over site B. Thus, in addition to the conventional key and lock mechanism, we have a specific solvent-induced effect that influences relative preferences for sites. In Figure 6b, we have a similar situation but without any key and lock mechanism. The polymer

0

A

a -

A

P

0

b

Figure 6. (a) Two identical binding sites A and B that interact with a ligand L with the same binding energy. However, the indirect solvent-induced effect causes site A to be more favorable for binding. (b) Same situation as in (a), but there is no key and lock mechanism. Here, only the indirect effect produces the preferential binding of L to the site A.

SOLVENT EFFECT ON BINDING THERMODYNAMICS

bp, and study the binding once in water and once in an inert solvent (or ideally in vacuum). Clearly, this suggestion seems impractical at present. In the study of binding of proteins to DNA, it is often claimed that the specificity of the binding region is provided by the specificity of the bps in the binding region and not due to the sugar-phosphate backbone. The reason is that the sequence of bps can be highly specific in terms of providing FGs in the grooves that will HB with the protein. On the other hand, the FGs on the phosphatesugar chain are relatively uniformly distributed and therefore cannot contribute to the specific binding region. This is strictly true only if the sugar -phosphate chain is completely uniform (in terms of the periodicity of appearance and orientations of the FGs). I t is conceivable that slight changes in twist of the helix backbone or small changes in the orientations of the phosphate-sugar FGs caused by the bps could lead to a more specific pattern of FGs that would contribute to the specificity for the indirect solvent-induced recognition. An origin of this effect could be sequence-specific dependence of DNA conformations, which has been clearly demonstrated in a B-form DNA dodecamer crystal. The large flexibility of DNA helices would facilitate such an effect, but it remains to be determined whether such an effect could be a large contributor to specific recognition.

may be split into two contributions M u(xa7xi)

UF(Xk,Xi)

-

= UH(X,,Xi)

k=l

where U H ( X , ,X , ) is the hard-core repulsive interaction between the solute a and the i t h water molecule in the configuration ( X , , X t ) . The second term on the rhs of Eq. (Al) consists of all the interactions between FGs on the surface of a and a solvent molecule.t U F ( X , , X , ) specifically denotes the interaction between the k t h FG on a and the i t h water molecule. The FG may be any group that is exposed to the solvent. These could be methyl, hydroxyl, carboxyl, etc., groups that interact with water through van der Waals, hydrogen-bond, or electrostatic interactions (excluding the repulsive hard-core interactions which are taken care of by U"). The main reason for making the distinction between the two terms in Eq. (Al) is that the former depends on the entire volume of the solute a, whereas the latter depends only on the surface. Corresponding to the division of U in Eq. (Al), we can define the total binding energy of a to the solvent by summation over all water molecules in the system. Thus N

B, APPENDIX A Classification of Regions on the Surface of a Polymer

In the third section we have looked a t the firstorder term in the density expansion of y (or SG). This term corresponds, from the formal point of view, to the case where only a single solvent molecule interacts with the complex PL. Because of its relative simplicity, we could pursue this case to the point of making approximate numerical estimates of the order of magnitude of this term. In this appendix we discuss the formal classification of various regions on the surface of the biopolymer. I t is shown that this classification leads to essentially the same three different effects that we have discussed in the third section. However, here no assumptions will be made regarding the solvent density. Let a be either P or L. We assume that a is a large solute, either a protein or a nucleic acid. We also assume that the solute-solvent interaction

915

h.1

u(x,,x,)

=

=

B,"

1=1

B,",k (A2)

f k= 1

where BZk is the total binding energy of the k t h FG on the surface a (excluding the hard interaction) to the solvent a t a specific configuration of the system; X , , X,, . . . , X N . Using a similar procedure as carried out in the fourth section, we write the solvation free energy of a as

A G =~ - k ln(exp[ ~ -PB,]),

'Here, by definition, any group that interacts with the solvent, not through the hard repulsive interaction. is presumed to belong to the surface of a.

916

BEN-NAIM, TING, AND JERNIGAN

+Jq \J

Corresponding to this classification, we write the binding energies of the various solutes to the solvent as

E

-E Figure 7. The three regions, I, J, and E, on the surfaces of P and L, discussed in appendix A.

Here, AG,* is split into two terms. The first AG,*H is the solvation free energy of the hard part of the interaction. This is essentially the work required to create a cavity suitable to accommodate the solute a. Clearly, this term depends on the volume of a , and one can show that this term is always positive and increases monotonically with the solute diameter6 for a given solvent. The second term on the rhs of Eq. (A3) is the conditional solvation free energy of all the FGs on the surface of a, given that the hard-core potential has already been solvated. The subscript H in ( ) H indicates the conditional character of this average, whereas the symbol ( )o stands for an average over all solvent molecules with the probability distribution of the pure solvent. We now classify the FGs that contribute to AG,*F/H into three regions (Figure 7):

Here, all the Bs refer to the binding through the FGs and not through the hard interaction [i.e., the second term on the rhs of Eq. (AZ)]. We refrained from adding a superscript F in Eq. (A4) for the sake of notational simplicity. Note also that, by definition, the coniplex PL does not contain a term BPL,I,since the regions I on P and L are not exposed to the solvent in PL. For the solvation free energy of P, we assume that the FGs in regions P, and PI (i.e., region E and I on P) are independent in the following sense

1. Region E (for external) consists of all the FGs

(either on P or on L), the solvation of which is not affected by the binding of P and L. Intuitively, all the FGs on P and L that are f a r away from the binding region will be included in region E. 2. Region I (for inner) includes all the FGs on either P or L that do contribute to AGp*F/H and to AGtFlH, respectively, but do not contribute to AG Intuitively, I includes FGs that are solvated when the solutes P and L are separated but are not exposed to the solvent after P and L bind to form the complex PL. 3. Region J (for joint) includes all the FGs that are not included in E and I. These groups are exposed to the solvent before and after the binding process, but the extent of the solvation might be different in these two states.

and, similarly, for the solute L

;l/H.

Similarly, for the complex PL, we assume that P, and L, are independent. [This does not follow directly from Eqs. (A5) and (A6). However, if P and L do not wrap around each other, it is likely that if E and I are independent in the separate solutes P

SOLVENT EFFECT ON BINDING THERMODYNAMICS

and L then the two E regions PE and LE on PL will also be independent.]

With these assumptions, the expression for SG *F/H can be simplified considerably, since all the terms that include the E region will cancel out. [Note, however, that the conditions in Eqs. (A5), (A6), and (A7) are different. Nevertheless, we assume that these conditions have relatively little effect on the binding energies of FGs in the E regions.] Thus, we may write the total solvent-induced effect as

The three contributions included in the squared brackets in Eq. (A8) correspond to the three terms that were discussed in relation to the first-order term in Eq. (23). Here, the first term essentially depends on the difference in the excluded volume between PL and P and L separately. This effect is probably always negative; i.e., it will enhance binding. I t is also an effect that appears for any solvent, not necessarily water, and is expected to be larger, the larger the contact region between P and L.

917

The second term is due to the loss of the solvation free energy of the I regions on P and L when they form the complex PL. This term is always positive; i.e., it will contribute to enhance the dissociation of P and L. Normally, the FGs in this region will bind with each other in the complex PL. This will have a negative contribution to AGB, so that the two contributions might roughly cancel each other. The third term on the rhs of Eq. (A8), due to region J, is probably the most important one when the solvent is water. A detailed examination of one case of this term has been discussed in the third section. Note that the average in the numerator can be written as an average of two factors, namely,

where each factor contains the binding energy of the J region in P and in L, respectively. This average of the product cannot be factored into a product of two averages (as appear in the denomi-

nators). This follows from the definition of region J, i.e., that the solvation of FGs in this region is affected by the binding of P and L (otherwise they would have belonged, by definition, to either region E or I). We can proceed a little further with the analysis of the last term if we assume that the pairs of FGs that can form bridges through water molecules are far apart so that they can be considered independent; i.e., we rewrite the last term on the rhs of Eq.

918

BEN-NAIM, TING, AND JERNIGAN

(A8) as

where the products FIk p,, and TI, I,, are over all the FGs in the P, and L, regions, respectively. Thus, the final sum in Eq. (A10) is over all pairs of FGs k E PtJand I E L, that are correlated in the complex PL. Each quantity cor(k, 1 ) can be positive or negative according to whether the solvation of each of the pairs of FGs becomes weaker or stronger upon binding of P and L. APPENDIX B Estimate of the Range of Configurations Favoring a Double H B in Integrals of Eq. (23)

Consider two FGs (these could be two hydroxyl groups on a polymer or two 0 - H groups on different water molecutes) that are a t a distance of R,, = IR, - R,I = 4.5 A and a t the correct orientations so that they can form two HBs with a third water molecule (Figure 8). Consider integrals of the form

where the integration is over all locations and orientations of molecule 3 / 4 3 )

=

/dR,JdQ,

(B2)

We choose functions f,, and f,, to be very sharply peaked a t some value of R,Q, so that the integral I can be approximated by

I=

1 ~

877

where TJ'is the range of configurations for which the integrand gets most of its contribution. We want to estimate TJ'in comparison with q, the relevant range of configurations for one HB. Let d R , = d3C3 dy3dz,. Clearly, if functions f13 and f,, are very sharp and symmetric with respect to the location of the maximum, the variations in Ax3, Ay;], and AzT3will have the same effect on both functions f I l 3 and f 2 3 ; i.e., deviation, say, in Ax,3that reduces considerably the function f13 will a t the same time reduce the function f2, to the same extent. Therefore, the range of locations of molecule 3 (Figure 8) is the same for integrals of the form of Eqs. (44) and (26). As for the orientations, suppose we choose the following three axes of rotations for molecule 3-the 0-0 line connecting (1) and (2), the 0-0 line connecting (2) and (3), and the x axis [which passes through the center of (3) and perpendicular to the plane of Figure 81. Clearly, rotations about the x axis will have the same effect on f13 and f23. Rotation about the 0-0 axis of (1) and (3) will have no effect on f13, so that in integrals of the form (26) we have the full span of rotations of 277. However, when considering integrals of the form (44),the rotation over this axis will be restricted to a range of angles Aa. We can conclude that TJ'is related to 9 by the approximate equation Aa

TJ= ' TJz

[ f13(max)12/dR, dQ3 Using the estimates of the range of configurations given by Dahl and Andersen,15 we have the

SOLVENT EFFECT ON BINDING THERMODYNAMICS

L

t

R12 = 4 . 5 A

Figure 8. Two water molecules, 1 and 2, that are at an optimal distance and orientations to form two HBs with a third water molecule.

following results: For the strong HB: 2.45 5 R 5 2.95 12.75T / 180 we have strong) q,(strong)

=

A,

Aa

5.27 X 1OP6cm3mol-' 12.75

'

= -

360

=

1.87 x 10-7cm3 mol-'

(B5)

For the weak HB: 2.35 5 R 5 3.085 A and Aa 28T / 180 q(weak)

=

8.56 x lOP4cm3mol-'

$(weak)

=

-9

28 360

=

=

6.66 x 10-5cm3 mol-'

=

(B6)

In the fifth section we have used the value of 77' for the strong HB.

REFERENCES

2. Takeda, Y. Ohlendorf, D. H., Anderson, W. F. & Matthews, B. W. (1983) Science 221, 1020-1026. 3. McGhee, E. D. & von Hippel, P. H. (1974) J . Mol. Biol. 86, 469-489. 4. Sarai, A., Jernigan, R. L., Kim, J. G. & Takeda, Y., submitted for publication. 5. Stryer, L. (1988) Biochemistry, 3rd ed., W. H. Freeman, New York. 6. Ben-Naim, A. (1987) Solvation Thermodynamics, Plenum Press, New York, in press. 7. Flory, P. J. (1969) Statistical Mechanics of Chain Molecules, Wiley-Interscience, New York. 8. Ben-Naim, A. (1980) Hydrophobic Interactions, Plenum Press, New York. 9. Ben-Naim, A., Biopolymers, in press. 10. Hill, T. L. (1955) Statistical Mechanics, McGrawHill, New York. 11. McQuanie, D. A. (1976) Statistical Mechanics, Harper & Row, New York. 12. Ben-Naim, A. (1974) Water and Aqueous Solutions, Plenum Press, New York. 13. Hirschfelder, J. O., Curtiss, C. F. & Bird, R. B. (1954) Molecular Theory of Gases and Liqulds, Wiley, New York. 14. Eisenberg, D. & Kauzmann, W. (1969) The Structure and Properties of Water, Oxford University Press. 15. Dahl, L. W. & Andersen, H. C. (1983) J . Chem. Phys. 78, 1980-1993. 16. Ben-Naim, A. (1989) J . Chem. Phys., 90,7412-7425. 17. Mezei, M. & Ben-Naim, A., to be published. 18. Bernstein, F. C., Koetzle, T. F., Williams, G. J. B., Meyer, E. F., Jr., Brice, M. D., Rodgers, J. R., Kennard, O., Shimanouchi, T. & Tasumi, M. (1977) J . Mol. Biol., 112, 535-542. 19. Ben-Naim, A., Ting, K. L. & Jernigan, R. L., (1989) Biopolymers, 28, 1309- 1325. 20. Drew, H. R., Wing, R. M., Takano, T., Broka, C., Tanaka, S., Itakura, K. & Dickerson, 11. E. (1981) Proc. Natl. Acad. Sci. USA 78, 2179-2183.

1. von Hippel, P. H. (1979) in Biological Regulation

and Development, Goldberg, R. F., Ed., Plenum Press, New York, pp. 279-347.

919

Received November 22, 1988 Accepted April 25, 1989