TERRELL L. HILL tenth that of the hydrolysis. This somewhat complicates calculation of the rate coefficients for the hydrolysis since it becomes desirable (although not essential, since the decarboxylation is relatively slow) to use a rate law for successive firstorder reactions of the type ester
ki kD + acid + hydrocarbon
+ COS
The applicable equation for calculation of kl is
[CONTRIBUTIONFROM
THE
Vol. 78
where a is initial concentration of ester. The value of k D is measured either by following the decarboxylation of the mesitoic acid which remains after the ester has reacted or by separate experiments with the acid. Incidentally, the decarboxylation is acid catalyzed, as is to be expected from the experiments of S c h ~ b e r t a~t ~higher ~ ' ~ acidities, and the rate varies with the HOacidity function. ITHACA. N. Y .
NAVAL MEDICAL RESEARCH INSTITUTE]
On Intermolecular and Intramolecular Interactions between Independent Pairs of Binding Sites in Proteins and Other Molecules BY TERRELL L. HILL RECEIVED JANUARY 28, 1956 The KirkwoodShumaker suggestion, that matching constellations of dissociable groups on two different protein molecules might lead to a significant intermolecular attractive force, is examined in terms of two models. The general conclusion reached is that in many cases the KirkwoodShumaker proposal is a very reasonable one. Two related topics also discussed are: (1)the intermolecular or intramolecular potential of average force between two groups whose charges are not permanent but fluctuate because of binding equilibria with ions in solution; and (2) the effect of pairs of interacting groups on titration curves of proteins and other molecules.
I. Introduction At the end of their paper on the force between two protein molecules, Kirkwood and Shumaker' suggest that ". . . steric matching of a constellation of basic groups on one molecule with a complementary constellation on the other could conceivably produce a redistribution of protons leading to a strong and specific attraction . . ..' ' The primary purpose of this paper is to examine this suggestion quantitatively for perhaps the simplest possible model (described in Section 11). Specifically, we assume that matching constellations on two neighboring protein molecules exist, and then calculate the extent to which the protons would, in fact, take advantage of these matching constellations by redistributing themselves in such a way as to produce an attractive force. The most extreme situation would arise when the protons are frozen in the particular distribution corresponding to the maximum possible attractive force between the protein molecules. Roughly speaking, such redistributions of protons would be favored as far as the energy of the systems is concerned, but opposed by configurational entropy considerations. The outcome of this energyentropy competition is not particularly obvious in advance. The treatment to be given applies also to intramolecular interactions and therefore to titration curves of proteins and other molecules. The results suggest that the usual discussion of electrostatic effects on titration curves of proteins in terms only of the net charge of the entire molecule may overlook important local electrostatic interactions. Some of the equations derived below are new only2 in their manner of derivation (via the grand partition function) and in their particular application to the problem outlined above. (1) J. G. Kirkwood a n d J. B. Shumaker, Proc. Nal. Acad. Sci., 98, 863 (1952) (2) See, for example. E. J. Cohn and J. T.Edsall, "Proteins, Amino Acids and Peptides," Reinhold I'ubl. Corl]., New York, PIT.I' , 1913.
11. The Model and General Relations Figure l a represents two protein or other molecules close together and with matching constellations of two types of sites or groups, 1 and 2 . Both sites of a pair are capable of binding the same kind of ion (or molecule), for example, a proton. The distance between each pair of sites is Y. The effects we are interested in here arise only when r is rather small (see Section 111); hence, as a first approximation, we neglect (1) interactions between sites on the same protein molecule, (2) second and higher neighbor interactions between sites on different protein molecules, and (3) interactions between the sites shown and other types of sites. Thus we are concerned here with a group of independent pairs of binding sites. With this model the discussion will apply also to Fig. l b , in which several equivalent but independent pairs of sites are distributed through a single large molecule. Because of the assumed independence of pairs, the average properties per pair will not depend on the number of pairs under consideration; hence this number need not be specified. For example, Figs. IC and Id, each with only one pair of sites, are included. The most straightforward way to derive the required equations is to consider each pair of sites as a system in a grand ensemble. The ensemble consists of a very large number of systems (pairs). Let jl be the partition function (including the binding energy) of a molecule or ion bound a t site 1 of a pair, and similarly let j 2 refer to a molecule or ion bound a t site 2. Let WAA(Y), WAB(Y), WBA(Y) and l v B B ( r ) be the free energies3 of interaction between the sites, separated by a distance r, when both sites are occupied (AA), when site 1is occupied and site 2 unoccupied (AB), etc., respectively. That is, the (:I) T h e corresponding energies are T V A A  T ! b l i ~ . k . d O T ) ,etc
INTERMEDIATE AND INTRAMOLECULAR INTERACTIONS
July 20, 1956
3331
w's are the potentials of average force4 for the pair of sites occupied in each of the four ways possible; the potential zero is chosen at r = 03 in each case. Then the canonical ensemble partition functions Qn (n is the number of molecules or ions bound on a pair) are Qa =
eWm/kT
+jZeWBA/kT
Q1 = j , e  W A B l k T
(1)
Qz = j l j 2 e   W A A I k T
and the grand partition function is E = Qo QiX QaX2 where
+
+
(2)
= edkT
(3)
and I( is the chemical potential of the bound molecules or ions. If c is the concentration (or activity) of these molecules or ions in solution, then p = psoln = p*
+ kT In c
(4)
where p* is a standard (c = 1) or reference chemical potential, independent of c. From wellknown properties of the grand partition f ~ n c t i o nthe , ~ numbers n A A of AA pairs, G A B of AB pairs, etc., out of a total of, say, M pairs in the ensemble ( M + ), are
(d) Fig. l.(a) Matching pairs of interacting sites on two large molecules; (b) pairs of interacting sites in the same large molecule; (c) pair of interacting sites on two small molecules; (d) pair of interacting sites in the same small molecule.
constants for the two sites as
0)
The probability cupied is e,
=
nA.4
el that a given site of type 1 is oc
+
n4B
+
= jlkwAB"T
e2 =
+
n ~ 4 = j2hwB41k1
+ QA2
For an AA pair, the interaction force is fAA = and similarly for other pairs. The average force f between a pair of sites (&., averaged over the four different manners of occupation) is then fAAnAA
f
fABnAR
f
fBAnBA
02
elo
jh'
+
(11)
(12)
jh
e p*/k T T , Kz =
K 1 =
31
bWAA/br,
s=
1 Kz = ~ ( ezO)
where the superscripts refer to r = m ( W S= 0). From eq. 6 and 7 el0 1 1  ea0  1 1 =
(6)
(7) M E The fraction of all sites occupied, 0, is given by6 ~~
el"),
el
H
and, for a type 2 site n4.4
41 
Hence, using eq. 4, eq. 11 becomes Q2X2
2
JLf
K1 =
fBBnBB
(9) M It is easy to verify by differentiation that the potential @of this averageforce7 (i.e.,f =   d @ / d 7 ) is
(4) The "average" here is over possible configurations of solvent molecules, electrolyte, etc.. for a fixed manner of occupation of the pair of sites. We encounter the further average over manners of occupation in eq. 10. ( 5 ) See, for example, T. L. Hill, "Statistical Mechanics," McGrawHill Book Co., New York, N. Y . , 1956. (6) Compare E a / G in eq. 67 of T. L.Hill, J . C h e m . P h y s . , 28, 623 (1955). (7) Compare eq. 2 of Kirkwood and Shumaker,' and eq. 87 of Hil1.a See also eq. 37.69 of €IIII,~with r for E .

(13)
32
and
111. Redistribution of Bound Ions Special Case I.In this special case the unoccupied (B) sites are equivalent ( j , = j 2 j, K1 = Kz = K ) and have a charge of  1. The ion being bound has a charge of f2. This situation might arise, for example, in the binding of a doubly charged metallic ion on negatively charged groups (say, COO) of a protein. Hence we have the symmetrical situation in Fig. 2 , with B
B
A
B
B
Fig. 2.Pairs
A
A
 +
+ 
 
A
+ +
of charges in Special Case I.
WAB = WBA =
with the potential zero a t 7 = m . is also the isothermal, reversible work, a t constant p , required to up to a separation r . bring a pair of sites from r = For use below, let us define intrinsic dissociation
eP*/kT

WAA =

WBB< 0
(15)
When r is small (WAR large), the bound ions will tend to redistribute themselves to favor AB or BA over AA or BB pairs. To discuss this tendency quantitatively, let us define a correlation coefficient (or order parameter ) by the relation nAB
+
nBA
 %AA  n B B
(16) M That is, each AB or BA pair contributes +1 and each A4 or BB pair contributes  1; El is the average contribution per pair. With perfect positive correlation or order (41 = +1) all pairs are AB or BA and we have the maximum possible attractive €1
=
3332
TERRELL L. HILL
force between the matching constellations in Fig. la. In fact, from eq. 9 and 15
I .o
(a) 8°=0.9901,x~100 (b)8O=O.9,x.9 (c)B0=0.75,x=3
O'*
Thus, the attractive force f falls off with increasing r not only because f A B falls off with r but also because of &. Since f A B is negative, f and 51 have opposite signs. We obtain from Section I1 and eq. 15 and 16
o,2

Note that x is defined in such a way that 0 = when x = 1. At r = 03 (WAB= 0 ) , these equations reduce to X 
80
l + x
Equation 18 and 20 can be used to calculate (1 and 8 as functions of r or WABa t constant c . This corresponds physically to varying the distance between the two protein surfaces in Fig. l a in the presence of a fixed concentration of ions in solution. On the other hand, 0 as a function of x a t constant Y or W A Bwould correspond to the binding isotherm for Figs. l b and Id (see Section V). Figures 3 and 4 give 4'1 and 8, respectively, as functions of WAB/kT for several choices of h: (which
0.2 
0.4

0.2

04

o,6
3
2 WAB/kTd r+
0
I
( r = 4
Fig. 4.Fraction of sites occupied 0 versus interaction free energy w A B / k T in Special Case 1.
Also
0.0
(f)eo=o.i,x=o.iii (g)8"= 0.0099,x.0.01 4
where
(1.
Vol. 78
is proportional to G) or 8O(x). These figures show that .$ approaches +1 for moderate values of  wAR/kT, and that this is accomplished by an adsorption or desorption of ions to give 8 ' / z together with a redistribution of the ions which are bound to favor AB and BA over AA and BB pairs. T o examine the dependence of 51 on r, we write

where DE is defined by this equation and is therefore an "effective" dielectric constant. In Fig. 3 , we see that, for values of x which are not too extreme,  W A B / ~ T > 3 is a rough criterion for 4, near f l . At 300°K., this is equivalent to D E r (in A.) < 185.7. For example, if Y = 5 A.,we DE < 46. These inhave DE < 37, or if r ='4 equalities could very well be satisfied in Fig. la, because of dielectric saturation, despite the possible presence of electrolyte ions; they are almost certainly satisfied in Figs. l b and Id, because in this case the medium between the charges has a relatively low dielectric constant.* We can conclude that, in this example, the KirkwoodShumaker proposal that matching constellations (as in Fig. la) might lead to an essentially "frozen" optimal bound ion distribution is certainly a reasonable one. Special Case 11.In this case unoccupied site 1 has zero charge (e.g., NH2 or N=), unoccupied site 2 has a charge 1 (e.g., COO or O), j 1 # j 2 (or K1 # Kz) in general, and the ion being bound 1 (e.g., H+). The charge distribuhas a charge tions are shown in Fig. 5 , with
A.,
+
WAB< 0 WBB = WBA= WA.A= 0
( a ) eo= 0 . 5 ;X = I
Here we define a correlation coefficient & by
(b) 8': 0.25,0.75; x = 0.333.3 ( c ) e 0 = 0 . 1 , 0 . 9 ; ~o= 111.9 (d)B'= 0.0099, 0.9901; X : 0.01,100
€2
1 4
3
2
WAg/kTo
I
1 B
2 B
0

0 (r =4
coefficient E1 wersus interaction free W A B / k T in Special Case I.
(25)
nAB/Jlf
That is, each AB pair contributes +1 and all other pairs make no contribution; i2is the average con
Fig. 6.Pairs
ro
Fig. 3.Correlation energy
(24)
(8)
T. G.
(1938).
1 A
2 B
+ 
1 B 0
2 A 0
1 A
+
2 A 0
of charges in Special Case 11.
Kirkwood and F. H. Westheimer,
J. C h e m .
P h y s . , 6 , JOG
July 20, 1956
INTERMEDIATE AND
3333
INTRAMOLECULAR INTERACTIONS
tribution per pair. I n the limit Y + 0, all pairs will be of the AB type, (2 = 1, and we will have the maximum possible attractive force between the matching constellations of Fig. la. From eq. 9 and 24
tzis always positive and f always negative. From Section I1 and eqs. 24 and 25
K,/K,=100 x = 1/100
where
cI
and
\
Figures 6 , 7 and 8 show tz,el, ez and t9 as functions of W A B I k T . Thengeneral tendency as Y c 0 (  W A B / k T + m) i s f o r & + 1 and&+Osothat AB (or ) pairs may be formed (tZ+ 1). Figure 6 (&I& = 100) represents a case of the type site 1 = NH2 or N= and site 2 = COO. Figure 8 (&I& = 100) might apply to site 1 = N= and site 2 = 0 or 4. Figure 7 (K1 = K2) is intermediate; for example, site 1 = NHz and site 2 =
+
06   3
0.4 2
0.2
 I
S
.
'\
0.0   4
0.6   3
0.4 2
0.2 I I
, .___ ......._....
.......
~
Q
1.0
'
\
'\ 0.0   4
\
0.6   3
0.4 2
0.2
re
Fig. 6.Special Case I1 with K2/KI = 100. The inner ordinate refers t o m / k T : (a) x = 1; (b) x = 1/10; the dotted curves and (e), (e,), (e,) refer, however, to x = 10; (c) x = 1/100.
I I
0
6
4 WAB/kT+
2
0 (r ==)
I+
Fig. 7.Special Case I1 with K I = Kz. The inner ordinate refers t o m / k T : (a) x = 1; ( b ) x = 1/10.
TERRELL L. HILL
3334
ions in solution. Some aspects of this subject have been discussed recently by Kirkwood and Shumaker’ and by The appropriate (intermolecular or intramolecular) potential of this latter type for the present model is given by eq. 10. We discuss the potential here instead of the force since this avoids any definite commitment concerning the dependence of D Eon r in eq. 23. Special Case 1.Equations 10 and 15 give
1.0
C
0.8 0.8
Vol. 78
h 4

e  m i j ! k T=
+ (1 f +
2xe”ABikT
il
52)f3WABij!kT
x)2
(331
m
06L3
m / k T is plotted against WAB,’kTin Fig. 9. has a minimum where E1 changes sign in Fig. 3, since is an integral off, a t constant p , in eq. 17. Also, the slope in Fig. 9 approaches  1as + 1in Fig. 3. After the slope in Fig. 9 has essentially attained the value 1 (as r + 0), the residual constant difference between WABand % is an accumulated (ie,on integrating fdr from to r ) reflection of the lower values of the correlation coefficient a t large r .
I
\
+
\
B
I
8
4
6
(a) x = I (b) x =0.333,3
\
\V
WAB/hTT9
2
10 4
(I
u/kT
r ‘4
Fig. 8 Specia ]Case I1 with K I / K *= 100. The inner ordinatereferstom/kT: (a) x = 1; (b) x = 1/10; (c) x = 1/100
The magnitude of  TVAB/kT necessary to give, say, tz> 0.9 varies rather widely depending on the particular choice of parameters. Table I lists the range in D Eassociated with difference minimum val;es of  WAB/kT, for T = 300’K. and r = 4 or 5 A. The KirkwoodShumaker suggestion of a “frozen distribution” is seen to be reasonable in some cases but not in others. However, in Figs. l b and Id, values of D E < 14 (see Table I) might be anticipated in general so that practically complete “freezing” would probably be the rule rather than the exception for r of the order of 5 A. or less. Incidentally, the curves in Figs. 6, 7 and 8 for x < 1 also apply, because of symmetry, to x > 1. That is Edl/x1 e,(x) = 1  el ( 1 l X ) O , ( x ) = 1  e,( l / x ) 6 ( ~= ) 1  e(l/x) &(XI
I
5
4
I
3
I
2
WAB/kT + r 4
I
I
0
(r4
Fig. 9.Potential of average force m / k T w r s m interaction free energy WaB/kT in Special Case 1.
Special Case 11.We
obtain from eq. 10 and 24
=
(32)
This is illustrated by the dotted curves in Fig. 6b. IV. Potential of Average Force The potential (of average force) between two permanent charges, as, for example, in eq. 28, is a familiar concept. Much less familiar but equally important in biology nucleic acids, etc.) . (proteins, . is ihe potential of average force between two groups whose charges are not permanent but fluctuate because of binding (or dissociation) equilibria with
where Z is given by eq. 30. %/kT is plotted against WAB/kT as the dashed curves in Figs. 6, 7 and 8. There is exhibited a wide range in the in these figures, as expected from magnitude of the connection between and $2 (eq. 26).
V. Titration Curves Eauation 8 is also of interest as a titration curve (i.e.,LOas a function of X or c a t constant r ) , when intramolecular interactions are under consideration (for example, Figs. l b and Id).
INTERMEDIATE AND INTRAMOLECULAR INTERACTIONS
July 20, 1956

3335
being a t x 1) when K ( l ) / K ( z> ) 16. There is only one inflection point (at x = 1) when K(1)/K(2)6 16. Theslopeatx = 1is
More explicitly, eq. 8 appears as 28 =
(35)
Alternatively we have the equivalent thermodynamic expression which follows2 from the conventional definition of successive dissociation constants K ( I )and K(2, 2c2
C
or
The latter relation might be useful in estimating
K ( I ) / K (from ~ ) an experimental curve when K ( I ) / K ( ~ ) is not large. Note that 0 6 s 6 ' / z . Ify>>l Let us again (see eq. 19 and 31) define x so that 6 = l / z when x = 1. Here (37)
and
(47)
and site 1fills (predominantly) first as x is gradually increased from x = 0. If y > K2 (see eq. 42 and Table I1 below). To put eq. 35 in the same form as eq. 38, we mulThe complete titration curve in the form 8 versus tiply numerator and denominator by eWBBIkT. On In x is determined by Kc1)/K(z,only, according to equating the terms in X2 and x 2 , using eq. 4 and 13, eq. 38. Hence, from eq. 40, there are an infinite we find number of combinations of KJK2 and the W's which will lead to the same titration curve. As an K(1)K(2) = KIKne(wrAWBB)IkT (39) illustration, consider Special Case I1 of Section IV, I n eq. 19 and 31, WAA= WBB,and hence K(1)Kt2, where = K1K2. Also, from the terms in X and x 2W = WAB
where and
Incidentally, in applications one will in general have a set of "conservation" equations
+ WSB WBA = WBB + WBX U'BB + WBX + WSB + W s s "AB
W.4.4 =
=
~
B
B
where WXXis the potential between two bound ions (or molecules) a t a distance Y apart, WXBis the potential between an ion a t site l and unoccupied site 2 , etc. Then 2w = Wxx
(44)
Ordinarily, of course, WXX 3 0. In this case, since y yl 3 2 , K(I)/K(2)3 4. However, if W X X< 0 (e.g., van der Waals attraction between < 4. uncharged molecules), we can have K(1,1K(2) The complete range possible is 0 6 K(1,/K(Z)6 03. By differentiation of eq. 38 i t is easy to show that there are three (symmetrically distributed) inflection points in a e versus In x curve (the central one
+
The same titration curve (log K(1)/K(2)= 4.978) is obtained, for example, in each of the cases shown in Table 11. Only in the first case in Table 11does site 2 fill first. Results of this type emphasize again the well recognized fact that the mutual interaction of close pairs of sites can modify profoundly the titration behavior which may be expected from the intrinsic constants alone. I n some cases, because of such complications, association of a particular dissociable group with a given inflection point on a titration curve may be quite hazardous. Also, the usual treatment of electrostatic effects in large molecules in terms of the net charge of the entire molecule ignores possible local interactions of this sort. This might be particularly serious in wellorganized molecules (e.g., DNA) where many equivalent and hence reinforcing pairs might be present. TABLE I LIMITING EFFECTIVE DIELECTRICCONSTANTS AT T = 300'K. r  4 A .
WAR ~ > 2 kT 4 6 8
7 = 5 A .
D E < 70
D E < 56
35 23 17
28 19 14
VOl. 7s
TERRELL L. HILL
3336
Symmetrical Dicarboxylic Acid :
TABLE I1
SETSOF PARAMETERS GIVINGTHE SAME TITRATION CURVE
 (WdkT)
log ( K l / K d
4.978 4.978 2.000 2.000 0.000
4eW~~!kT
(51)
Symmetrical Diamine :
8,000 3.428 5.727
__ K(l) = &W**/kT K(z)
Finally, several additional special cases of eq. 40 should be noted Special Case I of Section IV: K(z)
~
K(?)
0.000 0.000
!h = 4 e  4 W ~ ~ ! k T
K(l)
(50)
(52)
The latter two examples are of course. The author is indebted to Dr. Sidney Bernhard for a number of stimulating discussions. BETHESDA, MARYLAND