Jon Applequist Deoartment of Biochemistry and Biophysics low; state ~nivirsity Ames, Iowa 5001I
I Cooperative Ligand Binding to Linear I Chain Molecules
Suppose a linear chain molecule C has N sites to which a ligand X may bind in solution. The binding isotherm may be presented as a graph of the fraction f of sites occupied versus In(X), the logarithm of the activity of X (Fig. 1). If the sigmoidal curve so obtained is sharper than is observed for ordinary ligand hinding to molecules having only a single binding site, the binding is regarded as cooperative. If the curve is ahnormally broad, the binding is regarded as anticooperative. (The terms "positive cwperativity" and "negative cooperativity" are sometimes applied, respectively, to these two cases.) Cooperative binding results when a bound ligand increases the affinity of adjacent sites for the ligand. Anticwperative binding results from the opposite effect. If the interactions between sites are limited to nearest neighbors on the linear chain. then the bindine isotherm lends itself readilv to a quantitative interpretation. The mathematical prohlem can be reduced to that of the one-dimensional kine lattice ( I ). which was first solved in connection with the theory of f e k magnetism. The purpose of this article is to summarize the problem as it applies to cooperative binding to long chain molecules and to present some illustrations which help to visualize the connection between the interaction parameters and the shape of the binding isotherm. Let ko be the equilibrium constant for binding of X t o a site on C adjacent to unoccupied sites. Let kl he the equilibrium constant for binding to a site adjacent to one occupied site. Then k, = koe-'lRT
(1)
1.0
t
05
0 -
2
o
2
4
6
IWX)+ hk.
Fig.s
Bindng
la s , ~ ~~ .. , t ~that
8 1 0 1 2
-
tor (eqn, (g,), me nunber on each abscissa at me midpoim. mls isequal to me value
Equation (5) expresses P(n,v) in the notation used earlier (2) in the aoolication of the Isine model toconformational eouilibria i n h e a r chain molecul&. A knowledge of P(n,u) enahes one to calculate various averages over the species present. For example, the average number of occupied sites ( n ) is
where the summation is over all states (n,v). Defining the fraction of occu~iedsites f as ( n ) l N and calculatine this quantity by means of eqn. is), we find after some manipulation (2)that f takes the followinr! - relatively . simple . form in the limit of verylong chains
where t is the free energy of interaction between adjacent occupied sites. Consider the reaction where CX,, represents any of the possible species with n bound ligands arranged in v uninterrupted sequences. The equilibrium constant for reaction (2) is the product of the constants for the elementary steps
where g(n,u) is the number of ways of arranging n ligands in v sequences on a linear array of N sites. The form of g(n,v) plays a crucial role in determining the shape of the binding isotherm, as will be shown below. This function may be calculated by standard combinatorial methods ( I , 2), and is found to be (n - l)!(N - n + I)! (4) g(n8Y)= (n - l)!(n - v)!v!(N - n - u + l)!
In Figure 1,f is plotted as a function of l n ( ~ ) +Inko, which is equal to ins dRT,according to eqn. (6). The value of dRT (= Inn) is given on each curve. For r < 0 the binding is cooperative; for t = 0 it is noncooperative; for r > 0 it is anticooperative, and for large t the isotherm is seen to take on a double sigmoid shape. The significance of this will be discussed below. From eqn. (9) it is seen that the midpoint of the isotherm, f = #,occurs a t s = 1.From eqn. (6) we thus have the following relationship
+
+
(10) In(X), = -Ink0 dRT where (X), is the ligand activity when f = %. Hence the midpoint is shifted from -Inko toward smaller (X) in the cooperative case and toward larger (X) in the anticooperative case. The slope of the isotherm a t f = %isfound from eqn. (9) to be
The relative population P(n,u) of species of CX,, may be exoressed as follows P(n,u) = 0 = g(n,u)s"a" (C)
(5)
where, according to eqns. (1) and (3) s = (X)koe-*mT
(6)
= .dRT
(7)
and
Thus c can he determined from the observed slope. From eqn. (10) one can then determine ko from the observed (X),. We can also calculate an average number of ligands (1) in an uninterrupted bound sequence, defined as ( 1 ) = (n)l(v), where (v) is the average number of sequences as calculated by a formula analogous to eqn. (8). I t has been shown (2) that for very long chains at f = 'h the result is simply
+
( I ) = 1 e-e/*T
(12)
Volume 54. Number 7. July 1977 1 417
Thus, for the cooperative case the bound ligands tend to aggregate into long sequences, while for the anticooperativecase they tend to disperse into short sequences. The above results were obtained from a knowledge of the relative populations of species, P(n,v). Let us now reconsider this function to see how its behavior is related to the ohservable properties. From eqn. (5) we have InP(n,u) =In&,")
+ nlns + ulno
(13)
The Ing(n,u) portion of this function is represented as a surface in Figure 2. This figure was generated by a computer,
oarticular interest to note that. since Ins varies linearlv with in(x), an increase in (X) corrdsponds to an increase i n the done of this nlane in the n direction. As a result. the ~ e a in k l & t , u ) shifts toward larger n, as one might expeci for an increase in (XL . . Computer-generated surfaces representing lnP(n,u) are shown in Fieures 3-5 for three cases. The range of s covered in each case"is the same. By following the max&um point on the surface we can get a eood idea of the average behavior of all of the molecule~beca&efor long chains theaverage state is verv close to the most DoDulated state. In the cooperative case i ~ i ~ u 3) r eeach surface may be visualized as the sum of the surface in Figure 2 and a plane which has a strongly negative slope in the u direction: thus shifting the maximum toward a small value of u. This means that bonded sequences tend to be long in the cooperative case. The maximum shifts from the entirely unbound state at Ins = -0.9 to the nearly fully bound state at Ins = 0.9. The
Figve 2. Surfacerepresanunglng(n,u). Rr, r a n g of n is fmm 1 to N. The r a n g of u is from 1 to N/2. The biangular 'Wings" representthe zero plane. and are &Ide me ran- of possible states.
using eqn. (4) for the case N = 50. The general features of this surface have been described earlier (2). It was found that the overall appearance is insensitive to N when N is large. The peak of the surface occurs at that state (n,") whase population has the greatest contribution from the multiplicity of ways in which the state can be realized. The surfacelies entirelv within a triangular area of the n-v plane containing all of the possible states. According to eqn. (13), lnP(n,u) is-the sum of the function represented by Figure 2 and the function nlns + ulna. The latter function is simply a plane passing through the origin with a slope of Ins in the n direction and a slope of In0 in the v direction. The peak of the population surface then hecomes shifted toward the "uphill" region of this plane. It is of
Figure 3. Population surfaces f w a cooperative case. elRT = -2
418 1 Journal of Chemlcal Education
Figure 4. Population surfaces far Ibnonuroperalive case. e/RT = 0 .
Figure 5. Population surfaces far an antimperative case. e/RT = 2 .
sharpness of the biding isotherm, a characteristic of this case, is thus seen to be due to the small curvature of the surface in the vicinitv of the neak. In the n&coopeiative case (Fig. 4) the planar portion of the InP(n.v) function has zero slooe in the v direction. so that variatio& in s correspond to a rotation of this plane about the v axis. The curvature of the surface near the oeak is ereater and, accordingly, the shift in the degree of binding at t i e peak is less than in the cooperativecase. Also, the peak has shifted toward larger u values, i.e., toward shorter sequences. These trends are continued in the antimperative case (Fig. 5). where the maximum is strongly peaked and correspondinelv little shifted on variation of s. The double-arched structure a t the boundary of the lng(n,v) surface (Fig. 2) has a notable effect on the anticooperative case, because the mnximum population is located close to this boundary. The doublesipoid binding isotherm in Figure 1 for thecase tIRT = 4 is evidently a consequence of this simcture. The curve has the appearance of an isotherm for binding to a mixture of two types of independent sites with widely differing hinding constants. This is, in effect, the case when highly anticooperative interactions are nresent. When f < K . lieands do not occupy adjacent sites, a;ld only alterna'te sites with hinding constant ko hecome filled. When f > H, all available sites are adjacent to two occupied sites, and the hinding constant is k&'IRT. Thus, in Figure 1,the two points of maximum slope in the double sigmoid curve are separated by a distance of ao~roximatelv2cIRT on the abscissa. It has been shown (3) tL8t the hin&ng isotherm starts to pass from a single t i double sigmoid shape when a becomes greater than three.
-
I will close by referring the reader to several published applications of the Ising model to the hinding of ligands to linear chain molecules. The anticooperative hinding of ions to polyelectrolyte chains was treated by Marcus (4) and by Harris and Rice (5).Hi11 (6)derived the hindine isotherm in a form different from that given here and obtaiied eqn. (11) as well. Huane and Ts'o 17) and later Davies and Davidson 18) applied ~ill'sformula the cooperative binding of poly&: cleotides and their comolementarv monomers. Schneider e t al. (9)gave an alternative development of the theory and applied it to the cooperative amylose-iodine-iodide system. Damle (10) treated the hinding of oligonucleotides to polynucleotides in amoregeneral way, including the Isingmodel as a special case. Schwarz et al. (11) have applied the model to the cooperative binding of dyes to poly(g1utamic acid). Acknowledgment This investigation was supported by a research grant from the National Institute of General Medical Sciences G M 13684). Literature Clted (11 Iaing.E.,Z. Physik. 31.253(1925). (2) Applequid.J., J.Cham. Phya.. 38.934 (1963). 13) Aooleoui%t.J.. J Chem. Phv*..49.3327 119681
a
Volume 54, Number 7, July 1977 / 419
