1.9.1. INDEPENDENT BINDING SITES
The binding of a substrate (or other ligand) to an enzyme site, represented by the simple interaction of enzyme and substrate, E + S ES, is given by the following equation, assuming that all the sites bind the substrate independently with a disso- ciation constant equal to Kd:
Kd = (Ef)(Sf)/ES = (n – r)E(Sf)/rE = (n r)(Sf)/r (1.1) The terms Ef and Sf refer to the concentrations of unbound binding sites (not free E) and free ligand, respectively. For an enzyme containingn identical binding sites, as found in many oligomeric proteins, we can substitute (n – r) E for Ef and rE for ES, in which r is the average number of bound ligands per enzyme molecule at any particular substrate concentration and E is the enzyme concentration.
Rearrangement of the equation leads to the following Scatchard equation:
r/(Sf) = n/Kd – r/Kd (1.2)
A plot of r/(Sf) vs. r gives a straight line, as shown in the Scatchard plot (Figure 1.15), with the slope being equal to –1/Kd and the intercept on the abscissa being equal to n, the number of binding sites for the substrate or ligand. The Scatchard plot provides an excellent method for analyzing the binding interactions of any two components, provided that all the sites act independently and have the same disso- ciation constant.
Binding analyses using the Scatchard plot require determination of the enzyme concentration and the amount of bound and free ligand. This can be relatively difficult as both the concentration of free and bound ligand in the same solution must be determined under conditions in which a thermodynamic equilibrium is maintained.
One method involves equilibrium dialysis, in which the smaller binding partner (substrate) transfers freely through a dialysis membrane with the larger component (enzyme) remaining on one side of the membrane. The amount of ligand can now readily be measured on both sides of the membrane by using radiolabeled substrate or by deducing the absorption, fluorescence, or some other property of the ligand that is directly related to its concentration. The ligand concentration in the presence of the enzyme is equal to the sum of the concentrations of the free and bound ligand, whereas on the other side of the membrane, the ligand concentration corresponds only to that for the free ligand. Provided that the concentrations on the two sides of the membrane are significantly different, the concentration of the bound ligand can be determined and r calculated by dividing by the enzyme concentration. Measure- ments are then made at different substrate concentrations, and the values of r/(Sf) vs.r plotted over the appropriate range. Other methods exist, including gel filtration and centrifugation, which also partially separate free and bound ligands but still allow thermodynamic equilibrium to be maintained between the enzyme and sub-
strate. These latter methods are also particularly advantageous for studying the interactions of large molecules (e.g., for enzyme–enzyme interaction), as the Scat- chard equation can be applied to study the interaction of any two molecules.
There are also convenient methods to measure the fraction (y = r/n) of bound ligands. One of the simplest methods is to follow any perturbation of the spectral (absorption, fluorescence) or other properties of the enzyme or ligand on interaction.
In most instances in which a spectral property is altered on interaction of the two binding molecules, a plot of the signal change can be used to obtain the dissociation constant. The signal change (Dx) will be proportional to the amount of bound ligand (ES), with the maximum change (Dxmax) corresponding to the total concentration of enzyme binding sites (nE). The term Dx/Dxmax only gives the fraction of protein with bound ligand, (y = r/n), and thus it is not possible to determine n without independent data. Substitution of the bound and total sites by Dx and Dxmax, respectively, then gives a plot of Dx vs. Dx/(Sf) with a slope equal to –Kd. If the bound amount is much lower than the free ligand concentration at all points, as occurs in many experimental sets, then the total substrate concentration (S) can be used instead of Sf.
Kd = (Ef)(Sf)/ES = (nE – ES)(Sf)/ES
FIGURE 1.15 Scatchard plot for enzymes with identical binding sites for a ligand, in which the sites are independent or exhibit negative or positive cooperativity. The moles of a ligand bound per enzyme molecule (r) divided by the free ligand concentration (Sfree) is plotted vs. r.
r Sfree
r Sfree
r Sfree
r r r
n slope = –1 / Kd
ES = nE – Kd(ES)/(Sf)
y = ES/nE = r/n = Sf/(Kd+ Sf) = Dx/Dxmax
Dx = Dxmax – Kd(Dx)/(Sf) (1.3) 1.9.2 ALLOSTERIC BEHAVIOR — HOMOTROPIC INTERACTIONS
If binding at one site on the oligomeric enzyme affects the binding at another site, then the Scatchard plot will not be linear (Figure 1.15) and a plot of the fraction of bound ligand (y) vs. S will not give a hyperbolic plot (as shown in Figure 1.16) for enzymes with independent sites. These interactions between ligand binding sites may either increase or decrease the apparent binding of a ligand to a site, leading to positive or negative cooperativity, respectively.
This type of interaction has been called allosteric behavior to distinguish this effect from the interactions of two ligands at the same site. The interactions of identical ligands binding at two identical sites are called homotropic interactions, whereas heterotropic interactions refer to the interactions of different ligands binding at two different sites. For the allosteric enzyme ATCase in Figure 1.9, heterotropic interactions occur between the regulatory sites binding the CTP inhibitor and the catalytic sites binding the substrates, whereas homotropic interactions occur between the substrate-binding sites on the catalytic subunits.
FIGURE 1.16 Plot of fractional saturation (y) vs. S (top) and the Hill plot (bottom) for enzymes with multiple sites for a ligand, in which the sites are independent or exhibit positive or negative cooperativity.
Positive cooperativity arises when the binding of the first molecule to an enzyme enhances the interaction with the second, with subsequent molecules effectively causing an apparent decrease in the dissociation constant for that ligand. Figure 1.16 shows that a plot of y vs. S for positive cooperative binding gives a sigmoid-shaped curve and that a plot of the bound vs. the free ligand in the Scatchard plot in Figure 1.15 will initially rise to a maximum and then decrease as the sites become saturated.
There are two main models to account for positive cooperativity. The Monod–Wyman–Changeux (MWC) model of allosteric behavior proposed that there are two states of the oligomeric enzyme with different affinities for the ligand. As the initial molecules will primarily bind to the sites in the high-affinity state, the equilibrium between the low-affinity and high-affinity states will be shifted to the state with high affinity, in effect creating more high-affinity sites even though some of the sites will be filled. This increase in high-affinity sites results in an increase in the apparent affinity of the ligand for the enzyme, resulting in positive cooperative behavior. An alternative model, often referred to as the sequential or Koshland–Nem- ethy–Filmer (KNF) model, proposes that after the binding of the first ligand, the actual affinity at the second and subsequent sites increases, presumably due to a conformational change. The MWC model is both simple and elegant, and the number of parameters used to fit the data is relatively few. The KNF model uses different dissociation constants for each site on the enzyme and may give a closer fit of the data to the model simply because of the higher number of variable parameters.
The KNF model, however, is advantageous for understanding negative cooper- ativity, as the MWC model does not provide an explanation for this effect. Figure 1.16 (top) shows that for a negative cooperative enzyme, in a plot of y vs. S, the relative binding decreases at higher ligand concentrations compared with the hyper- bolic binding curve expected for an enzyme with independent binding sites. Simi- larly, the Scatchard plot for an enzyme exhibiting negative cooperative behavior between binding sites (Figure 1.15) will curve, with the slope decreasing as the enzyme becomes saturated. It is important to note, however, that this type of behavior is not only consistent with negative cooperativity but also with simple heterogeneity in enzyme preparations for ligand binding arising either from heterogeneous enzyme molecules or from heterogeneous sites within the same enzyme molecule or both.
Indeed, this latter cause may be the reason for some of the examples of enzymes with negative cooperative behavior in the literature.
The relative degree of cooperativity between sites can be obtained by using the Hill equation. The value n in the Hill equation corresponds to a hypothetical number of bound molecules needed to be bound to the enzyme to give the degree of cooperativity for full occupation of sites in any molecule (E +nS ESn). Conse- quently, Kd would be given by the equation.
Kd = (Ef)(S)n/(ESn) (1.4) Rearrangement and replacement of the dissociation constant by the affinity constant (Ka = 1/Kd) and substitution by y, the fraction of bound enzyme (ESn = yE;
Ef = (1 – y)E), gives the ratio of the bound enzyme to the free enzyme.
ESn/Ef = y/(1 – y) = (S)n/Kd = (S)n(Ka)
log (y/(1 – y)) = log Ka + n log S (1.5) A plot of the log of bound/free enzyme [log (y/(1 – y)) vs. log S] gives a slope corresponding to n. The substrate concentration given in the equation is generally the total substrate concentration, as most if not all analyses are conducted under conditions in which the amount of bound substrate is very low relative to the total substrate concentration. As binding is not completely cooperative, the plot generally curves. However, a relatively linear region can be obtained using data from the central region of the binding curve (from y = 0.1 to 0.9). Values of n will be above or below unity depending on whether there are positive or negative cooperative interactions between ligand binding sites, as shown in Figure 1.16 (bottom).
1.9.3 ALLOSTERIC INTERACTIONS BETWEEN TWO DIFFERENT
LIGANDS — HETEROTROPIC INTERACTIONS
The interaction of one type of ligand at one site of an enzyme can affect the binding of a different ligand at another site. These interactions are referred to as heterotropic interactions, in contrast to homotropic interactions involving binding at only one type of site. Consider the most common cases (ordered and random binding), in which the enzyme interacts with two different ligands, A and B, but a reaction does not take place and an equilibrium can therefore be established. The apparent dissociation constant for A, Ka*, is equal to the concentration of A times the sum of the concentration of all species without bound A divided by the concentration of all species with bound A, as indicated in the following equations for each set of pathways:
(1.6)
Ka* = (Ef)(A)/(EA + EAB) = (Ef)(A)/(EA)(1 + B/Kb¢) = Ka/(1 + B/Kb¢)
(1.7)
Ka* = (Ef + EB)(A)/(EAB) = EB(1 + Kb/B)(A)/(EB)(A)/Ka¢ = Ka¢(1 + Kb/B)
(1.8) E
A K
EA B K
a b EAB
¢
E B K
EB A K
b a EAB
¢
Ka E
EA
EB EAB
Kb¢ Ka¢
Kb
Ka* = (Ef + EB)(A)/(EA + EAB) = (Ef)(A)(1 + B/Kb)/EA(1 +B/Kb¢)
= Ka(1 + B/Kb)/(1 + B/Kb¢) Ka* = Ka (Kb¢/Kb) (Kb + B)/(Kb¢ + B)
In all cases, the presence of the second ligand, B, affects the interaction with A.
If A binds before B, then Ka* will decrease from Ka to zero as B increases in concentration because this ligand simply pulls the equilibrium toward EAB, causing an apparent increase in affinity for A. If B binds before A, then Ka* will decrease to Ka¢ as B is increased. For random binding of the two ligands, as B is increased, Ka* will change from Ka to Ka (Kb¢/Kb) = Ka¢, either decreasing if B has a higher affinity for EA than E or increasing if B binds tighter to E than EA. The heterotropic interactions described here for the binding of two substrates to the enzyme also represent the effects of other ligands including metal ions, anions, cations, and even protons that bind at one site and affect the binding of substrate at another site. The effects of these heterotropic interactions on binding will also be more complex if there are homotropic interactions for either of the different types of binding sites.
1.10 SPECIFICITY, PROTEIN ENGINEERING, AND