Enzymes (Second Edition)

360 ENZYME REACTIONS WITH MULTIPLE SUBSTRATES

11.4 ISOTOPE EXCHANGE STUDIES FOR DISTINGUISHING REACTION MECHANISMS

An alternative means of distinguishing among reaction mechanisms is to look at the rate of exchange between a radiolabeled substrate and a product molecule under equilibrium conditions (Boyer, 1959; Segel, 1975).

The first, and simplest mechanistic test using isotope exchange is to ask whether exchange of label can occur between a substrate and product in the presence of enzyme, but in the absence of the second substrate. Looking over the various reaction schemes presented in this chapter, it became obvious that such an exchange could take place only for a double-displacement reaction:

—A*

B

E A*X E·A*X EX · A* EX EX·B E · BX E BX

For random or compulsory ordered reactions, the need to proceed through the ternary complex before initial product release would prevent the incorporation of radiolabel into one product in the absence of the second substrate.

Next, let us consider what happens when the rate of isotope exchange is measured under equilibrium conditions for a general group transfer reaction:

AX B A BX

Under these conditions the forward and reverse reaction rates are equivalent, and the equilibrium constant is given by:

If under these conditions radiolabeled substrate B is introduced in an amount so small that it is insufficient to significantly perturb the equilibrium, the rate of formation of labeled BX can be measured. The measurement is repeated at increasing concentrations of A and AX, to keep the ratio [A]/[AX] constant (i.e., to avoid a shift in the position of the equilibrium). As the amounts of A and AX are changed, the rate of radiolabel incorporation into BX will be affected.

Suppose that the reaction proceeds through a compulsory ordered mechanism in which B is the first substrate to bind to the enzyme and BX is the last product to be released. If this is the case, the rate of radiolabel incorporation into BX will initially increase as the concentrations of A and AX are increased. As the concentrations of A and AX increase further, however, the formation of the ternary complexes E · AX · B and E · A · BX will be favored, while dissociation of the EB and EBX complexes will be disfavored. This will have the effect

362 ENZYME REACTIONS WITH MULTIPLE SUBSTRATES

of the EAX and EA forms. The EAX form will react with B, leading to formation of BX, while the EA form will not. Hence, the rate of radiolabel incorporation into BX will increase with increasing [AX] as a hyperbolic function (Figure 11.5B). The same hyperbolic relationship would also be observed for a reaction that proceeded through a random ordered mechanism. In this latter case, however, the hyperbolic relationship also would be seen for experiments performed with labeled AX and varying [B].

Thus isotope exchange in the absence of the second substrate is diagnostic of a double-displacement reaction, while compulsory ordered and random ordered reactions can be distinguished on the basis of the relation of the rate of radiolabel exchange between one substrate and product of the reaction to the concentration of the other substrate and product under equilibrium conditions. (See Segel, 1975, for a more comprehensive treatment of isotope exchange studies for multisubstrate enzymes.)

11.5 USING THE KING--ALTMAN METHOD TO DETERMINE VELOCITY EQUATIONS

The velocity equations for bi bi reactions can be easily related to the Henri—Michaelis—Menten equation described in Chapter 5. However, for more complex reaction schemes, such as those involving multiple intermediate species, it is often difficult to derive the velocity equation in simple terms. An alternative method, devised by King and Altman (1956), allows the derivation of a velocity equation for essentially any enzyme mechanism in terms of the individual rate constants of the various steps in catalysis. On the basis of the methods of matrix algebra, King and Altman derived empirical rules for writing down the functional forms of these rate constant relationships. We provide a couple of illustrative examples of their use and encourage interested readers to explore this method further.

To begin with, we shall consider a simple uni uni reaction as first encountered in Chapter 5:

E S ES E P

In the King and Altman approach we consider the reaction to be a cyclic process and illustrate it in a way that displays all the interconversions among the various enzyme forms involved:

For each step in the reaction we can define a term (kappa) which is the product of the rate constant for that step and the concentration of free substrate involved in the step. Next, we determine every pathway by which a particular enzyme species might be formed in the reaction scheme. For the simple uni uni reaction under consideration we have:

where [form] is the concentration of the particular enzyme form under

consideration, is the sum of the kappa products for that enzyme form,

and is the sum of the kappa products for all species. Applying this to our uni uni reaction we obtain:

Substituting the equalities given in Equations 11.12 and 11.13 into Equation 11.14, we obtain:

Inspecting Equation 11.15, we immediately see that k is equivalent to k , and (k k /k ) is equivalent to the Michaelis constant, K . If we invoke the further equality that V k [E], we see that the King—Altman approach results in the same velocity equation we had derived as Equation 5.24.

Now let us consider the more complex case of a double-displacement bi bi reaction using the King—Altman approach. Note here that the initial concentrations of the two products A and BX are zero, and the release of these

364 ENZYME REACTIONS WITH MULTIPLE SUBSTRATES

products from the enzyme is essentially irreversible. Hence, the cyclic form of the reaction scheme is:

Consideration of this reaction yields the relationships given in Table 11.4. The overall rate equation for a double-displacement reaction is:

Combining Equations 11.16 and 11.17, and performing a few rearrangements we obtain:

With the appropriate substitutions, Equation 11.18 can be recast, using the approach of Alberty, to yield the more familiar form first presented as Equation 11.8.

With similar considerations, the velocity equations for random ordered and compulsory ordered bi bi mechanisms can likewise be derived. With some practice, this seemingly cumbersome approach provides a clear and intuitive means of deriving the appropriate velocity equation for complex enzymatic systems. A more thorough treatment of the King—Altman approach can be found in the text by Segel (1975) as well as in the original contribution by King and Altman (1956).

11.6 SUMMARY

In this chapter we have briefly introduced the concept of multisubstrate enzyme reactions and have presented steady state equations to describe the

366 ENZYME REACTIONS WITH MULTIPLE SUBSTRATES

velocities for these reactions. We have seen that enzyme reactions involving two substrates and two products can proceed by at least three distinct mechanisms: random ordered, compulsory ordered, and double-displacement reactions. Experimental methods were presented to allow the investigator to distinguish among these mechanisms on the basis of kinetic measurements, product inhibition studies, and radioisotope exchange studies. We briefly described the method of King and Altman for deriving the velocity equation of complex enzymatic reaction, such as those involving multiple substrates.

The importance of multisubstrate enzymatic reactions can hardly be overstated. In fact, the vast majority of enzymatic reactions in nature proceed through the utilization of more than one substrate to yield more than one product.

REFERENCES AND FURTHER READING

Alberty, R. A. (1953) J. Am. Chem. Soc. 75, 1928.

Boyer, P. D. (1959) Arch. Biochem. Biophys. 82, 387.

Cleland, W. W. (1963) Biochim. Biophys. Acta, 67, 188.

Cornish-Bowden, A., and Wharton, C. W. (1988) Enzyme Kinetics, IRL Press, Oxford, pp. 25—33.

Dalziel, K. (1975) Kinetics and mechanism of nicotinamide-dinucleotide-linked dehydrogenases, in T he Enzymes, 3rd ed., P. D. Boyer, Ed., Academic Press, San Diego, CA, pp. 1—60.

King, E. L., and Altman, C. (1956) J. Phys. Chem. 60, 1375.

Palmer, T. (1981) Understanding Enzymes, Wiley, New York, pp. 170—189.

Segel, I. H. (1975) Enzyme Kinetics, Wiley, New York, pp. 506—883.

368 COOPERATIVITY IN ENZYME CATALYSIS

Also, ligand binding at the active site of the enzyme can be affected by binding of a structurally unrelated ligand at a distant separate site; this effect is known as heterotropic cooperativity. Thus small molecules can bind to sites other than the enzyme active site and, as a result of their binding, induce a conformational change in the enzyme that regulates the affinity of the active site for its substrate (or other ligands). Such molecules are referred to as allosteric effectors, and they can operate to enhance active site substrate affinity (i.e., serving as allosteric activators) or to diminish affinity (i.e., serving as allosteric repressors). Both types of allosteric effector are seen in biology, and they form the basis of metabolic control mechanisms, such as feedback loops.

In this chapter we shall describe some examples of cooperative and allosteric proteins that not only illustrate these concepts but also have historic significance in the development of the theoretical basis for understanding these effects. We shall then briefly describe two theoretical frameworks for describing the two effects. Finally, we shall discuss the experimental consequences of cooperativity and allostery, and appropriate methods for analyzing the kinetics of such enzymes.

The treatment to follow discusses the effects of cooperativity in terms of substrate binding to the enzyme. The reader should note, however, that ligands other than substrate also can display cooperativity in their binding. In fact, in some cases enzymes display cooperative inhibitor binding, but no cooperativity is observed for substrate binding to these enzymes. Such special cases are beyond the scope of the present text, but the reader should be aware of their existence. A relatively comprehensive treatment of such cases can be found in the text by Segel (1975).

12.1 HISTORIC EXAMPLES OF COOPERATIVITY AND ALLOSTERY IN PROTEINS

The proteins hemoglobin and the Trp repressor provide good examples of the concepts of ligand cooperativity and allosteric regulation, respectively. Hemoglobin is often considered to be the paradigm for cooperative proteins. This primacy is in part due to the wealth of information on the structural determinants of cooperativity in this protein that is available as a result of detailed crystallographic studies on the ligand-replete and ligand-free states of hemoglobin. Likewise, much of our knowledge of the regulation of Trp repressor activity comes from detailed crystallographic studies.

Hemoglobin, as described in Chapter 3, is a heterotetramer composed of two copies of the subunit and two copies of the subunit. These subunits fold independently into similar tertiary structures that provide a binding site for a heme cofactor (i.e., an iron-containing porphyrin cofactor: see Figure 3.19). The heme in each subunit is associated with the protein by a coordinate bond between the nitrogen of a histidine residue and the central iron atom of the heme. Iron typically takes up an octahedral coordination geometry

composed of six ligand coordination sites. In the heme groups of hemoglobin, four of these coordination sites are occupied by nitrogens of the porphyrin ring system and a fifth is occupied by the coordinating histidine, leaving the sixth coordination site open for ligand binding. This last coordination site forms the O binding center for each subunit of hemoglobin.

A very similar pattern of tertiary structure and heme binding motif is observed in the structurally related monomeric protein myoglobin, which also binds and releases molecular oxygen at its heme iron center. Based on the similarities in structure, one would expect each of the four hemes in the hemoglobin tetramer to bind oxygen independently, and with an affinity similar to that of myoglobin. In fact, however, when O binding curves for these two proteins are measured, the results are dramatically different, as illustrated in Figure 12.1. Myoglobin displays the type of hyperbolic saturation curve one would expect for a simple protein—ligand interaction. Hemoglobin, on the other hand, shows not a simple hyperbolic saturation curve but, instead, a sigmoidal dependence of O binding to the protein as a function of O concentration. This is the classic signature for cooperatively interacting binding sites. That is, the four heme groups in hemoglobin are not acting as independent oxygen binding sites, but instead display positive cooperativity in their binding affinities. The degree of cooperativity among these distant sites is such that the data for oxygen binding to hemoglobin are best described by a two-state model in which all the molecules of hemoglobin contain either 4 or

Figure 12.1 Plot of bound molecular oxygen as a function of oxygen concentration for the proteins hemoglobin (Hb) and myoglobin (Mb), illustrating the cooperativity of oxygen binding for hemoglobin.