The Quasi SteadyState Approximation

An alternative analysis of an enzymatic reaction was proposed by Briggs and Haldane (1925), and their analysis is now the basis for most present-day descriptions of enzyme reactions. Briggs and Haldane assumed that the rates of formation and breakdown of the complex were essentially equal at all times (except perhaps at the beginning of the reaction, as the complex is "filling up"). Thus, dc/dt should be approximately zero. With this approximation, it is relatively simple to determine the velocity of the reaction.

To give this approximation a systematic mathematical basis, it is useful to introduce dimensionless variables s c k_i + k2 eo k_i a =—, x = —, t = kieot, k =—--, e =—, a =--, (1.13)

so eo kiso so kiso in terms of which we obtain the system of two differential equations da _

There are usually a number of ways that a system of differential equations can be nondimensionalized. This nonuniqueness is often a source of great confusion, as it is often not obvious which choice of dimensionless variables and parameters is "best." In Section 1.4 we discuss this difficult problem briefly.

The remarkable effectiveness of enzymes as catalysts of biochemical reactions is reflected by their small concentrations needed compared to the concentrations of the substrates. For this model, this means that e is small, typically in the range of 10-2 to 10-7. Therefore, the reaction (1.15) is fast, equilibrates rapidly and remains in near-equilibrium even as the variable a changes. Thus, we take the quasi-steady-state approximation ejX = 0. Notice that this is not the same as taking jX = 0. However, because of the different scaling of x and c, it is equivalent to taking jc = o as suggested in the introductory paragraph. The quasi-steady-state approximation means that the variable x is changing while restricted to some manifold described by setting the right-hand side of (1.15) to zero. This assumption is valid, provided that e is small and jX is of order 1.

It follows from the quasi-steady-state approximation that a x =-, (1.16)

dT a + k where q = k — a = . Equation (1.17) describes the rate of uptake of the substrate and is called a Michaelis-Menten law. In terms of the original variables, this law is

V = dp = —ds = k e0S = VmaxS _ dt ~ dt ~ s + Km s + Km where Km = k—1+kL. In quasi-steady state, the concentration of the complex satisfies

Note the similarity between (1.12) and (1.18), the only difference being that the equilibrium approximation uses Ks, while the quasi-steady-state approximation uses Km. Despite this similarity of form, it is important to keep in mind that the two results are based on different approximations.

As with the law of mass action, the Michaelis-Menten law (1.18) is not universally applicable but is a useful approximation. It may be applicable even if e = e0/s0 is not small (see, for example, Exercise 6), and in model building it is often invoked without regard to the underlying assumptions.

While the individual rate constants are difficult to measure experimentally, the ratio Km is relatively easy to measure because of the simple observation that (1.18) can be written in the form

In other words, 1/V is a linear function of 1/s. Plots of this double reciprocal curve are called Lineweaver-Burk plots, and from such (experimentally determined) plots, Vmax and Km can be found.

Although a Lineweaver-Burk plot makes it easy to determine Vmax and Km from reaction rate measurements, it is not a simple matter to determine the reaction rate as a function of substrate concentration during the course of a single experiment. Substrate concentrations usually cannot be measured with sufficient accuracy or time resolution to permit the calculation of a reliable derivative. In practice, since it is more easily measured, the initial reaction rate is determined for a range of different initial substrate concentrations.

An alternative method to determine Km and Vmax from experimental data is the direct linear plot (Eisenthal and Cornish-Bowden, 1974; Cornish-Bowden and Eisenthal, 1974). First we write (1.18) in the form

s and then treat Vmax and Km as variables for each experimental measurement of V and s. (Recall that typically only the initial substrate concentration and initial velocity are used.) Then a plot of the straight line of Vmax against Km can be made. Repeating this for a number of different initial substrate concentrations and velocities gives a family of straight lines, which, in an ideal world free from experimental error, intersect at the single point Vmax and Km for that reaction. Of course, in reality, experimental error precludes an exact intersection, but Vmax and Km can be estimated from the median of the pairwise intersections.

1.2.3 Enzyme Inhibition

An enzyme inhibitor is a substance that inhibits the catalytic action of the enzyme. Enzyme inhibition is a common feature of enzyme reactions, and is an important means by which the activity of enzymes is controlled. Inhibitors come in many different types. For example, irreversible inhibitors, or catalytic poisons, decrease the activity of the enzyme to zero. This is the method of action of cyanide and many nerve gases. For this discussion, we restrict our attention to competitive inhibitors and allosteric inhibitors.

To understand the distinction between competitive and allosteric inhibition, it is useful to keep in mind that an enzyme molecule is usually a large protein, considerably larger than the substrate molecule whose reaction is catalyzed. Embedded in the large enzyme protein are one or more active sites, to which the substrate can bind to form the complex. In general, an enzyme catalyzes a single reaction or substrates with similar structures. This is believed to be a steric property of the enzyme that results from the three-dimensional shape of the enzyme allowing it to fit in a "lock-and-key" fashion with a corresponding substrate molecule.

If another molecule has a shape similar enough to that of the substrate molecule, it may also bind to the active site, preventing the binding of a substrate molecule, thus inhibiting the reaction. Because the inhibitor competes with the substrate molecule for the active site, it is called a competitive inhibitor.

However, because the enzyme molecule is large, it often has other binding sites, distinct from the active site, the binding of which affects the activity of the enzyme at the active site. These binding sites are called allosteric sites (from the Greek for "another solid") to emphasize that they are structurally different from the catalytic active sites. They are also called regulatory sites to emphasize that the catalytic activity of the protein is regulated by binding at this site. The ligand (any molecule that binds to a specific site on a protein, from Latin ligare, to bind) that binds at the allosteric site is called an effector or modifier, which, if it increases the activity of the enzyme, is called an allosteric activator, while if it decreases the activity of the substrate, it is called an allosteric inhibitor. The allosteric effect is presumed to arise because of a conformational change of the enzyme, that is, a change in the folding of the polypeptide chain, called an allosteric transition.

Competitive Inhibition

In the simplest example of a competitive inhibitor, the reaction is stopped when the inhibitor is bound to the active site of the enzyme. Thus, ki k S + E ^ Ci-4E + P, k3

From the law of mass action we get

As before, it is not necessary to write an equation for the accumulation of the product. Furthermore, we know that e+ci +c2 = e0. To be systematic, the next step is to introduce dimensionless variables, and identify those reactions that are rapid and equilibrate rapidly to their quasi-steady states. However, from our previous experience (or from a calculation on a piece of scratch paper), we know, assuming the enzyme-to-substrate ratios are small, that the fast equations are those for ci and c2. Hence, the quasi-steady states are found by (formally) setting dci/dt = dc2/dt = 0 and solving for ci and c2. Recall that this does not mean that ci and c2 are unchanging, rather that they are changing in quasi-steady-state fashion, keeping the right-hand sides of these equations nearly zero. This gives

Kmi + KiS + KmKi where Km = k-k+k2, Ki = k-3/k3. Thus the velocity of the reaction is

Notice that the effect of the inhibitor is to increase the effective equilibrium constant of the enzyme by the factor i + i/Ki, from Km to Km (i + i/Ki), thus decreasing the velocity of reaction, while leaving the maximum velocity unchanged.

k3i k-3

Figure 1.1 Diagram of the possible states of an enzyme with one allosteric and one catalytic binding site.

Allosteric Inhibitors

If the inhibitor can bind at an allosteric site, we have the possibility that the enzyme could bind both the inhibitor and the substrate simultaneously. In this case, there are four possible binding states for the enzyme, and transitions between them, as demonstrated graphically in Fig. 1.1.

The simplest analysis of this reaction scheme is the equilibrium analysis. (The more complicated quasi-steady-state analysis is posed as Exercise 2.) We define Ks = k-1/k1, Ki = k-3/k3, and let x,y, and z denote, respectively, the concentrations of ES, EI and EIS. Then, it follows from the law of mass action that in steady state,

where e0 = e + x + y + z is the total amount of enzyme. Notice that this is a linear system of equations for x,y, and z. Although there are four equations, one is a linear combination of the other three (the system is of rank three), so that we can determine x,y, and z as functions of i and s, finding e0Ki s x = Ke0T7Kss+s ■ (L33)

It follows that the reaction rate, V = k2x, is given by

where Vmax = k2e0. Thus, in contrast to the competitive inhibitor, the allosteric inhibitor decreases the maximum velocity of the reaction, while leaving Ks unchanged. (Of course, the situation is more complicated if the quasi-steady-state approximation is used, and no such simple conclusion follows.)

1.2.4 Cooperativity

For many enzymes, the reaction velocity is not a simple hyperbolic curve, as predicted by the Michaelis-Menten model, but often has a sigmoidal character. This can result from cooperative effects, in which the enzyme can bind more than one substrate molecule but the binding of one substrate molecule affects the binding of subsequent ones.

Originally, much of the theoretical work on cooperative behavior was stimulated by the properties of hemoglobin, and this is often the context in which cooperativity is discussed. A detailed discussion of hemoglobin and oxygen binding is given in Chapter 16, while here cooperativity is discussed in more general terms.

Suppose that an enzyme can bind two substrate molecules, so it can exist in one of three states, namely as a free molecule E, as a complex with one occupied center Ci, and as a complex with two occupied centers C2. The reaction mechanism is represented by

k3 k

Using the law of mass action, one can write down the rate equations for the 5 concentrations [S], [E], [Ci], [C2], and [P]. However, because the amount of product [P] can be determined by quadrature, and because the total amount of enzyme molecule is conserved, we only need three equations for the three quantities [S], [Ci], and [C2]. These are ds

dt dc

—— = kise — (k_ i + k2)ci — k3sci + (k4 + k_ 3)c2, (i.38)

dt where s = [S], ci = [Ci], c2 = [C2], and e + ci + c2 = e0.

Proceeding as before, we invoke the quasi-steady-state assumption that dc^dt = dc2/dt = 0, and solve for ci and c2 to get

K2eos

where Ki = k—ik+kl and K2 = k4+^—3. The reaction velocity is thus given by

It is instructive to examine two extreme cases. First, if the active sites act independently and identically, then k1 = 2k3 = 2k+, 2k—1 = k—3 = 2k— and 2k2 = k4, where k+ and k— are the forward and backward reaction rates for the individual binding sites. The factors of 2 occur because two identical binding sites are involved in the reaction, doubling the amount of the reactant. In this case,

where K = k—j+kl is the equilibrium constant for the individual binding site. As expected, the rate of reaction is exactly twice that for the individual binding site.

In the opposite extreme, suppose that the binding of the first substrate molecule is slow, but that with one site bound, binding of the second is fast (this is large positive cooperativity). This can be modeled by letting k3 ^to and k1 ^ 0, while keeping k1k3 constant, in which case K2 ^ 0 and K1 ^to while K1K2 is constant. In this limit, the velocity of the reaction is k4eos2 Vmaxs2 (1 44)

K2 + s2 K2 + s2 ' y ' Km + s Km + s where Km = K1K2, and Vmax = k4eo.

In general, if n substrate molecules can bind to the enzyme, there are n equilibrium constants, K1 through Kn. In the limit as Kn ^ 0 and K1 ^to while keeping K1Kn fixed, the rate of reaction is

V sn

Km + sn where Km = Wn= 1Ki. This rate equation is known as the Hill equation. Typically, the Hill equation is used for reactions whose detailed intermediate steps are not known but for which cooperative behavior is suspected. The exponent n and the parameters Vmax and Km are usually determined from experimental data. Observe that n ln s = n ln Km + ln( ——V—— ), (1.46)

so that a plot of ln( V V_V) against ln s (called a Hill plot) should be a straight line of slope n. Although the exponent n suggests an n-step process (with n binding sites), in practice it is not unusual for the best fit for n to be noninteger.

An enzyme can also exhibit negative cooperativity, in which the binding of the first substrate molecule decreases the rate of subsequent binding. This can be modeled by decreasing k3. In Fig. 1.2 we plot the reaction velocity against the substrate concentration for the cases of independent binding sites (no cooperativity), extreme positive cooperativity (the Hill equation), and negative cooperativity. From this figure it can be seen that with positive cooperativity, the reaction velocity is a sigmoidal function of the substrate concentration, while negative cooperativity primarily decreases the velocity.

o CO CD Substrate concentration, s

Figure 1.2 Reaction velocity plotted against substrate concentration, for three different cases. Positive cooperativity, K = 1000, K2 = 0.001; independent binding sites, K = 0.5, K2 = 2; and negative cooperativity, K1 = 0.5, K2 = 100. The other parameters were chosen as e0 = 1, k2 = 1, k4 = 2. Concentration and time units are arbitrary.

The Monod-Wyman-Changeux Model

Cooperative effects occur when the binding of one substrate molecule alters the rate of binding of subsequent ones. However, the above models give no explanation of how such alterations in the binding rate occur. The earliest mechanistic model proposed to account for cooperative effects in terms of the enzyme's conformation was that of Monod, Wyman, and Changeux (1965). Their model is based on the following assumptions about the structure and behavior of enzymes.

1. Cooperative proteins are composed of several identical reacting units, called protomers, that occupy equivalent positions within the protein.

2. Each protomer contains one binding site for each ligand.

3. The binding sites within each protein are equivalent.

4. If the binding of a ligand to one protomer induces a conformational change in that protomer, an identical conformational change is induced in all protomers.

5. The protein has two conformational states, usually denoted by R and T, which differ in their ability to bind ligands.

To illustrate how these assumptions can be quantified, we consider a protein with only two binding sites. Thus, the protein can exist in one of six states: R/, i = 0, 1, 2, or T{,i = 0,1, 2, where the subscript i is the number of bound ligands. For convenience, we also assume that Ri cannot convert directly to Ti, or vice versa, and similarly for

2sk3

Ti ski 2k-1

Figure 1.3 Diagram of the states of the protein, and the possible transitions, in a six-state Monod-Wyman-Changeux model.

R2 and T2. The general case is left for Exercise 3. The states of the protein and the allowable transitions are illustrated in Fig. 1.3.

We now assume that all the reactions are in equilibrium. We let a lowercase letter denote a concentration, and thus ri and ti denote the concentrations of chemical species R, and T, respectively. Also, as before, we let s denote the concentration of the substrate. Then, the fraction Y of occupied sites (also called the saturation function) is

Furthermore, with K = k-i/ki, for i = 1,2, 3, we find that ri = 2sK-1ro, r2 = s2K-2 ro, ti = 2sK3 1to, t2 = s2K3 2to. Substituting these into (1.47) gives sK-1(1 + sK-1) + K-1[sK-1(1 + sK-1)] = (1 + sK-1)2 + K-1(1 + sK3-1)2 , where we have used that r0/t0 = K2. More generally, if there are n binding sites, then

sK-1(1 + sK-1)n-1 + K2-1[sK3-1(1 + sK-1)n ] (1 + sK-1)n + K-1(1 + sK-1)n

In general, Y is a sigmoidal function of s.

Some special cases are of interest. For example, if K3 = to, so that the substrate cannot bind directly to the T conformation, then

or if K2 = œ, so that only the R conformation exists, then s

which is the Michaelis-Menten equation.

There are many other models of enzyme cooperativity, and the interested reader is referred to Dixon and Webb (1979) for a comprehensive discussion and comparison of other models in the literature.