Back to Contents!

More Mechanisms: A Model of Enzyme Kinetics

Sometimes it is expedient - even crucial - for a chemical reaction to occur in a time (i.e. "rapid") manner. The rate desired (required) is usually much quicker than the ambient temperature and prevailing conditions would allow. In these cases the mechanism must be coaxed to proceed more rapidly - but without error! Also, this added "coaxing agent" may not interfere with the reaction - other than to speed it up! A chemical that fulfills the above requirements is termed a catalyst. A catalyst is a substance that - when added to a chemical reaction - speeds it up - but without altering the final product(s). Thus, the catalyst cannot be consumed - it should be regenerated. Such species - when present in biological systems - are termed enzymes. These highly specific molecules are centrally responsible for the precise "clockwork-like" choreography of the myriad processes that occur in seconds in the human body. Because of this delicate precision, enzymes are very sensitive to prevailing conditions. In the human body, for example, the thousands of enzymes present are sensitive to acidity (measured as [H^+] or pH = log_{10}[H^+]) and sensitive to (body) temperature. Any fluctuation in either temperature or pH will wreak havoc - sometimes fatally. In this section we want to restrict our discussion of catalysts - which can occur in inorganic systems as well as in biological systems - to those that occur in biological systems - i.e. enzymes.

Enzymes are astounding catalysts that are highly specific for a particular reaction while being extremely versatile at times. We will not go into the structural features that give an enzyme its specificity/versatility. Rather we will explore a simple mechanistic model that rationalizes the behavior of most enzymes and explore some of its consequences. This simpler model was initially developed in 1913 by Lenor Michaelis and Maude Menten and is referred to as the Michaelis-Menten (M-M) model of enzyme kinetics. In 1925, a slightly more sophisticated model was developed by G.E. Briggs and James B.S. Haldane and was based on the Steady-State Approximation (SSA). We will concentrate on the model developed by Briggs and Haldane. In honor of the pioneering work of Michaelis and Menten, the refined model of Briggs and Haldane is still commonly referred to as the "Michaelis-Menten" mechanism.

The Beginning of Enzyme Kinetics - Adrian Brown

In 1902 the study of enzyme kinetics began with the work of Adrian Brown. He studied the rate of hydrolysis of sucrose (C_{12}H_{22}O_{11}) by water (H_2O) to yield glucose and its structural isomer fructose - both have formula C_6H_{12}O_6. Specifically, Brown studied the rate of hydrolysis of sucrose as catalyzed by the yeast enzyme invertase (now known as beta-fructofuranosidase).
Sucrose + Glucose + Fructose

Brown showed that when the sucrose concentration was much greater than the enzyme concentration, the reaction is zero order in sucrose concentration, i.e.,

	Rate = k [sucrose]°  = k         if [sucrose] >> [enzyme]

This "other" chemical species - which was introduced to the sucrose solution - but was not consumed, i.e., was regenerated at the end of the conversion, was the enzyme invertase. It was also found that without the enzyme, the reaction still occurred, but at a much slower rate. Thus, Brown demonstrated that this enzyme invertase: was not a required ingredient in the stoichiometric reaction, was not consumed as a result of the reaction, increased the rate of the reaction due to its presence. In brief, this enzyme was serving as a catalyst. In fact, the mechanism that Brown first proposed serves as a simple starting point for many general models of catalysis.

Brown proposed that the overall reaction is composed of two elementary reactions in which the reactant in the stoichiometric reaction - which was termed the "substrate" forms a complex with the enzyme that subsequently decomposes to products and the "regenerated" enzyme. The mechanism is often depicted as shown below.

In the above mechanism: E denotes the enzyme, S denotes the substrate, P denotes the products, and ES denotes Brown's hypothesized enzyme-substrate complex. Note that E is a catalyst and ES is an intermediate - be sure that you understand the difference between the two! Also, the elementary rate constants k_1, k_{-1}, and k_2 are defined as follows. k_1 denotes the elementary rate constant for the formation of ES from E and S, k_{-1} denotes the elementary rate constant for the return of ES to E and S ("unsuccessful" reaction), and k_2 denotes the elementary rate constant for the subsequent decomposition of ES to P and E. According to the model, when the substrate concentration becomes high enough to entirely convert the enzyme to the ES form, the second step of the reaction becomes the rate-determining ("slow") step and the rate of the overall reaction (S goes to P) becomes "insensitive" to any further increase in the substrate concentration. Let's proceed to develop the differential equations for this mechanism.

Mathematical development of the mechanism

In the above mechanism we have three (3) elementary reactions. The first reaction actually depicts two (2) elementary reactions, i.e. the reversible formation/breakdown of the enzyme-substrate complex. Note that just because this reaction is reversible, does not mean that it is at equilibrium! The final elementary reaction depicts the successful decomposition of the enzyme-substrate complex into the product(s) plus regenerated enzyme.

Commonly the rate of reaction is determined by measuring the rate of formation of product (P), i.e. +d[P]/dt. Biochemists commonly refer to +d[P]/dt as the "velocity" of the reaction and symbolize it by "v". From the mechanism above, it is easy to see (prove for yourself) that:

From our earlier discussion, you should be able to prove that the differential equation which expresses the net rate of increase of concentration of the intermediate ES - according to this mechanism - is:

What we want to do, of course, is to obtain an expression for [ES] in terms of the concentration of S and E that we introduce into the system - since these are the species that we can "control" in the laboratory. We can then substitute this expression into our velocity equation and thus obtain an expression for the (measured) velocity of reaction in terms of the substrate and enzyme concentrations. In order to do this we must solve the above differential equation. We must make a few simplifying assumptions first. The first of the two possibilities was first proposed by Michaelis and Minton (in 1913) and the second possibility was first proposed by Briggs and Haldane (in 1926). We list them below.

Equilibrium assumption

Lenor Michaelis and Maude Menten, in 1913, advancing earlier work of the chemist Victor Henri, assumed that k_{-1} » k_2. If this is true, then the reversible step in the mechanism does achieve equilibrium and we can write the law of chemical equilibrium for this reversible step and hence equate the ratio of the forward (k_1) to reverse (k_{-1}) rate constants to the equilibrium expression (see our earlier chemistry writeup for background):

Note that this actually is the expression for the ES « -- » E + S - dissociation of the ES into E and S - instead of the way it is listed in the mechanism (i.e. E + S « -- » ES). This is how it is commonly written however - so we will stick with it! The equilibrium constant above (K_S) is thus termed the dissociation constant. If we solve this expression for [ES] in terms of [E] and [S], we get:

Substituting this into our velocity expression, we arrive at:

Although the assumption that k_{-1} » k_2 is not often correct, the enzyme-substrate (ES) complex is known as the Michaelis complex - in recognition of this pioneering work.

Steady-State Approximation (SSA)

The figure below depicts the curves for concentration as a function of time for the various species in the originally proposed mechanism - at the physiologically common condition that substrate concentration is in large excess of the enzyme concentration, i.e. [S] » [E].

<IMG src="http://dept.physics.upenn.edu/courses/gladney/mathphys/images/michmencon.gif">

Except for initial build-up of [ES] at the beginning of the reaction (a few milliseconds) the [ES] remains approximately constant until substrate is nearly all used up. Hence, d[ES]/dt is approximately zero, i.e. the [intermediate] does not change appreciably with time. As explained previously, we can thus apply the steady-state approximation (SSA) to [ES]. This is what G.E. Briggs and James B.S. Haldane did in 1925. From our earlier expression for the time rate of change of [ES], we can write:

Note that all of the concentrations above are "instaneous" concentrations, i.e. concentrations at time "t" - not necessarily initial concentrations. So, although we can solve the above expression for [ES] as a function of [E] and [S], even though we can monitor [S] at time "t", it is usually much more difficult to measure [E] at time "t". Therefore, we introduce the more directly measurable total enzyme concentration "[E]"_T - which is the initial amount of enzyme introduced to the system. From conservation of mass (at constant volume), it is easy to see that:

So, if we substitute [ES] = [E]_T - [ES] into our steady-state expression above, prove for yourself that the result is:

Solving for [ES] yields:

.

Multiplying the right-hand side of the above equation by "1" in the form of k_1/k_1 gives:

The ratio of elementary rate constants, (k_{-1} + k_2)/k_1, is termed the Michaelis constant and is symbolized by K_M, i.e.,

substituting everything into our expression for the velocity of reaction (v), prove for yourself that the result is given by:

It turns out that the above expression is even a better approximation to the measured velocity of reaction (v) if v is measured "early" in the reaction, i.e., after [ES] reaches its "constant" value but before the production of "P" becomes reversible (why?). Thus, the above expression is commonly written in terms of the "initial velocity" (v_0) as:

The use of the initial velocity (v_0) - usually defined as the measured velocity before more than about 10 percent of the substrate (S) has been converted to product (P) minimizes complicating factors such as the reversible reaction mentioned above as well as inhibition of the enzyme by product and progressive inactivation of the enzyme.

As the [S] becomes very large, the velocity of reaction will not increase indefinitely, but - for a fixed amount of [E]_T - will reach a limiting value termed V_{max}, the maximal velocity. Prove for yourself - after looking at the above expression - that

Substituting V_{max} for k_2[E]_T in our boxed expression above gives the "usual" form - termed the Michaelis-Menten equation:

This expression is the basic equation of enzyme kinetics. This equation is plotted so that you can view it. We'll explore this expression and some of its consequences in a future section. Stay tuned!

As an example of a complex chemical reaction, you can consider the chain of chemical reactions that proceed when a photon hits the retina in your eye. Here's a Java applet that simulates the process. Enjoy!