ENZYMES - MICHAELIS-MENTON KINETICS

Michaelis-Menton Curv
Many clinically important enzyme-catalyzed reactions show hyperbolic kinetics as illustrated in this graph. As the concentration of substrate [S] increases, the rate of formation of product, V, also increases. When [S] is low V increases by a large amount for each increase in [S] but as [S] becomes large, the change in V for each increment in [S] lessens and V asymptotically approaches a maximum value, Vmax. Once this maximum reaction rate is achieved, further increases in [S] have no effect on V. The reaction thus combines features of a reaction with first-order kinetics (the reaction rate is proportional to substrate concentration) when [S] is low and zero-order kinetics (the reaction rate is constant, independent of [S]), when [S] is high. The significance of the constant KM will become clear below.

How can we describe the kinetics of such a reaction quantitatively in terms of parameters that can be measured easily? The model developed by Leonor Michaelis and Maud Menten a century ago does this. Its value lies in the facts that the assumptions underlying their model are simple enough to yield equations that are easy to manipulate, yet realistic enough to allow accurate quantitative predictions of useful features of catalyzed reactions in a clinical setting; e.g., the effects of drugs that act as competitive inhibitors. Here, we will briefly show how the equation is developed from these basic assumptions as a way of understanding these applications.

Steady-State Kinetics
The first assumption concerns the sequence of steps that make up one round of a catalyzed reaction: enzyme (E) and substrate (S) molecules interact rapidly but reversibly to form an enzyme-substrate complex (ES), which reacts slowly to yield a molecule of product (P) and regenerate the enzyme. Product is assumed to be very quickly released from the enzyme. Each of these steps is associated with a rate constant.

An additional assumption is that [S] is much greater than [E] (a plausible assumption for most enzymes in human cells), so the concentration of free substrate, [S], is essentially equal to the total substrate concentration. From this assumption and the ones about rates of the individual steps of the process comes the crucial steady-state assumption: as the reaction gets underway, a steady state is quickly reached in which the rate of formation of enzyme-substrate complex by binding of E and S is equal to the rates of loss of the complex by dissociation to re-form E and S and by reaction to form E and P. The model only considers reactions under conditions where little product has accumulated so, even if a reaction is chemically reversible, formation of ES by the reverse reaction can be ignored. (Note that this “little product” restriction is generally physiologically realistic: the individual reactions we will consider are parts of larger processes, so as fast as product is formed by one reaction, it is consumed by following ones. Indeed, we will often see that accumulation of a product functions as a signal to shut down the overall process.) These assumptions allow the derivation of the Michaelis-Menten equation:

Michaelis-Menton Equation
KM = (k2 + k3)/k1, the sum of the rate constants for loss of enzyme-substrate complex divided by the rate constant for the formation of the complex. KM is called the Michaelis constant.

A strong association between a substrate and an enzyme is reflected in a low KM value; a weaker association is reflected in a larger KM value; an enzyme with a low KM value is thus likely to be able to act on a substrate even when substrate concentrations are low, while one with a high KM value is likely to be active only when substrate concentrations are high. We will see that this provides a useful way of characterizing the various protein channels responsible for cellular glucose uptake and understanding their distinctive roles in various physiological states in the body.

The significance of KM in graph at the start of this section now also is clear. Unlike the individual rate constants, the value of KM can be measured. It has units of moles/liter, and equals the concentration of substrate needed for the reaction to proceed at half its maximum velocity.

What about reactions involving more than one substrate, like the hydrolysis of a peptide bond? In the test tube, this situation can be handled by keeping the concentration of one substrate constant while allowing the other to vary. This process is described as measuring the KM of the enzyme with respect to the variable substrate. This strategy has relevance in the body as well: we will encounter many cases in which one of the substrates of a reaction of interest is indeed nearly constant under physiological conditions while that of a second substrate can vary widely. The water needed for that hydrolysis reaction, for instance, it present at a constant high concentration in the body. One more piece of algebra is useful for describing quantitative properties of enzymes that show Michaelis-Menten kinetics. Taking the inverse of both sides of the Michaelis-Menten equation yields the Lineweaver-Burk equation.

Lineweaver Burk Equation
Lineweaver_burke Basic Graph

With the equation in this rearranged form, Vmax and KM values for an enzyme can be calculated from measurements of reaction rates at various substrate concentrations by simple linear extrapolation. More important for our purposes, this way of visualizing rate data provides an easy way of distinguishing two important kinds of enzyme inhibitors, competitive ones and noncompetitive ones.

Lineweaver Berk Competitive
As its name implies, a competitive inhibitor acts by binding the active site of the enzyme, blocking access by substrate. When the enzyme-inhibitor complex dissociates, the enzyme will be free to bind a molecule of substrate and function as a catalyst. The amount of inhibition observed will thus be determined by the concentrations of enzyme, substrate, and inhibitor and the relative affinities of the enzyme for substrate molecules versus inhibitor molecules. Consequently, the effects of an inhibitor can be reversed by increasing the concentration of substrate. If substrate is present in large enough concentrations, every time an enzyme molecule becomes free, its next encounter will almost always be with a substrate molecule and the reaction will proceed almost as though no inhibitor were present. More formally, in the presence of a competitive inhibitor, the apparent KM of an enzyme is increased (reflecting the reduced efficiency with which it binds substrate when relative amounts of substrate and inhibitor are comparable) but the Vmax of the enzyme is unchanged (reflecting the ability of large amounts of substrate to swamp out effects of the inhibitor).
Lineweaver Berk NonCompetitive
These kinetic properties are clear from the Lineweaver-Burk plot — the black line shows the plot obtained in the absence of inhibitor and the red line shows the plot obtained in its presence.

A non-competitive inhibitor, in contrast, binds to a site on the enzyme distinct from the enzyme’s active site. Inhibitor binding has no effect on the affinity of enzyme for substrate, but causes a conformational change in the enzyme that prevents it from catalyzing the conversion of bound substrate to product. Otherwise, binding of inhibitor and substrate are independent events, and no amount of substrate binding can dislodge inhibitor molecules from their site on the enzyme. Rather, the rate of dissociation of enzyme-inhibitor complexes is determined only by the affinity of the enzyme for the inhibitor. Consequently, the KM of enzyme for substrate is unchanged in the presence of a noncompetitive inhibitor, but the apparent Vmax of the reaction is reduced. In effect, while in the absence of inhibitor all enzyme molecules are available to act on substrate, in the presence of a noncompetitive inhibitor only some of them are available. The result is shown as a Lineweaver-Burk plot in the diagram.