THE KINETICS OF ENZYME CATALYZED REACTIONS WILLIAM H. R. SHAW The University of Texas, Austin, Texas
The Enzymes They light the firefly's tiny torch a t dusk Destroy and build great structures in the realm 01 molecules. While we with prying eyes And puny instruments attempt to watch, To marvel, and-perhaps-to understand.
A m H o w H a great variety of substances can hasten chemical reactions, one class of catalytic agents is endowed with such efficiency, specificity, and importance to life processes that it truly can be called unique. These catalysts are the enzymes. Enzymes occur in a diversity of places: in the venom of snakes (I), for example, where the enzyme hyaluronidase enables the poison to spread rapidly through the tissues of a victim; in the tails of fireflies (2), where the enzyme luciferase rearts with its substrate luciferin in the presence of oxygen, and light is produced; in the brain and nervous system where the enzyme cholinesterase is associated with the transmission of nerve impulses. The last-named enzyme is also found in high coucentration in the electric organ of the electric eel ( 1 ) . There is an enzyme (or enzymes) that hastens a Walden inversion (S), and another called hydrogenase (4) that catalyzes the transformation of ortho to para hydrogen. I t is small wonder, then, that the study of the intriguing b~rderlinefields t h a t lie between. Enzymes are biochemical catalysts that are synthesized by the living cell but are capable of action in uitro as well as i n uioo. A discussion of methods of isolation, general properties and descriptive material is beyond the scope of the immediate work. These important aspects of enzymology are moreover comprehensively treated in numerous reference works (I), (6),(6),and treatises (7), (8). Suffice it to say: that all enzymes thus far isolated are proteins, that many have been obtained in the crystalline state (a), (9),and that some eonsist of a protein moiety and an essential auxiliary dialyzable part called a prosthetic gronp. MICHAELIS-MENTEN MECHANISM
The rate of an enzyme catalyzed reaction is, in general, directly proportional to the total enzyme coneentration. The dependence of rate on substrate concentration, howver, is considerably more complex (Figure 1 ) . At lorn substrate concentration the reaction appears to be first order; hut, with increasing eoncentration, the apparent order decreases until, a t high substrate concentration, the rate passes through a region of essentially zero order kinetics on its asymtot~eapproach to a limiting value, V,,,.. A mechanistic explanation of this interesting behavior was first offered in 1913 by Michaelis and Menten (10) although the basic notion had been presented earlier by Brown (If),
Henri (12), and others, Briggs and Haldane (IS) extended and amplified the original treatment. The proposed mechanism can he summarized in the following form :
Equation (1) assumes that the enzyme, E, combines with its substrate, S, to form an intermediate compound the enzyme-substrate, ES, a t a rate characterized by the rate constant k,. Once formed the enzyme substrat,e has two alternative fates. It can reform the free enzyme and substrate (k2),or decompose ( 1 ~ to ~ ) form the product, P, and regenerate the enzyme. A conservation equation can also he written: e =
[El
+ IESI
(2)
That is-the total enzyme concentration, e, is equal to the concentration of the free enzyme plus the concentration of enzyme bound up in the enzyme-substrate complex. Since the net rate of formation of ES is the rate of formation minus. the rate of loss it becomes apparent from equation (1) that:
For E, 8, and P analogous rate expressions can be formulated as follom:
d[El = (k, + k,) [ES] - h[EI IS1 dt
(4)
A rigorous description of the system would necessitate solving the two simultaneous differential equations (3) and (5), but no simple analytical solution for these equations can he obtained. The situation is a common one in chemical kinetics and is routinely treated by the steady-state assumption of Bodenstein (14). His approximation assumes that the concentration of active intermediates may be considered to remain constant, during most of the reaction. Applied to the ES eompound the treatment reveals that: [ES]*
=
dIESl* = const., dt
,
(7)
The asterisk denotes the fact that the equation will only hold after a steady state has been established or, in other words, when the rate of formation of ES is equal to its rate of disappearance. From equation (3) it follows that:
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The collection of constants has been equated to Michaelis' constant K,. By combining equation (8) with equation (2) it is now possible to eliminate [El* and express [ES]* in terms of [ S ]* and e.
One additional assumption is necessary. Since the enzyme concentration is usually several orders of magnitude smaller than the substrate concentration, i t follows that very little substrate is bound up in the ES complex; and if the reaction is studied in its early stages [PI
