Enzyme Kinetics Simulation

Textbooks for biochemistry make much of the classical enzyme kinetics of W.W. Cleland and of the Michaelis-Menten expression. The equations for a single reaction catalyzed by an enzyme are:
           K1                   K3
A + Ea  <=========> Complex  ------->  P + Ea
           K2
The differential equations based on mass balances are:
dA/dt = - K1 [A] [Ea] + K2 [Complex]
d Ea/dt = - K1 [A] [Ea] + (K2 + K3) [Complex]
d Complex/dt = K1 [A] [Ea] - (K2 + K3) [Complex]
dP/dt = K3 [Complex]
where A is the concentration of some reactive biochemical
Ea is the concentration of an enzyme (note that it appears at two places in the reaction scheme)
[Complex] is the concentration of an intermediate formed by A and Ea
P is the concentration of product

The first reaction is reversible, but the equilibrium is assumed to greatly favor the products for the reaction with the rate K3. When the derivatives are set equal to zero and the reaction rate coefficients are consolidated, the Michaelis-Menten equation results (it was the inspiration for the Monod equation). The Lineweaver-Burk equation comes from taking reciprocals and rearranging. Stepwise explanation of transformation.


where V is the observed rate and V with the hat is the maximum rate (excess substrate)
Km, the Michaelis constant, accounts for the previous rate constants.

This set of simultaneous differential equations was used in a JAVA applet for you to experiment with concentrations and coefficients. Be sure to test the ratio of K1 / K2 = 1 while K3 has a relatively low value because this is used in most textbooks that discuss enzyme kinetics. The rates of substrate decline and product formation are nearly linear throughout much of the simulation. This alone has some teaching value because a student can verify that the textbook results make sense. Additional insight comes from tinkering with the rate coefficients to see their effect.

It must be appreciated that one reaction with one enzyme in vitro is not typical of living biochemistry where there are complicated pathways and intermediates tend not to accumulate. Actual metabolic pathways have branches and many steps, and the over accumulation of intermediates is prevented by feedback inhibition where a concentration of a biochemical further along in the pathway inhibits or represses the activity of the enzyme for an early step.

Reference

Bungay, H.R., "Basic Biochemical Engineering", 2nd edition, BiLine Assoc., Troy, NY, 1993. 

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