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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:
K_{1} K_{3
}A + E_{a} <=========> Complex -------> P + E_{a
} K_{2}

The differential equations based on mass balances are:

dA/dt = - K_{1} [A] [E_{a}] + K_{2} [Complex]

d E_{a}/dt = - K_{1} [A] [E_{a}] + (K_{2}
+ K_{3}) [Complex]

d Complex/dt = K_{1} [A] [E_{a}]
- (K_{2} + K_{3}) [Complex]

dP/dt = K_{3} [Complex]

where A is the concentration of some reactive biochemical

E_{a} 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 E_{a}

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 K_{3}.
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)

K_{m}, 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 K_{1} / K_{2} = 1 while K_{3}
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.**

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Reference

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

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