Enzyme Technology
Reversible reactions
A reversible enzymic reaction (e.g., the conversion of glucose to fructose, catalysed by glucose isomerase) may be
represented by the following scheme where the reaction goes through the
reversible stages of enzyme-substrate (ES) complex formation, conversion to
enzyme-product (EP) complex and finally desorption of the product. No step is
completely rate controlling.
[1.12]
Pairs of symmetrical equations may be obtained for the
change in the concentration of the intermediates with time:
(1.36)
(1.37)
Assuming that there is no
denaturation, the total enzyme concentration must remain constant and:
(1.38a)
therefore:
(1.38b)
gathering terms in [ES]
(1.39a)
(1.39b)
and,
(1.38c)
gathering terms in [EP]
(1.40a)
(1.40b)
Under similar conditions
to those discussed earlier for the Michaelis-Menten mechanism (e.g., under the
steady-state assumptions when both d[ES]/dt and d[EP]/dt are zero, or more
exactly when
(1.41)
and
(1.42)
are both
true. The following equations may be derived from equation 1.39b using the
approximation, given by equations 1.41 and
collecting terms.
(1.43a)
(1.43b)
(1.43c)
Also, the following equations (symmetrical to the above) may be derived from
equation
1.40b by using the approximation, given by equation 1.42, and
collecting terms.
(1.44a)
(1.44b)
(1.44c)
Substituting for [ES] from equation 1.44c into equation
1.43a
(1.43d)
(1.43e)
(1.43f)
Removing identical terms from both sides of the equation:
(1.43g)
Gathering all the terms in [EP]:
(1.43h)
(1.43i)
Also, substituting for [EP] from equation 1.43c into equation
1.44a
(1.44d)
(1.44e)
(1.44f)
Removing identical terms from both sides of the equation:
(1.44g)
Gathering all the terms in [ES]:
(1.44h)
(1.44i)
The net rate of reaction (i.e., rate at
which substrate is converted to product less the rate at which product is
converted to substrate) may be denoted by v where,
(1.45)
Substituting from equations 1.43i and
1.44i
(1.46a)
Simplifying:
(1.46b)
Therefore,
(1.47)
where:
(1.48)
(1.49)
(1.50)
(1.51)
At equilibrium:
(1.52)
and, because the numerator of equation 1.47 must equal
zero,
(1.53)
where
[S]¥ and [P]¥
are the equilibrium
concentrations of substrate and product (at infinite time). But by definition,
(1.54)
Substituting from
equation 1.53
(1.55)
This
is the Haldane relationship.
Therefore:
(1.56)
If KmS and
KmP are approximately equal (e.g., the commercial immobilised
glucose isomerase, Sweetase, has Km(glucose) of 840 mM and
Km(fructose) of 830 mM at 70°C), and noting that the total
amount of substrate and product at any time must equal the sum of the substrate
and product at the start of the reaction:
(1.57)
Therefore:
(1.58)
Therefore:
(1.59)
where:
(1.60)
K' is not a true kinetic constant as it is only
constant if the initial substrate plus product concentration is kept constant.
Also,
(1.61)
Substituting from equation 1.54,
(1.62)
Let [S#] equal the concentration difference
between the actual concentration of substrate and the equilibrium concentration.
(1.63)
Therefore:
(1.64)
Substituting in equation 1.47
(1.65)
Rearranging equation 1.55,
(1.66)
Therefore:
(1.67)
Therefore:
(1.68)
Where
(1.69)
Therefore:
(1.70)
Therefore:
(1.71)
and:
(1.72)
As in the case
of K' in equation 1.59, K is not a true kinetic constant as it
varies with [S]¥ and hence the sum of [S]0 and
[P]0. It is only constant if the initial substrate plus product
concentration is kept constant. By a similar but symmetrical argument, the net
reverse rate of reaction,
(1.73)
with constants
defined as above but by symmetrically exchanging KmP with
KmS, and Vr with Vf.
Both equations (1.59)
and (1.68) are useful when modelling reversible reactions, particularly the
technologically important reaction catalysed by glucose isomerase. They may be
developed further to give productivity-time estimates and for use in the
comparison of different reactor configurations (see Chapters 3 and
5).
Although
an enzyme can never change the equilibrium position of a catalysed reaction, as
it has no effect on the standard free energy change involved, it can favour
reaction in one direction rather than its reverse. It achieves this by binding
strongly, as enzyme-reactant complexes, the reactants in this preferred
direction but only binding the product(s) weakly. The enzyme is bound up with
the reactant(s), encouraging their reaction, leaving little free to catalyse the
reaction in the reverse direction. It is unlikely, therefore, that the same
enzyme preparation would be optimum for catalysing a reversible reaction in both
directions.
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This page was established in 2004 and last updated by Martin
Chaplin on
6 August, 2014
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