Solution of the Briggs-Haldane System
Exact Solution for Constant [S]
We consider the enzyme kinetics model with an intermediate enzyme-substrate and three reaction rate constants, depicted by the reaction
This reaction can be studies using the following system of differential equations:
Assuming that [S] does not change much over small times, we let Then the first two equations of the system become
This is a linear system of differential equations, which can be solved exactly for [E] and [ES]. There are two methods that can be used, a matrix method and that of writing this system as one linear, second order constant coefficient differential equation of the type seen by undergraduates in a first course on differential equations. We employ the latter method.
Differentiating the first equation, we obtain
Now, replace by the expression in the second equation of the system:
This equation can then be turned into an equation in [E] alone by using the first equation of the system. Solving for [ES] and substituting the new expression in the last equation, we get
This equation can be integrated to yield
where A is an arbitrary constant. Another integration gives the result.
where B is a second integration constant.
From equation one, we can solve for [ES] to obtain
where we have introduced
We can determine the integration constants from the initial conditions
These conditions determine a system of two algebraic equations for A and B:
Solving this system, we find
Inserting these expressions into the solutions for [E] and [ES], we obtain
We can now find [P]. Integrating the fourth equation of the system yields
where the integration constant C can be determined from the initial condition
One finds that so
Finally, we should note that there is a natural time scale in this problem. Namely, the time
Condition for Constant [S]
Finally, we need to check the third equation of the system. Since [S] is assumed to be constant, then
for the time of observation. This gives a condition on the assumption that [S] does not change very much. Using the solutions above and the typical magnitudes of the variables used in the program, we can establish a bound on the right hand side of this equation.
We can easily bound this expression. Note that
Typical values for the constants in this bound are
Then we find that
Scaling the System
In studying such systems, it is best to convert it to dimensionless quantities and determine which terms are the dominant terms in the system. We will introduce the following dimensionless variables:
where a and b are characteristics concentrations and T is a characteristic time. Furthermore, we relate the rate constants by
Then,
The system becomes
where Typical values of the parameters are given by
Each equation has a factor of In the first equation, the first term in the first factor is of order and the second is of order b. These terms are generally of the same order as indicated below. The overall contributions from each term in the first and second equations are of order 0.1 to 10 for times on the order of T.
/ / b (M) / / / /100 / / / 1.1 / 0.1 / / 0.1
1000 / / / 2.0 / 1.0 / / 1.0
10000 / / / 11. / 10. / / 10.
In the third equation, we see by a similar argument, that the terms are of the order of or This confirms that the change in [S] is small over this time scale.
Finally, the order of the term in the last equation is which varies on the orders of to 10.
These considerations further confirm the need to keep all the terms in the system studied above and confirm the assumption that [S] can be treated as a constant. Such an analysis can also be used to search for systems in which the nonlinear terms in the first three equation should be kept.
The Velocity
The velocity is given by
for the steady state solutions. Steady state solutions are obtained from the solutions obtained above by seeking the asymptotic values of the solutions as t gets large. We find that
and
Therefore, we have
Thinking of v as a function of this expression does not have an extremum, but it does level off as gets large. This asymptote is called and is easily seen to be
We now determine at what velocity , which is a well-known relation in the study of this system. From the above expressions for v and we have
So, when
Finally, the expression for v can be rewritten in terms of
In summary, we have obtained known expressions involving the velocity of this reaction.