The Computable Differential Equation Lecture ... - Bruce E. Shapiro

178 CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS
In this system the input (or substrate) A is converted to B (called the product) as
a result of interactions with an enzyme E. First, we have that A combines with E
to form a molecular complex that we call X, at a rate that is proportional to the
concentrations of both A and E, with constant of proportionality (called the rate
constant) α. Let A, B, E, and X denote the total concentrations of each of these
chemicals. Then for each molecule of X created, one molecule each of A and X are
consumed, e.g.,
X ′ = αAE + other reactions affecting X (9.124)
A ′ = −αAE + other reactions affecting A (9.125)
The second reaction tells us that X can be converted back to A and E at a rate β,
X ′ = −βX + other reactions affecting X (9.126)
A ′ = βX + other reactions affecting A (9.127)
The third reaction tells us that X can also be converted to B and E at the same
time:
X ′ = −γX + other reactions affecting X (9.128)
B ′ = γX + other reactions affecting B (9.129)
Finally, we observe that E and X are really different forms of the same molecule,
i.e., X is the same thing as E with A bound to it, so we have a conservation law
Putting it all together gives us the following DAE:
E + X = 1 (9.130)
A ′ = −αAE + βX (9.131)
B ′ = γX (9.132)
X ′ = αAE − (γ + β)X (9.133)
0 = E + X − 1 (9.134)
This particular system is semi-explicit, that is, we can clearly tell which equations
involve the explicit derivatives, and which equation involves the constraint. Thus
there are (at least) three ways we can go about solving this system numerically.
1. We could differentiate 9.157 and solve the fully explicit system of four ODE’s:
A ′ = −αAE + βX (9.135)
B ′ = γX (9.136)
X ′ = αAE − (γ + β)X (9.137)
E ′ = −αAE + (γ + β)X (9.138)
This method has the disadvantage that it obscures the constrain, i.e., by looking
at the equations it is not at all obvious that E + X = 1. In fact, this
system satisfies any constraint E + X = C for any constant C, which you can
see by adding the last two equations together to obtain d/dt(E + X) = 0.
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007