Biochemical Reactions and Enzyme Kinetics - Introduction to Biomedical Engineering - page 451

chemical reaction, we will write the differential equations directly from the stoichiometric
equation. Of course, Eq. (8.2) can be written in terms of concentrations by substituting
(concentration

volume) for quantity.
Eq. (8.2) is a nonlinear differential equation. In general, nonlinear equations cannot be
solved directly, but they must be simulated using a program like SIMULINK. Keep in mind
that there are some special nonlinear differential equations that can be solved, but these
are few in number, with a few presented in this chapter. Since

q A ¼ q B

, we can integrate

both sides of R t

0 dq A ¼ R t

0 dq b

and have q A q A ð 0 Þ¼ q B q B ð 0 Þ , which, when substituted into

Eq. (8.2), gives

q A ¼ Kq A q B ¼ Kq A q A q A ð ð 0 Þþ q B ð 0 Þ Þ ð 8 : 3 Þ

Rearranging Eq. (8.3) yields

dq A
q A q A q A ð Þ ¼ Kdt

ð 8 : 4 Þ

ð 0 Þþ q B ð 0 Þ

The left-hand side of Eq. (8.4) is rewritten using a technique called partial fraction expan-
sion (details are briefly described here for the unique roots case) 1 as

0 1

B dq A

Þ þ B

dq A
q A

@

1 2

A ¼ Kdt

ð q A q A ð 0 Þþ q B ð 0 Þ

0
@

1
A

0
@

1
A ¼ Kdt

ð 8 : 5 Þ

1 dq A

q A q A ð Þ dq A

q A ð 0 Þ q B ð 0 Þ ð 0 Þþ q B ð 0 Þ q A

where

q A ¼ q A ð0Þ q B ð0Þ ¼ q A ¼ q A ð0Þ q B ð0Þ ¼

1
q A q A q A ð

1
q A

1

B ¼ q A q A ð ð 0 Þþ q B ð 0 Þ Þ

1

ð 0 Þþ q B ð 0 Þ Þ q A ð 0 Þ q B ð 0 Þ

1 In general, if

1 1

þ a 1 s þ a 0 ¼ s n þ a n 1 s n 1 ð s s 1 Þ s s n ð Þ

and the roots are real, then

1

B

1 þ B n

1
s s

Þ ¼

ð s s Þ s s n ð s s n

1

where

s ¼ s i

1

B i ¼ s s i ð Þ

s s 1 ð Þ s s n ð Þ

It should be clear when evaluating the coefficient B i in the previous equation that the factor ð s s i Þ always

cancels with the same term in the denominator before calculating B i :