Environmental Engineering Reference
In-Depth Information
Both alternative systems are implemented in MATLAB
using the
ode15s
®
solver. After the input section the M-file looks like this:
%----------------------execution-----------------------------------
% substrate= y(1),enzyme2= y(2),intermediate= y(3), product= y(4)
options = odeset('AbsTol',1e-20);
[t,y] = ode15s(@detail,[0 T],[s0; e0; i0; p0],options,k);
[tt,z] = ode15s(@lumped,[0 T],s0,options,e0,i0,k);
%---------------------- graphical output --------------------------
plot(t,y(:,1:4)); hold on;
plot(tt,z(:,1),'--',tt,s0-z(:,1)+p0,'--');
legend ('substrate','enzyme','intermediate','product',…
'lumped substrate','lumped product');
grid;
%---------------------- functions ---------------------------------
function
dydt = detail(t,y,k)
r1 = k(1)*y(1)*y(2);
r2 = k(3)*y(3);
r3 = k(2)*y(3);
dydt = zeros(4,1);
dydt(1) = -r1 + r2;
dydt(2) = -r1 + r2 + r3;
dydt(3) = r1 - r2 - r3;
dydt(4) = r3;
function
dzdt = lumped(t,z,e0,i0,k)
dzdt(1) = -k(1)*k(2)*(e0+i0)*z/(k(2)+k(3)+k(1)*z);
The system of four species is solved in the
y
-vector, and the system is specified
in the
detail
-function. The substrate development according to the
Michaelis-Menton kinetics is given in variable
z
and specified in the
lumped
-
function.
The complete code is included in the accompanying software under the name
'MichaelisMenten.m'
For the input parameters
s
0
¼
1
; e
0
¼
0
:
2
; i
0
¼
0
:
1and
p
0
¼
0
:
3andtherate
coefficients
4 the result is depicted in Fig.
9.2
.The
strong initial change is obviously not captured by the Michaelis-Menten
approach. But for long time periods the common Michaelis-Menten model is
a good approximation, although several initial conditions and rate constants are
lumped into two parameters. The difference between the detailed and the lumped
model is visualized in Fig.
9.3
. For long times the relative error for the substrate is
almost 100%. That must not be crucial, because the high relative error coincides
with low concentrations, as the comparison between the two figures shows.
However, if the initial concentrations are several orders of magnitude above
a limit value, low concentrations may still be harmfull and the error of 100% is
not tolerable.
k
1
¼
1
; k
2
¼
0
:
15
; k
3
¼
0
:
