MATLAB Examples

Stationary Solutions of an Enzyme Model (Demo : enz)

The equations, that model a two-compartment enzyme system (Kernevez,1980), are given by

: $s_1' = (s_0 - s_1) + (s_2 - s_1) - \rho R (s_1)$,

: $s_2' = (s_0 + \mu - s_2) + (s_1 - s_2) - \rho R (s_2)$,

where

  • $R (s) = \frac{s}{1 + s + \kappa s^{2}}$.
  • The free parameter is $s_0$.
  • Other parameters are fixed.

This equation is also considered by (Doedel, Keller and Kern, 1991).

clear all

Create continuation object.

a{1}=auto;

Print function file to screen.

type(a{1}.s.FuncFileName);
function [f,o,dfdu,dfdp]= func(par,u,ijac)
%
% function file for enz demo
% 
f=[];
o=[];
dfdu=[];
dfdp=[];
%
s0=par(1);
rm=par(2);
rh=par(3);
rk=par(4);
%
s1=u(1);
s2=u(2);
%
rs1=s1/(1+s1+rk*s1.^2);
rs2=s2/(1+s2+rk*s2.^2);
%
f(1)=(s0-s1) + (s2-s1) - rh * rs1;
f(2)=(s0+rm-s2) + (s1-s2) - rh * rs2;



Set initial conditions.

[a{1}.s.Par0,a{1}.s.U0,a{1}.s.Out0]=stpnt;

Set constants.

a{1}.c=cenz1(a{1}.c);

Run continuation.

a{1}=runauto(a{1});
 
    --------------- DYNAMICAL SYSTEMS TOOLBOX ---------------------     
 
USER NAME      : ECOETZEE
DATE           : 26/10/2010 10:09:57
 
 
<
  BR    PT  TY  LAB      PAR(01)      L2-NORM         U(01)         U(02)
   1     1  EP    1   0.00000E+00   0.00000E+00   0.00000E+00   0.00000E+00
   1   141  LP    2   3.43569E+01   1.48261E+00   1.04836E+00   1.04836E+00
   1   145  BP    3   3.42229E+01   1.66053E+00   1.17417E+00   1.17417E+00
   1   214  BP    4   2.21816E+01   6.21472E+00   4.39447E+00   4.39447E+00
   1   244  LP    5   1.88871E+01   1.24765E+01   8.82220E+00   8.82220E+00
   1   250  EP    6   1.89864E+01   1.39722E+01   9.87985E+00   9.87985E+00
  BR    PT  TY  LAB      PAR(01)      L2-NORM         U(01)         U(02)
   2    84  LP    7   2.53727E+01   5.91179E+00   5.88408E+00   5.71738E-01
   2   104  LP    8   2.66221E+01   8.84510E+00   8.78280E+00   1.04795E+00
   2   172  LP    9   2.21816E+01   6.21472E+00   4.39452E+00   4.39443E+00
   2   173  BP   10   2.21825E+01   6.21537E+00   4.34437E+00   4.44492E+00
   2   227  LP   11   2.66221E+01   8.84511E+00   1.04796E+00   8.78281E+00
   2   250  EP   12   2.62164E+01   8.09079E+00   7.85840E-01   8.05254E+00

 Total Time    0.531E+00
>

Data is contained in the autof7 object.

a{1}.f7
ans = 

  autof7

  Properties:
       Ibr: [500x1 double]
      Mtot: [500x1 double]
       Itp: [500x1 double]
       Lab: [500x1 double]
       Par: [500x1 double]
    L2norm: [500x1 double]
         U: [500x2 double]
       Out: [500x0 double]


Special points are contained in the autof8 object.

a{1}.f8
ans = 

  autof8

  Properties:
       Ibr: [12x1 double]
      Mtot: [12x1 double]
       Itp: [12x1 double]
       Lab: [12x1 double]
      Nfpr: [12x1 double]
       Isw: [12x1 double]
      Ntpl: [12x1 double]
       Nar: [12x1 double]
    Nrowpr: [12x1 double]
      Ntst: [12x1 double]
      Ncol: [12x1 double]
     Nparx: [12x1 double]
      Ifpr: []
         T: [12x1 double]
        Tm: []
       Par: [12x36 double]
     Rldot: []
         U: [12x2 double]
       Ups: []
    Udotps: []


Create plot object and plot diagram.

  • Blue solid lines represent stable solutions
  • Red dashed lines represent unstable solutions
p=plautobj;
set(p,'xLab','Par','yLab','L2norm');
ploteq(p,a{1});