The following is a simplified reaction scheme for the competitive inhibitors with recombinant human renin (Morelock et al. 1995).
Output 19.10.1: Competitive Inhibition of Recombinant Human Renin
In Output 19.10.1, E=enzyme, D=probe, and I=inhibitor.
The differential equations that describe this reaction scheme are as follows:





For this system, the initial values for the concentrations are derived from equilibrium considerations (as a function of parameters) or are provided as known values.
The experiment used to collect the data was carried out in two ways; preincubation (type='disassoc') and no preincubation (type='assoc'). The data also contain repeated measurements. The data contain values for fluorescence F, which is a function of concentration. Since there are no direct data for the concentrations, all the differential equations are simulated dynamically.
The SAS statements used to fit this model are as follows:
title1 'Systems of Differential Equations Example'; proc sort data=fit; by type time; run; %let k1f = 6.85e6 ; %let k1r = 3.43e4 ; %let k2f = 1.8e7 ; %let k2r = 2.1e2 ; %let qf = 2.1e8 ; %let qb = 4.0e9 ; %let dt = 5.0e7 ; %let et = 5.0e8 ; %let it = 8.05e6 ;
proc model data=fit; parameters qf = 2.1e8 qb = 4.0e9 k2f = 1.8e5 k2r = 2.1e3 l = 0; k1f = 6.85e6; k1r = 3.43e4; /* Initial values for concentrations */ control dt 5.0e7 et 5.0e8 it 8.05e6; /* Association initial values */ if type = 'assoc' and time=0 then do; ed = 0; /* solve quadratic equation */ a = 1; b = (&it+&et+(k2r/k2f)); c = &it*&et; ei = (b(((b**2)(4*a*c))**.5))/(2*a); d = &dted; i = &itei; e = &etedei; end; /* Disassociation initial values */ if type = 'disassoc' and time=0 then do; ei = 0; a = 1; b = (&dt+&et+(&k1r/&k1f)); c = &dt*&et; ed = (b(((b**2)(4*a*c))**.5))/(2*a); d = &dted; i = &itei; e = &etedei; end; if time ne 0 then do; dert.d = k1r* ed  k1f *e *d; dert.ed = k1f* e *d  k1r*ed; dert.e = k1r* ed  k1f* e * d + k2r * ei  k2f * e *i; dert.ei = k2f* e *i  k2r * ei; dert.i = k2r * ei  k2f* e *i; end; /* L  offset between curves */ if type = 'disassoc' then F = (qf*(ded)) + (qb*ed) L; else F = (qf*(ded)) + (qb*ed); fit F / method=marquardt; run;
This estimation requires the repeated simulation of a system of 41 differential equations (5 base differential equations and 36 differential equations to compute the partials with respect to the parameters).
The results of the estimation are shown in Output 19.10.2.
Output 19.10.2: Kinetics Estimation
Systems of Differential Equations Example 
Nonlinear OLS Summary of Residual Errors  

Equation  DF Model  DF Error  SSE  MSE  Root MSE  RSquare  Adj RSq 
f  5  797  2525.0  3.1681  1.7799  0.9980  0.9980 
Nonlinear OLS Parameter Estimates  

Parameter  Estimate  Approx Std Err  t Value  Approx Pr > t 
qf  2.0413E8  681443  299.55  <.0001 
qb  4.2263E9  9133195  462.74  <.0001 
k2f  6451186  866998  7.44  <.0001 
k2r  0.007808  0.00103  7.55  <.0001 
l  5.76974  0.4138  13.94  <.0001 