## Coupled firstorder reactions

As a prelude to modelling sequences of enzyme-catalyzed reactions, consider a sequence of only two reactions. This configuration is said to be a coupled reaction.

From Section 1.3.1, and by analogy with Eqn [1.10], the rate equations that describe the kinetics of this coupled system are d [A]

Q: Is it possible to solve the system of differential equations defined by Eqns [1.16 - 1.18] by using the function DSolve?

A: Yes, the program is as follows:

Use DSolve to solve the first equation .

firstDE = DSolve [{a' [t] & - k1 a[t] , a & a0} , a[t] , t]

Use this solution to define a function for the time course of a[t].

Next, solve the second and third differential equations and define the appropriate functions for b[t] and c[t].

secondDE = DSolve[{b' [t] & k1 a[t] - k2 b[t] ,b & b0} ,b[t] , t] b[t_] = b[t] /. secondDE;

thirdDE = DSolve[{c' [t] & k2 b[t] , c & c0} ,c[t] , t] c[t_] =c[t] /. thirdDE;

(e—t (0-5i-0-52)-t0-52 (-a0 0.51 + a0 et (0-5i-0-52) 0.51 +

b0 et (0-5i—0-52L 0.51 - b0 et (0-5i-0-52L 0.52))}}

1 (e—t0-51 (a0 et0-5l0-51 + b0 et0-5l0-51 + c0 et0-5l0-51 —

a0 et (0-51—10-52) 0-51 — b0 et (0-51—1^^ 0-5x + a0 0-52 — a0 et0-51 0-52 — b0 et 0-51 0-52 — c0 et0-51 0-52 + b0 et (0-51—10-52 L 0-52 ))}}

Hence, after having defined the parameter values and initial conditions, we plot the time courses of a[t] , b[t] , and c[t] as follows:

AxesLabel -> { "Time ", "Concentration" }] ;

Concentration

Concentration Figure 1.4. Simulation of the time course of the reaction scheme described by Eqn [1.15] using DSolve, with the differential equations in Eqns [1.16 - 1.18]. The ordinate denotes concentration in mmol L-1 and the abscissa, time in seconds.

In the previous worked example each differential equation was solved sequentially, with the solution of the first giving the expression for the dependent variable [A]. This was then fed into the analysis for [B] and so on. Performing the integration in sequence like this was done only for pedagogical purposes. However, it is possible with the function DSolve to enter a complete list that contains all the relevant expressions for the differential equations and initial conditions, and to solve them simultaneously. This is done as follows, where we use a different set of parameter values from the example above so that different time courses are obtained.

Q: Can the differential equations in the previous question be solved simultaneously using DSolve?

A: Yes, DSolve can be used in the following way:

solution = DSolve[ {a' [t] & -k1 a[t] , b' [t] & k1 a [t] - k2 b[t] , c' [t] & k2b[t] , a & a0, b & b0, c & c0} , {a[t] , b[t] , c[t]}, t]

{{a[t] + a0 e-t °-5i, b[t] ^ _ 0. 5 0. ^ (e-t o.5l-t o.52

(-a0 et0'51 0.51 _ bO et0'51 0.51 + aO et0'52 0 . 51 + bO et0'51 0.52)) , c[t] ^ \$ \$ 1 \$ \$ (e_t0-51-t0-52 (a0 et0'51 0.51 + b0 et0'51 0.51 _ -0.51 + 0.52

a0 et0-51+t 0-52 0 . 51 _ b0 et0-51+t 0-52 0 . 51 _ c0 et0-51+t0-52 0 . 51 _ b0 et0'51 0.52 _ a0 et0'52 0 . 52 + a0 et0-51+t0-52 0.52 +

b0 "t0*51+t 0.52 0 . 52 + c0 "t0'51+t 0*52 0 . 52 ))= =

By specifying the appropriate functions, parameter values, and initial conditions, the results of the integration can be plotted.

a[t_] = a[t] /. solution; b[t_] = b [t] /. solution; c[t ] =c[t] /. solution;

AxesLabel -> {"Time ", "Concentration" }] ;

Concentration

Concentration Figure 1.5. Simulation of the time course of the reaction scheme described by Eqn [1.15] using DSolve with the differential equations in Eqns [1.16 - 1.18]; but unlike the analysis for Fig. 1.3, the equations were solved simultaneously. The ordinate denotes concentration in mmol L-1 and the abscissa, time in seconds.

Figure 1.5. Simulation of the time course of the reaction scheme described by Eqn [1.15] using DSolve with the differential equations in Eqns [1.16 - 1.18]; but unlike the analysis for Fig. 1.3, the equations were solved simultaneously. The ordinate denotes concentration in mmol L-1 and the abscissa, time in seconds.