Systems Biology (Methods in Molecular Biology)

concentration of the connected species are governed by the
generalized kinetic evolution equation:

∂Xi

∂t

¼FiðÞX 1 ;X 2 ;...;Xn,i¼1,...,n ð 7 Þ

where the corresponding vector form of Eq.7 is∂∂Xt¼FXðÞ.Fis a
vector of any nonlinear function including reaction and diffusion of
the species. Here, the partial, instead of ordinary, differential nota-
tion is used to consider diffusion processes, where the species’
concentrations could also vary with space. For simplicity, we will
now ignore the diffusion process and revert to the ordinary form,
assuming a well-mixed homogenous condition. Equation 7
becomes

dXi

dt

¼FiðÞX 1 ;X 2 ;...;Xn,i¼1,...,n ð 8 Þ

For illustration with an example, letn¼3 and each reaction be
governed by first-order mass action kinetics

X 1!

k 1

X 2!

k 2

X 3

Equation8 for the above can be written as

dX 1

dt

¼k 1 ½ŠðX 1 9 Þ

dX 2

dt

¼k 1 ½ŠX 1 k 2 ½ŠX 2 ð 10 Þ

dX 3

dt

¼k 2 ½ŠðX 2 11 Þ

In a closed system, as there is no exchange of species to external
environments, the total masses of all species at any time will be
constant. Also, the rates of reactions,kvalues, are assumed con-
stant. Therefore, the summation of the three differential equations
9 to 11 will be zero. This simplest linear system will remain stable
for all positive real values of rate constants or species concentration.
Figure3a shows the concentrations of species with time for a
selected initial condition and parameter values.
Let us now consider the second reaction,X 2 toX 3 , utilizes an
enzymatic catalyst (X 1 toX 2 remains unchanged), Eqs.10 and 11
become:

dX 2

dt

¼k 1 ½ŠX 1

Vmax½ŠX 2

KMþ½ŠX 2

ð 12 Þ

dX 3

dt

¼

Vmax½ŠX 2

KMþ½ŠX 2

ð 13 Þ

Although the reaction kinetics has changed and follows a
hyperbolic relation, the system still remains linear and stable for

Complex Biological Responses Using Simple Models 177