Systems Biology (Methods in Molecular Biology)

to make a change of variables from the dependent variablexto a

local variable. Now let:

xðtÞ¼xþEðtÞ, ð 23 Þ

where it is assumed thatE(t) 1, so that we can justify

dropping all terms of order two and higher in the expansion.

SubstitutingxðtÞ¼xþEðtÞinto the Right-Hand Side (RHS)

of the ODE yields

fðxðtÞÞ ¼ fðÞ¼xþEðÞt fðÞþx fðÞx EðÞþt fðÞx

E^2 ðtÞ

2

þ

¼ 0 þfðxÞEðtÞþOðE^2 Þ,

ð 24 Þ

and dropping higher order terms, we obtain

fðxÞfðxÞEðtÞ: ð 25 Þ

Note that dropping these higher order terms is valid sinceE(t)

1. Now substitutingxðtÞ¼xþEðtÞinto the Left-Hand Side

(LHS) of the ODE,

E^0 ðtÞ¼fðxÞEðtÞ: ð 26 Þ

The goal is to determine if we have growing or decaying solu-

tions. If the solutions grows, then the equilibrium point is

unstable. If the solution decays, then the fixed point is stable.

To determine whether or not the solution is stable or unstable

we simply solve the ODE and get the solution as

EðtÞ¼E 0 expððxÞEðtÞÞ, ð 27 Þ

whereE 0 is a constant. Hence, the solution is growing if

f_ðxÞ>0 and decaying iff_ðxÞ<0. As a result, the equilibrium

point is stable iff_ðxÞ<0, unstable iff_ðxÞ>0.

A first-order autonomous ODE with a parameterrhas the

general formdx/dt¼f(x,r). The fixed points are the values of

xfor whichf(x,r)¼0. A bifurcation occurs when the number

or the stability of the fixed points changes as system parameters

change. The classical types of bifurcations that occur in nonlin-

ear dynamical systems are produced from the following proto-

typical differential equations:

l saddle:dx/dt¼r+x^2. A saddle-node bifurcation or tangent

bifurcation is a collision and disappearance of two equilibria

in dynamical systems. In autonomous systems, this occurs

when the critical equilibrium has one zero eigenvalue. This

phenomenon is also called fold or limit point bifurcation. An

equilibrium solution (wherex¼0) is simplyx¼p\pm

ffiffiffi

r

p

.

Therefore, ifr<0, then we have no real solutions, ifr>

0, then we have two real solutions.

Inverse Problems in Systems Biology 85