Template:ETH Sensitivity Matrices

From 2006.igem.org

To analyze the sensitivity of the system for all concerned parameters, we compute the sensitivity matrix S:

S = (∂x/x) / (∂p/p) = (∂x/∂p) * (p/x)
 
S : sensitivity matrix, #x rows, #p columns
x : states (concentrations)
p : parameters

We use jacobian matrices of the system equations to compute the sensitivity matrix S. We therefore augment the set of differential equations by

dS(t)/dt = Jx(t) S(t) + Jp(t)

Jx(t) = ∂f(x,p,t)/∂x
Jp(t) = ∂f(x,p,t)/∂p

Since we are interested in the output sensitivities at steady state, we set dS(t)/dt = 0 and can solve for S:

S = − Jx \ Jp

\ : matrix left division