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
