Stability of open pathways

We consider the steady state of an open biochemical pathway, with controlled flow. Previously we have shown that the steady state of open enzyme catalysed reactions may be unstable, which discourages the application of the quasi-steady-state approximation (QSSA) [IEE Proc. – Syst. Biol. 153 (2006) 187]. Here we examine basic open biochemical pathway structures, to see the stability of their steady states. Following De Leenheer et al. [J. Math. Chem. 41 (2007) 295], we employ the Gershgorin circle theorem, which elegantly assesses stability. This is the key tool for our analysis. Once we have the linear stability matrix laid out in a suitable form, the application of the method is straightforward. We find that in open biochemical pathways, simple chains, branches and loops always have stable steady states. We conclude that simple open pathways are stable.