Steady states, eigenvalues and circuits

Stability at stationary states is usually studied in terms of eigenvalues of the Jacobian matrix. This is the most appropriate approach from the mathematical point of view, but not from the biological one, from which one considers interactions between different components, and especially feedback circuits. The following lemma relates matrix eigenvalues to qualitative, circuit-related properties.

Lemma 1   Let $M$ be a square, non-degenerate matrix of dimension $n$. If $\mathrm{sgn}(\det M)\ne (-1)^n$, then $M$ has a positive circuit. If $M$ is degenerate, then either $M$ has no decomposition in circuits or $M$ has a positive circuit.

The proof of lemma 1 is technical but simple; it has been given by Plahte et al. (1995). In some cases, it is possible to derive a lower bound on the weight of the positive circuit for the sign of the determinant to be different from $(-1)^n$. We do not show the detail of the demonstration because the result one gets is heavily dependent on the particular structure of the matrix. However, we will illustrate such a lower bound on positive feedback in section 6. The following corollary (whose proof can be found in the appendix) gives an application of lemma 1 which has an intuitive interpretation: if the system's motion in a certain direction is amplified (the direction being associated to a strictly positive eigenvalue), there must be some positive feedback in the matrix.

Corollary 1   Let $M$ be a square, non-degenerate matrix of dimension $n$. If $M$ has at least one real, strictly positive eigenvalue, then $M$ has a positive circuit.

Note however that the existence of an eigenvalue with a strictly positive real part is not a sufficient condition for the existence of a positive circuit, as illustrated by the matrix $(0,1,0;0,0,1;-1,0,0)$, which has a negative circuit as single circuit, but two complex eigenvalues with strictly positive real parts. Thus, instability at a steady state is not necessarily provided by a positive feedback circuit. Using the same mathematical approach as for the above lemma, and as was pointed out by Levins (1975), one can derive constraints on the weights of feedback circuits at a stable steady state:

Lemma 2   Let $s \in D$ be a stable ss, $M=J_{F}(s)$ be the Jacobian matrix of $F$ at point $s$. We have

$$\forall l = 1..n, (-1)^{l}\frac{\sum_{L\subset I,\mid L \mid=l}\det M_L}{(^{n}_{l})} > 0$$

One can see directly from the demonstration of lemma 1 that positive feedback circuits contribute negative terms to the sum in lemma 2. Other things being equal, positive feedback circuits are thus destabilising above a certain weight (in other cases, positive feedback can actually be stabilising and negative feedback destabilising, see Cinquin & Demongeot for details). This allows to derive constraints on the weight of positive feedback circuits at stable stationary states, but one must bear in mind that the weight of feedback circuits cannot usually be altered without displacing stationary states. Thus, if we consider a ss of $F$ $s$, we have the following possibilities:

• $s$ is unstable, with a real eigenvalue being at least partly responsible for the unstability; in this case, there is a positive circuit in the Jacobian matrix
• $s$ is unstable only because of pairs of complex eigenvalues with positive real parts. In this case, the unstability stems from groups of 2 state variables which oscillate in an expanding fashion. In many regulation systems there is a key "switch" which integrates diverse information sources, and activates effectors when it reaches a critical threshold. The switch has to be upregulated when it reaches the threshold, so that there is a clear distinction between the "on" and "off" states. If the source of unstability is the switch-molecule, it is therefore unlikely that it is associated to a complex eigenvalue, because there would be an oscillation above and under the threshold.
• $s$ is stable; in this case, it is possible to derive constraints on weights of the circuits, following the approach detailed by Levins (1975).

The fact that there can be unstable ss without real, positive eigenvalues is interesting per se, but will not affect the extent of the existence results presented in the following sections.