Preliminaries

This section provides the rigorous mathematical framework for the following discussion. It can be skipped by readers not interested in the mathematical detail. Our discussion applies to autonomous differential systems, whose mathematical formulation is as follows: we consider an open domain $D \subset \Re^{n}$, a function $F \in C^{\infty } (D,\Re^{n})$, a real interval $L$, and a function $x \in C^{1} (L,D)$ such that

$$\forall t \in L, \frac{\mathrm{d}x(t)}{\mathrm{d}t} = F(x(t))$$ (1)

Vector $x$ describes the state of the system, and vector $F(x)$ gives the direction in which the system will evolve when it starts at point $x$ of its state space. Points where $F$ vanishes, and where the system maintains an equilibrium, are stationary states. We suppose that no stationary state of $F$ is degenerate (which implies that stationary states are isolated). Let $J_{F} (x)$ be the Jacobian matrix of $F$ at $x \in D$. $J_{F} (x)$ can be considered to define an oriented, weighted interaction graph on the set $I=\{ 1, .., n\}$, in which an arc points from $i$ to $j$ if and only if $\frac{\partial F_{j}}{\partial x_{i}} (x) \ne 0$ (i.e. if variable $j$ depends on variable $i$), the associated weight being $\frac{\partial F_{j}}{\partial x_i} (x)$ (reciprocally, by reversing this process, a unique matrix $M$ is associated to a weighted interaction-graph, by setting the value of $M_{i,j}$ to the weight of the arc pointing from $j$ to $i$ if it exists, and 0 if it doesn't exist). There is a circuit involving nodes $i_1,..,i_k$ if and only if $\prod_{j=2}^{k+1} \frac{\partial F_{i_j}}{\partial x_{i_{j-1}}}$, where $i_{k+1}=i_1$. A same node is not allowed to appear more than once in the graph circuits considered in the following (i.e., in the previous example, $\forall\ p,q\ \in\ \{1..k\}\ \mathrm{s.t.}\ \ p\ne q,\ i_p\ne i_q$); circuits comprising a single arc (from a vertex to itself) are allowed, and correspond to diagonal, autocatalysis term in the Jacobian, and a variable exerting a direct influence (positive or negative) on itself. The sign of a circuit (positive or negative) is defined as the sign of the product of the weights of the associated arcs; i.e., a circuit is negative if it has an odd number of negative interactions, and positive otherwise. Finally, the weight of a circuit is defined as the product of the weights of its arcs.

A variable is part of a feedback circuit if its corresponding vertex in the interaction graph is part of at least one circuit; if there is such a positive (respectively negative) circuit, the variable is under positive (respectively negative) feedback control. It is possible for a variable to be part of a negative and a positive feedback circuit at the same time.

It should be noted that we consider general feedback circuits whose length can be greater than 1, and that the sign of the circuit is not necessarily the same as the sign of the interaction that links a later part of a pathway to the earlier part (for example, if one considers a biochemical synthesis pathway, the end-product P can inhibit an early enzyme E of the pathway, as in models considered in [2], but this inhibition can still be part of a positive feedback circuit, if E itself inhibits another enzyme that leads to the creation of P).

Note also that, except in the trivial linear case (which doesn't give rise to interesting behaviours such as non-degenerate multistationarity), the Jacobian matrix is not constant within the domain $D$; it is thus possible, and likely, that the weights and the signs of the circuits are not constant, and even that the structure of the interaction graph itself is not constant. Note also that if there are two distinct mechanisms by which a variable depends upon another (these mechanisms possibly being antagonistic), it is the sum of the corresponding derivatives which will appear in the relevant Jacobian term. For example, if a protein whose concentration corresponds to $x_i$ exerts a positive but saturable effect on its own synthesis (positive autocatalysis, or auto-activation), and undergoes exponential decay (negative autocatalysis, or auto-inhibition), such that

$$\frac{\mathrm{d}x_i}{\mathrm{d}t} = - x_i + v \frac{x_i}{K+x_i},\ v>0,\ K>0,$$

then the overall autocatalysis for $x_i$, corresponding to the Jacobian term $J_{F} (x)_{i,i},$ will be positive for low values of $x_i$ (provided that $\frac{v}{K}>1$), and negative for high values of $x_i$ (for example for $x_i>v$).

We will call stationary states stable if and only if they are locally asymptotically stable, i.e. if and only if all eigenvalues of the Jacobian matrix have strictly negative real parts. Due to the strong dependency of the characteristic polynomial of a matrix on its circuit structure, stability exclusively depends on interactions which are part of a feedback circuit; it is only the weight of a circuit which is important (not the weights of its particular elements), because only they appear in the development of the determinant giving the characteristic polynomial. In the following, we will consider stability of dynamical systems around steady states.

While studying the effect of feedback circuits on a particular system, one is confronted with the problem that changes in system parameters lead to changes in weights of feedback circuits at given points of the state space, but stationary states are also displaced as a result of the change of parameter values, and the weight of feedback circuits at the stationary states is thus not always readily controllable. It can thus be that increasing the strength of the interactions in the equations governing a system does not lead to increased feedback circuit weights at the stationary states of the system.