Preliminaries

In the following, 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. We will use the abbreviation ss for stationary states of the system, i.e. the points where $F$ vanishes, and where the system maintains an equilibrium. We will suppose throughout this paper that no ss of $F$ is degenerate. Let $J_{F}(x)$ be the Jacobian matrix of $F$ at $x \in D$ (i.e. the matrix of partial derivatives of the field components with respect to system variables), and $\mathrm{sgn}(J_{F}(x))$ the matrix obtained by taking the sign of each component of $J_{F}(x)$ ($+1$, 0, or $-1$). $\mathrm{sgn}(J_{F}(x))$ (respectively $J_{F}(x)$) 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 $\frac{\partial F_{j}}{\partial x_i}(x)$, i.e. the strength of the dependency of $F_{j}$ on $x_i$. There is a circuit involving nodes $i_1,..,i_k$ if and only if $\prod_{j=2}^{k+1} \frac{\partial F_{i_j}}{\partial x_{i_{j-1}}} \ne 0$, 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, $\forall p,q \in \{1..k\} \mathrm{s.t.} p\ne q, i_p\ne i_q$). In the interaction graph, a node is part of a circuit if and only if the corresponding variable is part of a feedback circuit, i.e. if its own value affects the way it will change, either directly or through a chain of dependencies. Circuits comprising a single arc (from a vertex to itself) are allowed, and correspond to a diagonal 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 (in which case the corresponding variables are under negative feedback control), and positive otherwise (in which case the corresponding variables are under positive feedback control). Finally, the weight of a circuit is defined as the product of the weights of its arcs; the higher the weight, the higher the feedback intensity. Note 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), and undergoes exponential decay (negative autocatalysis), 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 $x_i$. Finally, the following terminology will be useful in later sections:

• A matrix $M$ can be decomposed into circuits if there exists a subset of the arcs of its interaction graph such that considering this subset, every vertex is part of a single circuit; it can be easily shown that a matrix can be split into circuits if and only if there is a non-zero term in the development of its determinant
• $F$ is inward-pointing if and only if there exists a domain containing all steady states of the system, such that the frontier $\partial$ of the domain is diffeomorphic to the unit ball of $\Re^n$ (meaning intuitively that the frontier is smooth and its shape is sufficiently close to that of a ball), and such that $F$ points strictly toward the interior of the domain at points of $\partial$ (this means that if one chooses a smooth vector field $\chi$ normal to $\partial$ and respecting its canonical orientation, then $\forall x \in \partial, <\chi,F>(x)<0)$
• We will call a ss $s$ stable if and only if it is locally asymptotically stable, i.e. if all eigenvalues of $J_{F}(s)$ have a strictly negative real part.