These examples show that, while it is generally true that by adding enough negative feedback one can stabilise any given system (see below), and that by adding enough positive feedback one can destabilise any given system, the behaviour between extremes is more complicated (and it should be because the stability of a variable can depend on a coefficient very far away in the interaction graph). This fact becomes even more obvious when one calculates the Routh-Hurwitz equations for a given system. These equations provide necessary and sufficient inequalities on the coefficients of the Jacobian matrix of a system for this system to be stable. These inequalities, which with current mathematical knowledge can only marginally be generically simplified, are quite complex even if the underlying system is highly structured in a simple fashion. The Routh-Hurwitz inequalities for the system described in Figure 2 are shown in Table 1.
Relationships between Jacobian matrix structures and biological or chemical system behaviours have already been investigated extensively (see for example [12] or [13], or more recently [14] and [15]). The problem of the stability of qualitative matrices (matrices grouped according to the signs of their coefficients) has also received extensive attention in the past, before computers made their way into everyday scientific life; important results are summarised in [16], and we review quickly how those results apply to feedback circuits and stability. Results relevant to our discussion are that if a system comprises a positive autocatalysis loop, a positive circuit of length 2, or any circuit, positive or negative, of length 3 or more, then the system can be made unstable by altering weights of interactions while strictly conserving their signs (this is a direct application of lemma 5.1 in [16]). To make system stability a property robust against changes in values of its interaction weights, an interesting goal from the evolutionary standpoint because it frees the system from certain particular kinetic values, it thus seems necessary to limit to a strict minimum the number of negative feedback circuits of length greater than 2. Positive feedback, if present, can always destabilise the system if its amplitude is sufficiently augmented; however, as will be detailed below, the existence of positive feedback is a necessary condition for certain useful behaviours of regulation systems, and thus often cannot be avoided. Furthermore, it is remarkable that if one modifies a stable system by introducing positive autocatalysis on variables previously not autocatalysed, the resulting system can always be brought back to stability by altering interaction weights without changing their sign (theorem 5.5 in [16]).
The most practical way to deal with the Routh-Hurwitz inequalities seems to be by means of computerised, symbolic computing. We have developed an automated system based on the symbolic computing program MAPLE (and especially on the "Hurwitz" package of MAPLE), to test whether a system of given dimension, whose coefficients are kept symbolic, can be stable for a given set of values of its coefficients (details to be reported elsewhere). This system is useful for proving relationships between system structure and system stability for a given system dimension, and conjecturing that the relationships are also valid for higher dimensions. A conjecture we have come to is the following: if one wants to stabilise a given negative circuit of length strictly greater than two, and if one is allowed negative circuits of length of 2 or 1, then local, negative feedback has to be added to all of the variables but one. This tends to show that stabilisation is best dealt with locally. From an evolutionary standpoint, it should be less costly to provide each system variable with local, negative feedback, rather than to ensure that long feedback circuits are maintained in such a way as to stabilise variables with no local feedback. Local, negative feedback could be provided by specific and non-specific proteolysis (which would thus not only serve the purposes of automatically removing partially degraded proteins or transducing certain signals).
To conclude this discussion of the respective relationships of positive and negative feedback to stability, we point out that autocatalysis has a special role in that it must be negative on average for the system to be stable (the trace of a matrix is the sum of the real parts of its eigenvalues), and that the theorem of the "dominant diagonal" (a well-known theorem from linear algebra, also mentioned in [16]), has the intuitive interpretation that if for each variable the autocatalysis loop has a negative (respectively positive) weight sufficiently higher in absolute value than the sum of other interactions affecting the variable, then the system is stable (respectively unstable).
