LOGICAL IDENTIFICATION OF ALL STEADY-STATES - THE CONCEPT OF FEEDBACK LOOP CHARACTERISTIC STATES

Biological regulatory systems can be described in terms of non-linear differential equations or in logical terms (using an ''infinitely non-linear'' approximation). Until recently, only part of the steady states of a system could be identified on logical grounds. The reason was that steady states frequently have one or more variable located on a threshold (see below); those steady states were not detected because so far no logical status was assigned to threshold values. This is why we introduced logical scales with values 0, 1theta, 1, 2theta, 2, ..., in which 1theta, 2theta, ... are the logical values assigned to the successive thresholds of the scale. We thus have, in addition to the regular logical states, singular states in which one or more variables is located on a threshold. This permits identifying all the steady states on logical grounds. It was noticed that each feedback loop (or reunion of disjointed loops) can be characterized by a logical state located at the thresholds at which the variables of the loop operate. This led to the concept of loop-characteristic state, which, as we will see, enormously simplifies the analysis. The core of this paper is a formal demonstration that among the singular states of a system, only loop-characteristic states can be steady. Reciprocally, given a loop-characteristic state, there are parameter values for which this state is steady; in this case, the loop is effective (i.e. it generates multistationarity if it is a positive loop, homeostasis if it is a negative loop). This not only results in the above-mentioned radical simplification of the identification of the steady states, but in an entirely new view of the relation between feedback loops and steady states.