Multistationarity, the basis of cell differentiation and memory. II. Logical analysis of regulatory networks in terms of feedback circuits

Circuits and their involvement in complex dynamics are described in differe ntial terms in Part I of this work. Here, we first explain why it may be ap propriate to use a logical description, either by itself or in symbiosis wi th the differential description. The major problem of a logical description is to find an adequate way to involve time. The procedure we adopted diffe rs radically from the classical one by its fully asynchronous character. In Sec. II we describe our "naive" logical approach, and use it to illustrate the major laws of circuitry (namely, the involvement of positive circuits in multistationarity and of negative circuits in periodicity) and in a biol ogical example. Already in the naive description, the major steps of the lo gical description are to: (i) describe a model as a set of logical equation s, (ii) derive the state table from the equations, (iii) derive the graph o f the sequences of states from the state table, and (iv) determine which of the possible pathways will be actually followed in terms of time delays. I n the following sections we consider multivalued variables where required, the introduction of logical parameters and of logical values ascribed to th e thresholds, and the concept of characteristic state of a circuit. This ge neralized logical description provides an image whose qualitative fit with the differential description is quite remarkable. A major interest of the g eneralized logical description is that it implies a limited and often quite small number of possible combinations of values of the logical parameters. The space of the logical parameters is thus cut into a limited number of b oxes, each of which is characterized by a defined qualitative behavior of t he system. Our analysis tells which constraints on the logical parameters m ust be fulfilled in order for any circuit (or combination of circuits) to b e functional. Functionality of a circuit will result in multistationarity ( in the case of a positive circuit) or in a cycle (in the case of a negative circuit). The last sections deal with "more about time delays" and "revers e logic," an approach that aims to proceed rationally from facts to models. (C) 2001 American Institute of Physics.