Logical analysis of systems comprising feedback loops.

Abstract In genetic and other fields, one has more and more to deal with complex models involving feedback loops. The dynamic behaviour and final state(s) expected for such systems are often far from immediately obvious. In this work, a number of systems of increasing complexities have been analysed in terms of sequential logics; this allows a qualitative but rigorous treatment of systems in which the Boolean approximation is acceptable. For Boolean systems, we found the following generalizations: 1. (1) If one considers a system made of a linear, open (“chain”) or closed (“loop”) sequence of control elements, the final state(s) of the system depend essentially on the parity of the number of negative control elements involved. A chain or a loop will be qualified ”positive” or “negative” depending on whether it comprises an even or an odd number of negative elements. 2. (2) Positive loops can accommodate two stable steady states (and one unstable cycle which is of little practical interest). Negative loops are characterized by sustained oscillations. 3. (3) Grafting additional control chain(s) onto a feedback loop will let the final state of the system unchanged (two stable steady states for a positive loop, sustained oscillations for a negative loop) if one deals with a positive chain grafted with an AND connection, or a negative chain grafted with a OR connection. 4. (40 In contrast, if one grafts onto a feedback loop a positive control chain with a OR connection or a negative control chain with an AND connection, the. system will accommodate only a single stable steady state. 5. (5) Two conjugated loops, both positive or both negative, behave essentially like the corresponding simple loop; in the first case the system has a choice between two stable steady states, in the second case, one gets cycles. 6. (6) For two conjugated loops, one positive, one negative, the analysis predicts one stable steady state and cycles. A formal demonstration of these propositions can be found in Van Ham & Lasters (1978). With these simple rules, one can often predict the essential features of the behaviour of complex systems, just by reducing them to simpler “homologous” systems whose behaviour is known. So far, however, no simple rules have been found for multiple conjugated loops; in such cases, one has to proceed through the logical equations, matrices and graphs. This can be done “by hand” if the number of variables is not too high (say ⩽ 6); otherwise one can use an automatized procedure. The ranges of applicability of the discontinuous approximation used here and of the classical macroscopic continuous approximation (using differential equations) are discussed. The paper ends with a discussion about the biological interest of positive feedback loops, and especially their possible role in cell determination and differentiation.