In the field of biological regulation, models dictated by experimental work are usually complex networks comprising intertwined feedback loops. In this paper the biological roles of individual positive loops (multistationarity, differentiation) and negative loops (homeostasis, with or without oscillations, buffering of gene dosage effect) are discussed. The relationship between feedback loops and steady states is then clarified, and the problem: "How can one conveniently disentangle complex networks?" is then considered. Initiated long ago, logical descriptions have been generalized from various viewpoints; these developments are briefly discussed. The recent concept of the loop-characteristic state, defined as the logical state located at the level of the thresholds involved in the loop, together with its application, are then presented. Biological applications are also discussed.
Circuits and their involvement in complex dynamics are described in differential terms in Part I of this work. Here, we first explain why it may be appropriate to use a logical description, either by itself or in symbiosis with the differential description. The major problem of a logical description is to find an adequate way to involve time. The procedure we adopted differs 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 biological example. Already in the naive description, the major steps of the logical description are to: (i) describe a model as a set of logical equations, (ii) derive the state table from the equations, (iii) derive the graph of 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. In the following sections we consider multivalued variables where required, the introduction of logical parameters and of logical values ascribed to the thresholds, and the concept of characteristic state of a circuit. This generalized logical description provides an image whose qualitative fit with the differential description is quite remarkable. A major interest of the generalized 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 boxes, each of which is characterized by a defined qualitative behavior of the system. Our analysis tells which constraints on the logical parameters must be fulfilled in order for any circuit (or combination of circuits) to be 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 "reverse logic," an approach that aims to proceed rationally from facts to models. (c) 2001 American Institute of Physics.
A biological introduction serves to remind us that differentiation is an epigenetic process, that multistationarity can account for epigenetic differences, including those involved in cell differentiation, and that positive feedback circuits are a necessary condition for multistationarity and, by inference, for differentiation. The core of the paper is comprised of a formal description of feedback circuits and unions of disjoint circuits. We introduce the concepts of full-circuit (a circuit or union of disjoint circuits which involves all the variables of the system), and of ambiguous circuit (a circuit whose sign depends on the location in phase space). We describe the partition of phase space (a) according to the signs of the ambiguous circuits, and (b) according to the signs of the eigenvalues or their real part. We introduce a normalization of the system versus one of the circuits; in two variables, this permits an entirely general description in terms of a common diagram in the "circuit space." The paper ends with general statements concerning the requirements for multistationarity, stable periodicity, and deterministic chaos. (c) 2001 American Institute of Physics.
We investigate the dynamical properties of a simple four-variable model describing the interactions between the tumour suppressor protein p53, its main negative regulator Mdm2 and DNA damage, a model inspired by the work of Ciliberto et al. [2005. Steady states and oscillations in the p53/Mdm2 network. Cell Cycle 4(3), 488-493]. Its core consists of an antagonist circuit between p53 and nuclear Mdm2 embedded in a three-element negative circuit involving p53, cytoplasmic and nuclear Mdm2. A major concern has been to develop an integrated approach in which various types of descriptions complement each other. Here we present the logical analysis of our network and briefly discuss the corresponding differential model. Introducing the new notion of "logical bifurcation diagrams", we show that the essential qualitative dynamical properties of our network can be summarized by a small number of bifurcation scenarios, which can be understood in terms of the balance between the positive and negative circuits of the core network. The model displays a wide variety of behaviours depending on the level of damage, the efficiency of damage repair and, importantly, the DNA-binding affinity and transcriptional activity of p53, which are both stress- and cell-type specific. Our results qualitatively account for several experimental observations such as p53 pulses after irradiation, failure to respond to irradiation, shifts in the frequency of the oscillations, or rapid dampening of the oscillations in a cell population. They also suggest a great variability of behaviour from cell to cell and between different cell-types on the basis of different post-translational modifications and transactivation properties of p53. Finally, our differential analysis provides an interpretation of the high and low frequency oscillations observed by Geva-Zatorsky et al. [2006. Oscillations and variability in the p53 system. Mol. Syst. Biol. 2, 2006.0033] depending on the irradiation dose. A more detailed analysis of our differential model as well as its stochastic analysis will be developed in a next paper.
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