CURRENT AND EARLIER RESEARCH

1) Early work in experimental genetics.

a) discovery of DNA denaturation (1951).

Proper data about the extinction coefficients of bases, nucleosides and nucleotides became available in the late forties. We then constructed a theoretical spectrum of DNA, based on the spectral characteristics of its constituents. It was immediately apparent that the UV absorption of native DNA is by far (ca 40%) lower than expected. That this is not due to a mistake, appeared from the fact that the spectrum of hydrolyzed DNA fits perfectly with the expectation. It turned out soon that various mild treatment known to have no effect on the internucleotidic linkages result in a massive and partly irreversible increase of the UV extinction coefficient at and around 2600 A (hyperchromic effect). The only possible interpretation was that DNA has a secondary structure of weak bounds (hydrogen bounds or Van der Waals interactions) that collapses (denaturation) as a result of a variety of mild treatments. This was soon confirmed by Meselson and Stahl, who showed that our conditions result in a separation of the two strands of the Watson-Crick double helix.

b) Trans-activation and demonstration of the occurrence of positive controls in the regulation of transcription (1966).

Temperate bacteriophages are viruses that have a choice between two fates after infecting bacteria. Either they multiply, kill and lyse their host, or they establish a permanent symbiosis with it. In this case, the viral DNA becomes inserted in the bacterial chromosome. In this form, it is called a "prophage". The resulting organism, a so-called lysogenic bacterium, carries hereditarily the viral DNA as a prophage. In this condition, one of the viral genes (called cI) produces a repressor that blocks the expression of all the other genes of the virus, as shown by François Jacob. This explains why a lysogenic bacterium can live normally in spite of the presence of the viral genome. In addition, the presence of the repressor in its cytoplasm renders it immune to any further infection by a phage of the same specificity of immunity. There exist, however, so-called "heteroimmune" phages that are identical with the preceding one, except precisely for the specificity of immunity. These heteroimmune phage can infect successfully even a lysogenic bacterium. It turned out that when a lysogenic bacterium is infected by a heteroimmune phage, a number of prophege genes are switched on in spite of the persistence of the repressor (trans-activation, initially called "trans-induction"). This definitely shows that in a lysogenic bacterium most of the prophage genes are not blocked by the repressor directly. Rather, they are blocked because the repressor has prevented the synthesis of gene products required to switch them on. In other words, these "trans-inducible" genes are subject to positive control.

2) Asynchronous logical description of regulatory networks.

Admittedly, any biological process is exceedingly complex and depends on a number of variables. Yet, among these many variables, a small number are presumably determinant in the overall operation of the system. If this view is correct, a major aim of research should be to identify these crucial variables, to disentangle their interactions and in this way to try understanding the essential qualitative features of systems.

The most obvious way to model biological regulatory systems consists of a description in terms of differential equations. Unfortunately, even if one uses a limited number of variables, an analytical description is almost always precluded by the nonlinear character of some interactions. It is thus tempting to simplify the description thanks to an acceptable idealization. The linear idealization is catastrophic, except in close vicinity of steady states: all the interesting features (multiple steady states, stable periodicity,...) of the system vanish in this description. An alternative idealization is suggested by the fact that many biological interactions have a sigmoid profile: an effector is essentially inefficient below a threshold concentration value, its effect increases rapidly around this threshold value and saturates for higher concentrations. If this sigmoid is steep enough, it is legitimate to reason in all-or-none terms, as if the effector was totally inefficient below a threshold value and fully efficient above this value. This leads to a discrete ("logical") description, in which, at least initially, a gene is considered "on" or "off" and a product, "present" or "absent". It must be clear that "absent" does not really mean that the concentration of the product is zero, but rather that it is below its threshold concentration. It is rewarding to observe that, in contrast with the linear idealization, the logical description preserves the essential features of the dynamics of systems (see the beautiful papers published in 1972-1973 by Glass & Kaufman in J. Theoret. Biol.).

Clearly, in order to be applicable to the description, analysis and synthesis of biological processes, a logical language has to include time in an appropriate way, especially in view of the fact (see below) that essential elements of regulatory systems interact in a cyclic way. With this in mind, we developed a logical tool that was from the beginning an asynchronous description. (Thomas, 1973). Concretely, when two or more genes are switched on (vs off) by the same signal, their products reach (vs fall below) their efficient concentration threshold after time delays that have no reason whatsoever to be equal. Essentially, the logical description involves:

a) a graph of the interactions
b) logical equations, derived from (a)
c) a graph of all the possible sequences of states, derived from (b)
d) a logical analysis of which sequences will actually be followed, in terms of inequalities between time delays
e) a reverse logical analysis (synthetic, inductive approach) aims to build models in a rational way from the data (instead of deducing pathways from a preexisting model)

Initially presented in what we now call its "naïve" form, the logical method has been generalized in various ways:

a) binary ("boolean") variables are generalized into n-level variables (in collaboration with P. Van Ham) where required.
b) E.H. Snoussi has introduced logical parameters, that render the description at the same time much more general and much more flexible.
c) the threshold values (s1, s2, ..., sn) have been introduced into the logical scale (0, s1, 1, s2, 2, ..., n). This permitted to identify all the steady states (stable or not) of a system in logical terms.
d) if one tells noting about the time delays, the whole graph of the sequences of states remains open. If instead one gives a well-defined value to each time delay, the system has to follow a unique, well-defined pathway. In practice, in a population of cells, each time delay has an average value and a distribution. This introduced stochastic aspects in the logical description and, in practice, allows one to operate in terms of simulations. A recent account of our logical method can be found in Thomas & Kaufman (2001 a).

One of the interests of a logical method is that parameter space consists of a finite (and often small) number of boxes, within each of which the system behaves in a qualitatively similar way. The major shortcoming of the logical methods is that they apply only to systems whose major interactions are sufficiently steep sigmoids. This is usually acceptable for biological systems, but not in general. For this reason, we turned in more recent work to the more general approach of the theory of circuits.

3) Nonlinear dynamics seen in terms of the theory of circuits.

Suppose that element x, for example, the product of a gene, exerts an influence on the rate of production of element y (one can thus draw an arrow x->y), which exerts an influence on the rate of production of element z (y->z), which in turn exerts an influence on the rate of production of the initial element x (z->x). In such a case, one says that x (but y or z as well) exerts a feedback on its own production via the other elements. We say that there exists a circuit x->y->z->x. This type of situation (a circular causality) turned out not only to be extremely frequent, but to be an essential feature of the operation of all regulated systems, in biology and elsewhere.
The development of the logical tool, described just above, was of a great help to understand how circuits operate, because they can be described in such a simple, almost skeletal, way that a great number of systems can be treated in a short time. Let us mention here already that what had been learned about the functions of circuits using the logical method has been since fully confirmed by using other (notably differential) descriptions.

A circuit can be positive (if each element exerts a positive action on its own production) or negative. In fact, a circuit is positive or negative depending on whether it comprises an even or an odd number of negative interactions.
It could be shown that the two types of circuits have contrasting roles. A negative circuit functions as a thermostat: it tends to keep its variables at or near supposedly optimal values usually far from both the "top"(heater on without control) and the "bottom" (heater off) levels.
This is homeostasis, with or without oscillations.
In contrast, a positive circuit will oblige its variables to chose and durably keep a value close either to the top or to the bottom value ("multistationarity"). This may seem absurd at first view (as if a thermostat had been trafficked so that the heater goes on functioning when temperature is already high and stopped functioning when temperature is too low!) but it is in fact essential for cell differentiation and memory. In differentiation, crucial genes are switched on or off durably by transient signals, and this requires that they be part of a positive circuit. Neurons forming a positive circuit have two stable states and can switch from the dormant state to an active state (evocation of a memory). It turned out that negative and positive circuits are not only the simple ways to generate homeostasis and multistationarity, respectively, but that they are in fact necessary conditions for these processes. This was expressed initially as conjectures (Thomas,1981, the second one recently generalized):

- the presence of a positive circuit in the graph of a system is a necessary condition for multistationarity (and thus, in biology, for differentiation and memory)
- the presence of a negative circuit in the graph of a system is a necessary condition for the existence of an attractor, punctual (stable state), periodic (limit cycle) or chaotic, and thus for homeostasis, with or without oscillations.
These conjectures have been subject recently to several formal demonstrations (Plahte et al. (1995), Snoussi (1998), Gouzé (1998), Cinquin & Demongeot (2002), Soulé (2003)...).

Interactions and circuits can be defined in a more rigorous way from the Jacobian matrix of the system. When an element aij of the Jacobian matrix is non-zero, it means that variable xj influences the rate of production of variable xI , and one can write an arrow xj -> xi in the graph of the system. A circuit is defined from a set of non-zero elements of the Jacobian matrix such that that the row (i) and column (j) indices form a circular permutation. The role of circuits in the dynamics of systems is made clear by the fact that only those terms of the Jacobian matrix that belong to a circuit are represented in the characteristic equation. Consequently, only those terms involved in one or more circuit influence the eigenvalues at a given location of phase space.
Of special interest are those circuits that imply all the variables of the system. We call them "nuclei". The list of the nuclei of a system is given by the terms of the determinant of its Jacobian matrix. An isolated nucleus can generate one or more steady states, whose nature (node, focus, saddle point, ...) is entirely determined by the sign patterns of the nucleus. One begins to understand how nuclei interact with each other to yield the global behavior of the system. Much of the behavior of complex systems can be predicted by examining the sign patterns of the nuclei present in the Jacobian matrix (see Thomas & Kaufman, 2004).