Positive and negative cycles in Boolean networks.

We review and discuss some results about the influence of positive and negative feedback cycles in asynchronous Boolean networks. These results merge several ideas of Thomas: positive and negative feedback cycles have been largely emphasized by Thomas, through the so called Thomas' rules, and asynchronous Boolean networks have been introduced by Thomas as a model for the dynamics of gene networks, which is nowadays very popular.

On circuit functionality in Boolean networks.

It has been proved, for several classes of continuous and discrete dynamical systems, that the presence of a positive (resp. negative) circuit in the interaction graph of a system is a necessary condition for the presence of multiple stable states (resp. a cyclic attractor). A positive (resp. negative) circuit is said to be functional when it "generates" several stable states (resp. a cyclic attractor). However, there are no definite mathematical frameworks translating the underlying meaning of "generates." Focusing on Boolean networks, we recall and propose some definitions concerning the notion of functionality along with associated mathematical results.

Influence of Dispersion Interactions on the Polymorphic Stability of Crystalline Oxides.

The accurate determination of relative phase stabilities using DFT methods is a significant challenge when some of these can vary by only a few kJ/mol. Here, we demonstrate that for a selection of oxides (TiO2, MnO2, and ZnO) the inclusion of dispersion interactions, accomplished using the DFT-D3 correction scheme, allows for the correct ordering and an improved calculation of the energy differences between polymorphic phases. The energetic correction provided is of the same order of magnitude as the energy difference between phases. D3-corrected hybrid functionals systematically yield results closest to experiment. We propose that the inclusion of dispersion interactions makes a significant contribution to the relative energetics of polymorphic phases, especially those with different densities, and should therefore be included for calculations of relative energies using DFT methods.

Stable periodicity and negative circuits in differential systems.

We provide a counter-example to a conjecture of René Thomas on the relationship between negative feedback circuits and stable periodicity in ordinary differential equation systems (Kaufman et al. in J Theor Biol 248:675-685, 2007). We also prove a weak version of this conjecture by using a theorem of Snoussi.

Modelization of the regulation of protein synthesis following fertilization in sea urchin shows requirement of two processes: a destabilization of eIF4E:4E-BP complex and a great stimulation of the 4E-BP-degradation mechanism, both rapamycin-sensitive.

Fertilization of sea urchin eggs involves an increase in protein synthesis associated with a decrease in the amount of the translation initiation inhibitor 4E-BP. A highly simple reaction model for the regulation of protein synthesis was built and was used to simulate the physiological changes in the total 4E-BP amount observed during time after fertilization. Our study evidenced that two changes occurring at fertilization are necessary to fit with experimental data. The first change was an 8-fold increase in the dissociation parameter (koff1) of the eIF4E:4E-BP complex. The second was an important 32.5-fold activation of the degradation mechanism of the protein 4E-BP. Additionally, the changes in both processes should occur in 5 min time interval post-fertilization. To validate the model, we checked that the kinetic of the predicted 4.2-fold increase of eIF4E:eIF4G complex concentration at fertilization matched the increase of protein synthesis experimentally observed after fertilization (6.6-fold, SD = 2.3, n = 8). The minimal model was also used to simulate changes observed after fertilization in the presence of rapamycin, a FRAP/mTOR inhibitor. The model showed that the eIF4E:4E-BP complex destabilization was impacted and surprisingly, that the mechanism of 4E-BP degradation was also strongly affected, therefore suggesting that both processes are controlled by the protein kinase FRAP/mTOR.

Boolean models of biosurfactants production in Pseudomonas fluorescens.

Cyclolipopeptides (CLPs) are biosurfactants produced by numerous Pseudomonas fluorescens strains. CLP production is known to be regulated at least by the GacA/GacS two-component pathway, but the full regulatory network is yet largely unknown. In the clinical strain MFN1032, CLP production is abolished by a mutation in the phospholipase C gene (plcC) and not restored by plcC complementation. Their production is also subject to phenotypic variation. We used a modelling approach with Boolean networks, which takes into account all these observations concerning CLP production without any assumption on the topology of the considered network. Intensive computation yielded numerous models that satisfy these properties. All models minimizing the number of components point to a bistability in CLP production, which requires the presence of a yet unknown key self-inducible regulator. Furthermore, all suggest that a set of yet unexplained phenotypic variants might also be due to this epigenetic switch. The simplest of these Boolean networks was used to propose a biological regulatory network for CLP production. This modelling approach has allowed a possible regulation to be unravelled and an unusual behaviour of CLP production in P. fluorescens to be explained.

Application of formal methods to biological regulatory networks: extending Thomas' asynchronous logical approach with temporal logic.

Based on the discrete definition of biological regulatory networks developed by René Thomas, we provide a computer science formal approach to treat temporal properties of biological regulatory networks, expressed in computational tree logic. It is then possible to build all the models satisfying a set of given temporal properties. Our approach is illustrated with the mucus production in Pseudomonas aeruginosa. This application of formal methods from computer science to biological regulatory networks should open the way to many other fruitful applications.

Epigenesis and dynamic similarity in two regulatory networks in Pseudomonas aeruginosa.

Mucoidy and cytotoxicity arise from two independent modifications of the phenotype of the bacterium Pseudomonas aeruginosa that contribute to the mortality and morbidity of cystic fibrosis. We show that, even though the transcriptional regulatory networks controlling both processes are quite different from a molecular or mechanistic point of view, they may be identical from a dynamic point of view: epigenesis may in both cases be the cause of the acquisition of these new phenotypes. This was highlighted by the identity of formal graphs modelling these networks. A mathematical framework based on formal methods from computer science was defined and implemented with a software environment. It allows an easy and rigorous validation and certification of these models and of the experimental methods that can be proposed to falsify or validate the underlying hypothesis.