Cycles involving covalent modification of proteins are key components of the intracellular signaling machinery. Each cycle is comprised of two interconvertable forms of a particular protein. A classic signaling pathway is structured by a chain or cascade of basic cycle units in such a way that the activated protein in one cycle promotes the activation of the next protein in the chain, and so on. Starting from a mechanistic kinetic description and using a careful perturbation analysis, we have derived, to our knowledge for the first time, a consistent approximation of the chain with one variable per cycle. The model we derive is distinct from the one that has been in use in the literature for several years, which is a phenomenological extension of the Goldbeter-Koshland biochemical switch. Even though much has been done regarding the mathematical modeling of these systems, our contribution fills a gap between existing models and, in doing so, we have unveiled critical new properties of this type of signaling cascades. A key feature of our new model is that a negative feedback emerges naturally, exerted between each cycle and its predecessor. Due to this negative feedback, the system displays damped temporal oscillations under constant stimulation and, most important, propagates perturbations both forwards and backwards. This last attribute challenges the widespread notion of unidirectionality in signaling cascades. Concrete examples of applications to MAPK cascades are discussed. All these properties are shared by the complete mechanistic description and our simplified model, but not by previously derived phenomenological models of signaling cascades.

We address the issue of spatially localized periodic oscillations in coupled networks-so-called discrete breathers-in a general context. This context is concerned with general conditions which allow continuation of periodic solutions of vector fields. One advantage of our approach is to encompass in the same mathematical framework the cases of conservative and dissipative systems. An essential feature is that we allow the period to vary. In particular, we deduce existence of discrete breathers in networks where each site has an equilibrium and some sites have a limit cycle, and in Hamiltonian networks without requiring local anharmonicity. The latter case is dealt with by considering the persistence of families of periodic solutions in the more general context of systems with an integral, not just Hamiltonian ones.

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We consider simple examples illustrating some new features of the linear response theory developed by Ruelle for dissipative and chaotic systems [J. of Stat. Phys. 95 (1999) 393]. In this theory the concepts of linear response, susceptibility and resonance, which are familiar to physicists, have been revisited due to the dynamical contraction of the whole phase space onto attractors. In particular the standard framework of the "fluctuation-dissipation" theorem breaks down and new resonances can show up oustside the powerspectrum. In previous papers we proposed and used new numerical methods to demonstrate the presence of the new resonances predicted by Ruelle in a model of chaotic neural network. In this article we deal with simpler models which can be worked out analytically in order to gain more insights into the genesis of the "stable" resonances and their consequences on the linear response of the system. We consider a class of 2-dimensional time-discrete maps describing simple rotator models with a contracting radial dynamics onto the unit circle and a chaotic angular dynamics θ t+1 = 2θ t (mod 2π). A generalisation of this system to a network of interconnected rotators is also analysed and related with our previous studies [8,11]. These models permit us to classify the different types of resonances in the susceptibility and to discuss in particular the relation between the relaxation time of the system to equilibrium with the mixing time given by the decay of the correlation functions. Also it enables one to propose some general mechanisms responsible for the creation of stable resonances with arbitrary frequencies, widths, and dependency on the pair of perturbed/observed variables.

A dynamic mathematical model has been developed and validated to describe the synthesis of pectate lyases (Pels), the major virulence factors in Dickeya dadantii. This work focuses on the simultaneous modeling of the metabolic degradation of pectin by Pel enzymes and the genetic regulation of pel genes by 2-keto-3-deoxygluconate (KDG), a catabolite product of pectin that inactivates KdgR, one of the main repressors of pel genes. This modeling scheme takes into account the fact that the system is composed of two time-varying compartments: the extracellular medium, where Pel enzymes cleave pectin into oligomers, and the bacterial cytoplasm where, after internalization, oligomers are converted to KDG. Using the quasi-stationary state approximations, the model consists of some nonlinear differential equations for which most of the parameters could be estimated from the literature or from independent experiments. The few remaining unknown parameters were obtained by fitting the model equations against a set of Pel activity data. Model predictions were verified by measuring the time courses of bacterial growth, Pel production, pel mRNA accumulation, and pectin consumption under various growth conditions. This work reveals that pectin is almost totally consumed before the burst of Pel production. This paradoxical behavior can be interpreted as an evolutionary strategy to control the diffusion process so that as soon as a small amount of pectin is detected by the bacteria in its surroundings, it anticipates more pectin to come. The model also predicts the possibility of bistable steady states in the presence of constant pectin compounds. Dickeya dadantii (ex Erwinia chrysanthemi) is a soft rottingGram-negative bacterium that attacks a wide range of plant species, including many crops of economical importance. These bacteria are found on plant surfaces and in soil where they may enter the plant via wound sites or through natural openings. During infection, D. dadantii first colonizes the intercellular space (apoplast) where they can remain latent until conditions become favorable for the development of the disease. Soft rot, the visible symptom, is mainly due to the degradation of pectin present in the plant cell wall. D. dadantii can utilize pectin as its sole carbon and energy source (Fig.