Elimination of fast variables in stochastic nonlinear kinetics.

A reduced chemical scheme involving a small number of variables is often sufficient to account for the deterministic evolution of the concentration of the main species contributing to a reaction. However, its predictions are questionable in small systems used, for example in fluorescence correlation spectroscopy (FCS) or in explosive systems involving strong nonlinearities such as autocatalytic steps. We make precise dynamical criteria defining the validity domain of the quasi-steady-state approximation and the elimination of a fast concentration in deterministic dynamics.

Fisher-Kolmogorov-Petrovskii-Piskunov wave front as a sensor of perturbed diffusion in concentrated systems.

The sensitivity to perturbations of the Fisher and Kolmogorov, Petrovskii, Piskunov front is used to find a quantity revealing perturbations of diffusion in a concentrated solution of two chemical species with different diffusivities. The deterministic dynamics includes cross-diffusion terms due to the deviation from the dilution limit. The behaviors of the front speed, the shift between the concentration profiles of the two species, and the width of the reactive zone are investigated, both analytically and numerically.

How many enzyme molecules are needed for discrimination oriented applications?

Chemical reactions establish a molecular mechanism for information processing in living organisms. Here we consider a simple enzymatic reaction model that can be used to discriminate parameters characterizing periodic reagent inflow. Numerical simulations based on the kinetic equations show that there exist a range of inflow frequencies and amplitudes in which the time evolution of the system is very sensitive to small changes in the values of these parameters.

Modeling somite scaling in small embryos in the framework of Turing patterns.

The adaptation of prevertebra size to embryo size is investigated in the framework of a reaction-diffusion model involving a Turing pattern. The reaction scheme and Fick's first law of diffusion are modified in order to take into account the departure from dilute conditions induced by confinement in smaller embryos. In agreement with the experimental observations of scaling in somitogenesis, our model predicts the formation of smaller prevertebrae or somites in smaller embryos.

Discrimination of time-dependent inflow properties with a cooperative dynamical system.

Many physical, chemical, and biological systems exhibit a cooperative or sigmoidal response with respect to the input. In biochemistry, such behavior is called an allosteric effect. Here, we demonstrate that a system with such properties can be used to discriminate the amplitude or frequency of an external periodic perturbation.

Nanoscale Turing structures.

Formation of Turing patterns of nanoscopic length scale is simulated using molecular dynamics. Based on Fourier spectra of the concentrations of species, we compare stabilities of the structures of different wavelengths and for different intermolecular potentials. Long range attraction is shown to oppose the formation of structures.

Coherence resonances in an excitable thermochemical system with multiple stationary states.

A master equation approach is used to study the influence of internal fluctuations on the dynamics of three excitable thermochemical systems exhibiting continuous as well as discrete changes of temperature. The systems differ by the types of excitability. The dependences of the relative deviations from mean values of the interspike intervals and escape times from the stable stationary state on the size of the systems calculated from simulations of stochastic trajectories exhibit minima, which testify to the appearance of resonance phenomena.

Stochastic transitions through unstable limit cycles in a model of bistable thermochemical system.

The master equation approach is used to study transitions through an unstable limit cycle surrounding a stable focus in two-variable systems with three stationary states. The model considered describes a bistable thermochemical system. Two cases are studied.

Master equation simulations of bistable and excitable dynamics in a model of a thermochemical system.

The effect of fluctuations on the dynamics of a model of a bistable thermochemical system is studied by means of the master equation. The system has three stationary states and exhibits two types of bistability: the coexistence of two stable focuses and the coexistence of a stable focus with a stable limit cycle separated by a saddle point. Stochastic effects are important when the system is close to the bifurcation, in which the stable limit cycle disappears through a homoclinic orbit.

Multipeak distributions of first passage times in bistable dynamics in a model of a thermochemical system.

A master equation is used to study transitions between the stable limit cycle and stable focus in the two-variable bistable system. The distribution function of the mean first passage time between these attractors and the relative dispersion of the mean first return time from the stable focus to itself as a function of the intensity of fluctuations are calculated and discussed. A coherence resonance is observed for the return time from the focus to itself.

Master equation simulations of a model of a thermochemical system.

Master equation approach is used to study the influence of fluctuations on the dynamics of a model thermochemical system. For appropriate values of parameters, the deterministic description of the system gives the subcritical or supercritical Hopf bifurcations. For small systems (containing 100 000 particles) close to the supercritical Hopf bifurcation, the stochastic trajectories obtained from numerical simulations do not allow to distinguish between damped oscillations around a stable focus and sustained oscillations around a small stable limit cycle.