We propose a reaction-diffusion model that describes in detail the cascade of molecular events during blood coagulation. In a reduced form, this model contains three equations in three variables, two of which are self-accelerated. One of these variables, an activator, behaves in a threshold manner. An inhibitor is also produced autocatalytically, but there is no inhibitor threshold, because it is generated only in the presence of the activator. All model variables are set to have equal diffusion coefficients. The model has a stable stationary trivial state, which is spatially uniform and an excitation threshold. A pulse of excitation runs from the point where the excitation threshold has been exceeded. The regime of its propagation depends on the model parameters. In a one-dimensional problem, the pulse either stops running at a certain distance from the excitation point, or it reaches the boundaries as an autowave. However, there is a parameter range where the pulse does not disappear after stopping and exists stationarily. The resulting steady-state profiles of the model variables are symmetrical relative to the center of the structure formed. (c) 2001 American Institute of Physics.

In certain experimental conditions, bacteria form complex spatial-temporal patterns. A striking example of such kind was reported by Budrene and Berg (1991), who observed a wide variety of different colony structures ranging from arrays of spots to radially oriented stripes or arrangements of more complex elongated spots, formed by Escherichia coli. We discuss the relevant mechanisms of intercellular regulation in bacterial colony which may cause pattern formation, and formulate the corresponding mathematical model. In numerical experiments a variety of patterns, observed in real systems, is reproduced. The dynamics of their formation is investigated.

Two mathematical models of clot growth in the fluid flows have been considered. The first one is the model of embolus growth in a wall-adjacent flow. The effect of hydrodynamic flows on proceeding chemical reactions and the backward effect of the growing clot on the flow are taken into account. The growing thrombus is assumed to be porous and having low permeability, that is in good agreement with experimental data. The exact solutions determining the distribution of a fluid velocity close to the embolus have been used. Numerical analysis of these solutions have demonstrated that hydrodynamic flows can essentially affect the processes of blood coagulation, and consequently on the clot structure. Their presence might lead to the destruction of chemical fronts having a cylindrical symmetry and formation of the so-called chemical spots. The second model describes the initial stage of thrombus growing in the hemorrhage into a natural internal space. It permits accounting for vessel geometry and provides studying the effects of geometric parameters on fluid flows and coagulation processes. The process of thrombus growth is shown to depend on the ratio of typical values of blood velocity in the vessel and rate of chemical reactions.

We constructed a mathematical model of clotting, which is based on a current view of the molecular pathways of blood coagulation. Several hypothetical reactions are introduced to allow accurate description of the spatio-temporal dynamics of blood clotting. The resulting model describes well all spatio-temporal aspects of clotting, as well as data obtained in the homogeneous systems.

Biological membranes are complex environments whose physico-chemical properties are of utmost importance for the understanding of many crucial biological processes. Much attention has been given in the literature to the description of membranes along the z-axis perpendicular to the membrane. Here, we instead consider the lateral dynamics of lipids and peripheral proteins due to their electrostatic interaction. Previously, we constructed a Monte Carlo automaton capable of simulating mutual diffusive dynamics of charged lipids and associated positively charged peptides. Here, we derive and numerically analyze a system of Poisson-Boltzmann-Nernst-Planck (PBNP) equations that provide a mean-field approximation compatible with our Monte Carlo model. The thorough comparison between the mean-field PBNP equations and Monte Carlo simulations demonstrates that both the approaches are in a good qualitative agreement in all tested scenarios. We find that the two methods quantitatively deviate when the local charge density is high, presumably because the Poisson-Boltzmann formalism is applicable in the so-called weak coupling limit, whose validity is restricted to low charge densities. Nevertheless, we conclude that the mean-field PBNP approach provides a good approximation for the considerably more detailed Monte Carlo model at only a fraction of the associated computational cost and allows simulation of the membrane lateral dynamics on the space and time scales relevant for the realistic biological problems.