Vitaly Volpert

Traveling Wave Solutions of Parabolic Systems

Part I. Stationary waves: Scalar equation Leray-Schauder degree Existence of waves Structure of the spectrum Stability and approach to a wave Part II. Bifurcation of waves: Bifurcation of nonstationary modes of wave propagation Mathematical proofs Part III. Waves in chemical kinetics and combustion: Waves in chemical kinetics Combustion waves with complex kinetics Estimates and asymptotics of the speed of combustion waves Asymptotic and approximate analytical methods in combustion problems (supplement) Bibliography.

The effect of convection on a propagating front with a liquid product: Comparison of theory and experiments

This work is devoted to the investigation of propagating polymerization fronts converting a liquid monomer into a liquid polymer. We consider a simplified mathematical model which consists of the heat equation and equation for the depth of conversion for one-step chemical reaction and of the Navier-Stokes equations under the Boussinesq approximation. We fulfill the linear stability analysis of the stationary propagating front and find conditions of convective and thermal instabilities. We show that convection can occur not only for ascending fronts but also for descending fronts. Though in the latter case the exothermic chemical reaction heats the cold monomer from above, the instability appears and can be explained by the interaction of chemical reaction with hydrodynamics. Hydrodynamics changes also conditions of the thermal instability. The front propagating upwards becomes less stable than without convection, the front propagating downwards more stable. The theoretical results are compared with experiments. The experimentally measured stability boundary for polymerization of benzyl acrylate in dimethyl formamide is well approximated by the theoretical stability boundary. (c) 1998 American Institute of Physics.

Gap Junctional Blockade Stochastically Induces Different Species-Specific Head Anatomies in Genetically Wild-Type Girardia dorotocephala Flatworms

The shape of an animal body plan is constructed from protein components encoded by the genome. However, bioelectric networks composed of many cell types have their own intrinsic dynamics, and can drive distinct morphological outcomes during embryogenesis and regeneration. Planarian flatworms are a popular system for exploring body plan patterning due to their regenerative capacity, but despite considerable molecular information regarding stem cell differentiation and basic axial patterning, very little is known about how distinct head shapes are produced. Here, we show that after decapitation in G. dorotocephala, a transient perturbation of physiological connectivity among cells (using the gap junction blocker octanol) can result in regenerated heads with quite different shapes, stochastically matching other known species of planaria (S. mediterranea, D. japonica, and P. felina). We use morphometric analysis to quantify the ability of physiological network perturbations to induce different species-specific head shapes from the same genome. Moreover, we present a computational agent-based model of cell and physical dynamics during regeneration that quantitatively reproduces the observed shape changes. Morphological alterations induced in a genomically wild-type G. dorotocephala during regeneration include not only the shape of the head but also the morphology of the brain, the characteristic distribution of adult stem cells (neoblasts), and the bioelectric gradients of resting potential within the anterior tissues. Interestingly, the shape change is not permanent; after regeneration is complete, intact animals remodel back to G. dorotocephala-appropriate head shape within several weeks in a secondary phase of remodeling following initial complete regeneration. We present a conceptual model to guide future work to delineate the molecular mechanisms by which bioelectric networks stochastically select among a small set of discrete head morphologies. Taken together, these data and analyses shed light on important physiological modifiers of morphological information in dictating species-specific shape, and reveal them to be a novel instructive input into head patterning in regenerating planaria.

Mathematical model of evolutionary branching

This work is devoted to the study of an evolutionary system where similar individuals are competing for the same resources. Mathematically it is a Fisher equation with an integral term describing this non-local competition. Due to this competition, an initially monomorphic population may split into two distinct sub-populations, hence exhibiting a branching capacity. This framework can be applied to various contexts where recognizers are competing for some signals. The pattern formation capacity of this model is investigated analytically and numerically.

Modelling of thrombus growth in flow with a DPD-PDE method.

Hemostatic plug covering the injury site (or a thrombus in the pathological case) is formed due to the complex interaction of aggregating platelets with biochemical reactions in plasma that participate in blood coagulation. The mechanisms that control clot growth and which lead to growth arrest are not yet completely understood. We model them with numerical simulations based on a hybrid DPD-PDE model. Dissipative particle dynamics (DPD) is used to model plasma flow with platelets while fibrin concentration is described by a simplified reaction-diffusion-advection equation. The model takes into account consecutive stages of clot growth. First, a platelet is weakly connected to the clot and after some time this connection becomes stronger due to other surface receptors involved in platelet adhesion. At the same time, the fibrin mesh is formed inside the clot. This becomes possible because flow does not penetrate the clot and cannot wash out the reactants participating in blood coagulation. Platelets covered by the fibrin mesh cannot attach new platelets. Modelling shows that the growth of a hemostatic plug can stop as a result of its exterior part being removed by the flow thus exposing its non-adhesive core to the flow.

Mathematical study of feedback control roles and relevance in stress erythropoiesis.

This work is devoted to mathematical modelling of erythropoiesis. We propose a new multi-scale model, in which we bring together erythroid progenitor dynamics and intracellular regulatory network that determines erythroid cell fate. All erythroid progenitors are divided into several sub-populations according to their maturity. Two intracellular proteins, Erk and Fas, are supposed to be determinant for regulation of self-renewal, differentiation and apoptosis. We consider two growth factors, erythropoietin and glucocorticoids, and describe their dynamics. Several feedback controls are introduced in the model. We carry out computer simulations of anaemia and compare the obtained results with available experimental data on induced anaemia in mice. The main objective of this work is to evaluate the roles of the feedback controls in order to provide more insights into the regulation of erythropoiesis. Feedback by Epo on apoptosis is shown to be determinant in the early stages of the response to anaemia, whereas regulation through intracellular regulatory network, based on Erk and Fas, appears to operate on a long-term scale.

Solvability conditions for some non-Fredholm operators

We obtain solvability conditions for some elliptic equations involving non Fredholm operators with the methods of spectral theory and scattering theory for Schrodinger type operators. Though the Fredholm property is not satisfied, the solvability conditions are formulated in terms of orthogonality of the right-hand side to solutions of the homogeneous adjoint equation.

Properness and topological degree for general elliptic operators

The paper is devoted to general elliptic operators in Holder spaces in bounded or unbounded domains. We discuss the Fredholm property of linear operators and properness of nonlinear operators. We construct a topological degree for Fredholm and proper operators of index zero.

Modeling erythroblastic islands: Using a hybrid model to assess the function of central macrophage

The production and regulation of red blood cells, erythropoiesis, occurs in the bone marrow where erythroid cells proliferate and differentiate within particular structures, called erythroblastic islands. A typical structure of these islands consists of a macrophage (white cell) surrounded by immature erythroid cells (progenitors), with more mature cells on the periphery of the island, ready to leave the bone marrow and enter the bloodstream. A hybrid model, coupling a continuous model (ordinary differential equations) describing intracellular regulation through competition of two key proteins, to a discrete spatial model describing cell-cell interactions, with growth factor diffusion in the medium described by a continuous model (partial differential equations), is proposed to investigate the role of the central macrophage in normal erythropoiesis. Intracellular competition of the two proteins leads the erythroid cell to either proliferation, differentiation, or death by apoptosis. This approach allows considering spatial aspects of erythropoiesis, involved for instance in the occurrence of cellular interactions or the access to external factors, as well as dynamics of intracellular and extracellular scales of this complex cellular process, accounting for stochasticity in cell cycle durations and orientation of the mitotic spindle. The analysis of the model shows a strong effect of the central macrophage on the stability of an erythroblastic island, when assuming the macrophage releases pro-survival cytokines. Even though it is not clear whether or not erythroblastic island stability must be required, investigation of the model concludes that stability improves responsiveness of the model, hence stressing out the potential relevance of the central macrophage in normal erythropoiesis.

Mathematical modelling of atherosclerosis as an inflammatory disease

Atherosclerosis is an inflammatory disease. The atherosclerosis process starts when low-density lipoproteins (LDLs) enter the intima of the blood vessel, where they are oxidized (ox-LDLs). The anti-inflammatory response triggers the recruitment of monocytes. Once in the intima, the monocytes are transformed into macrophages and foam cells, leading to the production of inflammatory cytokines and further recruitment of monocytes. This auto-amplified process leads to the formation of an atherosclerotic plaque and, possibly, to its rupture. In this paper we develop two mathematical models based on reaction–diffusion equations in order to explain the inflammatory process. The first model is one-dimensional: it does not consider the intima’s thickness and shows that low ox-LDL concentrations in the intima do not lead to a chronic inflammatory reaction. Intermediate ox-LDL concentrations correspond to a bistable system, which can lead to a travelling wave that can be initiated by certain conditions, such as infection or injury. High ox-LDL concentrations correspond to a monostable system, and even a small perturbation of the non-inflammatory case leads to travelling-wave propagation, which corresponds to a chronic inflammatory response. The second model we suggest is two-dimensional: it represents a reaction–diffusion system in a strip with nonlinear boundary conditions to describe the recruitment of monocytes as a function of the cytokines’ concentration. We prove the existence of travelling waves and confirm our previous results, which show that atherosclerosis develops as a reaction–diffusion wave. The results of the two models are confirmed by numerical simulations. The latter show that the two-dimensional model converges to the one-dimensional one if the thickness of the intima tends to zero.