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In this paper, a two-dimensional model for the growth of multi-layer tumors is presented. The model consists of a free boundary problem for the tumor cell membrane and the tumor is supposed to grow or shrink due to cell proliferation or…
Cellular differentiation in a developping organism is studied via a discrete bistable reaction-diffusion model. A system of undifferentiated cells is allowed to receive an inductive signal emenating from its environment. Depending on the…
A dynamical systems scenario for developmental cell biology is proposed, based on numerical studies of a system with interacting units with internal dynamics and reproduction. Diversification, formation of discrete and recursive types, and…
We explore a mechanism of pattern formation arising in processes described by a system of a single reaction-diffusion equation coupled with ordinary differential equations. Such systems of equations arise from the modeling of interactions…
We study diffusion-controlled processes in nonequilibrium steady states, where standard rate theory assumptions break down. Using transition path theory, we generalize the relations between reactive probability fluxes and measures of the…
A general theory is developed for the evolution of the cell order (CO) distribution in planar granular systems. Dynamic equations are constructed and solved in closed form for several examples: systems under compression; dilation of very…
In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally…
Diffusion is a fundamental phenomenon that occurs ubiquitously in nature and remains the subject of continuous research interest. Understanding diffusion is a key to understanding leaving systems. In this Chapter, I discuss diffusion of…
The impact of quenched disorder on deterministic diffusion in chaotic dynamical systems is studied. As a simple example, we consider piecewise linear maps on the line. In computer simulations we find a complicated scenario of multiple…
The wavelength of a periodic spatial structure of Turing type is an intrinsic property of the considered reaction-diffusion dynamics and we address the question of its control at the microscopic scale for given dynamical parameters. The…
Cells coexist together in colonies or as tissues. Their behaviour is controlled by an interplay between intercellular forces and biochemical regulation. We develop a simple model of the cell cycle, the fundamental regulatory network…
Phase separation under non-equilibrium conditions is exploited by biological cells to organize their cytoplasm but remains poorly understood as a physical phenomenon. Here, we study a ternary fluid model in which phase-separating molecules…
The propagation of unstable interfaces is at the origin of remarkable patterns that are observed in various areas of science as chemical reactions, phase transitions, growth of bacterial colonies. Since a scalar equation generates usually…
We propose a simple mathematical model to describe the mechanical relaxation of cells within a curved epithelial tissue layer represented by an arbitrary curve in two-dimensional space. This model generalises previous one-dimensional models…
In this paper, we investigate a mathematical model describing the interactions between effector cells (E), cancer cells (T), and the IL-2 compound (IL). The model considered here is a generalization, taking into account some cross-diffusion…
Isologous diversification theory for cell differentiation is proposed, based on simulations of interacting cells with biochemical networks and cell division process following consumption of some chemicals. According to the simulations of…
We study the consequences of deterministic chaos for diffusion-controlled reaction. As an example, we analyze a diffusive-reactive deterministic multibaker and a parameter-dependent variation of it. We construct the diffusive and the…
In many biological processes heterogeneity within cell populations is an important issue. In this work we consider populations where the behavior of every single cell can be described by a system of ordinary differential equations.…
Nucleation and growth is studied in a system undergoing diffusion-controlled condensation under gradual changes in parameters, such as cooling. It is demonstrated that when Gibbs-Thompson effect becomes negligible, the system falls into a…
We propose a model for cell polarization as a response to an external signal which results in a system of PDEs for different variants of a protein on the cell surface and interior respectively. We study stationary states of this model in…