Existence of global solutions of a macroscopic model of cellular motion in a chemotactic field (original) (raw)
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A Multiscale Model of Cell Motion in a Chemotactic Field
2006
The Cellular Potts Model (CPM) has been used at a cellular scale for simulating various biological phenomena such as differential adhesion, fruiting body formation of the slime mold Dictyostelium discoideum, angiogenesis, cancer invasion, chondrogenesis in embryonic vertebrate limbs, and many others. Continuous models in the form of partial differential, integral or integro-differential equations are used for studying biological problems at large scale. It is crucial for developing multi-scale biological models to establish a connection between discrete stochastic models, including CPM, and continuous models. To demonstrate multiscale approach we derive in this paper continuous limit of a two dimensional CPM with the chemotactic interactions in the form of a Fokker-Planck equation describing evolution of the cell probability density function. This equation is then reduced to the classical macroscopic Keller-Segel model. Theoretical results are verified numerically by comparing Monte...
Physical Review E, 2006
The Cellular Potts Model (CPM) has been used for simulating various biological phenomena such as differential adhesion, fruiting body formation of the slime mold Dictyostelium discoideum, angiogenesis, cancer invasion, chondrogenesis in embryonic vertebrate limbs, and many others. In this paper, we derive continuous limit of discrete one dimensional CPM with the chemotactic interactions between cells in the form of a Fokker-Planck equation for the evolution of the cell probability density function. This equation is then reduced to the classical macroscopic Keller-Segel model. In particular, all coefficients of the Keller-Segel model are obtained from parameters of the CPM. Theoretical results are verified numerically by comparing Monte Carlo simulations for the CPM with numerics for the Keller-Segel model.
Well-posedness for chemotaxis dynamics with nonlinear cell diffusion
Journal of Mathematical Analysis and Applications, 2013
This paper investigates the well-posedness of a reaction-diffusion system of chemotaxis type, with a nonlinear diffusion coefficient and a dynamics (growth-death) of the cell population b, and a stationary equation for the chemoattractant c. With respect to other works in which a nonlinear diffusion for cells has been considered, we treat here two distinct cases for this diffusion coefficient, the first in which it is a positive bounded function on R, and the other in which it may display a singularity at a finite value of the cell density. Essentially, the latter model is new and describes the saturation of the cell population in the neighborhood of a critical value for its diffusion coefficient. The chemotactic sensitivity function is supposed to depend both on the cell and chemoattractant densities. For homogeneous Neumann boundary conditions for the cell population and chemoattractant we prove the existence of a local in time solution when the L 2-norm of the initial datum b 0 is sufficiently small and compute the maximum time interval for which the solution is bounded and smooth. Under a stronger assumption related to the chemotactic sensitivity we show that there exists a global in time solution for arbitrarily large initial data. In a case when the diffusion coefficient is singular, we focus on a model expressed by a variational inequality, describing the saturation of the cell population at the blowing-up diffusion value. Here, the proof requires the study of an intermediate problem with Robin boundary conditions, which may be interesting by itself. In all situations, uniqueness follows on a time interval included (not necessarily strictly) in that of the solution existence, under sufficient conditions.
Classification and stability of global inhomogeneous solutions of a macroscopic model of cell motion
Mathematical Biosciences, 2012
Many micro-organisms use chemotaxis for aggregation, resulting in stable patterns. In this paper, the amoeba Dictyostelium discoideum serves as a model organism for understanding the conditions for aggregation and classification of resulting patterns. To accomplish this, a 1D nonlinear diffusion equation with chemotaxis that models amoeba behavior is analyzed. A classification of the steady state solutions is presented, and a Lyapunov functional is used to determine conditions for stability of inhomogenous solutions. Changing the chemical sensitivity, production rate of the chemical attractant, or domain length can cause the system to transition from having an asymptotic steady state, to having asymptotically stable single-step solution and multi-stepped stable plateau solutions.
Continuous Macroscopic Limit of a Discrete Stochastic Model for Interaction of Living Cells
Physical Review Letters, 2007
We derive a continuous limit of a two-dimensional stochastic cellular Potts model (CPM) describing cells moving in a medium and reacting to each other through direct contact, cell-cell adhesion, and longrange chemotaxis. All coefficients of the general macroscopic model in the form of a Fokker-Planck equation describing evolution of the cell probability density function are derived from parameters of the CPM. A very good agreement is demonstrated between CPM Monte Carlo simulations and a numerical solution of the macroscopic model. It is also shown that, in the absence of contact cell-cell interactions, the obtained model reduces to the classical macroscopic Keller-Segel model. A general multiscale approach is demonstrated by simulating spongy bone formation, suggesting that self-organizing physical mechanisms can account for this developmental process.
arXiv (Cornell University), 2023
A general class of hybrid models has been introduced recently, gathering the advantages of multiscale descriptions. Concerning biological applications, the particular coupled structure fits to collective cell migrations and pattern formation phenomena due to intercellular and chemotactic stimuli. In this context, cells are modelled as discrete entities and their dynamics are given by ODEs, while the chemical signal influencing the motion is considered as a continuous signal which solves a diffusive equation. From the analytical point of view, this class of model has been recently proved to have a mean-field limit in the Wasserstein distance towards a system given by the coupling of a Vlasov-type equation with the chemoattractant equation. Moreover, a pressureless nonlocal Eulertype system has been derived for these models, rigorously equivalent to the Vlasov one for monokinetic initial data. For applications, the monokinetic assumption is quite strong and far from real experimental setting. The aim of this paper is to introduce a numerical approach to the hybrid coupled structure at the different scales, investigating the case of general initial data. Several scenarios will be presented, aiming at exploring the role of the different terms on the overall dynamics. Finally, the pressureless nonlocal Euler-type system is generalized by means of an additional pressure term.
Global convergence of a kinetic model of chemotaxis to a perturbed Keller–Segel model
Nonlinear Analysis-theory Methods & Applications, 2006
We consider a class of kinetic models of chemotaxis with two positive non-dimensional parameters coupled to a parabolic equation of the chemo-attractant. If both parameters are set equal zero, we have the classical Keller-Segel model for chemotaxis. We prove global existence of solutions of this two-parameters kinetic model and prove convergence of this model to models of chemotaxis with global existence when one of these two parameters is set equal zero. In one case, we find as a limit model a kinetic model of chemotaxis while in the other case we find a perturbed Keller-Segel model with global existence of solutions. á§
An Asymptotic Preserving Scheme for the Diffusive Limit of Kinetic Systems for Chemotaxis
Multiscale Modeling & Simulation, 2013
In this work we numerically study the diffusive limit of run & tumble kinetic models for cell motion due to chemotaxis by means of asymptotic preserving schemes. It is well-known that the diffusive limit of these models leads to the classical Patlak-Keller-Segel macroscopic model for chemotaxis. We will show that the proposed scheme is able to accurately approximate the solutions before blow-up time for small parameter. Moreover, the numerical results indicate that the global solutions of the kinetic models stabilize for long times to steady states for all the analyzed parameter range. We also generalize these asymptotic preserving schemes to two dimensional kinetic models in the radial case. The blow-up of solutions is numerically investigated in all these cases.
Eigenfunction Approach to Transient Patterns in a Model of Chemotaxis
Mathematical Modelling of Natural Phenomena, 2016
In the paper we examine solutions to a model of cell movement governed by the chemotaxis phenomenon derived in [14] and established via macroscopic limits of corresponding microscopic cell-based models with extended cell representations. The model is given by two PDEs for the density of cells and the concentration of a chemical. To avoid singularities in cell density, the aggregating force of chemotaxis phenomenon is attenuated by a density dependent diffusion of cells, which grows to infinity with density tending to a certain critical value. In this paper we recover the quasi-periodic structures provided by this model by means of (local in time) expansion of the solution into a basis of eigenfunctions of the linearized system. Both planar and spherical geometries are considered.
Kinetic Models for Chemotaxis and their Drift-Diffusion Limits
Monatshefte Fur Mathematik, 2004
Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemo-attractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a drift-diffusion model is proven. The drift-diffusion models derived in this way include the classical Keller-Segel model. Furthermore, sufficient conditions for kinetic models are given such that finite-time-blow-up does not occur. Examples are given satisfying these conditions, whereas the macroscopic limit problem is known to exhibit finite-time-blow-up. The main analytical tools are entropy techniques for the macroscopic limit as well as results from potential theory for the control of the chemo-attractant density.