Keller Segel Equation Research Articles

Solving a system of fractional partial differential equations arising in the model of HIV infection of CD4+ cells and attractor one-dimensional Keller-Segel equations

In this paper, we make use of the relatively new analytical technique, the homotopy decomposition method (HDM), to solve a system of fractional nonlinear differential equations that arise in the model for HIV infection of CD4+ T cells and attractor one-dimensional Keller-Segel equations. The technique is described and illustrated with a numerical example. The reliability of HDM and the reduction in computations give HDM a wider applicability. In addition, the calculations involved in HDM are very simple and straightforward.

Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller–Segel equation

Starting from the quantitative stability result of Bianchi and Egnell for the 2-Sobolev inequality, we deduce several different stability results for a Gagliardo-Nirenberg-Sobolev inequality in the plane. Then, exploiting the connection between this inequality and a fast diffusion equation, we get a quantitative stability for the Log-HLS inequality. Finally, using all these estimates, we prove a quantitative convergence result for the critical mass Keller-Segel system.

A multi-cellular network of metabolically active E. coli as a weak gel of living Janus particles

Bioluminescence images of nutrient rich liquid cultures of lux-gene reporter Escherichia coli were recorded for several hours after being placed into small diameter cylindrical containers (glass tubes and microtiter plate wells). It was found that luminous cells distribute near the three-phase contact line forming an irregular array of clumps, channels and plumes at the solid–liquid interface. The experimentally observed quasi-2-dimensional spatiotemporal patterns (‘venation patterns’) were simulated fairly well by the mean field Keller–Segel equations of chemotactic aggregation. The equations use a logistic cell growth term whose carrying capacity depends on oxygen concentration. The experimental and numerical results are interpreted as follows: (1) the patterns of bioluminescence form due to arrested phase separation in the culture; (2) the luminous phase formed along the upper part of the solid–liquid interface is a weak gel-like network of metabolically active bacteria which exhibit ligand–receptor type cell–cell adhesion and high rates of oxidative phosphorylation; (3) active bacteria in this gel can be viewed as self-generated Janus particles, i.e., polarized particles whose surface is divided into two chemically varying regions, the adhering region and the non-adhering region; (4) the life-time of the metabolically active bacterium (living Janus particle) in the weak gel phase is estimated to be ∼25 s; (5) the observed network of luminous bacteria can be viewed as a biofilm, in which bacteria move due to self-phoresis by generating pH gradients via metabolic reactions; and (6) the reversible clustering of cells near interfaces is attributed to the bacterial energy-taxis phenomenon.

Existence of smooth solutions to coupled chemotaxis-fluid equations

We consider a system coupling the parabolic-parabolic Keller-Segel equations to the in- compressible Navier-Stokes equations in spatial dimensions two and three. We establish the local existence of regular solutions and present some blow-up criteria. For two dimensional Navier-Stokes-Keller-Segel equations, regular solutions constructed locally in time are, in reality, extended globally under some assumptions pertinent to experimental observation in [20] on the consumption rate and chemotactic sensitivity. We also show the existence of global weak solutions in spatially three dimensions with rather restrictive consumption rate and chemotactic sensitivity.

On spectra of linearized operators for Keller–Segel models of chemotaxis

We consider the phenomenon of collapse in the critical Keller–Segel equation (KS) which models chemotactic aggregation of micro-organisms underlying many social activities, e.g. fruiting body development and biofilm formation. Also KS describes the collapse of a gas of self-gravitating Brownian particles. We find the fluctuation spectrum around the collapsing family of steady states for these equations, which is instrumental in the derivation of the critical collapse law. To this end we develop a rigorous version of the method of matched asymptotics for the spectral analysis of a class of second order differential operators containing the linearized Keller–Segel operators (and as we argue linearized operators appearing in nonlinear evolution problems). We explain how the results we obtain are used to derive the critical collapse law, as well as for proving its stability.

Functional inequalities, thick tails and asymptotics for the critical mass Patlak–Keller–Segel model

We investigate the long time behavior of the critical mass Patlak–Keller–Segel equation. This equation has a one parameter family of steady-state solutions ϱ λ , λ > 0 , with thick tails whose second moment is unbounded. We show that these steady-state solutions are stable, and find basins of attraction for them using an entropy functional H λ coming from the critical fast diffusion equation in R 2 . We construct solutions of Patlak–Keller–Segel equation satisfying an entropy–entropy dissipation inequality for H λ . While the entropy dissipation for H λ is strictly positive, it turns out to be a difference of two terms, neither of which needs to be small when the dissipation is small. We introduce a strategy of controlled concentration to deal with this issue, and then use the regularity obtained from the entropy–entropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards ϱ λ .

BREZIS–MERLE INEQUALITIES AND APPLICATION TO THE GLOBAL EXISTENCE OF THE CAUCHY PROBLEM OF THE KELLER–SEGEL SYSTEM

We discuss the existence of the global solution for two types of nonlinear parabolic systems called the Keller–Segel equation and attractive drift–diffusion equation in two space dimensions. We show that the system admits a unique global solution in [Formula: see text]. The proof is based upon the Brezis–Merle type inequalities of the elliptic and parabolic equations. The proof can be applied to the Cauchy problem which is describing the self-interacting system.

A one-dimensional model for the interaction between cell-to-cell adhesion and chemotactic signalling

We develop and analyse a discrete, one-dimensional model of cell motility which incorporates the effects of volume filling, cell-to-cell adhesion and chemotaxis. The formal continuum limit of the model is a non-linear generalisation of the parabolic-elliptic Keller–Segel equations, with a diffusivity which can become negative if the adhesion coefficient is large. The consequent ill-posedness results in the appearance of spatial oscillations and the development of plateaus in numerical solutions of the underlying discrete model. A global-existence result is obtained for the continuum equations in the case of favourable parameter values and data, and a steady-state analysis, which, amongst other things, accounts for high-adhesion plateaus, is carried out. For ill-posed cases, a singular Stefan-problem formulation of the continuum limit is written down and solved numerically, and the numerical solutions are compared with those of the original discrete model.

A one-dimensional Keller–Segel equation with a drift issued from the boundary

We investigate in this Note the dynamics of a one-dimensional Keller–Segel type model on the half-line. On the contrary to the classical configuration, the chemical production term is located on the boundary. We prove, under suitable assumptions, the following dichotomy which is reminiscent of the two-dimensional Keller–Segel system. Solutions are global if the mass is below the critical mass, they blow-up in finite time above the critical mass, and they converge to some equilibrium at the critical mass. Entropy techniques are presented which aim at providing quantitative convergence results for the subcritical case. This Note is completed with a brief introduction to a more realistic model (still one-dimensional).

The volcano effect in bacterial chemotaxis

A population-level model of bacterial chemotaxis is derived from a simple bacterial-level model of behavior. This model, to be contrasted with the Keller–Segel equations, exhibits behavior we refer to as the “volcano effect”: steady-state bacterial aggregation forming a ring of higher density some distance away from an optimal environment. The model is derived, as in Erban and Othmer (2004) [1], from a transport equation in a state space including the internal biochemical variables of the bacteria and then simplified with a truncation at low moments with respect to these variables. We compare the solutions of the model to stochastic simulations of many bacteria, as well as the classic Keller–Segel model. This model captures behavior that the Keller–Segel model is unable to resolve, and sheds light on two different mechanisms that can cause a volcano effect.

Convergence of the Mass-Transport Steepest Descent Scheme for the Subcritical Patlak–Keller–Segel Model

Variational steepest descent approximation schemes for the modified Patlak–Keller–Segel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated with this equation for subcritical masses. As a consequence, we recover the recent result about the global in time existence of weak solutions to the modified Patlak–Keller–Segel equation for the logarithmic interaction kernel in any dimension in the subcritical case. Moreover, we show how this method performs numerically in dimension one. In this particular case, this numerical scheme corresponds to a standard implicit Euler method for the pseudoinverse of the cumulative distribution function. We demonstrate its capabilities to reproduce the blow-up of solutions for supercritical masses easily without the need of mesh-refinement.

Strong-Coupling Dynamics of a Multicellular Chemotactic System

Chemical signaling is one of the ubiquitous mechanisms by which intercellular communication takes place at the microscopic level, particularly via chemotaxis. Such multicellular systems are popularly studied using continuum, mean-field equations. In this Letter we study a stochastic model of chemotactic signaling. The Langevin formalism of the model makes it amenable to calculation via nonperturbative analysis, which enables a quantification of the effect of fluctuations on both the weak and the strongly coupled biological dynamics. In particular, we show that the (i) self-localization due to autochemotaxis is impossible. (ii) When aggregation occurs, the aggregate performs a random walk with a renormalized diffusion coefficient D(R) proportiuonal to epsilon-2N-3. (iii) The stochastic model exhibits sharp transitions in cell motile behavior for negative chemotaxis, behavior that has no parallel in the mean-field Keller-Segel equations.

Many-body theory of chemotactic cell-cell interactions

We consider an individual-based stochastic model of cell movement mediated by chemical signaling fields. This model is formulated using Langevin dynamics, which allows an analytic study using methods from statistical and many-body physics. In particular we construct a diagrammatic framework within which to study cell-cell interactions. In the mean-field limit, where statistical correlations between cells are neglected, we recover the deterministic Keller-Segel equations. Within exact perturbation theory in the chemotactic coupling epsilon , statistical correlations are non-negligible at large times and lead to a renormalization of the cell diffusion coefficient D(R)--an effect that is absent at mean-field level. An alternative closure scheme, based on the necklace approximation, probes the strong coupling behavior of the system and predicts that D(R) is renormalized to zero at a critical value of epsilon, indicating self-localization of the cell. Stochastic simulations of the model give very satisfactory agreement with the perturbative result. At higher values of the coupling simulations indicate that D(R) approximately epsilon(-2) , a result at odds with the necklace approximation. We briefly discuss an extension of our model, which incorporates the effects of short-range interactions such as cell-cell adhesion.

Optimal control problems governed by some semilinear parabolic equations

This paper is concerned with the optimal control problem for the Keller–Segel equations. That is, as a generalization of Ryu and Yagi (J. Math. Anal. Appl. 256 (2001) 45–66) we derive the optimality conditions for the optimal control problem governed by a semilinear abstract equation of nonmonotone type. Moreover, we prove the uniqueness of the optimal control under some conditions.

Optimal Control of Keller–Segel Equations

Optimal Control of Keller–Segel Equations

Chemotaxis and chemokinesis in eukaryotic cells: the Keller-Segel equations as an approximation to a detailed model.

More than 20 years after its proposal, Keller and Segel's model (1971, J. theor. Biol., 30, 235-248) remains by far the most popular model for chemical control of cell movement. However, before the Keller-Segel equations can be applied to a particular system, appropriate functional forms must be specified for the dependence on chemical concentration of the cell transport coefficients and the chemical degradation rate. In the vast majority of applications, these functional forms have been chosen using simple intuitive criteria. We focus on the particular case of eukaryotic cell movement, and derive an approximation to the detailed model of Sherratt et al. (1993, J. theor. Biol., 162, 23-40). The approximation consists of the Keller-Segel equations, with specific forms predicted for the cell transport coefficients and chemical degradation rate. Moreover, the parameter values in these functional forms can be directly measured experimentally. In the case of the much studied neutrophil-peptide system, we test our approximation using both the Boyden chamber and under-agarose assays. Finally, we show that for other cell-chemical interactions, a simple comparison of time scales provides a rapid check on the validity of our Keller-Segel approximation.

Chemotaxis and chemokinesis in eukaryotic cells: The keller-segel equations as an approximation to a detailed model

Chemotaxis and chemokinesis in eukaryotic cells: The keller-segel equations as an approximation to a detailed model

Chemosensory responses of microorganisms: Asymmetric diffusion as the limit of a biased random walk

A macroscopic asymmetric diffusion equation to model the responses of microbial populations to chemical attractants and repellents is derived from a biased random walk model for the motion of individual cells. The equation is different from the well-known Keller-Segel equation which contains a Fickian diffusion term. The implications of this difference for selected problems of biological interest are considered. In particular the aggregation of a population of microorganisms in a region of high concentration of attractant is discussed. Some similarities and limitations of both models are noted.

Convergence of stochastic interacting particle systems in probability under a Sobolev norm

Abstract In this paper, we consider particle systems with interaction and Brownian motion. We prove that when the initial data is from the sampling of Chorin’s method, i.e., the initial vertices are on lattice points hi ∈ Rd with mass ρ0(hi)h, where ρ0 is some initial density function, then the regularized empirical measure of the interacting particle system converges in probability to the corresponding mean-field partial differential equation with initial density ρ0, under the Sobolev norm of L ∞(L2) ∩ L2(H1). Our result is true for all those systems when the interacting function is bounded, Lipschitz continuous and satisfies certain regular condition. And if we further regularize the interacting particle system, it also holds for some of the most important systems of which the interacting functions are not. For systems with repulsive Coulomb interaction, this convergence holds globally on any interval [0, t]. And for systems with attractive Newton force as interacting function, we have convergence within the largest existence time of the regular solution of the corresponding Keller-Segel equation.

Propagation of chaos for the Keller–Segel equation with a logarithmic cut-off

Propagation of chaos for the Keller–Segel equation with a logarithmic cut-off