Chemotaxis Model Research Articles

Analytical Solution of One-Dimensional Keller-Segel Equations via New Homotopy Perturbation Method

For solving a system of nonlinear partial differential equations (PDE) emerging in an attractor one-dimensional chemotaxis model, we used a relatively new analytical method called the new modified homotopy perturbation method (NMHPM). We use NMHPM for solving one-dimensional Keller-Segel models for different types. Some properties show biologically acceptable dependency on parameter values, and numerical solutions are provided. NMHPM’s stability and reduced computing time provide it with a broader range of applications. The algorithm provides analytical approximations for different types of Keller-Segel equations. Some numerical illustrations are given to show the efficiency of the algorithm.

Global boundedness and asymptotic stabilization in a chemotaxis system with density-suppressed motility and nonlinear signal production

In this paper, we study the following chemotaxis model with density-suppressed motility and nonlinear production{ut=Δ(φ(v)u)+ru−μuα,x∈Ω,t>0,vt=Δv−v+wβ,x∈Ω,t>0,wt=Δw−w+uγ,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn(n≥2) with smooth boundary, where r∈R,μ,β,γ>0 and α>1. The positive motility function satisfies φ(s)∈C3([0,∞)) and φ′(s)≤0 for all s≥0. It is showed that the system admits a globally bounded and classical solution under some conditions on α,β and γ. Then, under stricter constraints on φ, we obtained that the parameter β has a wider range than before, which is enough to ensure the global boundedness of the solution. Furthermore, if μ is sufficiently large, we proved that the solution converges to ((r+μ)1α−1,(r+μ)βγα−1,(r+μ)γα−1) in L∞(Ω) as t→∞ for all r∈R,μ,β,γ>0 and α>1, where r+:=max⁡{r,0}. Finally, we showed that the convergence is exponential in cases r>0 and r<0.

Dynamics for a chemotaxis model with general logistic damping and signal dependent motility

Dynamics for a chemotaxis model with general logistic damping and signal dependent motility

Sensory adaptation in a continuum model of bacterial chemotaxis—working range, cost-accuracy relation, and coupled systems

Sensory adaptation enables organisms to adjust their perception in a changing environment. A paradigm is bacterial chemotaxis, where the output activity of chemoreceptors is adapted to different baseline concentrations via receptor methylation. The range of internal receptor states limits the stimulus magnitude to which these systems can adapt. Here, we employ a highly idealized, Langevin-equation based model to study how the finite range of state variables affects the adaptation accuracy and the energy dissipation in individual and coupled systems. Maintaining an adaptive state requires constant energy dissipation. We show that the steady-state dissipation rate increases approximately linearly with the adaptation accuracy for varying stimulus magnitudes in the so-called perfect adaptation limit. This result complements the well-known logarithmic cost-accuracy relationship for varying chemical driving. Next, we study linearly coupled pairs of sensory units. We find that the interaction reduces the dissipation rate per unit and affects the overall cost-accuracy relationship. A coupling of the slow methylation variables results in a better accuracy than a coupling of activities. Overall, the findings highlight the significance of both the working range and collective operation mode as crucial design factors that impact the accuracy and energy expenditure of molecular adaptation networks.

The fully parabolic multi-species chemotaxis system in

Abstract This article is devoted to the analysis of the parabolic–parabolic chemotaxis system with multi-components over $\mathbb{R}^2$ . The optimal small initial condition on the global existence of solutions for multi-species chemotaxis model in the fully parabolic situation had not been attained as far as the author knows. In this paper, we prove that under the sub-critical mass condition, any solutions to conflict-free system exist globally. Moreover, the global existence of solutions to system with strong self-repelling effect has been discussed even for large initial data. The proof is based on the modified free energy functional and the Moser–Trudinger inequality for system.

Direct measurement of dynamic attractant gradients reveals breakdown of the Patlak-Keller-Segel chemotaxis model.

Chemotactic bacteria not only navigate chemical gradients, but also shape their environments by consuming and secreting attractants. Investigating how these processes influence the dynamics of bacterial populations has been challenging because of a lack of experimental methods for measuring spatial profiles of chemoattractants in real time. Here, we use a fluorescent sensor for aspartate to directly measure bacterially generated chemoattractant gradients during collective migration. Our measurements show that the standard Patlak-Keller-Segel model for collective chemotactic bacterial migration breaks down at high cell densities. To address this, we propose modifications to the model that consider the impact of cell density on bacterial chemotaxis and attractant consumption. With these changes, the model explains our experimental data across all cell densities, offering insight into chemotactic dynamics. Our findings highlight the significance of considering cell density effects on bacterial behavior, and the potential for fluorescent metabolite sensors to shed light on the complex emergent dynamics of bacterial communities.

Phenotypic Variability Shapes Bacterial Responses to Opposing Gradients

Although bacterial chemotaxis behavior in a single gradient has been well studied, chemotaxis of bacterial population in complex environments is not well understood. Here, we discovered four distinct behaviors of populations in microfluidic experiments with different opposing gradients of MeAsp and serine. By using a population chemotaxis model based on the dynamics of intracellular signaling pathways, we found that the nongenetic variability of the relevant chemoreceptors (Tar and Tsr) within the population is responsible for the diverse population behaviors. Through analyses combining the phenotype-to-performance mapping and Tar/Tsr ratio distribution, we predicted the phase diagram of population chemotaxis behaviors under varying chemical gradients and the effect of growth period on population behaviors, which were verified by additional experiments. Our study suggests that phenotypic heterogeneity in chemoreceptors enables diverse chemotactic strategies, which cells may adopt to improve their population fitness in complex environments. Published by the American Physical Society 2024

Different Morphotypes of Physarum polycephalum as Models for Chemotaxis and Locomotion.

The acellular slime mold Physarum polycephalum is a large, unicellular amoeba, which, due to its huge size, is well suited to investigate chemotaxis and cellular locomotion. The myxomycete has an astonishing behavioral repertoire and is highly responsive to changes in its environment, which map to changes in its tubular network, internal cytoplasm flow, and cytoskeleton. The behavioral repertoire includes problem-solving, decision-making, and memory. P. polycephalum's chemo- and phototaxis are especially well studied. This chapter describes how to cultivate different morphotypes of P. polycephalum (micro-, meso-, and macroplasmodia). Furthermore, the setup of a chemotaxis experiment and the acquisition and analysis of chemotaxis data is described.

Pattern formation of a volume-filling chemotaxis model with a bistable source

&lt;p&gt;In this paper, the pattern formation of a volume-filling chemotaxis model with bistable source terms was studied. First, it was shown that self-diffusion does not induce Turing patterns, but chemotaxis-driven instability occurs. Then, the asymptotic behavior of the chemotaxis model was analyzed by weakly nonlinear analysis with the method of multiple scales. When the chemotaxis coefficient exceeded a threshold value and there was a single unstable mode, the supercritical and subcritical bifurcation of the model was discussed. The amplitude equations and the asymptotic expressions of the patterns were obtained. When the chemotaxis coefficient was large enough, the two-mode competition behavior of the model with two unstable modes was analyzed, and the corresponding amplitude equations and the asymptotic expressions of the patterns were obtained. Finally, numerical simulations were provided to further illuminate the above analytical results.&lt;/p&gt;

Dirac-type aggregation with full mass in a chemotaxis model

Dirac-type aggregation with full mass in a chemotaxis model

Boundary layer problem on the chemotaxis model with Robin boundary conditions

Boundary layer problem on the chemotaxis model with Robin boundary conditions

Remark on the entropy production of adaptive run-and-tumble chemotaxis

Chemotactic active particles, such as bacteria and cells, exhibit an adaptive run-and-tumble motion, giving rise to complex emergent behaviors in response to external chemical fields. This motion is generated by the conversion of internal chemical energy into self-propulsion, allowing each agent to sustain a steady-state far from thermal equilibrium and perform works. The rate of entropy production serves as an indicates of how extensive these agents operate away from thermal equilibrium, providing a measure for estimating maximum obtainable power. Here we present the general framework for calculating the entropy production rate created by such population of agents from the first principle, using the minimal model of bacterial adaptive chemotaxis, as they execute the most basic collective action – the mass transport.

Synchronized rotations of active particles on chemical substrates.

Many microorganisms use chemical 'signaling' - a quintessential self-organizing strategy in non-equilibrium - that can induce spontaneous aggregation and coordinated motion. Using synthetic signaling as a design principle, we construct a minimal model of active Brownian particles (ABPs) having soft repulsive interactions on a chemically quenched patterned substrate. The interplay between chemo-phoretic interactions and activity is numerically investigated for a proposed variant of the Keller-Segel model for chemotaxis. Such competition not only results in a chemo-motility-induced phase-separated state, but also results in a new cohesive clustering phase with synchronized rotations. Our results suggest that rotational order can emerge in systems by virtue of activity and repulsive interactions alone without an explicit alignment interaction. These rotations can also be exploited by designing mechanical devices that can generate reorienting torques using active particles.

Well-posedness of a chemotaxis model with positive and negative cross-diffusion

Well-posedness of a chemotaxis model with positive and negative cross-diffusion

Reassessing the Standard Chemotaxis Framework for Understanding Biased Migration in Helicobacter pylori.

Helicobacter pylori infections are a major cause of peptic ulcers and gastric cancers. The development of robust inflammation in response to these flagellated, motile bacteria is correlated with poor prognosis. Chemotaxis plays a crucial role in H. pylori colonization, enabling the bacteria to swim toward favorable chemical environments. Unlike the model species of bacterial chemotaxis, Escherichia coli, H. pylori cells possess polar flagella. They run forward by rotating their flagella counterclockwise, whereas backward runs are achieved by rotating their flagella clockwise. We delve into the implications of certain features of the canonical model of chemotaxis on our understanding of biased migration in polarly flagellated bacteria such as H. pylori. In particular, we predict how the translational displacement of H. pylori cells during a backward run could give rise to chemotaxis errors within the canonical framework. Also, H. pylori lack key chemotaxis enzymes found in E. coli, without which sensitive detection of ligands with a wide dynamic range seems unlikely. Despite these problems, H. pylori exhibit robust ability to migrate toward urea-rich sources. We emphasize various unresolved questions regarding the biophysical mechanisms of chemotaxis in H. pylori, shedding light on potential directions for future research. Understanding the intricacies of biased migration in H. pylori could offer valuable insights into how pathogens breach various protective barriers in the human host.

Boundedness Through Nonlocal Dampening Effects in a Fully Parabolic Chemotaxis Model with Sub and Superquadratic Growth

This work deals with a chemotaxis model where an external source involving a sub and superquadratic growth effect contrasted by nonlocal dampening reaction influences the motion of a cell density attracted by a chemical signal. We study the mechanism of the two densities once their initial configurations are fixed in bounded impenetrable regions; in the specific, we establish that no gathering effect for the cells can appear in time provided that the dampening effect is strong enough. Mathematically, we are concerned with this problem ut=Δu-χ∇·(u∇v)+auα-buα∫ΩuβinΩ×(0,Tmax),τvt=Δv-v+uinΩ×(0,Tmax),uν=vν=0on∂Ω×(0,Tmax),u(x,0)=u0(x)≥0,v(x,0)=v0(x)≥0,x∈Ω¯,◊\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t=\\Delta u-\\chi \ abla \\cdot (u\ abla v)+au^\\alpha -bu^\\alpha \\int _\\Omega u^\\beta &{}\ extrm{in}\\ \\Omega \ imes (0, T_{max}),\\\\ \ au v_t=\\Delta v-v+u &{}\ extrm{in}\\ \\Omega \ imes (0, T_{max}),\\\\ u_\ u =v_\ u =0 &{}\ extrm{on}\\ \\partial \\Omega \ imes (0, T_{max}),\\\\ u(x, 0)=u_0(x)\\ge 0, v(x,0)=v_0(x)\\ge 0, &{}x \\in {\\bar{\\Omega }}, \\end{array}\\right. } \\quad {\\Diamond } \\end{aligned}$$\\end{document}for τ=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au =1$$\\end{document}, n∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document}, χ,a,b>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi ,a,b>0$$\\end{document} and α,β≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha , \\beta \\ge 1$$\\end{document}. Herein u stands for the population density, v for the chemical signal and Tmax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{max}$$\\end{document} for the maximal time of existence of any nonnegative classical solution (u, v) to system (◊\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Diamond $$\\end{document}). We prove that despite any large-mass initial data u0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_0$$\\end{document}, whenever(The subquadratic case) 1≤α<2andβ>n+42-α,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le \\alpha <2 \\quad \ ext {and} \\quad \\beta >\\frac{n+4}{2}-\\alpha ,$$\\end{document}(The superquadratic case) β>n2and2≤α<1+2βn,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta >\\frac{n}{2} \\quad \ ext {and} \\quad 2\\le \\alpha < 1+ \\frac{2\\beta }{n},$$\\end{document} actually Tmax=∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{max}=\\infty $$\\end{document} and u and v are uniformly bounded. This paper is in line with the result in Bian et al. (Nonlinear Anal 176:178–191, 2018), where the same conclusion is established for the simplified parabolic-elliptic version of model (◊\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Diamond $$\\end{document}), corresponding to τ=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au =0$$\\end{document}; more exactly, this work extends the study to the fully parabolic case Bian et al. (Nonlinear Anal 176:178–191, 2018).

Global bounded classical solutions to a parabolic–elliptic chemotaxis model with local sensing and asymptotically unbounded motility

AbstractGlobal existence and boundedness of classical solutions are shown for a parabolic–elliptic chemotaxis system with local sensing when the motility function is assumed to be unbounded at infinity. The cornerstone of the proof is the derivation of ‐estimates on the second component of the system and is achieved by various comparison arguments.

Traveling waves and their spectral instability in volume–filling chemotaxis model

In this paper, I consider a volume-filling chemotaxis model with a small cell diffusion coefficient and chemotactic sensitivity. By the geometric singular perturbation theory together with the center-stable and center unstable manifolds, one gets the existence of a positive traveling wave connecting the two constant steady states (0,0) and (b,αbβ) with a small wave speed ϵc. In addition, the traveling wave is monotone for b≥1 and is not monotone for 0<b<1. Moreover, by the spectral analysis it shows that the above traveling wave is spectrally unstable in some exponentially weighted spaces.

Finite-time blow-up of solution for a chemotaxis model with singular sensitivity and logistic source

Finite-time blow-up of solution for a chemotaxis model with singular sensitivity and logistic source

A simple reaction–diffusion system as a possible model for the origin of chemotaxis

Chemotaxis is a directed cell movement in response to external chemical stimuli. In this paper, we propose a simple model for the origin of chemotaxis – namely how a directed movement in response to an external chemical signal may occur based on purely reaction–diffusion equations reflecting inner working of the cells. The model is inspired by the well-studied role of the rho-GTPase Cdc42 regulator of cell polarity, in particular in yeast cells. We analyse several versions of the model to better understand its analytic properties and prove global regularity in one and two dimensions. Using computer simulations, we demonstrate that in the framework of this model, at least in certain parameter regimes, the speed of the directed movement appears to be proportional to the size of the gradient of signalling chemical. This coincides with the form of the chemical drift in the most studied mean field model of chemotaxis, the Keller–Segel equation.