Multiscale dynamics of biological cells with chemotactic interactions: From a discrete stochastic model to a continuous description

Alber, Mark; Chen, Nan; Glimm, Tilmann; Lushnikov, Pavel M.

doi:10.1103/physreve.73.051901

Cited by 60 publications

(60 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Near singularity the Keller-Segel model is not applicable when typical distance between bacteria is about size of bacteria. In that regime modification of the Keller-Segel equation was derived from microscopic stochastic dynamics of bacteria which prevents collapse due to excluded volume constraint (different bacteria cannot occupy the same volume) [22,23,24]. Here however the original Keller-Segel model without regularization is considered.…”

mentioning

confidence: 99%

Logarithmic-type Scaling of the Collapse of Keller-Segel Equation

Dyachenko¹,

Lushnikov²,

Vladimirova³

2011

AIP Conference Proceedings

Self Cite

View full text Add to dashboard Cite

Abstract. Keller-Segel equation (KS) is a parabolic-elliptic system of partial differential equations with applications to bacterial aggregation and collapse of self-gravitating gas of brownian particles. KS has striking qualitative similarities with nonlinear Schrodinger equation (NLS) including critical collapse (finite time point-wise singularity) in two dimensions. The self-similar solutions near blow up point are studied for KS in two dimensions together with time dependence of these solutions. We found logarithmic-type modifications to (t 0 − t) 1/2 scaling law of self-similar solution in qualitative analogy with log-log modification for NLS. We found very good agreement between the direct numerical simulations of KS and the analytical results obtained by developing a perturbation theory for logarithmic-type modifications. It suggests that log-log modification in NLS also could be verified in a similar way. was derived for macroscopically averaged dynamics of bacteria with the bacterial density ρ(r,t) and the chemoattractant concentration c(r,t) at spatial point r and time t. Bacteria and biological cells often communicate through chemotaxis, when bacteria move towards chemical gradient of substance called chemoattractant. Bacteria can also secrete the same chemoattractant which creates nonlocal attraction between them through secretion, diffusion and detection of chemoattractant by other bacteria of the same type. Bacteria are self-propelled and without chemotactic clue center of mass of each bacteria typically experiences random walk. That random walk is described in ( where we assumed that D, D c , α and k are constants and rewrote all variables in dimensionless form as t → t 0 t,and t 0 is a typical timescale of the dynamics of ρ in Eq. (1). Eq. (2) also describes a dynamics of a gas of self-gravitating Brownian particles which has applications in astrophysics [10]. In that case the second Eq. in (2) is a Poisson equation for the gravitational potential −c while ρ is the gas density (all units are dimensionless). The first Eq. in (2) is a Smoluchowski equation for ρ in the gravitational potential −c. Below we refer to ρ and c as the density of bacteria and the concentration of chemoattractant, respectively, but all results below are equally true for a gravitational collapse of the gas of self-gravitating Brownian particles.In dimension one (D=1) the solution of RKS exist globally in time while for D > 1 (e.g. in dimensions two and three ) the finite time singularity is possible [6] provided bacterial density is large enough. E.g. for D = 2 finite time

show abstract

mentioning

confidence: 99%

Logarithmic-type Scaling of the Collapse of Keller-Segel Equation

Dyachenko¹,

Lushnikov²,

Vladimirova³

2011

AIP Conference Proceedings

Self Cite

View full text Add to dashboard Cite

show abstract

“…A direct simulation of an overdamped Langevin model (1) with dv i (t) = 0 may provide better quantitative agreement with the soliton. Note that we do not have to select the velocity of the new tip branching from a given one in the overdamped case and that the injecting boundary condition (16) becomes −∂p/∂x = 2βj 0 given by (19) at x = 0 [12]. With a few exceptions [5][6][7]9], most models in the literature are overdamped.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In more detailed models, endothelial cells change form and move over the extracellular matrix by Monte Carlo dynamics of a cellular Potts model [18]. In appropriate long time limits, the corresponding discrete time random walk reduces to Langevin equations [17,19]. Thus, the ideas presented in this paper might be useful when considering other related angiogenesis models.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Ensemble Averages, Soliton Dynamics and Influence of Haptotaxis in a Model of Tumor-Induced Angiogenesis

Bonilla

Carretero

Terragni

2017

Entropy

View full text Add to dashboard Cite

Abstract:In this work, we present a numerical study of the influence of matrix degrading enzyme (MDE) dynamics and haptotaxis on the development of vessel networks in tumor-induced angiogenesis. Avascular tumors produce growth factors that induce nearby blood vessels to emit sprouts formed by endothelial cells. These capillary sprouts advance toward the tumor by chemotaxis (gradients of growth factor) and haptotaxis (adhesion to the tissue matrix outside blood vessels). The motion of the capillaries in this constrained space is modelled by stochastic processes (Langevin equations, branching and merging of sprouts) coupled to continuum equations for concentrations of involved substances. There is a complementary deterministic description in terms of the density of actively moving tips of vessel sprouts. The latter forms a stable soliton-like wave whose motion is influenced by the different taxis mechanisms. We show the delaying effect of haptotaxis on the advance of the angiogenic vessel network by direct numerical simulations of the stochastic process and by a study of the soliton motion.

show abstract

“…Nevertheless, the constitutive equations that characterize e.g., migration dynamics, cell adhesion, and stress-strain in the tissue and ECM will determine the results that will be obtained. The determination of those equations is a non-trivial task and in general they should be determined from experiments or might be inferred from agent-based models using averaging techniques [49,83,149,[226][227][228][229]. This branch is usually refered to as multiscale models within the mathematical and engineering community, and has to be distinguished from multi-level model which integrate components, mechanisms and information from different scales in one model [230].…”

Section: Hybrid Discrete-continuum Modelsmentioning

confidence: 99%

Simulating tissue mechanics with agent-based models: concepts, perspectives and some novel results

et al. 2015

View full text Add to dashboard Cite

In this paper we present an overview of agentbased models that are used to simulate mechanical and physiological phenomena in cells and tissues, and we discuss underlying concepts, limitations, and future perspectives of these models. As the interest in cell and tissue mechanics increase, agent-based models are becoming more common the modeling community. We overview the physical aspects, complexity, shortcomings, and capabilities of the major agent-based model categories: lattice-based models (cellular automata, lattice gas cellular automata, cellular Potts models), off-lattice models (center-based models, deformable cell models, vertex models), and hybrid discrete-continuum models. In this way, we hope to assist future researchers in choosing a model for the phenomenon they want to model and understand. The article also contains some novel results.

show abstract

Multiscale dynamics of biological cells with chemotactic interactions: From a discrete stochastic model to a continuous description

Cited by 60 publications

References 29 publications

Logarithmic-type Scaling of the Collapse of Keller-Segel Equation

Logarithmic-type Scaling of the Collapse of Keller-Segel Equation

Ensemble Averages, Soliton Dynamics and Influence of Haptotaxis in a Model of Tumor-Induced Angiogenesis

Simulating tissue mechanics with agent-based models: concepts, perspectives and some novel results

Contact Info

Product

Resources

About