Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact

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

doi:10.1103/physreve.78.061904

Cited by 76 publications

(96 citation statements)

References 56 publications

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“…This matter is of interest in foreseeing possible stability conditions of biological interfaces. This approach has been considered in the multiscale models [20,21] in which continuous non-linear transport equations have been derived from microscopic cell level properties, tackled by the CPM formalism.…”

Section: Cell-based Models Describing Experimental Datamentioning

confidence: 99%

“…Accordingly, the description of these bio-systems comprised cell motility under chemotaxis with randomly fluctuating cell shapes and continuous deterministic equations related to cellular density distributions. In addition, from microscopic dynamics utilizing the excluded volume approach, non-linear diffusion equations with diffusion coefficients depending on cellular volume fraction to prevent the collapse of cellular density were obtained [20]. These versatile models have been applied for interpreting different aspects of cell physiology [20][21][22][23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

“…Another approach to these complex mechanisms started from the behavior of individual cells as in cellular Potts models (CPMs) [18,19] that were grounded on dynamic data derived from either individual or collective cells at different colony regions [18][19][20][21]. Accordingly, the description of these bio-systems comprised cell motility under chemotaxis with randomly fluctuating cell shapes and continuous deterministic equations related to cellular density distributions.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, from microscopic dynamics utilizing the excluded volume approach, non-linear diffusion equations with diffusion coefficients depending on cellular volume fraction to prevent the collapse of cellular density were obtained [20]. These versatile models have been applied for interpreting different aspects of cell physiology [20][21][22][23][24][25][26][27][28]. Thus, an extended CPM version provided a strong correlation of cell migration directionality with topological surrounding extracellular matrix (ECM) distribution and a biphasic dependence of migration on the matrix structure and stiffness, cell density, and adhesion [27].…”

Section: Introductionmentioning

confidence: 99%

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Spatio-temporal morphology changes in and quenching effects on the 2D spreading dynamics of cell colonies in both plain and methylcellulose-containing culture media

et al. 2016

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To deal with complex systems, microscopic and global approaches become of particular interest. Our previous results from the dynamics of large cell colonies indicated that their 2D front roughness dynamics is compatible with the standard Kardar-Parisi-Zhang (KPZ) or the quenched KPZ equations either in plain or methylcellulose (MC)-containing gel culture media, respectively. In both cases, the influence of a non-uniform distribution of the colony constituents was significant. These results encouraged us to investigate the overall dynamics of those systems considering the morphology and size, the duplication rate, and the motility of single cells. For this purpose, colonies with different cell populations (N) exhibiting quasi-circular and quasi-linear growth fronts in plain and MC-containing culture media are investigated. For small N, the average radial front velocity and its change with time depend on MC concentration. MC in the medium interferes with cell mitosis, contributes to the local enlargement of cells, and increases the distribution of spatio-temporal cell density heterogeneities. Colony spreading in MC-containing media proceeds under two main quenching effects, I and II; the former mainly depending on the culture medium composition and structure and the latter caused by the distribution of enlarged local cell domains. For large N, colony spreading occurs at constant velocity. The characteristics of cell motility, assessed by measuring their trajectories and the corresponding velocity field, reflect the effect of enlarged, slowmoving cells and the structure of the medium. Local average cell size distribution and individual cell motility data from plain and MC-containing media are qualitatively consistent with the predictions of both the extended cellular Potts models and the observed transition of the front roughness dynamics from a standard KPZ to a quenched KPZ. In this case, quenching J Biol Phys (2016) 42:477-502 DOI 10.1007

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Section: Cell-based Models Describing Experimental Datamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spatio-temporal morphology changes in and quenching effects on the 2D spreading dynamics of cell colonies in both plain and methylcellulose-containing culture media

et al. 2016

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“…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

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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

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Convergence analysis of structure‐preserving numerical methods for nonlinear Fokker–Planck equations with nonlocal interactions

Duan

Chen

Liu

et al. 2021

Math Methods in App Sciences

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A class of nonlinear Fokker–Planck equations with nonlocal interactions may include many important cases, such as porous medium equations with external potentials and aggregation–diffusion models. The trajectory equation of the Fokker–Plank equation can be derived based on an energetic variational approach. A structure‐preserving numerical scheme that is mass conservative, energy stable, uniquely solvable, and positivity preserving at a theoretical level has also been designed in the previous work. Moreover, the numerical scheme is shown to satisfy the discrete energetic dissipation law and preserve steady states and has been observed to be second order accurate in space and first‐order accurate time in various numerical experiments. In this work, we give the rigorous convergence analysis for the highly nonlinear numerical scheme. A careful higher order asymptotic expansion is needed to handle the highly nonlinear nature of the trajectory equation. In addition, two step error estimates (a rough estimate and a refined estimate) are necessary in the convergence proof. Different from a standard error estimate, the rough estimate is performed to control the nonlinear term. A few numerical results are also presented to verify the optimal convergence order and the preservation of equilibria.

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Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact

Cited by 76 publications

References 56 publications

Spatio-temporal morphology changes in and quenching effects on the 2D spreading dynamics of cell colonies in both plain and methylcellulose-containing culture media

Spatio-temporal morphology changes in and quenching effects on the 2D spreading dynamics of cell colonies in both plain and methylcellulose-containing culture media

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

Convergence analysis of structure‐preserving numerical methods for nonlinear Fokker–Planck equations with nonlocal interactions

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