Continuous Macroscopic Limit of a Discrete Stochastic Model for Interaction of Living Cells

Abstract

In the development of multiscale biological models it is crucial to establish a connection between discrete microscopic or mesoscopic stochastic models and macroscopic continuous descriptions based on cellular density. In this paper a continuous limit of a two-dimensional Cellular Potts Model (CPM) with excluded volume is derived, describing cells moving in a medium and reacting to each other through both direct contact and long range chemotaxis. The continuous macroscopic model is obtained as a Fokker-Planck equation describing evolution of the cell probability density function. All coefficients of the general macroscopic model are derived from parameters of the CPM and a very good agreement is demonstrated between CPM Monte Carlo simulations and 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. General multiscale approach is demonstrated by simulating spongy bone formation from loosely packed mesenchyme via the intramembranous route suggesting that self-organizing physical mechanisms can account for this developmental process. A large literature exists studying continuous limits of point-wise discrete microscopic models for biological systems. For example, the classic Keller-Segel PDE model of chemotaxis [1] was derived from a discrete model with point-wise cells undergoing random walk [2][3][4][5]. However, many biological phenomena require taking into account the finite size of biological cells, and much less work has been done on deriving macroscopic limits of microscopic models which treat cells as extended objects. The mesoscopic Cellular Potts Model (CPM), first introduced by Glazier and Graner [6,7], has been used as a component of multiscale, experimentally motivated hybrid approaches, combining discrete and macroscopic continuous representations, to simulate, among others, morphological phenomena in the cellular slime mold Dictyostelium discoideum [8], vascular development [9] and the proximo-distal increase in the number of skeletal elements in the developing avian limb [10].One of the earliest attempts at combining mesoscopic and macroscopic levels of description of cellular dynamics was described in [11] where the diffusion coefficient for a collection of noninteracting randomly moving cells was derived from a one-dimensional CPM. Recently a microscopic limit of subcellular elements model [12] was derived in the form of an advection-diffusion partial differential equation for cellular density. In previous papers [13,14] we studied the continuous limit of 1D and 2D models of individual cell motion in a medium, in the presence of an external field but without contact cell-cell interactions.This paper describes a theoretical analysis leading to a continuous macroscopic limit of the two-dimensional

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

“…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.…”

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

“…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].…”

“…"Space-free" agent-based models have been considered for example in hematology and are reported elsewhere [40]. Spatial agent-based models can roughly been distinguished by those operating on space fixed lattices [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60] and those operating without a lattice ("lattice-free", "off-lattice"; [52,[61][62][63][64][65][66]). Among lattice models, either (i) a lattice site may be occupied by many cells (e.g., [67]) permitting modeling of large cell population sizes, (ii) cells may occupy at most a single lattice site (e.g., [68]), or (iii) many lattice sites may be occupied by one biological cell (e.g., [53]).…”

Macroscopic model of self-propelled bacteria swarming with regular reversals

“…Second effect is due to the frequent collision of cells during each period 2T making that effect essentially collective one. These two regimes make Myxobacteria quite distinct from bacteria like E. Coli or amoeba Dictyostelium discoideum which experience diffusion as random walk of Brownian-like particles [8][9][10] without any periodic motion.…”

“…Recently continuous limit models describing dynamics of cellular density were derived from the microscopic motion of randomly moving cells exhibiting volume exclusion and chemotaxis [8][9][10]. In particular, the Ref.…”