Modelling collective cell behaviour

Markham, Deborah C.; Baker, Ruth E.; Maini, Philip K.

doi:10.3934/dcds.2014.34.5123

Cited by 11 publications

(12 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Increasing recognition of these benefits has stimulated efforts to connect discrete and continuum models in a variety of biological contexts [38][39][40][41][42][43][44][45]; see the review by Codling et al [46]. Typically, a lattice-based approach and a mean-field approximation, in which the occupancy of lattice sites is assumed to be independent, are used to derive continuum approximations to discrete models, though corrections to mean-field models have been investigated [47][48][49]. In the context of angiogenesis, Spill et al [50] used a one-dimensional mesoscale lattice-based model to derive macroscopic descriptions of angiogenesis based on the snail-trail approach.…”

Section: Introductionmentioning

confidence: 99%

Modeling angiogenesis: A discrete to continuum description

2017

View full text Add to dashboard Cite

Angiogenesis is the process by which new blood vessels develop from existing vasculature. During angiogenesis, endothelial tip cells migrate via diffusion and chemotaxis, loops form via tip-to-tip and tip-to-sprout anastomosis, new tip cells are produced via branching, and a vessel network forms as endothelial cells follow the paths of tip cells. The latter process is known as the snail trail. We use a mean-field approximation to systematically derive a continuum model from a two-dimensional lattice-based cellular automaton model of angiogenesis in the corneal assay, based on the snail-trail process. From the two-dimensional continuum model, we derive a one-dimensional model which represents angiogenesis in two dimensions. By comparing the discrete and one-dimensional continuum models, we determine how individual cell behavior manifests at the macroscale. In contrast to the phenomenological continuum models in the literature, we find that endothelial cell creation due to tip cell movement (vessel formation via the snail trail) manifests as a source term of tip cells on the macroscale. Further, we find that phenomenological continuum models, which assume that endothelial cell creation is proportional to the flux of tip cells in the direction of increasing chemoattractant concentration, qualitatively capture vessel formation in two dimensions, but must be modified to accurately represent vessel formation. Additionally, we find that anastomosis imposes restrictions on cell density, which, if violated, leads to ill-posedness in our continuum model. We also deduce that self-loops should be excluded when tip-to-sprout anastomosis is active in the discrete model to ensure propagation of the vascular front.

show abstract

Section: Introductionmentioning

confidence: 99%

Modeling angiogenesis: A discrete to continuum description

2017

View full text Add to dashboard Cite

show abstract

“…The short-range interactions experienced by cells often lead to a self-generated spatial structure which can, in turn, have a significant impact on the dynamics of the cell population [14][15][16]. For instance, many cell types are known to form clusters or aggregates as a result of attractive interactions [17,18].…”

Section: Introductionmentioning

confidence: 99%

Spatial moment dynamics for collective cell movement incorporating a neighbour-dependent directional bias

Binny

Plank

James

2015

J. R. Soc. Interface.

View full text Add to dashboard Cite

The ability of cells to undergo collective movement plays a fundamental role in tissue repair, development and cancer. Interactions occurring at the level of individual cells may lead to the development of spatial structure which will affect the dynamics of migrating cells at a population level. Models that try to predict population-level behaviour often take a mean-field approach, which assumes that individuals interact with one another in proportion to their average density and ignores the presence of any small-scale spatial structure. In this work, we develop a lattice-free individual-based model (IBM) that uses random walk theory to model the stochastic interactions occurring at the scale of individual migrating cells. We incorporate a mechanism for local directional bias such that an individual's direction of movement is dependent on the degree of cell crowding in its neighbourhood. As an alternative to the mean-field approach, we also employ spatial moment theory to develop a population-level model which accounts for spatial structure and predicts how these individual-level interactions propagate to the scale of the whole population. The IBM is used to derive an equation for dynamics of the second spatial moment (the average density of pairs of cells) which incorporates the neighbour-dependent directional bias, and we solve this numerically for a spatially homogeneous case.

show abstract

“…A slightly higher order scheme is the Kirkwood superposition approximation that retains 〈 s i s j 〉 p ( t ) ( k = 2) but approximates triple correlations ( k = 3) by a function of second-order correlations, is derivable from [22, 22] a constrained maximum-entropy sense of approximation ≈, and has been used in multi-cellular stochastic modeling [23]. Fully dynamic higher order cutoffs to the moment hierarchy for k > 2 include setting higher cumulants to zero [24], dropping higher order terms from the Kramers–Moyal expansion [25] using moment closure functions that would be correct for log-normal rather than Gaussian distributions [26], and ‘equation-free’ moment closure [27] by sparingly invoking fine-scale simulations.…”

Section: Theorymentioning

confidence: 99%

Model reduction for stochastic CaMKII reaction kinetics in synapses by graph-constrained correlation dynamics

et al. 2015

View full text Add to dashboard Cite

Astochastic reaction network model of Ca2+ dynamics in synapses (Pepke et al PLoS Comput. Biol. 6 e1000675) is expressed and simulated using rule-based reaction modeling notation in dynamical grammars and in MCell. The model tracks the response of calmodulin and CaMKII to calcium influx in synapses. Data from numerically intensive simulations is used to train a reduced model that, out of sample, correctly predicts the evolution of interaction parameters characterizing the instantaneous probability distribution over molecular states in the much larger fine-scale models. The novel model reduction method, ‘graph-constrained correlation dynamics’, requires a graph of plausible state variables and interactions as input. It parametrically optimizes a set of constant coefficients appearing in differential equations governing the time-varying interaction parameters that determine all correlations between variables in the reduced model at any time slice.

show abstract

Modelling collective cell behaviour

Cited by 11 publications

References 37 publications

Modeling angiogenesis: A discrete to continuum description

Modeling angiogenesis: A discrete to continuum description

Spatial moment dynamics for collective cell movement incorporating a neighbour-dependent directional bias

Model reduction for stochastic CaMKII reaction kinetics in synapses by graph-constrained correlation dynamics

Contact Info

Product

Resources

About