Interacting motile agents: taking a mean-field approach beyond monomers and nearest-neighbor steps

Comparative analysis of continuum angiogenesis models

Although discrete approaches are increasingly employed to model biological phenomena, it remains unclear how complex, population-level behaviours in such frameworks arise from the rules used to represent interactions between individuals. Discrete-to-continuum approaches, which are used to derive systems of coarse-grained equations describing the mean-field dynamics of a microscopic model, can provide insight into such emergent behaviour. Coarse-grained models often contain nonlinear terms that depend on the microscopic rules of the discrete framework, however, and such nonlinearities can make a model difficult to mathematically analyse. By contrast, models developed using phenomenological approaches are typically easier to investigate but have a more obscure connection to the underlying microscopic system. To our knowledge, there has been little work done to compare solutions of phenomenological and coarse-grained models. Here we address this problem in the context of angiogenesis (the creation of new blood vessels from existing vasculature). We compare asymptotic solutions of a classical, phenomenological “snail-trail” model for angiogenesis to solutions of a nonlinear system of partial differential equations (PDEs) derived via a systematic coarse-graining procedure (Pillay et al. in Phys Rev E 95(1):012410, 2017. https://doi.org/10.1103/PhysRevE.95.012410). For distinguished parameter regimes corresponding to chemotaxis-dominated cell movement and low branching rates, both continuum models reduce at leading order to identical PDEs within the domain interior. Numerical and analytical results confirm that pointwise differences between solutions to the two continuum models are small if these conditions hold, and demonstrate how perturbation methods can be used to determine when a phenomenological model provides a good approximation to a more detailed coarse-grained system for the same biological process.

Evaluating snail-trail frameworks for leader-follower behavior with agent-based modeling

Branched networks constitute a ubiquitous structure in biology, arising in plants, lungs, and the circulatory system; however, the mechanisms behind their creation are not well understood. A commonly used model for network morphogenesis proposes that sprouts develop through interactions between leader (tip) cells and follower (stalk) cells. In this description, tip cells emerge from existing structures, travel up chemoattractant gradients, and form new networks by guiding the movement of stalk cells. Such dynamics have been mathematically represented by continuum "snail-trail" models in which the tip cell flux contributes to the stalk cell proliferation rate. Although snail-trail models constitute a classical depiction of leader-follower behavior, their accuracy has yet to be evaluated in a rigorous quantitative setting. Here, we extend the snail-trail modeling framework to two spatial dimensions by introducing a novel multiplicative factor to the stalk cell rate equation, which corrects for neglected network creation in directions other than that of the migrating front. Our derivation of this factor demonstrates that snail-trail models are valid descriptions of cell dynamics when chemotaxis dominates cell movement. We confirm that our snail-trail model accurately predicts the dynamics of tip and stalk cells in an existing agent-based model (ABM) for network formation [Pillay et al., Phys. Rev. E 95, 012410 (2017)10.1103/PhysRevE.95.012410]. We also derive conditions for which it is appropriate to use a reduced, one-dimensional snail-trail model to analyze ABM results. Our analysis identifies key metrics for cell migration that may be used to anticipate when simple snail-trail models will accurately describe experimentally observed cell dynamics in network formation.

From a discrete model of chemotaxis with volume-filling to a generalized Patlak-Keller-Segel model

We present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased random walk. In order to take into account possible alterations in cellular motility observed at high cell densities (i.e. volume-filling), we let the probabilities of cell movement be modulated by a decaying function of the cell density. We formally show that a general form of the celebrated Patlak-Keller-Segel (PKS) model of chemotaxis can be formally derived as the appropriate continuum limit of this discrete model. The family of steady-state solutions of such a generalized PKS model are characterized and the conditions for the emergence of spatial patterns are studied via linear stability analysis. Moreover, we carry out a systematic quantitative comparison between numerical simulations of the discrete model and numerical solutions of the corresponding PKS model, both in one and in two spatial dimensions. The results obtained indicate that there is excellent quantitative agreement between the spatial patterns produced by the two models. Finally, we numerically show that the outcomes of the two models faithfully replicate those of the classical PKS model in a suitable asymptotic regime.