Grid-free models of multicellular systems, with an application to large-scale vortices accompanying primitive streak formation.

This paper is comprised of two parts. In the first we provide a brief overview of grid-free methods for modeling multicellular systems. We focus on an approach based on Langevin equations, in which inertia is ignored, and stochastic effects on cell motion are included. The discussion starts with simpler models, in which cells are modeled as adhesive spheres. We then turn to more sophisticated approaches in which nontrivial cell shape is accommodated, including the recently introduced Subcellular Element Model, in which each cell is described as a cluster of adhesively coupled over-damped subcellular elements, representing patches of cytoskeleton. In the second part of the paper we illustrate the use of a standard grid-free cell-based model to computationally probe interesting new features associated with primitive streak formation in the chick embryo. Streak formation is a key developmental step in amniotes (i.e., birds, reptiles, and mammals), and can be observed in detail in the chick embryo, where the streak extends across a tightly-packed two-dimensional sheet (the epiblast) comprised of about 50,000 cells. The Weijer group [Cui, Yang, Chuai, Glazier, and Weijer, Dev. Biol. 284 (2005) 37-47] recently observed that streak formation is accompanied by coordinated cell movement lateral to the streak, resulting in two large counter-rotating vortices. We study a mechanism based on cell polarity (in the plane of the epiblast) that provides an explanation for these vortices, and test it successfully using computer simulations. This mechanism is robust, since the emergent vortex formation depends only on the gross features of the initial spatial distribution of planar polarity in the epiblast.