Mathematical and computational multiscale models are becoming increasingly important investigative tools in developmental biological research. During certain developmental processes cells that start out as independent entities interact to form multicellular structures. Cells of the embryonic vertebrate limb in high-density culture undergo chondrogenic pattern formation, which results in the formation of regularly-spaced “islands” of cartilage analogous to the cartilage primordia of the developing limb skeleton. In this article we describe a discrete, agent-based stochastic model for studying the behavior of limb bud precartilage mesenchymal cells in vitro. This model, like an earlier one, employs a biologically motivated reaction-diffusion process and cell-matrix adhesion as the basis of self-organizing pattern formation, but constitutes an improvement in biological fidelity over previous descriptions in that it is multiscale (i.e., cell and molecular dynamics occur on distinct scales), and the cells are represented as spatially extended objects. The improved model reproduces a broader set of results of the micromass culture system than the previous one and discloses multiple dynamical regimes that suggest new biological experiments.

Cells of the embryonic vertebrate limb in high-density culture undergo chondrogenic pattern formation, which results in the production of regularly spaced "islands" of cartilage similar to the cartilage primordia of the developing limb skeleton. The first step in this process, in vitro and in vivo, is the generation of "cell condensations," in which the precartilage cells become more tightly packed at the sites at which cartilage will form. In this paper we describe a discrete, stochastic model for the behavior of limb bud precartilage mesenchymal cells in vitro. The model uses a biologically motivated reaction-diffusion process and cell-matrix adhesion (haptotaxis) as the bases of chondrogenic pattern formation, whereby the biochemically distinct condensing cells, as well as the size, number, and arrangement of the multicellular condensations, are generated in a self-organizing fashion. Improving on an earlier lattice-gas representation of the same process, it is multiscale (i.e., cell and molecular dynamics occur on distinct scales), and the cells are represented as spatially extended objects that can change their shape. The authors calibrate the model using experimental data and study sensitivity to changes in key parameters. The simulations have disclosed two distinct dynamic regimes for pattern self-organization involving transient or stationary inductive patterns of morphogens. The authors discuss these modes of pattern formation in relation to available experimental evidence for the in vitro system, as well as their implications for understanding limb skeletal patterning during embryonic development.

We present COMPUCELL3D, a software framework for three-dimensional simulation of morphogenesis in different organisms. COMPUCELL3D employs biologically relevant models for cell clustering, growth, and interaction with chemical fields. COMPUCELL3D uses design patterns for speed, efficient memory management, extensibility, and flexibility to allow an almost unlimited variety of simulations. We have verified COMPUCELL3D by building a model of growth and skeletal pattern formation in the avian (chicken) limb bud. Binaries and source code are available, along with documentation and input files for sample simulations, at http:// compucell.sourceforge.net.

MOTIVATION: CompuCell is a multi-model software framework for simulation of the development of multicellular organisms known as morphogenesis. It models the interaction of the gene regulatory network with generic cellular mechanisms, such as cell adhesion, division, haptotaxis and chemotaxis. A combination of a state automaton with stochastic local rules and a set of differential equations, including subcellular ordinary differential equations and extracellular reaction-diffusion partial differential equations, model gene regulation. This automaton in turn controls the differentiation of the cells, and cell-cell and cell-extracellular matrix interactions that give rise to cell rearrangements and pattern formation, e.g. mesenchymal condensation. The cellular Potts model, a stochastic model that accurately reproduces cell movement and rearrangement, models cell dynamics. All these models couple in a controllable way, resulting in a powerful and flexible computational environment for morphogenesis, which allows for simultaneous incorporation of growth and spatial patterning. RESULTS: We use CompuCell to simulate the formation of the skeletal architecture in the avian limb bud. AVAILABILITY: Binaries and source code for Microsoft Windows, Linux and Solaris are available for download from http://sourceforge.net/projects/compucell/