We discuss theoretical and experimental approaches to three distinct developmental systems that illustrate how theory can influence experimental work and vice-versa. The chosen systems – Drosophila melanogaster, bacterial pattern formation, and pigmentation patterns – illustrate the fundamental physical processes of signaling, growth and cell division, and cell movement involved in pattern formation and development. These systems exemplify the current state of theoretical and experimental understanding of how these processes produce the observed patterns, and illustrate how theoretical and experimental approaches can interact to lead to a better understanding of development. As John Bonner said long ago
`We have arrived at the stage where models are useful to suggest experiments, and the facts of the experiments in turn lead to new and improved models that suggest new experiments. By this rocking back and forth between the reality of experimental facts and the dream world of hypotheses, we can move slowly toward a satisfactory solution of the major problems of developmental biology.'

During vertebrate development cells acquire different fates depending largely on their location in the embryo. The definition of a cell's developmental fate relies on extensive intercellular communication that produces positional information and ultimately generates an appropriately proportioned anatomy. Here we place reaction-diffusion mechanisms in the context of general concepts regarding the generation of positional information during development and then focus on these mechanisms as parsimonious systems for positioning anatomical structures relative to one another. In particular, we discuss the evidence for reaction-diffusion systems operating in the developing skin to yield the periodic arrangements of hairs and feathers and discuss how best to bring together experimental molecular biology and numerical simulations to yield a more complete understanding of the mechanisms of development and natural variation.

The maintenance of a stable stem cell population in the epidermis is important for robust regeneration of the stratified epithelium. The population size is usually regulated by cell secreted extracellular signalling molecules as well as intracellular molecules. In this paper, a simple model incorporating both levels of regulation is developed to examine the balance between growth and differentiation for the stem cell population. In particular, the dynamics of a known differentiation regulator c-Myc, its threshold dependent differentiation, and feedback regulation on maintaining a stable stem cell population are investigated.

The circulatory system is one of the first to function during development. The earliest event in the system's development is vasculogenesis, whereby vascular progeniter cells form clusters called blood islands, which later fuse to form capillary networks. There exists a very good in vitro system that mimics this process. When HUVECs (Human Umbilical Vein Endothelial Cells) are cultured on Matrigel, they spontaneously form a capillary network structure. Two theoretical models have been proposed to explain the pattern formation of this in vitro system. Both models utilize chemotaxis to generate spatial instability, and one model specifies VEGF as the chemoattractant. However, there are several unknown factors concerning the experimental model. First, the pattern formation process occurs at the interface between the liquid medium and Matrigel, and it is unclear whether diffusion in the liquid or gel is critical. Second, the diffusion coefficient of VEGF, which determines the spatial scale of the capillary structure, has not been properly measured. In the present study, we modified the experimental system to clarify the effect of diffusion in Matrigel, and experimentally measured the diffusion coefficient of VEGF in this system. The relationship with the spatial scale of the pattern generated is discussed.

Skeletal patterning in the vertebrate limb, i.e., the spatiotemporal regulation of cartilage differentiation (chondrogenesis) during embryogenesis and regeneration, is one of the best studied examples of a multicellular developmental process. Recently [Alber et al., The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb, Bulletin of Mathematical Biology, 2008, v70, pp. 460-483], a simplified two-equation reaction-diffusion system was developed to describe the interaction of two of the key morphogens: the activator and an activator-dependent inhibitor of precartilage condensation formation. A discontinuous Galerkin (DG) finite element method was applied to solve this nonlinear system on complex domains to study the effects of domain geometry on the pattern generated [Zhu et al., Application of Discontinuous Galerkin Methods for reaction-diffusion systems in developmental biology, Journal of Scientific Computing, 2009, v40, pp. 391-418]. In this paper, we extend these previous results and develop a DG finite element model in a moving and deforming domain for skeletal pattern formation in the vertebrate limb. Simulations reflect the actual dynamics of limb development and indicate the important role played by the geometry of the undifferentiated apical zone.

Cell-based, mathematical models help make sense of morphogenesis—i.e. cells organizing into shape and pattern—by capturing cell behavior in simple, purely descriptive models. Cell-based models then predict the tissue-level patterns the cells produce collectively. The first step in a cell-based modeling approach is to isolate sub-processes, e.g. the patterning capabilities of one or a few cell types in cell cultures. Cell-based models can then identify the mechanisms responsible for patterning in vitro. This review discusses two cell culture models of morphogenesis that have been studied using this combined experimental-mathematical approach: chondrogenesis (cartilage patterning) and vasculogenesis (de novo blood vessel growth). In both these systems, radically different models can equally plausibly explain the in vitro patterns. Quantitative descriptions of cell behavior would help choose between alternative models. We will briefly review the experimental methodology (microfluidics technology and traction force microscopy) used to measure responses of individual cells to their micro-environment, including chemical gradients, physical forces and neighboring cells. We conclude by discussing how to include quantitative cell descriptions into a cell-based model: the Cellular Potts model.