We discuss theoretical and experimental approaches to three distinctdevelopmental systems that illustrate how theory can influence experimental workand vice-versa. The chosen systems – Drosophila melanogaster,bacterial pattern formation, and pigmentation patterns – illustrate thefundamental physical processes of signaling, growth and cell division, and cellmovement involved in pattern formation and development. These systems exemplifythe current state of theoretical and experimental understanding of how theseprocesses produce the observed patterns, and illustrate how theoretical andexperimental approaches can interact to lead to a better understanding ofdevelopment. As John Bonner said long ago
`We have arrived at the stage where models are useful to suggestexperiments, and the facts of the experiments in turn lead to new and improvedmodels that suggest new experiments. By this rocking back and forth between thereality of experimental facts and the dream world of hypotheses, we can moveslowly toward a satisfactory solution of the major problems of developmentalbiology.'

During vertebrate development cells acquire different fates depending largely ontheir location in the embryo. The definition of a cell's developmentalfate relies on extensive intercellular communication that produces positionalinformation and ultimately generates an appropriately proportioned anatomy. Herewe place reaction-diffusion mechanisms in the context of general conceptsregarding the generation of positional information during development and thenfocus on these mechanisms as parsimonious systems for positioning anatomicalstructures relative to one another. In particular, we discuss the evidence forreaction-diffusion systems operating in the developing skin to yield theperiodic arrangements of hairs and feathers and discuss how best tobring together experimental molecular biology and numerical simulations to yielda more complete understanding of the mechanisms of development and naturalvariation.

The maintenance of a stable stem cellpopulation in the epidermis is important for robust regenerationof the stratified epithelium. The population size is usuallyregulated by cell secreted extracellular signalling molecules aswell as intracellular molecules. In this paper, a simple modelincorporating both levels of regulation is developed to examinethe balance between growth and differentiation for the stem cellpopulation. In particular, the dynamics of a known differentiationregulator c-Myc, its threshold dependent differentiation, andfeedback regulation on maintaining a stable stem cell populationare investigated.

The circulatory system is one of thefirst to function during development. The earliest event in thesystem's development is vasculogenesis, whereby vascularprogeniter cells form clusters called blood islands, which laterfuse to form capillary networks. There exists a very goodin vitro system that mimics this process. When HUVECs(Human Umbilical Vein Endothelial Cells) are cultured on Matrigel,they spontaneously form a capillary network structure. Twotheoretical models have been proposed to explain the patternformation of this in vitro system. Both models utilizechemotaxis to generate spatial instability, and one modelspecifies VEGF as the chemoattractant. However, there are severalunknown factors concerning the experimental model. First, thepattern formation process occurs at the interface between theliquid medium and Matrigel, and it is unclear whether diffusion inthe liquid or gel is critical. Second, the diffusion coefficientof VEGF, which determines the spatial scale of the capillarystructure, has not been properly measured. In the present study,we modified the experimental system to clarify the effect ofdiffusion in Matrigel, and experimentally measured the diffusioncoefficient of VEGF in this system. The relationship with thespatial 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 oneof the best studied examples of a multicellular developmental process.Recently [Alber et al., The morphostatic limit for a model ofskeletal pattern formation in the vertebrate limb, Bulletin ofMathematical Biology, 2008, v70, pp. 460-483], a simplified two-equationreaction-diffusion system was developed to describe the interaction of two ofthe key morphogens: the activator and an activator-dependent inhibitor ofprecartilage condensation formation. A discontinuous Galerkin (DG)finite element method was applied to solve this nonlinear system on complexdomains to study the effects of domain geometry on the pattern generated [Zhu etal., Application of Discontinuous Galerkin Methods for reaction-diffusionsystems in developmental biology, Journal of Scientific Computing, 2009, v40,pp. 391-418]. In this paper, we extend these previous results and develop a DGfinite element model in a moving and deforming domain for skeletal patternformation in the vertebrate limb. Simulations reflect the actual dynamics oflimb development and indicate the important role played by the geometryof the undifferentiated apical zone.

Cell-based, mathematical models helpmake sense of morphogenesis—i.e. cells organizing intoshape and pattern—by capturing cell behavior in simple, purelydescriptive models. Cell-based models then predict thetissue-level patterns the cells produce collectively. The firststep in a cell-based modeling approach is to isolatesub-processes, e.g. the patterning capabilities of one or afew cell types in cell cultures. Cell-based models can thenidentify the mechanisms responsible for patterning in vitro.This review discusses two cell culture models of morphogenesisthat have been studied using this combinedexperimental-mathematical approach: chondrogenesis (cartilagepatterning) and vasculogenesis (de novo blood vessel growth). Inboth these systems, radically different models can equallyplausibly explain the in vitro patterns. Quantitativedescriptions of cell behavior would help choose betweenalternative models. We will briefly review the experimentalmethodology (microfluidics technology and traction forcemicroscopy) used to measure responses of individual cells to theirmicro-environment, including chemical gradients, physical forcesand neighboring cells. We conclude by discussing how to includequantitative cell descriptions into a cell-based model: theCellular Potts model.