A major goal of developmental biology is to understand how the precise locations of structures and cell types in the embryo are specified. Much of this tissue patterning is orchestrated by morphogens, diffusible molecules that exert qualitatively different effects on cell fate at different concentrations. In many cases, tissue patterning is consistent with an action of morphogen gradients that instruct cells to adopt different fates as a function of their location in space. Such position-dependent assignment of fates is an essential feature of a patterning system. Here we propose to investigate morphogen signaling and pattern formation through an integrated program of mathematical and experimental approaches. We will go beyond phenomenological modeling and explicitly incorporate specific biological processes-receptor binding and dissociation, endo/exocytosis of receptors, degradation, feedback regulation of morphogen and receptor synthesis, etc.--that are known from experiments to influence morphogen-mediated patterning. Our ultimate goal is to develop quantitative, mechanistic, and experimentally testable theories of the signaling and gene transcription that underlie pattern formation. [unreadable] [unreadable] Quantitative theories of morphogen-mediated patterning that are based on known biological processes will advance our understanding of fundamental developmental biology, and may also lead to applications to other complex biological systems, such as signaling and gene-regulatory networks. The results from these applications may help to understand and address many types of human developmental abnormalities.