  A central question in developmental biology is how size and position are
determined. The genetic code carries instructions on how to control these
properties in order to regulate the pattern and morphology of structures in the
developing organism. Transcription and protein translation mechanisms implement
these instructions. However, this cannot happen without some manner of sampling
of epigenetic information on the current patterns and morphological forms of
structures in the organism. Any rigorous description of space- and time-varying
patterns and morphological forms reduces to one among various classes of
spatio-temporal partial differential equations. Reaction-transport equations
represent one such class. Starting from simple Fickian diffusion, the
incorporation of reaction, phase segregation and advection terms can represent
many of the patterns seen in the animal and plant kingdoms. Morphological form,
requiring the development of three-dimensional structure, also can be
represented by these equations of mass transport, albeit to a limited degree.
The recognition that physical forces play controlling roles in shaping tissues
leads to the conclusion that (nonlinear) elasticity governs the development of
morphological form. In this setting, inhomogeneous growth drives the elasticity
problem. The combination of reaction-transport equations with those of
elasto-growth makes accessible a potentially unlimited spectrum of patterning
and morphogenetic phenomena in developmental biology. This perspective
communication is a survey of the partial differential equations of mathematical
physics that have been proposed to govern patterning and morphogenesis in
developmental biology. Several numerical examples are included to illustrate
these equations and the corresponding physics, with the intention of providing
physical insight wherever possible.
