Process Algebra and Membrane Computing have produced a number of formalisms for
describing the hierarchical (topological) organization of complex systems. In
applications to biological modeling, those approaches are sufficient to characterize
well-mixed systems subdivided into nested compartments, which are commonly found in
(sub-)cellular biology. There are many situations in which, however, geometry is
necessary: both at the subcellular level and at higher levels of cellular organization.
Developmental biology, in particular, deals with dynamic spatial arrangements of cells,
and with forces and interactions between them. Many approaches have been developed for
discrete geometric organization, including Cellular Automata, and graph models, but few
cover both complex geometry and complex interaction.
In the context of Process Algebra, to begin to move from topology to geometry, we have
developed a calculus of processes located in 3-dimensional geometric space. While it may
seem in principle easy to ‘add a position to each process’, naive attempts result in
awkward formal systems with too many features: coordinates, position, velocity, identity,
force, collision, communication, and so on. Moreover, developmental biology is peculiar
in that the coordinate space is not fixed: it effectively expands, moves, and warps as
the organism is developing, making approaches based on fixed grids also awkward.
In this work we present a ?-calculus extended with a single new geometric construct,
frame shift, which consists of applying a 3-dimensional affine transformation to a whole
evolving process. This calculus is sufficient to express many dynamic geometric
behaviors, thanks to the combined power of Affine Geometry and Process Algebra. It
remains a relatively simple ?-calculus, technically formulated in a familiar way, with a
large but standard and fairly orthogonal geometric subsystem. From a Process Algebra
point of view, the calculus adds powerful geometric data structures and transformations.
From an Affine Geometry point of view, the calculus preserves key geometric invariants.