The modeling and simulation of morphogenetic phenomena require to take into account the coupling between the processes that take place in a space and the modification of that space due to those processes, leading to a chicken-and-egg problem. To cope with this issue, we propose to consider a growing structure as the byproduct of a multitude of interactions between its constitutive elements. An interaction-based model of computation relying on spatial relationships is then developed leading to an original style of programming implemented in the MGS programming language. While MGS seems to be at first glance a domain specific programming language, its underlying interaction-based paradigm is also relevant to support the development of generic programming mechanisms. We show how the specification of space independent computations achieves polytypism and we develop a direct interpretation of well-known differential operators in term of data movements.