A Computational Approach of the French Flag Model to Connect Growth and Specification in Developmental Biology

Abstract

The study of the mechanisms that link the global system behavior to the interaction of its parts is an important issue to the field of complex systems. This occurs in many natural systems in biology, ecology and physics. Nature-inspired computation tries to capture and exploit natural solutions in order to solve real problems in many different fields such as telecommunication networks, architecture and urban spaces design, sociology or economy planning. The main purpose of the present research is the creation of a computational path connecting a set of low-level interactions with high-level emerging behaviors. The low-level interactions are modeled by very simple neighbor binary rules. The upper level highlights a regular pattern formation. This framework has been applied to approach the pattern formation in the vertebrate limb development under the scope of the French Flag model. We have developed a four-level theoretical framework. The lowest level defines a set of neighbor binary rules. The second level comes from the iterative application of these rules on sequences of binary values. The resulting sequences are interpreted as arithmetic function values, in binary or decimal representation. The third level highlights a set of computational primitives that have the capability to build directly the function values. Any function can be carried out by means of a particular combination of these primitives, denoted seed behavior. In the fourth level, the emergent behavioral effect is highlighted as a scaling process of the seed behavior. Some functions that have been defined by the four-level framework have been successfully applied to model the French Flag model (progressive specification, intercalary specification and early specification) which is a well-known approach to the pattern formation in the vertebrate limb development. Our empirical research highlights it is possible to develop a computational model which connects very simple rules to complex behavioral patterns. The way consists of a hierarchical organization of computing embedded steps. Our model provides a numerical and geometrical counterpart to the French Flag model and therefore proves its capability to be a formal basis to the case of vertebrate limb development. These encouraging results suggest as a future work the improvement in this model. The composition of functions must be explored as well as new combinations of building primitives in order to identify more emergent effects. More, the relevance of quantitative emergent effects must be highlighted in future applications to natural cases. It is expected that the development of these operations will help addressing new issues such as truncation and graft in vertebrate limbs.