## Abstract

In paper I a theory of functional organization in terms of functional interactions was proposed for a formal biological system (FBS). A functional interaction was defined as the product emitted by a structural unit, i.e. an assembly of molecules, cells, tissues or organs, which acts on another. We have shown that a self-association hypothesis could be an explanation for the source of these functional interactions because it is consistent with increased stability of the system after association. The construction of the set of interactions provides the topology of the biological system, called (O-FBS), incontrast to the (D-FBS) which describes the dynamics of the processes associated with the functional interactions. In this paper, an optimum principle is established, due to the non-symmetry of functional interactions, which could explain the stability of an FBS, and a criterion of evolution for the hierarchical topological organization of a FBS during development is deduced from that principle. The combinatorics of the (O-FBs) leads to the topological stability of the related graph. It is shown that this problem can be expressed as the re-distribution of sources and sinks, when one of them is suppressed, given the constraint of the invariance of the physiological function. Such an optimum principle could be called a 'principle of increase in functional order by hierarchy'. The first step is the formulation of a 'potential' for the functional organization, which describes the ability of the system to combine functional interactions, such that the principle of vital coherence (paper I) is satisfied. This function measures the number of potential functional interactions. The second step is to discover the maximum of this function. Biological systems in such a state of maximum organization are shown to satisfy particular dynamics, which can be experimentally verified: either the number of sinks decreases, or this number increases, in a monotonic way. The class of systems considered here is assumed to satisfy such an extremum hypothesis. The third step is a study of the variation of the degree of organization (paper I), i.e. the number of structural units when the biological system is growing. We establish an optimum principle for a new function called 'orgatropy'. By adding a criterion of specialization to the system we show the emergence of a level of organization with a re-organization of the system. A 'hierarchization operator' is defined, which leads to the fourth step of this theory, i.e. the conditions of the variation in time of the (O-FBS) sub-system, as described by a 'functional order' mathematical function. It is shown that this function is a Lyapounov function, and drives the system towards a stable state with maximum specialization. Two consequences of the theory are studied: (i) the monotonic phases which are observed during development of the nervous system, and (ii) the similarities and differences observed between physical systems and biological systems.