Evolution of Biological Information Processing Systems
National Science Foundation, Arlington, Va
January 6-7, 1995
23. MODELING OF COMPLEX SYSTEMS USING CATEGORY THEORY
Andrée Ehresmann and Jean-Paul Vanbremeersch
Université de Picardie Jules Veme, Amiens, France
Information processing in biological systems depends on complex biochemical interactions between components of the same or of different levels (from atoms to molecules, to cells, to tissues...), and the global dynamics follows from a balance between overlapping and possibly conflicting regulations, each one operating at its own time-scale. This characteristics of biological systems differentiates them from Systems Studied in Physics.
However, most mathematical models developed for Biology inherit their methods from Theoretical Physics, mostly based on classical differential equations, on information theory, or thermodynamics, on dissipative systems or chaos,.... so they give valuable results, but only for specific, temporarily and locally circumscribed mechanisms. For instance a recent model (by Novak and Tyson) of the regulation of the cell cycle reduces it to the 2 feedback loops between cyclins and a protein kinase and does not account for the coupling with the cell environment; even so it already necessitates a set of 13 differential equations and the initial determination of 13 parameters.
These models, tailored for a specific process in a well-delimited environment, are not
flexible enough to deal with more global information processing, nor even with some contradictory experimental results involving simple regulations in a variable environment. For instance, Nunez has shown that more systemist approaches allow to explain e.g. the ambivalent effects of Alpha-foetoprotein (AFP) on immuno-competent cells (they depend on the concentration of free unsaturated fatty acids, because these acids bind to AFP and have immuno-modulatory properties of their own).
So new mathematical tools seem necessary, and such a tool could be Category Theory, as we have proposed in a series of papers since 1986. This theory, introduced by Eilenberg and Mac Lane in 1945, has been designed for the study of interrelations between different structures. Let us recall that a category consists of objects and links (represented by arrows) between them, forming an oriented graph on which there is given a law to combine a path of successive arrows from N to N' into a well-defined arrow from N to N'.
Thc interest of categories in the study of complexity had already been underlined by R. Rosen in 1958, but his models (e.g., its metabolism-repair systems), as well as those of biomathematicians following his lead, do not exploit the whole strength of category theory for they make only use of 'large' categories as an overall frame. To cope with complex information processing in a dynamical manner and to support the main biochemical mechanisms (in particular those singled out in the C8 taxonomy of J. Chandler), we must resort to more specific categorical constructions (such as the complexification process), and account for temporal and energetical considerations, in particular through an enrichment of the categories by a ponderation of their links (representing their strengths and their propagation delays). The model we propose (called Memory Evolutive Systems) has several characteristics which could make it amenable to the study of evolutionary information processing in biological systems:
1. Components of several complexity levels can be handled simultaneously to study their interactions. This relies on an adequate usage of the well-known categorical colimit operation that allows to represent in the same category a hierarchy of objects of increasing complexity with a fractal-like property: an object C of level n has its own structure of category, which is a sub-category (or pattern) of level n-1 describing its internal conformation; and conversely, C is obtained as the colimit of this pattern (concatenation process).
For example, the closure and conformation of a cell at a given time will be modeled by a category in which the objects are all the components of the cell from its atoms, to its macromolecules, to populations of molecules and organelles, ..., and the links describe the topological or energetical interactions between these objects. In this category, the DNA is a particular object of the sub-cellular level; but, considered with its geometric molecular conformation, it is also represented by the sub-category (or pattern) of the molecular level of which the objects are the successive DNA bases, and the links are paths of oriented chemical bonds between them; the force of each link is correlated to its length and the combination law takes into account the angle between successive bonds. The DNA as an entity is obtained by concatenating this pattern through a 2-step colimit process, the first giving its primary structure, the second leading to its temary structure.
2. The dynamics of a biological system, say a cell, consists in the ingestion of extracellular products (endocytosis), the assembly of new components (synthesis of proteins,...) , the destruction or dis-assembÌy of some constituents. In the categorical setting, this dynamics will be represented by the complexification process with respect to a strategy aiming at the addition or suppression of certain objects or colimits to a category. This process depends on both local and global information; for instance the formation of a colimit consists in a local strengthening of the links of a pattern, but once the pattern takes an identity of its own as a colimit, it acquires emergent properties with global implications.
3. The robustness and plasticity of biological systems can be related to the multiplicity (or degeneracy) property of the colimit operation, that measures the extent of modifications a pattern may tolerate while its colimit remains unchanged. We have shown that this multiplicity property is also at the root of the emergence of more and more complex objects by iteration of the complexification process. In particular it helps explain the formation of higher order cognitive processes in a neural system, which are represented by iterated colimits of coherent Hebb assemblies of synchronous neurons. It could also lead to applications in Evolution Theory.
To account for the multiplicity of overlapping internaÌ regulations in an organism, we introduce a net of competitive internal regulatory centers (CR) of different levels. Each CR locally controls a stepwise process at its own time-scale, in which a cycle decomposes into several phases: processing of accessible information to form its actual landscape, selection of a strategy on this landscape, command of this strategy through messages sent to effectors, evaluation of the result and possibly its memorization for later use.
5. The possible conflicts between the various CRs and the repair pathways can be studied in this frame. The conflicts ensue from the fact that, at a given time, the current strategies independently devised by the CRs in their landscapes are not necessarily coherent. To overcome the conflict, some strategies will be discarded, thus causing a fracture for the corresponding CRs. The repair of the fractures may come from the same CR, or from other CRs which impose a new strategy, with possible repercussions later on. Thus the global dynamics is modulated by the dialectics between CRs with heterogeneous complexity levels and time-scales. For instance, the DNA replication of a bacterium is interrupted if the usual simultaneous repair mechanisms are disrupted because of too much damage in a strand; but then the higher cell level can impose to pursue the replication, possibly with a mutation, through the activation of the SOS system.
.The need for a sufficient coordination between the interconnected regulatory mechanisms (CRs) operating on different time-scales imposes stringent temporal constraints on a biological system. An analysis of the situation at the level of a specific CR exhibits some structural and temporal constraints that must be respected for its cycle to be completed in due time; they are expressed in the form of inequalities correlating the period of the CR with the mean propagation delays of the links in its landscape and with the stability spans of its objects (the stability span of a complex object is related to the rate of renewal of its internal organization, think of the half-life of a protein). If these constraints cannot be realized during a long enough time, there is a de-resynchronization consisting in a change of period for the CR. In particular, we have proposed a theory of aging based on such a cascade of de-resynchronizations at increasing levels; this theory seems to unify the various physiological theories.
The MES model gives a general frame, emphasizing the time dimension, to study emergent properties and intertwined complex regulations. However, it is more qualitative than quantitative, and a collaboration with biologists, in particular to provide data about the time-schedules of specific reactions and information processing, would be necessary to develop it up to concrete applications in Biology.