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.