Abstract Chemistry:
Organization as a self-maintaining network of transformations

Leo Buss (Yale) and I have written extensively about the motivation underlying this project (Fontana and Buss, 1996,1994a,b): Biology lacks a theory of the possible, because it lacks a theory of organization. What do we mean by organization? We mean a self-maintaining system of molecular transformations. This, of course, is not all there is to the notion of organization, but it constitutes one fundamental aspect. How can it be formalized? Our answer was to seek a formal level of abstraction for chemistry that emphasized molecules as agents of transformation, to define a population dynamics of such agents that would yield self-maintaining ensembles as fixed-``points'' and to characterize these ensembles as kinetically persistent algebraic structures.

In chemistry, molecules combine and interact by transforming one another into further molecules. In mathematics, a theory of transformations is given by lambda-calculus, a formal language that provides a notation and rules to express functions that act on the very expressions representing them. The significance of lambda-calculus derives from the mathematical meaning of its rules. This is best conveyed by a variant of lambda-calculus in which expressions also possess an abstract ``shape'', known as type, that constrains who can interact with whom<sup>3</sup>. It is well-known in logic that lambda-calculus corresponds to a theory of proofs. In this correspondence, a type stands for a logical proposition and a lambda-expression of that type represents a proof of that proposition. The rules of transformation correspond to the rules of logical inference<sup>4</sup>. This notion of formal truth has an appealing physical flavor: something is true, if it can be built. To the extent that lambda-calculus abstracts an essential aspect of chemistry, we are led to a new perspective on chemistry. At one level of description, a molecule is a quantum mechanical entity. Yet, at another level, a molecule may be the proof of a logical proposition which we call its shape<sup>5</sup>. The synthesis of a molecule is like proving a theorem by using lemmas (other molecules) according to rules of a symbolic logic yet to be specified. A self-maintaining system of theorems that mutually prove one another, feeding on axioms, is what logicians call a ``theory'' - and biologists call a metabolism.

Buss and I pursued this idea by implementing a computer model of a well-stirred stochastic flow reactor aimed at exploring the generic behavior of many applicatively interacting lambda-expressions. This constructive dynamical system indeed converges on self-maintaining networks of transformation characterized by syntactical and functional invariances, a capacity for self-repair upon removal of network components, strong constraints to network extension upon the addition of new expressions, and a ``center'', defined as the smallest kinetically persistent and self-maintaining generator set of the network. Such networks we called ``organizations''. The invariances underlying these organizations permit their abstract description as algebras, independently of their concrete instantiation in lambda-calculus. Imposition of different boundary conditions on the stochastic flow reactor generates different levels of organization, and a diversity of organizations within each level. Level 0 is defined by self-copying expressions or simple ensembles of copying expressions. Level 1 denotes self-maintaining organizations in the above sense, and Level 2 is defined by self-maintaining metaorganizations composed of Level 1 organizations.

The wealth of phenomena generated by this simple abstraction of chemistry and the prospect of a formal framework to describe them, encouraged us to seek modifications of extant logics and calculi in an attempt to move closer to the chemistry we know. So far we have not succeeded in a mathematically meaningful way. I continue, however, to seek ways of reviving this project along with applications to other domains, since the idea of organization as a self-sustaining network of transformations is hardly unique to chemistry.

Historically, this project is an ambitious extension of the ``autocatalysis'' and ``autopoiesis'' themes pioneered by Francisco Varela (Maturana and Varela, 1980), Stuart Kauffman (Kauffman, 1993), Norman Packard, Doyne Farmer (Farmer et al., 1982), John McCaskill and Otto Rössler. It is ambitious in that it imputes an analogy between the structure of logic and chemistry. It is an extension in that it plays with these themes at the level of abstract functional ensembles.

Walter Fontana, Santa Fe Institute