Chemical Organization Theory
A set of molecules is called an organization if the following two properties are satisfied: closure and self-maintenance. A set of molecular species is closed when all reaction rules applicable to the set cannot produce a molecular species that is not in the set. This is similar to the algebraic closure of an operation in set theory.
- Given an algebraic chemistry , a set of molecular species is closed, if for every reaction with , also holds.
The second important property, self-maintenance, assures, roughly speaking, that all molecules that are consumed within a self-maintaining set can also be produced by some reaction pathways within the self-maintaining set. The general definition of self-maintenance is more complicated than the definition of closure because the production and consumption of a molecular species can depend on many molecular species operating as a whole in a complex pathway.
- Given an algebraic chemistry , let i denote the i-th molecular species of and the j-th reaction rules is . Given the stoichiometric matrix that corresponds to where mi,j denotes the number of molecules of species i produced or used up in reaction j, a set of molecular species is self-maintaining, if there exists a flux vector satisfying the following three conditions:
- if where .
These three conditions can be read as follows: When the j-th reaction is applicable to the set S, the flux must be positive (Condition 1). All other fluxes are set to zero (Condition 2). Finally, the production rate fi for all the molecular species must be nonnegative (Condition~3). Note that we have to find only one such flux vector in order to show that a set is self-maintaining.
Taking closure and self-maintenance together, we arrive at an organization:
- A set of molecular species that is closed and self-maintaining is called an organization.