BioSysBio:abstracts/2007/Naoki Matsumaru/Appendix

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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.

Closure
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.

Self-maintenance
Given an algebraic chemistry , let denote the -th molecular species of and the -th reaction rules is . Given the stoichiometric matrix that corresponds to where denotes the number of molecules of species produced or used up in reaction , a set of molecular species is self-maintaining, if there exists a flux vector satisfying the following three conditions:
  • if
  • if
  • if where .

These three conditions can be read as follows: When the -th reaction is applicable to the set , the flux must be positive (Condition 1). All other fluxes are set to zero (Condition 2). Finally, the production rate 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:

Organization
A set of molecular species that is closed and self-maintaining is called an organization.