# 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
*i*denote the*i*-th molecular species of and the*j*-th reaction rules is . Given the stoichiometric matrix that corresponds to where*m*_{i,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
- if
- 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 *f*_{i}
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.