BioSysBio:abstracts/2007/Thorsten Lenser/AppendixA

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 $$\langle {\mathcal M},{\mathcal R} \rangle$$, a set of molecular species $$C \subseteq {\mathcal M}$$ is closed, if for every reaction $$(A \rightarrow B) \in {\mathcal R}$$ with $$A \in \mathcal{P}_M(C)$$, also $$B \in \mathcal{P}_M(C)$$ 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 $$\langle {\mathcal M},{\mathcal R} \rangle$$, let $$i$$ denote the $$i$$-th molecular species of $${\mathcal M}$$ and the $$j$$-th reaction rules is $$(A_j \rightarrow B_j) \in {\mathcal R}$$. Given the stoichiometric matrix $$\mathbf{M} = (m_{i,j})$$ that corresponds to $$\langle {\mathcal M},{\mathcal R} \rangle$$ where   $$m_{i,j}$$ denotes the number of molecules of species $$i$$ produced or used up in reaction $$j$$, a set of molecular species $$S \subseteq {\mathcal M}$$ is self-maintaining, if there exists a flux vector $$\mathbf{v} = (v_{A_1 \rightarrow B_1}, \dots, v_{A_j \rightarrow B_j}, \dots, v_{A_{|{\mathcal R}|} \rightarrow B_{|{\mathcal R}|}})^T$$ satisfying the following three conditions:


 * $$v_{A_j \rightarrow B_j} > 0$$ if $$A_j \in \mathcal{P}_M(S)$$
 * $$v_{A_j \rightarrow B_j} = 0$$ if $$A_j \notin \mathcal{P}_M(S)$$
 * $$f_i \geq 0$$ if $$s_i \in S$$ where $$(f_1, \dots, f_i, \dots, f_{|{\mathcal M}|})^T = \mathbf{M v}$$.

These three conditions can be read as follows: When the $$j$$-th reaction is applicable to the set $$S$$, the flux $$v_{A_j \to B_j}$$ 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 $$s_i \in S$$ 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 $$O \subseteq {\mathcal M}$$ that is closed and self-maintaining is called an organization.