# BioSysBio:abstracts/2007/Naoki Matsumaru/Appendix

## 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 mi,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 fi 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.