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 \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.
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