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 [math]\displaystyle{ \langle {\mathcal M},{\mathcal R} \rangle }[/math], a set of molecular species [math]\displaystyle{ C \subseteq {\mathcal M} }[/math] is closed, if for every reaction [math]\displaystyle{ (A \rightarrow B) \in {\mathcal R} }[/math] with [math]\displaystyle{ A \in \mathcal{P}_M(C) }[/math], also [math]\displaystyle{ B \in \mathcal{P}_M(C) }[/math] 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 [math]\displaystyle{ \langle {\mathcal M},{\mathcal R} \rangle }[/math], let [math]\displaystyle{ i }[/math] denote the [math]\displaystyle{ i }[/math]-th molecular species of [math]\displaystyle{ {\mathcal M} }[/math] and the [math]\displaystyle{ j }[/math]-th reaction rules is [math]\displaystyle{ (A_j \rightarrow B_j) \in {\mathcal R} }[/math]. Given the stoichiometric matrix [math]\displaystyle{ \mathbf{M} = (m_{i,j}) }[/math] that corresponds to [math]\displaystyle{ \langle {\mathcal M},{\mathcal R} \rangle }[/math] where [math]\displaystyle{ m_{i,j} }[/math] denotes the number of molecules of species [math]\displaystyle{ i }[/math] produced or used up in reaction [math]\displaystyle{ j }[/math], a set of molecular species [math]\displaystyle{ S \subseteq {\mathcal M} }[/math] is self-maintaining, if there exists a flux vector [math]\displaystyle{ \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 }[/math] satisfying the following three conditions:
  • [math]\displaystyle{ v_{A_j \rightarrow B_j} \gt 0 }[/math] if [math]\displaystyle{ A_j \in \mathcal{P}_M(S) }[/math]
  • [math]\displaystyle{ v_{A_j \rightarrow B_j} = 0 }[/math] if [math]\displaystyle{ A_j \notin \mathcal{P}_M(S) }[/math]
  • [math]\displaystyle{ f_i \geq 0 }[/math] if [math]\displaystyle{ s_i \in S }[/math] where [math]\displaystyle{ (f_1, \dots, f_i, \dots, f_{|{\mathcal M}|})^T = \mathbf{M v} }[/math].

These three conditions can be read as follows: When the [math]\displaystyle{ j }[/math]-th reaction is applicable to the set [math]\displaystyle{ S }[/math], the flux [math]\displaystyle{ v_{A_j \to B_j} }[/math] must be positive (Condition 1). All other fluxes are set to zero (Condition 2). Finally, the production rate [math]\displaystyle{ f_i }[/math] for all the molecular species [math]\displaystyle{ s_i \in S }[/math] 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 [math]\displaystyle{ O \subseteq {\mathcal M} }[/math] that is closed and self-maintaining is called an organization.