File:Organizational-structure-mis-n04-01.png

Organizational structure found in a chemical reaction network to solve the maximal independent set problem in an undirected graph with four vertexes and four edges. There are ten organizations in total. The smallest organization is the empty set at the bottom, and the biggest organizations contain four species at the top.

Focusing on the biggest organizations of size four, we have:

[math]\displaystyle{ \{s_1^0, s_2^1, s_3^0, s_4^1\}, \{s_1^0, s_2^0, s_3^1, s_4^0\}, \{s_1^1, s_2^0, s_3^0, s_4^1\} }[/math]

(Note: species [math]\displaystyle{ s_1^0 }[/math] is denoted as [math]\displaystyle{ \mathsf{s10} }[/math] in the figure.)

Each organization can be interpreted as a set of vertexes:

[math]\displaystyle{ \{v_2^{},v_4^{}\}, \{v_3^{}\}, \{v_1^{}, v_4^{}\} }[/math],

and these are the maximal independent sets. Note that not every organization can be mapped to a set of vertexes. There are organizations of size less than four, and those represent an undefined state, neither a vertex is included in nor excluded from the set.

When using reaction networks for computation, we should specify reaction kinetics and construct dynamical reaction systems (e.g., ordinary differential equations). The theory of chemical organization states that, if the dynamical reaction system has a form of ordinary differential equations and there exists a steady state, the species with positive concentrations in the steady state constitutes an organization. In other words, the combination of species in the organization is more persistent dynamically than other non-organizational sets. Hence, the organizational analysis of a reaction network lists possible computational outputs resulting from dynamical reaction processes. Here, only qualitative presence of species is considered as outcomes of the computation.