Top-Down Analysis of Temporal Hierarchy in Biochemical Reaction Networks
Figure 1
Subspaces of the Jacobian matrix and different approaches for decomposing it into dynamically independent interactions between metabolites.
(A) The Jacobian acts as a linear operator mapping the dynamics onto the deviation variable. (B) The Modal Matrix maps network dynamics onto independent time scales. Panels C and D illustrate two approaches to understanding the interactions between metabolites on the different time scales. (C) Beginning from the fastest time scale and moving forward, components that move together on subsequent time scales are lumped into an aggregate pool variable. The pooling pictorially for three different time scales in glycolysis and the Rapoport-Leubering shunt in the red cell. The large blue dots indicate pool formation between two metabolites, signifying that these two metabolites become coupled or correlated on slower times scales. In this case, glyceraldehyde 3-phosphate and dihydroxyacetone phosphate pool together after the first time scale, the hexose phosphates pool together after the fourth time scale, and the triose phosphates pool together after the eighth time scale. (D) Each time scale is analyzed independently and the interactions are defined in terms of the coefficients in the model and their contribution to the cumulative sum of the modal coefficients. Analyzing all of the modes in this manner allows the identification of variables that are dominant across multiple modes and identifying the time scales across which they are most active. Four fundamental subspaces are associated with J and its mapping onto its time derivative. The key to temporal decomposition is the time-ordered removal of dynamic motions that lead to the step-by-step increase in the null and left null spaces of J.