Figure 4

Calculating microstate probabilities at steady state. (A) On the left, a labelled, directed graph G; on the right, the linear differential equation obtained by taking each edge to be a chemical reaction under mass-action kinetics with the edge label as the rate constant. The resulting matrix is the Laplacian matrix, $\mathcal{ℒ}\left(G\right)$, of G. (B) Illustration of Equation 7. On the left, a strongly connected graph; on the right, the spanning trees of the graph, each rooted at the circled vertex. Because the graph is strongly connected, each vertex has at least one spanning tree rooted there. The basis vector ${\rho }^G\in ker\mathcal{ℒ}\left(G\right)$ is calculated from the spanning trees using Equation 7. Probabilities of microstates are then given by normalising the entries of ρ ^G, as in Equation 4. (C) On the left, the non-strongly connected graph in (A) is shown along with its three strongly connected components (SCCs) demarcated by the dotted lines. The two terminal SCCs are marked with an asterisk and denoted T ₁ and T ₂. Each terminal SCC gives rise to a basis vector in $ker\mathcal{ℒ}\left(G\right)$ using Equation 7, as in (B), and then forming a normalised vector, as shown by following the curved arrows. Note that vertices that are not in a terminal SCC (i.e., vertices 1, 2 and 3) have zero entries in each basis vector. Any steady state, x ^∗, can be expressed as a linear combination of these basis vectors, as in Equation 9 SCC, strongly connected component.