Transition path theory¶

TPT gives a probability-flux decomposition of an MSM between two state sets A (source) and B (target). gpvolve-v2 implements the Berezhkovskii, Hummer, Szabo (2009) formulation. No msmtools dependency.

The four quantities¶

Forward committor. \(q^{+}_{i} = \mathbb{P}(\text{chain reaches } B \text{ before } A \mid X_0 = i)\). Solved as a linear system with boundary conditions \(q^{+}_{A} = 0\) and \(q^{+}_{B} = 1\):

\[ (I - P_{\text{off}}) \, q \;=\; b \]

where \(P_{\text{off}}\) is \(P\) with rows in \(A\) and \(B\) zeroed.

Backward committor. \(q^{-}_{i} = \mathbb{P}(\text{chain came from } A \text{ before } B \mid X_0 = i)\). Equal to \(1 - q^{+}_{i}\) only for reversible chains. For non-reversible chains gpvolve-v2 builds the time-reversed transition matrix explicitly and runs the forward solver with swapped boundary sets.

Reactive flux.

\[ f_{ij} \;=\; \pi_i \, q^{-}_{i} \, P_{ij} \, q^{+}_{j}, \qquad i \neq j \]

Nonnegative and supported on the off-diagonal of the graph.

Rate.

\[ k_{AB} \;=\; \sum_{i \in A,\; j \notin A} \pi_i \, P_{ij} \, q^{+}_{j} \]

API¶

from gpvolve import forward_committor, backward_committor, reactive_flux, rate

q_plus  = forward_committor(P, A=source, B=target)
q_minus = backward_committor(P, A=source, B=target)
F       = reactive_flux(P, A=source, B=target)
k       = rate(P, A=source, B=target)

All four accept either a single state or an iterable of states for both A and B. They must be disjoint.

Dominant pathways¶

dominant_pathways(flux, A, B, top_k=10) returns the top-k bottleneck pathways through the reactive flux. The decomposition is the standard Dijkstra-on-(-log(flux)) maximum-bottleneck-path algorithm, repeated with the bottleneck flux subtracted at each step (the Karp 1980 procedure). Pathways are returned as PathEnsemble instances sorted by bottleneck flux descending.