# Transition Path Theory¶

Transition path theory is a mathematical framework to analyze MSMs to find transition pathways.

## Computing Mean First Passage Times (MFPTs)¶

Enspara implements fast, raw-matrix transition path theory (i.e. there is no dependence on any enspara-specific objects) for use in extracting various parameters derived in TPT. Among these parameters are Mean First Passage Times, which represent the mean time required to reach a particular state from some other specific state.

First, you’ll need a transition probability matrix and (optionally) equilibrium probabilities. For this recipe, we’ll use the enspara MSM class, but any transition probability matrix and equilibrium probability distribution (as a numpy array) works.

```
from enspara import tpt
from enspara.msm import MSM, builders
msm = MSM(lag_time=10, method=builders.transpose)
msm.fit(a)
# mfpts is an array where mfpts[i, j] gives you the mfpt from i to j
mfpts = tpt.mfpts(tprob=msm.tprobs_, populations=msm.eq_probs_)
```

## Extracting maximum flux pathways¶

You can also extract a maximum flux pathway. First need a transition probability matrix and (optionally) equilibrium probabilities.

```
# assuming we've fit an MSM, as above in the MFPT example
source_state = 1
sink_state = 100
# compute the net flux matrix from our msm
nfm = tpt.net_fluxes(
msm.tprobs_,
source_state, sink_state,
populations=msm.eq_probs_)
path, flux = tpt.top_path(maximizer_ind, minimizer_ind, nfm.todense())
```