We use a one-dimensional lattice to model a coarse-grained version of the DNA template. On this lattice, particles representing polymerase molecules can enter on one side, move along the lattice and exit on the right side. These transitions occur stochastically with exponential waiting time whenever the target site is vacant. As the positions of the mRNA to which fluorescent molecules attach are known, this model allows to predict observed fluorescence over time.

The discussed model is an example of a pure Markov jump process. We propose an efficient way to evaluate the log-likelihood of the model by integrating the master equation using a Krylov subspace method for matrix exponentials.  The corresponding gradient with respect to the model parameters is evaluated similarly by integrating a backward master equation. Our approach was implemented in C++, parallelized with OpenMP and wrapped as a Python extension. This allows to integrate our approach into probabilistic programming languages such as Pyro. The combination with Pyro facilitates the exploration of different model configurations and gives access to powerful inference algorithms such as Hamiltonian Monte Carlo and Automatic Differentiation Variational Inference. In particular, it allows straightforward pooling of multiple traces to improve inference accuracy.