  The chemical master equation (CME) is a fundamental description of
interacting molecules commonly used to model chemical kinetics and noisy gene
regulatory networks. Exact time-dependent solutions of the CME -- which
typically consists of infinitely many coupled differential equations -- are
rare, and are valuable for numerical benchmarking and getting intuition for the
behavior of more complicated systems. Jahnke and Huisinga's landmark
calculation of the exact time-dependent solution of the CME for monomolecular
reaction systems is one of the most general analytic results known; however, it
is hard to generalize, because it relies crucially on properties of
monomolecular reactions. In this paper, we rederive Jahnke and Huisinga's
result on the time-dependent probability distribution and moments of
monomolecular reaction systems using the Doi-Peliti path integral approach,
which reduces solving the CME to evaluating many integrals. While the
Doi-Peliti approach is less intuitive, it is also more mechanical, and hence
easier to generalize. To illustrate how the Doi-Peliti approach can go beyond
the method of Jahnke and Huisinga, we also find an explicit and exact
time-dependent solution to a problem involving an autocatalytic reaction that
Jahnke and Huisinga identified as not solvable using their method. We also find
a formal exact time-dependent solution for any CME whose list of reactions
involves only zero and first order reactions, which may be the most general
result currently known.
