  The discrete chemical master equation (dCME) provides a fundamental framework
for studying stochasticity in mesoscopic networks. Because of the multi-scale
nature of many networks where reaction rates have large disparity, directly
solving dCMEs is intractable due to the exploding size of the state space. It
is important to truncate the state space effectively with quantified errors, so
accurate solutions can be computed. It is also important to know if all major
probabilistic peaks have been computed. Here we introduce the Accurate CME
(ACME) algorithm for obtaining direct solutions to dCMEs. With multi-finite
buffers for reducing the state space by O(n!), exact steady-state and
time-evolving network probability landscapes can be computed. We further
describe a theoretical framework of aggregating microstates into a smaller
number of macrostates by decomposing a network into independent aggregated
birth and death processes, and give an a priori method for rapidly determining
steady-state truncation errors. The maximal sizes of the finite buffers for a
given error tolerance can also be pre-computed without costly trial solutions
of dCMEs. We show exactly computed probability landscapes of three multi-scale
networks, namely, a 6-node toggle switch, 11-node phage-lambda epigenetic
circuit, and 16-node MAPK cascade network, the latter two with no known
solutions. We also show how probabilities of rare events can be computed from
first-passage times, another class of unsolved problems challenging for
simulation-based techniques due to large separations in time scales. Overall,
the ACME method enables accurate and efficient solutions of the dCME for a
large class of networks.
