  A minimal model for gene expression, consisting of a switchable promoter
together with the resulting messenger RNA, is equivalent to a Poisson channel
with a binary Markovian input process. Determining its capacity is an
optimization problem with respect to two parameters: the average sojourn times
of the promoter's active (ON) and inactive (OFF) state. An expression for the
mutual information is found by solving the associated filtering problem
analytically on the level of distributions. For fixed peak power, three
bandwidth-like constraints are imposed by lower-bounding (i) the average
sojourn times (ii) the autocorrelation time and (iii) the average time until a
transition. OFF-favoring optima are found for all three constraints, as
commonly encountered for the Poisson channel. In addition, constraint (i)
exhibits a region that favors the ON state, and (iii) shows ON-favoring local
optima.
