  Far-from equilibrium molecular templating networks, like those that maintain
the populations of RNA and protein molecules in the cell, are key biological
motifs. These networks share the general property that assembled products are
produced and degraded via complex pathways controlled by catalysts, including
molecular templates. Although it has been suggested that the information
propagated from templates to products sets a lower bound on the thermodynamic
cost of these networks, this bound has not been explored rigorously to date. We
show that, for an arbitrarily catalytic reaction network in steady state, the
specificity with which a single product can dominate the ensemble is upper
bounded, and the entropy of the product ensemble lower bounded, by a function
of $\Delta G$, the difference between the maximal and minimal free-energy
changes along pathways to assembly. These simple bounds are particularly
restrictive for systems with a smaller number of possible products $M$.
Remarkably, however, although $\Delta G$ constrains the information propagated
to the product distribution, the systems that saturate the bound operate in a
pseudo-equilibrium fashion, and there is no minimal entropy production rate for
maintaining this non-equilibrium distribution. Moreover, for large systems, a
vanishingly small subset of the possible products can dominate the product
ensemble even for small values of $\Delta G/\ln M$.
