Title: Information propagation in far-from-equilibrium molecular templating networks is optimised by pseudo-equilibrium systems with negligible dissipation

Abstract: 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 $\Delta G/\ln M$.