Leaderless deterministic chemical reaction networks

This paper answers an open question of Chen et al. (DNA 2012: proceedings of the 18th international meeting on DNA computing and molecular programming, vol 7433 of lecture notes in computer science. Springer, Berlin, pp 25-42, 2012), who showed that a function f : N-k -> N-l is deterministically computable by a stochastic chemical reaction network (CRN) if and only if the graph of f is a semilinear subset of Nk+l. That construction crucially used "leaders": the ability to start in an initial configuration with constant but non-zero counts of species other than the k species X-1, ... , X-k representing the input to the function f. The authors asked whether deterministic CRNs without a leader retain the same power. We answer this question affirmatively, showing that every semilinear function is deterministically computable by a CRN whose initial configuration contains only the input species X-1, ... , X-k, and zero counts of every other species, so long as f (0) = 0. We show that this CRN completes in expected time O(n), where n is the total number of input molecules. This time bound is slower than the O(log(5) n) achieved in Chen et al. (2012), but faster than the O(n log n) achieved by the direct construction of Chen et al. (2012).