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 \(f:\mathbb {N}^k\rightarrow \mathbb {N}^l\) is deterministically computable by a stochastic chemical reaction network (CRN) if and only if the graph of \(f\) is a semilinear subset of \(\mathbb {N}^{k+l}\). 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 \(X_1,\ldots ,X_k\) 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 \(X_1,\ldots ,X_k\), and zero counts of every other species, so long as \(f({\bf 0})={\bf 0}\). 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).