Leaderless deterministic chemical reaction networks

Abstract

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).