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 $$f:\mathbb {N}^k\rightarrow \mathbb {N}^l$$f:Nk→Nl is deterministically computable by a stochastic chemical reaction network (CRN) if and only if the graph of $$f$$f is a semilinear subset of $$\mathbb {N}^{k+l}$$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 $$k$$k species $$X_1,\ldots ,X_k$$X1,…,Xk representing the input to the function $$f$$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 $$X_1,\ldots ,X_k$$X1,…,Xk, and zero counts of every other species, so long as $$f({\bf 0})={\bf 0}$$f(0)=0. We show that this CRN completes in expected time $$O(n)$$O(n), where $$n$$n is the total number of input molecules. This time bound is slower than the $$O(\log ^5 n)$$O(log5n) achieved in Chen et al. (2012), but faster than the $$O(n \log n)$$O(nlogn) achieved by the direct construction of Chen et al. (2012).