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 ƒ : N^k → N^l is deterministically computable by a stochastic chemical reaction network (CRN) if and only if the graph of ƒ is a semilinear subset of 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,…,X_k representing the input to the function ƒ. 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 ƒ(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^5n) achieved in Chen et al. (2012), but faster than the O(nlog n) achieved by the direct construction of Chen et al. (2012).

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