Tight Bounds for Active Self-Assembly Using an Insertion Primitive

Abstract

We prove two limits on the behavior of a model of self-assembling particles introduced by Dabby and Chen (Proceedings of 24th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1526---1536, 2013), called insertion systems, where monomers insert themselves into the middle of a growing linear polymer. First, we prove that the expressive power of these systems is equal to context-free grammars, answering a question posed by Dabby and Chen. Second, we prove that systems of k monomer types can deterministically construct polymers of length $$n = 2^{\varTheta (k^{3/2})}$$n=2ź(k3/2) in $$O(\log ^{5/3}(n))$$O(log5/3(n)) expected time, and that this is optimal in both the number of monomer types and expected time.