On the Computational Power of Phosphate Transfer Reaction Networks

Abstract

Phosphate transfer reactions (Principles of biochemistry, Prentice Hall, Upper Saddle River, 1996) involve the transfer of a phosphate group from a donor molecule to an accepter, which is ubiquitous in biochemistry. Besides natural systems, some synthetic molecular systems such as seesaw gates are also equivalent to (subsets of) phosphate transfer reaction networks. In this paper, we study the computational power of phosphate transfer reaction networks (PTRNs). PTRNs are chemical reaction networks (CRNs) with only phosphate transfer reactions. Previously, it is known (Nat Comput 13:517–534, 2014) that a function can be deterministically computed by a CRN if and only if it is semilinear. However, the computational power of programmable phosphate transfer networks is unknown. In this paper, we present a formal model to describe PTRNs and study the computational power of these networks. We prove that when each molecule can only carry one phosphate group, the output must be the total initial count in a subset \(S_1\) minus the total initial count of another subset \(S_2\). On the other hand, when every molecule can carry up to three phosphate groups, or two phosphate groups with different functions, PTRNs can “simulate” arbitrary CRNs. Finally, when each molecule can carry up to two functionally identical phosphate groups (or, equivalently, two phosphate groups which must be added/removed in a sequential manner), we prove that the computational power is strictly stronger than PTRNs with at most one phosphate group per molecule.