Verifying polymer reaction networks using bisimulation

Abstract

The Chemical Reaction Network model has been proposed as a programming language for molecular programming. Methods to implement arbitrary CRNs using DNA strand displacement circuits have been investigated, as have methods to prove the correctness of those or other implementations. However, the stochastic Chemical Reaction Network model is provably not deterministically Turing-universal, that is, it is impossible to create a stochastic CRN where a given output molecule is produced if and only if an arbitrary Turing machine accepts. A DNA stack machine that can simulate arbitrary Turing machines with minimal slowdown deterministically has been proposed, but it uses unbounded polymers that cannot be modeled as a Chemical Reaction Network. We propose an extended version of a Chemical Reaction Network that models unbounded linear polymers made from a finite number of monomers. This Polymer Reaction Network model covers the DNA stack machine, as well as copy-tolerant Turing machines and some examples from biochemistry. We adapt the bisimulation method of verifying DNA implementations of Chemical Reaction Networks to our model, and use it to prove the correctness of the DNA stack machine implementation. We define a subclass of single-locus Polymer Reaction Networks and show that any member of that class can be bisimulated by a network using only four primitives, suggesting a method of DNA implementation. Finally, we prove that deciding whether an implementation is a bisimulation is Π20-complete, and thus undecidable in the general case, although it is tractable in many special cases of interest. We hope that the ability to model and verify implementations of Polymer Reaction Networks will aid in the rational design of molecular systems.