Impossibility of Sufficiently Simple Chemical Reaction Network Implementations in DNA Strand Displacement

Abstract

DNA strand displacement (DSD) has recently become a common technology for constructing molecular devices, with a number of useful systems experimentally demonstrated. To help with DSD system design, various researchers are developing formal definitions to model DNA strand displacement systems. With these models a DSD system can be defined, described by a Chemical Reaction Network, simulated, and otherwise analyzed. Meanwhile, the research community is trying to use DSD to do increasingly complex tasks, while also trying to make DSD systems simpler and more robust. I suggest that formal modeling of DSD systems can be used not only to analyze DSD systems, but to guide their design. For instance, one might prove that a DSD system that implements a certain function must use a certain mechanism. As an example, I show that a physically reversible DSD system with no pseudoknots, no effectively trimolecular reactions, and using 4-way but not 3-way branch migration, cannot be a systematic implementation of reactions of the form $A \rightleftharpoons B$ that uses a constant number of toehold domains and does not crosstalk when multiple reactions of that type are combined. This result is a tight lower bound in the sense that, for most of those conditions, removing just that one condition makes the desired DSD system possible. I conjecture that a system with the same restrictions using both 3-way and 4-way branch migration still cannot systematically implement the reaction $A + B \rightleftharpoons C$.