Automated Design and Verification of Localized DNA Computation Circuits

Simple computations can be performed using the interactions between single-stranded molecules of DNA. These interactions are typically toehold-mediated strand displacement reactions in a well-mixed solution. We demonstrate that a DNA circuit with tethered reactants is a distributed system and show how it can be described as a stochastic Petri net. The system can be verified by mapping the Petri net onto a continuous time Markov chain, which can also be used to find an optimal design for the circuit. This theoretical machinery can be applied to create software that automatically designs a DNA circuit, linking an abstract propositional formula to a physical DNA computation system that is capable of evaluating it. Computation with DNA has been the subject of much interest from the points of view of both pure computer science and nanomedicine. A 2009 paper by Andrew Phillips and Luca Cardelli showed how DNA strand displacement can be thought of as a formal computing language [8]. Further work by Matthew Lakin and colleagues produced Microsoft Visual DSD, a computational tool for the design and analysis of such reactions [6]. In the field of nanomedicine, Benenson et al. created a biomolecular DNA computing system that can produce an mRNA inhibitor to control gene expression [2]. These papers all consider DNA strands as freely floating reactants in a well-mixed solution. There are no topological or geometric constraints that prevent two species from interacting. Such constraints can be introduced by tethering DNA reactants to rigid structures. Yin and colleagues designed a DNA Turing machine that operates by DNA walkers moving on a rigid lattice [11]. Another method utilizes the tethering of walkers to a DNA origami tile [9]. In a 2005 paper, Jonathan Bath and colleagues introduced a DNA walker powered by a nicking enzyme that is capable of traversing a track of single-stranded DNA anchorages (Figure 1) [1]. Shelley Wickham and colleagues built on this design in a 2011 paper that demonstrated how the walker could be programmed to navigate a series of tracks on an origami tile [10]. The result was a DNA walker that could perform a computation, namely a binary decision tree. This is a “local” computation that is performed by reactants that are tethered to the origami tile. Localized DNA computation has also been the subject of theoretical and computational study. A recent paper by Dannenberg et al. analyzed the computational potential of localized DNA circuits [3]. Lakin and colleagues incorporated tethered