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, allowing the quantitative prediction and design of structures and interactions. The use of DNA as an engineering material has also been aided by the exponentially decreasing cost of oligonucleotide preparation and purification 3 . These developments have led to new non-biological uses of DNA as a material for self-assembly 4,5 and molecular computation 6 , and provided the foundation for the field of DNA nanotechnology. DNA nanotechnology uses DNA strands to manipulate the spatial and temporal distribution of matter, and can be broadly divided into structural and dynamic DNA nanotechnology. Structural DNA nanotechnology has achieved the construction of two-and three-dimensional objects of varying sizes and complexity using 'bottom-up' DNA self-assembly, and has culminated in the development of macroscopic materials with nanometre-scale addressability Here, we review dynamic DNA devices whose operation is based on DNA strand displacement. We show how the systematic use of this simple and robust mechanism makes it possible to produce molecular logic circuits, catalytic amplifiers, autonomous molecular walkers and reprogrammable DNA nanostructures. Although we focus on work using strand displacement in which no covalent bonds are modified, dynamic DNA devices have also been engineered using ribozymes and deoxyribozymes of molecular species, and with the complex spatial and temporal dynamics that can arise from interactions among them. Dna strand displacement Strand displacement is the process through which two strands with partial or full complementarity hybridize to each other, displacing one or more pre-hybridized strands in the process. Strand displacement can be initiated at complementary single-stranded domains (referred to as toeholds) and progresses through a branch migration process that resembles a random walk (Box 1). By varying the strength (length and sequence composition) of toeholds, the rate of strand-displacement reactions can be quantitatively controlled over a factor of 10 6 (refs 17-19). Importantly, this feature allows engineering control over the kinetics of synthetic DNA devices. In molecular biology, strand displacement frequently denotes a process mediated by enzymes such as polymerases switchable devices and structures The systematic use of toehold-mediated strand displacement in DNA nanotechnology was pioneered by Yurke et al. 29 , who observed that the same strand of DNA can undergo multiple hybridization and strand-displacement cycles through the use of toeholds. Using this crucial idea, Yurke demonstrated a set of DNA tweezers-two double-helical arms connected by a single-stranded flexible hingethat could be repeatedly cycled between an open and a closed state through successive additions of two specific single-stranded DNA 'fuel' molecules (inputs A and B in Yurke's tweezers showed that DNA hybridization and strand displacement can be used to engineer molecular-scale changes in structure. In contrast to previous demonstrations of molecular devices that were switched by changes in environmental conditions (salt, pH, temperature)

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Biological organisms use complex molecular networks to navigate their environment and regulate their internal state. The development of synthetic systems with similar capabilities could lead to applications such as smart therapeutics or fabrication methods based on self-organization. To achieve this, molecular control circuits need to be engineered to perform integrated sensing, computation and actuation. Here we report a DNA-based technology for implementing the computational core of such controllers. We use the formalism of chemical reaction networks as a ’programming language’ and our DNA architecture can, in principle, implement any behaviour that can be mathematically expressed as such. Unlike logic circuits, our formulation naturally allows complex signal processing of intrinsically analogue biological and chemical inputs. Controller components can be derived from biologically synthesized (plasmid) DNA, which reduces errors associated with chemically synthesized DNA. We implement several building-block reaction types and then combine them into a network that realizes, at the molecular level, an algorithm used in distributed control systems for achieving consensus between multiple agents. M olecular devices have captured the imagination of chemists and engineers for at least 30 years1. Rationally designed ‘active ’ molecules include nanoparticles for the targeted

We investigate the computing power of a restricted class of DNA strand displacement structures: those that are made of double strands with nicks (interruptions) in the top strand. To preserve this structural invariant, we impose restrictions on the single strands they interact with: we consider only two-domain single strands consisting of one toehold domain and one recognition domain. We study fork and join signal-processing gates based on these structures, and we show that these systems are amenable to formalization and to mechanical verification. 1

We present a process algebra for DNA computing, discussing compilation of other formal systems into the algebra, and compilation of the algebra into DNA structures.

Reversible structures are computational units that may progress forward and backward and are primarily inspired by dna circuits. We demonstrate a standardization theorem that bears a quadratic algorithm for reachability when units have unique id. We also discuss the encoding of a reversible concurrent calculus into reversible structures.

DNA nanotechnology has emerged as a reliable and programmable way of controlling matter at the nanoscale through the specificity of Watson–Crick base pairing, allowing both complex self-assembled structures with nanometer precision and complex reaction networks implementing digital and analog behaviors. Here we show how two well-developed frameworks, DNA tile self-assembly and DNA strand-displacement circuits, can be systematically integrated to provide programmable kinetic control of self-assembly. We demonstrate the triggered and catalytic isothermal self-assembly of DNA nanotubes over 10 mm long from precursor DNA double-crossover tiles activated by an upstream DNA catalyst network. Integrating more sophisticated control circuits and tile systems could enable precise spatial and temporal organization of dynamic molecular structures.

Abstract. Synthetic biology focuses on the re-engineering of living organisms for useful purposes while DNA computing targets the construction of therapeutics and computational circuits directly from DNA strands. The complexity of biological systems is a major engineering challenge and their modeling relies on a number of diverse formalisms. Moreover, many applications are "mission-critical" (e.g. as recognized by NASA's Synthetic Biology Initiative) and require robustness which is difficult to obtain. The ability to formally specify desired behavior and perform automated computational analysis of system models can help address these challenges, but today there are no unifying scalable analysis frameworks capable of dealing with this complexity. In this work, we study pertinent problems and modeling formalisms for DNA computing and synthetic biology and describe how they can be formalized and encoded to allow analysis using Satisfiability Modulo Theories (SMT). This work highlights biological engineering as a domain that can benefit extensively from the application of formal methods. It provides a step towards the use of such methods in computational design frameworks for biology and is part of a more general effort towards the formalization of biology and the study of biological computation.

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