*authors contributed equally to this work 2 DNA origami is a robust molecular assembly technique by which a single-stranded DNA template is folded by annealing it with hundreds of short 'staple' strands. 1-4 The guiding design principle of nanofabrication by DNA self-assembly is that the target structure is the single most stable configuration; 5 however, the pathway and kinetics of origami assembly are poorly understood. The folding transition is cooperative 4,6,7 , and there is a strong analogy with protein folding: both are governed by information encoded in polymer sequence. [8][9][10][11] Misfolded structures are kinetic traps. The yield of well-folded DNA origami can be low: 2 yield is improved by titration of cations 2,12 or by following empirical design rules, 12,13 but it is frequently necessary to separate wellfolded origami from misfolded objects. 2, 3, 14-16 Here, we present an origami structure that is designed to reveal the assembly process. Our system has the unusual property of having a small set of distinguishable, well-folded shapes that represent discrete and approximately degenerate energy minima in a vast folding landscape. We obtain a high yield of well-folded origami structures, demonstrating the existence of efficient folding pathways. The distribution of shapes provides information about individual trajectories through the folding landscape. We show that the assembly pathway can be steered by rational design and identify similarities to protein folding: assembly is highly cooperative; reversible bond-formation is important in recovering from transient misfoldings; and the early formation of long-range connections can be very effective in forcing particular folds. Expanding the rational design process to include the assembly pathway is the key to reproducible synthesis, which is essential if nucleic acid selfassembly is to continue its rapid development 1-3,17-19 and become a reliable manufacturing technology. 20 3This study is based on a simplified version of the archetypal origami tile 1 and, in particular, on the distribution of observed folds of a 'dimer' variant which contains two copies of the template sequence in head-to-tail repeat. The 'monomer' tile ( Fig. . 1) (Fig. 1c); approximately 80% of tiles appear to be well folded.The 'dimer' template is also circular. It contains two identical copies of the monomer joined head-to-tail and can therefore bind two copies of each staple (Fig. 2). Each pair of body and seam staples can bind in one of two configurations (Fig. 2a) to form either an internal link within each copy of the monomer sequence or a pair of cross-links between the two copies.The total number of possible domain pairings is 2 76 ≈ 10 23 . Although many of these configurations are sterically inaccessible it is clear that the result of reducing the specificity of staple binding is that, as in the case of protein folding, the number of possible states of the system is overwhelmingly greater than the number of well-folded structures. However, in contrast to proteins (and t...

We present a modelling framework, and basic model parameterization, for the study of DNA origami folding at the level of DNA domains. Our approach is explicitly kinetic and does not assume a specific folding pathway. The binding of each staple is associated with a free-energy change that depends on staple sequence, the possibility of coaxial stacking with neighbouring domains, and the entropic cost of constraining the scaffold by inserting staple crossovers. A rigorous thermodynamic model is difficult to implement as a result of the complex, multiply connected geometry of the scaffold: we present a solution to this problem for planar origami. Coaxial stacking of helices and entropic terms, particularly when loop closure exponents are taken to be larger than those for ideal chains, introduce interactions between staples. These cooperative interactions lead to the prediction of sharp assembly transitions with notable hysteresis that are consistent with experimental observations. We show that the model reproduces the experimentally observed consequences of reducing staple concentration, accelerated cooling and absent staples. We also present a simpler methodology that gives consistent results and can be used to study a wider range of systems including non-planar origami.

Abstract. Unlike their traditional, silicon counterparts, DNA computers have natural interfaces with both chemical and biological systems. These can be used for a number of applications, including the precise arrangement of matter at the nanoscale and the creation of smart biosensors. Like silicon circuits, DNA strand displacement systems (DSD) can evaluate non-trivial functions. However, these systems can be slow and are susceptible to errors. It has been suggested that localised hybridization reactions could overcome some of these challenges. Localised reactions occur in DNA 'walker' systems which were recently shown to be capable of navigating a programmable track tethered to an origami tile. We investigate the computational potential of these systems for evaluating Boolean functions. DNA walkers, like DSDs, are also susceptible to errors. We develop a discrete stochastic model of DNA walker 'circuits' based on experimental data, and demonstrate the merit of using probabilistic model checking techniques to analyse their reliability, performance and correctness.

We consider the problem of synthesising rate parameters for stochastic biochemical networks so that a given time-bounded CSL property is guaranteed to hold, or, in the case of quantitative properties, the probability of satisfying the property is maximised or minimised. Our method is based on extending CSL model checking and standard uniformisation to parametric models, in order to compute safe bounds on the satisfaction probability of the property. We develop synthesis algorithms that yield answers that are precise to within an arbitrarily small tolerance value. The algorithms combine the computation of probability bounds with the refinement and sampling of the parameter space. Our methods are precise and efficient, and improve on existing approximate techniques that employ discretisation and refinement. We evaluate the usefulness of the methods by synthesising rates for three biologically motivated case studies: infection control for a SIR epidemic model; reliability

Unlike their traditional, silicon counterparts, DNA computers have natural interfaces with both chemical and biological systems. These can be used for a number of applications, including the precise arrangement of matter at the nanoscale and the creation of smart biosensors. Like silicon circuits, DNA strand displacement systems (DSD) can evaluate non-trivial functions. However, these systems can be slow and are susceptible to errors. It has been suggested that localised hybridization reactions could overcome some of these challenges. Localised reactions occur in DNA 'walker' systems which were recently shown to be capable of navigating a programmable track tethered to an origami tile. We investigate the computational potential of these systems for evaluating Boolean functions. DNA walkers, like DSDs, are also susceptible to errors. We develop a discrete stochastic model of DNA walker 'circuits' based on experimental data, and demonstrate the merit of using probabilistic model checking techniques to analyse their reliability, performance and correctness.