It was first suggested 1 more than 30 years ago that Watson-Crick base pairing might be used to rationally design nanoscale structures from nucleic acids. Since then, and especially since introduction of the origami technique 2 , DNA nanotechnology has seen astonishing developments and increasingly more complex structures are being produced [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] . But even though general approaches for creating DNA origami polygonal meshes and design software are available 14,16,17,[19][20][21] , constraints arising from DNA geometry and sense/antisense pairing still impose important restrictions and necessitate a fair amount of manual adjustment during the design process. Here we present a general method for folding arbitrary polygonal digital meshes in DNA that readily produces structures that would have been very difficult to realize with previous approaches. This is achieved with a high level of automation of the design process, which uses a routing algorithm based on graph theory and a relaxation simulation to trace scaffold strands through the target structures. Moreover, unlike conventional origami designs built from closed-packed helices, our structures have a more open conformation with one helix per edge and are thus stable in salt conditions commonly used in biological assays.The starting point of the present method is a 3D mesh representing the geometry one wishes to realize at the nanoscale. Focusing only on polyhedral meshes, i.e. meshes which enclose a volume inflatable to a ball, and in contrast to several previous approaches 14,17,19 (see Extended Data Fig. 1) we aim to replace the edges of the mesh by single DNA double helices such that the scaffold strand traverses each of these edges once. This problem is closely related to the Chinese Postman Tour problem 22 in graph theory, which we use to find solutions as doing so by hand would be practically impossible for most meshes. The main principles underpinning our design paradigm are that the technique should allow meshes to be triangulated to optimize structural rigidity; that each edge should be represented by one double helix to enable construction of large structures using as little DNA as possible (though some meshes require two helices to render certain edges as discussed below); and that vertices should be nonBenson et. al, -DNA Rendering of Polyhedral Meshes at the Nanoscale Confidential 2 crossing (i.e. the scaffold should not cross itself in the vertices to ensure non-knotted paths with fewer topological-and kinetic traps during folding, and planar vertex junctions that avoid mesh protrusions due to stacking of crossing helices at each vertex).The overall design scheme is split into four discrete steps: i) Drawing of a 3D polygon mesh in a 3D software, Fig. 1a. ii) Generating an appropriate routing of the long scaffold strand through all the edges of the mesh, Fig. 1b-e. iii) Determining the least strained DNA helix arrangement realizing the 3D mesh, Fig. 1f-i. And iv), Optional fine tuning of the des...

A local search algorithm solving an NP-complete optimisation problem can be viewed as a stochastic process moving in an 'energy landscape' towards eventually finding an optimal solution. For the random 3-satisfiability problem, the heuristic of focusing the local moves on the presently unsatisfied clauses is known to be very effective: the time to solution has been observed to grow only linearly in the number of variables, for a given clauses-to-variables ratio α sufficiently far below the critical satisfiability threshold α c ≈ 4.27. We present numerical results on the behaviour of three focused local search algorithms for this problem, considering in particular the characteristics of a focused variant of the simple Metropolis dynamics. We estimate the optimal value for the "temperature" parameter η for this algorithm, such that its linear-time regime extends as close to α c as possible. Similar parameter optimisation is performed also for the well-known WalkSAT algorithm and for the less studied, but very well performing Focused Record-to-Record Travel method. We observe that with an appropriate choice of parameters, the linear time regime for each of these algorithms seems to extend well into ratios α > 4.2 -much further than has so far been generally assumed. We discuss the statistics of solution times for the algorithms, relate their performance to the process of "whitening", and present some conjectures on the shape of their computational phase diagrams.

The use of DNAa sananoscale construction material has been ar apidly developing field since the 1980s, in particular since the introduction of scaffolded DNAorigami in 2006. Although software is available for DNAo rigami design, the user is generally limited to architectures where finding the scaffold path through the object is trivial. Herein, we demonstrate the automated conversion of arbitrary twodimensional sheets in the form of digital meshes into scaffolded DNAn anostructures.W ei nvestigate the properties of DNA meshes based on three different internal frameworks in standardf olding buffer and physiological salt buffers.W e then employ the triangulated internal framework and produce four 2D structures with complex outlines and internal features. We demonstrate that this highly automated technique is capable of producing complex DNAn anostructures that fold with high yield to their programmed configurations,c overing around 70 %more surface area than classic origami flat sheets.Since its introduction in the 1980s, [1] DNAn anotechnology has been arapidly growing and diversifying field. This growth has accelerated since the introduction of scaffolded DNA origami in 2006.[2] In aDNA origami structure,along strand, called the scaffold, traverses the entire structure pairing with hundreds of oligonucleotides,c alled staple strands,t hat hold the structure together.The structures are often based around as quare or honeycomb lattice [3] where finding the scaffold path and designing staples is relatively easy,e specially when using software like caDNAno.[4] DNAn anostructures based on small polyhedra have been demonstrated with both scaffolded [5] and non-scaffolded [6] designs.S caffolded DNA nanostructures based on meshwork designs have also been demonstrated with crossing four-arm junctions, [7] with others containing meshes with two DNAd ouble helices per edge. [8] However,n og eneral strategy for producing arbitrary wireframe 2D structures has been demonstrated.Am ajor branch of research has been the addition of functional elements to DNAn anostructures to give them novel properties.C arbon nanotubes and metal nanoparticles have been added for electronic [9] and plasmonic [10] applications.P roteins have been added for templating enzymatic reactions [11] or cell signaling studies. [12] Fluorophores have been added to study energy transfer [13] and to create nanoscale barcodes.[14] DNAo rigami structures have also been used to control the shape of metal particles [15] and graphene sheets.[16] Demonstrations of drug loading [17] and lipid encapsulation [18] indicate that DNAn anostructures could serve as drug delivery tools.M any applications rely on single layer DNAo bjects as they offer the largest 2D canvas for functionalization and are rigid when immobilized on surfaces.Building on Rothemunds method [2] fors caffolded DNA nanostructures,w er ecently developed am ethod for automatically generating wireframe structures from polyhedral meshes. [19] This method relies on an algorithm for finding an E...

We study the performance of stochastic local search algorithms for random instances of the K-satisfiability (K-SAT) problem. We present a stochastic local search algorithm, ChainSAT, which moves in the energy landscape of a problem instance by never going upwards in energy. ChainSAT is a focused algorithm in the sense that it focuses on variables occurring in unsatisfied clauses. We show by extensive numerical investigations that ChainSAT and other focused algorithms solve large K-SAT instances almost surely in linear time, up to high clause-to-variable ratios ␣; for example, for K ‫؍‬ 4 we observe linear-time performance well beyond the recently postulated clustering and condensation transitions in the solution space. The performance of ChainSAT is a surprise given that by design the algorithm gets trapped into the first local energy minimum it encounters, yet no such minima are encountered. We also study the geometry of the solution space as accessed by stochastic local search algorithms.geometry of solutions ͉ local search ͉ performance ͉ random K-SAT

We survey and summarize the literature on the computational aspects of neural network models by presenting a detailed taxonomy of the various models according to their complexity theoretic characteristics. The criteria of classification include the architecture of the network (feedforward versus recurrent), time model (discrete versus continuous), state type (binary versus analog), weight constraints (symmetric versus asymmetric), network size (finite nets versus infinite families), and computation type (deterministic versus probabilistic), among others. The underlying results concerning the computational power and complexity issues of perceptron, radial basis function, winner-take-all, and spiking neural networks are briefly surveyed, with pointers to the relevant literature. In our survey, we focus mainly on the digital computation whose inputs and outputs are binary in nature, although their values are quite often encoded as analog neuron states. We omit the important learning issues.