Different models have been proposed to understand natural phenomena at the molecular scale from a computational point of view. Oritatami systems are a model of molecular co-transcriptional folding: the transcript (the "molecule") folds as it is synthesized according to a local energy optimisation process, in a similar way to how actual biomolecules such as RNA fold into complex shapes and functions. We introduce a new model, called turedo, which is a self-avoiding Turing machine on the plane that evolves by marking visited positions and that can only move to unmarked positions. Any oritatami can be seen as a particular turedo. We show that any turedo with lookup radius 1 can conversely be simulated by an oritatami, using a universal bead type set. Our notion of simulation is strong enough to preserve the geometrical and dynamical features of these models up to a constant spatio-temporal rescaling (as in intrinsic simulation). As a consequence, turedo can be used as a readable oritatami "higher-level" programming language to build readily oritatami "smart robots", using our explicit simulation result as a compiler. As an application of our simulation result, we prove two new complexity results on the (infinite) limit configurations of oritatami systems (and radius-1 turedos), assembled from a finite seed configuration. First, we show that such limit configurations can embed any recursively enumerable set, and are thus exactly as complex as aTAM limit configurations. Second, we characterize the possible densities of occupied positions in such limit configurations: they are exactly the Π₂-computable numbers between 0 and 1. We also show that all such limit densities can be produced by one single oritatami system, just by changing the finite seed configuration. None of these results is implied by previous constructions of oritatami embedding tag systems or 1D cellular automata, which produce only computable limit configurations with constrained density.

Since the 1990s, increasingly complex nanostructures have been reliably obtained out of self-assembled DNA strands: from "simple" 2D shapes to 3D gears and articulated nano-objects, and even computing structures. The success of the assembly of these structures relies on a fine tuning of their structure to match the peculiar geometry of DNA helices. Various softwares have been developed to help the designer. These softwares provide essentially four kind of tools: an abstract representation of DNA helices (e.g. cadnano, scadnano, DNApen, 3DNA, Hex-tiles); a 3D view of the design (e.g., vHelix, Adenita, oxDNAviewer); fully automated design (e.g., BScOR, Daedalus, Perdix, Talos, Athena), generally dedicated to a specific kind of design, such as wireframe origami; and coarse grain or thermodynamical physics simulations (e.g., oxDNA, MrDNA, SNUPI, Nupack, ViennaRNA,...). MagicDNA combines some of these approaches to ease the design of configurable DNA origamis. We present our first step in the direction of conciliating all these different approaches and purposes into one single reliable GUI solution: the first fully usable version (design from scratch to export) of our general purpose 3D DNA nanostructure design software ENSnano. We believe that its intuitive, swift and yet powerful graphical interface, combining 2D and 3D editable views, allows fast and precise editing of DNA nanostructures. It also handles editing of large 2D/3D structures smoothly, and imports from the most common solutions. Our software extends the concept of grids introduced in cadnano. Grids allow to abstract and articulated the different parts of a design. ENSnano also provides new design tools which speeds up considerably the design of complex large 3D structures, most notably: a 2D split view, which allows to edit intricate 3D structures which cannot easily be mapped in a 2D view, and a copy, paste & repeat functionality, which takes advantage of the grids to design swiftly large repetitive chunks of a structure. ENSnano has been validated experimentally, as proven by the AFM images of a DNA origami entirely designed in ENSnano. ENSnano is a light-weight ready-to-run independent single-file app, running seamlessly in most of the operating systems (Windows 10, MacOS 10.13+ and Linux). Precompiled versions for Windows and MacOS are ready to download on ENSnano website. As of writing this paper, our software is being actively developed to extend its capacities in various directions discussed in this article. Still, its 3D and 2D editing interface is already meeting our usability goals. Because of its stability and ease of use, we believe that ENSnano could already be integrated in anyone’s design chain, when precise editing of a larger nanostructure is needed.

We study the oritatami model for molecular co-transcriptional folding. In oritatami systems, the transcript (the "molecule") folds as it is synthesized (transcribed), according to a local energy optimisation process, which is similar to how actual biomolecules such as RNA fold into complex shapes and functions as they are transcribed. We prove that there is an oritatami system embedding universal computation in the folding process itself. Our result relies on the development of a generic toolbox, which is easily reusable for future work to design complex functions in oritatami systems. We develop "low-level" tools that allow to easily spread apart the encoding of different "functions" in the transcript, even if they are required to be applied at the same geometrical location in the folding. We build upon these low-level tools, a programming framework with increasing levels of abstraction, from encoding of instructions into the transcript to logical analysis. This framework is similar to the hardware-to-algorithm levels of abstractions in standard algorithm theory. These various levels of abstractions allow to separate the proof of correctness of the global behavior of our system, from the proof of correctness of its implementation. Thanks to this framework, we were able to computerise the proof of correctness of its implementation and produce certificates, in the form of a relatively small number of proof trees, compact and easily readable/checkable by human, while encapsulating huge case enumerations. We believe this particular type of certificates can be generalised to other discrete dynamical systems, where proofs involve large case enumerations as well.

We study the difference between the standard seeded model of tile self-assembly, and the "seedless" two-handed model of tile self-assembly. Most of our results suggest that the two-handed model is more powerful. In particular, we show how to simulate any seeded system with a two-handed system that is essentially just a constant factor larger. We exhibit finite shapes with a busy-beaver separation in the number of distinct tiles required by seeded versus two-handed, and exhibit an infinite shape that can be constructed two-handed but not seeded. Finally, we show that verifying whether a given system uniquely assembles a desired supertile is co-NP-complete in the two-handed model, while it was known to be polynomially solvable in the seeded model.

This paper introduces a simple formalism for dealing with deterministic, non-deterministic and stochastic cellular automata in an unified and composable manner. This formalism allows for local probabilistic correlations, a feature which is not presen

We consider Kleinberg's celebrated small-world model (2000). This model is based on a d-dimensional grid graph of n nodes, augmented by a constant number of long-range links per node. It is known that this graph has diameter O(log n), and that a simple greedy search algorithm visits an expected number of O(log2 n) nodes, which is asymptotically optimal over all decentralized search algorithms. Besides the number of nodes visited, a relevant measure is the length of the path constructed by the search algorithm. A decentralized algorithm by Lebhar and Schabanel (2003) constructs paths of expected length O(log n (loglog n)2) by visiting the same number of nodes as greedy search. A natural question, posed by Kleinberg (2006), is whether there are decentralized algorithms that construct paths of length O(log n) while visiting only a poly-logarithmic number of nodes.In this paper we resolve this question. For grid dimension d=1, we answer the question in the negative, by showing that any decentralized algorithm that visits a poly-logarithmic number of nodes constructs paths of expected length O(log n loglog n). Further we show that this bound is tight; a simple variant of the algorithm by Lebhar and Schabanel matches this bound. For dimension de2, however, we answer the question in the affirmative; the bound is achieved by essentially the same algorithm we used for d=1. This is the first time that such a dichotomy, based on the dimension d, has been observed for an aspect of this model. Our results may be applicable to the design of peer-to-peer networks, where the length of the path along which data are transferred is critical for the network's performance.

We investigate the problem of optimizing the routing performance of a virtual network by adding extra random links. Our asynchronous and distributed algorithm ensures, by adding a single extra link per node, that the resulting network is a navigable small world, i.e., in which greedy routing, using the distance in the original network, computes paths of polylogarithmic length between any pair of nodes with probability 1-O(1/n). Previously known small world augmentation processes require the global knowledge of the network and centralized computations, which is unrealistic for large decentralized networks. Our algorithm, based on a careful multi-layer sampling of the nodes and the construction of a light overlay network, bypasses these limitations. For bounded growth graphs, i.e., graphs where, for any node u and any radius r the number of nodes within distance 2r from u is at most a constant times the number of nodes within distance r, our augmentation process proceeds with high probability in O(log n log D) communication rounds, with O(log n log D) messages of size O(log n) bits sent per node and requiring only O(log n log D) bit space in each node, where n is the number of nodes, and D the diameter. In particular, with the only knowledge of original distances, greedy routing computes, between any pair of nodes in the augmented network, a path of length at most O(log2 n log2 D) with probability 1 - O(1/n), and of expected length O(log n log2 D). Hence, we provide a distributed scheme to augment any bounded growth graph into a small world with high probability in polylogarithmic time while requiring polylogarithmic memory. We consider that the existence of such a lightweight process might be a first step towards the definition of a more general construction process that would validate Kleinberg's model as a plausible explanation for the small world phenomenon in large real interaction networks.

The data broadcast problem is to find a schedule for broadcasting a given set of messages over multiple channels. The goal is to minimize the cost of the broadcast plus the expected response time to clients who periodically and probabilistically tune in to wait for particular messages. The problem models disseminating data to clients in asymmetric communication environments, where there is a much larger capacity from the information source to the clients than in the reverse direction. Examples include satellites, cable TV, internet broadcast, and mobile phones. Such environments favor the ``push-based'' model where the server broadcasts (pushes) its information on the communication medium and multiple clients simultaneously retrieve the specific information of individual interest. This paper presents the first polynomial-time approximation scheme (PTAS) for data broadcast with O(1) channels and when each message has arbitrary probability, unit length and bounded cost. The best previous polynomial-time approximation algorithm for this case has a performance ratio of 9/8.