To understand and engineer biological and artificial nucleic acid systems, algorithms are employed for prediction of secondary structures at thermodynamic equilibrium. Dynamic programming algorithms are used to compute the most favoured, or Minimum Free Energy (MFE), structure, and the Partition Function (PF) - a tool for assigning a probability to any structure. However, in some situations, such as when there are large numbers of strands, or pseudoknotted systems, NP-hardness results show that such algorithms are unlikely, but only for MFE. Curiously, algorithmic hardness results were not shown for PF, leaving two open questions on the complexity of PF for multiple strands and single strands with pseudoknots. The challenge is that while the MFE problem cares only about one, or a few structures, PF is a summation over the entire secondary structure space, giving theorists the vibe that computing PF should not only be as hard as MFE, but should be even harder. We answer both questions. First, we show that computing PF is #P-hard for systems with an unbounded number of strands, answering a question of Condon Hajiaghayi, and Thachuk [DNA27]. Second, for even a single strand, but allowing pseudoknots, we find that PF is #P-hard. Our proof relies on a novel magnification trick that leads to a tightly-woven set of reductions between five key thermodynamic problems: MFE, PF, their decision versions, and #SSEL that counts structures of a given energy. Our reductions show these five problems are fundamentally related for any energy model amenable to magnification. That general classification clarifies the mathematical landscape of nucleic acid energy models and yields several open questions.

Molecular programmers and nanostructure engineers use domain-level design to abstract away messy DNA/RNA sequence, chemical and geometric details. Such domain-level abstractions are enforced by sequence design principles and provide a key principle that allows scaling up of complex multistranded DNA/RNA programs and structures. Determining the most favoured secondary structure, or Minimum Free Energy (MFE), of a set of strands, is typically studied at the sequence level but has seen limited domain-level work. We analyse the computational complexity of MFE for multistranded systems in a simple setting were we allow only 1 or 2 domains per strand. On the one hand, with 2-domain strands, we find that the MFE decision problem is NP-complete, even without pseudoknots, and requires exponential time algorithms assuming SAT does. On the other hand, in the simplest case of 1-domain strands there are efficient MFE algorithms for various binding modes. However, even in this single-domain case, MFE is P-hard for promiscuous binding, where one domain may bind to multiple as experimentally used by Nikitin [Nat Chem., 2023], which in turn implies that strands consisting of a single domain efficiently implement arbitrary Boolean circuits.

Abstract Molecular robotics is challenging, so it seems best to keep it simple. We consider an abstract molecular robotics model based on simple folding instructions that execute asynchronously. Turning Machines are a simple 1D to 2D folding model, also easily generalisable to 2D to 3D folding. A Turning Machine starts out as a line of connected monomers in the discrete plane, each with an associated turning number. A monomer turns relative to its neighbours, executing a unit-distance translation that drags other monomers along with it, and through collective motion the initial set of monomers eventually folds into a programmed shape. We provide a suite of tools for reasoning about Turning Machines by fully characterising their ability to execute line rotations: executing an almost-full line rotation of $$5\pi /3$$ 5 π / 3 radians is possible, yet a full $$2\pi$$ 2 π rotation is impossible. Furthermore, line rotations up to $$5\pi /3$$ 5 π / 3 are executed efficiently, in $$O(\log n)$$ O ( log n ) expected time in our continuous time Markov chain time model. We then show that such line-rotations represent a fundamental primitive in the model, by using them to efficiently and asynchronously fold shapes. In particular, arbitrarily large zig-zag-rastered squares and zig-zag paths are foldable, as are y-monotone shapes albeit with error (bounded by perimeter length). Finally, we give shapes that despite having paths that traverse all their points, are in fact impossible to fold, as well as techniques for folding certain classes of (scaled) shapes without error. Our approach relies on careful geometric-based analyses of the feats possible and impossible by a very simple robotic system, and pushes conceptional hardness towards mathematical analysis and away from molecular implementation.

Polynomial time dynamic programming algorithms play a crucial role in the design, analysis and engineering of nucleic acid systems including DNA computers and DNA/RNA nanostructures. However, in complex multistranded or pseudoknotted systems, computing the minimum free energy (MFE), and partition function of nucleic acid systems is NP-hard. Despite this, multistranded and/or pseudoknotted systems represent some of the most utilised and successful systems in the field. This leaves open the tempting possibility that many of the kinds of multistranded and/or pseudoknotted systems we wish to engineer actually fall into restricted classes, that do in fact have polynomial time algorithms, but we've just not found them yet. Here, we give polynomial time algorithms for MFE and partition function calculation for a restricted kind of multistranded system called the 1D scaffolded DNA computer. This model of computation thermodynamically favours correct outputs over erroneous states, simulates finite state machines in 1D and Boolean circuits in 2D, and is amenable to DNA storage applications. In an effort to begin to ask the question of whether we can naturally compare the expressivity of nucleic acid systems based on the computational complexity of prediction of their preferred energetic states, we show our MFE problem is in logspace (the complexity class L), making it perhaps one of the simplest known, natural, nucleic acid MFE problems. Finally, we provide a stochastic kinetic simulator for the 1D scaffolded DNA computer and evaluate strategies for efficiently speeding up this thermodynamically favourable system in a constant-temperature kinetic regime.

Molecular robotics is challenging, so it seems best to keep it simple. We consider an abstract molecular robotics model based on simple folding instructions that execute asynchronously. Turning Machines are a simple 1D to 2D folding model, also easily generalisable to 2D to 3D folding. A Turning Machine starts out as a line of connected monomers in the discrete plane, each with an associated turning number. A monomer turns relative to its neighbours, executing a unit-distance translation that drags other monomers along with it, and through collective motion the initial set of monomers eventually folds into a programmed shape. We fully characterise the ability of Turning Machines to execute line rotations, and to do so efficiently: computing an almost-full line rotation of 5π/3 radians is possible, yet a full 2π rotation is impossible. We show that such line-rotations represent a fundamental primitive in the model, by using them to efficiently and asynchronously fold arbitrarily large zig-zag-rastered squares and y-monotone shapes.

We prove a Pumping Lemma for the noncooperative abstract Tile Assembly Model, a model central to the theory of algorithmic self-assembly since the beginning of the field. This theory suggests, and our result proves, that small differences in the nature of adhesive bindings between abstract square molecules gives rise to vastly different expressive capabilities. In the cooperative abstract Tile Assembly Model, square tiles attach to each other using multi-sided cooperation of one, two or more sides. This precise control of tile binding is directly exploited for algorithmic tasks including growth of specified shapes using very few tile types, as well as simulation of Turing machines and even self-simulation of self-assembly systems. But are cooperative bindings required for these computational tasks? The definitionally simpler noncooperative (or Temperature 1) model has poor control over local binding events: tiles stick if they bind on at least one side. This has led to the conjecture that it is impossible for it to exhibit precisely controlled growth of computationally-defined shapes. Here, we prove such an impossibility result. We show that any planar noncooperative system that attempts to grow large algorithmically-controlled tile-efficient assemblies must also grow infinite non-algorithmic (pumped) structures with a simple closed-form description, or else suffer blocking of intended algorithmic structures. Our result holds for both directed and nondirected systems, and gives an explicit upper bound of (8|T|)4|T|+1(5|σ| + 6), where |T| is the size of the tileset and |σ| is the size of the seed assembly, beyond which any path of tiles is pumpable or blockable.

The field of algorithmic self-assembly is concerned with the computational and expressive power of nanoscale self-assembling molecular systems. In the well-studied cooperative, or temperature 2, abstract tile assembly model it is known that there is a tile set to simulate any Turing machine and an intrinsically universal tile set that simulates the shapes and dynamics of any instance of the model, up to spatial rescaling. It has been an open question as to whether the seemingly simpler noncooperative, or temperature 1, model is capable of such behaviour. Here we show that this is not the case by showing that there is no tile set in the noncooperative model that is intrinsically universal, nor one capable of time-bounded Turing machine simulation within a bounded region of the plane.

In the field of small universal Turing machines, Yurii Rogozhin holds a special prize: he was first to close off an infinite number of open questions by drawing a closed curve that separates the infinite set of Turing machines that are universal from a finite set of small machines for which we don't yet know. Rogozhin did this by finding the smallest known universal Turing machines at the time, both in terms of number of states and number of symbols. This brief note summarises this and a few of Yurii's other contributions to the field, including his work with Manfred Kudlek on small circular Post machines.

We prove a negative result on the power of a model of algorithmic self-assembly for which finding general techniques and results has been notoriously difficult. Specifically, we prove that Winfree's abstract Tile Assembly Model is not intrinsically universal when restricted to use noncooperative tile binding. This stands in stark contrast to the recent result that the abstract Tile Assembly Model is indeed intrinsically universal when cooperative binding is used (FOCS 2012). Noncooperative self-assembly, also known as “temperature 1”, is where all tiles bind to each other if they match on at least one side. On the other hand, cooperative self-assembly requires that some tiles bind on at least two sides. Our result shows that the change from non-cooperative to cooperative binding qualitatively improves the range of dynamics and behaviors found in these models of nanoscale self-assembly. The result holds in both two and three dimensions; the latter being quite surprising given that three-dimensional noncooperative tile assembly systems simulate Turing machines. This shows that Turing universal behavior in self-assembly does not imply the ability to simulate all algorithmic self-assembly processes. In addition to the negative result, we exhibit a three-dimensional noncooperative self-assembly tile set capable of simulating any two-dimensional noncooperative self-assembly system. This tile set implies that, in a restricted sense, non-cooperative self-assembly is intrinsically universal for itself.

We investigate computing models that are presented as families of finite computing devices with a uniformity condition on the entire family. Examples of such models include Boolean circuits, membrane systems, DNA computers, chemical reaction networks

We describe a computational model for studying the complexity of self-assembled structures with active molecular components. Our model captures notions of growth and movement ubiquitous in biological systems. The model is inspired by biology's fantastic ability to assemble biomolecules that form systems with complicated structure and dynamics, from molecular motors that walk on rigid tracks and proteins that dynamically alter the structure of the cell during mitosis, to embryonic development where large scale complicated organisms efficiently grow from a single cell. Using this active self-assembly model, we show how to efficiently self-assemble shapes and patterns from simple monomers. For example we show how to grow a line of monomers in time and number of monomer states that is merely logarithmic in its length. Our main results show how to grow arbitrary connected two-dimensional geometric shapes and patterns in expected time polylogarithmic in the size of the shape plus roughly the time required to run a Turing machine deciding whether or not a given pixel is in the shape. We do this while keeping the number of monomer types logarithmic in shape size, plus monomers required by the Kolmogorov complexity of the shape or pattern. This work thus highlights the fundamental efficiency advantage of active self-assembly over passive self-assembly and motivates experimental effort to construct self-assembly systems with active molecular components.

We prove that the abstract Tile Assembly Model (aTAM) of nanoscale self-assembly is intrinsically universal. This means that there is a single tile assembly system U that, with proper initialization, simulates any tile assembly system T. The simulation is "intrinsic" in the sense that the self-assembly process carried out by U is exactly that carried out by T, with each tile of T represented by an m × m "super tile" of U. Our construction works for the full aTAM at any temperature, and it faithfully simulates the deterministic or nondeterministic behavior of each T. Our construction succeeds by solving an analog of the cell differentiation problem in developmental biology: Each super tile of U, starting with those in the seed assembly, carries the "genome" of the simulated system T. At each location of a potential super tile in the self-assembly of U, a decision is made whether and how to express this genome, i.e., whether to generate a super tile and, if so, which tile of T it will represent. This decision must be achieved using asynchronous communication under incomplete information, but it achieves the correct global outcome(s).

We show that the Tile Assembly Model exhibits a strong notion of universality where the goal is to give a single tile assembly system that simulates the behavior of any other tile assembly system. We give a tile assembly system that is capable of simulating a very wide class of tile systems, including itself. Specifically, we give a tile set that simulates the assembly of any tile assembly system in a class of systems that we call \emph{locally consistent}: each tile binds with exactly the strength needed to stay attached, and that there are no glue mismatches between tiles in any produced assembly. Our construction is reminiscent of the studies of \emph{intrinsic universality} of cellular automata by Ollinger and others, in the sense that our simulation of a tile system $T$ by a tile system $U$ represents each tile in an assembly produced by $T$ by a $c \times c$ block of tiles in $U$, where $c$ is a constant depending on $T$ but not on the size of the assembly $T$ produces (which may in fact be infinite). Also, our construction improves on earlier simulations of tile assembly systems by other tile assembly systems (in particular, those of Soloveichik and Winfree, and of Demaine et al.) in that we simulate the actual process of self-assembly, not just the end result, as in Soloveichik and Winfree's construction, and we do not discriminate against infinite structures. Both previous results simulate only temperature 1 systems, whereas our construction simulates tile assembly systems operating at temperature 2.

We present universal Turing machines with state-symbol pairs of (5, 5), (6, 4), (9, 3) and (15, 2). These machines simulate our new variant of tag system, the bi-tag system and are the smallest known single-tape universal Turing machines with 5, 4, 3 and 2-symbols, respectively. Our 5-symbolmachine uses the same number of instructions (22) as the smallest known universal Turing machine by Rogozhin. Also, all of the universalmachines we present here simulate Turing machines in polynomial time.

We present three small universal Turing machines that have 3 states and 7 symbols, 4 states and 5 symbols, and 2 states and 13 symbols, respectively. These machines are semi-weakly universal which means that on one side of the input they have an infinitely repeated word, and on the other side there is the usual infinitely repeated blank symbol. This work can be regarded as a continuation of early work by Watanabe on semi-weak machines. One of our machines has only 17 transition rules, making it the smallest known semi-weakly universal Turing machine. Interestingly, two of our machines are symmetric with Watanabe's 7-state and 3-symbol, and 5-state and 4-symbol machines, even though we use a different simulation technique.

We show that 2-tag systems efficiently simulate Turing machines. As a corollary we find that the small universal Turing machines of Rogozhin, Minsky and others simulate Turing machines in polynomial time. This is an exponential improvement on the previously known simulation time overhead and improves a forty year old result in the area of small universal Turing machines