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The relationship between the thermodynamic and computational properties of physical systems has been a major theoretical interest since at least the 19th century. It has also become of increasing practical importance over the last half-century as the energetic cost of digital devices has exploded. Importantly, real-world computers obey multiple physical constraints on how they work, which affects their thermodynamic properties. Moreover, many of these constraints apply to both naturally occurring computers, like brains or Eukaryotic cells, and digital systems. Most obviously, all such systems must finish their computation quickly, using as few degrees of freedom as possible. This means that they operate far from thermal equilibrium. Furthermore, many computers, both digital and biological, are modular, hierarchical systems with strong constraints on the connectivity among their subsystems. Yet another example is that to simplify their design, digital computers are required to be periodic processes governed by a global clock. None of these constraints were considered in 20th-century analyses of the thermodynamics of computation. The new field of stochastic thermodynamics provides formal tools for analyzing systems subject to all of these constraints. We argue here that these tools may help us understand at a far deeper level just how the fundamental thermodynamic properties of physical systems are related to the computation they perform., Competing Interests: Competing interests statement:The authors declare no competing interest.

Strand displacement reactions underlie dynamic nucleic acid nanotechnology. The kinetic and thermodynamic features of DNA-based displacement reactions are well understood and well predicted by current computational models. By contrast, understanding of RNA/DNA hybrid strand displacement kinetics is limited, restricting the design of increasingly complex RNA/DNA hybrid reaction networks with more tightly regulated dynamics. Given the importance of RNA as a diagnostic biomarker, and its critical role in intracellular processes, this shortfall is particularly limiting for the development of strand displacement-based therapeutics and diagnostics. Herein, we characterise 22 RNA/DNA hybrid strand displacement systems, alongside 11 DNA/DNA systems, varying a range of common design parameters including toehold length and branch migration domain length. We observe  that differences in stability between RNA-DNA hybrids and DNA-DNA duplexes have large effects on strand displacement rates, with rates for equivalent sequences differing by up to 3 orders of magnitude. Crucially, however, this effect is strongly sequence-dependent, with RNA invaders strongly favoured in a system with RNA strands of high purine content, and disfavoured in a system when the RNA strands have low purine content. These results lay the groundwork for more general design principles, allowing for creation of de novo reaction networks with novel complexity while maintaining predictable reaction kinetics.

Kinetic proofreading is used throughout natural systems to enhance the specificity of molecular recognition. At its most basic level, kinetic proofreading uses a supply of chemical fuel to drive a recognition interaction out of equilibrium, allowing a single free-energy difference between correct and incorrect targets to be exploited two or more times. Despite its importance in biology, there has been little effort to incorporate kinetic proofreading into synthetic systems in which molecular recognition is important, such as nucleic acid nanotechnology. In this article, we introduce a DNA strand displacement-based kinetic proofreading motif, showing that the consumption of a DNA-based fuel can be used to enhance molecular recognition during a templated dimerization reaction. We then show that kinetic proofreading can enhance the specificity with which a probe discriminates single nucleotide mutations, both in terms of the initial rate with which the probe reacts and the long-time behavior.

Protein-based fluorescent reporters have been widely used to characterize and localize biological processes in living cells. However, these reporters may have certain drawbacks for some applications, such as transcription-based studies or biological interactions with fast dynamics. In this context, RNA nanotechnology has emerged as a promising alternative, suggesting the use of functional RNA molecules as transcriptional fluorescent reporters. RNA-based aptamers can bind to nonfluorescent small molecules to activate their fluorescence. However, their performance as reporters of gene expression in living cells has not been fully characterized, unlike protein-based reporters. Here, we investigate the performance of three RNA light-up aptamers─F30-2xdBroccoli, tRNA-Spinach, and Tornado Broccoli─as fluorescent reporters for gene expression in Escherichia coli and compare them to a protein reporter. We examine the activation range and effect on the cell growth of RNA light-up aptamers in time-course experiments and demonstrate that these aptamers are suitable transcriptional reporters over time. Using flow cytometry, we compare the variability at the single-cell level caused by the RNA fluorescent reporters and protein-based reporters. We found that the expression of RNA light-up aptamers produced higher variability in a population than that of their protein counterpart. Finally, we compare the dynamical behavior of these RNA light-up aptamers and protein-based reporters. We observed that RNA light-up aptamers might offer faster dynamics compared to a fluorescent protein in E. coli . The implementation of these transcriptional reporters may facilitate transcription-based studies, gain further insights into transcriptional processes, and expand the implementation of RNA-based circuits in bacterial cells.

Polymers consisting of more than one type of monomer, known as copolymers, are vital to both living and synthetic systems. Copolymerization has been studied theoretically in a number of contexts, often by considering a Markov process in which monomers are added or removed from the growing tip of a long copolymer. To date, the analysis of the most general models of this class has necessitated simulation. We present a general method for analyzing such processes without resorting to simulation. Our method can be applied to models with an arbitrary network of sub-steps prior to addition or removal of a monomer, including non-equilibrium kinetic proofreading cycles. Moreover, the approach allows for a dependency of addition and removal reactions on the neighboring site in the copolymer and thermodynamically self-consistent models in which all steps are assumed to be microscopically reversible. Using our approach, thermodynamic quantities such as chemical work; kinetic quantities such as time taken to grow; and statistical quantities such as the distribution of monomer types in the growing copolymer can be directly derived either analytically or numerically from the model definition.

Landauer's principle places a fundamental lower limit on the work required to perform a logically irreversible operation. Logically reversible gates provide a way to avoid these work costs and also simplify the task of making the computation as a whole thermodynamically reversible. The inherent reversibility of mechanical logic gates would make them good candidates for the design of practical logically reversible computing systems if not for the relatively large size and mass of such systems. In this paper we outline the design and simulation of reversible molecular mechanical logic gates that come close to the limits of thermodynamic reversibility even under the effects of thermal noise, and outline associated circuit components from which arbitrary combinatorial reversible circuits can be constructed and simulated. We demonstrate that isolated components can be operated in a thermodynamically reversible manner, and explore the complexities of combining components to implement more complex computations. Finally, we demonstrate a method to construct arbitrarily large reversible combinatorial circuits using multiple external controls and signal boosters with a working half-adder circuit.

The kinetics of DNA hybridization are fundamental to biological processes and DNA-based technologies. However, the precise physical mechanisms that determine why different DNA sequences hybridize at different rates are not well understood. Secondary structure is one predictable factor that influences hybridization rates but is not sufficient on its own to fully explain the observed sequence-dependent variance. In this context, we measured hybridization rates of 43 different DNA sequences that are not predicted to form secondary structure and present a parsimonious physically justified model to quantify our observations. Accounting only for the combinatorics of complementary nucleating interactions and their sequence-dependent stability, the model achieves good correlation with experiment with only two free parameters. Our results indicate that greater repetition of Watson-Crick pairs increases the number of initial states able to proceed to full hybridization, with the stability of those pairings dictating the likelihood of such progression, thus providing new insight into the physical factors underpinning DNA hybridization rates., (© The Author(s) 2022. Published by Oxford University Press on behalf of Nucleic Acids Research.)

Synthetic biology research and its industrial applications rely on deterministic spatiotemporal control of gene expression. Recently, electrochemical control of gene expression has been demonstrated in electrogenetic systems (redox-responsive promoters used alongside redox inducers and electrodes), allowing for the direct integration of electronics with biological processes. However, the use of electrogenetic systems is limited by poor activity, tunability, and standardization. In this work, we developed a strong, unidirectional, redox-responsive promoter before deriving a mutant promoter library with a spectrum of strengths. We constructed genetic circuits with these parts and demonstrated their activation by multiple classes of redox molecules. Last, we demonstrated electrochemical activation of gene expression under aerobic conditions using a novel, modular bioelectrochemical device. These genetic and electrochemical tools facilitate the design and improve the performance of electrogenetic systems. Furthermore, the genetic design strategies used can be applied to other redox-responsive promoters to further expand the available tools for electrogenetics.

The production of sequence-specific copolymers using copolymer templates is fundamental to the synthesis of complex biological molecules and is a promising framework for the synthesis of synthetic chemical complexes. Unlike the superficially similar process of self-assembly, however, the development of synthetic systems that implement templated copying of copolymers under constant environmental conditions has been challenging. The main difficulty has been overcoming product inhibition or the tendency of products to adhere strongly to their templates-an effect that gets exponentially stronger with the template length. We develop coarse-grained models of copolymerization on a finite-length template and analyze them through stochastic simulation. We use these models first to demonstrate that product inhibition prevents reliable template copying and then ask how this problem can be overcome to achieve cyclic production of polymer copies of the right length and sequence in an autonomous and chemically driven context. We find that a simple addition to the model is sufficient to generate far longer polymer products that initially form on, and then separate from, the template. In this approach, some of the free energy of polymerization is diverted into disrupting copy-template bonds behind the leading edge of the growing copy copolymer. By additionally weakening the final copy-template bond at the end of the template, the model predicts that reliable copying with a high yield of full-length, sequence-matched products is possible over large ranges of parameter space, opening the way to the engineering of synthetic copying systems that operate autonomously.

Synthetic RNA systems offer unique advantages such as faster response, increased specificity, and programmability compared to conventional protein-based networks. Here, we demonstrate an in vitro RNA-based toggle switch using RNA aptamers capable of inhibiting the transcriptional activity of T7 or SP6 RNA polymerases. The activities of both polymerases are monitored simultaneously by using Broccoli and malachite green light-up aptamer systems. In our toggle switch, a T7 promoter drives the expression of SP6 inhibitory aptamers, and an SP6 promoter expresses T7 inhibitory aptamers. We show that the two distinct states originating from the mutual inhibition of aptamers can be toggled by adding DNA sequences to sequester the RNA inhibitory aptamers. Finally, we assessed our RNA-based toggle switch in degrading conditions by introducing controlled degradation of RNAs using a mix of RNases. Our results demonstrate that the RNA-based toggle switch could be used as a control element for nucleic acid networks in synthetic biology applications.

The oxDNA model of Deoxyribonucleic acid has been applied widely to systems in biology, biophysics and nanotechnology. It is currently available via two independent open source packages. Here we present a set of clearly documented exemplar simulations that simultaneously provide both an introduction to simulating the model, and a review of the model's fundamental properties. We outline how simulation results can be interpreted in terms of-and feed into our understanding of-less detailed models that operate at larger length scales, and provide guidance on whether simulating a system with oxDNA is worthwhile., Competing Interests: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest., (Copyright © 2021 Sengar, Ouldridge, Henrich, Rovigatti and Šulc.)

One of the main objectives of synthetic biology is the development of molecular controllers that can manipulate the dynamics of a given biochemical network that is at most partially known. When integrated into smaller compartments, such as living or synthetic cells, controllers have to be calibrated to factor in the intrinsic noise. In this context, biochemical controllers put forward in the literature have focused on manipulating the mean (first moment) and reducing the variance (second moment) of the target molecular species. However, many critical biochemical processes are realized via higher-order moments, particularly the number and configuration of the probability distribution modes (maxima). To bridge the gap, we put forward the stochastic morpher controller that can, under suitable timescale separations, morph the probability distribution of the target molecular species into a predefined form. The morphing can be performed at a lower-resolution, allowing one to achieve desired multi-modality/multi-stability, and at a higher-resolution, allowing one to achieve arbitrary probability distributions. Properties of the controller, such as robustness and convergence, are rigorously established, and demonstrated on various examples. Also proposed is a blueprint for an experimental implementation of stochastic morpher.

The use of templates is a well-established method for the production of sequence-controlled assemblies, particularly long polymers. Templating is canonically envisioned as akin to a self-assembly process, wherein sequence-specific recognition interactions between a template and a pool of monomers favor the assembly of a particular polymer sequence at equilibrium. However, during the biogenesis of sequence-controlled polymers, template recognition interactions are transient; RNA and proteins detach spontaneously from their templates to perform their biological functions and allow template reuse. Breaking template recognition interactions puts the product sequence distribution far from equilibrium, since specific product formation can no longer rely on an equilibrium dominated by selective copy-template bonds. The rewards of engineering artificial polymer systems capable of spontaneously exhibiting nonequilibrium templating are large, but fields like DNA nanotechnology lack the requisite tools; the specificity and drive of conventional DNA reactions rely on product stability at equilibrium, sequestering any recognition interaction in products. The proposed alternative is handhold-mediated strand displacement (HMSD), a DNA-based reaction mechanism suited to producing out-of-equilibrium products. HMSD decouples the drive and specificity of the reaction by introducing a transient recognition interaction, the handhold. We measure the kinetics of 98 different HMSD systems to prove that handholds can accelerate displacement by 4 orders of magnitude without being sequestered in the final product. We then use HMSD to template the selective assembly of any one product DNA duplex from an ensemble of equally stable alternatives, generating a far-from-equilibrium output. HMSD thus brings DNA nanotechnology closer to the complexity of out-of-equilibrium biological systems.

Synthetic molecular circuits implementing DNA or RNA strand-displacement reactions can be used to build complex systems such as molecular computers and feedback control systems. Despite recent advances, application of nucleic acid-based circuits in vivo remains challenging due to a lack of efficient methods to produce their essential components, namely, multistranded complexes known as gates, in situ , i.e., in living cells or other autonomous systems. Here, we propose the use of naturally occurring self-cleaving ribozymes to cut a single-stranded RNA transcript into a gate complex of shorter strands, thereby opening new possibilities for the autonomous and continuous production of RNA strands in a stoichiometrically and structurally controlled way.

Biology demonstrates how a near infinite array of complex systems and structures at many scales can originate from the self-assembly of component parts on the nanoscale. But to fully exploit the benefits of self-assembly for nanotechnology, a crucial challenge remains: How do we rationally encode well-defined global architectures in subunits that are much smaller than their assemblies? Strain accumulation via geometric frustration is one mechanism that has been used to explain the self-assembly of global architectures in diverse and complex systems a posteriori . Here we take the next step and use strain accumulation as a rational design principle to control the length distributions of self-assembling polymers. We use the DNA origami method to design and synthesize a molecular subunit known as the PolyBrick, which perturbs its shape in response to local interactions via flexible allosteric blocking domains. These perturbations accumulate at the ends of polymers during growth, until the deformation becomes incompatible with further extension. We demonstrate that the key thermodynamic factors for controlling length distributions are the intersubunit binding free energy and the fundamental strain free energy, both which can be rationally encoded in a PolyBrick subunit. While passive polymerization yields geometrical distributions, which have the highest statistical length uncertainty for a given mean, the PolyBrick yields polymers that approach Gaussian length distributions whose variance is entirely determined by the strain free energy. We also show how strain accumulation can in principle yield length distributions that become tighter with increasing subunit affinity and approach distributions with uniform polymer lengths. Finally, coarse-grained molecular dynamics and Monte Carlo simulations delineate and quantify the dominant forces influencing strain accumulation in a molecular system. This study constitutes a fundamental investigation of the use of strain accumulation as a rational design principle in molecular self-assembly.

Enzymes are central to both metabolism and information processing in cells. In both cases, an enzyme's ability to accelerate a reaction without being consumed in the reaction is crucial. Nevertheless, enzymes are transiently sequestered when they bind to their substrates; this sequestration limits activity and potentially compromises information processing and signal transduction. In this article, we analyse the mechanism of enzyme-substrate catalysis from the perspective of minimizing the load on the enzymes through sequestration, while maintaining at least a minimum reaction flux. In particular, we ask: which binding free energies of the enzyme-substrate and enzyme-product reaction intermediates minimize the fraction of enzymes sequestered in complexes, while sustaining a certain minimal flux? Under reasonable biophysical assumptions, we find that the optimal design will saturate the bound on the minimal flux and reflects a basic trade-off in catalytic operation. If both binding free energies are too high, there is low sequestration, but the effective progress of the reaction is hampered. If both binding free energies are too low, there is high sequestration, and the reaction flux may also be suppressed in extreme cases. The optimal binding free energies are therefore neither too high nor too low, but in fact moderate. Moreover, the optimal difference in substrate and product binding free energies, which contributes to the thermodynamic driving force of the reaction, is in general strongly constrained by the intrinsic free-energy difference between products and reactants. Both the strategies of using a negative binding free-energy difference to drive the catalyst-bound reaction forward and of using a positive binding free-energy difference to enhance detachment of the product are limited in their efficacy.

Toehold-mediated strand displacement is the most abundantly used method to achieve dynamic switching in DNA-based nanotechnology. An "invader" strand binds to the "toehold" overhang of a target strand and replaces a target-bound "incumbent" strand. Here, the complementarity of the invader to the single-stranded toehold provides the free energy bias of the reaction. Despite the widespread use of strand displacement reactions for realizing dynamic DNA nanostructures, variants on the basic motif have not been completely characterized. Here we introduce a simple thermodynamic model, which is capable of quantitatively describing the kinetics of strand displacement reactions in the presence of mismatches, using a minimal set of parameters. Furthermore, our model highlights that base pair fraying and internal loop formation are important mechanisms when involving mismatches in the displacement process. Our model should provide a helpful tool for the rational design of strand-displacement reaction networks.

Recent years have seen great advances in the development of synthetic self-assembling molecular systems. Designing out-of-equilibrium architectures, however, requires a more subtle control over the thermodynamics and kinetics of reactions. We propose a mechanism for enhancing the thermodynamic drive of DNA strand-displacement reactions whilst barely perturbing forward reaction rates: the introduction of mismatches within the initial duplex. Through a combination of experiment and simulation, we demonstrate that displacement rates are strongly sensitive to mismatch location and can be tuned by rational design. By placing mismatches away from duplex ends, the thermodynamic drive for a strand-displacement reaction can be varied without significantly affecting the forward reaction rate. This hidden thermodynamic driving motif is ideal for the engineering of non-equilibrium systems that rely on catalytic control and must be robust to leak reactions.

DNA cyclization is a powerful technique to gain insight into the nature of DNA bending. While the wormlike chain model provides a good description of small to moderate bending fluctuations, it is expected to break down for large bending. Recent cyclization experiments on strongly bent shorter molecules indeed suggest enhanced flexibility over and above that expected from the wormlike chain. Here, we use a coarse-grained model of DNA to investigate the subtle thermodynamics of DNA cyclization for molecules ranging from 30 to 210 base pairs. As the molecules get shorter, we find increasing deviations between our computed equilibrium j -factor and the classic wormlike chain predictions of Shimada and Yamakawa for a torsionally aligned looped molecule. These deviations are due to sharp kinking, first at nicks, and only subsequently in the body of the duplex. At the shortest lengths, substantial fraying at the ends of duplex domains is the dominant method of relaxation. We also estimate the dynamic j -factor measured in recent FRET experiments. We find that the dynamic j -factor is systematically larger than its equilibrium counterpart-with the deviation larger for shorter molecules-because not all the stress present in the fully cyclized state is present in the transition state. These observations are important for the interpretation of recent cyclization experiments, suggesting that measured anomalously high j -factors may not necessarily indicate non-WLC behavior in the body of duplexes.

Recently proposed information-exploiting systems extract work from a single heat bath by using temporal correlations on an input tape. We study how enforcing time-continuous dynamics, which is necessary to ensure that the device is physically realizable, constrains possible designs and drastically diminishes efficiency. We show that these problems can be circumvented by means of applying an external, time-varying protocol, which turns the device from a "passive," free-running machine into an "actively" driven one.

Living systems produce "persistent" copies of information-carrying polymers, in which template and copy sequences remain correlated after physically decoupling. We identify a general measure of the thermodynamic efficiency with which these nonequilibrium states are created and analyze the accuracy and efficiency of a family of dynamical models that produce persistent copies. For the weakest chemical driving, when polymer growth occurs in equilibrium, both the copy accuracy and, more surprisingly, the efficiency vanish. At higher driving strengths, accuracy and efficiency both increase, with efficiency showing one or more peaks at moderate driving. Correlations generated within the copy sequence, as well as between template and copy, store additional free energy in the copied polymer and limit the single-site accuracy for a given chemical work input. Our results provide insight into the design of natural self-replicating systems and can aid the design of synthetic replicators., Competing Interests: The authors declare no conflict of interest.

During the last decade coarse-grained nucleotide models have emerged that allow us to study DNA and RNA on unprecedented time and length scales. Among them is oxDNA, a coarse-grained, sequence-specific model that captures the hybridisation transition of DNA and many structural properties of single- and double-stranded DNA. oxDNA was previously only available as standalone software, but has now been implemented into the popular LAMMPS molecular dynamics code. This article describes the new implementation and analyses its parallel performance. Practical applications are presented that focus on single-stranded DNA, an area of research which has been so far under-investigated. The LAMMPS implementation of oxDNA lowers the entry barrier for using the oxDNA model significantly, facilitates future code development and interfacing with existing LAMMPS functionality as well as other coarse-grained and atomistic DNA models.

Inspired by recent successes using single-stranded DNA tiles to produce complex structures, we develop a two-step coarse-graining approach that uses detailed thermodynamic calculations with oxDNA, a nucleotide-based model of DNA, to parametrize a coarser kinetic model that can reach the time and length scales needed to study the assembly mechanisms of these structures. We test the model by performing a detailed study of the assembly pathways for a two-dimensional target structure made up of 334 unique strands each of which are 42 nucleotides long. Without adjustable parameters, the model reproduces a critical temperature for the formation of the assembly that is close to the temperature at which assembly first occurs in experiments. Furthermore, the model allows us to investigate in detail the nucleation barriers and the distribution of critical nucleus shapes for the assembly of a single target structure. The assembly intermediates are compact and highly connected (although not maximally so), and classical nucleation theory provides a good fit to the height and shape of the nucleation barrier at temperatures close to where assembly first occurs.

We present a detailed coarse-grained computer simulation and single molecule fluorescence study of the walking dynamics and mechanism of a DNA bipedal motor striding on a DNA origami. In particular, we study the dependency of the walking efficiency and stepping kinetics on step size. The simulations accurately capture and explain three different experimental observations. These include a description of the maximum possible step size, a decrease in the walking efficiency over short distances and a dependency of the efficiency on the walking direction with respect to the origami track. The former two observations were not expected and are non-trivial. Based on this study, we suggest three design modifications to improve future DNA walkers. Our study demonstrates the ability of the oxDNA model to resolve the dynamics of complex DNA machines, and its usefulness as an engineering tool for the design of DNA machines that operate in the three spatial dimensions.

Improved understanding of molecular systems has only emphasised the sophistication of networks within the cell. Simultaneously, the advance of nucleic acid nanotechnology, a platform within which reactions can be exquisitely controlled, has made the development of artificial architectures and devices possible. Vital to this progress has been a solid foundation in the thermodynamics of molecular systems. In this pedagogical review and perspective, we discuss how thermodynamics determines both the overall potential of molecular networks, and the minute details of design. We then argue that, in turn, the need to understand molecular systems is helping to drive the development of theories of thermodynamics at the microscopic scale.

We introduce new Langevin-type equations describing the rotational and translational motion of rigid bodies interacting through conservative and non-conservative forces and hydrodynamic coupling. In the absence of non-conservative forces, the Langevin-type equations sample from the canonical ensemble. The rotational degrees of freedom are described using quaternions, the lengths of which are exactly preserved by the stochastic dynamics. For the proposed Langevin-type equations, we construct a weak 2nd order geometric integrator that preserves the main geometric features of the continuous dynamics. The integrator uses Verlet-type splitting for the deterministic part of Langevin equations appropriately combined with an exactly integrated Ornstein-Uhlenbeck process. Numerical experiments are presented to illustrate both the new Langevin model and the numerical method for it, as well as to demonstrate how inertia and the coupling of rotational and translational motion can introduce qualitatively distinct behaviours.

We consider the challenge of operating a reliable bit that can be rapidly erased. We find that both erasing and reliability times are non-monotonic in the underlying friction, leading to a trade-off between erasing speed and bit reliability. Fast erasure is possible at the expense of low reliability at moderate friction, and high reliability comes at the expense of slow erasure in the underdamped and overdamped limits. Within a given class of bit parameters and control strategies, we define 'optimal' designs of bits that meet the desired reliability and erasing time requirements with the lowest operational work cost. We find that optimal designs always saturate the bound on the erasing time requirement, but can exceed the required reliability time if critically damped. The non-trivial geometry of the reliability and erasing time scales allows us to exclude large regions of parameter space as suboptimal. We find that optimal designs are either critically damped or close to critical damping under the erasing procedure., Competing Interests: We declare we have no competing interests.

Producing a polymer copy of a polymer template is central to biology, and effective copies must persist after template separation. We show that this separation has three fundamental thermodynamic effects. First, polymer-template interactions do not contribute to overall reaction thermodynamics and hence cannot drive the process. Second, the equilibrium state of the copied polymer is template independent and so additional work is required to provide specificity. Finally, the mixing of copies from distinct templates makes correlations between template and copy sequences unexploitable, combining with copying inaccuracy to reduce the free energy stored in a polymer ensemble. These basic principles set limits on the underlying costs and resource requirements, and suggest design principles, for autonomous copying and replication in biological and synthetic systems.

The modeling of complex reaction-diffusion processes in, for instance, cellular biochemical networks or self-assembling soft matter can be tremendously sped up by employing a multiscale algorithm which combines the mesoscopic Green's Function Reaction Dynamics (GFRD) method with explicit stochastic Brownian, Langevin, or deterministic molecular dynamics to treat reactants at the microscopic scale [A. Vijaykumar, P. G. Bolhuis, and P. R. ten Wolde, J. Chem. Phys. 143, 214102 (2015)]. Here we extend this multiscale MD-GFRD approach to include the orientational dynamics that is crucial to describe the anisotropic interactions often prevalent in biomolecular systems. We present the novel algorithm focusing on Brownian dynamics only, although the methodology is generic. We illustrate the novel algorithm using a simple patchy particle model. After validation of the algorithm, we discuss its performance. The rotational Brownian dynamics MD-GFRD multiscale method will open up the possibility for large scale simulations of protein signalling networks.

We propose a physically realizable information-driven device consisting of an enzyme in a chemical bath, interacting with pairs of molecules prepared in correlated states. These correlations persist without direct interaction and thus store free energy equal to the mutual information. The enzyme can harness this free energy, and that stored in the individual molecular states, to do chemical work. Alternatively, the enzyme can use the chemical driving to create mutual information. A modified system can function without external intervention, approaching biological systems more closely.

By using oxDNA, a coarse-grained nucleotide-level model of DNA, we are able to directly simulate the self-assembly of a small 384-base-pair origami from single-stranded scaffold and staple strands in solution. In general, we see attachment of new staple strands occurring in parallel, but with cooperativity evident for the binding of the second domain of a staple if the adjacent junction is already partially formed. For a system with exactly one copy of each staple strand, we observe a complete assembly pathway in an intermediate temperature window; at low temperatures successful assembly is prevented by misbonding while at higher temperature the free-energy barriers to assembly become too large for assembly on our simulation time scales. For high-concentration systems involving a large staple strand excess, we never see complete assembly because there are invariably instances where two copies of the same staple both bind to the scaffold, creating a kinetic trap that prevents the complete binding of either staple. This mutual staple blocking could also lead to aggregates of partially formed origamis in real systems, and helps to rationalize certain successful origami design strategies.

The rupture of double-stranded DNA under stress is a key process in biophysics and nanotechnology. In this article, we consider the shear-induced rupture of short DNA duplexes, a system that has been given new importance by recently designed force sensors and nanotechnological devices. We argue that rupture must be understood as an activated process, where the duplex state is metastable and the strands will separate in a finite time that depends on the duplex length and the force applied. Thus, the critical shearing force required to rupture a duplex depends strongly on the time scale of observation. We use simple models of DNA to show that this approach naturally captures the observed dependence of the force required to rupture a duplex within a given time on duplex length. In particular, this critical force is zero for the shortest duplexes, before rising sharply and then plateauing in the long length limit. The prevailing approach, based on identifying when the presence of each additional base pair within the duplex is thermodynamically unfavorable rather than allowing for metastability, does not predict a time-scale-dependent critical force and does not naturally incorporate a critical force of zero for the shortest duplexes. We demonstrate that our findings have important consequences for the behavior of a new force-sensing nanodevice, which operates in a mixed mode that interpolates between shearing and unzipping. At a fixed time scale and duplex length, the critical force exhibits a sigmoidal dependence on the fraction of the duplex that is subject to shearing.

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.

DNA origami is a robust assembly technique that folds a single-stranded DNA template into a target structure by annealing it with hundreds of short 'staple' strands. Its guiding design principle is that the target structure is the single most stable configuration. The folding transition is cooperative and, as in the case of proteins, is governed by information encoded in the polymer sequence. A typical origami folds primarily into the desired shape, but misfolded structures can kinetically trap the system and reduce the yield. Although adjusting assembly conditions or following empirical design rules can improve yield, well-folded origami often need to be separated from misfolded structures. The problem could in principle be avoided if assembly pathway and kinetics were fully understood and then rationally optimized. To this end, here we present a DNA origami system with the unusual property of being able to form a small set of distinguishable and well-folded shapes that represent discrete and approximately degenerate energy minima in a vast folding landscape, thus allowing us to probe the assembly process. The obtained high yield of well-folded origami structures confirms the existence of efficient folding pathways, while the shape distribution provides information about individual trajectories through the folding landscape. We find that, similarly to protein folding, the assembly of DNA origami is highly cooperative; that reversible bond formation is important in recovering from transient misfoldings; and that the early formation of long-range connections can very effectively enforce particular folds. We use these insights to inform the design of the system so as to steer assembly towards desired structures. Expanding the rational design process to include the assembly pathway should thus enable more reproducible synthesis, particularly when targeting more complex structures. We anticipate that this expansion will be essential if DNA origami is to continue its rapid development and become a reliable manufacturing technology.

The effect of secondary structure on DNA duplex formation is poorly understood. Using oxDNA, a nucleotide level coarse-grained model of DNA, we study how hairpins influence the rate and reaction pathways of DNA hybridzation. We compare to experimental systems studied by Gao et al. (1) and find that 3-base pair hairpins reduce the hybridization rate by a factor of 2, and 4-base pair hairpins by a factor of 10, compared to DNA with limited secondary structure, which is in good agreement with experiments. By contrast, melting rates are accelerated by factors of ∼100 and ∼2000. This surprisingly large speed-up occurs because hairpins form during the melting process, and significantly lower the free energy barrier for dissociation. These results should assist experimentalists in designing sequences to be used in DNA nanotechnology, by putting limits on the suppression of hybridization reaction rates through the use of hairpins and offering the possibility of deliberately increasing dissociation rates by incorporating hairpins into single strands., (© The Author(s) 2015. Published by Oxford University Press on behalf of Nucleic Acids Research.)

We introduce an extended version of oxDNA, a coarse-grained model of deoxyribonucleic acid (DNA) designed to capture the thermodynamic, structural, and mechanical properties of single- and double-stranded DNA. By including explicit major and minor grooves and by slightly modifying the coaxial stacking and backbone-backbone interactions, we improve the ability of the model to treat large (kilobase-pair) structures, such as DNA origami, which are sensitive to these geometric features. Further, we extend the model, which was previously parameterised to just one salt concentration ([Na(+)] = 0.5M), so that it can be used for a range of salt concentrations including those corresponding to physiological conditions. Finally, we use new experimental data to parameterise the oxDNA potential so that consecutive adenine bases stack with a different strength to consecutive thymine bases, a feature which allows a more accurate treatment of systems where the flexibility of single-stranded regions is important. We illustrate the new possibilities opened up by the updated model, oxDNA2, by presenting results from simulations of the structure of large DNA objects and by using the model to investigate some salt-dependent properties of DNA.

Advances in DNA nanotechnology have stimulated the search for simple motifs that can be used to control the properties of DNA nanostructures. One such motif, which has been used extensively in structures such as polyhedral cages, two-dimensional arrays, and ribbons, is a bulged duplex, that is, two helical segments that connect at a bulge loop. We use a coarse-grained model of DNA to characterize such bulged duplexes. We find that this motif can adopt structures belonging to two main classes: one where the stacking of the helices at the center of the system is preserved, the geometry is roughly straight, and the bulge is on one side of the duplex and the other where the stacking at the center is broken, thus allowing this junction to act as a hinge and increasing flexibility. Small loops favor states where stacking at the center of the duplex is preserved, with loop bases either flipped out or incorporated into the duplex. Duplexes with longer loops show more of a tendency to unstack at the bulge and adopt an open structure. The unstacking probability, however, is highest for loops of intermediate lengths, when the rigidity of single-stranded DNA is significant and the loop resists compression. The properties of this basic structural motif clearly correlate with the structural behavior of certain nano-scale objects, where the enhanced flexibility associated with larger bulges has been used to tune the self-assembly product as well as the detailed geometry of the resulting nanostructures. We further demonstrate the role of bulges in determining the structure of a "Z-tile," a basic building block for nanostructures.

We introduce two new thermostats, one of Langevin type and one of gradient (Brownian) type, for rigid body dynamics. We formulate rotation using the quaternion representation of angular coordinates; both thermostats preserve the unit length of quaternions. The Langevin thermostat also ensures that the conjugate angular momenta stay within the tangent space of the quaternion coordinates, as required by the Hamiltonian dynamics of rigid bodies. We have constructed three geometric numerical integrators for the Langevin thermostat and one for the gradient thermostat. The numerical integrators reflect key properties of the thermostats themselves. Namely, they all preserve the unit length of quaternions, automatically, without the need of a projection onto the unit sphere. The Langevin integrators also ensure that the angular momenta remain within the tangent space of the quaternion coordinates. The Langevin integrators are quasi-symplectic and of weak order two. The numerical method for the gradient thermostat is of weak order one. Its construction exploits ideas of Lie-group type integrators for differential equations on manifolds. We numerically compare the discretization errors of the Langevin integrators, as well as the efficiency of the gradient integrator compared to the Langevin ones when used in the simulation of rigid TIP4P water model with smoothly truncated electrostatic interactions. We observe that the gradient integrator is computationally less efficient than the Langevin integrators. We also compare the relative accuracy of the Langevin integrators in evaluating various static quantities and give recommendations as to the choice of an appropriate integrator.

We study the thermodynamics and kinetics of an RNA toehold-mediated strand displacement reaction with a recently developed coarse-grained model of RNA. Strand displacement, during which a single strand displaces a different strand previously bound to a complementary substrate strand, is an essential mechanism in active nucleic acid nanotechnology and has also been hypothesized to occur in vivo. We study the rate of displacement reactions as a function of the length of the toehold and temperature and make two experimentally testable predictions: that the displacement is faster if the toehold is placed at the 5' end of the substrate; and that the displacement slows down with increasing temperature for longer toeholds., (Copyright © 2015 Biophysical Society. Published by Elsevier Inc. All rights reserved.)

We study the dynamics of DNA hairpin formation using oxDNA, a nucleotide-level coarse-grained model of DNA. In particular, we explore the effects of the loop stacking interactions and non-native base pairing on the hairpin closing times. We find a nonmonotonic variation of the hairpin closing time with temperature, in agreement with the experimental work of Wallace et al. (Proc. Natl. Acad. Sci. U.S.A. 2001, 98, 5584-5589). The hairpin closing process involves the formation of an initial nucleus of one or two bonds in the stem followed by a rapid zippering of the stem. At high temperatures, typically above the hairpin melting temperature, an effective negative activation enthalpy is observed because the nucleus has a lower enthalpy than the open state. By contrast, at low temperatures, the activation enthalpy becomes positive mainly due to the increasing energetic cost of bending a loop that becomes increasingly highly stacked as the temperature decreases. We show that stacking must be very strong to induce this experimentally observed behavior, and that the existence of just a few weak stacking points along the loop can substantially suppress it. Non-native base pairs are observed to have only a small effect, slightly accelerating hairpin formation.

Double phosphorylation of protein kinases is a common feature of signaling cascades. This motif may reduce cross-talk between signaling pathways because the second phosphorylation site allows for proofreading, especially when phosphorylation is distributive rather than processive. Recent studies suggest that phosphorylation can be pseudo-processive in the crowded cellular environment, since rebinding after the first phosphorylation is enhanced by slow diffusion. Here, we use a simple model with unsaturated reactants to show that specificity for one substrate over another drops as rebinding increases and pseudo-processive behavior becomes possible. However, this loss of specificity with increased rebinding is typically also observed if two distinct enzyme species are required for phosphorylation, i.e., when the system is necessarily distributive. Thus the loss of specificity is due to an intrinsic reduction in selectivity with increased rebinding, which benefits inefficient reactions, rather than pseudo-processivity itself. We also show that proofreading can remain effective when the intended signaling pathway exhibits high levels of rebinding-induced pseudo-processivity, unlike other proposed advantages of the dual phosphorylation motif.

DNA is used to construct synthetic systems that sense, actuate, move and compute. The operation of many dynamic DNA devices depends on toehold-mediated strand displacement, by which one DNA strand displaces another from a duplex. Kinetic control of strand displacement is particularly important in autonomous molecular machinery and molecular computation, in which non-equilibrium systems are controlled through rates of competing processes. Here, we introduce a new method based on the creation of mismatched base pairs as kinetic barriers to strand displacement. Reaction rate constants can be tuned across three orders of magnitude by altering the position of such a defect without significantly changing the stabilities of reactants or products. By modelling reaction free-energy landscapes, we explore the mechanistic basis of this control mechanism. We also demonstrate that oxDNA, a coarse-grained model of DNA, is capable of accurately predicting and explaining the impact of mismatches on displacement kinetics.

We present a new, nucleotide-level model for RNA, oxRNA, based on the coarse-graining methodology recently developed for the oxDNA model of DNA. The model is designed to reproduce structural, mechanical, and thermodynamic properties of RNA, and the coarse-graining level aims to retain the relevant physics for RNA hybridization and the structure of single- and double-stranded RNA. In order to explore its strengths and weaknesses, we test the model in a range of nanotechnological and biological settings. Applications explored include the folding thermodynamics of a pseudoknot, the formation of a kissing loop complex, the structure of a hexagonal RNA nanoring, and the unzipping of a hairpin motif. We argue that the model can be used for efficient simulations of the structure of systems with thousands of base pairs, and for the assembly of systems of up to hundreds of base pairs. The source code implementing the model is released for public use.

To simulate long time and length scale processes involving DNA it is necessary to use a coarse-grained description. Here we provide an overview of different approaches to such coarse-graining, focussing on those at the nucleotide level that allow the self-assembly processes associated with DNA nanotechnology to be studied. OxDNA, our recently-developed coarse-grained DNA model, is particularly suited to this task, and has opened up this field to systematic study by simulations. We illustrate some of the range of DNA nanotechnology systems to which the model is being applied, as well as the insights it can provide into fundamental biophysical properties of DNA.