Is stochastic thermodynamics the key to understanding the energy costs of computation?

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.

Kinetic Proofreading Can Enhance Specificity in a Nonenzymatic DNA Strand Displacement Network.

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.

The stability and number of nucleating interactions determine DNA hybridization rates in the absence of secondary structure.

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.

A universal method for analyzing copolymer growth.

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.

Minimal mechanism for cyclic templating of length-controlled copolymers under isothermal conditions.

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.

Nonequilibrium correlations in minimal dynamical models of polymer copying.

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.

In situ Generation of RNA Complexes for Synthetic Molecular Strand-Displacement Circuits in Autonomous 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.

Guiding the folding pathway of DNA 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.

Modeling DNA-Strand Displacement Reactions in the Presence of Base-Pair Mismatches.

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.

Optimizing enzymatic catalysts for rapid turnover of substrates with low enzyme sequestration.

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.

DNA bipedal motor walking dynamics: an experimental and theoretical study of the dependency on step size.

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.

Coarse-grained simulation of DNA using LAMMPS : An implementation of the oxDNA model and its applications.

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.

Multi-scale coarse-graining for the study of assembly pathways in DNA-brick self-assembly.

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.

The importance of thermodynamics for molecular systems, and the importance of molecular systems for thermodynamics.

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.

Fundamental Costs in the Production and Destruction of Persistent Polymer Copies.

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.

Publisher's Note: Biochemical Machines for the Interconversion of Mutual Information and Work [Phys. Rev. Lett. 118, 028101 (2017)].

This corrects the article DOI: 10.1103/PhysRevLett.118.028101.

Biochemical Machines for the Interconversion of Mutual Information and Work.

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.

Multiscale simulations of anisotropic particles combining molecular dynamics and Green's function reaction dynamics.

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.

DNA hairpins destabilize duplexes primarily by promoting melting rather than by inhibiting hybridization.

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.

Modelling toehold-mediated RNA strand displacement.

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.