Verifying chemical reaction network implementations: A pathway decomposition approach

The emerging fields of genetic engineering, synthetic biology, DNA computing, DNA nanotechnology, and molecular programming herald the birth of a new information technology that acquires information by directly sensing molecules within a chemical environment, stores information in molecules such as DNA, RNA, and proteins, processes that information by means of chemical and biochemical transformations, and uses that information to direct the manipulation of matter at the nanometer scale. To scale up beyond current proof-of-principle demonstrations, new methods for managing the complexity of designed molecular systems will need to be developed. Here we focus on the challenge of verifying the correctness of molecular implementations of abstract chemical reaction networks, where operation in a well-mixed “soup” of molecules is stochastic, asynchronous, concurrent, and often involves multiple intermediate steps in the implementation, parallel pathways, and side reactions. This problem relates to the verification of Petri nets, but existing approaches are not sufficient for providing a single guarantee covering an infinite set of possible initial states (molecule counts) and an infinite state space potentially explored by the system given any initial state. We address these issues by formulating a new theory of pathway decomposition that provides an elegant formal basis for comparing chemical reaction network implementations, and we present an algorithm that computes this basis. Our theory naturally handles certain situations that commonly arise in molecular implementations, such as what we call “delayed choice,” that are not easily accommodated by other approaches. We further show how pathway decomposition can be combined with weak bisimulation to handle a wider class that includes most currently known enzyme-free DNA implementation techniques. We anticipate that our notion of logical equivalence between chemical reaction network implementations will be valuable for other molecular implementations such as biochemical enzyme systems, and perhaps even more broadly in concurrency theory

Predicting Minimum Free Energy Structures of Multi-Stranded Nucleic Acid Complexes Is APX-Hard

Given multiple nucleic acid strands, what is the minimum free energy (MFE) secondary structure that they can form? As interacting nucleic acid strands are the basis for DNA computing and molecular programming, e.g., in DNA self-assembly and DNA strand displacement systems, determining the MFE structure is an important step in the design and verification of these systems. Efficient dynamic programming algorithms are well known for predicting the MFE pseudoknot-free secondary structure of a single nucleic acid strand. In contrast, we prove that for a simple energy model, the problem of predicting the MFE pseudoknot-free secondary structure formed from multiple interacting nucleic acid strands is NP-hard and also APX-hard. The latter result implies that there does not exist a polynomial time approximation scheme for this problem, unless ? = NP, and it suggests that heuristic methods should be investigated

Minimum Free Energy, Partition Function and Kinetics Simulation Algorithms for a Multistranded Scaffolded DNA Computer

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\u27ve 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

The role of thermal fluctuations in the motion of a free body

The motion of a rigid body is described in Classical Mechanics with the venerable Euler's equations which are based on the assumption that the relative distances among the constituent particles are fixed in time. Real bodies, however, cannot satisfy this property, as a consequence of thermal fluctuations. We generalize Euler's equations for a free body in order to describe dissipative and thermal fluctuation effects in a thermodynamically consistent way. The origin of these effects is internal, i.e. not due to an external thermal bath. The stochastic differential equations governing the orientation and central moments of the body are derived from first principles through the theory of coarse-graining. Within this theory, Euler's equations emerge as the reversible part of the dynamics. For the irreversible part, we identify two distinct dissipative mechanisms; one associated with diffusion of the orientation, whose origin lies in the difference between the spin velocity and the angular velocity, and one associated with the damping of dilations, i.e. inelasticity. We show that a deformable body with zero angular momentum will explore uniformly, through thermal fluctuations, all possible orientations. When the body spins, the equations describe the evolution towards the alignment of the body's major principal axis with the angular momentum vector. In this alignment process, the body increases its temperature. We demonstrate that the origin of the alignment process is not inelasticity but rather orientational diffusion. The theory also predicts the equilibrium shape of a spinning body.Comment: 24 pages, 1 figure with Supplemental Materia

Effective design principles for leakless strand displacement systems

Artificially designed molecular systems with programmable behaviors have become a valuable tool in chemistry, biology, material science, and medicine. Although information processing in biological regulatory pathways is remarkably robust to error, it remains a challenge to design molecular systems that are similarly robust. With functionality determined entirely by secondary structure of DNA, strand displacement has emerged as a uniquely versatile building block for cell-free biochemical networks. Here, we experimentally investigate a design principle to reduce undesired triggering in the absence of input (leak), a side reaction that critically reduces sensitivity and disrupts the behavior of strand displacement cascades. Inspired by error correction methods exploiting redundancy in electrical engineering, we ensure a higher-energy penalty to leak via logical redundancy. Our design strategy is, in principle, capable of reducing leak to arbitrarily low levels, and we experimentally test two levels of leak reduction for a core “translator” component that converts a signal of one sequence into that of another. We show that the leak was not measurable in the high-redundancy scheme, even for concentrations that are up to 100 times larger than typical. Beyond a single translator, we constructed a fast and low-leak translator cascade of nine strand displacement steps and a logic OR gate circuit consisting of 10 translators, showing that our design principle can be used to effectively reduce leak in more complex chemical systems

Verifying chemical reaction network implementations: A pathway decomposition approach

The emerging fields of genetic engineering, synthetic biology, DNA computing, DNA nanotechnology, and molecular programming herald the birth of a new information technology that acquires information by directly sensing molecules within a chemical environment, stores information in molecules such as DNA, RNA, and proteins, processes that information by means of chemical and biochemical transformations, and uses that information to direct the manipulation of matter at the nanometer scale. To scale up beyond current proof-of-principle demonstrations, new methods for managing the complexity of designed molecular systems will need to be developed. Here we focus on the challenge of verifying the correctness of molecular implementations of abstract chemical reaction networks, where operation in a well-mixed “soup” of molecules is stochastic, asynchronous, concurrent, and often involves multiple intermediate steps in the implementation, parallel pathways, and side reactions. This problem relates to the verification of Petri nets, but existing approaches are not sufficient for providing a single guarantee covering an infinite set of possible initial states (molecule counts) and an infinite state space potentially explored by the system given any initial state. We address these issues by formulating a new theory of pathway decomposition that provides an elegant formal basis for comparing chemical reaction network implementations, and we present an algorithm that computes this basis. Our theory naturally handles certain situations that commonly arise in molecular implementations, such as what we call “delayed choice,” that are not easily accommodated by other approaches. We further show how pathway decomposition can be combined with weak bisimulation to handle a wider class that includes most currently known enzyme-free DNA implementation techniques. We anticipate that our notion of logical equivalence between chemical reaction network implementations will be valuable for other molecular implementations such as biochemical enzyme systems, and perhaps even more broadly in concurrency theory

Compiler-aided systematic construction of large-scale DNA strand displacement circuits using unpurified components

Biochemical circuits made of rationally designed DNA molecules are proofs of concept for embedding control within complex molecular environments. They hold promise for transforming the current technologies in chemistry, biology, medicine and material science by introducing programmable and responsive behaviour to diverse molecular systems. As the transformative power of a technology depends on its accessibility, two main challenges are an automated design process and simple experimental procedures. Here we demonstrate the use of circuit design software, combined with the use of unpurified strands and simplified experimental procedures, for creating a complex DNA strand displacement circuit that consists of 78 distinct species. We develop a systematic procedure for overcoming the challenges involved in using unpurified DNA strands. We also develop a model that takes synthesis errors into consideration and semi-quantitatively reproduces the experimental data. Our methods now enable even novice researchers to successfully design and construct complex DNA strand displacement circuits

Linewidths and shifts of very low temperature CO in He: A challenge for theory or experiment?

The pressure broadening and shifting coefficients for pure rotational transitions of CO in a He bath gas at very low temperatures are calculated from the best available potential energy surfaces, and compared with very recent measurements by M. M. Beaky, T. M. Goyette, and F. C. De Lucia ͓J. Chem. Phys. 105, 3994 ͑1996͔͒. The results obtained for two recent empirical potentials determined from fits to Van der Waals spectra, and for a recent high quality purely ab initio surface, are consistent with one another. The best of the spectroscopic potentials also yields good agreement with high temperature virial coefficients and transport properties. Predictions from all three of these potentials agree with linebroadening and shifting measurements at temperatures above ϳ20 K, but are in substantial disagreement with the measurements at temperatures below 4 K. At present, the source of this discrepancy is not known