Deterministic Function Computation with Chemical Reaction Networks

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.

Provenance of life: Chemical autonomous agents surviving through associative learning

We present a benchmark study of autonomous, chemical agents exhibiting associative learning of an environmental feature. Associative learning systems have been widely studied in cognitive science and artificial intelligence but are most commonly implemented in highly complex or carefully engineered systems, such as animal brains, artificial neural networks, DNA computing systems, and gene regulatory networks, among others. The ability to encode environmental information and use it to make simple predictions is a benchmark of biological resilience and underpins a plethora of adaptive responses in the living hierarchy, spanning prey animal species anticipating the arrival of predators to epigenetic systems in microorganisms learning environmental correlations. Given the ubiquitous and essential presence of learning behaviors in the biosphere, we aimed to explore whether simple, nonliving dissipative structures could also exhibit associative learning. Inspired by previous modeling of associative learning in chemical networks, we simulated simple systems composed of long- and short-term memory chemical species that could encode the presence or absence of temporal correlations between two external species. The ability to learn this association was implemented in Gray-Scott reaction-diffusion spots, emergent chemical patterns that exhibit self-replication and homeostasis. With the novel ability of associative learning, we demonstrate that simple chemical patterns can exhibit a broad repertoire of lifelike behavior, paving the way for in vitro studies of autonomous chemical learning systems, with potential relevance to artificial life, origins of life, and systems chemistry. The experimental realization of these learning behaviors in protocell or coacervate systems could advance a new research direction in astrobiology, since our system significantly reduces the lower bound on the required complexity for autonomous chemical learning.

Programming and training rate-independent chemical reaction networks

To program complex behavior in environments incompatible with electronic controllers such as within bioreactors or engineered cells, we turn to chemical information processors. While chemical reactions can perform computation in the stoichiometric exchange of reactants for products (how many molecules of which reactants result in how many molecules of which products), the control of reaction rates is usually thought to allow more complex computation. Motivated by the fact that correct stoichiometry is easier to ensure than reaction rates, we provide a method for programming and training chemical computation by stoichiometry. We show that such computation can be straightforwardly programmed in a manner analogous to sequential programming, and demonstrate the execution of neural networks capable of complex machine learning tasks.

Embedding computation in biochemical environments incompatible with traditional electronics is expected to have a wide-ranging impact in synthetic biology, medicine, nanofabrication, and other fields. Natural biochemical systems are typically modeled by chemical reaction networks (CRNs) which can also be used as a specification language for synthetic chemical computation. In this paper, we identify a syntactically checkable class of CRNs called noncompetitive (NC) whose equilibria are absolutely robust to reaction rates and kinetic rate law, because their behavior is captured solely by their stoichiometric structure. In spite of the inherently parallel nature of chemistry, the robustness property allows for programming as if each reaction applies sequentially. We also present a technique to program NC-CRNs using well-founded deep learning methods, showing a translation procedure from rectified linear unit (ReLU) neural networks to NC-CRNs. In the case of binary weight ReLU networks, our translation procedure is surprisingly tight in the sense that a single bimolecular reaction corresponds to a single ReLU node and vice versa. This compactness argues that neural networks may be a fitting paradigm for programming rate-independent chemical computation. As proof of principle, we demonstrate our scheme with numerical simulations of CRNs translated from neural networks trained on traditional machine learning datasets, as well as tasks better aligned with potential biological applications including virus detection and spatial pattern formation.

Composable Rate-Independent Computation in Continuous Chemical Reaction Networks

Biological regulatory networks depend upon chemical interactions to process information. Engineering such molecular computing systems is a major challenge for synthetic biology and related fields. The chemical reaction network (CRN) model idealizes chemical interactions, allowing rigorous reasoning about the computational power of chemical kinetics. Here we focus on function computation with CRNs, where we think of the initial concentrations of some species as the input and the equilibrium concentration of another species as the output. Specifically, we are concerned with CRNs that are rate-independent (the computation must be correct independent of the reaction rate law) and composable ( f°g can be computed by concatenating the CRNs computing f and g). Rate independence and composability are important engineering desiderata, permitting implementations that violate mass-action kinetics, or even "well-mixedness", and allowing the systematic construction of complex computation via modular design. We show that to construct composable rate-independent CRNs, it is necessary and sufficient to ensure that the output species of a module is not a reactant in any reaction within the module. We then exactly characterize the functions computable by such CRNs as superadditive, positive-continuous, and piecewise rational linear. Thus composability severely limits rate-independent computation unless more sophisticated input/output encodings are used.

Principles of Information Storage in Small-Molecule Mixtures

Molecular data systems have the potential to store information at dramatically higher density than existing electronic media. Some of the first experimental demonstrations of this idea have used DNA, but nature also uses a wide diversity of smaller non-polymeric molecules to preserve, process, and transmit information. In this paper, we present a general framework for quantifying chemical memory, which is not limited to polymers and extends to mixtures of molecules of all types. We show that the theoretical limit for molecular information is two orders of magnitude denser by mass than DNA, although this comes with different practical constraints on total capacity. We experimentally demonstrate kilobyte-scale information storage in mixtures of small synthetic molecules, and we consider some of the new perspectives that will be necessary to harness the information capacity available from the vast non-genomic chemical space.

Synthesizing and tuning stochastic chemical reaction networks with specified behaviours

Methods from stochastic dynamical systems theory have been instrumental in understanding the behaviours of chemical reaction networks (CRNs) arising in natural systems. However, considerably less attention has been given to the inverse problem of synthesizing CRNs with a specified behaviour, which is important for the forward engineering of biological systems. Here, we present a method for generating discrete-state stochastic CRNs from functional specifications, which combines synthesis of reactions using satisfiability modulo theories and parameter optimization using Markov chain Monte Carlo. First, we identify candidate CRNs that have the possibility to produce correct computations for a given finite set of inputs. We then optimize the parameters of each CRN, using a combination of stochastic search techniques applied to the chemical master equation, to improve the probability of correct behaviour and rule out spurious solutions. In addition, we use techniques from continuous-time Markov chain theory to analyse the expected termination time for each CRN. We illustrate our approach by synthesizing CRNs for probabilistically computing majority, maximum and division, producing both known and previously unknown networks, including a novel CRN for probabilistically computing the maximum of two species. In future, synthesis techniques such as these could be used to automate the design of engineered biological circuits and chemical systems.

A Uniform Molecular Low-Density Parity Check Decoder

Error correction codes, such as low-density parity check (LDPC) codes, are required to be larger scale to meet the increasing demands for reliable and massive data transmission. However, the construction of such a large-scale decoder will result in high complexity and hinder its silicon implementation. Thanks to the advantages of natural computing in high parallelism and low power, we propose a method to synthesize a uniform molecular LDPC decoder by implementing the belief-propagation algorithm with chemical reaction networks (CRNs). This method enables us to flexibly design the LDPC decoder with arbitrary code length, code rate, and node degrees. Compared with existing methods, our proposal reduces the number of reactions to update the variable nodes by 42.86% and the check nodes by 47.37%. Numerical results are presented to show the feasibility and validity of our proposal.

Computing with biological switches and clocks

The complex dynamics of biological systems is primarily driven by molecular interactions that underpin the regulatory networks of cells. These networks typically contain positive and negative feedback loops, which are responsible for switch-like and oscillatory dynamics, respectively. Many computing systems rely on switches and clocks as computational modules. While the combination of such modules in biological systems leads to a variety of dynamical behaviours, it is also driving development of new computing algorithms. Here we present a historical perspective on computation by biological systems, with a focus on switches and clocks, and discuss parallels between biology and computing. We also outline our vision for the future of biological computing.

Design and Analysis of Compact DNA Strand Displacement Circuits for Analog Computation Using Autocatalytic Amplifiers

A main goal in DNA computing is to build DNA circuits to compute designated functions using a minimal number of DNA strands. Here, we propose a novel architecture to build compact DNA strand displacement circuits to compute a broad scope of functions in an analog fashion. A circuit by this architecture is composed of three autocatalytic amplifiers, and the amplifiers interact to perform computation. We show DNA circuits to compute functions sqrt(x), ln(x) and exp(x) for x in tunable ranges with simulation results. A key innovation in our architecture, inspired by Napier's use of logarithm transforms to compute square roots on a slide rule, is to make use of autocatalytic amplifiers to do logarithmic and exponential transforms in concentration and time. In particular, we convert from the input that is encoded by the initial concentration of the input DNA strand, to time, and then back again to the output encoded by the concentration of the output DNA strand at equilibrium. This combined use of strand-concentration and time encoding of computational values may have impact on other forms of molecular computation.

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.

Chemical reaction network designs for asynchronous logic circuits

Chemical reaction networks (CRNs) are a versatile language for describing the dynamical behaviour of chemical kinetics, capable of modelling a variety of digital and analogue processes. While CRN designs for synchronous sequential logic circuits have been proposed and their implementation in DNA demonstrated, a physical realisation of these devices is difficult because of their reliance on a clock. Asynchronous sequential logic, on the other hand, does not require a clock, and instead relies on handshaking protocols to ensure the temporal ordering of different phases of the computation. This paper provides novel CRN designs for the construction of asynchronous logic, arithmetic and control flow elements based on a bi-molecular reaction motif with catalytic reactions and uniform reaction rates. We model and validate the designs for the deterministic and stochastic semantics using Microsoft's GEC tool and the probabilistic model checker PRISM, demonstrating their ability to emulate the function of asynchronous components under low molecular count.

Programming discrete distributions with chemical reaction networks

We explore the range of probabilistic behaviours that can be engineered with Chemical Reaction Networks (CRNs). We give methods to "program" CRNs so that their steady state is chosen from some desired target distribution that has finite support in [Formula: see text], with [Formula: see text]. Moreover, any distribution with countable infinite support can be approximated with arbitrarily small error under the [Formula: see text] norm. We also give optimized schemes for special distributions, including the uniform distribution. Finally, we formulate a calculus to compute on distributions that is complete for finite support distributions, and can be compiled to a restricted class of CRNs that at steady state realize those distributions.

Thermodynamics of random number generation

We analyze the thermodynamic costs of the three main approaches to generating random numbers via the recently introduced Information Processing Second Law. Given access to a specified source of randomness, a random number generator (RNG) produces samples from a desired target probability distribution. This differs from pseudorandom number generators (PRNGs) that use wholly deterministic algorithms and from true random number generators (TRNGs) in which the randomness source is a physical system. For each class, we analyze the thermodynamics of generators based on algorithms implemented as finite-state machines, as these allow for direct bounds on the required physical resources. This establishes bounds on heat dissipation and work consumption during the operation of three main classes of RNG algorithms-including those of von Neumann, Knuth, and Yao and Roche and Hoshi-and for PRNG methods. We introduce a general TRNG and determine its thermodynamic costs exactly for arbitrary target distributions. The results highlight the significant differences between the three main approaches to random number generation: One is work producing, one is work consuming, and the other is potentially dissipation neutral. Notably, TRNGs can both generate random numbers and convert thermal energy to stored work. These thermodynamic costs on information creation complement Landauer's limit on the irreducible costs of information destruction.

Modular verification of chemical reaction network encodings via serializability analysis

Chemical reaction networks are a powerful means of specifying the intended behaviour of synthetic biochemical systems. A high-level formal specification, expressed as a chemical reaction network, may be compiled into a lower-level encoding, which can be directly implemented in wet chemistry and may itself be expressed as a chemical reaction network. Here we present conditions under which a lower-level encoding correctly emulates the sequential dynamics of a high-level chemical reaction network. We require that encodings are transactional, such that their execution is divided by a "commit reaction" that irreversibly separates the reactant-consuming phase of the encoding from the product-generating phase. We also impose restrictions on the sharing of species between reaction encodings, based on a notion of "extra tolerance", which defines species that may be shared between encodings without enabling unwanted reactions. Our notion of correctness is serializability of interleaved reaction encodings, and if all reaction encodings satisfy our correctness properties then we can infer that the global dynamics of the system are correct. This allows us to infer correctness of any system constructed using verified encodings. As an example, we show how this approach may be used to verify two- and four-domain DNA strand displacement encodings of chemical reaction networks, and we generalize our result to the limit where the populations of helper species are unlimited.

Reachability bounds for chemical reaction networks and strand displacement systems

Chemical reaction networks (CRNs) and DNA strand displacement systems (DSDs) are widely-studied and useful models of molecular programming. However, in order for some DSDs in the literature to behave in an expected manner, the initial number of copies of some reagents is required to be fixed. In this paper we show that, when multiple copies of all initial molecules are present, general types of CRNs and DSDs fail to work correctly if the length of the shortest sequence of reactions needed to produce any given molecule exceeds a threshold that grows polynomially with attributes of the system.