Chemical Reaction Networks for Computing Polynomials

Supervised Learning in a Multilayer, Nonlinear Chemical Neural Network

The development of programmable or trainable molecular circuits is an important goal in the field of molecular programming. Multilayer, nonlinear, artificial neural networks are a powerful framework for implementing such functionality in a molecular system, as they are provably universal function approximators. Here, we present a design for multilayer chemical neural networks with a nonlinear hyperbolic tangent transfer function. We use a weight perturbation algorithm to train the neural network which uses a simple construction to directly approximate the loss derivatives required for training. We demonstrate the training of this system to learn all 16 two-input binary functions from a common starting point. This work thus introduces new capabilities in the field of adaptive and trainable chemical reaction network (CRN) design. It also opens the door to potential future experimental implementations, including DNA strand displacement reactions.

A Theoretical Framework for Implementable Nucleic Acids Feedback Systems

Chemical reaction networks can be utilised as basic components for nucleic acid feedback control systems' design for Synthetic Biology application. DNA hybridisation and programmed strand-displacement reactions are effective primitives for implementation. However, the experimental validation and scale-up of nucleic acid control systems are still considerably falling behind their theoretical designs. To aid with the progress heading into experimental implementations, we provide here chemical reaction networks that represent two fundamental classes of linear controllers: integral and static negative state feedback. We reduced the complexity of the networks by finding designs with fewer reactions and chemical species, to take account of the limits of current experimental capabilities and mitigate issues pertaining to crosstalk and leakage, along with toehold sequence design. The supplied control circuits are quintessential candidates for the first experimental validations of nucleic acid controllers, since they have a number of parameters, species, and reactions small enough for viable experimentation with current technical capabilities, but still represent challenging feedback control systems. They are also well suited to further theoretical analysis to verify results on the stability, performance, and robustness of this important new class of control systems.

Computing mathematical functions with chemical reactions via stochastic logic

This paper presents a novel strategy for computing mathematical functions with molecular reactions, based on theory from the realm of digital design. It demonstrates how to design chemical reaction networks based on truth tables that specify analog functions, computed by stochastic logic. The theory of stochastic logic entails the use of random streams of zeros and ones to represent probabilistic values. A link is made between the representation of random variables with stochastic logic on the one hand, and the representation of variables in molecular systems as the concentration of molecular species, on the other. Research in stochastic logic has demonstrated that many mathematical functions of interest can be computed with simple circuits built with logic gates. This paper presents a general and efficient methodology for translating mathematical functions computed by stochastic logic circuits into chemical reaction networks. Simulations show that the computation performed by the reaction networks is accurate and robust to variations in the reaction rates, within a log-order constraint. Reaction networks are given that compute functions for applications such as image and signal processing, as well as machine learning: arctan, exponential, Bessel, and sinc. An implementation is proposed with a specific experimental chassis: DNA strand displacement with units called DNA "concatemers".

Exponential Function Computation Based on DNA Strand Displacement Circuits

Due to its high programmability and storage, DNA circuits have been widely used in biological computing. In this paper, the addition, subtraction, multiplication, division, n-order and 1/n-order gates are built through DNA strand displacement reactions. The chemical reaction networks of the exponential function are established by using the six DNA analog computation gates. The integrated DNA strand displacement circuits are built through the chemical reaction networks. The exponential function polynomials can be computed through the integrated DNA strand displacement circuits. Finally, through visual DSD software verification, this design can realise the computation of exponential function polynomials, which provides a reference for solving exponential function equations and neural network computations in the future.

DNA Memristors and Their Application to Reservoir Computing

This paper introduces memristors realized by molecular and DNA reactions. Molecular memristors process one input molecule, generate two output molecules, and are realized using two molecular reactions with two different rate constants. The DNA memristors are realized using five DNA strand displacement (DSD) reactions with two effective rate constants. The hysteresis behavior is preserved in the proposed memristors, and this behavior can be altered by changing the ratios of the rate constants. The state of the memristor can be computed from the concentrations of the two output molecules using bipolar fractional coding. We describe how the proposed memristors can be used to learn the spatial and temporal properties of data via the reservoir computing (RC) model. An RC system can be divided into two parts: reservoir and readout layer. The first part transfers the information from the input space to a high-dimensional spatiotemporal feature space represented by the state of reservoirs. The connectivity structure of the reservoir will remain fixed through the dynamical evaluations. The readout layer effectively maps the projected features to the target output. A dynamical memristor array is used to implement an RC system that exploits the internal dynamical processes of the memristors. The readout layer implements a matrix-vector multiplication using molecular reactions, also based on bipolar fractional coding. All molecular reactions are mapped to DSD reactions. The RC system based on the DNA reservoir and the DNA readout layer is used to solve a handwritten digit recognition task and a second-order time series prediction task. The performance of the DNA RC system is comparable to that of an electronic memristor RC system for both tasks.

Reservoir Computing Using DNA Oscillators

This paper presents novel implementations for reservoir computing (RC) using DNA oscillators. An RC system consists of two parts: reservoir and readout layer. The reservoir projects input signals into a high-dimensional feature space which is formed by the state of the reservoir. The internal connectivity structure of the reservoir remains unchanged throughout computation. After training, the readout layer maps the projected features into the desired output. It has been shown in prior work that coupled deoxyribozyme oscillators can be used as the reservoir. In this paper, we utilize the n-phase molecular oscillator (n ≥ 3) presented in our prior work. The readout layer implements a matrix-vector multiplication using molecular reactions based on molecular analog multiplication. All molecular reactions are mapped to DNA strand displacement (DSD) reactions. We also introduce a novel encoding method that can significantly reduce the reaction time. The feasibility of the proposed RC systems based on the DNA oscillator is demonstrated for the handwritten digit recognition task and a second-order nonlinear prediction task.

Demonstration of elementary functions via DNA algorithmic self-assembly

Target-oriented cellular automata with computation are the primary challenge in the field of DNA algorithmic self-assembly in connection with specific rules. We investigate the feasibility of using the principle of cellular automata for mathematical subjects by using specific logic gates that can be implemented into DNA building blocks. Here, we connect the following five representative elementary functions: (i) enumeration of multiples of 2, 3, and 4 (demonstrated via R094, R062, and R190 in 3-input/1-output logic rules); (ii) the remainder of 0 and 1 (R132); (iii) powers of 2 (R129); (iv) ceiling function for n/2 and n/4 (R152 and R144); and (v) analogous pattern of annihilation (R184) to DNA algorithmic patterns formed by specific rules. After designing the abstract building blocks and simulating the generation of algorithmic lattices, we conducted an experiment as follows: designing of DNA tiles with specific sticky ends, construction of DNA lattices via a two-step annealing method, and verification of expected algorithmic patterns on a given DNA lattice using an atomic force microscope (AFM). We observed representative patterns, such as horizontal and diagonal stripes and embedded triangles, on the given algorithmic lattices. The average error rates of individual rules are in the range of 8.8% (R184) to 11.9% (R062), and the average error rate for all the rules was 10.6%. Interpretation of elementary functions demonstrated through DNA algorithmic patterns could be extended to more complicated functions, which may lead to new insights for achieving the final answers of functions with experimentally obtained patterns.

Solution of Simultaneous Higher Order Equations Based on DNA Strand Displacement Circuit

Currently, DNA strand displacement is often used to build neural networks or solve logical problems. While there are few studies on the use of DNA strand displacement to solve the higher order equations. In this paper, the catalysis, degradation, annihilation and adjusted reaction modules are built through DNA strand displacement. The chemical reaction networks of the corresponding higher order equations and simultaneous equations are established through these modules, and these chemical reaction networks can be used to build analog circuits to solve binary primary simultaneous equations and binary quadratic simultaneous equations. Finally, through Visual DSD software verification, this design can realize the solution of binary primary simultaneous equations and binary quadratic simultaneous equations, which provides a reference for DNA computation in the future.

Implementing parallel arithmetic via acetylation and its application to chemical image processing

Chemical mixtures can be leveraged to store large amounts of data in a highly compact form and have the potential for massive scalability owing to the use of large-scale molecular libraries. With the parallelism that comes from having many species available, chemical-based memory can also provide the physical substrate for computation with increased throughput. Here, we represent non-binary matrices in chemical solutions and perform multiple matrix multiplications and additions, in parallel, using chemical reactions. As a case study, we demonstrate image processing, in which small greyscale images are encoded in chemical mixtures and kernel-based convolutions are performed using phenol acetylation reactions. In these experiments, we use the measured concentrations of reaction products (phenyl acetates) to reconstruct the output image. In addition, we establish the chemical criteria required to realize chemical image processing and validate reaction-based multiplication. Most importantly, this work shows that fundamental arithmetic operations can be reliably carried out with chemical reactions. Our approach could serve as a basis for developing more advanced chemical computing architectures.

Molecular and DNA Artificial Neural Networks via Fractional Coding

This article considers implementation of artificial neural networks (ANNs) using molecular computing and DNA based on fractional coding. Prior work had addressed molecular two-layer ANNs with binary inputs and arbitrary weights. In prior work using fractional coding, a simple molecular perceptron that computes sigmoid of scaled weighted sum of the inputs was presented where the inputs and the weights lie between [-1,1]. Even for computing the perceptron, the prior approach suffers from two major limitations. First, it cannot compute the sigmoid of the weighted sum, but only the sigmoid of the scaled weighted sum. Second, many machine learning applications require the coefficients to be arbitrarily positive and negative numbers that are not bounded between [-1,1]; such numbers cannot be handled by the prior perceptron using fractional coding. This paper makes four contributions. First molecular perceptrons that can handle arbitrary weights and can compute sigmoid of the weighted sums are presented. Thus, these molecular perceptrons are ideal for regression applications and multi-layer ANNs. A new molecular divider is introduced and is used to compute sigmoid(ax) where . Second, based on fractional coding, a molecular artificial neural network (ANN) with one hidden layer is presented. Third, a trained ANN classifier with one hidden layer from seizure prediction application from electroencephalogram is mapped to molecular reactions and DNA and their performances are presented. Fourth, molecular activation functions for rectified linear unit (ReLU) and softmax are also presented.

Repeated Reuse of Deoxyribozyme-Based Logic Gates

Deoxyribozymes (DNAzymes) have demonstrated a significant capacity for biocomputing and hold promise for information processing within advanced biological devices if several key capabilities are developed. One required capability is reuse-having DNAzyme logic gates be cyclically, and controllably, activated and deactivated. We designed an oligonucleotide-based system for DNAzyme reuse that could (1) remove previously bound inputs by addition of complementary oligonucleotides via toe-hold mediated binding and (2) diminish output signal through the addition of quencher-labeled oligonucleotides complementary to the fluorophore-bound substrate. Our system demonstrated, for the first time, the ability for DNAzymes to have their activity toggled, with activity returning to 90-125% of original activity. This toggling could be performed multiple times with control being exerted over when the toggling occurs, with three clear cycles observed before the variability in activity became too great. Our data also demonstrated that fluorescent output of the DNAzyme activity could be actively removed and regenerated. This reuse system can increase the efficiency of DNAzyme-based logic circuits by reducing the number of redundant oligonucleotides and is critical for future development of reusable biodevices controlled by logical operations.

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 Mathematical Functions using DNA via Fractional Coding

This paper discusses the implementation of mathematical functions such as exponentials, trigonometric functions, the sigmoid function and the perceptron function with molecular reactions in general, and DNA strand displacement reactions in particular. The molecular constructs for these functions are predicated on a novel representation for input and output values: a fractional encoding, in which values are represented by the relative concentrations of two molecular types, denoted as type-1 and type-0. This representation is inspired by a technique from digital electronic design, termed stochastic logic, in which values are represented by the probability of 1's in a stream of randomly generated 0's and 1's. Research in the electronic realm has shown that a variety of complex functions can be computed with remarkably simple circuitry with this stochastic approach. This paper demonstrates how stochastic electronic designs can be translated to molecular circuits. It presents molecular implementations of mathematical functions that are considerably more complex than any shown to date. All designs are validated using mass-action simulations of the chemical kinetics of DNA strand displacement reactions.

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.

Design of an embedded inverse-feedforward biomolecular tracking controller for enzymatic reaction processes

Feedback control is widely used in chemical engineering to improve the performance and robustness of chemical processes. Feedback controllers require a 'subtractor' that is able to compute the error between the process output and the reference signal. In the case of embedded biomolecular control circuits, subtractors designed using standard chemical reaction network theory can only realise one-sided subtraction, rendering standard controller design approaches inadequate. Here, we show how a biomolecular controller that allows tracking of required changes in the outputs of enzymatic reaction processes can be designed and implemented within the framework of chemical reaction network theory. The controller architecture employs an inversion-based feedforward controller that compensates for the limitations of the one-sided subtractor that generates the error signals for a feedback controller. The proposed approach requires significantly fewer chemical reactions to implement than alternative designs, and should have wide applicability throughout the fields of synthetic biology and biological engineering.

Chemical reaction networks for computing logarithm

Living cells constantly process information from their living environment. It has recently been shown that a number of cell signaling mechanisms (e.g. G protein-coupled receptor and epidermal growth factor) can be interpreted as computing the logarithm of the ligand concentration. This suggests that logarithm is a fundamental computation primitive in cells. There is also an increasing interest in the synthetic biology community to implement analog computation and computing the logarithm is one such example. The aim of this article is to study how the computation of logarithm can be realized using chemical reaction networks (CRNs). CRNs cannot compute logarithm exactly. A standard method is to use power series or rational function approximation to compute logarithm approximately. Although CRNs can realize these polynomial or rational function computations in a straightforward manner, the issue is that in order to be able to compute logarithm accurately over a large input range, it is necessary to use high-order approximation that results in CRNs with a large number of reactions. This article proposes a novel method to compute logarithm accurately in CRNs while keeping the number of reactions in CRNs low. The proposed method can create CRNs that can compute logarithm to different levels of accuracy by adjusting two design parameters. In this article, we present the chemical reactions required to realize the CRNs for computing logarithm. The key contribution of this article is a novel method to create CRNs that can compute logarithm accurately over a wide input range using only a small number of chemical reactions.