From bacteria following simple chemical gradients1 to the brain distinguishing complex odour information2, the ability to recognize molecular patterns is essential for biological organisms. This type of information-processing function has been implemented using DNA-based neural networks3, but has been limited to the recognition of a set of no more than four patterns, each composed of four distinct DNA molecules. Winner-take-all computation4 has been suggested5,6 as a potential strategy for enhancing the capability of DNA-based neural networks. Compared to the linear-threshold circuits7 and Hopfield networks8 used previously3, winner-take-all circuits are computationally more powerful4, allow simpler molecular implementation and are not constrained by the number of patterns and their complexity, so both a large number of simple patterns and a small number of complex patterns can be recognized. Here we report a systematic implementation of winner-take-all neural networks based on DNA-strand-displacement9,10 reactions. We use a previously developed seesaw DNA gate motif3,11,12, extended to include a simple and robust component that facilitates the cooperative hybridization13 that is involved in the process of selecting a ‘winner’. We show that with this extended seesaw motif DNA-based neural networks can classify patterns into up to nine categories. Each of these patterns consists of 20 distinct DNA molecules chosen from the set of 100 that represents the 100 bits in 10 × 10 patterns, with the 20 DNA molecules selected tracing one of the handwritten digits ‘1’ to ‘9’. The network successfully classified test patterns with up to 30 of the 100 bits flipped relative to the digit patterns ‘remembered’ during training, suggesting that molecular circuits can robustly accomplish the sophisticated task of classifying highly complex and noisy information on the basis of similarity to a memory.
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We thank R. M. Murray for sharing an acoustic liquid-handling robot. We thank C. Thachuk and E. Winfree for discussions and suggestions. K.M.C. was supported by a NSF Graduate Research Fellowship. L.Q. was supported by a Career Award at the Scientific Interface from the Burroughs Wellcome Fund (1010684), a Faculty Early Career Development Award from NSF (1351081), and the Shurl and Kay Curci Foundation.Reviewer information
Nature thanks R. Schulman and the other anonymous reviewer(s) for their contribution to the peer review of this work.
Extended data figures and tables
The winner-take-all computation is broken into five subfunctions: weight multiplication, summation, pairwise annihilation, signal restoration and reporting. In the chemical reactions listed next to the five subfunctions, the species in black are needed as part of the function, the species in grey are needed to facilitate the reactions and the waste species are not shown. kf and ks are the rate constants of the pairwise-annihilation and signal-restoration reactions, respectively. In the DNA-strand-displacement implementation, weight multiplication and signal restoration are both catalytic reactions. The grey circle with an arrow indicates the direction of the catalytic cycle. Representative, but not all possible, states are shown for the pairwise-annihilation reaction. Zigzag lines indicate short (5 or 7 nucleotide) toehold domains and straight lines indicate long (15 or 20 nucleotide) branch-migration domains in DNA strands, with arrowheads marking their 3′ ends. Each domain is labelled with a name, and asterisks in the names indicate sequence complementarity. Black-filled and white-filled arrowheads indicate the forwards and backwards directions of a reaction step, respectively. All DNA sequences are listed in Supplementary Table 1.
a, Same as Fig. 1a. b, Seesaw circuit diagram11 for implementing the winner-take-all neural network. Each black number indicates the identity of a seesaw node. A total of n + 3m nodes are required for implementing a winner-take-all neural network with m memories that each has n bits. The location and absolute value of each red number indicates the identity and relative initial concentration of a DNA species, respectively. A red number on a wire connected to a node (or between two nodes) indicates a free signal molecule, which can be an input or fuel strand. A red number inside a node indicates a gate molecule, which can be a weight, summation gate or restoration gate. A red number on a wire that stops perpendicularly at two wires indicates an annihilator molecule. A negative red number inside a half node with a zigzag arrow indicates a reporter molecule.
a, Two-species winner-take-all behaviour. The experimental data (left, same as Fig. 2a) were used to identify the reverse rate constant kr = 0.4 s−1 of the annihilation reaction in simulations (right). All fluorescence kinetics data and simulation are shown over the course of 2.5 h. The standard concentration is 50 nM (1×). Initial concentrations of the annihilator, restoration gates, fuels and reporters are 75 nM (1.5×), 50 nM (1×), 100 nM (2×) and 100 nM (2×), respectively. b, A 4-bit pattern recognition circuit with input concentration varying from 50 nM to 500 nM. In each output trajectory plot, dotted lines indicate fluorescence kinetics data and solid lines indicate simulation. The patterns to the left and right of the arrow indicate input signal and output classification, respectively. c, Applying thresholding to clean up noisy input signals. The thresholding mechanism has been reported previously in work on seesaw DNA circuits11. The extended toehold in threshold molecule has 7 nucleotides. In b and c, to compare the range of inputs, the concentration of each input strand is shown relative to 50 nM. The initial concentration of each weight molecule is either 0 or 50 nM; weight fuels are twice the concentration of weight molecules. The initial concentrations of the summation gates, annihilator, restoration gates, restoration fuels and reporters are 100 nM (1×), 400 nM (4×), 100 nM (1×), 200 nM (2×) and 200 nM (2×), respectively, with a standard concentration of 100 nM. Source Data
Extended Data Fig. 4 A winner-take-all DNA neural network that recognizes 9-bit patterns as ‘L’ or ‘T’.
In each output trajectory plot, dotted lines indicate fluorescence kinetics data and solid lines indicate simulation. The standard concentration is 50 nM (1×). The initial concentration of each input strand is either 0 or 50 nM (1×). The initial concentration of each weight molecule is either 0 or 10 nM (0.2×); weight fuels are twice the concentration of weight molecules. The initial concentrations of the summation gates, annihilator, restoration gates, restoration fuels and reporters are 50 nM (1×), 75 nM (1.5×), 50 nM (1×), 100 nM (2×) and 100 nM (2×), respectively. The patterns to the left and right of the arrow indicate input signal and output classification, respectively. In addition to the perfect inputs, 28 example input patterns with 1–5 corrupted bits were tested. Note that 5 is the maximum number of corrupted bits, because an ‘L’ with more than 5-bit corruption will be as similar as or more similar to a ‘T’, and vice versa. Source Data
Extended Data Fig. 5 A winner-take-all DNA neural network that recognizes 100-bit patterns as one of two handwritten digits.
a, Choosing the test input patterns on the basis of their locations in the weighted-sum space. b, Overlap between the two memories: ‘6’ and ‘7’. c, 36 test patterns with the number of flipped bits shown next to their weighted sums. d, Recognizing handwritten digits with up to 30 flipped bits compared to the perfect digits. Dotted lines indicate fluorescence kinetics data and solid lines indicate simulation. The standard concentration is 100 nM. Initial concentrations of all species are listed in Extended Data Fig. 10. The input pattern is shown in each plot. Note that 40 is the maximum number of flipped bits because all patterns have exactly 20 1s. Source Data
Extended Data Fig. 6 Three-species winner-take-all behaviour and rate measurements for selecting DNA sequences in winner-take-all reaction pathways.
a, Fluorescence kinetics data for a three-species winner-take-all circuit. Initial concentrations of the three weighted-sum species are shown on top of each plot as a number relative to a standard concentration of 50 nM (1×). The initial concentrations of the annihilator, restoration gates, fuels and reporters are 75 nM (1.5×), 50 nM (1×), 100 nM (2×) and 100 nM (2×), respectively. b, Measuring the rates of 15 catalytic gates. Fluorescence kinetics data (dotted lines) and simulations (solid lines) of the signal restoration reaction are shown, with a trimolecular rate constant (k) fitted using a Markov chain Monte Carlo package (https://github.com/joshburkart/mathematica-mcmc). The reporting reaction was needed for the fluorescence readout. Initial concentrations of all species are listed as a number relative to a standard concentration of 50 nM. c, The 15 catalytic gates sorted and grouped on the basis of their rate constants. All rate constants are within ±65% of the median. The two coloured groups of three rate constants are within ±5% of the median. These two groups of catalytic gates were selected for signal restoration in the winner-take-all DNA neural networks that remember two to nine 100-bit patterns (Methods section ‘Sequence design’). Source Data
Extended Data Fig. 7 A winner-take-all DNA neural network that recognizes 100-bit patterns as one of three handwritten digits.
a, Circuit diagram. b, Choosing the test input patterns on the basis of their locations in the weighted-sum space. c, Overlap between the three memories: ‘2’, ‘3’ and ‘4’. d, Recognizing handwritten digits with up to 28 flipped bits compared to the ‘remembered’ digits. Dotted lines indicate fluorescence kinetics data and solid lines indicate simulation. The standard concentration is 100 nM. Initial concentrations of all species are listed in Extended Data Fig. 10. The input pattern is shown in each plot. Note that 40 is the maximum number of flipped bits because all patterns have exactly 20 1s. Source Data
The compiler21 is a software tool for designing DNA-based winner-take-all neural networks. Users start by uploading a file that describes a winner-take-all neural network. Alternatively, the weight matrix and test patterns can be drawn graphically. Next, a plot of the weighted-sum space provides a visual representation of the classification decision boundaries. The kinetics of the system can be simulated using Mathematica code downloaded from the compiler website, and the set of reaction functions are displayed online. Finally, the compiler produces a list of DNA strands that are required to experimentally demonstrate the network as designed by the user.
a, Logic circuits that determine whether a 9-bit pattern is more similar to ‘L’ or ‘T’. b, Logic circuits that recognize 100-bit handwritten digits. To find a logic circuit that produces correct outputs for a given set of inputs, with no constraint on other inputs, we first created a truth table including all experimentally tested inputs and their corresponding outputs. The outputs for all other inputs were specified as ‘don’t care’, meaning the values could be 0 or 1. The truth table was converted to a Boolean expression and minimized in Mathematica, and then minimized again jointly for multiple outputs and mapped to a logic circuit in Logic Friday (https://download.cnet.com/Logic-Friday/3000-20415_4-75848245.html). In the minimized truth tables shown here, ‘X’ indicates a specific bit of the input on which the output does not depend. For comparison, minimized logic circuits were also generated from training sets with a varying total number of random examples from the MNIST database. The performance of each logic circuit, defined as the percentage of correctly classified inputs, was computed using all examples in the database. To make the minimization and mapping to logic gates computable in Logic Friday, the size of the input was restricted to the 16 most significant bits, determined on the basis of the weight matrix of the neural networks.
Extended Data Fig. 10 Species and their initial concentrations in all neural networks that recognize 100-bit patterns.
a, List of species and strands. Reporters were annealed with top strands (that is, Rep[j]-t) in 20% excess. All other two-stranded complexes were annealed with a 1:1 ratio of the two strands and then PAGE-purified (Methods section ‘Purification’). b, Weights and example inputs in the neural network that recognizes ‘6’ and ‘7’. c, Weights in the neural network that recognizes ‘1’–‘9’. Weights and inputs used in all experiments are listed in Supplementary Table 2. Detailed protocols for all experiments are listed in Supplementary Table 3.