Deep convolutional and fully-connected DNA neural networks

Domain: earth_science
Published: 2025-11-27
Editorial outcome: major_revision

Abstract
DNA molecules can be used to build “neural networks” that function like the brain, enabling them to perform complex computational tasks. However, a fundamental limitation of existing DNA networks is that their most basic computing units cannot perform true continuous and precise analog calculations, which restricts their ability to process complex information effectively. To address this, here we develop a DNA computing unit called CALCUL. This system successfully achieves fully analog computation, where all inputs, weighting parameters, and outputs are continuous and precise values. It performs the core operations of a neural network rapidly with high accuracy and is reusable. By integrating magnetic bead technology, we also enable modular operations and the construction of multilayer networks. Ultimately, we use this technology to construct a deep DNA neural network that correctly identifies complex color images with 100% accuracy. These developments provide a robust foundation for building more powerful and precise molecular computers.

Introduction
Currently, DNA has become a widely used macromolecule material for molecular computing, capable of storing and processing information at the molecular scale 1–7. Numerous DNA logic circuits with specific functions have been established 8–11, such as binary square root circuit, half adder and full adder. Although the field of DNA logic circuits has been developing rapidly, it is primarily unsuitable for dealing with complex and analog input information, such as recognizing images with numerous pixels. Under such scenario, neural networks are much better options as they possess the advantages of large-scale parallel computing and weighted processing of input data 12–19. Therefore, in recent years, researchers have been endeavoring to use DNA to construct the foundational element of a neural network: the weighted summation unit, and based on which, a variety of DNA neural networks were further developed. Cherry and Qian proposed a “winner take all” neural network to recognize nine types of digital figures, showcasing the feasibility of using DNA to construct complex neural networks 13. Okumura et al. invented the enzyme driven DNA neural networks, which showed much faster computing speed and thereby implemented spatial partitioning of nonlinear separable regions 20. Xiong et al. further integrated the weighted summation operations into convolutional networks and implemented recognition of 32 12×12 images with 100% correctness, which revealed the computational potential of DNA neural networks 21. For biomedical applications, Zhang et al. developed a DNA neural network based miRNA classifiers for cancer diagnosis in serum samples 22. Overall, due to the capability of highly paralleled computing and processing complex analog information, DNA neural networks are promising and powerful systems that might bring molecular computing to a much more diverse and functionalized level.

However, none of the current DNA neural networks are fully analog. This is because the fundamental elements of DNA neural networks—the weighted summation units—cannot achieve complete analog computing. In other words, their inputs, outputs, and weights cannot simultaneously be continuous and accurate. For instance, in the DNA weighted summation unit developed by Xiong et al., the input can only be a discrete value of 0 or 1 21. In another weighted summation unit developed by Zhang et al., the input and output can both be continuous. However, the weights are discrete and finite integer values (1, 2, 3, 4 and 5) 22. In the weighted summation units developed by Genot et al., the inputs, outputs, and weights could theoretically be continuous, but their accuracy has only been characterized in the simplest case of two-input weighted summation. The accuracy of more complex weighted summation has not been tested, let alone that of convolution or deep neural networks 23. More importantly, we believe that their sequence and structural design cannot ensure accurate analog computation due to the stochastic nature of their designed chemical reactions.

Unfortunately, the foundation of DNA circuits is concentration-based DNA chemical reactions, which are naturally continuous/analog. This is fundamentally different from electronic circuits, whose basic reactions are inherently digital. The absence of suitable analog computing units necessitates quantization of the analog DNA concentrations during weighted summation. This not only reduces the computational accuracy of the neural network but also necessitates the design of additional nodes to maintain the overall performance of the neural network. Furthermore, it requires the addition of extra quantization modules between nodes, which in turn causes the number of DNA strands in the network to proliferate exponentially with depth. As a result, constructing complex DNA neural networks, especially those with deep layers, becomes challenging.

Given these challenges, none of the current DNA weighted summation units can achieve fully analog computations where the inputs, weights and outputs are not only continuous but also accurate. The lack of such operation units block the construction of complex and deep DNA neural networks and limit the field’s development.

To over come these fundamental limitations, here we develop the CALCUL unit. Our system represents an advance in DNA-based neural computing, demonstrating high-performance weighted summation operations for multiple inputs within short timeframes. The system successfully implements both full-connection and convolution operations with high accuracy while maintaining functional reusability. Through integration of magnetic-bead techniques, we expand the system’s computational capabilities to enable multi-layer neural network operations. This enhanced CALCUL platform incorporates weighted summation with ReLU activation, negative weight implementation, and operation concatenation- all while preserving computational precision. The system’s capabilities are rigorously validated through image recognition tasks, including successful classification of handwritten digits using convolutional network architectures.

Building upon these capabilities, we construct a deep DNA neural network architecture comprising two convolutional layers and one fully-connected layer, demonstrating robust performance in classifying pixel-based images with continuous value inputs. Collectively, these achievements establish CALCUL units as a versatile and powerful platform for implementing DNA-based computing systems and neural network operations.

Results
Working principles
Herein, we have established a Classified allosteric-toehold based continuous and ultra-accurate (CALCUL) computing unit for weighted summation, in which the inputs, outputs and weights all present as continuous and accurate values. Based on the CALCUL unit, analog convolutional and fully-connected DNA neural networks were further established. As shown in Fig. 1A, B, we have designed two modes of CALCUL units: the cis-CALCUL mode for fully-connected operations and the trans-CALCUL mode for convolution operations. The difference is that in the cis-mode, the P-strand containing branch-migration domain B serves as the input and the Q-strand containing toehold domain A* serves as the weight, and vice versa for the trans-mode. Taking cis-CALCUL mode as an example, the CALCUL unit can be divided into two sections: the weight multiplication section and the summation section. The input strands (denoted as P-strand) are all single-stranded DNAs consisting of domain A and B. Although the sequences of domain A in different input strands are different, the sequences of their domain B remain consistent.

In fact, the primary driving force in our computational reactions originates from the temperature-annealing process. For each input, it will undergo a classified allosteric-toehold (CAT) annealing process as illustrated in Fig. 1A: there are two corresponding weight strands; When one weight strand (Q-strand), the input P-strand, and the T-strand/O-strand duplex are present together, thermal annealing leads to the formation of an effective PQT triplex (fluorescent); When another weight strand (Q’-strand), the input P-strand, and the T’-strand/O-strand duplex are present together, thermal annealing leads to the formation of an effective PQ’T’ triplex (non-fluorescent). Notably, during the isothermal process following annealing, the classified allosteric-toehold design in this system facilitates the convergence of the reaction outcome toward a stable state of complete reaction (the reaction mechanism is illustrated in Supplementary Fig. 2). Then, we can flexibly adjust the percentage of P-strand entering the effective downstream pathway at any value between 0 and 1 by setting the ratio of [Q-strand]/([Q-strand]+[Q’-strand]) at the corresponding value. As we know, the amount of P-strand entering into effective downstream pathway is exactly Iiwi, wherein the [P-strand] is Ii and the [Q-strand]/([Q-strand]+[Q’-strand]) ratio is wi. So far, the CALCUL unit has finished the weight multiplication operation (Iiwi) with the Ii and wi at any designated value. Then, to fully accomplish weighted summation, i.e. ∑Iiwi, we need a summation section that adds up all the Iiwi. As we have described above, in our sequence design, the domain B of P-strand and the domain D of Q-strand will cooperatively bind to T-strand, and the domain B of all inputs are the same and so are the domain D of all Q-strands. Therefore, all the input strands bound to their corresponding Q-strands will react with the same T-strand/O-strand duplex during the annealing process, and such integration of CAT reactions is exactly the summation operation of ∑Iiwi. Meanwhile, all the input strands that bound to corresponding Q’-strands will react with the same T’-strand/O-strand duplex during the annealing process, producing non-fluorescent PQ’T’ triplex. So far, the CALCUL unit has realized its inputs, weights and outputs to be continuous simultaneously.

While, as shown in Fig. 1B, in the trans-CALCUL unit, Q-strands serve as inputs and P-strands serve as weights. Alongside P-strand, a P’-strand labeled with a quencher is designed, and the weight value equals to the percentage of [P-strand]/([P-strand]+[P’-strand]). Different to the cis-CALCUL, in which a pair of effective module and void module is designed, there is only one module in the trans-CALCUL. Since the P’-strand is labeled with a quencher, the P’QT triplex will not emit signal and is classified as void output; the PQT triplex will emit signal and is classified as effective output, achieving the weight multiplication operations. Similar to the cis-CALCUL, all P-strands and Q-strands will react with the same OT duplex, realizing accumulated weight multiplication (weighted summation). We would like to note that the trans-CALCUL unit omit the OT’ duplex, so the reaction system is even more concise than the cis-CALCUL units, this is particularly crucial for constructing ConvNet (Convolutional neural Network).

However, aside from continuity, the weighted summation unit, as the foundation of DNA neural networks, must also be highly accurate. As for this, the computing accuracy of previously reported weighted summation units was – to the best of our knowledge - never thoroughly demonstrated 13,20–23. They focused on the units’ linear responses to inputs and the correctness in classifications, and ignored the absolute errors of the weighted summation operation itself. This will not cause conspicuous computing deviations when they are only one-layer networks. Yet with the neural network goes deeper, the deviation would accumulate and greatly affect the network’s functionality. We believe the previously reported weighted summation units require too many DNA strands and involve multiple pathways of strand displacement reactions, so the thermodynamics, kinetics and crosstalk cannot be robustly controlled, leading to inevitable considerable deviations.

Most DNA circuits aim to optimize thermodynamics for better performance, but kinetic barriers often cause output signals to deviate from theoretical values. For instance, basic toehold reactions may plateau prematurely, and more complex reactions are similarly affected. Thus, addressing kinetic barriers is crucial for DNA analog circuit construction (validation experiments are shown in Supplementary Fig. 5). Our designs consider these barriers, ensuring compatibility with thermal annealing. To achieve temperature-annealing-based computational reactions with high uniformity and accuracy, our mechanism incorporates sophisticated designs in both the reaction process and structural architecture. As shown in Fig. 1C, the CALCUL units utilize an identical strand displacement pathway for all inputs, ensuring uniform reaction thermodynamics and enhancing computing accuracy. To bolster reaction robustness, three key designs were implemented. Firstly, a 18-nt toehold long domain D was employed to favor the forward reaction, which means that the Q strand will bind to the T strand with robust and sufficient stability, ensuring near-complete conversion and reproducibility. Secondly, an allosteric domain X/X* was incorporated into the modules to prevent leakage between the P-strand and modules in the absence of the Q-strand, safeguarding against common issues in strand displacement systems that compromise accuracy. It is particularly noteworthy that the allosteric domain X/X* design differs significantly from the dissociation region in conventional toehold-mediated strand displacement reactions. This architecture not only effectively suppresses signal leakage but also ensures thermodynamic irreversibility - a critical feature for implementing temperature-annealing-based computational reactions (details are shown in Supplementary Fig. 2). Lastly, the longer T-strand was designated as the output strand to mitigate crosstalk between OT duplex and OT’ duplex, ensuring that fluorescent signals remain unchanged despite potential crosstalk, thereby preserving computing accuracy. In the fluorescent detection system design, we employed a modification scheme with the fluorophore conjugated to the long T-strand and the quencher attached to the short O-strand. By maintaining the quencher strand concentration at a 5% excess over the fluorophore strand, this strategy not only compensates for errors introduced during concentration measurement and sample loading, but also minimizes the impact of crosstalk reactions on signal output (This scheme is also applicable to magnetic bead assisted computational reactions). Notably, using the longer T-strand as output not only reduces crosstalk but also aids annealing, as annealing can accelerate crosstalk, but our design prevents it from affecting the output signal. The common approach of using the shorter O-strand as output is not suitable for annealing and cannot achieve precise analog computing, the experimental validation results for this section are presented in Supplementary Fig. 14. More detailed elaboration of the origin of the high accuracy analog computing of the CALCUL units could be found in Supplementary Section S1 of the Supplementary Information.

Integration of multiple pathways of allosteric toehold strand displacement
From the working principle described above, the Integration of multiple pathways of allosteric-toehold strand displacement reaction serves as the foundation for both cis- and trans- CACUL units: In cis-CALCUL units, weighted summation is achieved by two parallel integrations (effective and ineffective pathways) sharing the same input strands; in trans-CALCUL units, weighted summation corresponds to the integration with the input/output strand roles swapped. We therefore prioritized testing the feasibility and accuracy of such integration (Fig. 2A). We synthesized 8 inputs (P-strand-1 to 8) and their corresponding weights (Q-strand-1 to 8). According to the sequence design, all P-strands are led to invade the same T-strand-1/O-strand-1 duplex. To simplify, we set the P-strands to be binary (150 nM of input represented 1; no input represented 0) and the Q-strands to be analog values between 0 and 1, normalized by 150 nM. Then, we performed the operations of integrating 2, 4 and 8 inputs/weights, respectively. As shown in Fig. 2A, for each experimental group, we firstly conducted the operations on 4 designated standard sets of inputs/weights to obtain the standard curve, and the linearities in all groups were over 0.999. Then, we conducted the computation on two testing sets of inputs/weights, and based on the standard curves, we could transform the fluorescent signals to output values. The experimental results closely matched to the theoretical values, with all deviations remaining below 2.5%. It was worth noting that we could add strands (denoted as c-Q-strand) complementary to the Q-strand to reset the whole computing process and then restart it by adding W-strands again, and by doing so, we implemented 2 rounds of calculation with the computing accuracies ranging from 90.7% to 100.0%, median 98.6% (Supplementary Figs. 4, 6 and 7).

To further demonstrate the feasibility and accuracy, we tested multiplexed and paralleled integration (Fig. 2B). Firstly, two P-strands were designed to be integrated into two different T-strand/O-strand duplexes, simultaneously and respectively. Similarly, P-strands were binary and the weights were values between 0 and 1, normalized by 120 nM. Experimental results in Fig. 2C showed that paralleled integration of two allosteric toehold strand displacement reactions in one tube was completely feasible: the standard curves of two reaction pathways were both highly linear (Supplementary Fig. 8) and the computing accuracies were 98.9% and 98.0%. Furthermore, we tried integrating three paralleled reactions, simultaneously and respectively. The standard curves were highly linear (Supplementary Fig. 10) and the computation results were all highly accurate, with the deviations less than 2.9% (Fig. 2C). Notably, we also tested recycling of the paralleled multiplexed integration. Shown in Supplementary Figs. 9, 11, the above duplex and triplex integration were recycled for 2 rounds, and the linearity of standard curves remained higher than 0.99 and the computing accuracy kept over 95.3%. Overall, we have demonstrated that integration of multiple allosteric toehold pathways and paralleling such integration are both feasible and accurate (Fig. 2D).

Construction of the CALCUL units
Having demonstrated the feasibility and accuracy of integrating multiple pathways of allosteric toehold strand displacement, we then constructed complete cis- and trans- CALCUL units for weighted summation (∑Iiwi). The Ii were analog values between 0 and 1, derived from normalizing the concentration of the corresponding input strand by 120 nM. The wi was equal to [Weight-strand]/([Weight-strand]+[Weight-strand’]). As shown in Fig. 1A, B, in the cis-CALCUL units, P-strands serve as inputs, Q-strands and Q‘-strands serve as weights, while in the trans-CALCUL units, Q-strands serve as inputs, P-strands and P’- strands serve as weights. We then conducted the weighted summation operations on 2, 4 and 8 inputs (n=2, 4, 8) (Supplementary Figs. 12–17), respectively. For each n, we designed a set of higher weights and a set of lower weights so that we could examine the computing accuracy at a wider range. As was shown in Fig. 3, for all tests, both the cis- and trans- CALCUL units could implement the weighted summation operations within 40 min. And the computing accuracy was from 96.0% to 100.0% (median 98.9%) and 94.7% to 99.4% (median 98.0%) for cis- and trans- CALCUL units respectively. In addition, both the two types of CALCUL units could be recycled. The computing accuracy remained over 96.6% and 90.2% for cis- and trans- CALCUL units after 2 rounds of computations (Supplementary Figs. 18, 19). To the best of our knowledge, the performance of previously reported DNA weighted summation tools has been primarily evaluated through qualitative assessments. While these studies successfully demonstrated functional classification capabilities, a quantitative characterization of computing accuracy-specifically, the direct comparison between theoretical and experimental values-remains to be fully established. Our work addresses this gap by providing comprehensive data that rigorously validates the computational precision of the CALCUL unit. Considering its fast speed, distinct simplicity and high accuracy, we believe the CALCUL unit was a leading DNA tool for weighted summation in the field of DNA neural network (A systematic and comprehensive comparison is provided in Supplementary Table 27).

The cis-CALCUL units based full-connection and the trans-CALCUL units based convolution operations
As shown in Fig. 4A, for the full-connection operations based on cis-CALCUL units, we designed fully connected layers with 2 inputs/2 outputs, 3 inputs/3 outputs and 4 inputs/3 outputs. Experimental results showed that the cis-CALCUL unit based full-connection network successfully implemented the operations with the computing accuracy ranging from 96.1% to 99.3% (median 97.7%) and the operation time was less than 30 min. It is worth noting that for full connection operations, the cis-CALCUL unit is more compatible because full connection system requires assigning separate weights to each input in different pathways.

We then wanted to utilize our trans-CALCUL units to implement convolution. The schematic illustration of the convolution operation with a 2×2 stride 2 kernel by the trans-CALCUL unit was shown in Fig. 4B. In each calculation, m×n inputs are split into k×k regions of interest (with the same size as the kernel). Each input cell is represented by Qij strand, and each kernel cell by a combination of Pij and P’ij. Element-wise multiplication and summation (the convolution operation) is performed to give the output cell. In our case, strides is identical with the kernel size k, the output matrix size is thus [(n−k)/s+1]×[(m−k)/s+1]. It is worth noting that in the trans-CALCUL unit, the sequences of the A domains in different P and P’ strand pairs exhibit diversity, serving to determine the specific positions of the weights within the convolutional kernel. Meanwhile, the sequences of the B domains remain identical across all P and P’ strand pairs. This arrangement allows different convolutional kernels to share the same weight strands, thereby significantly reducing the number of strands required for a convolutional operation. For experimental verification, we designed the trans-CALCUL unit based ConvNet to recognize two 16-pixel (4×4) images depicting parallel lines in opposite directions by 2×2 kernels. The four modules were labeled with the same quencher (BHQ-2) but with four different fluorophores (FAM, HEX, TAMRA, ROX), so the four values of the 2×2 output feature map could be reported individually and simultaneously. We firstly ran the ConvNet on four standard sets of inputs to obtain four standard curves for all channels and then on the two images. As shown in Fig. 4B and Supplementary Fig. 24, the four channels all exhibited high linearity (R2>0.99). The system achieved perfect recognition, with the output feature maps showing high accuracy (95.2–100%; median 96.5%) relative to theoretical results.

The trans-CALCUL unit based ConvNet for image recognition
To be more convincing, we further used the trans-CALCUL unit based ConvNet to recognize six 64-pixel (8×8) images depicting digit 8 and 9 by 4×4 kernels (Fig. 5). The output 2×2 feature map was further pooled by summating each of the columns to give 2 outputs indicating the result for each category. Similarly, the linearities were over 0.99 (Supplementary Fig. 26), the recognition was fully correct, and the final output values were highly accurate (90.3% to 99.4%, median 94.7%). We would like to note that each of the recognition/convolution process was implemented within a single tube, without segmenting into paralleled tubes at all.

Magnetic bead assisted CALCUL units
In the above experiments, we have thoroughly characterized the CALCUL unit and the DNA neural networks based on it. However, we must emphasize that if deep DNA neural networks were to be constructed exclusively using the current CALCUL units, the interlayer reactions would remain isolated. Under such conditions, the layer-to-layer connections would operate through abstract numerical transfer rather than physical DNA strand, which to some extent weakened its practicability. Therefore, we would like to further improve the CALCUL unit so that the output strands enter into the downstream layer and thereby the whole computing process could be completely implemented by DNA strands. As we know, the inevitable DNA crosstalk and leakage has long been the restraint of the field; if adding all of the upstream strands into the downstream layer, the undesired side reactions would be significant 24,25. To overcome this, we introduced magnetic bead to “purify” the upstream strands and only allow the effective output strands to enter into the downstream layer, and by doing this, we have further extended the capability of the CALCUL units.

The first improvement was the incorporation of the cis-CALCUL unit and trans-CALCUL unit based ReLU module, which made our CALCUL unit more compatible for building neural networks. For the cis-CALCUL unit based ReLU module was shown in Fig. 6A, in the upstream CALCUL unit, the input P-strand had four domains: domain A and B were for allosteric toehold reaction, domain M and domain block were for ReLU. The O-strand was labeled with a biotin instead of a quenching group, and the T-strand was extended with an extra K* domain while the T’-strand remained unchanged. After weighted summation, magnetic beads coated with streptavidin were added and the magnetic field was applied. Consequently, the biotin labeled free O-strand, OWT triplex and OQ’T’ triplex would be absorbed onto the magnetic bead and isolated from the solution. The supernatant comprising of PQT triplex and PQ’T’ triplex was then transferred to another tube (Box 1), and the amount of PQT triplex was denoted as x. Then, Tr-strand and TrPDR triplex were added. The Tr-strand contained a domain K that was fully complementary to the domain K* of PQT triplex. The PN-strand was the input strand of the downstream CALCUL unit and the R-strand was labeled with a biotin. Since Tr-strand and TrPDR triplex contained the domain K, they both can hybridize with PQT triplex. However, the Tr-strand had much higher reaction priority with PQT because it was free and fully exposed whereas the domain K within TrPDR triplex was relatively blocked. Therefore, the PQT triplex would firstly bind to Tr-strand and produced PQTTr quadruplex. The amount of free Tr-strand added was denoted as b, and we could deduce that the amount of residual free PQT triplex was x-b. Then, the residual PQT triplex would react with the TrPDR triplex and produce identical amount (x-b) of PDR duplex. Overall, after reaction, the system consisted of PQTTr quadruplex, PQ’T’ triplex, excess TrPDR triplex and PDR duplex of amount x-b (Box 2). Then, the magnetic separation was applied and the supernatant was discarded. After resuspending the precipitate, the system contained excess TrPDR triplex and IDR duplex of amount x-b (Box 3). The last step was adding c-R-strand that was fully complementary to the R-strand and applied magnetic separation. Finally, the supernatant was transferred to another tube (Box 4) and the solution consisted of TrPD duplex and free ID-strand of amount x-b. Since the TrPD duplex was inert for downstream CALCUL unit, the final output of the upstream CALCUL unit was PN-strand of amount x-b, and the concentration of PN-strand was a×(x-b), where a was the dilution factor determined by the final solution volume. Overall, throughout the above process, we have achieved a ReLU module with the following expression:
y = {a(x-b), if x>b; 0, if x≤b} (1)

The ReLU and subtraction operations are considerably simpler when implemented with the magnetic bead-assisted trans-CALCUL unit than with the cis-CALCUL unit, due to the omission of the OT’ duplex. As shown in Fig. 6C, similarly, the O-strand and P’-strand are labeled with biotin instead of quenching group, and the T-strand is extended with an additional K* domain. The subsequent reaction principle and magnetic separation operation are consistent with the ReLU based on the cis-CALCUL unit, and finally, the ReLU module based on the trans-CALCUL unit can also be constructed.

For experimental verification, we implemented the weighted summation operation on 4 standard sets of inputs/weights to obtain the standard curve and 1 testing set of inputs/weights. For the cis-CALCUL unit based ReLU module the constant b in the ReLU was set at 0.2 by controlling the amount of Tr-strand. As shown in Fig. 6B, the standard curve demonstrated high linearity (R2=0.9977), and the computing result closely matched the theoretical value, with an accuracy of 98.9%. For the trans-CALCUL unit based ReLU module was shown in Fig. 6D, the constant b in the ReLU was set at 0.3 by controlling the amount of Tr-strand. The standard curve demonstrated high linearity (R2=0.9903), and the computed results closely matched the theoretical values, with accuracies of 96.2% and 99.4%. The computing results for both modes collectively demonstrate the feasibility and accuracy of incorporating a magnetic bead-assisted ReLU into the CALCUL unit.

The second improvement was the introduction of negative weights. The subtraction operation principle based on cis- and trans- CALCUL modes is the same. For the convenience of expression, IWT is uniformly used to represent PQT triplex in the cis-CALCUL mode and QPT triplex in the trans-CALCUL mode, and IWNTN is uniformly used to represent PQNTN triplex in the cis-CALCUR mode and QPNTN triplex in the trans-CALCUL mode.

As shown in Fig. 7A, the inputs multiplied by positive weights would be integrated into T-strand/O-strand duplex by the weight strands, producing IWT triplex and biotinylated O-strand, whereas the inputs multiplied by negative weights would be into TN-strand/O-strand duplex by WN-strand, producing IWNTN triplex and biotinylated O-strand. It was worth noting that the T-strand was extended with overhang T+ domain and S+ domain, and the TN-strand was also extended with overhang T- domain and S- domain. After adding magnetic bead and applying magnetic field, the supernatant comprising of IWT triplex and IWNTN triplex was transferred to the downstream tube, wherein the overhang 10-nt T- domain of IWNTN triplex and the overhang 6-nt T+ domain of IWT triplex would cooperatively lead the two triplexes to hybridize with U+-strand/U--strand duplex, producing IWTU+ quadruplex and IWNTNU- quadruplex. According the sequence design, unless united, IWT triplex or IWNTN triplex alone could not stably bind to the U+U- duplex. Therefore, the U+U- duplex served as an annihilator (Supplementary Fig. 1C, 3), consuming IWT triplex by the amount of IWNTN triplex, which, in all, presented as subtracting IWNTN from IWT (Fig. 7B and Supplementary Fig. 27A). Finally, the biotinylated U+-strand and U--strand would isolate the IWTU+ quadruplex and IWNTNU- quadruplex from the system and leave free IWT triplex, as the final output, to enter into the downstream layer. We then implemented the above weighted summation and subtraction operations on 4 standard sets of inputs/weights to obtain the standard curve and 2 testing sets of inputs/weights. As shown in Fig. 7C, Supplementary Figs. 27B, 28, the standard curves for both CALCUL modes exhibited high linearity (0.9999 and 0.9937), and the accuracies of the computing results were favorable (99.3% and 98.8% for cis-CALCUL; 97.0% and 95.0% for trans-CALCUL). Notably, the use of negative weights expanded the available weight range to −1≤w≤1. While the weighted summation result could be negative, such negative values have limited utility in multi-layer networks, as they would not be propagated through the subsequent ReLU process (Y=X for X≥b; Y=0 for X<b).

Magnetic bead assisted CALCUL unit based deep DNA neural networks
Having demonstrated the functionality and accuracy of magnetic bead assisted CALCUL unit, we next wanted to cascade the units to further build deep DNA neural networks. As shown in Fig. 7D, we firstly cascade two convolutional layers: the 16 inputs in a 4×4 pattern were convolved by a 2×2 kernel and activated by ReLU to produce a 2×2 feature map. The feature map was then convolved by a 2×2 kernel containing 2 positive weights and 2 negative weights and then subtracted to produce the final output. We ran the ConvNet on 3 standard sets of inputs/kernels to obtain the standard curve and 2 testing sets of inputs/kernels. As shown in Fig. 7D, Supplementary Figs. 29, 30, the linearity of the standard curves of two CALCUL modes were 0.9928 and 0.9980. The computing accuracies were 88.8% and 91.4% for the cis-CALCUL, while 95.6% and 100% for the trans-CALCUL, demonstrating the cascading potential of magnetic bead assisted CALCUL units.

Finally, we built another deep DNA neural network to recognize and classify 144-pixel color images in the 12×12 pattern. As shown in Fig. 7E, the network consisted of a trans-CALCUL based convolution layer with 3×3 kernel and trans-ReLU, a trans-CALCUL based convolution layer with 2×2 kernel and cis-ReLU and one cis-CALCUL based fully connected layer with 4 inputs/2 outputs. This network is constructed on the subset of extended MNIST (EMNIST 26) dataset, classifying handwritten letters A and B (Supplementary Fig. 31). At last, we used the CALCUL based deep neural network to recognize 12 144-pixel images depicting either “A’ or “B”. Experimental results in Fig. 7E showed that our ConvNet classified all the tested color images with 100% correctness, exhibiting the feasibility and accuracy of building CALCUL based deep DNA neural network.

Methods
Sequence design
The detailed structural design of the strands and complexes used in this work was illustrated in Supplementary Figs. 1–3, and their sequences were shown in Supplementary Tables 4–26. To reduce synthesis errors, no more than four consecutive ‘A’s or ‘T’s (excluding polyA used for connecting biotin) and no more than three consecutive “C“s were designed. The CG% was set to approximately 40% to ensure appropriate melting temperatures. DNA strands used in the same reaction do not contain obvious secondary structures and undesired crosstalks at 37°C. The sequence design was verified by NUPACK 34 to confirm their binding energy and specificity.
Beyond NUPACK simulations, the designed DNA strands were experimentally screened using multiplication operations performed in both cis and trans configurations of the CALCUL unit (see Supplementary Figs. 36–39 and Supplementary Tables 124–135). Based on quantitative fluorescence measurements, candidate strand groups with fluorescence signals within ±5% of the median value were selected. Experimental groups falling outside this range were excluded.

DNA oligonucleotides synthesis
Based on the design, unlabeled oligonucleotides purified by high affinity purification (HAP) and labeled oligonucleotides purified by high-performance liquid chromatography (HPLC) were provided by Sangon Biotech and used without further purification. All strands were shipped lyophilized and resuspended at 20 μM in 1× Tris-EDTA buffer, and stored at 4°C for further use.

Annealing protocol and buffer condition
The buffer used in all the experiments was 1× ThermoPol reaction buffer with 2 mM Mg2+ at pH 8.8 (New England Biolabs). All DNA strands were added to the target concentration according to experimental setting and mixed in 1× buffer solution. The strands were annealed by heating to 85°C for 10 min, cooling to 65°C, 55°C and 45°C at a rate of 4°C per second, holding for 1 min at each temperature, repeating this entire cycle twice, and finally cooling to 37°C.

Detailed experimental procedures
Single-layer computation experiments. Premixes of inputs, weights and modules were first prepared in accordance with the computational design. The solutions were then combined, subjected to an annealing procedure, and finally placed into a qPCR machine for data acquisition. Comprehensive details of the experimental configurations are systematically provided in Supplementary Tables 28–82.

Reset and recycling of the CALCUL units. Upon completion of a computational cycle, an equimolar amount of complementary strands relative to the weight strands was introduced into the reaction mixture. After annealing, the CALCUL unit was reset to its initial state. The premix weights solution was then reintroduced, and the system underwent another annealing process prior to fluorescence detection. A detailed description of the experimental setups can be found in Supplementary Tables 52–55, 65–68.

Multi-layer computation experiments. In multi-layer DNA neural networks, to facilitate the use of upstream DNA outputs as inputs for downstream layers, streptavidin-coated magnetic beads were employed to purify upstream DNA strands via affinity-based capture. The hierarchical interconnection protocol consists of the following steps:
(1) weighted summation;
(2) introduction of magnetic beads and incubation;
(3) collection of the supernatant for activation function;
(4) a second bead-based purification to isolate DNA inputs for downstream processing.
Detailed experimental procedures are systematically provided in Supplementary Fig. 40, Supplementary Tables 83–123.

Fluorescence acquisition
The output signal value were recorded by QuantReady K9600 qPCR machine and LineGene 4800 qPCR machine at 37°C with the fluorescence intensity measured. Four Fluorophore-Quencher pair were used: FAM (excitation 494 nm, emission 518 nm) - BHQ1 (quenching range 480–580 nm), HEX (excitation 535 nm, emission 556 nm) - BHQ1 (quenching range 480–580 nm), ROX (excitation 585 nm, emission 605 nm) - BHQ2 (quenching range 550–640 nm) and TAMRA (excitation 565 nm, emission 580 nm) - BHQ2 (quenching range 550–640 nm).
Considering efficiency (rapid computation completion), consistency (synchronized termination across replicates), and operational feasibility, we evaluated fluorescence signals at 10-min intervals for plateau verification. The termination criteria were applied as follows:
1) For all curves, calculate the fluorescence increase magnitude during the 5-min window preceding each check timepoint. If none exceed 1% growth, the experiment terminates.
2) In a set of experiments, if any curve fail to meet Criterion (1), the experiment will be reassessed after 10 min using the same standard. This iterative process is repeated until all curves satisfy the 1% threshold to determine a unified experimental termination time.
We also would like to note that in our experiment, we indeed observed some high-fluorescence curves (e.g., the full-connection operation with 4 inputs/3 outputs shown in Fig. 4A) continuing to exhibit exceptionally slow signal increases even after meeting the aforementioned experimental termination criteria. This phenomenon is likely caused by slowed reaction kinetics near thermodynamic equilibrium. Importantly, this behavior occurred synchronously in high-fluorescence standard curves. Since we convert fluorescence signals to computational results based on standard curves, the observed effect does not significantly impact the accuracy of CALCUL units. Experimental results confirm this conclusion (shown in Supplementary Figs. 32–35).

Statistics and reproducibility
Fluorescence signal data were exported from the qPCR instrument and analyzed using ORIGIN software. Quantitative outputs were derived from fluorescence signals using a pre-calibrated standard curve. For each weighted summation experiment, a standard curve was constructed by processing a minimum of three input/weight combinations designed to yield a gradient of output signals, under the same reaction conditions as the experimental samples. A scatterplot was generated with fluorescence intensity on the y-axis and the theoretical weighted summation value on the x-axis for these standards. The standard curve was then generated by performing linear regression (requiring R²>0.99) on these data points. The equation derived from this regression was applied to convert the fluorescence intensity of experimental samples into quantitative output values. No statistical method was used to predetermine sample size. No data were excluded from the analyses. The experiments were not randomized, and the investigators were not blinded to allocation during experiments and outcome assessment. All data generated in this study are provided in the Source Data file.

Streptavidin coated magnetic beads
The streptavidin coated magnetic beads were provided by Beaver Biosciences Inc, with a diameter of 300 nm. In the experiment, the concentration of magnetic beads was set at 6.7 mg per μmol of biotinylated DNA, and they were incubated with shaking at 37°C for 2 h, and magnetic separation was performed using a magnetic separator provided by Beaver Biosciences Inc. Notably, a conservative bead usage was adopted (3×) to ensure complete capture. We ensured K A, K B, W C, W D, b A, b B, b C, b D and C to be within [0, 1]. K B is a 2×2 convolution kernel with stride 2. For the image in the 8×8 pattern, K A is of size 2×2 with stride of 2. For the image in the 12×12 pattern, K A is of size 3×3 with stride of 3. We reduce the number of the fully connected layer in the classification task between letter A and B.
E = W E B (6)
Since the direct output D or E are not necessarily between 0 and 1, we apply the softmax function before the training loss, after multiplying back the scaling factor that we have divided previously.
y 1,2 = K A ⊗ b A K B ⊗ b B D (7)
Or in the case of classification of English letters,
y 1,2 = K A ⊗ b A K B ⊗ b B E (8)
pi = exp yi / (exp y1 + exp y2) (9)
We use the sparse categorical cross-entropy loss function
L = 1/N ∑N i pi log pi + (1-pi) log (1-pi) (10)
The training is performed on PyTorch 36 platform, learning rate is set to be 0.0001 with stochastic gradient descent (SGD) with a weight decay of 0.0001 and a momentum of 0.5. The network is trained for 100k iterations, and after each iteration, the weights and biases are clamped within [0, 1]. The weight matrices are initialized by Gaussian distribution with a mean of 0.5 and a standard deviation of 0.15. The performance of the network on the training and testing set is similar, indicating no significant overfitting. All the parameters are provided in the figures (Figs. 5 and 7).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Liu et al present a study on a new method for constructing DNA-based neural networks, based on an associative toehold mechanism. The authors demonstrate that their two units (cis- and trans-) can perform weighted linear sums with a high degree of accuracy. They show that these weighted summations can be \"reset\" (in that the inputs can be used again with a new set of weights) and that multiple summations can be performed in parallel. They demonstrate the possibility of using these networks to classify MNIST digits, and then extend their framework to include negative weights in the summation, non-linear thresholding and the ability to chain together multiple layers through a (somewhat cumbersome) multistep procedure. Within this framework, the authors use a two-layer network to distinguish handwritten letters from the MNIST database. Building chemical neural networks is an exciting topic and the motifs presented have promise. The authors have done *a lot* of work, and extensively characterised their motifs. The quantitative accuracy of the linear summation is also very impressive. The paper, however, is hard to read and the methods/approaches are not very transparent. In addition, the claims made in the title/abstract/introduction are perhaps not well-justified by the results presented, and could be misleading. We therefore feel that a substantial re-writing is required before the paper could be published in Nature Communications.\n\nSerious Comments.\n1. We do not understand how the cis- unit operates when there are multiple parallel outputs (eg. Fig 3b). The input species P_i are presumably shared between all weights, and so the output signal depends not just on the relative amounts of Q and Q' for the output in question, but also on how many other species the input couples to. How the system can nonetheless produce multiple responses in parallel is not discussed. Indeed, the design in Figure 3b shows no Q' strands at all (just different Q) so how it relates to the basic design in Fig. 1 is unclear.\n2. The methodology is unclear at multiple points.\n(a) Experimental protocols are nof described in detail. How were the actual experiments performed? What volumes were used, in which device, and with what settings? Which species were added and which species were initially present?\n(b) In many experiments, the fluorescence curves do not seem to have reached final steady state values. See, for example, Fig 3C (but it is generally quite common, particularly for the calibrations in the SI). How then was it decided when to stop experiments? They don't all seem to take the same amount of time. This factor is very important in establishing how well the computation match the expected value.\n(c) Presumably the linear fluorescence responses in Eg. Fig 3a are used to convert fluorescence signals in experiment to output signals, but how that is done when the neural nets are being applied is not discussed. Indeed, there is nothing at all in the methods on fluorescence measurments. This clarification is very important, because the apparent excellent performance of the summation gates depend on how exactly \"Fluorescence Intensity\" is mapped to \"Output value\". Indeed, we are surprised at the degree of precision for two reasons:\ni. Typically speaking, one would expect errors in the initial concentration of stocks, and from pippetting, to compromise the numerical accuracy of the summation.\nii. The approach seems to assume that reaction rates of inputs with their complementary weights are identical. However, while the strands are similar and may well have similar rates, a deviation of 10% or so (perhaps due to differences in secondary structure) would hardly be extraordinary.\nWhether the authors take special measures to eliminate these errors prior to the experiment, or whether they are effectively handled by the data processing in some way, is unclear.\n(d) Which exact experiments are performed is often unclear in the manuscript. It is very difficult for the reader, given how short the captions are, to interpret what is meant by the schematics in eg. Fig. 5, 6, 8, which we believe are meant to convey the experiment performed. What do the numbers in the circle actually mean, for example?\n(e) It seems like resetting is often (but not always) performed with different weights. In all cases this fact is not discussed, and in the SI the two separate sets of weights are not even provided.\n3. Relation between what is claimed and what is presented.\nThe authors make a number of claims in the paper that are not necessarily justified by the results - or at least, not without being substantially more precise about the claims.\n(a) positioning relative to the literature is somewhat imprecise. We think there is certainly novelty within the system proposed, but it needs to be stated more accurately. The authors describe their system as the first \"fully analogue\" DNA neural network, but then go on to describe the system of Genot et al, which is also analog. They criticise aspects of this work, but then they need to be careful about how they position their work with respect to it.\n(b) The authors describe their system as the \"first deep\" DNA-based neural network - but what does \"deep\" mean in this context? They demonstrate two layers, and their setup doesn't in fact naturally allow for multiple layers. Instead, it is necessary to use a convoluted intervening step involving magnetic beads. This is still an achievement, but the authors should be sure not to mislead readers with the claims they present in the introduction and abstract.\n(c) The authors should be precise about what their summation units can do - initially, they can only implement weights in the range 0<=w<=1 (not all real numbers). Negative weights are introduced later, but then it is a bit unclear what the range of allowable weights is. Furthermore, although negative weights can be implemented, we do not believe that the author's scheme can represent negative results of the summation (although it is hard to tell exactly how the negative weights operate). Does this matter?\n(d) The authors use the trans- version of their design to build \"convolutional\" neural networks. However, these convolutional neural networks use a stride length equal to the kernel size, which is not typically how convolutional neural networks operate (they would employ a smaller stride length). This fact seems, to us, to be a fundamental result of the way the DNA-based system works, rather than a simple choice that could be changed. The authors should therefore clarify what their system is capable of.\nSignificant Comments:\n4. It is very hard to understand the behaviour that the authors expect from their reaction networks. It would help a lot if, in the SI, the authors wrote down the ODEs that they believe their various primitives should obey, and outline the assumptions under which the steady state concentrations reach the desired values.\n5. The authors present the need to use the somewhat convoluted magnetic bead protocol in order to demonstrate multi-layer operation as a result of the general existence of cross-talk in DNA-based networks (lines 381-383). This seems to imply that it is the result of unitended reactions between partially matched toeholds, for example. However, we believe that there is a deeper issue in the specific context of the author's system, which is that their design inherently requires *the same* strands to act as inputs at multiple layers of the system. If we are correct, this is a challenge of a different type; it cannot be solved by making toeholds more orthogonal, etc. Again, the authors need to be clearer about what their system can and cannot do.\n6. We were a bit surprised that the authors didn't demonstrate the accurate functioning of a single RELU in Fig. 7, only the linear summation part. Indeed, it is not that clear which steps of the overall protocol in Fig. 7 are actually being checked. The authors are still referring to testing the linearity (eg. line 496) but surely the point is to show non-linearity?\n7. The final results in Fig. 8 are quite impressive, but they do show a fairly small differentiation in many cases, and there is no clear distinction between output levels that correspond to positive and negative results. It is unclear whether this is a failing of the DNA-based system, or of the neural net that it is implementing. In Fig S5.1 and S7, the authors present charts showing the \"accuracy of trained neural network model(s)\". We *think* (although we are not sure) that these are the in silico models that are then implemented with DNA. They seem to have a high failure rate - is their weakness what feeds through into the DNA system? Is the DNA system accurately reproducing the flawed neural net? Moreover, if the in silico neural nets frequently make mistakes, how were the 12 letters in Fig. 8 and the digits in Fig. 6 selected? Again, the authors need to be more explicit about what they have successfully shown.\n8. There is a lot of data associated with this manuscript and we would encourage the authors to upload it to a public repository with a well-constructed README so that it can be accessed and understood in perpetuity.\nMinor/technical comments:\n9. The abstract is extremely long, and contains a lot of detail/repetitive language that is probably not necessary and very hard to understand without having read the paper.\n10. The manuscript is understandable, but does need proofreading for English grammar and use of informal terms such as \"Nowadays\".\n11. Figure 1B: Why are there two identical gates? And why do D' toeholds interact with one of them, whereas D interact with the other? There seems to be some kind of mistake here. We think D' should just be D in the gate.\n12. The basic gate design is a version of the associative toehold, introduced in 2011 by Xi Chen. However, the authors call it \"allosteric toehold\" and do not cite this original paper.\n13. Figure 2 reads more like a table of contents figure than a scientific figure (as evidenced by the absence of a meaningful caption).\n14. We were unable to parse the discussion on lines 188-194.\n15. Lines 195-217 seem to just repeat much of what has already been said.\n16. Line 243 - As we understand, these experiments were performed without the Q' strands (unlike the illustration in Figure 1). But the difference here is not explained.\n17. Line 255 - Strands are introduced that can reset the inputs, but they rely on additional toeholds that are not in Fig. 1 (they are not introduced until Fig. 4, which is yet to be described at this point).\n18. Lines 295-297 - Is there a typo here? Isn't this what the authors do? Do they mean to say that they only focussed on the non-linear responses to inputs?\n19. The discussion in lines 461-486 refers to triplex (IWNTN) and quadruplex (IWNTnU-) complexes, but we can't see where these structures appear in the figures\n20. We are totally confused about how the subtraction motif is supposed to work, and how it is intended to give negative weights. See also the significant comment above re:describing the models that the authors believe their system follows.\n21. Although, the authors demonstrate discrimination on MNIST digits they only do this for two characters ('A' and 'B') and this is tested over a small number of samples (12) relative to in silico machine learning. We appreciate that this already required extensive experimental work and demonstrates an important proof-of-concept, however, it may be worth the authors discussing their confidence in the generalization of this system to more data and multiclassification of MNSIT characters.\n22. The authors should justify why image processing is useful in the context of DNA computing (apart from just being a benchmark of computational power). Convolutional neural networks were developed specifically to represent spatial features of images, but it is not clear from the manuscript how this would be useful in molecular programming applications.\n23. Many of the figures in the SI are low quality/low resolution.\n24. What value of \"a\" is used in the various experiments in the SI?\n25. It might be helpful to colour the DNA strands in the SI by domain identity", "reviewer_2": "In this manuscript, Liu, et al. demonstrate the capabilities of a DNA convolutional neural network in which inputs, weights and outputs are both continuous values. The unique molecular design of the cis- and trans- CALCUL computing units ensures operational stability and accuracy, allowing high-parallelism convolutional operations to be performed in a single reaction tube. This offers invaluable guidance for enhancing the scalability and computational accuracy of DNA neural networks in the future. Eventually, they successfully constructed a deep DNA neural network containing convolutional layers and fully connected layers to recognize 144-pixel images.\nThe paper is scientifically accurate and the authors present their findings with critical and effective analysis. The methods and the statistical analysis are clearly described. The figures and tables are informative. I suggest accepting this manuscript after they address the following concerns.\nComments\n1．For terms that may be unfamiliar to non-professional readers, such as “ConvNet”, it is recommended to provide a brief explanation when they first appear to improve the readability of the paper.\n2．The article proposes the CALCUL computing unit based on the allosteric-toehold domain, but the explanation of its operational principles and how it enhances computational stability and accuracy is insufficient. It would be beneficial if the authors could provide a more comprehensive description of the CALCUL unit's strand reaction process, specifically addressing how continuity and accuracy are maintained in the inputs, weights, and outputs. Additionally, it is important to clearly illustrate the role of allosteric-toehold domain design in this process.\n3．The article mentions that previous DNA weighted summation units have not been fully characterized in terms of computational accuracy, while CALCUL units have improved in this aspect. However, is the proof of the computational accuracy of CALCUL units rigorous? We encourage the authors to discuss in more detail how the standard curve is obtained, and to address whether the calculation accuracy based on this method is rigorous and representative.\n4．The article does not currently address the cost implications of using CALCUL units in practical applications. Given the relatively high costs associated with DNA synthesis and magnetic bead separation, we encourage the authors to include a discussion on the economic feasibility of large-scale applications. Potential cost-saving strategies, such as optimizing DNA dosage or reducing reaction steps, could also be explored.\n5．It is suggested that the author thoroughly revise the manuscript to improve the standardization of symbol usage, including but not limited to the correction of “ij” on line 328 and “ºC” on line 578.\n6．To improve the clarity and consistency of the figures, we recommend that the authors standardize the size, scale, and formatting of the graphs within Figures 3, 5, and 7. Specifically, the authors should ensure that axis labels, legends, and data points are uniformly presented across all panels, and consider using consistent color schemes and font sizes for better readability.\n7．The link provided on line 753 ('https://github.com/Xornent/dnaconv') appears to be invalid. As the training code is essential for reproducibility, the authors should update the link to ensure readers can access the necessary materials.\n8．The image recognition accuracy in Supplementary Figures S5.1 and S7 appears to be relatively low. The authors should discuss potential factors contributing to the low accuracy, such as model architecture, training data quality, or other limitations, and propose strategies for optimization.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Dear Reviewers,\nWe appreciate all of you for positive comments and constructive suggestions to our manuscript.\nWith this valuable opportunity, we have again obtained additional experimental data chart and\ncarefully and revised our manuscript according to the comments all reviewers raised. All the\nchanges were highlighted in RED in the Revised Manuscript with Highlighted Changes. The\nfollowing is the point-by-point response (R) to the reviewers’ comments (Q).\nReviewers’ Comments:\nReviewer #1\nSummary: Liu et al present a study on a new method for constructing DNA-based neural\nnetworks, based on an associative toehold mechanism. The authors demonstrate that their two\nunits (cis- and trans-) can perform weighted linear sums with a high degree of accuracy. They\nshow that these weighted summations can be \"reset\" (in that the inputs can be used again with\na new set of weights) and that multiple summations can be performed in parallel. They\ndemonstrate the possibility of using these networks to classify MNIST digits, and then extend\ntheir framework to include negative weights in the summation, non-linear thresholding and the\nability to chain together multiple layers through a (somewhat cumbersome) multistep procedure.\nWithin this framework, the authors use a two-layer network to distinguish handwritten letters\nfrom the MNIST database.\nBuilding chemical neural networks is an exciting topic and the motifs presented have promise.\nThe authors have done *a lot* of work, and extensively characterised their motifs. The\nquantitative accuracy of the linear summation is also very impressive. The paper, however, is\nhard to read and the methods/approaches are not very transparent. In addition, the claims made\nin the title/abstract/introduction are perhaps not well-justified by the results presented, and could\nbe misleading. We therefore feel that a substantial re-writing is required before the paper could\nbe published in Nature Communications.\nQ1: We do not understand how the cis- unit operates when there are multiple parallel outputs\n(eg. Fig 3b). The input species P_i are presumably shared between all weights, and so the output\nsignal depends not just on the relative amounts of Q and Q' for the output in question, but also\non how many other species the input couples to. How the system can nonetheless produce\nmultiple responses in parallel is not discussed. Indeed, the design in Figure 3b shows no Q' strands at all (just different Q) so how it relates to the basic design in Fig. 1 is unclear.\nResponse: Thank you for raising this question regarding Figure 3. As shown in the schematic\ndiagram of Figure 3B, each input strand connects to its corresponding set of weight strands\nthrough a unique A* domain, and these weight strands are then directed to specific OT-\ncomplexes via distinct D domains to achieve output pathway separation. This correspondence\nfollows a rigorous and orderly system where the suffix \"ij\" (where i=1,2,...,n; j=1,2,...,m) in each\nweight strand identifier precisely indicates its functional role in parallel\nreactions. Here, n equals the total number of input strands (all weight strands that interact with\nthe nth input strand labeled n1, n2 through nm), while m equals the number of output pathways\n(all weight strands that direct to the mth output pathway labeled 1m, 2m through nm). The final\nparallel computation results precisely match the equations presented in Figure 3B,\ndemonstrating the system's robust organization. We have revised Figure 3B to correct the\nnumbering error and have expanded the figure caption with additional explanatory details. We\nregret any confusion this may have caused.\nIn summary, our design for parallel integration of multiple allosteric-toehold strand displacement\npathways utilizes distinct D/D* domains to direct computational signals into different output\nchannels, where corresponding OT-complexes generate distinct fluorescent signals. This\nstructural design inherently enables multiple integrated reaction pathways to operate in parallel\nwhile sharing the same set of input strands.Reasonable domain design and sample\nconcentration ensure stable parallel reactions. (sample concentration settings were shown in\nsupplementary information S2)\nWe named Figure 3 \"Integration of multiple pathways of allosteric-toehold strand displacement\nreaction\" because this section solely presents preliminary experiments on the core reaction\nprinciples of this study—the cis-CALCUL units and trans-CALCUL units. Its characteristics\ninclude fewer parallel reactions, binary inputs limited to \"0\" or \"1\", and continuous weight values\nranging from \"0-1\". Consequently, the experiments do not involve the Q' or P' strands designed\nfor multiplicative operations between continuous values. By validating the computational\naccuracy and stability in this section, we directly demonstrate the feasibility of employing\nallosteric-toehold strand displacement reactions to design weighted summation elements.\nWe would like to highlight that the input strand concentration was systematically scaled\naccording to the number of output channels in our experimental design. For the binary input\nexperiments shown in Figure 3 (where inputs were limited to \"0\" or \"1\" without continuous value\nmultiplication), this scaling ensured each reaction pathway received sufficient input strands. In\nthe fully-connected network experiments demonstrated in Figure 5A, the same concentration\nscaling principle was applied. This approach worked effectively because the domain where Wij\nand W'ij strands are combined with Ii is the same, while maintaining a constant total\nconcentration of [Wij] + [Wij'] at 1.25 times the unit concentration (where unit concentration\ncorresponds to the input strand concentration entering a single output channel when the input\nvalue equals \"1\"). Through this carefully calibrated design, the input strands were automatically\ndistributed equally across all pathways, thereby achieving precise correspondence with the\npredetermined input values.(The specific concentration settings are displayed in the\nSupplementary Information, Section S4)\nAnd we would like to clarify that Figure 3 primarily presents part of our proof-of-concept\nexperimental validation, while Figure 1 serves as the actual schematic diagram corresponding to\nthe working principle of the cis-CALCUL unitsand trans-CALCUL units illustrated in Figure 4. To\nbetter address your concern, we have now included more detailed schematic diagrams in\nFigure R1, which comprehensively demonstrate the underlying mechanisms discussed in the\nmain text.\nFigure R1. Principle of allosteric toehold strand displacement reaction. The fundamental\nreactions of both cis-unit and trans-unit are allosteric toehold strand displacement reactions\ninvolving the P-strand, Q-strand, and OT complex. The distinction lies in that the cis-unit uses\nthe P strand as Inputs and the Q-strand as weights, while the trans-unit adopts the opposite\nconfiguration. The unique X domain design of this structure ensures that the reaction is both\nirreversible and leakage-free. The reaction occurs if and only when both Q and P-strands are\npresent: the Q-strand binds to the OT complex to activate the X domain, enabling the P-strand\nto successfully displace the O-strand. This reaction process eliminates the dissociative region\nused in traditional toehold strand displacement reactions to prevent leakage, making it\nthermodynamically irreversible. The binding of the Q-strand alone to the OT complex can only\nactivate the X domain, theoretically producing no leakage reactions. The P-strand alone does\nnot react with the OT complex. Conversely, when the PT complex and free O-strand coexist, a\nreverse reaction occurs to eliminate any leakage signals. This principle ensures that the system\nrequires no pre-incubation during sample loading. After direct heating and annealing, the signals\nexhibit excellent stability and accuracy. This figure has been added to the supplementary\ninformation as Figure S1.2.\nQ2. The methodology is unclear at multiple points.\nQ2a. Experimental protocols are nof described in detail. How were the actual experiments\nperformed? What volumes were used, in which device, and with what settings? Which species\nwere added and which species were initially present?\nResponse: Thanks to the suggestions you provided for adding details to the experiments, we\nhave placed the detailed experimental operation process in Figure R2 and R3. Since allosteric-\ntoehold strand displacement reactions are thermodynamically irreversible, our system can\nsignificantly enhance reaction speed and computational stability through thermal\nannealing. This also eliminates the need for pre-incubation of gate complexes or other structures\nduring sample preparation. All strands are added to the reaction tube as single strands at their\ndesignated concentrations, with no specific order of addition required.\nRegarding the volume of test samples and instrument specifications, such details have now\nbeen incorporated into the Methods section of the manuscript. We sincerely appreciate your\nfeedback in identifying these essential omissions in our original submission.\nFigure R2. Single-layer calculation experiments. The experimental procedure for single-\nlayer calculation involves only the following steps: dissolving DNA strands and measuring\ntheir concentrations, adding samples according to the concentration settings for the\nexperiment, performing thermal annealing, and monitoring fluorescence. The specific\nconcentration settings for each experiment are provided in the Supplementary Information.\nDetailed conditions for DNA dissolution and thermal annealing are described in the Methods\nsection.\nFigure R3. Calculation experiments assisted with magnetic beads. In the magnetic bead-\nassisted calculation experiments, the operational procedures and experimental design\nprinciples for the weighted summation process remain identical to those of the single-layer\ncalculation experiments. The magnetic separation operation is employed for ReLU or\nsubtraction operations, with specific procedures including: adding an appropriate amount of\nmagnetic beads, vortex incubation, and magnetic separation. The subsequent step of retaining\neither the beads or supernatant is determined by experimental requirements. When setting\nspecific loading concentrations, the unit DNA content remains consistent with upstream\ncalculations to ensure computational validity. More information about magnetic beads is\ndisplayed in the Methods section.\nQ2b. In many experiments, the fluorescence curves do not seem to have reached final steady\nstate values. See, for example, Fig 3C (but it is generally quite common, particularly for the\ncalibrations in the SI). How then was it decided when to stop experiments? They don't all seem\nto take the same amount of time. This factor is very important in establishing how well the\ncomputation match the expected value.\nResponse: Thank you for raising this critical methodological consideration. We would like to\nclarify our three guiding principles for determining the termination time of the experiments:\nFluorescence Stability: The fluorescence signal variation within 10 minutes must be less\nthan 2%.\nConsistency: Termination times are standardized for experiments of the same type within a\nsingle figure to ensure comparability.\nEfficiency: While adhering to Principle 1, we prioritize minimizing the total detection time.\nRegarding the weighted calculation process, deviations in fluorescence values (even if a\nperfect plateau phase is not fully reached) are corrected using the pre-established standard\ncurve, ensuring robust quantification accuracy.\nFor the 2-, 4-, and 8-input systems referenced in Figure 4a, the specific parameters (e.g.,\nreaction time, cycle numbers) applied to both cis- and trans-experiments are summarized in the\nfollowing table.\ntemperature Time Number of cycles\n85°C 20min\n70°C 1min 1\n55°C 1min\n37°C cycle 70 times, 40S each time\nTable R1. Annealing programme\nQ2c. Presumably the linear fluorescence responses in Eg. Fig 3a are used to convert\nfluorescence signals in experiment to output signals, but how that is done when the neural nets\nare being applied is not discussed. Indeed, there is nothing at all in the methods on fluorescence\nmeasurments. This clarification is very important, because the apparent excellent performance\nof the summation gates depend on how exactly \"Fluorescence Intensity\" is mapped to \"Output\nvalue\". Indeed, we are surprised at the degree of precision for two reasons:\ni. Typically speaking, one would expect errors in the initial concentration of stocks, and from\npippetting, to compromise the numerical accuracy of the summation.\nii. The approach seems to assume that reaction rates of inputs with their complementary\nweights are identical. However, while the strands are similar and may well have similar rates, a\ndeviation of 10% or so (perhaps due to differences in secondary structure) would hardly be\nextraordinary.\nWhether the authors take special measures to eliminate these errors prior to the experiment, or\nwhether they are effectively handled by the data processing in some way, is unclear.\nResponse: We appreciate your question, which has helped us identify the omission of critical\ninformation in the Methods section, and providing us with an opportunity to elaborate more\nthoroughly on the core principles of our study. We recognize that the original manuscript lacked\ndetails regarding fluorescence detection. We have now addressed this by adding a new\nsubsection titled 'Fluorescence Acquisition and Data Processing' to provide the necessary\nsupplemental information.\nRegarding the question about computational accuracy, we have organized our response into\nThree parts to systematically address your concerns:\nFluorescence-to-Output Signal Conversion\nThe conversion of fluorescence signals to quantitative outputs is achieved through a pre-\ncalibrated standard curve. For each weighted calculation experiment, at least three standard\ninput/weight combinations (spanning the expected summation range) are processed under\nidentical reaction conditions as the experimental groups. A scatter plot is generated by assigning\nthe weighted summation values to the x-axis and the corresponding fluorescence intensities to\nthe y-axis. The standard curve is then derived from linear regression of these data points. This\napproach ensures traceable and reproducible quantification, even for non-saturating\nfluorescence signals. The specific fitting curve and R2 value are presented in\nboth the main text and supplementary information.\nMitigation of Accuracy Limitations\nRegarding how our design achieves high computational accuracy and stability, we have\nprovided a systematic explanation in Figure 2 of the main text. Below we offer additional\nclarifications and further elaboration.\nWe acknowledge potential confounding factors such as initial strand concentrations, pipetting\nvariability, and secondary structure interference. To address these:\nOptimize DNA sequence design: We employed NUPACK to comprehensively predict and\neliminate potential secondary structures and crosstalk arising from DNA sequence design.\nMoreover, both the strand length and CG content of each specific domain are fixed, ensuring\nreaction parallelism and relatively uniform reaction kinetics.\nStandardized Protocols: Stringent procedures for strand dissolution, concentration\nmeasurement, dilution, and pipetting were implemented to minimize potential errors arising from\nhuman factors in DNA concentration quantification and sample loading.\nThermal annealing procedure: As illustrated in Figure 2 and Figure R1, The allosteric toehold\nstrand displacement reaction is thermodynamically irreversible. This means that after sufficient\nreaction with the corresponding substrate, the products will remain in a stable state rather than\na relatively unstable chemical equilibrium state. Building upon this principle, the thermal\nannealing process directly helps the reaction overcome kinetic barriers, enabling uniform and\nstable DNA strand binding while efficiently driving the system toward its stable state. This\nprotocol ensures consistent reaction kinetics and high computational fidelity. For further\nvalidation, we have included control experiments without thermal annealing in Figure R4. Under\nidentical sample loading and detection conditions, the Figure R4B group subjected to thermal\nannealing procedure demonstrated significantly improved reaction kinetics, better parallel\nreproducibility, and faster plateau attainment compared to Figure R4A group, confirming\ncomplete reaction progression.\nAdditional error-mitigation strategies are detailed in: (1) our interpretation of Figure 2 in the\nresponse to Q3a, and (2) the rationale for selecting the T-strand as the output strand in our\nresponse to Q14.\nFig. R4. Reaction results for convolutional calculation process with and without thermal\nannealing procedure. A) No annealing procedure, fluorescence is collected directly after\nsample addition. B) Annealing programs are available. The specific thermal annealing\nprocedure are shown in the Table R1.\nQ2d. Which exact experiments are performed is often unclear in the manuscript. It is very difficult\nfor the reader, given how short the captions are, to interpret what is meant by the schematics in\neg. Fig. 5, 6, 8, which we believe are meant to convey the experiment performed. What do the\nnumbers in the circle actually mean, for example?\nResponse: Thank you for your constructive feedback on Figures 5, 6, and 8. We have\nsummarized the following points to address your concerns:\n1. The numbers inside the circles in Figures 5 and 6 represent the concentration of the\ncorresponding strand in the system. For example, 41 represents 0.41X, where 1X = 100 nM.\nFigure 5 shows the results of the full connectivity experiment for the cis-CALCUL unit and the\nresults of the left-right direction slash recognition experiment for the trans-CALCUL unit, and\nFigure 6 shows the results of the digit recognition for the trans-CALCUL unit. The steps of adding\nstrands for their experiments are the same as in Fig. R2, except that they differ in the type and\nnumber of input strands, weight strands, and report strands. Please see Table S8.6.1 and Table\nS8.7.1 in the supplementary materials for specific strand sequences.\n2. We have provided additional experimental principles and details for the relevant content in\nFigure 8. The experimental principles for the subtraction experiment are detailed in\nSupplementary Material S1.3, and the experimental steps for the subtraction experiment after\nReLU are shown in Figure R3.\nQ2e. It seems like resetting is often (but not always) performed with different weights. In all\ncases this fact is not discussed, and in the SI the two separate sets of weights are not even\nprovided.\nResponse: Thank you for raising your concerns regarding the resetting process. We chose to\nuse different weight strands in the resetting process because identical weight strands would not\nadequately represent the diversity required in the resetting system. To ensure that different reset\noutcomes could be fully reflected in the resetting process, we opted for distinct weight strands\nto achieve this goal. Being able to reset and handle completely new computing situations is also\none of the important significance of the scientific problem of \"reuse\" in the field of DNA\ncomputing.\nIn addition, we have provided weight strand sequences related to the process reset in the\nsupplementary materials, specifically in Tables S8.1.1, S8.2.1, S8.3.1, S8.4.1, and S8.5.1,\nwhich show the W-r and c-W-r sequences. We presented only one set of weights because the\nreset experiments were all performed using the experimental groups that had already completed\nthe first round of computation. The results and experimental settings of these groups' initial\ncalculations have already been presented in the main text figures, and therefore were not\nrelabeled in the Supplementary Information.\nQ3: Relation between what is claimed and what is presented.\nThe authors make a number of claims in the paper that are not necessarily justified by the results\n- or at least, not without being substantially more precise about the claims.\nQ3a. positioning relative to the literature is somewhat imprecise. We think there is certainly\nnovelty within the system proposed, but it needs to be stated more accurately. The authors\ndescribe their system as the first \"fully analogue\" DNA neural network, but then go on to describe\nthe system of Genot et al, which is also analog. They criticise aspects of this work, but then they\nneed to be careful about how they position their work with respect to it.\nResponse: Thank you for your question, which gave us the opportunity to further improve the\nargument in the article. We would like to clarify and elaborate on the refined and innovative\ndesign of the CALCUL unit's molecular structure. All of these design choices were specifically\nimplemented to achieve unprecedented analog computation precision. Additionally, these\nfeatures inadvertently distinguish our CALCUL unit from previously reported weighted\nsummation tools. We have summarized these design elements in Main text Figure 2.\nFigure 2. Schematic illustration of the sources of the unprecedented computational\naccuracy of the simulation of the CALCUL unit\nThe most fundamental point is the highly concise and thermodynamically homogeneous design\nof our effective and null reaction paths. In a CALCUL cell, all P and Q-strands will enter the\nsame pair of gates/null gates. Regardless of the number of inputs and weights in the weighted\nsummation process, there is always only one gate/null gate, which makes our CALCUL cell\nthe most concise DNA tool for weighted summation. This simplicity brings two significant\nbenefits to computational accuracy: First, crosstalk between DNA strands greatly affects the\naccuracy of analog computations; the fewer strands in the system, the less crosstalk there is,\nwhich improves computational accuracy. Second, accurate weighted summation operations\nrequire that the reaction thermodynamics of each additive (i.e., the input strand) be highly\napproximate. In our CALCUL cell, all inputs assume the same strand substitution path; thus,\ntheir reaction thermodynamics are identical, which greatly contributes to the computational\naccuracy.\nIn addition to simplicity, another important molecular design is the thermodynamically identical\nnull gate. In the CALCUL unit, whether cis or trans, the thermodynamics of forming the PQT\ntriplex (the effective pathway) is absolutely equal to the thermodynamics of forming the P'QT\ntriplex (the null pathway); therefore, the weighting values will be very\naccurate. It is worth noting that in the Angew. Chem. Int. Ed. paper1 , they designed only empty\nweight strands (called matrix strands in their paper) but no empty gates (called reporters in their\npaper), so the empty paths are in fact rather different from the effective paths. This difference\nmay not lead to serious bias in simple computations (such as the simplest two-input weighted\nsummation shown in their work), but as the computations become more complex, the\ncomputational bias undoubtedly becomes larger. We also wish to point out that due to limitations\nin their molecular design, they were unable to introduce null reporters in their system, which we\nwill explain in point 2 below.\nWe have made three key designs to enhance the robustness of the involved responses.\nFirst, the toehold domain D was set to be 18 nucleotides long. Such a long length allows the\nforward reaction (i.e., the formation of the PQT, PQ'T, or P'QT triplet) to be extremely\nthermodynamically favorable, so that the conversion rate will be close to 100%, and thus the\nentire forward reaction can be robustly repeated.\nSecond, a degenerate domain X and X are incorporated into the gate, and the P-strand\nhybridizes only to domain B of the gate. Thus, domain X acts as an inhibitor, preventing leakage\nbetween the P-strand and the gate in the absence of the Q-strand. This kind of leakage is very\ncommon in strand replacement systems, and it would greatly impair the accuracy of the\nsimulation calculations.\nThird, we set the longer T-strand as the output strand to eliminate the crosstalk between gates\nand null gates. If we set the shorter O-strand as the output, which is commonly used in other\nworks 1, then the output O-strand in the gate should be fluorescently labeled, while the one in\nthe empty gate should be unlabeled. Since their sequences are identical, there will inevitably\nbe crosstalk between the O-strand and the gate, which will bias the signal and affect the\ncomputational accuracy. Here, we set the longer strands as outputs, while the shorter O-strands\nin both gates and empty gates are labeled with a bursting agent; thus, although crosstalk\nbetween the O-strands and gates still exists, the fluorescence signals do not change as if the\ncrosstalk has been eliminated.\nWe wish to point out that none of the three strategies described above were developed by Genot\net al. in their Angew paper. Specifically, as we mentioned above, they did not design null\nreporters because their outputs are shorter strands, and thus null reporters would crosstalk with\neffective reporters; the negative effects of such crosstalk may outweigh the positive effects of\nthermodynamically identical null reporters. Overall, Genot et al. did not develop any practical\nstrategy to improve the computational accuracy of their simulations, and it is no wonder that\ntheir absolute computational bias is already close to 10% in the simplest two-input weighted\nsummation operation.\nTo summarize, in fact our FIRST is used for highly accurate positive weighted summation and\nnot for DNA neural networks, and we are not exaggerating on this point, but given that such a\ndescription would still be controversial, we decided to change it to NEWEST.\nREFERENCES:\nApplication to Matrix Multiplication and Weighted Sums. Angewandte Chemie\nInternational Edition 52 (2013).\nQ3b. The authors describe their system as the \"first deep\" DNA-based neural network - but what\ndoes \"deep\" mean in this context? They demonstrate two layers, and their setup doesn't in fact\nnaturally allow for multiple layers. Instead, it is necessary to use a convoluted intervening step\ninvolving magnetic beads. This is still an achievement, but the authors should be sure not to\nmislead readers with the claims they present in the introduction and abstract.\nResponse: Thank you for raising your concerns regarding the depth of the convolutional\nexperiment. We would like to clarify that our convolutional process consists of three layers:\nConvolutional Layer I, Convolutional Layer II, and a fully connected layer (see Figure 8E). In the\nfield of DNA neural networks, the understanding of convolutional depth is still evolving. If we\nwere to follow the standards proposed by some researchers2, our convolutional process would\nactually achieve a depth of five layers (weighted summation + RELU + weighted summation +\nRELU + weighted summation). However, according to the standard definition of neural\nnetworks, the information processing following the weighted summation calculation, whether it\nis simple addition and subtraction or complex RELU operations, is merged with the weighted\nsummation and considered as one layer. Therefore, our convolutional process is fully three\nlayers deep, whereas previous related literature only reported single-layer convolutional\nprocesses. Given this comparison, we claim that our work represents the \"first deep\" DNA-\nbased neural network.\nREFERENCES:\nNat Mach Intell 4, 625-+ (2022).\nQ3c. The authors should be precise about what their summation units can do - initially, they can\nonly implement weights in the range 0<=w<=1 (not all real numbers). Negative weights are\nintroduced later, but then it is a bit unclear what the range of allowable weights is. Furthermore,\nalthough negative weights can be implemented, we do not believe that the author's scheme can\nrepresent negative results of the summation (although it is hard to tell exactly how the negative\nweights operate). Does this matter?\nResponse: Thank you for your discussion on the functional scope of the summation unit. In\nsummation operations, we introduce negative weights solely to achieve effective weight\ndistribution, Expand the range of weight settings to -1<=w<=1. Regarding the ‘negative sum\na ( x b ), if x b\n(b is a positive number). In multi-layer\nwith its expression given by: y 0 , if x b\nresult,’ this is related to the ReLU process. Typically, the ReLU computation is defined as (Y = X\n(X ≥ 0), 0 (X < 0)), while the ReLU computation implemented in this study is more complex,\nneural networks, expressing negative output values from weighted summation has limited\npractical utility, as these negative signals would not be propagated downstream after ReLU\ncalculations.\nQ3d. The authors use the trans- version of their design to build \"convolutional\" neural networks.\nHowever, these convolutional neural networks use a stride length equal to the kernel size, which\nis not typically how convolutional neural networks operate (they would employ a smaller stride\nlength). ? This fact seems, to us, to be a fundamental result of the way the DNA-based system\nworks, rather than a simple choice that could be changed. The authors should therefore clarify\nwhat their system is capable of.\nResponse: Thank you for your in-depth consideration of the convolutional computing structure.\nIn computer-based convolutional neural network models, the stride length is typically freely\nadjustable, which helps enhance machine learning efficiency and performance. However, when\nconstructing DNA neural networks, the constraints of DNA reactions compel us to prioritize\nmaintaining simplicity in reaction quantity, DNA strand complexity, and structural design while\nachieving computational functionality. We must acknowledge that implementing freely\nconfigurable stride lengths in DNA neural networks remains highly challenging at present. After\ncomprehensive consideration, we therefore selected a specific stride length setting that allows\nthe convolutional kernel to precisely cover each input once. This design approach also\nreferences published work on DNA convolutional neural networks2, which similarly utilized a\nstride equivalent to the kernel size.\nREFERENCES:\nNature Machine Intelligence 2022 4:7 4 (2022-07-04).\nQ4. It is very hard to understand the behaviour that the authors expect from their reaction\nnetworks. It would help a lot if, in the SI, the authors wrote down the ODEs that they believe\ntheir various primitives should obey, and outline the assumptions under which the steady state\nconcentrations reach the desired values.\nResponse: Thank you for your insightful comments regarding the reaction process. We\nacknowledge that your concerns may stem from an incomplete understanding of the\nthermodynamic design underlying the experimental reaction process. The heating and annealing\nprocess is not a simple reaction kinetics process occurring at room temperature, but rather a\nrapid thermodynamic process. Therefore, we are unable to obtain the initial ODEs.\nIn the design of the gate strands, as previously addressed, we have elongated the X region,\nmaking the thermodynamics extremely favorable. Once the kinetic barriers are removed through\nheating and annealing, the strand displacement reaction rates in our system become very close\nto 100% and remain stable. This represents the core innovation of our work, and all our\nexperimental results have validated our design and hypothesis.\nQ5. The authors present the need to use the somewhat convoluted magnetic bead protocol in\norder to demonstrate multi-layer operation as a result of the general existence of cross-talk in\nDNA-based networks (lines 381-383). This seems to imply that it is the result of unitended\nreactions between partially matched toeholds, for example. However, we believe that there is a\ndeeper issue in the specific context of the author's system, which is that their design inherently\nrequires *the same* strands to act as inputs at multiple layers of the system. If we are correct, this\nis a challenge of a different type; it cannot be solved by making toeholds more orthogonal, etc.\nAgain, the authors need to be clearer about what their system can and cannot do.\nResponse: Thank you for your insightful question regarding the design of shared strands. We\nhope Figure R5 can clarify how multi-layer CALCUL is designed and reacts without bead-\nassisted conditions. When examining Figure R5 alongside Figure R1, it becomes evident that\nsimply extending the 3' end of the original O-strand with new domains to serve as input strands\nfor downstream reactions can fulfill multi-layer CALCUL operations.\nIn the cis-unit, the P-strand (domains A* and B) acts as the input strand, requiring the lower-\nlevel input strand to adopt a similar structure. Thus, the original O-strand can be redesigned as\na P2-strand (domains X1, B, and C), where the B and C domains functionally serve as\ndownstream input for the new allosteric toehold strand displacement reaction. Here, the B\ndomain assumes the role of the original A* domain, while the C domain replaces the original B\ndomain. Similarly, in the trans-unit, the original O-strand can be redesigned as a Q2-strand\n(domains X1, B, X2, and C), with the B, X2, and C domains acting as downstream inputs. The\nB domain replaces the original D domain, the X2 domain substitutes the original X1 domain,\nand the C domain takes over the function of the original A domain.\nKinetically, the single-stranded overhangs of the P2T-complex and Q2T-complex do not\ndirectly react with downstream gate complexes. Leakage reactions only occur when the B\ndomain becomes transiently exposed due to strand breathing and the downstream input forms\na Q2P2 complex with the weight strand. Notably, the X1 domain significantly suppresses such\nleakage.\nThermodynamically, the most significant change is that we can no longer employ immediate\nheating-annealing after sample loading to accelerate reactions, as this would cause the terminal\ngate strands to react uncontrollably without upstream signal regulation. Consequently, the\nloading protocol must adopt traditional toehold DNA computing methods—pre-incubating gate\ncomplexes and maintaining isothermal conditions. This will significantly prolong reaction time,\nreduce computational accuracy, and impose stricter requirements on sequence design and\npreliminary strand screening.\nIn summary, multi-layer CALCUL units remain computationally feasible without magnetic bead\nassistance, eliminating the need for the same strands to act as inputs across multiple layers.\nHowever, we must acknowledge that this design compromises the simplicity of strand reactions\nand the stability and accuracy gained from heating-annealing protocols. These considerations\noriginally motivated our decision to introduce magnetic beads into the\nmulti-layer CALCUL system. We appreciate your question for prompting us to revisit and\nsystematically articulate these insights.\nFigure R5. Molecular design of muti-layer CALCUL units without magnetic bead-assisted.\nQ6. We were a bit surprised that the authors didn't demonstrate the accurate functioning ? of a\nsingle RELU in Fig. 7, only the linear summation part. Indeed, it is not that clear which steps of\nthe overall protocol in Fig. 7 are actually being checked. The authors are still referring to testing\nthe linearity (eg. line 496) but surely the point is to show non-linearity?\nResponse: Thank you for your question regarding Figure 7. In this figure, we aimed to\ndemonstrate the computational performance of the ReLU function in both the cis-CALCUL and\ntrans-CALCUL units. We would like to clarify that the data presented in Figure 7 represent not\nthe intermediate weighted summation results but the final outputs after the complete ReLU\ncomputation. Specifically, if we denote the weighted summation output as O, the final computed\nvalue Y follows the ReLU equation: Y = O - b,if O ≥ b, and Y = 0, if O < b.\nFor the cis-CALCUL unit in Figure 7A, we set the threshold b at 0.20. Both the standard curve\nand computational detection results underwent this ReLU operation. The standard curve values\nof 0.00, 0.80, 1.80, and 3.80 were obtained by applying the ReLU function to the corresponding\nweighted summation outputs (O) of 0.00, 1.00, 2.00, and 4.00, respectively. Similarly, the\nweighted summation result of Pi and Qi was 3.00 in computational testing, which after ReLU\nprocessing yielded the theoretical value of 2.80. The trans-CALCUL unit followed an identical\ncomputational framework, with the only difference being the adjustment of the threshold b to\n0.30.\nThese experimental standard curves and calculated values collectively provide a\ncomprehensive validation of the ReLU equation's computational results. Therefore, we did not\nconduct additional experiments specifically targeting nonlinearity verification. We appreciate this\nopportunity to clarify our methodology and hope this explanation adequately addresses your\nconcerns regarding Figure 7.\nQ7. The final results in Fig. 8 are quite impressive, but they do show a fairly small differentiation\nin many cases, and there is no clear distinction between output levels that correspond to positive\nand negative results. It is unclear whether this is a failing of the DNA-based system, or of the\nneural net that it is implementing. In Fig S5.1 and S7, the authors present charts showing the\n\"accuracy of trained neural network model(s)\". We *think* (although we are not sure) that these are the in\nsilico models that are then implemented with DNA. They seem to have a high failure rate - is their\nweakness what feeds through into the DNA system? Is the DNA system accurately reproducing the\nflawed neural net? Moreover, if the in silico neural nets frequently make mistakes, how were\nthe 12 letters in Fig. 8 and the digits in Fig. 6 selected? Again, the authors need to be more\nexplicit about what they have successfully shown.\nResponse: Thank you for raising the question regarding the accuracy of the neural network\nmodel. To maintain the relative simplicity of the experimental system, we employed a simplified,\nshallow network that, while smaller in scale, incorporates all the functionalities we were able to\nimplement and those essential for a convolutional neural network (CNN). Specifically, the letter\nrecognition network shown in Figure 8 utilized only 23 parameters, while the digit recognition\nnetwork shown in Figure 6 used merely 16 parameters. For handwritten letter recognition tasks,\nit is evident that a CNN with fewer than 30 parameters and at most two layers would not perform\nwell in terms of overall recognition accuracy. Typically, CNN models that achieve good\nperformance on the EMNIST dataset require thousands of parameters, which are currently\nchallenging to implement in our neural network framework.\nHowever, our focus in this study is not on the recognition performance of the model itself, but\nrather on whether our chemical system can precisely replicate the results obtained in silico. To\naddress this, we selected several examples that the in-silico model could correctly distinguish\n(e.g., digits 8 and 9, and letters A and B) and conducted multiple chemical experiments. By\nanalyzing the results, we demonstrated that our chemical system is capable of accurately\nreproducing the outcomes predicted by the model.\nWe hope this clarification addresses your concerns. Please let us know if further adjustments\nare needed.\nQ8. There is a lot of data associated with this manuscript and we would encourage the authors\nto upload it to a public repository with a well-constructed README so that it can be accessed\nand understood in perpetuity.\nResponse: We thank the reviewer for their suggestion. The complete annotated dataset has\nbeen provided to the editorial office as supplementary material, ensuring full transparency and\nreproducibility. We believe this fulfills the journal's data-sharing requirements.\nMinor/technical comments:\nQ9. The abstract is extremely long, and contains a lot of detail/repetitive language that is\nprobably not necessary and very hard to understand without having read the paper.\nResponse: Thank you for your comments on the abstract section, we have streamlined the\nlanguage and re-done the abstract section.\nQ0. The manuscript is understandable, but does need proofreading for English grammar and\nuse of informal terms such as \"Nowadays\".\nResponse: Thank you for your suggestions regarding the formatting issues in the manuscript.\nWe have thoroughly reviewed and revised the entire text to ensure it meets the required\nstandards. The formatting has been adjusted throughout the document to address the concerns\nraised.\nQ1. Figure 1B: Why are there two identical gates? And why do D' toeholds interact with one\nof them, whereas D interact with the other? There seems to be some kind of mistake here. We\nthink D' should just be D in the gate.\nResponse: Thank you for your meticulous observation and rigorous logical thinking. We\nacknowledge the inaccuracy in the schematic representation in Figure 1B. The \"D' domain\" in\nthe P'Q-complex should indeed be replaced with the blue \"D domain\" throughout, and this\ncorrection has been implemented in the revised Figure 1. Regarding the trans-CALCUL units,\ntheir design for effective and ineffective weights strands is indeed unique: the P and P' strands\nactually share identical DNA sequences, differing only in the 3' terminal modification (quencher\ngroup present in P' but absent in P). This deliberately simple and uniform design ensures precise\nexecution of the weight values according to the [P]/([P]+[P']) ratio. We should clarify that the two\nOT-gate complexes in Figure 1B are identical in structure; they are shown separately only to\ndemonstrate the two distinct reaction pathways with QP and QP'.\nWe sincerely appreciate your careful observation that helped us identify this oversight.\nQ2. The basic gate design is a version of the associative toehold, introduced in 2011 by Xi\nChen. However, the authors call it \"allosteric toehold\" and do not cite this original paper.\nResponse: We sincerely appreciate your thoughtful question regarding our implementation of\nallosteric toeholds. In our design, we incorporated allosteric toeholds to enable X/X allosteric\nmodification of the substrate (as illustrated in Figure R1). This strategic approach serves two\ncritical purposes: (1) the allosteric domain effectively prevents spurious interactions between\nthe P-strand and gate strand when the input strand Q is absent, and (2) it significantly mitigates\nthe accuracy reduction commonly observed in conventional strand displacement systems\n(demonstrated in Figure 2 of the main text). For a more detailed discussion of the specific\nadvantages of this architecture, we kindly refer you to our response to Question 3, where we\nelaborate further on the structural rationale.\nIn fact, this study was inspired by our group's 2021 work on clip toehold3 (Ref. 10 in the main\ntext), which served as the foundation for our current innovative design and applications.\nTo clarify, the allosteric toehold mechanism in our design is fundamentally distinct from the\nassociative toehold, primarily due to the inclusion of the X-domain (allosteric module). This\nstructural innovation serves as a critical contributor to the gate’s computational accuracy,\nwhich is why we explicitly term it \"allosteric toehold.\"\nWe recognize that the allosteric toehold may exhibit superficial similarities to associative\ntoehold4 systems in appearance. To provide proper context and acknowledge prior work,\nwe have included a citation to this reference in the revised manuscript.\nREFERENCES:\nand Construction of Complex DNA Nanodevices. ACS Nano 15, 11573-11584 (2021).\nAm Chem Soc 134, 263-271 (2012).\nQ3. Figure 2 reads more like a table of contents figure than a scientific figure (as evidenced\nby the absence of a meaningful caption).\nResponse: We sincerely appreciate your valuable suggestion regarding Figure 2, which plays\na pivotal role in demonstrating the experimental principles and highlighting the core innovations\nof our study, as well as elucidating the system's operational mechanism. In consideration of the\nexceptional computational accuracy achieved in this work, we have incorporated an additional\ndetailed schematic (Figure R1) to provide readers with enhanced clarity regarding the specific\nreaction processes involved.\nQ4. We were unable to parse the discussion on lines 188-194.\nResponse: We sincerely appreciate your thorough and insightful comments regarding the\nexperimental principles. To ensure clarity, we will now provide a more detailed explanation of\nthe concepts presented in this paragraph.\n\"Notably, using the longer T-strand as output not only reduces crosstalk but also aids annealing,\nas annealing can accelerate crosstalk, but our design prevents it from affecting the output\nsignal.The common approach of using the shorter O-strand as output is not suitable for\nannealing and cannot achieve precise analogue computing.\"\nIn traditional DNA gate designs for molecular computing, the OT-complex conventionally uses\nthe short strand O as the output signal strand with a 3'-fluorophore modification, while the longer\nstrand T bearing a 5'-quencher typically remains uninvolved in signal output. Our CALCUL unit\ndesign intentionally inverts this architecture by modifying the short O strand with a 3'-quencher\nand the long T strand with a 5'-fluorophore. This strategic reversal allows us to introduce the O\nstrand in slight excess (specifically at 1.05 times the total concentration of T and T' strands, as\ndocumented in the Supplementary Information), which provides multiple experimental\nadvantages without interfering with the primary reaction.\nThe modest excess of O strand effectively quenches fluorescence fluctuations arising from\nconcentration measurement inaccuracies and pipetting variations, thereby enhancing both the\nreliability and precision of our computational results. Furthermore, our experimental\nobservations revealed that systems using short O strands as inputs exhibited significantly\ndegraded performance in both standard curve linearity and computational accuracy following\nthermal annealing procedures. We attribute this phenomenon to the increased susceptibility of\nshort O strands to random dissociation and crosstalk during elevated temperature cycles - an\nissue that is effectively circumvented by our alternative design employing the O strand as the\nquenching component.\nWe have conducted systematic experiments to validate this design rationale, with the\ncorresponding methodology and results presented in Supplementary Information Figure\nS3.2.1. We acknowledge that the original version of the manuscript did not discuss these\nvalidation data in this particular section, an omission that may have created interpretative\nchallenges. To address this, we have incorporated these essential experimental findings in the\nrevised manuscript to ensure comprehensive understanding.\nQ5. Lines 195-217 seem to just repeat much of what has already been said.\nResponse: We appreciate your constructive suggestions regarding the textual presentation.\nWe recognize that this section did contain excessive technical details and redundant\ndescriptions. In the revised manuscript, we have substantially streamlined this content by\nremoving repetitive elements and presenting a more concise yet comprehensive summary of\nthese findings.\nQ6. Line 243 - As we understand, these experiments were performed without the Q' strands\n(unlike the illustration in Figure 1). But the difference here is not explained.\nResponse: Thank you for your further question regarding the Q’-strands. We apologize for not\nproviding a more detailed explanation of the Q’-strands in the manuscript. Similar to the first\nquestion you raised, Figure 3 only represents a part of the proof-of-concept experiment, while\nFigure 1 serves as the schematic for Figure 4. We have addressed this matter in both our\nresponses to Q1 and Q11, which we hope will satisfactorily resolve your inquiry.\nWe hope this additional information clarifies the role and application of the Q’-strands in our\nstudy. Please let us know if further details are required.\nQ7. Line 255 - Strands are introduced that can reset the inputs, but they rely on additional\ntoeholds that are not in Fig. 1 (they are not introduced until Fig. 4, which is yet to be described\nat this point).\nResponse: Thank you for your question about the reset part. In the reset experiment, we added\na reset toehold to the weight strand solely to achieve the reuse functionality through strand\nreplacement. During the entire reuse process, only the weight strand is reset.\nAdditionally, the inclusion of the reset toehold region does not affect the accuracy of the\nconvolution calculations. To maintain clarity in illustrating the core principles, we intentionally\nomitted the R domain from schematic diagrams where strand reuse was not involved.\nFurthermore, in experiments that did not require testing reuse functionality, we deliberately\nexcluded the R domain from the sequence design.\nQ8. Lines 295-297 - Is there a typo here? Isn't this what the authors do? Do they mean to say\nthat they only focussed on the non-linear responses to inputs?\nResponse: The original intention of this passage was to emphasize that our study not only\nexamines linear reactions and classification outcomes, but also imposes rigorous requirements\non computational accuracy. We sincerely apologize for the linguistic ambiguity that caused\nconfusion during your review. To improve readability, we have revised the statement to:\n'We would like to note that for the previously reported DNA weighted summation tools,\nresearchers never thoroughly characterized their computing accuracies but only focus on their\n on their\nqualitative results in classifications, which, in our perspective, was a compromised and indirect\ndemonstration of the functionality of a weighted summation tool.'\nQ9. The discussion in lines 461-486 refers to triplex (IWNTN) and quadruplex (IWNTnU-) complexes,\nbut we can't see where these structures appear in the figures\nResponse: We appreciate the opportunity to clarify our naming conventions for the molecular\ncomponents in this study. Structurally, we designated strands containing A* and B domains as\nP-strands, while those composed of D, X, and A domains were termed Q-strands. Functionally,\nwe abbreviated input strands as I-strands and weight strands as W-strands for descriptive\nconvenience.\nSpecifically, in cis-CALCUL experiments, the P-strands functions as the input strands (I) with Q\nserving as the weight -strands (W), whereas this relationship is reversed in trans-CALCUL\nexperiments where P acts as the weight strands (W) and Q as the input strands (I).\nA similar description has been included in the corresponding section of the main text:\n'For the convenience of expression, IWT is uniformly used to represent PQT triplex in the cis-\nCALCUL mode and QPT triplex in the trans-CALCUL mode, and IWNTN is uniformly used to\nrepresent PQNTN triplex in the cis-CALCUR mode and QPNTN triplex in the trans-CALCUL\nmode.'\nWe sincerely apologize for any confusion caused by this nomenclature system.\nQ10. We are totally confused about how the subtraction motif is supposed to work, and how it\nis intended to give negative weights. See also the significant comment above redescribing the\nmodels that the authors believe their system follows.\nResponse: Thank you for your suggestion. We have further explained the principle of the\nsubtraction experiment in Figure R6.\nOur subtraction experiment model is designed for winner-take-all computation. Winner-take-all\ncomputation is one of the simplest competitive neural network models5, inspired by the lateral\ninhibition and competition observed in biological neurons in the brain. This design not only\nfulfills the experimental purpose of our subtraction experiment but also enhances the\ncomputational capabilities of our neural network.\nFig. R6. The principle of subtraction experiment assisted by magnetic beads. The core\ncomponent for subtraction operation is an annihilator formed by the U+U- complex. The U+U-\ncomplex remains inert when encountering either O+ or O− strands individually. However, it\nspecifically reacts with equal concentrations of O+ and O− strands through strand\ndisplacement, producing inert waste complexes. After complete reaction with a certain\nconcentration of O+ and O− strands, and the waste complex is removed via magnetic bead\nseparation. Subsequent addition of the O+ responsive Reporter enables detection of the\nsubtraction result, as shown in the equation in the figure.This figure has been added to the\nsupplementary information as Figure S1.2.\nREFERENCES:\nbased\nwinner-take-all neural networks. Nature 559, 370-376 (2018).\nQ21. Although, the authors demonstrate discrimination on MNIST digits they only do this for two\ncharacters ('A' and 'B') and this is tested over a small number of samples (12) relative to in silico\nmachine learning. We appreciate that this already required extensive experimental work and\ndemonstrates an important proof-of-concept, however, it may be worth the authors discussing\ntheir confidence in the generalization of this system to more data and multiclassification of\nMNSIT characters.\nResponse: Thank you for your concern regarding the system's ability to generalize over more\ndata and multiple classification tasks. In this article, we only picked a small subset of samples\nfrom a relatively easy bifurcation classification task. However, with the data we acquired on this\nsmall benchmarking set, we can show that our system design is capable of accomplishing more\ngeneral and difficult task that rely on a larger network.\nWe first measure the mean absolute errors produced by our summation units, with the data\ngenerated in the previous experiments, and find two properties of the errors:\n(1) The errors are independent of network depth, and will not accumulate through multiple\nrounds of calculation pipeline.\n(2) The errors are linearly correlated with the number of weights that participated in a single\nround of summation, and are not correlated with the exact values of the weights.\n(3) Absolute errors are normally distributed.\nWe quantify the errors and find out that they scatter normally around a small value (mean -\n0.0036, standard variance 0.0158 per weight). At the first glance, this is small enough and\nshould be tolerable for weights and inputs that are mainly between 0 and 1.\nIn this study, the networks we adopt are rather small and have few parameters. This directly\nresult in the poor performance even in-silico. One would not expect high performance on MNIST\ndataset with only less than 20 parameters in a network model. So, to prove the efficacy and\ngeneralizability of our computing system, we decide to construct a greater model that achieve\nbetter accuracy on the full 10-class MNIST classification problem(Table R2). By taking the\nabsolute errors measured in real experiments into consideration, we simulate the same normally\ndistributed random error and propagate them into the prediction task. We believe that this will\nyield an estimate of the accuracy if we run the model in our chemical system.\nFor the model simulated, we add more parallel convolutions and build them with greater\nparameter size. The model architecture is as follows:\nLayer Input shape Output shape # Parameters\nConv2d.1 [N, 1, 12, 12] [N, 5, 5, 5] 85\nReLU.1 [N, 5, 5, 5]\nConv2d.2 [N, 5, 3, 3] 230\nReLU.2 [N, 5, 3, 3]\nFlatten [N, 45]\nLinear.1 [N, 20] 920\nLinear.2 [N, 10] 210\nTotal [N, 1, 12, 12] [N, 10] 1445\nFigure R2. Adjustment of parameters related to the structure of computer-simulated\nnetworks\nAt epoch 900, the overall accuracy of the test set reaches 95.2%, and the accuracy of each\ncategory ranges from 91.8% to 98.2% (e.g., category 0: 98.16%, category 9: 91.87%), which\nindicates that the model is capable of distinguishing between 10 categories of characters. By\nepoch 900, we do not observe much difference between the training and testing set, indicating\nno over-fitting of the model.\nWhen we incorporate the errors into every round of weighted summation (-0.0036 +- 0.0158,\nmean +- sd), the accuracy of prediction on test set decreased to 74.8%, This result reveals the\nimpact of the actual experimental environment (e.g., signal noise, device fluctuations) on the\nsystem. However, such results are acceptable for such complex system (involving 1445\nparameters and 4 layers of neural network) that runs in a test tube workflow.\nSimulated accuracy after\nMNSIT class Computer accuracy incorporating experimental\nerror\n0 98.1633% 88.2653%\n1 98.0617% 61.5859%\n2 95.2519% 71.7054%\n3 94.8515% 71.6832%\n4 94.8065% 84.3177%\n5 92.8251% 73.3184%\n6 97.2860% 88.7265%\n7 95.6226% 70.3307%\n8 92.7105% 69.6099%\n9 91.8731% 71.3578%\nMean 95.2000% 74.8000%\nTable R3.Computer simulation accuracy and accuracy of simulation considering\nexperimental error when ep = 900.\nQ22. The authors should justify why image processing is useful in the context of DNA computing\n(apart from just being a benchmark of computational power). Convolutional neural networks\nwere developed specifically to represent spatial features of images, but it is not clear from the\nmanuscript how this would be useful in molecular programming applications.\nResponse: Thank you for your insightful discussion on the specific applications of DNA\ncomputing in image processing. DNA computing offers several unique advantages in this field:\nParallel Processing: DNA molecules can simultaneously perform a vast number of chemical\nreactions, making them inherently suitable for parallel operations in image processing. For\nexample, the sliding of convolutional kernels and weight-sharing mechanisms in convolutional\nneural networks (ConvNets) can be efficiently implemented through parallel DNA reactions.\nMolecular-Scale Pattern Recognition: DNA molecules can mimic the pattern recognition\ncapabilities of biological neural networks through specific interactions. For instance, DNA strand\ndisplacement reactions can simulate the multiplication and summation of weights between local\nreceptive fields of input images and convolutional kernels.\nFault Tolerance: DNA reactions exhibit inherent fault tolerance, enabling the processing of noisy\nimage data. Even when input images are similar, DNA circuits can accurately classify them.\nBiocompatibility: The molecular nature of DNA computing allows it to interface directly with\nbiological systems. This feature holds great potential for future applications in biomedical image\nanalysis, such as molecular marker recognition in disease diagnosis.\nImage processing not only serves as a testbed for validating DNA computing capabilities but\nalso highlights its unique advantages. The parallelism, molecular-level analog capabilities, noise\nresistance, and biocompatibility of DNA computing make it promising for tasks such as pattern\nrecognition and medical image analysis, paving the way for the development of future molecular\nintelligent systems.\nMany related studies are currently exploring this field (e.g., DNA-based biocomputing circuits\nfor sensing and processing biomolecular information, and DNA circuits for real-time computing\nand control inside living cells). Building on these efforts, we continue to develop new\nfoundational tools and aim to demonstrate their functionality in image recognition.\nWe hope this explanation clarifies the potential applications of DNA computing in image\nprocessing. Please let us know if further details are needed.\nQ23. Many of the figures in the SI are low quality/low resolution.\nResponse: Thanks to your suggestion, we have replaced the low-definition images in the\nsupplement with high-definition images.\nQ24. What value of \"a\" is used in the various experiments in the SI?\nResponse: Thank you for your question. In the supplementary material diagram, a represents\nthe values taken by different W in the matrix below. For example, in the supplementary material\nFig. S2.1.2, a represents the concentrations of the four weight strands in the system as 0.36X,\n0.60X, 0.48X, and 0.36X, respectively.\nQ25. It might be helpful to colour the DNA strands in the SI by domain identity\nResponse: Thanks to your suggestion, we have color swapped the DNA in the supplemental\nmaterial to more clearly distinguish the domain structure.\nReviewer #2 (Remarks to the Author):\nIn this manuscript, Liu, et al. demonstrate the capabilities of a DNA convolutional neural network\nin which inputs, weights and outputs are both continuous values. The unique molecular design\nof the cis- and trans- CALCUL computing units ensures operational stability and accuracy,\nallowing high-parallelism convolutional operations to be performed in a single\nreaction tube. This offers invaluable guidance for enhancing the scalability and computational\naccuracy of DNA neural networks in the future. Eventually, they successfully constructed a deep\nDNA neural network containing convolutional layers and fully connected layers to recognize\n144-pixel images.\nThe paper is scientifically accurate and the authors present their findings with critical and\neffective analysis. The methods and the statistical analysis are clearly described. The figures\nand tables are informative. I suggest accepting this manuscript after they address the following\nconcerns.\nComments\nQ1.For terms that may be unfamiliar to non-professional readers, such as “ConvNet”, it is\nrecommended to provide a brief explanation when they first appear to improve the readability\nof the paper.\nResponse: Thank you for your suggestion. We have provided more detailed explanations for\nthe technical terms that appear for the first time in the original text.\nQ2. The article proposes the CALCUL computing unit based on the allosteric-toehold domain,\nbut the explanation of its operational principles and how it enhances computational stability and\naccuracy is insufficient. It would be beneficial if the authors could provide a more comprehensive\ndescription of the CALCUL unit's strand reaction process, specifically addressing how continuity\nand accuracy are maintained in the inputs, weights, and outputs. Additionally, it is important to\nclearly illustrate the role of allosteric-toehold domain design in this process.\nResponse: We sincerely appreciate both Reviewer 1 and Reviewer 2 for raising this important\nquestion regarding computational accuracy. As detailed in our response to Reviewer 1 (Q2c,\nQ3a and Q14) and systematically illustrated in Figure R1 and Figure 2 of the main text, we\nhave implemented multiple design strategies to ensure high precision. For clarity, we summarize\nthe key points here:\nWe attribute the significant improvement in computational precision to three design features and\none reaction advantage:\n1. Design Features\n(1) Optimized Toehold Domain\nAn 18-nt toehold domain (D) was implemented to maximize forward reaction efficiency,\nachieving near-complete conversion and high reproducibility.\n(2) Leakage Prevention\nThe *X/X** allosteric domain in the gate complex eliminates spurious interactions between the\nP-strand and gate in the absence of the Q-strand, addressing a key accuracy-limiting factor in\nstrand displacement systems.\n(3) Crosstalk Mitigation\nDesignating the longer T-strand as the output strand ensures fluorescence signal stability\nmitigate potential crosstalk between functional and void gates, preserving computational\nfidelity.\n2. Reaction Advantage\nThermal Annealing Compatibility\nThe allosteric toehold strand displacement reaction is thermodynamically irreversible, enabling\nour system to achieve rapid equilibrium through thermal annealing. By mixing all DNA strands\nin a single step and applying thermal annealing, the reaction reaches completion within minutes\nwhile eliminating kinetic traps - a process that simultaneously maintains computational accuracy\nand stability.\nThese synergistic features collectively enable the demonstrated high-precision performance\n(The validation data are shown in Figure 4D).\nAdditionally, in response to Reviewer 2 s emphasis on achieving continuous computational\nvalues, we would like to supplement the explanation with Figure 1 in the main text. The core\nprinciple enabling continuous-value multiplication operation in our study relies on mixing\neffective and ineffective weight strands, allowing input strands to partition into different reaction\npathways according to the concentration ratios of weight strands. Here, the effective pathway\ngenerates fluorescent signal output, while the ineffective pathway merely consumes substrate\nwithout producing output (in multi-layer neural network construction, fluorescent signals are\nreplaced by domain signals transmitted through biotin modification and magnetic bead\nseparation).\nIn cis-CALCUL units, pathway partitioning is achieved through combinations of:\nEffective components: Q strands with OT-complex gates\nIneffective components: Q strands with OT -complex gates\nThese pathways feature distinct structural characteristics:\nEffective pathway: Shares identical D/D* domains with 5 -fluorophore modification on T-strand\nIneffective pathway: Shares identical D /D * domains without 5 -end modification\nFor trans-CALCUL units, pathway partitioning requires only:\nEffective P strands (unmodified 3 end)\nIneffective P strands (3 -quencher modification)\nDespite their identical DNA sequences, P and P strands achieve functional differentiation solely\nthrough this terminal modification, while sharing the same OT-complex.\nAs demonstrated in Figure 1, stable parallel reactions producing fixed product ratios emerge\nwhen specific concentrations of P, Q, O, and T strands are introduced. This reaction framework\nserves as the foundation for our weighted summation computational units.\nFig. 1. The working principles of the cis-CALCUL and trans-CALCUL units. A) Schematic\nillustration of the weighted summation operation of the cis-CALCUL unit. Multiplexed CAT\nreactions are undergoing within one tube, realizing “real” weighted summation operation with\nthe inputs, weights and outputs all being continuous and accurate values. B) Schematic\nillustration of the weighted summation operation of the trans-CALCUL unit. Similarly, based on\nMultiplexed CAT reactions, but with different roles played by the domains compared to cis-\nCALCUL units, which helps it to be more suitable for convolution operations.\nQ3. The article mentions that previous DNA weighted summation units have not been fully\ncharacterized in terms of computational accuracy, while CALCUL units have improved in this\naspect. However, is the proof of the computational accuracy of CALCUL units rigorous? We\nencourage the authors to discuss in more detail how the standard curve is obtained, and to\naddress whether the calculation accuracy based on this method is rigorous and representative.\nResponse: Our standard curve is obtained by setting different input values and corresponding\nweight values, such that the product of each pair forms an arithmetic sequence, e.g., 0, 0.3, 0.6,\n0.9 (see Supplementary Material Fig. S2.1.2). We then perform convolution experiments to\nobtain the corresponding fluorescence values for each calculated value. Finally, we plot the\ncalculated values and fluorescence values in Excel as a scatter plot, add a trend line, and\ncalculate R2 to obtain the corresponding standard curve.\nQ4. The article does not currently address the cost implications of using CALCUL units in\npractical applications. Given the relatively high costs associated with DNA synthesis and\nmagnetic bead separation, we encourage the authors to include a discussion on the economic\nfeasibility of large-scale applications. Potential cost-saving strategies, such as optimizing DNA\ndosage or reducing reaction steps, could also be explored.\nResponse: Thank you for your concern about the issue of computational costs. We have\ncompiled a detailed breakdown of the costs for this experiment, using the two-layer\nconvolutional network based on magnetic bead assisted trans-CALCUL units experiment as an\nexample(Figure 8D), as shown in the Table R4 below:\nConsumables Fees (¥)\nMagnetic beads (1 time) 120.01\nDNA strands 314.15\nOther (TE buffer, enzyme-free water, etc.) 7.06\nTable R4. The cost of three-layer convolution\nThe total cost of a single neural network calculation is ¥441.2. Among the total computational\ncosts, DNA synthesis constitutes the major expense. For large-scale applications, in-house\nsynthesis or biological production could serve as cost-effective alternatives to commercial\norders.\nRegarding the magnetic bead costs, while they may appear high, we intentionally used a\nconservative bead concentration (3× the recommended amount) to minimize incubation time.\nHowever, the cost can be further reduced by optimizing the protocol—decreasing bead usage\nwhile extending incubation time—without compromising separation efficiency.\nQ5. It is suggested that the author thoroughly revise the manuscript to improve the\nstandardization of symbol usage, including but not limited to the correction of “ij” on line 328\nand “ºC” on line 578.\nResponse: Thank you for your suggestion. We have revised the original text where standard\nsymbols were used incorrectly.\nQ10. To improve the clarity and consistency of the figures, we recommend that the authors\nstandardize the size, scale, and formatting of the graphs within Figures 3, 5, and 7. Specifically,\nthe authors should ensure that axis labels, legends, and data points are uniformly presented\nacross all panels, and consider using consistent color schemes and font sizes for better\nreadability.\nResponse: Thank you for your suggestion. We have further adjusted and optimized the relevant\ncontent in Figure 3, 5, and 7 of the main text. Changes have been marked in red.\nQ11. The link provided on line 753 ('https://github.com/Xornent/dnaconv') appears to be invalid.\nAs the training code is essential for reproducibility, the authors should update the link to ensure\nreaders can access the necessary materials.\nResponse: Thank you for your suggestion. We have added the new link.\nQ8. The image recognition accuracy in Supplementary Figures S5.1 and S7 appears to be\nrelatively low. The authors should discuss potential factors contributing to the low accuracy,\nsuch as model architecture, training data quality, or other limitations, and propose strategies for\noptimization.\nResponse: Thank you for your suggestion, in response to your concern about the accuracy of\nimage recognition in Supplementary Figures S5.1 and S7, We measured the mean absolute\nerrors produced by our summation units, with the data generated in the previous experiments,\nand find three properties of the errors:\n(1) The errors are independent of network depth, and will not accumulate through multiple\nrounds of calculation pipeline.\n(2) The errors are linearly correlated with the number of weights that participated in a single\nround of summation, and are not correlated with the exact values of the weights.\n(3) Absolute errors are normally distributed.\nWe quantify the errors and find out that they scatter normally around a small value (mean -\n0.0036, standard variance 0.0158 per weight). This is small enough and should be tolerable for\nweights and inputs that are mainly between 0 and 1.\nWe decide to construct a greater model that achieve better accuracy on the full 10-class MNIST\nclassification problem. By taking the absolute errors measured in real experiments into\nconsideration, we simulate the same normally distributed random error and propagate them into\nthe prediction task. We believe that this will yield an estimate of the accuracy if we run the model\nin our chemical system.\nThe new model consists of around 1500 parameters, and reached acceptable accuracy: At\nepoch 900, the overall accuracy of the test set reaches 95.2%, and the accuracy of each\ncategory ranges from 91.8% to 98.2% (e.g., category 0: 98.16%, category 9: 91.87%), which\nindicates that the model is capable of distinguishing between 10 categories of characters. By\nepoch 900, we do not observe much difference between the training and testing set, indicating\nno over-fitting of the model.\nWhen we incorporate the errors into every round of weighted summation (-0.0036 +- 0.0158,\nmean +- sd), the accuracy of prediction on test set reached 74.8%, This result reveals the impact\nof the actual experimental environment (e.g., signal noise, device fluctuations) on the system.\nSuch results are acceptable for such complex system (involving 1445 parameters and 4 layers\nof neural network) that runs in a test tube workflow.\nYours sincerely,\nProf. Xianjin Xiao,\nTongji Medical College,\nHuazhong University of Science and Technology,\nWuhan 430030, PR China,\nEmail: xiaoxianjin@hust.edu.cn"}, {"round": 2, "reviewer_comments": {"reviewer_1": "We have reviewed the revised manuscript and the authors responses. The manuscript has not been changed substantially, instead, the authors have focused on providing a lengthy response letter. Where the authors have answered our questions, they have for the most part not updated the manuscript accordingly, likely leaving future readers with similar concerns. In its current form we cannot justify supporting the publication of the manuscript in Nature Communications.\n\nOur most serious remaining reservations are:\n1. We feel that the answer to our question about the selection of termination time (see Q2b) is sufficiently weak to raise serious concerns about a major claim in the paper (i.e. the supposed accuracy of linear summation). This concern could be allayed if the authors provide a rigorous and independently repeatable procedure by which they terminated their experiments.\n2. More generally, the authors still have not provided a rigorous methods section that would allow others to reproduce their experiments exactly, even beyond this question of termination time. What are the protocols followed, step by step? This should be a fundamental requirement of any published paper, let alone one in Nature Communications.\n3. The use of a total concentration of input equal to number of output channels * total weight per output channel/1.25. Using this approach assumes not only that the positive and void weights within each output channel compete for the input with equal thermodynamic strength, but that all weights across all channels do so. If this were not the case, one channel could sequester more than its share of inputs, affecting the output from other channels. Although the system is designed so that binding of inputs to weights and summation gates is supposed to be equally stable, different dangling sequences may have metastable hairpins etc or stacking interactions that can change reaction dGs by a couple of kT, potentially causing problems in such a sensitive calculation. It isn't shown that this issue isn't a problem in practice.\n\nMore presentational comments, less related to scientific merit:\n3. The paper remains very confusing and hard to follow! We only now fully appreciate that the approach is based on thermal annealing rather than strand displacement kinetics. This si an important and interesting part of their work! However, in part because the work is presented like one relying on strand displacement. For example, the figures include \"gates\" that aren't actually pre-prepared as metastable duplexes. See for example Fig R1. This is really confusing.\n4. We still take issue with the position of this work in the literature. They need to provide a much clearer comparative analysis of existing literature to make such strong claims. We would suggest rather than making sensationist statements such as ‘unprecedented’ and ‘the most powerful’ they instead rank their system against prior literature according to metrics. The reader can then appreciate the quantitative advantage of their system. We also maintain that the mechanism described would indeed be a variant of associate strand displacement (with an added \"clamp region\") were it performed at constant temperature (with metastable gates) rather than annealing.\n5. The shortcoming of having stride-length equal to kernel size should be explained, as should the fact that its an unavoidable part of their system", "reviewer_2": "I think this revised paper has been in good shape.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Response to reviewers’ comments\nReviewer #1:\nWe have reviewed the revised manuscript and the authors responses. The manuscript\nhas not been changed substantially, instead, the authors have focused on providing a\nlengthy response letter. Where the authors have answered our questions, they have for\nthe most part not updated the manuscript accordingly, likely leaving future readers with\nsimilar concerns. In its current form we cannot justify supporting the publication of the\nmanuscriptin NatureCommunications.\nResponse: We sincerely appreciate your valuable comments and fully acknowledge the\ninadequaciesinourpreviousrevision, wherewefailedtosufficiently incorporateourresponse\tothereviewers'suggestionsintothemanuscript.\nIn this resubmission, we have standardized all experimental procedures, supplemented\nnecessary experimental results to strengthen our work, and systematically revised both the\nmainmanuscriptandsupplementarymaterials,withallrevisionsclearlyhighlightedin“BLUE”\tforyourreview.Modificationsin thefirstround ofreview responses arestill markedin“RED”.\nAsummaryofourrevisionlocationsisprovidedinTableR1.\nTableR1.Summaryofreviewers'commentsandauthors'revisions\nFirst-round review\nReviewer #1andReviewer #3\nQuestion\nQuestion Revisionlocation\nNumber\nWe do not understand how the cis- unit operates\nwhen there are multiple parallel outputs (eg. Fig\n3b). The input species P_i are presumably shared\nbetween all weights, and so the output signal\ndependsnotjust ontherelativeamountsofQ and\nQ'fortheoutputinquestion,butalsoonhowmany MainarticleFigure1A,1Band2B，\nQ1\notherspeciestheinputcouplesto.Howthesystem Lines109-120,245-251\ncan nonetheless produce multiple responses in\nparallel is not discussed. Indeed, the design in\nFigure 3b shows no Q' strands at all (just different\nQ) sohowitrelatestothebasicdesigninFig.1is\nunclear.\nExperimental protocols are nof described in detail.\nMainarticleLines643-668,\nHow were the actual experiments performed?\nQ2a SupplementaryInformation\nWhat volumes were used, in which device, and\nSectionS12\nwithwhatsettings?Whichspecieswereaddedand\nwhichspecieswereinitiallypresent?\nIn many experiments, the fluorescence curves do\nnotseemtohavereachedfinalsteadystatevalues.\nSee, for example, Fig 3C (but it is generally quite MainarticleFigure2,3,4,5,6and7,\ncommon,particularlyforthecalibrations intheSI).\nLines677-697,\nQ2b How then was it decided when to stop SupplementaryInformation\nexperiments?Theydon'tallseemtotakethesame SectionS2,S3,S4,S5,S6andS9\namount of time. This factor is very important in\nestablishing how well the computation match the\nexpectedvalue.\nPresumably the linear fluorescence responses in\nEg.Fig3aareusedtoconvertfluorescencesignals\nin experiment to output signals, but how that is\ndonewhentheneuralnetsarebeingappliedisnot\ndiscussed. Indeed, there is nothing at all in the\nmethods on fluorescence measurments. This\nclarification is very important, because the\napparent excellent performance of the summation\ngates depend on how exactly \"Fluorescence\nIntensity\" ismappedto \"Output value\". Indeed, we\nare surprised at the degree of precision for two\nMainarticleFigure1C,Lines\nreasons:\n163-202and698-707\ni.Typicallyspeaking,onewouldexpecterrorsinthe\nQ2c SupplementaryInformation\ninitial concentration of stocks, and from pippetting,\nSectionS1,FigureS2.1.3and\nto compromise the numerical accuracy of the\nSectionS10\nsummation.\nii. The approach seems to assume that reaction\nrates of inputs with their complementary weights\nareidentical.However,whilethestrandsaresimilar\nandmaywellhavesimilarrates,adeviationof10%\nor so (perhaps due to differences in secondary\nstructure)wouldhardlybeextraordinary.\nWhether the authors take special measures to\neliminate these errors prior to the experiment, or\nwhether they are effectively handled by the data\nprocessinginsomeway,isunclear.\nWhich exact experiments are performed is often\nunclear in the manuscript. It is very difficult for the\nMainarticleLines387-390and\nreader, given how short the captions are, to\n643-668,\nQ2d interpret what is meant by the schematics in eg.\nSupplementaryInformation\nFig.5,6,8,whichwebelievearemeanttoconvey\nSectionS12\ntheexperimentperformed.Whatdothenumbersin\nthecircleactuallymean,forexample?\nQ2e It seems like resetting is often (but not always) MainarticleFigure3,\nperformed with different weights. In all cases this SupplementaryInformation\nfactisnotdiscussed,andintheSIthetwoseparate SectionS2.1.2,S2.2.2,S2.3.2,\nsetsofweightsarenotevenprovided. S2.4.2,S2.5.2,S3.4andS4.2.2\npositioning relative to the literature is somewhat\nimprecise.Wethinkthereiscertainlynoveltywithin\nthe system proposed, but it needs to be stated\nMainarticleLines163-202,214-217\nmoreaccurately.Theauthorsdescribetheirsystem\n311-314and599-601\nQ3a as the first \"fully analogue\" DNA neural network,\nSupplementaryInformation\nbut then go on to describe the system of Genot et\nTableS11\nal, which is also analog. They criticise aspects of\nthis work, but then they need to be careful about\nhowtheypositiontheirworkwithrespecttoit.\nThe authors describe their system as the \"first\ndeep\" DNA-based neural network - but what does\n\"deep\" mean in this context? They demonstrate\ntwo layers,andtheir setupdoesn't infactnaturally MainarticleLine214-219,393-406\nallowformultiplelayers.Instead,itisnecessaryto andFigure6,7\nQ3b\nuse a convoluted intervening step involving SupplementaryInformation\nmagnetic beads. This is still an achievement, but SectionS6\nthe authors should be sure not to mislead readers\nwiththeclaimstheypresentintheintroductionand\nabstract.\nThe authors should be precise about what their\nsummation units can do - initially, they can only\nimplement weights in the range 0<=w<=1 (not all\nreal numbers). Negative weights are introduced\nlater, but then it is a bit unclear what the range of\nQ3c allowable weights is. Furthermore, although MainarticleLines506-510\nnegative weights can be implemented, we do not\nbelieve that the author's scheme can represent\nnegative results of the summation (although it is\nhard to tell exactly how the negative weights\noperate).Doesthismatter?\nThe authors use the trans- version of their design\nto build \"convolutional\" neural networks. However,\nthese convolutional neural networks use a stride\nlength equal to the kernel size, which is not\ntypically how convolutional neural networks\nQ3d operate (they would employ a smaller stride MainarticleLines553-565\nlength). ？ This fact seems, to us, to be a\nfundamental result of the way the DNA-based\nsystem works, rather than a simple choice that\ncould be changed. The authors should therefore\nclarifywhattheirsystemiscapableof.\nItisveryhardtounderstandthebehaviourthatthe\nauthors expect from their reaction networks. It\nMainarticleFigure1C,Lines\nwouldhelpalotif,intheSI,theauthorswrotedown\n109-120and163-202,\nQ4 the ODEs that they believe their various primitives\nSupplementaryInformation\nshould obey, and outline the assumptions under\nSectionS1,FigureS2.1.3\nwhich the steady state concentrations reach the\ndesiredvalues.\nTheauthorspresenttheneedtousethesomewhat\nconvoluted magnetic bead protocol in order to\ndemonstratemulti-layeroperationasaresultofthe\ngeneral existence of cross-talk in DNA-based\nnetworks(lines381-383).Thisseemstoimplythat\nit is the result of unitended reactions between\nMainarticleLines392-406\npartially matched toeholds, for example. However,\nand623-628\nwe believe that there is a deeper issue in the\nQ5 SupplementaryInformation\nspecific context of the author's system, which is\nSectionS10and\nthat their design inherently requires *the same*\nSectionS12.28toS12.31\nstrands to act as inputs at multiple layers of the\nsystem. If we are correct, this is a challenge of a\ndifferent type; it cannot be solved by making\ntoeholds more orthogonal, etc. Again, the authors\nneedtobecleareraboutwhattheirsystemcanand\ncannotdo.\nWe were a bit surprised that the authors didn't\ndemonstrate the accurate functioning ？ of a\nsingle RELU in Fig. 7, only the linear summation\nMainarticleFigure6andLines\npart. Indeed, it is not that clear which steps of the\nQ6 442-456\noverall protocol in Fig. 7 are actually being\nchecked. The authors are still referring to testing\nthe linearity(eg.line496) butsurelythepoint isto\nshownon-linearity?\nThe final resultsin Fig.8 are quite impressive, but\nthey do show a fairly small differentiation in many\ncases, and there is no clear distinction between\noutput levels that correspond to positive and\nnegative results. It is unclear whether this is a\nfailing of the DNA-based system, or of the neural MainarticleFigure7\nQ7 netthatitisimplementing.InFigS5.1andS7,the Lines588-596\nauthors present charts showing the \"accuracy of\ntrained neural network model(s)\". We *think*\n(although we are not sure) that these are the in\nsilicomodelsthatarethenimplementedwithDNA.\nThey seem to have a high failure rate - is their\nweakness what feeds through into the DNA\nsystem?IstheDNAsystemaccuratelyreproducing\nthe flawed neural net? Moreover, if the in silico\nneural nets frequently make mistakes, how were\nthe 12 letters in Fig. 8 and the digits in Fig. 6\nselected? Again, the authors need to be more\nexplicitaboutwhattheyhavesuccessfullyshown.\nThere is a lot of data associated with this\nmanuscriptandwewouldencouragetheauthorsto\nQ8 upload it to a public repository with a Data\nwell-constructed README so that it can be\naccessedandunderstoodinperpetuity.\nTheabstractisextremelylong,andcontainsalotof\ndetail/repetitive language that is probably not\nQ9 MainarticleLines5-26\nnecessary and very hard to understand without\nhavingreadthepaper.\nThe manuscript is understandable, but does need\nMainarticleLines28,30,32,96and\nQ10 proofreading for English grammar and use of\n263\ninformaltermssuchas\"Nowadays\".\nFigure1B:Whyaretheretwoidenticalgates?And\nwhy do D' toeholds interact with one of them,\nQ11 whereasDinteractwiththeother?Thereseemsto MainarticleFigure1B\nbe some kind of mistake here. We think D' should\njustbeDinthegate.\nThe basic gate design is a version of the\nassociativetoehold,introducedin2011byXiChen.\nQ12 References11\nHowever,theauthorscallit\"allosterictoehold\"and\ndonotcitethisoriginalpaper.\nFigure 2 reads more like a table of contentsfigure\nQ13 than a scientific figure (as evidenced by the MainarticleFigure1C\nabsenceofameaningfulcaption).\nMainarticleLines185-200,\nWe were unable to parse the discussion on lines\nQ14 SupplementaryInformationFigure\n188-194.\nS3.2.1\nLines 195-217 seem to just repeat much of what\nQ15 Deleted\nhasalreadybeensaid.\nLine 243 - As we understand, these experiments\nwere performed without the Q' strands (unlike the MainarticleFigure1,2and3,Lines\nQ16\nillustration in Figure 1). But the difference here is 245-251\nnotexplained.\nLine255-Strandsareintroducedthatcanresetthe\ninputs,buttheyrelyonadditionaltoeholdsthatare SupplementaryInformationFigure\nQ17\nnot in Fig. 1 (they are not introduced until Fig. 4, S1.1\nwhichisyettobedescribedatthispoint).\nQ18 Lines295-297-Isthereatypohere?Isn'tthiswhat MainarticleLines305-309\ntheauthorsdo?Dotheymeantosaythattheyonly\nfocussedonthenon-linearresponsestoinputs?\nThe discussion in lines 461-486 refers to triplex\n(IWNTN) and quadruplex (IWNTnU-) complexes,\nQ19 MainarticleLines476-481\nbut we can't see where these structures appear in\nthefigures.\nWeare totallyconfused about howthe subtraction\nmotifissupposedtowork,andhowitisintendedto\nSupplementaryInformationFigure\nQ20 give negative weights. See also the significant\nS1.3\ncomment above redescribing the models that the\nauthorsbelievetheirsystemfollows.\nAlthough, the authors demonstrate discrimination\nonMNISTdigitstheyonlydothisfortwocharacters\n('A'and'B') andthisistestedover asmallnumber\nof samples (12) relative to in silico machine\nlearning. We appreciate that this already required\nQ21 MainarticleLines601-607\nextensive experimental work anddemonstrates an\nimportant proof-of-concept, however, it may be\nworththeauthorsdiscussingtheirconfidenceinthe\ngeneralization of this system to more data and\nmulticlassificationofMNSITcharacters.\nTheauthorsshouldjustifywhyimageprocessingis\nusefulinthecontextofDNAcomputing(apartfrom\njust being a benchmark of computational power).\nConvolutional neural networks were developed\nQ22 MainarticleLines601-610\nspecificallytorepresentspatialfeaturesofimages,\nbut it is not clear from the manuscript how this\nwould be useful in molecular programming\napplications.\nSupplementaryInformationFigure\nMany of the figures in the SI are low quality/low\nQ23 S3.1.2,S3.2.2,S3.3.2,S4.4,S5.2,\nresolution.\nS6.2,S6.3.1andS6.3.2\nWhat value of \"a\" is used in the various\nQ24 SupplementaryInformationLine59\nexperimentsintheSI?\nItmightbehelpfultocolourtheDNAstrandsinthe SupplementaryInformation\nQ25\nSIbydomainidentity. SectionS8\nReviewer #2\nQN Question Modifylocation\nFor terms that may be unfamiliar to\nnon-professional readers, such as “ConvNet”, it is\nQ1 recommended to provide a brief explanation when MainarticleLines95-96,110-111\nthey first appear to improve the readability of the\npaper.\nThe article proposes the CALCUL computing unit\nbased on the allosteric-toehold domain, but the\nexplanation of its operational principles and how it\nenhances computational stability and accuracy is\ninsufficient. It would be beneficial if the authors MainarticleFigure1,Lines109-120\ncouldprovideamorecomprehensivedescriptionof and163-202,\nQ2\nthe CALCUL unit's strand reaction process, SupplementaryInformation\nspecifically addressing how continuity and SectionS1\naccuracyaremaintainedintheinputs,weights,and\noutputs. Additionally, it is important to clearly\nillustrate the role of allosteric-toehold domain\ndesigninthisprocess.\nThe article mentions that previous DNA weighted\nsummationunits have not been fully characterized\nintermsofcomputationalaccuracy,whileCALCUL\nunitshaveimprovedinthisaspect.However,isthe MainarticleFigure1C,Lines\nproof of the computational accuracy of CALCUL 163-202,698-707\nQ3\nunits rigorous? We encourage the authors to SupplementaryInformation\ndiscuss in more detail how the standard curve is SectionS9andS10\nobtained, and to address whether the calculation\naccuracy based on this method is rigorous and\nrepresentative.\nThe article does not currently address the cost\nimplications of using CALCUL units in practical\napplications. Given the relatively high costs\nassociatedwithDNAsynthesisandmagneticbead\nQ4 separation, we encourage theauthors to include a MainarticleLines713-716\ndiscussion on the economic feasibility of\nlarge-scale applications. Potential cost-saving\nstrategies, such as optimizing DNA dosage or\nreducingreactionsteps,couldalsobeexplored.\nItissuggestedthattheauthorthoroughlyrevisethe\nmanuscript to improve the standardization of MainarticleLines634,639,640-642\nQ5\nsymbol usage, including but not limited to the and712\ncorrectionof“ij”online328and“ºC”online578.\nTo improve the clarity and consistency of the\nfigures, we recommend that the authors\nstandardize the size, scale, and formatting of the\ngraphs within Figures 3, 5, and 7. Specifically, the MainarticleFigure2,4and6\nQ6\nauthors should ensure that axis labels, legends,\nand data points are uniformly presented across all\npanels, and consider using consistent color\nschemesandfontsizesforbetterreadability.\nQ7 The link provided on line 753 MainarticleLine884\n('https://github.com/Xornent/dnaconv') appears to\nbe invalid. As the training code is essential for\nreproducibility,theauthorsshouldupdatethelinkto\nensure readers can access the necessary\nmaterials.\nThe image recognition accuracy in Supplementary\nFigures S5.1 and S7 appears to be relatively low.\nThe authors should discuss potential factors\nQ8 MainarticleLines588-596\ncontributing to the low accuracy, such as model\narchitecture, training data quality, or other\nlimitations,andproposestrategiesforoptimization.\nSecond-round review\nWe feel that the answer to our question about the\nselection of termination time (see Q2b) is\nsufficientlyweaktoraiseseriousconcernsabouta MainarticleFigure2,3,4,5,6and7,\nmajor claim in the paper (i.e. the supposed Lines677-697,\nQ1\naccuracyof linear summation).This concerncould SupplementaryInformation\nbe allayed if the authors provide a rigorous and SectionS2-S6andS9\nindependentlyrepeatableprocedurebywhichthey\nterminatedtheirexperiments.\nMore generally, the authors still have not provided\narigorousmethodssectionthatwouldallowothers\nto reproduce their experiments exactly, even MainarticleLines643-668,\nQ2 beyondthisquestionofterminationtime.Whatare SupplementaryInformation\nthe protocols followed, step by step? This should SectionS12\nbe a fundamental requirement of any published\npaper,letaloneoneinNatureCommunications.\nThe use of a total concentration of input equal to\nnumberofoutputchannels*totalweightperoutput\nchannel/1.25. Using this approach assumes not\nonlythat thepositive and void weightswithin each\noutput channel compete for the input with equal\nthermodynamicstrength,butthatallweights\nacrossallchannelsdoso.Ifthiswerenotthecase,\nMainarticleLines623-628,\nonechannelcouldsequestermorethan itsshareofinputs,\naffectingtheoutputfromotherchannels.Although SupplementaryInformation\nQ3 thesystemisdesignedsothatbindingofinputs SectionS10and\ntoweightsandsummationgatesis SectionS12.28toS12.31\nsupposed to be equally stable, different dangling\nsequences may have metastable hairpins etc or\nstackinginteractionsthatcanchangereactiondGs\nby a couple of kT, potentially causing problems in\nsucha sensitivecalculation.Itisn't shown that this\nissueisn'taprobleminpractice.\nMorepresentationalcomments, less relatedto scientificmerit\nThe paper remains very confusing and hard to\nfollow! We only now fully appreciate that the\napproach is based on thermal annealing rather\nthan strand displacement kinetics. This si an\nimportant and interesting part of their work! MainarticleFigure1,2,3,4,6and7,\nQ3\nHowever, in part because the work is presented Lines142,178-179,365,and645,\nlike one relying on strand displacement. For SupplementaryInformationFigure\nexample, the figures include \"gates\" that aren't S1.2,SectionS6,S10andS12\nactuallypre-preparedasmetastableduplexes.See\nforexampleFigR1.Thisisreallyconfusing.\nWe still take issue with the position of this work in\ntheliterature.Theyneedtoprovideamuchclearer\ncomparative analysis of existing literature to make\nsuch strong claims. We would suggest rather than\nmaking sensationist statements such as\n‘unprecedented’ and ‘the most powerful’ they MainarticleLines163-202,311-314\ninstead rank their system against prior literature and597-599,\nQ4\naccording to metrics. The reader can then SupplementaryTableS11\nappreciate the quantitative advantage of their\nsystem. We also maintain that the mechanism\ndescribed would indeed be a variant of associate\nstrand displacement (with an added \"clamp\nregion\")wereitperformedatconstanttemperature\n(withmetastablegates) rather thanannealing.\nTheshortcomingofhavingstride-lengthequalto\nQ5 kernelsizeshouldbeexplained,asshouldthefact MainarticleLines553-565\nthatitsanunavoidablepartoftheirsystem\nQ1. We feel that the answer to our question about the selection of termination time (see\nQ2b) is sufficiently weak to raise serious concerns about a major claim in the paper (i.e.\nthe supposed accuracy of linear summation).This concern could be allayed if the authors\nprovidearigorousandindependentlyrepeatableprocedurebywhichtheyterminatedtheir\nexperiments.\nResponse: Considering efficiency (short computation duration), consistency (synchronized\ntermination rules across experiments), and operational feasibility, we evaluated fluorescence\nsignalsevery10minutestoassessiftheyreachedplateauandtherebywecouldterminatethe\nfluorescencerecording.Theterminationcriteriawereasfollows:\n1) For all curves, calculate the magnitudes of fluorescence increase during the 5-minute\nwindow preceding each check timepoint. If none exceed 1% growth, the experiment are\nterminated.\n2)IfanycurvefailstomeetCriterion(1),theexperimentiscontinuedforanother10min,and\nthecurvedwillbereassessedafter10minutesusingthesamestandard.Thisiterativeprocess\nisrepeateduntilallcurvessatisfythe1%requirement.\nFollowingtheaboverules,wedevotedeffortstoreviewingalldatainourmanuscriptandfor\nasmallportionofcurvesthatdidnotmeettheaforementionedcriteria,wereanalyzedthedata\norconductedadditionalexperiments.Specifically:\nThe termination timepoint for the experiments shown in Figure 6, Supplementary\nInformationSectionS6.1,andSectionS6.2is10min.\nThe termination timepoint for the experiments shown in Figure 7 and Supplementary\nInformationSectionS6.3is20min.\nThe termination timepoint for the experiments shown in Figure 2 and Supplementary\nInformationSectionsS2.4.1andS2.5.1is30min.\nTheterminationtimepointfortheexperimentsshowninFigures3,4,5andSupplementary\nInformation Sections S3, Section S4, Section S5, as well as the reused experiments in\nSectionS2is40min.\nThe termination timepoint for the experiments shown in Figure 3 and the related reused\nexperimentsinSupplementaryInformationSectionS4is50min.\nWe also would like to note that in our experiment, we indeed observed some\nhigh-fluorescence curves (e.g., the full-connection operation with 4 inputs/3 outputs shown in\nFigure 4A) exhibited slow signal increases even after meeting our termination criteria, likely\ndue to slowed kinetics near thermodynamic equilibrium. Fortunately, this phenomenon\noccurred synchronously in high-fluorescence standard curves. Since the final computational\noutputsarederivedfromstandardcurve-convertedfluorescence,weassumedthattheseslow\nsignalincreases onlyimposedminimalimpactonthecomputingaccuracy.Tovalidatethis,we\nrepeatedthreecurveswithslowsignalincreasesattheend:\n①Thehighvaluecurveof8-wayintegrationofFigure2A\n②TheHEXcurveof4-input/3-outputfull-connectionofFigure4A\n③TheROXcurveof4-input/3-outputfull-connectionofFigure4A\nThe experiments were terminated at 60 min, and we calculated the output values using the\nfluorescence data of 40 min, 50 min and 60 min respectively.As shown in Figure R1 (Figure\nS.9.2intherevisedSI)herein,althoughthefluorescencesignalsstillincreasedslowlyafter40\nmin,thecomputingoutputsremainedalmostunchanged.\nWe have added the above description and supplementary results to Main Article (Lines\n677-697)andSupplementaryInformationSectionS9.\nFigure R1. Validation of computational output consistency across different sampling\ntimes. Test 1:The highvalue curveof 8-way integrationof Figure2A;Test 2:The HEX curve\nof 4-input/3-output full-connection of Figure 4A; Test 3: The ROX curve of 4-input/3-output\nfull-connection of Figure 4A. Computational outputs show minimal variation at different\nsamplingtimes.\nQ2. More generally, the authors still have not provided a rigorous methods section that\nwould allow others to reproduce their experiments exactly, even beyond this question of\ntermination time. What are the protocols followed, step by step? This should be a\nfundamental requirement of any published paper, let alone one in Nature\nCommunications.\nResponse: We express our gratitude for your valuable comments and acknowledge the\nabsence of a systematic and comprehensive description of the experimental procedures. To\naddress this, we have meticulously described the step-by-step procedures for all experiments\nSupplementary Information Section S12 and the Methods section of the Main Article\n(Lines643-668).\nQ3. The use of a total concentration of input equal to number of output channels * total\nweight per output channel/1.25. Using this approach assumes not only that the positive\nand void weights within each output channel compete for the input with equal\nthermodynamicstrength,butthatallweightsacrossallchannelsdoso.Ifthiswerenotthe\ncase,onechannelcouldsequestermorethan itsshareofinputs,affectingtheoutputfrom\nother channels.Although the system is designed so that binding of inputs to weights and\nsummation gates is supposed to be equally stable, different dangling sequences may\nhave metastable hairpins etc or stacking interactions that can change reaction dGs by a\ncouple of kT, potentially causing problems in such a sensitive calculation. It isn't shown\nthatthisissue isn'taproblem inpractice.\nResponse: After carefully considering your comments, we identify your main concern as\nspecifically regarding uneven strand allocation when inputs are distributed across multiple\npathways.Weclarifythatthispotentialissueisrestrictedtocis-CALCUL-basedfull-connection\noperations where inputs distribute across different output channels. In contrast,\ntrans-CALCUL-based convolution operations inherently avoid such concerns because inputs\nonlyenterintooneoutputchannel.\nRegarding the sequence design of cis-CALCUL-based full-connection operations, as noted\nby the reviewer, the binding domains of one input across different weights and summation\ngates remain identical (domain A and D). Consequently, the free energy transitions (ΔG) of\none input entering different output channels are theoretically equivalent. However, we fully\nrecognize the crucial impact of the dangling sequence issue. In fact, we mitigate such\ninfluence through both sequence design and experimental validation.At the sequence design\nphase, NUPACK was employed to predict potential secondary structures, and strands with\ndanglingsequenceswereexcluded.Subsequently,aquantityofweightsandsummationgates\nexceedingtherequirednumberoftypeswassynthesized,andeachwasreactedwiththeinput\nseparately. As depicted in Figure R2, weights and summation gates exhibiting highly\nconsistent fluorescence increase rates and plateau values were selected for subsequent\nexperiments (for more specific details of this part, please refer to revised Supplementary\nInformation Section S10.2 and S12.29). Moreover, in the case of cis - CALCUL - based\nweightedsummation,whiletheproblemofnon-uniformdistributionacrosschannelsisabsent,\nthe dangling sequence has the potential to influence the distribution between positive gates\nand void gates. Consequently, during the design of sequences for cis - CALCUL - based\nweighted summation, NUPACK prediction and experimental screening were also utilized\n(SupplementarySectionS10.1andS12.28).\nFigure R2. The experimental results of DNA sequence screening experiment of\nfull-connectionoperationsbasedoncis-CALCULunits.Thedottedlineshowsthemedian\nfluorescent intensity, and the light red area represents the median ± 5% range.\nFluorescence trajectories for Inputs 1-4 (red, yellow, blue, green, respectively) depict\ncomputationalresponsesacrossthreeindependentchannelsperinput.\nIt is also worthy of note that, in the trans - CALCUL - based weighted summation and\nconvolution operations, when multiple inputs enter a single summation gate, the problem of\nuneven input distribution across channels does not arise. However, there may be an issue of\nuneven distribution of summation gates across different input - weight duplexes. If one\npathway is influenced, its partial outputs will be changed, leading to the propagation of\nerrorstothefinalsummationou .Asaresult,duringthesequencedesignfortrans\n(IiWi)\n-CALCUL-basedweightedsummationandconvolutionoperations,identicalstrandscreening\ntput( IiWi)\nprocedures (NUPACK prediction and experimental validation) were implemented (see\nSupplementary Information Section S10.3 to S10.4 and S12.30 to S12.31). This ensures\nthe comprehensive alleviation of sequence - dependent artifacts across all computational\nparadigms.\nWeapologizeforthelackofelaborationonthisprotocolintheoriginalmanuscript.Inthe\nrevisedversion,theabove-mentioneddescriptionhasbeenadded(MainarticleLines\n623-628,SupplementaryInformationSectionS10andSectionS12.28toS12.31).\nMorepresentationalcomments, less relatedto scientificmerit:\nQ3. The paper remains very confusing and hard to follow! We only now fully appreciate\nthattheapproachisbasedonthermalannealingrather thanstranddisplacementkinetics.\nThis si an important and interesting part of their work! However, in part because the work\nis presented like one relying on strand displacement. For example, the figures include\n\"gates\"thataren'tactuallypre-preparedasmetastableduplexes.SeeforexampleFigR1.\nThis isreallyconfusing.\nResponse: We appreciate your comments, as they have enabled us to identify the\ndeficiencies in the readability and focal expression of our schematic diagrams. We have\nundertaken an extensive revision of the schematics (Figures 1, 2, 3, 4 and 6). This revision\nincludes altering specific diagram representations, incorporating explanatory labels, and\nexplicitly emphasizing the critical function of thermal annealing process through the use of\ndedicatedsymbols.\nAs illustrated in Figure 1 and Supplementary Information Section S9, our experimental\nprotocol initially involved preparing a mixture of strands O and T.At room temperature, these\nstrands predominantly formed OT duplexes, which are termed \"gates\". Although thermal\nannealing was subsequently employed for DNA computations, rendering the OT duplexes\nnon-functionalasgates,forclarityandreadercomprehension,wehavestillincorporatedthem\nin the scheme. With regard to Supplementary Information Figure S1.2, Figure S1.2A\ndisplays the final structures after thermal annealing, while Figures S1.2B and C depict the\ncomplexes formed during the premix preparation. By presenting these detailed structural\ninformation, we aim to provide readers with a clearer understanding of the hybridization\ncorrespondencebetween domains during thermalannealing andthe thermodynamicchanges\nbefore and after the reaction. Concurrently, to prevent any potential confusion, we have\nuniformlyreplacedallinstancesoftheterm\"gates\"(includinglabelsandtextualmentions)with\n\"modules\" throughout the manuscript (Figure 1,4,6,7, Lines 142, 178-179, 365, and 645,\nSupplementaryInformationSectionS6,S10,S12).\nQ4.Westill take issue with the position ofthis work inthe literature.They need to provide\namuchclearer comparativeanalysisofexistingliteraturetomakesuchstrong claims.We\nwould suggest rather than making sensationist statements such as ‘unprecedented’ and\n‘the most powerful’ they instead rank their system against prior literature according to\nmetrics. The reader can then appreciate the quantitative advantage of their system. We\nalsomaintainthatthemechanismdescribedwouldindeedbeavariantofassociatestrand\ndisplacement (with an added \"clamp region\") were it performed at constant temperature\n(withmetastablegates) rather thanannealing.\nResponse: In the Introduction section (Lines 28-66), we conducted an analysis and\ncomparison with several published works in the area of DNA neural networks. Nevertheless,\nthepreviousdiscussionmainlycenteredoncomputationalaccuracyaswellasthecontinuityof\ninputs,weights,andoutputs.Hereintherevisedversion,wehaveprovidedamoresystematic\nand comprehensive analysis in Supplementary Table S11. This table shows that our method\nhas advantages in terms of the numerical continuity of inputs, weights, and outputs,\ncomputationalaccuracy,andtheconstructionofdeepneuralnetworks.However,thelanguage\ninourmanuscript doesneedtobemorepreciseandrigorous.Therefore, wehaverevised the\nrelevantexpressions(Lines201,311-314and597-599).\nTable.R2.ComparativeanalysisofresearchesinthefieldofDNAneuralnetworks\nNumber Maximum Image Calculation Continuity Continuity of Continuity\nofneural weighted recognition accuracy (2 ofinputs weights ofoutputs\nnetwork summation matrixsize inputs)\nlayers* scale\nGenotetal. 1 2inputs - ＞85% √ ?† √\n(2013)[1]\nCherryetal. 1 100inputs 10×10 0or1 √ √\n-\n(2018)[2]\nZhang et al. 1 4inputs - ＞93% √ Consecutive √\n(2020)[3] integers\nOkumuraet 1 2inputs - ＞91% √ √ 0or1\nal.(2022)\n[4]\nXiongetal. 1 36inputs 12×12 ＞75% 0or1 √ √\n(2022)[5]\nOurWork 3 16inputs 12×12 ＞95% √ √ √\n*One layer of the DNA neural network involves the execution of at least one complete weighted\nsummationoperationontheinputdata.\n†Theoreticallyfeasible,withoutsufficientexperimentalvalidation.\nRegarding the associate strand displacement you mentioned, as you stated, our\ncomputation is accomplished via thermal annealing instead of strand displacement.\nConsequently, in terms of the fundamental reaction pathway, our approach is completely\ndistinct fromtheassociate stranddisplacement. Moreover,we wouldliketo emphasize that to\nfacilitate this thermal annealing process, we integrated several intricate design elements.\nThese elements essentially differentiate our mechanism from associate strand displacement\nsystemsinboththereactionpathwayandstructuraldesign.Thefineddesignsconsistof:the\nlong domain D, allosteric domains X/X*, and the configuration of the T - strand as the output\nstrand, all of which are depicted in Figure 1C. Wehave presented a moredetailed discussion\ninthemaintext(Lines163-202).\nQ5. The shortcoming of having stride-length equal to kernel size should be explained, as\nshouldthe factthatits an unavoidablepartoftheir system\nResponse: In computer-based convolutional neural network models, the stride length is\ntypicallyfreelyadjustable, whichhelps enhancemachinelearningefficiencyandperformance.\nHowever, we must acknowledge that implementing freely configurable stride lengths in DNA\nneuralnetworksremainshighlychallengingatpresent.Aftercomprehensiveconsideration,we\nselected a specific stride length setting that allows the convolutional kernel to precisely cover\neach input once. This design approach was also adopted by a previous work on DNA\nconvolutional neural networks5. As you rightly pointed out, this indeed constitutes an\nindispensable aspect of our system. Tothis end, we have supplemented relevant discussions\nin the manuscript (Lines 553-565) to facilitate the readers to understand our work\ncomprehensivelyandengageindeepercontemplation.\nREFERENCES:\napplication to matrix multiplication and weighted sums. Angew Chem Int Ed Engl 52,\n1189-1192(2013).\nwinner-take-allneuralnetworks.Nature559,370-376(2018).\n709-715(2020).\n610,496-501(2022).\ncircuits.NatureMachineIntelligence4,625-+(2022)."}]

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