Fractional coding and Matrix multiplication are two important operations for the computing system. Fractional coding is the foundation of other complex arithmetical operations. It is a numeric format for representing numbers with a decimal part. While matrix multiplication is vital for the implementation of deep networks. Matrix operations are widely used in deep learning because they are an efficient way to represent and manipulate complex mathematical relationships.

Salehi et al. (2018) constructed two types of fractional coding. One is the unipolar fractional coding, the other is the bipolar fractional coding. The definition is illustrated as formulas 1, 2: x = X 1 X 0 + X 1 (1)

x is the variable. A pair of molecular X 0 and X 1 is assigned to x . The value of the variable is determined by the ratio of the concentration for the assigned pair. Therefore, the value of x is confined to unit interval [0, 1]. This strategy is unipolar fractional coding. x = X 1 − X 0 X 0 + X 1 (2)

Similarly, the value of the variable x locates in the range of [−1, 1], represented by formula 2. This is the definition of bipolar fractional encoding. Further, they implemented mathematical functions, including exponentials, sigmoid, sine, cosine, and tanh functions (Salehi et al., 2018). They defined the five basic DSD reactions as the multiplication (Mult), Nor-Mult (NMult), multiplexer, bipolar Mult, and bipolar NMult units. Then, mathematical functions can be transformed into Taylor expansions, and constructed by these five units, Figure 3A.

Matrix multiplication can be performed via DSD reactions. Reference (Genot et al., 2013) is the first research that explicitly illustrated the implementation of matrix multiplication with DSD. Genot et al. (2013) designed the typical multiplication of a 2 × 2 vector by a 2 × 1 matrix. They utilized the combinations of toehold and migration domains, and implemented the multiplication of a 2 × 2 vector by a 2 × 1 matrix. In this research, toehold and migration domains were dynamically and combinatorially linked to form DNA complexes, which represent the matrices. n toehold domains and n substitution domains can form n 2 substitution complexes, which greatly reduce the number of required chains and increase the computational power. The operation processes are shown in Figure 3B, the multiplication M 11 M 12 M 21 M 22 × X 1 X 2 = Y 1 Y 2 (3)

The elements in Metrix M were represented by strands { d 1 b 1 , d 1 b 2 , d 2 b 1 , d 2 b 2 }, Metrix X was represented by { b ¯ 1 t 1 , b ¯ 2 t 2 }, the cap means its complementary sequences. The output of this matrix multiplication is matrix Y, represented by two strands { Y 1 , Y 2 }.

DSD-based chemical reaction networks can be designed in a programmable language for solving mathematical problems. Tang et al. (2021) designed weighted reactions, sum reactions, threshold modules via DSD, and solved a three-parameter 0–1 knapsack problem. Lopiccolo et al. (2021) implemented a last-in, first-out stack structure via DSD. This stacked structure stores two signals, and signals are released into the solution by order of the input strand. Later, the stack can be rebooted by the activation strand.

Logic gates are the primary unit of integrated circuits. They are the foundation of modern computer systems. DSD and modified DSD reactions can be programmed and utilized as logic gates. An outstanding research of improved DSD reaction is the seesaw gate, proposed by Qian and Winfree (2011). The mechanism of the seesaw gate is the reversible DSD reaction. Figure 4A depicts the reaction principle. The input strand will first react with a threshold gate until all the thresholds are consumed. As the threshold chain has a longer toehold domain, which guarantees the first step. Then, excessive input can react with the output gate and produce the output strand. Last, the output strand will react with the report gate and give the fluorescent report signal. Because the fuel strand and input strand have the same toehold and migration domain. Fuel strand can replace the input strand out from the byproduct of second step.

The renewable gThere are many topics involving DSD logic gates, including callability, signal restoration, time-responsive, etc. ate is one of them. Eshra et al. (2017) modified the seesaw gate motif into a hairpin, called the “DNA hairpin-seesaw gate”. As shown in Figure 4B, the inner two hairpin motifs. The sequences of them from 5′ to 3′ are {S₁, T₂, S₃, T₁ ^*, S₁ ^*, T₁ ^*} and {S₂, T₂, S₃, T₁ ^*, S₂ ^*, T₁ ^*} (cap “*”represents the complementary sequences). S₁ pairs with S₁ ^* to form the hairpin, S₂ pairs with S₂ ^* to form the other hairpin. The two toehold domains T₁ ^* and T₂ locate at the two sides of these motifs, similar to the original seesaw gate. Further, they added two extractors to initialize the hairpin-seesaw gate, which realized the renewable process. This hairpin gate can be reused more than three times in consecutive calculations. They constructed a 2-input renewable circuit via this motif (Eshra et al., 2019), as shown in Figure 4B. Based on DSD’s AND gate and XOR gate, half-adder or full-adder circuits can be assembled. Xiao et al. (2020) constructed a three bits full-adder . Xie et al. (2022) used three dual-track logic gates and assembled a four bits full-adder.

In 2020, Wang et al. (2020) designed DNA switching circuits (DSCs). The input chain interacts with the gate to generate an output strand. Then the output strand will propagate and arrive at the next gate. Gates will change their states according to the output strand. Therefore, the ON and OFF states are switched, which can represent Yes and No. The output strand propagates like a current passing through the DSCs’ gate. Besides, the DSC scheme does not use the dual-track strategy to design NOT gates. (Dual-track strategy refers to a design methodology used in digital circuits that employs two parallel design approaches or paths. The two paths operate in parallel and regularly interact with each other so that both are progressing toward the final goal. For example, the two paths of full-adder are add-path and carry-path. Furthermore, the full-adder is the fundamental component of digital computing.) As a result, its required chains are reduced by 3/4 compared to current DSD circuits. The implementation scheme is shown in Figure 4C.

Liu et al. (2020) created a cross-inhibit gate and then used it to perform four-input time-sensitive circuits. Signal strand A can react with detector DA to produce output OA and kill strand KA, as shown in Figure 4D. Afterward, KA can react with detector DB, which inhibits the reaction of DB and signal strand B. As a result, signal A inhibited signal B. If signal B is first added, then the situation is the opposite. This DSD-based cross-inhabit strategy is a simple and effective method for constructing time-sensitive circuits. Fern et al. (2017) also designed time-delayed DSD circuits. They created a simple DSD circuit with eight strands that can release target DNA strands into solution at a constant rate for hours to days. The result is illustrated in Figure 4E.

Machine learning algorithms and artificial neural networks can be implemented via DSD reactions. The decision tree is a classical classification ML algorithm. It processes the classification pathway based on the known probability of occurrence situations. Its classification pathway can be executed via DSD reactions. Chen et al. (2022b) designed a domino-like DSD sequential system, which can execute four steps of the decision pathway. As shown in Figure 5A.

Support Vector Machine (SVM) is another powerful ML algorithm. It is a two-class classification algorithm. Its basic approach is to find the maximum classification interval in the feature space. Lopez et al. (2018) performed a gene classification SVM system. The selected genes for each class are pretrained on a silicon computer. Then, these genes transcript to RNA and reaction with DNA complexes under the guidance of SVM algorithm. The weights of each gene are implemented by transcript times, as shown in Figure 5B. Using seesaw gates and the dual-track strategy, Qian et al. 2011 further constructed a Hopfield network. This network contains twenty-four circuits and “remembers” four patterns.

In 2018, Qian et al. extended the computational border of the seesaw gate. They constructed a winner-take-all neural network without using the dual-track strategy (Cherry and Qian, 2018). This network is a three-layer, fully connected artificial network. The inputs are 100 bits DNA strands in 10 × 10 patterns. The hidden layer consists of 20 distinct DNA molecules that can react with these 100 bits DNA strands via DSD reactions. The output layer shows the classification results of handwritten digits ‘1’ to ‘9’. This is the first research attempt using DSD to construct complex ANN.

Convolutional Neural Network (CNN) can also be implemented via DSD reactions. The essence of the convolution operation is matrix multiplication. Xiong et al. (2022) take the same matrix multiplication strategy as Genot et al. (2013). They designed the Metrix operation DSD system, including the multiplication of two matrices, matrix Addition, and matrix Subtraction.

The matrix multiplication is X 2 × 2 × W 2 × 1 , results are y 1 and y 2 . Then, y 1 and y 2 are added or subtracted to produce the final results Y which is the result of this convolution process. They implemented this kernel W 2 × 1 to convolute the input ‘image’ (image is transferred to 144 bits DNA strands in 12 × 12 patterns). Then the image indentation result can be illustrated by ∑ 1 n Y (n is the convolutional times). As shown in Figure 5C. Using this DSD-based CNN method, they successfully identified oracle bones as well as English letters and Arabic numerals.

CRISPR (Clustered Regularly Interspaced Short Palindromic Repeats) was first found from Escherichia coli bacteria by Ishino et al. (1987). The CRISPR system consists of an artificially designed sgRNA (single-guide RNA) and a Cas protein, which combine to form a complex. sgRNA has a guide sequence of about 20 nt that matches the target gene, then the PAM sequence in the upstream can cleaved, repressed, and activate the target gene with the help of Cas protein. As sgRNA is a segment of RNA around 100 nt, it is possible to program it.

The combination of CRISPR and DSD enables some in-cell logical circuits, even intracellular gene regulation can be realized.Jin et al. (2019) added the toehold domain for sgRNA at the 5′ end (guide sequence) and 3′ end (scaffold structure) without affecting its activity. Because the guide sequence of sgRNA is sensitive to its formation, the binding and unbinding of the toehold domain can be utilized for controlling its activity. Then, they used this designed CRISPR as the switch to control a DSD system.

Li et al. (2019) designed the scaffold structure of sgRNA, as shown in Figure 6A. Regular sgRNA consists of a guide sequence (the yellow part) and a scaffold sequence (the black part). They added the mRNA sensing sequence (the green part) and the toehold domain (the red part) in the sgRNA, which is named msgRNA. This structure was designed to disrupt the scaffold of sgRNA, making it impossible to bind with the Cas9 protein. When mRNA is added, the strand displacement reaction opens the hairpin of msgRNA, restoring sgRNA activity. Hao et al. disrupted the structure of sgRNA through a blocking strand. Then they added a replacement strand to react with the blocking strand, thus restoring the activity of sgRNA (Hao et al., 2020).

The fully paired double-stranded DNA (dsDNA) is generally considered the waste product of the DSD reaction. However, Roser et al. made the waste strands useful through CRISPR technology (Montagud-Martínez et al., 2021), as shown in Figure 6B. The regular dsDNA is cut by CRISPR Cas9 and produces output2. Then output2 will react with prehybridized ssDNAs through DSD reaction and produce output3. Last, output3 can regulate the target gene with the help of T7 pol. The whole system includes CRISPR-mediated reactions, toehold-mediated reactions, and the transcript process.

From the above-mentioned designs, we can conclude that DSD reactions can be used as the switch to control the CRISPR process. These above-mentioned studies have a common strategy, which is to program the DSD reaction domain in sgRNA sequences. This strategy is direct and efficient.

Based on DSD reactions, DNA structures can be designed as molecular robots, molecular machines, and many other molecular devices. These devices can be precisely controlled via the DSD reaction. DSD combines with various molecules can make these devices efficient for many applications.

The ATGC four nucleotides can be used as 4-bit coding information, and synthetic DNA strands have been utilized as data storage material. Generally, a strip of DNA data consists of an addressing domain, an information domain, and a correction domain. DSD reactions can be deployed on these domains. The input strand of the DSD can be utilized as the information encryption, information reader, initiator, etc. The DSD-based information processing is simple and effective.

Lin et al. (2020) inserted a T7 promoter and a single overhang strand into the information domain. The overhanging strand can accomplish the tagging, locking, replacement, and deletion processes for the DNA information with the help of the T7 promoter. Additionally, the DNA data can be read out through the PCR process without disrupting the original DNA strand. Lin increases the information capacity and reduces coding complexity by using the DSD system, which avoids the impact of primers on DNA coding regions. As illustrated in Figure 8A.

Kim et al. (2020) used the DSD reactions to extract image data that was stored in DNA strands. They encoded a false message-encoded strand that has the same primers and length as the true message-encoded strand. Correct information can be extracted from the true strands through DSD reactions and the PCR process, while the false strand cannot. However, if the whole system is heated to above 95° and kept for 5 min, the correct information and the error information will be mixed together, resulting in decoding failure, and the related data cannot be recovered, which plays the role of fast erasure of the target information. As shown in Figure 8B.

Banal et al. (2021) sealed DNA information in impervious silica capsules. The surface of silica gel is labeled with single overhanging DNA strands, which represent the characteristics of the stored information. When information retrieval is required, a complementary strand with magnetic bead-modified DNA strands will be added to the solution. These two strands will react through the DSD reaction, and the silica capsule data will be captured by magnetic adsorption.

Zhu et al. (2022) used the DSD reactions as the encryption approach. This encryption approach includes the conversion of information to DNA sequences through Huffman coding. Then, the key to this DNA information is transferred through a degradation reaction. Finally, the DSD transferred key is extended via a catalysis reaction, which increases the decryption complexity. The key transfer reactions are illustrated in Figure 8C.

DSD can be designed as a tool for cancer detection. DNA strands modified with aptamers can target the cancer cell surface, and the detection signals or drugs can be released through the DSD reaction. In these applications, the ligands target specific cells, and the DSD system is utilized as the switch to turn on the drug release process or cancer detection process.

Peng et al. (2018) constructed a three-dimensional DNA-based nanomachine that can target cancer cells. This nanomachine can identify the DNA aptamers that are produced by SELEX cancer cells. When the nanomachine binds to the surface of cancer cells, it will produce the fluorescent signal through the DSD reaction. As shown in Figure 9A. This DSD-based cancer detection method is ultra-sensitive to cancer cells and can be used as a tool for early cancer identification.

Molecular profiling of tumor-derived extracellular vesicles (tEVs) is a vital cancer biomarker. Li et al. (2021) used the thermophoresis mediated DNA computing device to identify tEVS. They constructed an AND gate that consists of EpCAM-S-T2 and HER2-S-T2 proteins. tEVS causes the overexpression of two proteins. The whole tEVSs are binding to a Microbead via CD63 aptamer. Last, they added the DNA hairpin strands into the solution, and then the AND gate produced the fluorescent signals, which can represent the expression level of tEVs.

Chang et al. (2019) identified and narrowed down a cancer cell-type subpopulation from large populations of similar cells through the DSD logic device. They programmed two cancer-expression proteins, aptamers Sgc8c-S-T1 and TCO1-S-T2, on an AND logic gate. These two aptamers can identify the target cancer cell and attach themselves to its surface. Then, input strands can react with the AND gate and produce fluorescent signals. The scheme of this method is illustrated in Figure 9B. These two methods (Li et al., 2021)- (Chang et al., 2019) have a similarity. They both used two aptamers to identify cancer and then assembled an AND gate. The final signal reports were accomplished via DSD AND gate reactions.