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In order to take advantage of the merits of DNA computing, researchers do not only use DNA computing to implement Turing machine but also try to employ it for specific applications, especially those involving complex problems. There are two basic questions arising when applying DNA computing for specific applications. One is encoding the real‐world signals to the input variables for CRNs and then decoding them back into real‐world signals after computation (Figure 3.5). The other one is how to design chemical reactions for specific functions.

Figure 3.5 The system performing encoding, computing, and decoding signals in CRNs.

Source: From Salehi et al. [17]. Reproduced with the permission of American Chemical Society.

In order to use the concentrations of molecules to represent variables' values, researchers have considered three types of encoding – the direct representation, the dual‐rail representation [18], and the fractional representation [17]. In the direct representation, values of all variables are indicated by concentrations of molecular types. In the dual‐rail representation, the difference between concentrations of two species represents the value of a variable. In the fractional representation, values of variables are determined by ratios of two molecular species in the reaction system. To be specific, e.g. if (X₀, X₁) is the fractional representation for a variable x, its value is x = [X₁]/([X₀] + [X₁]), where [·] denotes concentrations of molecular types.

After defining the various input signals in a biochemical system with one of the three types of encoding representations mentioned above, the system can be solved through ordinary differential equations (ODEs). For CRN analysis, mass action kinetics is considered as a proper kinetic scheme [19]. For mass action kinetics, the rate of a chemical reaction is proportional to the product of concentrations of reactants. For instance, consider a reaction given by

Since the reaction fires at a rate proportional to [X₁][X₂], or [rate of reaction] ∝ k[X₁][X₂], where k is rate constant associated with the reaction, we can model the reaction by ODEs as follows:

ODE simulation is a continuous deterministic model of chemical kinetics.

An alternative approach to achieving mass action kinetics modeling is referred to as stochastic simulation [20]. Compared with deterministic modeling, stochastic simulation is discrete and stochastic, and the computation is based on probabilities.

Many researchers have investigated methods to implement digital logic with molecular reactions, including combinational components and sequential components. For combinational components, the inverter is the simplest but very important logic gate since other more complicated structures such as NAND gates, adders, and multipliers will make use of it. In a biochemical system, it can be implemented by implementing the transfers between the molecular types representing 0 and 1, respectively [21]. The simplification methods for digital combinational logic have been studied in [22].

Take the AND gate as an example for two‐input logic gates implemented by molecular reactions [21]. Suppose the inputs of the gate are X and Y and the output is Z, respectively. The inputs and output signals are represented by the concentration of X₀/X₁, Y₀/Y₁, and Z₀/Z₁. If the value of X is 0, then all X₁ will be transferred to X₀. According to the target logic function, the chemical reactions are designed as

where represents an indicator to generate Z₁ and can be transferred to an external sink indicated by the fourth reaction. The analysis of inverter and AND gate implementation with molecular reactions can be applied to explain other complex gates such as NOR, XOR, and NAND [21], which form the basic blocks for modules like multipliers and digital signal processors.

For sequential components, all modules are under control of the clock signals. Sequential digital logic circuits can be classified into two categories, synchronous circuits and asynchronous circuits, depending on whether the circuits are governed by a global clock or not. A sustained‐chemical‐oscillator‐based synchronization mechanism is introduced to implement synchronous circuits with molecular reactions, which have been widely studied by the synthetic biology community. An example is “red‐green‐blue” (RGB) oscillator [23–25], which is first proposed by [23] and can be used to establish an order for the transformation of molecular quantities in the counter implemented by molecular reactions. The RGB oscillator is also useful for generating a global clock as the designers wish. Reactions in an RGB oscillator are assigned to one of the three categories – red, green, and blue. Quantities are transformed between color categories according to the absence of molecules in the third category as (Figure 3.6a)

Here, R, G, and B are introduced molecular types. And r, g, and b are the “absence indicators” corresponding to R, G, and B, respectively, and are continually generated as

The feature of indicators quickly consumed by corresponding signal molecules assures that the succeeding phase cannot begin unless all reactions in a given phase have completed (Figure 3.6b). With the aid of such clock signals, analog circuits for basic arithmetic, like addition, subtraction, multiplication, and division, can be implemented with molecular reactions [27].

Figure 3.6 (a) Sequence of reactions for the three‐phase clock based on the RGB oscillator.

Source: Adapted from Kharam et al. [26]

. (b) ODE‐based simulation of the chemical kinetics of the proposed N‐phase clock (here N = 2), where the amplitude and frequency of oscillation waves can be adjusted.

Source: From Jiang et al. [25]. Reproduced with the permission of American Chemical Society.

Asynchronization circuits are implemented by locking the computation of biochemical modules. In asynchronous circuit designs, it is analogous to handshaking mechanisms. By introducing a specific molecular type, the module's key, to each module, the sequence of reactions is prevented from firing without the key, thus under proper control [28].

Several researchers have turned their attention to the implementation of more complicated functions in specific systems [29–32]. Salehi et al. [29] first points out that stochastic logic could be converted to molecular designs that can be readily utilized in the design of molecular filters and channel decoders. Using fractional coding [17], DNA computing‐based vector machine and artificial neural networks have been proposed in [30–32]. To resolve several complex design issues raised by nonlinear functions, Taylor series are applied to approximate a function with a polynomial [17]. A polynomial is a mathematical expression involving variables and coefficients; its operations are limited to multiplication, subtraction, addition, and nonnegative integer exponents of variables. The nucleus of designing functions with chemical reactions is to implement addition, subtraction, and multiplication with molecular reactions.

Take the addition a + b = c as an example. The corresponding chemical reactions are designed as

where the concentrations of molecular types A, B, and C represent the values of a, b, and c, respectively. As both the inputs A and B are transferred to C, the concentration of C is the sum of the initial concentrations of A and B, namely, the values of a and b.

Digital signal processing (DSP) modules such as filters and fast Fourier transform (FFT) processors are of good importance and perform a wide variety of functions. CRN is a novel alternative to traditional application‐specific integrated circuits (ASICs) to implement DSP algorithms since they are also applicable to the field of molecular computing. Methods for implementing DSP algorithms using synchronous, RGB, and asynchronous schemes have been demonstrated in detail by [23,24,33,34] (Figures 3.7 and 3.8).

Figure 3.7 A folded eight‐point four‐parallel real‐valued FFT processor.

Source: From Jiang et al. [25]. Reproduced with the permission of American Chemical Society.

Figure 3.8 Molecular reactions for each element of the FFT processor in Figure 3.7.

Source: From Jiang et al. [25]. Reproduced with the permission of American Chemical Society.

In the field of synthetic chemical circuits, DNA‐based logic gates played a crucial role. One interesting idea is to develop DNA circuit construction techniques and scale it up, which can help us to build larger molecular circuits systematically. In [35], starting from simple building blocks called DNA gate motif, the authors developed an abstract model for the design of large‐scale DNA circuits. The authors show that following the proposed method, circuits such as feed forward digital circuits can be effectively constructed. Stemming from this construction method, Qian and Winfree [36] presents a design of relatively complex digital logic circuits in DNA computing systems. The design is based on dual‐rail representation [18] and “seesaw” gate motif [35,36], which enables logic operations such as AND, OR, and NOT to be implemented by DNA strand displacement reactions. Such logic gates are proved cascadable, and a four‐bit square root DNA circuit is constructed as a design case to validate the scalability and computing capability of DNA logic circuits.

In 2011, an experimental method to implement neural network computation was proposed [37]. With DNA displacement reactions that implement linear threshold function, traditional computation in neural networks can be performed in molecular computing systems. A simple but interesting experiment called Hopfield associative memory validates the functionality of the system. In 2018, a more powerful DNA reaction implementation of neural network computing was provided [38]. Based on winner‐take‐all mechanism [39], computation in neural networks such as weighted summation and thresholding of binary input data can be performed in DNA reactions with high accuracy. An example of recognizing handwritten numbers from Modified National Institute of Standards and Technology (MNIST) database [40] shows that such seemingly complex computations are able to be handled by DNA strand displacement reactions. Though training in these two works is performed in silicon‐based computers, the applications show the potential of DNA materials in building scaled functional computing systems. In 2018, implementation of probabilistic switching circuits based on DNA strand displacement reactions was proposed [41]. In the fabricated system, input signals can be converted to output signals with predefined probabilities, and experiments proved the functionality of such DNA circuits.

To conclude, the designs of molecular computing systems present a design hierarchy. Starting from employing data representation methods such as fractional encoding, researchers build basic molecular circuit modules such as logic gates and clock generator. Based on that, more complex functions, e.g. DSP and neural network computation, can be realized. There are various applications in silicon‐based hardware that can be implemented by the interaction of chemical materials, which will be the main focus of future research.