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Figure 3.4 Diagrams of the DNA implementation models. (a) Mapping model of bimolecular reactions in [6]. The two reactants are taken into the system using the first two reactions: one reversible and one irreversible, and the third irreversible reaction displaces the products of the formal reaction.

      Source: Modified from Soloveichik et al. [6].

      (b) Reaction design of A + B → C. By cascading several displacement reactions, the output is eventually displaced, and kinetic features are well reserved.

      Source: Adapted from Chen et al. [7].

      (c) Implementation of bimolecular reversible formal reaction.

      Source: Adapted from Qian et al. [16].

      In conclusion, from the perspective of theoretical computer science, chemical materials are powerful computing tools. It is capable of performing universal computing, and its programmability can be utilized by designers to create computing systems with desired functions. To fully exploit the computing power of chemical materials, there are researches focusing on the features and organization methods of CRNs and the wet experimental implementation of such systems. More applications of molecular computing are expected in future works.

3.2 Application‐Specific DNA Circuits

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.

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