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
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
