Computational Molecular Biology 2025, Vol.15, No.4, 193-207 http://bioscipublisher.com/index.php/cmb 202 2020). Meanwhile, they also carefully controlled the degradation rate of the protein, keeping the oscillation period on a scale of several hours, which was convenient for experimental observation. However, the original Repressilator was not perfect. In Escherichia coli, although periodic oscillations have indeed been observed, the phases between cells are not synchronized, and the amplitude of the oscillations also decays over time. Later, many improved versions emerged. For instance, some people have attached degradation labels to the repressor proteins to make them break down more quickly, thereby reducing the cycle fluctuations. Some people have joined the positive feedback pathway or used group communication signals (such as acyl high-serine lactone) to achieve oscillation synchronization between cells (Zhang et al., 2022). These adjustments make the system more stable, neater and closer to the ideal periodic behavior. 7.2 Mathematical equations and parameter settings Based on the design topology of Repressilator, we can write the corresponding dynamic equation for each gene. Let $[A]$, $[B]$, and $[C]$represent the concentrations of repressor proteins encoded by genes A, B, and C, respectively. Then their generation and degradation can be described by the following ordinary differential equation: Among them, $\alpha_A,\alpha_B,\alpha_C$represent the maximum synthesis rate of each gene in the absence of repression (determined by promoter strength and translation efficiency), and $\beta$is the assumed constant of the same protein degradation (dilution) rate. The function $f_X([\text{repressor}])$describes the inhibitory effect of repressor proteins on promoter activity. Common Hill function forms: Here, $K_C, K_A, and K_B$represent the concentrations at which the repressor protein reduces the promoter activity to half (a parameter for measuring repressor affinity), and $n$is the Hill coefficient, reflecting the degree of the synergistic effect. For dimer repression, generally, $n=2$is taken. The above system of equations constitutes the basic model of Repressilator. In terms of parameter selection, to achieve oscillation, it is necessary to make the closed-loop gain large enough and introduce an effective delay. The general experience for adjusting parameters is that high expression intensity (large $\alpha$) and high repression efficiency (small $K$and large $n$) help to meet the Hopf bifurcation conditions of oscillations (Verdugo, 2018), while a moderate degradation rate determines that the oscillation period is within an appropriate range. If the degradation is too slow and the system lag is too strong, the cycle will be too long and even chaos may occur. The degradation is too rapid and reduces the delay, which is not conducive to continuous oscillation (Sun et al., 2023). Researchers accelerated degradation by adding ssrA tags at the end of the protein sequence in their experiments, which was equivalent to increasing beta, and ultimately achieved an oscillation period of approximately 2 to 3 hours. In the model, the corresponding $\beta$is set to be of the same order as the cell division rate (approximately $0.5\ \text{h}^{-1}$). The maximum transcription rate $\alpha$can be estimated based on the strength of the promoter and the available amount of RNA polymerase, such as dozens of protein molecules per minute. To reflect qualitative behavior, parameters are often dimensionless. In Elowitz's original paper,
RkJQdWJsaXNoZXIy MjQ4ODYzNA==