Computational Molecular Biology 2025, Vol.15, No.4, 193-207 http://bioscipublisher.com/index.php/cmb 203 dimensionless parameters such as $T=\alpha/\beta$(steady-state yield multiple without repression) and $\mu=\text{Protein_decay/mRNA_decay}$were used. Dimensionless models can more clearly analyze the influence of parameter combinations on dynamics. By numerically simulating the above model, the phase diagrams of the system under different parameters can be obtained. For instance, when $K$is fixed and $\alpha$is added, the simulation shows that the system converges to a steady state when $\alpha$is less than a certain threshold. Self-oscillation occurs when the threshold is exceeded, which is consistent with the results of Hopf bifurcation analysis. If $\alpha$continues to increase too much, the oscillation may become irregular or even disappear, corresponding to the occurrence of high-order bifurcations in the model. To improve the consistency between the model and the experiment, some model improvements were proposed in subsequent studies. For instance, by incorporating dissipative processes or RNA dynamics: expand the equation to six (three mrnas + three proteins) to make the phase relationship of oscillations more accurate. However, adding mRNA variables also introduces new parameters such as the degradation rate of mRNA. Before obtaining the corresponding experimental data, typical values in the literature (such as the half-life of mRNA in a few minutes) are generally adopted. Overall, a reasonable set of parameter Settings for the Repressilator model might be: $n = 2 $, $100 \ \ \ alpha \ sim text {nM/h} $, $0.5 \ \ \ beta \ sim text {h} ^ {1} $, $K \ sim 40 \ \ text {nM} $, etc. By numerically integrating the model, the oscillation curve of protein concentration over time can be obtained. Adjusting parameters can change the period and amplitude of oscillation. Model equations and parameters can not only reproduce qualitative behavior but also be used to guide loop improvement. 7.3 Comparison and analysis of simulation results with experimental data When conducting numerical simulations with Repressilator's model, the first thing that can be observed is that familiar sense of rhythm. Under appropriate parameter conditions, the system will automatically enter a nearly sinusoidal periodic oscillation state. The three represser proteins will reach their peaks in sequence, with a temporal difference of approximately $120^\circ$. For instance, after A rises, it quickly reaches its peak, then B starts to rise and A drops. Then C comes in to drive the next cycle. The simulated curve seems to be constantly cycling around a restrictive loop. Based on Fourier analysis, the period $T$is approximately 2.5 hours, which is roughly the same as the 2-3 hours in the single-cell experiments with Escherichia coli. In terms of amplitude, the model can also show similar differences in the experiment - when the parameters change, the oscillation intensity changes (Figure 2) (Park et al., 2024). When approaching the Hopf bifurcation boundary, the oscillation is easily worn away by the noise, and after several rounds, the amplitude decays to a steady state. In the deeper oscillation region, the model provides continuous fixed-amplitude oscillation, corresponding to the kind of "durable" oscillation observed under the improved experimental conditions. To see if the model is reliable or not, we compared the simulation results with the experimental data point by point. Potvin-Trottier et al. (2016) once tracked the fluorescence variation trajectory of single cells. We superimposed the A protein concentration curve predicted by the model on their data. The periods and waveforms of both were quite consistent - both were quasi-sine waves with a period of approximately 200 minutes. It's just that the amplitude of the model is slightly smaller, possibly because there are inherent differences among different cells in the experiment, and some cells oscillate even more vigorously. Later, we added a noise term to the model and ran random simulations using the Gillespie algorithm. As a result, the period and amplitude showed intercellular variations, which were closer to the degree of dispersion observed in the experiment. 8 Application and Future Outlook Synthetic genetic circuits are no longer just conceptual devices in laboratories; they are now being regarded as "molecular logic units" capable of performing specific tasks. In the fields of medicine and bioengineering, its potential is regarded as extremely huge. The most typical example is cell therapy, especially CAR-T therapy. In the past, once T cells were activated, it was like a floodgate that had been opened; it was extremely difficult to close. Later, researchers attempted to incorporate synthetic gene circuits into it and use logical control to make it work "with a brain". The circuit can identify tumor-specific miRNA signals. Only when the correct "tumor
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