Computational Molecular Biology 2025, Vol.15, No.4, 193-207 http://bioscipublisher.com/index.php/cmb 201 However, negative feedback is not omnipotent. Theoretically, it accelerates the system's return to a steady state by changing the roots of the system's characteristic equation, pulling the real part to a negative value. However, if the feedback is too strong and there is a time delay, it may cause the system to over-correct and even start oscillating - at this point, the characteristic roots may pass through the virtual axis, resulting in Hopf bifurcation. That is to say, feedback that is "too frequent" may instead make the system unstable (Hu and Murray, 2019). Therefore, when designing, it is necessary to strike a proper balance between feedback gain and delay. On the contrary, positive feedback is like "adding fuel to the fire", which amplifies the deviation of the system. When a certain gene product can promote its own expression, even a small fluctuation may be rapidly amplified, and the system is pushed in a certain direction. Weak positive feedback can increase the signal output strength, but if it is too strong, it often leads to bistability or multistability, that is, there are two or more stable points in the system - high and low expression states coexist (Sun et al., 2022). This type of mechanism is precisely suitable for constructing gene switches, but from a control perspective, it is very "dangerous" because the system can be easily pushed to another steady state by noise or initial conditions. For this reason, in engineering, a small negative feedback is sometimes added to the strong positive feedback to form compound regulation, so that the system is not too extreme (Loman et al., 2025). 6.3 Bifurcation analysis and multistable phenomena When studying genetic circuits, people often find that the behavior of the system will suddenly "change", as if a certain switch has been activated. This phenomenon is not accidental but rather a result of parameter changes triggering what is called a "bifurcation". The purpose of bifurcation analysis is to figure out when and how a system jumps from one dynamic state to another when its parameters change slowly. For synthetic genetic circuits, there are mainly two common types of bifurcations: saddle bifurcation and Hopf bifurcation. The former usually implies the emergence or disappearance of a new steady state, while the latter marks the birth or extinction of oscillations. Take bistable switches as an example. This system will exhibit different stable structures under different concentrations of inducers. At the beginning, the system had only one steady state. When the concentration of the inducer gradually increases, two stable equilibrium points and one unstable equilibrium point will suddenly appear after a certain critical point - a typical saddle junction bifurcation (Leon et al., 2016). Experimentally, this corresponds to the different states that cells exhibit when the concentration of the inducer is increased or decreased, which is known as the "lag" phenomenon. The reaction trajectories of cells are different when they rise and fall. Through bifurcation analysis, the width of this lag interval can be precisely identified. When designing the gene switch, engineers hope that this area is wide enough so that the system switches more cleanly and is more noise-resistant (Dey and Barik, 2021). 7 Case Study: Repressilator Model for Synthetic Genes 7.1 The biological design and modeling framework of repressilator Repressilator can almost be regarded as the "pioneering work" of synthetic biology. It was the first artificially designed gene oscillator, successfully constructed by Elowitz and Leibler in 2000. The structure of this system is actually very simple - the three genes are connected end to end and inhibit each other. The product of gene A inhibits B, the product of B inhibits C, and C in turn inhibits A, forming a closed loop. Each promoter is controlled by the repressor protein of the upstream gene, and thus the entire circuit forms a delayed negative feedback. It is precisely this delay that makes the system never "catch up with itself", thus causing continuous oscillation (Tyler et al., 2019). The process is roughly as follows: when A begins to be expressed in large quantities, its protein gradually accumulates, and after a period of delay, it inhibits B. After B was suppressed, C took the opportunity to rise. Then the product of C, in turn, inhibits A, causing A to decline. This cycle repeats itself. The system can never stay stable at a certain point but keeps circling along a periodic trajectory, with the concentrations of the three proteins fluctuating one after another. Researchers selected three different sources of repressor proteins - TetR, λcI and LacI - and their respective exclusive promoters back then, so as to avoid cross-interference (Henningsen et al.,
RkJQdWJsaXNoZXIy MjQ4ODYzNA==