Computational Molecular Biology 2025, Vol.15, No.4, 193-207 http://bioscipublisher.com/index.php/cmb 195 oscillations. Otherwise, the system will soon "lose steam" and fall into a steady state. To make oscillations more stable, researchers often need to artificially increase delays or enhance nonlinearity during the design process. The behavior of the feedforward loop is more like a "brief flash". Take the incoherent feedforward loop (I1-FFL) as an example. The upstream regulatory factor X, on the one hand, directly activates the target gene Z, and on the other hand, it activates an intermediate regulatory factor Y, which in turn inhibits Z. So, the expression of Z is first rapidly pushed up by X, but as Y gradually accumulates, it will then push Z down. The result is a brief "pulse-like" expression. Both experiments and models show that this structure will only trigger a one-time response under continuous signals and then automatically shut down, making it very suitable for cells to recognize transient stimuli. 3 Theoretical Basis of Mathematical Modeling 3.1 Application of ordinary differential equations (Odes) in genetic circuit modeling Ordinary differential equations (Odes) are almost the "old tools" for studying the dynamics of genetic circuits. Its idea is very straightforward. It regards gene expression as a series of reactions and uses equations to describe the changes in mRNA or protein concentrations over time (Turpin et al., 2023; Spartalis et al., 2024). The advantage of the ODE model lies in its ability to clearly demonstrate which parameters in the system are the most critical, such as synthesis rate, degradation rate, and regulation intensity, and how they jointly determine behavior. However, the initial assumptions of such models were actually quite idealized. It regards molecular concentration as a continuous quantity, ignoring the random fluctuations when the number of molecules is small, nor does it take into account the dilution effect brought about by cell growth or division. However, despite this, ODE remains the most commonly used modeling method, featuring fast calculation, clear form, and easy analysis of results. 3.2 Stochastic models and noise analysis In the real cellular world, gene expression does not strictly follow the average value. When the number of molecules is small, even tiny thermal noise can be amplified, resulting in significant expression differences among different cells (Goetz et al., 2025). To describe this randomness clearly, researchers usually do not rely solely on deterministic models but introduce stochastic modeling methods. The most commonly used theoretical framework is the master equation of chemistry (CME), which describes the time evolution of a system in a certain state (such as the number of molecules) in a probabilistic way. However, this equation is too complex to be solved directly. Therefore, people more often use various approximate or numerical methods, such as Monte Carlo simulation, Gillespie algorithm, and fluctuation decomposition approximation, etc. The Gillespie algorithm is the most classic one. It simulates molecular events one by one according to the probability of each reaction occurring, allowing the evolution trajectory of the system to naturally "emerge". With it, one can observe the random fluctuations of gene circuits on a time scale. For instance, in a negative feedback loop, feedback can be used to suppress the fluctuation range of the target protein expression, thereby verifying the robustness of the system (Müller et al., 2025). To theoretically analyze these fluctuations, researchers have also developed analytical methods such as linear noise approximation (LNA), breaking down the system into average trends and adding small random disturbances, and then linearizing to solve for noise intensity. This method can calculate indicators such as the Fano factor and correlation function to measure the noise level of the system. Through these analyses, it was found that negative feedback often effectively reduces noise, while positive feedback tends to amplify noise and even cause differentiation in cell populations. But noise is not always a bad thing. For instance, some cell switching phenomena, where gene expression randomly jumps between high and low states, can only be explained by random models. 3.3 System dynamics model based on network topology In addition to describing molecular changes using reaction rates, researchers often look at the problem from a structural perspective - no longer focusing on the details of each reaction, but studying the network shape of "who regulates whom" among genes. Such a model pays more attention to the overall organization of the system rather than the numerical value of each parameter. Several common methods include Boolean networks, qualitative network models, and topology analysis based on graph theory. The idea of Boolean networks is very straightforward: simplify each gene state to "on (1)" or "off (0)", and use logical rules to determine the state at the
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