Computational Molecular Biology 2025, Vol.15, No.4, 193-207 http://bioscipublisher.com/index.php/cmb 198 they can only be calculated step by step through "integration" by a computer. The most commonly used ones are the time-domain integration algorithms for ordinary differential equations, such as the simplest Euler method or the more refined Runge-Kutta method. The Euler method is quite straightforward in its approach, using the current derivative as a linear approximation to move forward one step. However, it is not very accurate, and once the equation becomes too "rigid", the result is prone to divergence. In contrast, the Runge-Kutta series, especially the fourth-order RK4, is more stable and accurate. It calculates the slope four times at each step and takes the weighted average. In this way, the error is much smaller, and even if the step size is slightly larger, a stable result can be achieved (Manninen et al., 2006). However, not all systems are honest. Some genetic circuits are very "stubborn", featuring both fast-changing processes, such as protein degradation which is completed within a few minutes, and slow-changing processes, such as cell growth dilution which takes several hours. When encountering rigid systems with such huge time scale differences, explicit algorithms often fail and implicit methods have to be adopted, such as the implicit Euler or Gear method (BDF method). They are more troublesome to calculate, but in this case, they can keep the values stable and won't accidentally crash. For non-rigid cases, Runge-Kutta is sufficient. However, if efficiency is to be pursued, some people also use adaptive step size algorithms, such as Runge-Kutta-Fehlberg. It can automatically adjust the step size according to the curvature of the solution. Take small steps where the changes are drastic and large steps when it is stable. This is fast and accurate (Pájaro et al., 2017). In addition to the integration of the ODE, there is another type of stochastic simulation. For instance, the well-known Gillespie algorithm simulates each event one by one with great precision, but at the cost of an explosive amount of computation. This led to the emergence of approximate algorithms such as the τ -transition method and the virtual response method, which package and process many small events, making long-term simulations feasible. If the model takes spatial distribution into account, it becomes even more complex and requires the use of partial differential equations (Pdes) or spatial stochastic models for calculation. The finite difference method and the finite element method can all be put into use. Or use grid random walks to simulate molecular diffusion. If one wants to study spatial patterns within a group, such as how bacteria self-organize to form fringes, cellular automata or Agent-based models must be used. Such methods can make complex spatial behaviors clear at a glance. 5.2 Software platforms and computing tools (MATLAB, COPASI, BioNetGen, etc.) To simulate and analyze genetic circuits more conveniently, researchers have long developed a complete set of software "toolboxes". Different platforms have their own preferences. Some pursue flexibility, some focus on visualization, and others are specifically designed to serve automation. For instance, MATLAB is almost one of the most common numerical computing environments. The Simulink and SimBiology toolboxes that come with it enable users to build models directly using reaction diagrams. The system automatically generates differential equations, making it very convenient to run simulations, perform parameter scans, and conduct sensitivity analyses (Figure 1) (Sequeiros et al., 2023). What many researchers like about MATLAB is that it can not only perform theoretical calculations but also be used to verify experimental data. A complete process does not require a change of environment. However, MATLAB is, after all, commercial software, and there are quite a few free options available. COPASI (Complex Pathway Simulator) is a fully functional open-source option. It supports ordinary differential equations and stochastic simulation, and can perform parameter fitting, steady-state analysis, and even bifurcation analysis. The greatest advantage is the user-friendly graphical interface. Modeling, importing SBML files, and viewing the results directly without writing code (Welsh et al., 2018). Due to its low operational threshold, COPASI is particularly widely used in the modeling of metabolic networks and gene regulatory networks. Another situation is when the model is too complex, involving molecular binding or multi-site modification, and it is almost impossible to write equations manually. At this time, BioNetGen comes in handy. It enables users to define interactions in a regular way, automatically generate reaction networks and equations, and is highly suitable for the design of signal pathways or molecular interaction classes.
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