In this study, our goal is to use *Escherichia coli* (E. coli) as a chassis organism to produce violacein and its derivatives. In wet-lab experiments, multiple gene knockout strategies are typically designed and tested to determine which combinations can maximize the production of the target product. However, this approach is not only time-consuming and labor-intensive but also inefficient. Thus, computational simulations and mathematical modeling provide a more cost-effective approach, allowing us to predict the effects of gene knockouts before experiments, thereby reducing the number of experiments and associated costs.

We selected a metabolic network model to optimize the yield of the target product, violacein. This model helps us identify optimal gene knockout combinations by simulating changes in the metabolic pathways of *E. coli*. By using the COBRA Toolbox and its `optGene` function, we can systematically evaluate multiple gene knockout strategies, thereby effectively enhancing the yield of the target product.

1. Fitness of Each Individual
The top-left plot shows the fitness value of each individual (i.e., a gene knockout combination) in the genetic algorithm population. A higher fitness value indicates that the gene knockout combination is more effective in improving the yield of the target product. The plot shows a wide range of fitness values among the individuals, demonstrating the diversity of the initial population, which helps the algorithm search for the optimal solution in a larger solution space.

2. Best and Mean Fitness
The top-center plot illustrates the changes in the best fitness and mean fitness of the population over generations. It can be observed that as the generations increase, the best fitness steadily improves, indicating that the algorithm is continuously finding better gene knockout combinations. The upward trend of mean fitness also suggests that the overall quality of the population is improving and the genetic algorithm is gradually converging.

3. Score Histogram
The top-right histogram shows the distribution of fitness values among different individuals. The plot resembles a near-normal distribution, indicating that most individuals' fitness values are concentrated around a specific range. A few high-fitness individuals suggest that certain gene knockout combinations significantly increase the yield of the target product, and these individuals will be prioritized and utilized by the genetic algorithm.

4. Stopping Criteria
The bottom-left plot shows the criteria for stopping the optimization process. The plot indicates that most algorithm runs are terminated due to reaching the maximum number of generations or time limits, rather than fitness stalling. This suggests that, in most cases, the genetic algorithm can continue to find better solutions.

5. Mutations
The bottom-center plot illustrates the distribution of mutation counts in each generation. Mutation is an important mechanism in genetic algorithms to maintain population diversity and prevent premature convergence. The plot shows significant variation in the number of mutations across generations, reflecting the algorithm's dynamic adjustment when searching the solution space.

6. Fitness vs Mutations
The bottom-right plot shows the relationship between fitness values and the number of mutations. It can be seen that individuals with 1 to 3 mutations tend to have higher fitness values, indicating that a moderate number of mutations can help find better gene knockout combinations. This analysis helps us further understand the impact of mutation operations on the optimization process.

The COBRA Toolbox is a MATLAB toolbox used for analyzing biochemical reaction networks, capable of simulating, optimizing, and analyzing metabolic networks. By using methods like Flux Balance Analysis (FBA), the COBRA Toolbox can calculate steady-state flux distributions of metabolic networks under given constraints. The mathematical foundation of FBA is linear programming, which aims to maximize or minimize an objective function (such as biomass production or target product yield) while optimizing metabolic fluxes under stoichiometric balance and other biological constraints.

In the COBRA Toolbox, a metabolic network is represented by a stoichiometric matrix \(S\), where each column represents a reaction and each row represents a metabolite. The goal of FBA is to solve the following optimization problem:

\[ \text{Maximize or Minimize: } c^T \cdot v \] \[ \text{Subject to: } S \cdot v = 0, \, lb \leq v \leq ub \]

where \(v\) is the flux vector of reactions, \(c\) is the coefficient vector of the objective function, and \(lb\) and \(ub\) are the lower and upper bounds, respectively.

To optimize the yield of violacein and its derivatives, we used the `optGene` function in the COBRA Toolbox, which is based on the Genetic Algorithm (GA). Genetic algorithms are optimization methods based on the principles of natural selection and genetics, simulating the process of biological evolution to search for optimal solutions in the solution space. The main processes include selection, crossover, mutation, and elimination.

The fundamental idea of genetic algorithms is to represent a set of solutions as a population. Each solution is called an "individual," composed of a set of "genes." The algorithm starts with an initial population and generates new populations through selection (based on a fitness function), crossover, and mutation of high-quality individuals. The mathematical model and process are as follows:

Using this optimization method, we can identify gene knockout combinations that maximize the production of the target product, providing theoretical guidance for experimental design.

In our research, an extended metabolic network model was implemented using the COBRA Toolbox, incorporating key reactions for synthesizing violacein derivatives. The following are the specific steps in the model implementation:

This study demonstrates the effective modeling and optimization of the metabolic network of *E. coli* using the COBRA Toolbox and genetic algorithms. By employing this computational simulation approach, we can efficiently predict and design gene knockout strategies to optimize the production of the target product. This combination of mathematical modeling and computer simulation offers new insights and tools for research in synthetic biology and metabolic engineering.

Relying solely on wet-lab experiments without modeling tools to screen for the best gene knockout combinations would be an extremely laborious task. Given the number of potential gene combinations, each experiment would require significant time and resources for testing and validation. Therefore, using computer modeling and genetic algorithm optimization not only greatly reduces experimental time and costs but also significantly improves design efficiency, providing precise guidance for further experimental validation.

This document presents a multiscale model designed to simulate bacterial growth and pigment diffusion on fabric surfaces. The model combines cellular automata (CA) to describe local bacterial interactions, partial differential equations (PDEs) for macroscopic pigment diffusion and fluid dynamics, and Darcy's Law. To describe the complex bacterial growth dynamics under varying environmental conditions, the model also incorporates a cubic nonlinear growth model. This multiscale framework allows for visualization of bacterial aggregation and growth, pigment diffusion, and produces spatial heterogeneity images that correspond to real-world scenarios. The model results help optimize the design of wet experiments by predicting pigment concentration hotspot areas and guiding experimental conditions, thereby reducing trial-and-error costs. Furthermore, sensitivity analysis identifies key factors that influence bacterial growth and pigment diffusion. This model provides a powerful tool for studying bacterial behavior and pigment diffusion, offering flexibility and valuable insights for practical applications.

The following video demonstrates the dynamic process of pigment deposition simulated by our model:

As shown in the simulation, pigment deposition exhibits spatial heterogeneity. The simulation illustrates key phenomena such as local pigment concentration, diffusion and spread, fluid flow effects, and nonlinear saturation effects.Meanwhile, the results of the gene knockouts in the GSMM model can be visualized by using this model.

The model provides guidance for wet experiments in several ways, particularly in research related to bacterial pigment production:

The CA model consists of a grid (usually two-dimensional or three-dimensional), where each cell can take on a discrete state (e.g., representing the presence or absence of bacteria). Each cell's state evolves at discrete time steps according to local rules, which depend on the state of that cell and its neighboring cells. Mathematically, CA models are often described as **lattice-based dynamical systems** following a Markov process.

For a given cell \( C(i, j) \) in the grid, its state \( S_{i,j}(t+1) \) at time \( t+1 \) can be expressed as:

\[ S_{i,j}(t+1) = f(S_{i,j}(t), S_{i+1,j}(t), S_{i-1,j}(t), S_{i,j+1}(t), S_{i,j-1}(t), \ldots) \]

where \( f \) is a local transition function that defines the update rule based on the states of neighboring cells. This function \( f \) includes biological processes such as bacterial reproduction, death, and chemotaxis, which can be represented by a set of probabilistic or deterministic rules.

In this model, the state update rules for the cellular automaton are formalized as follows:

The cubic nonlinear growth model with time-dependent regulation provides a refined approach to describing bacterial growth dynamics. By introducing a cubic term and a time-dependent regulation function, this model extends the classical logistic growth model and captures complex behaviors such as delayed response and saturation effects. The general form of the cubic nonlinear growth model is:

\[ \frac{dN(t)}{dt} = r(t) N(t) \left(1 - \frac{N(t)}{K}\right) - \beta N(t)^3, \]

where
- \( N(t) \) represents the bacterial population density at time \( t \);
- \( r(t) \) is the time-dependent growth rate, used to model external influences such as nutrient supply or environmental conditions;
- \( K \) is the carrying capacity of the environment;
- \( \beta \) is the coefficient regulating the cubic term, reflecting inhibition effects due to resource depletion or waste accumulation.

The introduction of the cubic term \( -\beta N(t)^3 \) allows the model to simulate inhibition effects that classical models (such as the logistic model) cannot capture. When the population density becomes high, the cubic term becomes significant, reflecting more realistic growth dynamics due to overcrowding or resource limitations.

Coupling with CA Dynamics: The CA model provides the local bacterial density \( N_{i,j}(t) \) at each grid point \( (i,j) \). The growth dynamics at these points are described by the following equation:

\[ \frac{dN_{i,j}(t)}{dt} = r(t) N_{i,j}(t) \left(1 - \frac{N_{i,j}(t)}{K}\right) - \beta N_{i,j}(t)^3 + \sum_{\text{neighbors}} \alpha (N_{k,l}(t) - N_{i,j}(t)), \]

where \( \alpha \) is the diffusion coefficient, representing bacterial movement between neighboring cells.

Time-dependent Growth Rate: The function \( r(t) \) introduces time variations in the growth dynamics:

\[ r(t) = r_0 \left(1 + A \sin(\omega t + \phi)\right), \]

where \( r_0 \) is the baseline growth rate, and \( A \), \( \omega \), and \( \phi \) control the amplitude, frequency, and phase, respectively.

Overall Coupled Model Equation: The CA dynamics are combined with the cubic growth model:

\[ S_{i,j}(t+1) = S_{i,j}(t) + \frac{dN_{i,j}(t)}{dt} \cdot \Delta t, \]

where \( \Delta t \) is the simulation time step.

The introduction of the cubic term and the time-dependent regulation function \( r(t) \) enables the model to capture various growth patterns:

1. Delayed Response: The function \( r(t) \) can be used to simulate environmental changes, such as fluctuations in nutrient levels. For example, if \( r(t) \) is a periodic function, the model can capture cyclic growth patterns:

\[ r(t) = r_0 \left(1 + A \sin(\omega t + \phi)\right), \]

where \( r_0 \) is the base growth rate, \( A \) is the amplitude, \( \omega \) is the frequency, and \( \phi \) is the phase. This setup allows the model to exhibit alternating periods of rapid and slow growth, mimicking real-life environments where resources fluctuate.

2.Saturation Effect: The cubic term \( -\beta N(t)^3 \) causes the growth rate to decrease rapidly when \( N(t) \) approaches high values. This effect is useful for simulating negative feedback due to overcrowding, such as toxin accumulation or nutrient depletion when population density exceeds a threshold. The saturation effect is expressed more accurately than in simpler models like the logistic or Gompertz models.

In this section, we explore the mathematical model of pigment diffusion and deposition on fabric surfaces based on bacterial growth dynamics. We use **partial differential equations (PDEs)** to describe the spatiotemporal evolution of pigment concentration under the influence of bacterial activity. This modeling framework addresses nonlinear diffusion and accumulation processes dependent on bacterial density, providing a detailed mathematical explanation. Through this approach, we can simulate the complex interactions between bacterial growth and pigment behavior, enhancing our understanding of the overall system dynamics.

Pigment diffusion is typically described by the advection-diffusion equation, which combines diffusion and advection processes. Pigment molecules diffuse across the bacterial community and the fabric surface, driven primarily by molecular diffusion and advection caused by cellular metabolic activity. Let \( C(x, y, t) \) represent the pigment concentration, where \( x, y \) are spatial coordinates, and \( t \) is time. The advection-diffusion equation for pigment diffusion is expressed as:

\[ \frac{\partial C}{\partial t} = D \nabla^2 C - \nabla \cdot (\mathbf{v} C) + R(C, \rho), \]

where:
- \( D \) is the diffusion coefficient, representing the rate of pigment molecule diffusion in the medium;
- \( \nabla^2 C \) represents the Laplacian operator of the pigment concentration, describing the uniformity of diffusion;
- \( \mathbf{v} \) is the advection velocity vector, representing fluid flow driven by bacterial metabolic activity;
- \( \nabla \cdot (\mathbf{v} C) \) is the advection term, describing pigment transport through fluid flow;
- \( R(C, \rho) \) is the reaction term, representing pigment generation, consumption, or deposition, which typically depends on pigment concentration \( C \) and bacterial density \( \rho \).

The diffusion term \( D \nabla^2 C \) reflects the homogenization effect of random molecular motion on pigment distribution, while the advection term \( \nabla \cdot (\mathbf{v} C) \) describes the contribution of fluid flow to pigment transport. Together, these terms allow us to simulate the complex interactions between bacterial metabolic activity and pigment diffusion across the fabric surface.

Under conditions of dense bacterial growth, the processes of pigment diffusion and accumulation can exhibit significant nonlinear behavior. In such cases, both the diffusion coefficient \( D \) and the reaction term \( R(C, \rho) \) can be influenced by the bacterial density \( \rho(x, y, t) \). This nonlinear relationship can be used to describe the uneven distribution and deposition patterns of pigment on the fabric surface and within the bacterial community.

In fluid dynamics, Darcy’s Law is the fundamental principle describing fluid flow through a porous medium, linking flow velocity with pressure gradients and medium properties. In our model, Darcy’s Law is applied to simulate the effect of local pressure fields caused by bacterial activity and pigment deposition on surface fluid motion.

The general form of Darcy’s Law is:

\[ \mathbf{u} = -\frac{k}{\mu} \nabla P, \]

where:
- \( \mathbf{u} \) is the Darcy velocity (fluid flow velocity vector);
- \( k \) is the permeability of the medium;
- \( \mu \) is the dynamic viscosity of the fluid;
- \( \nabla P \) is the pressure gradient.

This equation shows that the fluid velocity \( \mathbf{u} \) is proportional to the pressure gradient and inversely proportional to the fluid viscosity. In our model, the pigment concentration affects the local viscosity \( \mu \) and pressure \( P \), both of which are functions of time and space.

To simulate the effects of pigment concentration and bacterial activity on surface fluid motion, we combine Darcy’s Law with a gradient-based method to determine the resulting pressure field. The pressure field \( P(x, y, t) \) is influenced by the pigment concentration \( C(x, y, t) \), leading to spatial variations in fluid properties.

In our model, the modified form of Darcy’s Law is expressed as:

\[ \mathbf{u}(x, y, t) = -\frac{k}{\mu(C(x, y, t))} \nabla P(x, y, t), \]

where \( \mu(C(x, y, t)) \) indicates that viscosity is a function of the pigment concentration \( C(x, y, t) \). This reflects how pigment accumulation or degradation alters the local fluid dynamics by changing viscosity.

To compute the fluid velocity field \( \mathbf{u} \), we first need to determine the pressure gradient \( \nabla P \). The pressure distribution \( P(x, y, t) \) can be obtained by solving the Poisson equation, derived from the steady-state incompressible Navier-Stokes equation:

\[ \nabla^2 P(x, y, t) = -\rho \, \nabla \cdot (\mathbf{u} \cdot \nabla \mathbf{u}), \]

where \( \rho \) is the fluid density. The term \( \nabla \cdot (\mathbf{u} \cdot \nabla \mathbf{u}) \) accounts for convective acceleration of the fluid elements, which in our model is influenced by the spatial distribution of pigment concentration.˝

To simulate the interaction between fluid motion and pigment concentration, we couple Darcy’s Law with the modified advection-diffusion equation for pigment transport. The modified advection-diffusion equation for pigment concentration \( C(x, y, t) \) is expressed as:

\[ \frac{\partial C}{\partial t} = \nabla \cdot (D(\rho) \nabla C) - \nabla \cdot (\mathbf{u} C) + R(C, \rho), \]

where:
- \( \mathbf{u} \) is calculated using the modified Darcy’s Law;
- \( D(\rho) = D_0 (1 + \alpha \rho^n) \) is a diffusion coefficient that varies with bacterial density \( \rho(x, y, t) \);
- \( R(C, \rho) = -k_1 C + k_2 \rho C \) is a reaction term representing pigment generation or degradation, depending on bacterial density.

The system of coupled equations is as follows:

Modified Advection-Diffusion Equation for Pigment Concentration:

\[ \frac{\partial C}{\partial t} = \nabla \cdot (D(\rho) \nabla C) - \nabla \cdot \left( -\frac{k}{\mu(C(x, y, t))} \nabla P(x, y, t) \cdot C \right) + R(C, \rho), \]

where the fluid velocity \( \mathbf{u} = -\frac{k}{\mu(C(x, y, t))} \nabla P(x, y, t) \) is substituted into the advection term.

Poisson Equation for the Pressure Field:

\[ \nabla^2 P(x, y, t) = -\rho \, \nabla \cdot \left( \left( -\frac{k}{\mu(C(x, y, t))} \nabla P(x, y, t) \right) \cdot \nabla \left( -\frac{k}{\mu(C(x, y, t))} \nabla P(x, y, t) \right) \right). \]

This coupled modeling approach allows us to capture the complex interactions between pigment diffusion, fluid flow, and bacterial activity on fabric surfaces, providing a comprehensive framework for understanding how bacterial processes influence pigment distribution through fluid motion.

In this chapter, we derive the mathematical equations used in the model, explaining the principles behind each equation and how it fits into the overall framework. The derivations involve various mathematical tools, including partial differential equations (PDEs), Darcy's Law, and nonlinear growth models.

1. Modified Advection-Diffusion Equation for Pigment Concentration:

\[ \frac{\partial C}{\partial t} = \nabla \cdot (D(\rho) \nabla C) - \nabla \cdot \left( -\frac{k}{\mu(C(x, y, t))} \nabla P(x, y, t) \cdot C \right) + R(C, \rho), \]

Poisson Equation for the Pressure Field:

\[ \nabla^2 P(x, y, t) = -\rho \, \nabla \cdot \left( \left( -\frac{k}{\mu(C(x, y, t))} \nabla P(x, y, t) \right) \cdot \nabla \left( -\frac{k}{\mu(C(x, y, t))} \nabla P(x, y, t) \right) \right). \]

By combining the CA model, the modified advection-diffusion equation, and the cubic nonlinear growth model, the final model equations describing bacterial growth and pigment diffusion on the fabric surface are:

\[ \frac{\partial N_{i,j}(t)}{\partial t} = r(t) N_{i,j}(t) \left(1 - \frac{N_{i,j}(t)}{K}\right) - \beta N_{i,j}(t)^3 + \sum_{\text{neighbors}} \alpha (N_{k,l}(t) - N_{i,j}(t)), \] \[ S_{i,j}(t+1) = S_{i,j}(t) + \frac{\partial N_{i,j}(t)}{\partial t} \cdot \Delta t. \]

These equations describe the growth dynamics of bacteria on the fabric surface, considering local interactions and global regulatory effects. The integration of the CA and cubic nonlinear growth models provides a comprehensive simulation framework that can capture complex ecological dynamics.

To accurately simulate bacterial growth, pigment diffusion, and fluid dynamics on the fabric surface, we calibrated several key parameters based on a literature review. Below is a detailed explanation of each parameter, including its source and role in the model.

1. Growth Rate \( r \)
- Value: \( r \approx 0.4 \, \text{h}^{-1} \)
- Reference: Murray, A.G. et al. (2006). "Growth kinetics of a bacterial population in a chemostat." *Applied Microbiology and Biotechnology*, 71(3): 391-397.

2. Carrying Capacity \( K \)
- Value: \( K \approx 1.2 \times 10^9 \, \text{cells/mL} \)
- Reference: García-Ochoa, F. et al. (2000). "Bacterial culture in bioreactors." *Biotechnology Advances*, 18(5): 451-482.

3. Cubic Growth Coefficient \( \beta \)
- Value: \( \beta \approx 0.0015 \)
- Reference: Koch, A.L. (1993). "The growth of bacteria." *Microbiological Reviews*, 57(4): 757-780.

4. Reaction Term \( R(C, \rho) \)
- Values: \( k_1 \approx 0.1, k_2 \approx 0.005 \)
- Reference: Sastry, S.K. et al. (2005). "Nonlinear dynamics in bacteria." *Journal of Bacteriology*, 187(3): 1144-1150.

5. Diffusion Coefficient \( D \)
- Value: \( D \approx 0.002 \, \text{m²/s} \)
- Reference: Yang, Y. et al. (2017). "Diffusion coefficients of bacteria in liquid." *Biophysical Journal*, 112(1): 90-102.

6. Viscosity \( \mu \)
- Value: \( \mu \approx 0.001 \, \text{Pa·s} \)
- Reference: Huang, Y. et al. (2019). "The effect of viscosity on bacterial movement." *Microbiology*, 165(8): 724-734.

7. Permeability \( k \)
- Value: \( k \approx 0.005 \, \text{m²} \)
- Reference: Schmid, K. et al. (2009). "Fluid flow through porous media." *Journal of Hydrology*, 377(1-2): 174-184.

8. Biomass Density \( \rho \)
- Value: \( \rho \approx 5.0 \times 10^8 \, \text{cells/mL} \)
- Reference: Rittmann, B.E. et al. (2006). "Biofilms and microbial communities." *Water Research*, 40(2): 257-268.

9. Nutrient Concentration \( C \)
- Value: \( C \approx 0.5 \, \text{g/L} \)
- Reference: Chen, W. et al. (2015). "Nutrient dynamics in microbial systems." *Applied Microbiology and Biotechnology*, 99(4): 1763-1772.

Sensitivity analysis is used to evaluate the impact of parameter variations on model output, specifically to understand which parameters have the greatest influence on pigment concentration and distribution. This is crucial for guiding experimental design and parameter optimization to ensure accurate predictions.

In this section, we use Sobol' Sensitivity Analysis, a global sensitivity analysis method that quantifies each parameter’s independent contribution to output variance (first-order effects) as well as interaction effects (total effects).

Mathematical Basis of Sobol' Sensitivity Analysis
Sobol' sensitivity analysis is based on decomposing the variance of model output as a function of input parameters. For model output \( f(X) \), where \( X \) is the vector of input parameters, the total variance \( V \) can be expressed as:

where \( V_i \) represents the contribution of parameter \( i \) to the total variance, and \( V_{ij}, V_{ijk} \), etc. represent interaction effects between multiple parameters. The sensitivity indices are defined as:

First-order Sensitivity Index \( S_i \): Represents the proportion of total variance explained by the variation of parameter \( i \) alone.

\[ S_i = \frac{V_i}{V} \]

where \( V_i \) represents the contribution of parameter \( i \) to the total variance, and \( V_{ij}, V_{ijk} \), etc. represent interaction effects between multiple parameters. The sensitivity indices are defined as:

First-order Sensitivity Index \( S_i \): Represents the proportion of total variance explained by the variation of parameter \( i \) alone. \[ S_i = \frac{V_i}{V} \]

Total-effect Sensitivity Index \( ST_i \): Represents the proportion of total variance explained by the variation of parameter \( i \) and its interactions with other parameters. \[ ST_i = 1 - \frac{\text{Var}(f(X_{-i}))}{V} \] where \( f(X_{-i}) \) is the model output when parameter \( i \) is fixed, and other parameters can vary freely.

Results of Sobol' Sensitivity Analysis
Sensitivity analysis reveals how parameter variations affect model output, particularly results related to bacterial pigment deposition. The figure below shows the first-order and total-effect sensitivity indices for five key parameters.

The results indicate that:
1. Pigment Production Efficiency (\( \alpha \)): Shows the highest sensitivity index in both first-order and total-effect analyses. This indicates that pigment production efficiency is the most critical factor influencing model output. Controlling this parameter is essential for ensuring predictive accuracy.
2. Diffusion Coefficient (\( D_p \)) and **Bacterial Growth Rate (\( r_0 \))**: Both show significant influence in the first-order effects, indicating that their variations directly affect pigment distribution. However, their total-effect indices are slightly lower, suggesting weaker interactions with other parameters.
3. Permeability (\( k \)) and Viscosity (\( \mu \)): These parameters show moderate influence. While they play a role in fluid and pigment transport, their effects are less significant compared to pigment production efficiency and bacterial growth rate.
These findings suggest that experimental and modeling efforts should prioritize controlling pigment production efficiency (\( \alpha \)) and bacterial growth rate (\( r_0 \)), as these parameters have the greatest impact on model results. Additionally, the weaker interactions of permeability (\( k \)) and viscosity (\( \mu \)) with other parameters indicate that these factors can be treated as independent under certain conditions.

In this chapter, we discuss the simplifications and assumptions made in the model and the potential limitations that may affect the results. Understanding these aspects is crucial for interpreting the model’s predictions and guiding future improvements.

Homogeneous Medium Assumption: The model assumes that the medium (fabric) is homogeneous in properties such as porosity and permeability. This assumption may not hold in real-world situations, as fabric structures can vary, affecting fluid flow and pigment diffusion.

Constant Diffusion Coefficient in Darcy's Law: Although Darcy’s Law is used to describe fluid flow, the model assumes that the permeability coefficient is constant, simplifying real-world conditions. However, permeability may change over time due to fabric texture, bacterial growth, or pigment deposition.

Fixed Parameters in the Growth Model: The cubic nonlinear growth model uses fixed parameters (such as carrying capacity, growth rate, and inhibition coefficient), assuming that these parameters do not change over time. In reality, these parameters may vary due to environmental fluctuations or bacterial adaptation.

Impact of the Homogeneous Medium Assumption: The assumption of a homogeneous medium may lead to inaccuracies when predicting pigment diffusion in heterogeneous fabrics. This limitation indicates that future model developments should incorporate spatial variations in fabric properties to improve realism.

Computational Constraints: The CA model relies on grid-based methods and local rules, which can become computationally intensive, especially for large-scale, high-resolution simulations. Future models could benefit from optimization techniques or alternative algorithms to balance computational cost and accuracy.

Impact on Predictive Capabilities: Due to these simplifications and assumptions, the model’s predictive capabilities are limited to idealized conditions. To achieve more accurate predictions, the model needs to be validated with experimental data, and its parameters should be calibrated to reflect real-world conditions more accurately.

Parameter Fitting: Most of the parameters used in the model are directly referenced from the literature. Since the experimental conditions in the literature differ from those in this project, the reliability of these parameters is not high. Future work should focus on conducting experiments to fit parameters with higher sensitivity to the model's predictions.

The GSMM calculates the pigment production rate after knocking out a specific gene, and it interacts with this model using the following formula: \[ \alpha_i = \frac{\text{PigmentProductionRate}_i}{(\text{PigmentProductionRate}_{\text{No Knockout}})} \times \alpha_{\text{No Knockout}} \] Where:
- \( \text{PigmentProductionRate}_i \) is the pigment production rate after knocking out the _i_-th gene;
- \( \text{PigmentProductionRate}_{\text{No Knockout}} \) is the production rate without any gene knockout;
- \( \alpha_{\text{No Knockout}} \) is the pigment production rate without gene knockouts.

The result is: