Modeling
1 Model
In bacteria, c-di-GMP is an important second messenger[1] that plays a crucial role in regulating the transition from a motile state to an adhesive state. The concentration of c-di-GMP affects bacterial motility, the synthesis of extracellular polymers, and biofilm formation[2]. By regulating the synthesis and degradation of c-di-GMP, bacteria can modulate the formation and stability of their biofilms. This model aims to describe how the c-di-GMP pathway regulates biofilm formation in Acidithiobacillus caldus. The model is structured around four components: the synthesis and degradation of c-di-GMP, the regulation of DGC and PDE, bacterial motility, and the synthesis of EPS, along with bacterial adhesion and biofilm formation.
2.1 Model Assumptions
Synthesis and Degradation of c-di-GMP It is assumed that the synthesis and degradation processes of c-di-GMP are rapid and follow first-order kinetic reactions. The synthesis rate of c-di-GMP is proportional to the concentration of DGC enzymes, while the degradation rate is proportional to both the concentration of PDE enzymes and the concentration of c-di-GMP.
Regulation of DGC and PDE It is assumed that the activity or expression levels of DGC and PDE are regulated by external signals, with changes in these signals linearly affecting the levels of DGC and PDE, which are directly controlled by environmental signals.
Bacterial Motility Assumption It is assumed that bacterial motility is negatively correlated with c-di-GMP concentration, and that motility is the sole determinant of the bacterial adhesion rate, independent of other factors.
EPS Synthesis Assumption It is assumed that the synthesis rate of EPS solely depends on c-di-GMP concentration, and the influence of other metabolic pathways within the bacteria on EPS synthesis rate can be ignored.
Bacterial Adhesion and Biofilm Formation Assumption It is assumed that the rate of bacterial adhesion is determined by bacterial motility and is proportional to the number of available adhesion sites on the surface. It is assumed that the biofilm formation process is limited by the number of bacteria and the EPS synthesis rate and can be described using a logistic growth model. Additionally, it is assumed that the distribution of bacteria and EPS within the biofilm is uniform across regions.
2.2 Sign explanation
$\frac{dG}{dt}$：The rate of change of c-di-GMP concentration $G$ over time $t$
$\alpha_D$：Rate constant for DGC catalyzing c-di-GMP synthesis.
$D$：Activity of DGC.
$\alpha_P$：Rate constant for PDE degrading c-di-GMP.
$P$：Activity of PDE.
$\gamma_G$：Natural degradation rate of c-di-GMP.
$D_0$：Maximum activity of DGC.
$\beta_G$：Inhibition coefficient of c-di-GMP on DGC activity.
$V$：Bacterial swimming speed.
$V_{max}$：Maximum swimming speed of bacteria.
$K_V$：Half-saturation constant, where V is half of its maximum speed when c-di-GMP concentration reaches K_V.
$\frac{dE}{dt}$：Rate of change of EPS synthesis rate $E$ over time $t$
$V_{E_max}$：Maximum rate of EPS synthesis.
$K_E$：Half-saturation constant, where the EPS synthesis rate is half of its maximum value when c-di-GMP concentration reaches $K_E$
$\frac{dA_1}{dt}$：Rate of change of the initial attached bacterial count $A_{1\ }$over time $t$
$\alpha_1$：Rate constant for initial attachment.
$S_{avail}$：Number of available adhesion sites.
$N$：Total number of bacteria in a motile state.
$S_{total}$：Total number of adhesion sites.
$\frac{dA_2}{dt}$：Total number of adhesion sites.$A_2$ over time $t$
$\alpha_2$：Rate constant for irreversible attachment.
$\frac{dB}{dt}$：Rate constant for irreversible attachment.$B$ over time $t$
$r_B$：Growth rate constant of biofilm formation.
$B_{max}$：Maximum thickness of the biofilm.
$r_{B0}$：Basal biofilm growth rate.
$\beta_B$：Coefficient of c-di-GMP concentration affecting biofilm growth rate.
$B_{0max}$：Maximum thickness of the biofilm in the absence of EPS.
$\gamma_B$：Coefficient affecting the maximum thickness of the biofilm due to EPS.
2.3 Model Establishment
2.3.1 Synthesis and Degradation of c-di-GMP
The synthesis of c-di-GMP is primarily mediated by DGC. DGC increases the intracellular c-di-GMP concentration by catalyzing the conversion of GTP to c-di-GMP[3]. The activity of DGC is typically regulated by negative feedback from intracellular c-di-GMP concentration, which can be described by the following equation:
$$ \begin{equation}\frac{dG}{dt}=\alpha_DD-\alpha_PPG-\gamma_GG\end{equation} \tag{2.1} $$2.3.2 Regulation of DGC and PDE
DGC is responsible for synthesizing c-di-GMP, and its activity is negatively regulated by c-di-GMP feedback inhibition. When the intracellular c-di-GMP concentration is high, the activity of DGC decreases. This feedback mechanism primarily functions to prevent the excessive accumulation of c-di-GMP, thereby maintaining system stability. The activity of DGC is modeled as follows:
$$ D=\frac{D_0}{1+\beta_GG} \tag{2.2} $$PDE is responsible for degrading c-di-GMP. In this model, the activity of PDE is assumed to be constant, meaning its degradation effect is stable and does not change with variations in c-di-GMP concentration. This setting helps ensure that c-di-GMP can be effectively degraded over time, supporting system stability and control of c-di-GMP concentration. The specific formula is described as follows：
$$ P=P_0 \tag{2.3} $$2.3.3 Bacterial Motility and EPS Synthesis
The concentration of c-di-GMP has significant regulatory effects on bacterial motility and EPS synthesis. High levels of c-di-GMP typically inhibit motility and promote EPS synthesis, aiding in biofilm formation[4].
1、Regulation of Bacterial Motility
Motility can be measured by the swimming speed of bacteria, and swimming speed V is negatively correlated with c-di-GMP concentration. This can be expressed by the following equation[5]:
$$ V=V_{max}\times(1-\frac{G}{K_V+G}) \tag{2.4} $$2、Regulation of EPS Synthesis
The synthesis rate of EPS is positively correlated with c-di-GMP concentration and can be represented using an equation similar to Michaelis-Menten kinetics:
$$ \frac{dE}{dt}=V_{E_max}\times\frac{G}{K_E+G} \tag{2.5} $$Through the regulation of c-di-GMP concentration, bacterial motility and EPS synthesis maintain a dynamic equilibrium[4]. When c-di-GMP concentration is low, bacteria tend to maintain motility, leading to less EPS synthesis and reduced likelihood of surface adhesion.
2.3.4 Bacterial Adhesion and Biofilm Formation
Biofilm formation is a multi-stage process that begins with the attachment of bacteria to surfaces, gradually forming a three-dimensional structure through EPS synthesis and bacterial proliferation. The biofilm formation model encompasses initial attachment of bacteria, irreversible attachment, EPS synthesis, and the formation of the biofilm's three-dimensional structure.
1、Bacterial Adhesion Model
Bacterial adhesion is the first step in biofilm formation and can be divided into two main stages: initial adhesion and irreversible adhesion. These two stages can be described using the following mathematical models.
a. Initial Adhesion Model
During the initial adhesion phase, the interaction between bacteria and the surface is temporary and reversible. This phase is influenced by bacterial motility and the availability of surface adhesion sites. This process can be described using the following kinetic equation:
$$ \frac{dA_1}{dt}=\alpha_1\times S_{avail}\times V\times N \tag{2.6} $$As time progresses, more bacteria adhere to the surface, leading to a scarcity of adhesion sites, affecting the initial adhesion rate. The availability of adhesion sites decreases over time and can be expressed as:
$$ S_{avail}(t)=S_{total}-A_1 \tag{2.7} $$b. Irreversible Adhesion Model
During the irreversible adhesion phase, bacteria firmly attach to the surface by secreting EPS. The rate of irreversible adhesion can be expressed by the following equation:
$$ \frac{dA_2}{dt}=\alpha_2\times A_1\times E \tag{2.8} $$2、Biofilm Formation Model
The formation of biofilms can be viewed as a subsequent process after bacterial attachment. Over time, the attached bacteria grow and proliferate on the surface, and through EPS, they anchor nearby bacteria together, ultimately forming a complex three-dimensional structure.
a. Growth Model of Biofilm Thickness or Density
The growth of biofilms is typically described by a logistic growth model, which initially exhibits exponential growth. As the biofilm thickens, nutrient and oxygen limitations slow down the growth, eventually stabilizing:
$$ \frac{dB}{dt}=r_B\times B\times(1-\frac{B}{B_{max}}) \tag{2.9} $$To account for the effect of c-di-GMP on biofilm growth, further assumptions can be made:
$$ r_B=r_{B0}+\beta_B\times G \tag{2.10} $$b. Role of EPS in Biofilm Formation
EPS serves as a matrix material in the biofilm structure, stabilizing the biofilm. As the concentration of EPS increases, the density and stability of the biofilm also increase. The role of EPS can be reflected through direct impacts:
$$ B_{max}=B_{0max}\times(1+\gamma_B\times\frac{E}{E+K_E}) \tag{2.11} $$2 Model Solution
We use the Runge-Kutta method to solve the model.
2.1 Introduction to the Runge-Kutta Method
The fourth-order Runge-Kutta method is a numerically accurate approach for solving ordinary differential equations. It estimates the next solution by calculating the slope multiple times. The general form is as follows:
Given the initial value problem:
$$ \frac{dy}{dt}\ =\ f\left(t,\ y\right),\ y\left(t_0\right)\ =\ y_0 \tag{2.12} $$The update formulas for the Runge-Kutta method are:
$$ k_1=hf\left(t_n,y_n\right) \tag{2.13} $$ $$ k_2=hf\left(t_n+\frac{h}{2},y_n+\frac{k_1}{2}\right) \tag{2.14} $$ $$ k_3=hf\left(t_n+\frac{h}{2},y_n+\frac{k_2}{2}\right) \tag{2.15} $$ $$ k_4=hf\left(t_n+h,y_n+k_3\right) \tag{2.16} $$ $$ y_{n+1}=y_n+\frac{1}{6}\left(k_1+2k_2+2k_3+k_4\right) \tag{2.17} $$2.2 Application of the Fourth-Order Runge-Kutta Method to Solve the Model
According to the equations provided in the document, we will take the synthesis and degradation of c-di-GMP concentration G as an example and apply the fourth-order Runge-Kutta method to solve it.
2.2.1 Model Equations
The rate of change of c-di-GMP concentration is described by the following formula:
$$ \frac{dG}{dt}=\alpha_DD-\alpha_PPG-\gamma_GG \tag{2.18} $$Based on these formulas, the process to solve forG\left(t\right)is as follows:
1. Initial Conditions: Assume at the initial moment $t_0=0$, $G\left(0\right)=G_0$.
2. Step Size Selection: Choose an appropriate step size $h$, for example, $h\ =\ 0.1$ seconds.
3. Calculate $k_1$ to $k_4$
$$ k_1=h\left(\alpha_DD-\alpha_PPG-\gamma_GG\right) \tag{2.19} $$ $$ k_2=h\left(\alpha_DD-\alpha_PP\left(G+\frac{k_1}{2}\right)-\gamma_G\left(G+\frac{k_1}{2}\right)\right)] \tag{2.20} $$ $$ k_3=h\left(\alpha_DD-\alpha_PP\left(G+\frac{k_2}{2}\right)-\gamma_G\left(G+\frac{k_2}{2}\right)\right)] \tag{2.21} $$ $$ k_4=h\left(\alpha_DD-\alpha_PP\left(G+k_3\right)-\gamma_G\left(G+k_3\right)\right)] \tag{2.22} $$4. Update$\ G_{n+1}$：
$$ G_{n+1}=G_n+\frac{1}{6}\left(k_1+2k_2+2k_3+k_4\right) \tag{2.23} $$2.2.2 Numerical Solution Steps
The numerical solution process can be implemented using programming, iteratively calculating the change of $G\left(t\right)$ over time.
2.2.3 Application to Other Model Equations
Similarly, the fourth-order Runge-Kutta method can be applied to solve other differential equations in the document, such as: By applying the fourth-order Runge-Kutta method to solve these model equations, the bacterial biofilm formation process can be accurately simulated.
2.3 Results
The experimental results, as shown in the figure, display the variation of c-di-GMP concentration over time. The experiment used the Runge-Kutta method for numerical solution of the model. From the figure, we can observe that the concentration of c-di-GMP increases over time, showing a relatively linear upward trend.
The analysis indicates that during the 0 to 10-second time frame, c-di-GMP concentration increases from approximately 0.1 to 0.45, suggesting that DGC (Diguanylate Cyclase) synthesis activity predominates over PDE (Phosphodiesterase) degradation. This results in the continuous accumulation of c-di-GMP. The observed trend implies that PDE degradation has not yet significantly impacted the concentration, but as time progresses, an increase in the degradation rate could lead to a leveling off or decline in the curve. The short 10-second time window of this experiment reflects early-stage dynamics, and extending the time frame may provide additional insights into long-term concentration changes. This model is useful for predicting intracellular c-di-GMP concentration fluctuations, aiding in understanding its roles in cell signaling, adhesion, and biofilm formation. By fine-tuning the DGC and PDE parameters, the model can be tested under various conditions to explore its dynamic behavior further.
In summary, the experimental results indicate that within the simulated time frame, the synthesis of c-di-GMP exceeds its degradation, showing continuous accumulation.
Reference
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