Adhesion Module
Modeling Background and Core Objectives
In the flowing environment of wastewater treatment, engineered bacteria need to adhere to carrier material surfaces to form stable biofilms, enabling continuous degradation functions and avoiding bacterial loss due to water flow. Bacterial adhesion is a dynamic coupling process of "growth-expression-binding": it involves the proliferation of free bacteria, synthesis and secretion of adhesion proteins, and then protein-mediated binding to material surfaces, ultimately forming biofilms.
This modeling focuses on the mechanistic differences between two adhesion schemes, quantifying the following processes through mathematical simulation:
- Dynamic changes in the number of free and bound bacteria;
- Impact of adhesion protein expression on bacterial growth metabolic burden;
- Stability and anti-scouring ability during biofilm formation;
- Performance differences between the two schemes in long-term operation, providing quantitative basis for carrier material selection and bacterial culture strategies in wet experiments.
Core Biological Process Modeling
Bacterial State Classification and Transition Rules
Both schemes involve three bacterial states, with transitions between states following these rules:
- Free Unexpressed State (U): Free bacteria that have not synthesized adhesion proteins, only proliferating without participating in adhesion;
- Free Expressed State (E): Free bacteria that have synthesized and secreted adhesion proteins, capable of binding to material surfaces through protein mediation;
- Bound State (B): Bacteria bound to material surfaces through adhesion proteins, located in biofilms, capable of proliferation but subject to spatial constraints.
State transition pathways:
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Figure 1: Bacterial state transition pathways for both adhesion schemes
Scheme 1: Dynamic Model of Fusion Protein System
Expression Product Characteristics: High molecular weight fusion protein, whose secretion depends on complex post-translational processing, resulting in low secretion efficiency;
Metabolic Burden: Synthesis of high molecular weight fusion proteins requires consumption of large amounts of energy and resources, leading to significant decrease in bacterial basal metabolic rate;
Binding Efficiency: Fusion proteins directly bind to material surfaces without relying on intermolecular diffusion, resulting in higher binding efficiency;
Binding Constraints: No limitation on the number of material surface sites, but subject to feedback inhibition from the biofilm microenvironment.
Parameter Definitions
Table 1: Fusion Protein System Parameters
| Parameter | Description | Value |
|---|---|---|
| rU1 | Basal growth rate of free unexpressed state (U) bacteria | 0.8 d⁻¹ |
| rE1 | Growth rate of free expressed state (E) bacteria (affected by metabolic burden) | 0.4 d⁻¹ (50% of rU1) |
| rB1 | Growth rate of bound state (B) bacteria (subject to spatial constraints) | 0.3 d⁻¹ |
| kU→E1 | Transition rate constant from U to E (expression initiation efficiency) | 0.2 d⁻¹ |
| kE→B1 | Binding rate constant from E to B (direct binding efficiency high) | 0.6 d⁻¹ |
| kB→U1 | Detachment rate constant from B to U (caused by biofilm disturbance) | 0.1 d⁻¹ |
| K1 | Environmental capacity of fusion protein system (limited by metabolic burden) | 8000 (dimensionless, representing relative bacterial count) |
| f(B1) | Biofilm microenvironment feedback inhibition factor (increases with B) | - |
Differential Equations
Free Unexpressed State (U1):
$$\frac{dU_{1}}{dt} = r_{U1}U_{1}\left( 1 - \frac{U_{1} + E_{1} + B_{1}}{K_{1}} \right) - k_{U \rightarrow E1}U_{1} + k_{B \rightarrow U1}B_{1}$$
Free Expressed State (E1):
$$\frac{dE_{1}}{dt} = r_{E1}E_{1}\left( 1 - \frac{U_{1} + E_{1} + B_{1}}{K_{1}} \right) + k_{U \rightarrow E1}U_{1} - k_{E \rightarrow B1}E_{1}\left( 1 - f\left( B_{1} \right) \right)$$
Bound State (B1):
$$\frac{dB_{1}}{dt} = r_{B1}B_{1}\left( 1 - \frac{B_{1}}{K_{1}} \right) + k_{E \rightarrow B1}E_{1}\left( 1 - f\left( B_{1} \right) \right) - k_{B \rightarrow U1}B_{1}$$
Scheme 2: Dynamic Model of Spy System
Expression Product Characteristics: Low molecular weight SpyTag protein, relying on simple signal peptide-mediated secretion without complex post-translational processing, resulting in high secretion efficiency;
Metabolic Burden: Synthesis of low molecular weight SpyTag protein requires fewer resources, leading to only mild decrease in bacterial basal metabolic rate;
Binding Efficiency: SpyTag and SpyCatcher bind through diffusion collision, limited by diffusion rate, concentration gradient and other factors, resulting in low binding efficiency;
Binding Constraints: Strictly limited by the number of SpyCatcher sites on material surfaces; when bound bacteria count approaches Smax, binding rate tends to saturate.
Parameter Definitions
Table 2: Spy System Parameters
| Parameter | Description | Value |
|---|---|---|
| rU2 | Basal growth rate of free unexpressed state (U) bacteria | 0.8 d⁻¹ |
| rE2 | Growth rate of free expressed state (E) bacteria (minimally affected by metabolic burden) | 0.7 d⁻¹ |
| rB2 | Growth rate of bound state (B) bacteria (subject to spatial constraints) | 0.35 d⁻¹ |
| kU→E2 | Transition rate constant from U to E (high expression initiation efficiency) | 0.3 d⁻¹ |
| kE→B2 | Binding rate constant from E to B (limited by diffusion, low efficiency) | 0.2 d⁻¹ |
| kB→U2 | Detachment rate constant from B to U (better biofilm stability) | 0.08 d⁻¹ |
| K2 | Environmental capacity of Spy system (light metabolic burden, high capacity) | 10000 (dimensionless) |
| Smax | Maximum capacity of SpyCatcher sites on material surface | - |
Differential Equations
Free Unexpressed State (U2):
$$\frac{dU_{2}}{dt} = r_{U2}U_{2}\left( 1 - \frac{U_{2} + E_{2} + B_{2}}{K_{2}} \right) - k_{U \rightarrow E2}U_{2} + k_{B \rightarrow U2}B_{2}$$
Free Expressed State (E2):
$$\frac{dE_{2}}{dt} = r_{E2}E_{2}\left( 1 - \frac{U_{2} + E_{2} + B_{2}}{K_{2}} \right) + k_{U \rightarrow E2}U_{2} - k_{E \rightarrow B2}E_{2}\left( 1 - \frac{B_{2}}{S_{max}} \right)$$
Bound State (B2):
$$\frac{dB_{2}}{dt} = r_{B2}B_{2}\left( 1 - \frac{B_{2}}{S_{max}} \right) + k_{E \rightarrow B2}E_{2}\left( 1 - \frac{B_{2}}{S_{max}} \right) - k_{B \rightarrow U2}B_{2}$$
Simulation Results Comparison and Key Conclusions
Total Bacteria Dynamics Comparison
Simulation results show that the Spy system (Scheme 2) achieves higher total bacterial count in the long term, while the fusion protein system (Scheme 1) has lower total bacterial count due to metabolic burden.
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Figure 2: Comparison of total bacterial count dynamics between the two schemes
Key observations:
- Spy system has higher environmental capacity (K2 = 10000 vs K1 = 8000);
- Spy system maintains higher free bacterial count, providing continuous adhesion potential;
- Fusion protein system's total bacterial count is limited by metabolic burden, with slower growth rate.
Adhered Bacteria Dynamics Comparison
Although the fusion protein system has higher binding efficiency, the Spy system achieves higher adhered bacterial count in the long term due to its higher total bacterial count and better biofilm stability.
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Figure 3: Comparison of adhered bacterial count dynamics between the two schemes
Key observations:
- Fusion protein system has faster initial adhesion rate but lower long-term stability;
- Spy system has slower initial adhesion but achieves higher adhered bacterial count in the long term;
- Spy system's biofilm has better anti-scouring ability (lower detachment rate).
State Distribution Comparison
The distribution of bacterial states differs significantly between the two schemes, reflecting their different adhesion mechanisms and metabolic characteristics.
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Figure 4: Fusion scheme state distribution
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Figure 5: Spy scheme state distribution
Key observations:
- Fusion protein system has higher proportion of expressed state bacteria but lower overall count;
- Spy system maintains higher proportion of unexpressed state bacteria, providing growth potential;
- Spy system achieves more balanced state distribution, beneficial for long-term stability.
Key Conclusions
Through mathematical modeling and simulation, we draw the following key conclusions:
- Metabolic Burden Impact: The fusion protein system's high metabolic burden significantly limits bacterial growth and total count, while the Spy system's light metabolic burden allows for higher bacterial density;
- Long-term Stability: Although the fusion protein system has higher binding efficiency, the Spy system achieves better long-term stability and higher adhered bacterial count due to its higher total bacterial count and lower detachment rate;
- Anti-scouring Ability: The Spy system's biofilm has stronger anti-scouring ability, making it more suitable for flowing wastewater treatment environments;
- Practical Application Recommendation: Considering long-term operational stability and bacterial density, the Spy system is more suitable for practical wastewater treatment applications.
These modeling results provide important theoretical basis for subsequent wet experiments and practical application scheme selection.