Plant bacterial diseases pose a significant threat to agricultural production. Typical diseases such as bacterial wilt and rice bacterial blight exhibit the following characteristics:
•Rapid spread: They spread quickly through various pathways, including soil transmission, seed-borne transmission, and insect vectors.
•Severe harm: They directly affect crop yields, and in severe cases, can lead to large-scale crop failure.
•Difficult prevention and control: Pathogens mutate rapidly, resulting in a gradual decrease in the effectiveness of traditional chemical control methods.
•Huge economic losses: Globally, the annual economic losses caused by bacterial diseases exceed tens of billions of US dollars.
Limitations of traditional prevention and control methods:
1.Drug resistance issue: Long-term use of chemical pesticides leads to the development of drug resistance in pathogens, significantly reducing the control effect.
2.Environmental pollution: Pesticide residues pollute soil and water bodies, disrupting the ecological balance.
3.Food safety: Pesticide residues pose a threat to food safety and affect human health.
4.Rising costs: The continuous development of new pesticides leads to increasing research, development, and application costs.

Erucamide is a long-chain fatty acid amide with a unique disease-resistant mechanism:
1.2.1 Molecular Mechanism of Action
1.Direct bacteriostatic effect
•Destroy the integrity of the pathogen cell membrane.
•Interfere with the cell wall synthesis process.
•Affect intracellular energy metabolism.
2.Blocking the type III secretion system (T3SS)
•Specifically bind to the HrcC protein.
•Interfere with the assembly of the T3SS injectisome.
•Prevent the injection of virulence proteins into plant cells.
3.Activating the plant immune system
•Induce the expression of defense genes.
•Enhance the thickness of the plant cell wall.
•Promote the accumulation of antimicrobial compounds.
4.Broad-spectrum resistance characteristics
•Effective against a variety of Gram-negative bacteria.
•Less likely to induce drug resistance.
•Environmentally friendly and easily degradable.

This project adopts synthetic biology methods to construct a "living biological factory" by modifying Escherichia coli:
Project Innovation Points:
1.Design-Build-Test-Learn (DBTL) cycle
•Rational design of synthetic pathways.
•Modular construction of genetic circuits.
•Systematic testing of functions.
•Iterative optimization of performance.
2.Multi-level regulation strategy
•Transcription level: Regulation of promoter strength.
•Translation level: Optimization of ribosome binding sites.
•Metabolic level: Redistribution of metabolic fluxes.
•Secretion level: Modification of extracellular transport systems.

The biosynthesis of erucamide follows the complete metabolic pathway as below:
Glucose → Pyruvate (via glycolysis) → Acetyl-CoA (via oxidative decarboxylation)
Acetyl-CoA → β-Ketobutyryl-ACP (catalyzed by FabH, initiation reaction) → Butyryl-ACP (via reduction, dehydration, and reduction catalyzed by FabG, FabZ, Fabl)
Butyryl-ACP → Palmitic acid (via cycle elongation) → Erucic acid (via further elongation catalyzed by KCS)
Erucic acid → Erucamide (via amidation)

Table 1. Detailed Functional Analysis of Three Key Proteins

3.1.1 Michaelis-Menten Equation
The classic kinetic model for a single enzyme-catalyzed reaction:

Under the steady-state assumption, the reaction rate is:

Where

3.1.2 Multi-Substrate Reaction Kinetics
For reactions involving multiple substrates, a two-substrate reaction model is used:

3.2.1 Tandem Enzyme Reactions
For an n-step continuous enzyme reaction:

The total reaction rate is limited by the slowest step (rate-limiting step principle):

3.2.2 Metabolic Control Analysis
The flux control coefficient is defined as:

It satisfies the summation theorem:

The mathematical model is constructed based on the following assumptions:
Model Assumptions:
1.Each enzyme reaction follows Michaelis-Menten kinetics.
2.Protein concentrations do not show saturation inhibition within the experimental range.
3.Substrate supply is sufficient and does not become a limiting factor.
4.Reaction conditions are constant (temperature, pH, ionic strength).
5.No feedback inhibition from the product.
6.There are synergistic effects between proteins. 4.2.1 Single Protein Contribution Function
The contribution of each protein to the total yield adopts the form of a saturation function:

Where Ki is the half-saturation constant of each protein.

4.2.2 Modeling of Synergistic Effects
Considering the synergistic effect between proteins, the synergistic effect factor is defined as:

Where:
•γ: Synergistic strength coefficient (0 < γ < 2)
•ε: A small constant to avoid zero denominator (usually set to 1)

Comprehensively considering various factors, the complete three-protein synergistic model is:

4.4.1 Literature Data Analysis
Based on published enzyme kinetics studies, the parameter values are as follows:
Table 2. Biological Basis and Values of Model Parameters

4.4.2 Parameter Sensitivity Analysis
The sensitivity coefficient is used to evaluate the impact of parameters on the model output:

5.1.1 Yield Range Analysis
Within the protein concentration range of Ci ∈ [0, 2] mg/ml, the model prediction results are as follows:
•Theoretical minimum yield: Ymin = 0 g/g (when all protein concentrations are 0)
•Theoretical maximum yield: Ymax = 28.47 g/g (when all protein concentrations are 2 mg/ml)
•Practical yield range: 5-25 g/g (corresponding to the effective disease-resistant concentration)
•Economically optimal range: 15-22 g/g (best cost-effectiveness)
5.1.2 Comparison with Literature Data

Table 3. Comparison between Model Predicted Values and Literature-Reported Erucamide Contents

5.2.1 Multi-Objective Optimization Problem
A multi-objective optimization model is established:

Where wi is the relative cost coefficient of each protein.

5.2.2 Solution Using Lagrange Multiplier Method
Construct the Lagrangian function:

The first-order necessary conditions are:

5.2.3 Optimal Solution
The optimal ratio is obtained through numerical calculation:

The corresponding optimal yield: Y* = 23.8 g/g
Optimal ratio: CPtsG : CFabH : CGlnA = 1.2 : 1.0：1.5

The degradation of calcium alginate relies on the decomposition of soil microorganisms. A higher concentration leads to a denser colloidal network, making it more difficult for microorganisms to penetrate and thus slower degradation (this conforms to the properties of polymer materials, referring to relevant studies in Progress in the Application of Biodegradable Polymer Materials). It is assumed that under a natural soil environment (25℃, 60% humidity, with 90% mass loss regarded as "complete degradation"), the relationship between the complete degradation time  t  and concentration is as follows (fitted to a reasonable trend):

The core of the model is to standardize the "pressure-bearing capacity (a positive indicator, the higher the better)" and "degradation rate (a negative indicator, the shorter the time the better)", and obtain a comprehensive score through weighted calculation. The concentration with the highest score is the optimal ratio.

1. Variable Definition
•Independent variable: Sodium alginate concentration x (%), with a value range of [0.5, 2.0];
•Positive indicator: Pressure-bearing value y(x) (g), whose measured data is fitted to a linear relationship (error < 5%): y(x) = 110x + 345 (Verification: When x = 0.5 , y = 400 ; when x = 2.0 , y = 565 , which is close to the measured value of 580 and acceptable);
•Negative indicator: Degradation time t(x) (days), fitted to an exponential relationship (the higher the concentration, the faster the degradation rate increases): t(x) = 8✖️1.3x (Verification: When x = 0.5 , t ≈11.8 ; when x = 2.0 , t ≈33.8 , which is close to the assumed value and acceptable);
•Comprehensive score : Weighted after standardization, with weights α (weight for pressure-bearing capacity) and β (weight for degradation rate), and α + β = 1 .

2. Standardization (Eliminating the Influence of Dimensions)
•Standardization of pressure-bearing capacity:

where ymin = 400 (at 0.5%) and ymax = 580 (at 2.0%), with the result range [0, 1];
•Standardization of degradation time:

where tmin= 12 (at 0.5%) and tmax= 35 (at 2.0%), with the result range [0, 1] (a higher value indicates faster degradation and better environmental friendliness).

3. Comprehensive Scoring Function
Weights are set according to the core product requirements:
•For agricultural/ecological scenarios (degradation as priority, pressure-bearing capacity only needs to meet transportation requirements): α = 0.4 , β = 0.6 ;
•For daily chemical/industrial scenarios (pressure-bearing capacity as priority, degradation only needs to meet the standard): α= 0.6 , β = 0.4 ;
•For general balanced scenarios: α = 0.5 , β = 0.5 .
Scoring formula:

The optimal ratio fully depends on the core product requirements, and is derived based on practical application scenarios:
Ecology-prioritized scenarios (e.g., agricultural slow-release fertilizers):The 0.5% concentration has the highest score (
S=0.600
), with the fastest degradation (12 days). Its pressure-bearing capacity of 400g can meet transportation needs (the weight of a single bead is only ≈4.7g, so the pressure-bearing capacity far exceeds the standard). However, it should be noted that beads with 0.5% concentration may be relatively soft, and their impact resistance needs to be tested.
Balanced scenarios (e.g., daily chemical cleaning beads):The 1.0% concentration has the highest score (
S=0.560
), with a pressure-bearing capacity of 470g (17.5% higher than that of 0.5%, resulting in better anti-breakage performance) and a degradation time of 18 days (only 6 days longer than that of 0.5%, still environmentally friendly). It is the optimal cost-performance solution.
Pressure-bearing-prioritized scenarios (e.g., industrial heavy-duty beads):The 2.0% concentration has the highest score (
S=0.600
), with the highest pressure-bearing capacity (580g). However, its degradation time is 35 days (23 days longer than that of 0.5%, leading to poor environmental friendliness). It is only recommended for scenarios where "high pressure-bearing capacity is a rigid demand + slow degradation is acceptable" (e.g., long-term storage products in sealed packaging).

Limitations: The degradation time is an assumed value; in practice, it needs to be measured through soil burial experiments (e.g., regular weighing and structural observation via electron microscopy);
Optimization directions: For higher accuracy, a "cost variable" can be added (higher concentration of sodium alginate leads to higher cost) to expand the model into a "pressure-bearing capacity-degradation-cost" three-dimensional model;
Verification method: Prepare samples with the predicted optimal concentration (e.g., 1.0%), test the pressure-bearing capacity (repeated 3 times to take the average value) and degradation rate (measure mass loss after 20 days of burial) simultaneously to verify the model’s rationality.
Final general recommendation for the optimal ratio: 1.0% sodium alginate concentration (balances pressure-bearing capacity, degradation, and cost, and is suitable for most civil scenarios).