Model

Model 1

Modeling Economic Losses Caused by Cadmium Heavy Metal Contamination Due to Increased Significance of Constructing a Heavy Metal Detection and Recovery System

As human activities become increasingly frequent, urban pollution has intensified, with urban wastewater discharge and the use of agricultural chemicals being the main contributors to soil heavy metal contamination. When the concentration of heavy metals in the soil exceeds the environmental capacity, it not only affects the normal activities of soil microorganisms but also directly impacts plant growth and development. Moreover, these heavy metals can accumulate in plant tissues, subsequently transferring through the food chain to animals and humans, posing a significant threat to their health. In China, soil heavy metal contamination is a severe issue, with nearly one-fifth of arable land affected by heavy metal pollutants like cadmium, arsenic, chromium, and lead. This contamination leads to a reduction of over 10 million tons of food production, including 12 million tons of toxic crops, resulting in direct economic losses exceeding 30 billion RMB. The potential health and environmental issues could lead to even greater unaccounted losses. Hence, enhancing the monitoring capabilities for soil heavy metals and strengthening ecological restoration techniques are critical for preventing, detecting, and controlling the risks associated with heavy metal pollution sources. Therefore, the construction of Escherichia coli capable of detecting and recovering heavy metal cadmium holds significant importance. In the following sections, we employ a mathematical model to illustrate the substantial economic losses resulting from cadmium heavy metal contamination in the soil.

Choosing a Mathematical Model for Environmental Damage Caused by Heavy Metal Contamination

Environmental pollution involves various pollutants, and their effects on the environment do not follow a simple linear relationship. When pollutant concentrations are very low, their destructive effects on the environment can be negligible. However, once pollutant concentrations reach a certain threshold, they can pose a significant hazard to the environment. Studies have indicated that the losses resulting from environmental pollution exhibit an "S"-shaped non-linear relationship with pollutant concentrations. In other words, pollutant damage to the environment is not readily apparent at low doses, but as the pollutant concentration reaches a critical threshold, the degree of environmental damage increases sharply. After reaching a certain level of pollutant concentration, the degree of environmental damage gradually increases, ultimately reaching the limit of pollution harm. Scholars from both domestic and international contexts have proposed several models to address soil heavy metal pollution issues, including single-factor index models, geometric mean models, fuzzy mathematics models, among others. Of these, the pollution loss rate model has shown promising results. This report is based on the pollution loss rate model to model and assess the economic losses resulting from cadmium contamination in the environment.

When there are n types of heavy metals in the soil, and the pollution loss rate of the i-th heavy metal to the soil is denoted as Ri, a differential equation is established to describe the relationship between heavy metal concentration and soil environmental economic losses:

Solving the Differential Equation to Obtain:

Ri = 1 / [1 + ai * exp(-bici)] ............ Equation (1) calculates the loss rate that heavy metal i incurs on the soil at concentration ci.

S = KRi .................... Equation (2) calculates the economic losses incurred by heavy metal i at concentration ci.

R = ............ Equation (3) calculates the overall loss rate incurred by multiple heavy metals.

Note: exp(-bici) = e^(-bici), where e is the natural logarithm base.

If there are two heavy metals, A and B, in the soil, and their respective individual heavy metal pollution loss rates are RA and RB, then the probability of A and B occurring is equal to the sum of the probabilities of these two events minus the product of their probabilities, i.e., RAB = RA + RB - RA * RB = 1 - (1 - RA) * (1 - RB). This can be extended to n types of heavy metals, and the comprehensive heavy metal pollution loss R can be calculated in a similar manner.

In the formulas, Ri represents the damage rate of heavy metal i to the soil environment, ai is a constant term, bi is the proportionality coefficient of heavy metal i in the soil, S represents the economic losses of heavy metal pollution to the soil when the mass concentration of heavy metal i is ci in terms of millions of yuan, and K represents the economic value achieved after soil utilization in millions of yuan. When there are n types of heavy metal pollution in the soil, the comprehensive loss rate incurred by the soil is denoted as R.

Establishing Economic Losses Resulting from Cadmium (Cd) Heavy Metal Pollution Using the Model

In the model described above, we consider the economic losses resulting from a single heavy metal, cadmium (Cd). To determine the parameters a and b in the model, we should establish the loss rates R1 and R2 at concentrations c1 and c2 under experimental conditions and substitute them into Equation (1) to obtain a system of linear equations:

R1 = 1 / (1 + a * e^(-b * c1))

R2 = 1 / (1 + a * e^(-b * c2))

R1 - 1 - 1 = a * e^(-b * c1)

R2 - 1 - 1 = a * e^(-b * c2)

Dividing the two equations results in (R1 - 1 - 1) / (R2 - 1 - 1) = e^(-b * (c1 - c2))

Taking the natural logarithm of both sides gives ln[(R1 - 1 - 1) / (R2 - 1 - 1)] = -b * (c1 - c2)

Therefore, b = [ln[(R1 - 1 - 1) / (R2 - 1 - 1)]] / (c2 - c1).

Substituting the obtained b value into either of the equations yields the value of a.

In the absence of experimental data, reference can be made to numerical values for the constant term a of various heavy metals as provided in literature [2]. For heavy metal cadmium, a = 160.58, and b = 4.837.

The loss rate caused by the heavy metal cadmium in soil at concentration c is given by: R = 1/[1 + 160.58exp(-4.837c)], and its function graph is shown in Figure 1.

R=1/[1+160.58exp(-4.837c)]

The cadmium ion concentration in the soil should ideally be obtained through sampling and measurement. Due to the absence of experimental data, reference has been made to sampled measurement values from other literature (Table 2)

Obtaining an average cadmium heavy metal concentration of 1.97 mg/kg in contaminated soil and applying it to the model formula R=1/[1+160.58exp(-4.837c)], we calculate a soil loss rate of R=98.83%. Rice cultivation covering an area of 1.0 km^2 can yield benefits of 3.675 million, which, in the formula S=KR, results in a constant value of K=367.5 thousand yuan per km^2. The economic losses caused by cadmium heavy metal pollution reach 363.2 thousand yuan per km^2.

Modeling Explanation:

• The cadmium heavy metal content in the soil should ideally be determined through experimental measurements. In the absence of experimental data, calculations were based on referenced literature data.
• In future assessments, with the implementation of the cadmium ion detection and recovery system developed by our team, changes in cadmium content in the soil before and after usage can be compared to determine how much economic loss can be mitigated by our team's developed cadmium detection and recovery system.

Reference：

[1]王丽智,吴攀.贵州兴仁煤矿废水灌溉区的土壤重金属污染评价及经济损失估算[J].北方环境, 2012(2):32-34.DOI:0.3969/j.issn.1007-0370.2012.02.026.

[2]徐欣,马建华,韩晋仙. 基于污染损失率法的土壤重金属污染评价——以开封市化肥河污灌区为例[J]. 河南大学

Model 2

his is an idea validation simulation: Our idea is to leverage the high efficiency of T7 RNA polymerase in transcription to amplify biological signals. Before conducting experiments, we want to verify the feasibility of our logic through modeling.

Idea: CadR is a constitutively expressed protein that activates the expression of T7 RNA polymerase in the presence of divalent cadmium ions, leading to efficient expression of yellow fluorescent protein.

This process converts the cadmium ion concentration signal into the signal of yellow fluorescent protein concentration, ultimately translating the concentration signal into fluorescence intensity through detection instruments. This is a type of biosensor that could potentially become a more intuitive method for detecting water pollution. However, due to the complexity of biological systems, we cannot intuitively determine whether the addition of T7 amplification will significantly change the strength of the reporting signal. T7 primarily enhances the transcription process of yellow fluorescent protein, but protein expression is also influenced by factors such as translation rate, degradation rate, and biological feedback. Therefore, I have developed a conceptual model to explore the potential impact of T7 amplification in the biosensor.

We assume that in the absence of the T7 amplification system, the transcription rate of yellow fluorescent protein (a) is 100 units per hour, the translation rate (b) is 100 units per hour, RNA degradation rate (d1) is 0.1, protein degradation rate (d2) is 0.2, plasmid copy number (c) is 10, the strength of biological feedback is 1, and the response time to contact the signal source is 24 hours. To compare the impact of protein expression on the biosensor, this model does not consider the growth of bacterial populations.

By simulating the protein expression process through a system of ordinary differential equations, we mainly focus on the effects of transcription, translation, degradation, and feedback on the biosensor. The equations are as follows:

By programming in MATLAB, it can be determined that the conceptual sensor exhibits a change in the expression intensity of yellow fluorescent protein within 24 hours of contacting the signal source. The maximum value can reach up to 300 units, as shown in the following figure:

While keeping the conceptual sensor unchanged, we introduce the T7 amplification system. Assuming that the maximum reaction rate catalyzed by T7 is ten times that of the original conceptual system, the expression level of T7 increases from 0 to its maximum value. However, the catalytic efficiency will not increase infinitely, and the maximum catalytic efficiency is tenfold. This model is treated as a positive feedback effect.

The equations are as follows, building upon the existing expression model with the addition of the influence of T7, as shown in the third equation:

The results obtained through MATLAB calculations, as shown in the figure above, indicate a relatively significant enhancement in the T7 amplification effect. The maximum expression of yellow fluorescent protein can reach up to 500 units when compared to the previous calculation.

In summary, our idea appears to be highly feasible, and we can proceed with further experiments.

Additionally, during the computational process, we observed that under a constant strength of biological negative feedback, the positive feedback of T7 may be inhibited. If the amplification effect is suboptimal in subsequent experiments, it could be due to the negative feedback resulting from transcription itself. However, if we appropriately increase the translation rate, the amplification effect of T7 will significantly improve and can overcome the inhibitory effects of negative feedback.

Model3

First, we used Colab's Alphafold2 to model the structure of the fusion protein, as shown in the figure. It is very clear that our fusion protein did not alter the overall structure of CsgA, and our MBP was successfully displayed, indicating that this construct aligns with our design and should theoretically function as intended.

On the left side is the CsgA protein predicted by the official Alphafold2, and on the right side is the CsgA-MBP fusion protein predicted using an online tool. In the fusion, the MBP portion is represented in orange. It can be observed that the fusion, expressing MBP, does not disrupt the structure of CsgA, and MBP is successfully displayed.