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Introduction

Compounds such as arsenic oxides and nitrates pose significant environmental hazards. To address this pressing issue, our research team has developed a biosensor based on electroactive biofilms. This innovative sensor relies on the biological principle of relieving inhibition of the expression of downstream genes, including gfp (green fluorescent protein) and mtrC, associated with extracellular electron transfer, when the ArsR protein in the bacterium binds to trivalent arsenic. By monitoring the levels of gfp or mtrC expression, we can accurately predict arsenic concentrations within the system.

Modeling

2.1 Biological Principles

Shewanella oneidensis MR-1 stands as prominent representative of electroactive bacteria, demonstrating its capability to transfer the intracellular electrons to extracellular electron acceptors. These electron acceptors can include electrodes within electrochemical systems, facilitated through the process of extracellular electron transfer (EET). MtrC is one of key proteins involved in EET in S. oneidensis MR-1.

In our pursuit of constructing an electroactive biofilm-based biosensor for arsenic detection, we undertook a two-fold strategy. First, we initiated a knockout of the mtrC gene, a pivotal component EET within S.oneidensis MR-1. Subsequently, we complemented its expression by integrating the mtrC gene at the downstream of arsenic-responsive transcription system sourced from Escherichia coli. Simultaneously, we incorporated gfp-producing biosensors into the system for ease of testing within S. oneidensis MR-1.

Within E. coli, the ArsR binds to the arsR promoter, forming a biologically less active arsR-P_ars complex. This complex inhibits the transcription of downstream gfp and mtrC genes, resulting in reduced expression of gfp and mtrC. The activity of the P_ars promoter and the concentration of ArsR are both influenced by the environmental arsenic concentration. Higher arsenic concentrations inhibit the binding of ArsR protein to the P_ars promoter, reducing the production of the arsR-P_ars complex and consequently increasing the expression levels of gfp and mtrC genes. Therefore, we can predict arsenic concentration in the system by detecting the expression levels of gfp or mtrC. The GFP fluorescence protein levels are represented by fluorescence intensity, while MtrC, a membrane protein typically involved in electron transfer between the outer and inner membranes of bacteria, is represented using EET efficiency which can be quantified by ferrozine assay-measured ferrous ion concentration or voltage intensity^[2].

Fig. 1 Principle of electroactive biofilm-based biosensor for arsenic detection

2.2 Model Assumptions

To simplify the problem background and establish a reasonable mathematical model, we have made the following assumptions after reviewing relevant literature and analyzing experimental results:

1. Environmental arsenic concentration affects the binding and dissociation of ArsR to P_ars.

2. The complex formed by the binding of ArsR to P_ars affects the transcription of downstream gfp and mtrC genes.

3. We ignore systematic errors and human measurement errors in the experimental process, assuming that the data obtained from experiments are true and reliable.

2.3 Symbol Description

2.4 Mathematical Model

The binding of ArsR to P_ars promoter can form an ArsR-P_ars complex that represses gfp and mtrC gene transcription and produces less gfp and mtrC:

where$$ M_{\left\{ arsR \right\}}\text{、}M_{\left\{ Pars \right\}}\text{、}M_{arsR+Pars} $$ denote the activity of ArsR, the activity of the P_ars promoter, and the activity of the complex formed by the binding of arsR and P_ars , respectively, where the activity of the P_ars promoter is affected by arsenic concentration, and denotes the arsenic concentration in the system.

ArsR activity was affected by arsenic concentration:

mtrC gene expression is regulated by the ArsR and P_ars complexes:

Expression of gfp is regulated by the ArsR and P_ars complexes:

$$M_{\left\{ mtrC \right\}}\text{、}M_{gfp}$$denotes the expression of mtrC and gfp.

2.4.1 Logistic Model Establishment

The logistic model is a commonly used mathematical model in various fields, including biology, medicine, and economics^[3]. It is used to describe the probability of an event occurring or the growth and inhibition of a phenomenon based on one or more explanatory variables. The fundamental idea of the logistic model is to model the relationship between the probability of an event occurring (or the growth/inhibition of a phenomenon) and one or more explanatory variables as an S-shaped curve, also known as a sigmoid curve. Logistic curves are typically used to describe growth, saturation, and inhibition phenomena in biology.

Based on experimental results, we found that the concentration of arsenic elements in the system can promote the expression of gfp and mtrC to a certain extent. When the arsenic concentration exceeds a certain threshold, it begins to inhibit the expression of these proteins. Therefore, we have used the logistic model to model this relationship and explore the differences in gfp and mtrC expression levels under different arsenic concentrations. When exploring the relationship between gfp expression and arsenic concentration, we consider gfp fluorescence intensity as the response variable and time as the explanatory variable. When exploring the relationship between mtrC expression and arsenic concentration, we consider ferrous ion concentration or voltage intensity as the response variable and time as the explanatory variable.

(1) Logistic Model for gfp

The mathematical expression for the logistic Stee model of the gfp is given below:

where $$f_{gfp}$$represents the fluorescence intensity of the gfp affected by the activity of the ArsR- P_ars complex at a given moment in time; $$Q_{gfp}$$ is the maximum fluorescence intensity (RFU/OD₆₀₀) produced in that system; $$\theta _{gfp}$$ is the slope parameter of the curve, in other word, the steepness of the curve; is the time per unit, and $$t_{gfp}$$ is the midpoint of the curve of the change in the gfp, which indicates the time corresponding to the time at which the intensity of the fluorescent protein reaches half of its value.

(2) Logistic Model for mtrC

MtrC is a membrane protein that typically participates in the electron transfer process between the outer and inner membranes of bacteria. It can accelerate the reduction of Fe(Ⅲ) to Fe(Ⅱ) in the system. Therefore, we establish a logistic model for the effect of MtrC on iron reduction. The mathematical expression for the logistic model of MtrC affecting iron reduction is as follows:

Here, $$f_{mtrC}$$ represents the influence of the ArsR- P_ars complex activity on the expression of mtrC at a certain time (representing ferrous ion concentration or voltage intensity, denoted as $$f_{mtrC}$$ in the following discussions); $$Q_{mtrC}$$ is the maximum ferrous ion concentration（）; $$\theta _{mtrC}$$ is the slope parameter of the curve, representing the steepness of the curve; $$t_2$$ is the unit of time (explanatory variable); $$t_{mtrC}$$ is the midpoint of the mtrC change curve, indicating the time at which the mtrC intensity reaches half.

2.4.2 Modeling Process

Taking the modeling process for gfp as an example, the specific steps are as follows:

Step1 Collect Experimental Data

First, conduct a series of experiments to collect data points for gfp intensity and corresponding times.

Step2 Data Preparation

Organize the experimental data into two columns: one for gfp intensity($$f_{gfp}$$)and the other for time($$t_1$$).

Step3 Initial Parameter Estimation

Estimate initial parameter values based on the approximate shape of the experimental data. For example, you can estimate $$\theta _{gfp}$$ as the steepness of the gfp intensity change curve, estimate $$Q_{gfp}$$ as the maximum gfp intensity, and estimate $$t_{gfp}$$ as the midpoint of the curve.

Step4 Model Fitting

Use maximum likelihood estimation (MLE) to estimate model parameters $$\theta _{gfp}\text{、}Q_{gfp}$$ and $$t_{gfp}$$. MLE aims to find a set of parameter values that minimize $$W_{gfp}$$, the loss function of the observed data. Specifically, for the logistic model, the loss function is a complex nonlinear function of the parameters and is typically solved using numerical optimization methods such as gradient descent or Newton's method. Parameters are iteratively updated until the maximum likelihood of the likelihood function is reached.

2.4.3 Solving the Logistic Model

(1) Basic Idea of Logistic Regression

Solving these problems by identifying statistical regularities using previous correlation data essentially translates into a multiple regression analysis problem $$Y=f(x)+\varepsilon $$, where $$x=[x_1,x_2,\cdots ,x_k]^T$$ and $$\varepsilon $$ is a random variable. Since the dependent variable Q takes on only two states: suppressed expression $$\left( Y=1 \right) $$ and promoted expression $$\left( Y=0 \right) $$. It is very difficult to find the relationship between the dependent variable $$Y$$ and the independent variable directly. Thus, instead of analyzing the relationship between $$Y$$ and x directly, the research problem can be shifted to analyze the relationship between the conditional probability $$P\{Y=1|x\}$$ and x, which is equivalent to finding a continuous function $$p(x)=P\{Y=1|x\}$$ that takes values between 0 and 1.

Mathematically, there are several functions that satisfy this condition, and the logistic regression is one of them. Similar to linear regression analysis, the basic principle of logistic regression is to fit a logistic model to a set of observed data and then use this model to reveal the dependence relationship between several independent variables and the probability of the dependent variable taking each value. It evaluates the accuracy of simulating the change law of related things using this model. Specifically, logistic regression analysis can determine if changes in independent variables affect the probability of a certain value for the dependent variable, while considering other variables. It also estimates the impact of each independent variable when other variables are kept constant.

Before using logistic regression analysis, certain conditions should be met:

(1) The dependent variable must be a categorical variable, including ordinal and nominal variables. Both types of variables should be represented by numbers. For example, we can use$$Y=1$$ to represent inhibited expression of protein and $$Y=0$$ to represent promoted expression of protein.

(2) The independent variables can be numeric continuous variables, ordinal variables, or nominal variables.

(2) Definition of Multinomial Logistic Regression Model

Since the concentration of fluorescent protein is influenced by multiple factors, we introduce a multinomial logistic regression model. Assuming that the dependent variable $$Y$$ is a binary variable taking values 1 and 0, $$x=[x_1,x_2,\cdots ,x_k]^T$$ are k factors influencing it, and $$\beta =[\beta _0,\beta _1,\cdots ,\beta _k]^T$$ are the regression coefficients. Then, the multinomial logistic regression model regarding the effects of on Y is defined as:

From the above equation, we can derive:

(3) Parameter Estimation for Logistic Regression

We use the maximum likelihood estimation (MLE) method to find the parameters of the model^[4].

Assume that we draw a random sample of size $$n_1+n_2$$ from the population $$(Y,x)$$, where (1,x₁), (1,x₂),..., (1,x_n1), (0,x_n1+1) and (0,x_n1+2), ..., (0,x_n1+n2) . The likelihood function of is given as follows:

Taking the logarithm of both sides and simplifying, we get:

Expressing this in vector form, we have:

To find the stationing point of equation (12), the system of likelihood equations for the log-likelihood function is given by:

In vector form, this is:

Equations (14) represent a system of nonlinear equations, which generally does not have an analytical solution. It can be solved numerically using methods such as the Newton-Raphson iteration method.

Then the Jacobian matrix with respect to is:

In vector form, this is:

According to the principle of the Newton-Raphson method^[5], the iterative formula for the parameter $$\beta $$ can be obtained as:

The algorithm concerning the Newton-Raphson method for calculating the parameter $$\beta $$ is as follows:

Step 1: Given the initial parameters $$\beta ^{(0)}$$ and the error tolerance accuracy $$\varepsilon$$ for parameter $$\beta $$, let n = 0;

Step 2:Calculate$$ \beta ^{(\mathrm{n}+1)}=\beta ^{(\mathrm{n)}}-[J(\beta ^{(n)})]^{-1}F(\beta ^{(n)}),n=0,1,2,\cdots $$;

Step 3:If $$ ||[J(\beta ^{(n)})]^{-1}F(\beta ^{(n)})||<\varepsilon $$ or $$ ||F(\beta ^{(n)})||<\varepsilon $$, i.e., the allowed precision is satisfied, then end, otherwise update the parameters $$\beta ^{(\mathrm{n)}}=\beta ^{(\mathrm{n}+1)}$$, $$n=n+1$$ , and go to Step 2.

According to the above algorithm, the maximum likelihood estimation of the parameters can be found.

2.5 Results Analysis

2.5.1 Analysis of the Logistic Model for gfp

Following the calculation steps described above and using computer programming, we obtained the expressions for the logistic model of gfp under the influence of different arsR - Pars complex activities. When the arsenic concentration is very high, fluorescence intensity can burst generated by gfp due to the chemical properties of arsenic. Therefore, we have only established models for gfp expression over time for arsenic concentrations ranging from 0.5 to 100 μM (see Table 1).

Table 1 Expression of gfp expression versus time at different arsenic concentrations

From the expression, a plot of gfp expression versus time at different arsenic concentrations can be obtained as shown below:

Fig. 2 Predicted gfp expression of biosensor at different concentrations of arsenic

The figure above, along with the provided expression, reveals that when the concentration of arsenic in the system increases, the expression of gfp increases with time, in other words, the expression of gfp is promoted. However, when the concentration of arsenic in the environment is 0~0.5 μM, the expression level of gfp does not change significantly, which indicates that when the concentration of arsenic is small, it has a weak influence on the expression level of gfp. When the concentration of arsenic exceeds 1 μM , it can be found that the gfp increases significantly. According to the logistic model, the final protein expression will tend to a stable value (in other words, the maximum expression of gfp, and will not increase indefinitely. At the same time, it's important to note that a higher arsenic concentration doesn't necessarily yield better results. Due to arsenic's characteristics, excessively high concentrations can actually lead to the death of gfp. In such cases, the protein's expression decreases gradually until it reaches zero. This conclusion is supported by experiments conducted at an arsenic concentration of 500 μM.

Fitting the experimental data to the obtained ratio, the figures is as follows:

Fig. 3 gfp logistic Steele model fitting data

In order to more clearly see the comparison between the logistic model and the original data, the above image is enlarged and compared：

Fig. 4 Zoom in on the comparison plot in the [0,10] interval

According to Fig. 4 , computer programming was used to obtain the following goodness of fit for the logistic model of gfp at different arsenic concentrations:

Table 2 Goodness of fit of the logistic model for gfp at different arsenic concentrations

From the above table, it can be found that the goodness-of-fit of the logistic Steele model under different arsenic concentrations is higher than 90%, indicating that the logistic Steele model can better describe the changes in gfp expression over time under different arsenic concentrations. After getting the fitted model, we can use it to predict the change of gfp intensity with time under different arsenic concentrations, explore the effect of arsenic concentration on gfp expression, and provide theoretical support for us to detect the concentration of arsenic elements in the system.

2.5.2 Analysis of logistic Steele modeling results for mtrC

MtrC is a membrane protein, involved in the electron transfer process between the outer and inner membranes of bacterial cells. Therefore, its expression is an important factor to affect the EET efficiency of S. oneidensis MR-1. The EET efficiency can be reflected by the reduction rate of Fe(Ⅲ) and electrical outputs in electrochemical systems, such as microbial fuel cells (MFCs). Thereby, by combining mtrC expression with arsenic-responsive transcription system, the arsenic concentration can be converted to the expression of mtrC which promotes the reduction of ferric iron or electricity generation. Based on the above calculation steps, computer programming was used to find the expression for the change in Fe(Ⅱ) concentration with time in different arsenic concentration systems.

Table 3 Expression of Fe(Ⅱ) versus time for different mtrC intensities

According to the expression, we obtained the figure of mtrC reducing Fe(Ⅲ) over time in the system of different arsenic concentration concentrations as follows:

Fig. 5 mtrC reducing Fe(Ⅲ) over time in different arsenic concentration systems

It can be found that the concentration of arsenic in the system increases accelerates the reduction rate of Fe(Ⅲ) to Fe(Ⅱ) in the system, in other words high concentration of arsenic promotes the expression of mtrC to accelerate the electron transfer process between the outer and inner membranes of bacterial cells.

The experimental data were fitted to the obtained ratio and the figures are as follows:

Fig. 6 Fitted data for the logistic model of Fe(Ⅲ) reduction by mtrC at different arsenic concentrations

The goodness-of-fit of the logistic model for the reduction of trivalent iron by mtrC at different arsenic concentrations can be obtained using computer programming as follows:

Table 4 Goodness-of-fit of the logistic Steele model for Fe(Ⅲ) reduction by mtrC at different arsenic concentrations

From the above table, it can be found that the goodness of fit of the logistic Steele model for the reduction of trivalent iron by mtrC under different arsenic concentrations is basically higher than 90%. This suggests that the logistic Steele model effectively describes the gradual increase in mtrC levels at different arsenic concentrations, facilitating the conversion of Fe(Ⅲ) to Fe(Ⅱ). However, it's important to note that the protein's expression is not infinite; instead, it gradually reaches a stable value. Consequently, the reduction of Fe(Ⅲ) to Fe(Ⅱ) also approaches a stable value.

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