Overview

The production and delivery of CsgA·MBP and CsgA·MT

In our design, only when MBP/MT is out of the cell, they can function as the absorbing units. Due to the equipment and time constraints, we need to model the process of the production and delivery of CsgA·MBP/MT to estimate the concentration of these protein which can be used to adsorb the metal ion and also find the optimal value of input concentration of IPTG.

Fig.2.1 Abstract of the production and delivery of the CsgA·MBP/MT

In our construction, we use tac(lac) promoter to control the production of CsgA·MBP and CsgA·MT. Tac promoter is hybridized by trp promoter and lac promoter^[3]. The operon of the tac promoter is almost the same with that of lac promoter. Therefore, the tac promoter can be inhibited by the binding of LacI in the Escherichia coli. In order to adjust the transcription activity of tac promoter, IPTG(Isopropyl β-D-Thiogalactoside) is used to bind with LacI so that LacI will not limit the function of tac promoter.

Aiming to build the model of this process, we makes some assumptions:
1, The model is used to simulate the production and transport of CsgA·MBP/MT.
2, Since the situation of cells in the device is defaulted to be in the stationary phase, the unstable growth rate of E.coli is not considered in these models.
3, LacI can either binds with $n_1$ molecules of IPTG to form the complex($LacI·n_1 IPTG$) or unbound so that intermediate states where less than $n_1$ molecules are bound can be neglected^[4].
4, The transporter can export $CsgA·MBP/MT$ due to the existence of CsgA.
5, The model ignores the detailed step that CsgB nucleates CsgA and assumes that CsgA will be continuously exported by transporters, so the export rate of CsgA·MBP and CsgA·MT is directly proportional to the concentration of CsgA·MBP/MT in the cells.
6，The model neglects the diffusion of IPTG into and out of cells, but put some adjustments on the values of input concentration of IPTG.

Based on the above mechanism and assumptions, two equations are combined to model the process that inducer binds repressor to prevent the inhibition on tac promoter. The first equation is to describe the relationship between IPTG and LacI. According to the third assumption, Hill function can be used to determine active LacI which can bind with tac promoter.

Fig.2.2 The equation of the concentration active LacI

In this equation, $[LacI^*]$ is the concentration of active LacI,$[LacI]_t$ represents the total concentration of LacI (including LacI binding with IPTG and active LacI), $[IPTG]$ is the concentration of IPTG input, $n$ is the hill coefficient and $K_{R1}$ is the half-saturation of IPTG.

The other equation denotes the transcription by the tac promoter. Although the reaction of the binding between LacI and tac promoter is not the process of enzyme kinetics, Michaelis-Menten equation can be used to describe the repression on tac promoter.

Fig.2.3 The equation of promoter activity

$\beta$ represents the maximal transcription rate of tac promoter and $K_{X1}$ is the half-saturation of active LacI.

By combining the two equations, we can understand the relationship between IPTG and tac promoter activity.

Fig.2.4 The equation of promoter activity combined with the two above equations

Here, we can predict the optimal concentration of input IPTG.

Fig.2.5 The prediction of the promoter activity under different concentrations of IPTG

As shown in the Fig.2.5, with the increase of the concentration of IPTG input, the promoter activity grows up but reaches the peak after [IPTG] = 1 mM (approximately) and remians maximum after 1 mM. Since too high concentration of IPTG will affect the growth of cells which cannot be expressed in this model, the optimal concentration of IPTG needs to cause the promoter activity at maximum and is not too high. Therefore, 1 mM may be a suitable concentration of IPTG input.(Experiment group had experiments related to it in the page).

The following equations are used in the production and secretion of CsgA·MBP/MT

Equations	Description
$\frac{d[mRNA·CsgA·MBP]}{dt} = \frac{\beta_1}{1+\frac{[LacI]_t}{K_{x1}}·\frac{1}{1+(\frac{[IPTG]}{K_{R1}})^{n_1}}} - r_1[mRNA·CsgA·MBP]$	Change of the concentration of the mRNA·CsgA·MBP based on IPTG binding LacI to reduce the binding between LacI and tac promoter according to Hill Equation and Michaelis–Menten Equation. $\beta_1$ represents the maximal transcription rate of tac promoter.$[LacI]_t$ is the normal concentration of LacI in one Escherichia coli. $K_{x1}$ is the unbinding LacI concentration at which transcription rate of tac promoter is at half-maximum. $n_1$ is the hill coefficient of IPTG and LacI. $K_{R1}$ stands for IPTG concentration at which the concentration of active LacI is at half-maximum. $r_1$ is the degradation rate of $mRNA·CsgA·MBP$
$\frac{d[CsgA·MBP]}{dt} = k_1[mRNA·CsgA·MBP] - r_2[CsgA·MBP]$	Change of concentration of $CsgA·MBP$ due to translation and degradation. $k_1$ is the translation rate of $CsgA·MBP$. $r_2$ stands for the degradation rate of $CsgA·MBP$
$\frac{d[mRNA·CsgA·MT]}{dt} = \frac{\beta_1}{1+\frac{[LacI]_t}{K_{x1}}·\frac{1}{1+(\frac{[IPTG]}{K_{R1}})^{n_1}}} - r_3[mRNA·CsgA·MT]$	Change of the concentration of the mRNA·CsgA·MT based on IPTG binding LacI to reduce the binding of LacI and tac promoter according to Hill Equation and action of mass. The process and values of most parameters are similar to that of $CsgA·MBP$. $r_3$ is the degradation rate of $CsgA·MT$ which is the only value different from that of $CsgA·MBP$.
$\frac{d[CsgA·MT]}{dt} = k_2[mRNA·CsgA·MT] - r_4[CsgA·MT]$	Change of concentration of $CsgA·MT$ due to translation and degradation. $k_2$ is the translation rate of $CsgA·MT$. $r_4$ stands for the degradation rate of $CsgA·MT$
$\frac{d[CsgA·MBP/MT]_{ex}}{dt} = k_3[CsgA·MBP/MT]$	Transport of CsgA·MBP/MT into outside of the cell. $k_3$ is the export rate of $CsgA·MBP/MT$

To be noticed, units of most of the above equations is $\mu M·s^{-1}$ while the unit of the export rate of CsgA·MBP/MT is $\mu mol·s^{-1}$. This is because we standardize the export rate $k_4$. According to law of mass action, the speed of one reaction is connected with the concentration of each substrate because the the higher concentration of substrate is, the higher probability molecules of substrates have to collide and react with each other. This means that the determination of value of the concentration of CsgA·MBP/MT is based on the volume of the container where cells live in actual situations. Therefore, the export rate is standardized to adapt different cases.

These are the parameters we used:

Parameters	Description	Source
$\beta_1 = 0.0014 (\mu M·s^{-1})$	Maximal transcription rate of tac promoter	Estimated from [5]
$[LacI]_t = 0.01 (\mu M)$	The normal concentration of LacI in one Escherichia coli	Estimated from [6]
$K_{x1} = 5E-5(\mu M)$	The unbinding LacI concentration at which transcription rate of tac promoter is at half-maximum	Estimated from [5:1]
$K_{R1} = 1(\mu M)$	Dissociation constant of IPTG and LacI	Estimated from [7]
$n_1 = 2$	Hill coefficient of IPTG and LacI	Estimated from [7:1]
$r_1 = 4.4E-3(s^{-1})$	The degradation rate of $mRNA·CsgA·MBP$	Estimated from [8]
$k_1 = 0.57(s^{-1})$	The translation rate of $mRNA·CsgA·MBP$	Estimated from [8:1]
$k_2 = 0.44(s^{-1})$	The translation rate of $mRNA·CsgA·MT$	Estimated from [8:2]
$r_2 = 6.3E-5(s^{-1})$	The degradation rate of $CsgA·MBP$	Estimated from [9]
$r_3 = 3.4E-3(s^{-1})$	The degradation rate of $mRNA·CsgA·MT$	Estimated from [8:3]
$r_4 = 4.88E-5(s^{-1})$	The degradation rate of $CsgA·MT$	Estimated from [9:1]
$k_3 = 1.8E-19(L·s^{-1})$	The export rate of CsgA·MBP/MT	Estimated from Team:TU_Delft (2015)

According to values of parameters and equations in this model, we can predict the amount of substance of CsgA·MBP and CsgA·MT:

Fig.2.6 The changes of amount of substance of CsgA·MBP/MT with time under 1 mM IPTG

According to the Fig.2.6, with certain time period and the size of contianer, the model can estimate the concentration of CsgA·MBP and CsgA·MT in preparation of fitting the experimetal data in absorption.

The absorption of silver ion

In this part, we build a model about the absorption of silver ions and verify this model according to the fitting of the experimental data. The values of key parameters acquired from fitting lay a foundation for further development in simulation of water-purified device.

The mechanism of the absorption implemented by MBP is that firstly MBP changes silver ion into AgNP (This process is irreversible) and then absorbs the AgNP(This process is reversible)^[10].

Fig.2.7 The biological pathway of the absorption of AgNP

Since the principle of the reaction mainly results from that the tyrosine in the MBP transfer its electron to the silver ion, we use $[CsgA·MBP]$ to represent the concentration of $CsgA·MBP$ which does not react with silver ions and $[CsgA·MBP^+]$ stands for the concentration of $CsgA·MBP$ which has been reacted with silver ions. Moreover, the part in the $CsgA·MBP$ functioning as adsorption is different from the part functioning as reaction in the protein, so we use $[CM]$ to stand for the concentration of CsgA·MBP which does not absorb silver nanoparticles.

However, in the process of building of the models, we encounter some troubles. From experiment, we can only know how many silver ions are absorbed by MBP (whether all these silver ion become AgNP can not be known). The other difficulty is that we are unable to acquire the values of key parameters about the reaction and absorption from previous articles. As a result, in our model, we combine the two mechanisms into one absorption process.

Fig.2.8 The modified biological pathway of the absorption of AgNP

Here, $[CsgA·MBP]$ stands for CsgA·MBP which does not chelate silver ions or AgNP. $[CsgA·MBP·n_2 AgNP]$ represents the CsgA·MBP which chelates silver ions or AgNP.

Considering the simplification and the whole process, assumptions are made:
1, For the process of absorption, we neglect the reaction which changes silver ions into silver nanoparticles and mix it into the reaction of absorption.
2, $CsgA·MBP$ can either binds with $n_2$ molecules of AgNP(silver ions) to form the complex($CsgA·MBP·n_2 AgNP/Ag^+$) or unbound so that intermediate states where less than $n_2$ molecules are bound can be neglected^[4:1].
3, When we consider the irreversible absorption, we do not matter whether the units absorbed are AgNPs or silver ions. The binding affinity is used to describe both of the objectives.
4,The concentration of AgNP in our model is more likely to be referred to the concentration of Ag(0)(simple substrate of silver) rather than how many nanoparticles(whose dimeter is larger than 1 nm) produced in the system, as it is difficult to obtain the number of nanoparticles.
5,This model do not consider the process that cells themselves transport silver ions into their bodies.

The following equations are used in the absorption of silver ions

Equations	Description
$\frac{d[CsgA·MBP·n_2 AgNP]}{dt} = k_{on1}[CsgA·MBP][Ag^{+}]^{n_2} - k_{off1}[CsgA·MBP·n_2 AgNP]$	The production of mixture of CsgA·MBP and AgNP($Ag^{+}$). $k_{on1}$ represents the reaction rate of association between CsgA·MBP and AgNp($Ag^{+}$). $k_{off1}$ stands for the reaction rate of dissociation between CsgA·MBP and AgNp($Ag^{+}$). $n_2$ is the hill coefficient of CsgA·MBP and AgNP($Ag^{+}$)
$\frac{d[AgNP·absorbed]}{dt} = n_2(k_{on1}[CsgA·MBP][Ag^{+}]^{n_2} - k_{off1}[CsgA·MBP·n_2 AgNP])$	The change of the concentration of the AgNP($Ag^{+}$) absorbed. The values of parameters are the same with the first equation
$\frac{d[CsgA·MBP]}{dt} = -(k_{on1}[CsgA·MBP][Ag^{+}]^{n_2} - k_{off1}[CsgA·MBP·n_2 AgNP])$	The change of the concentration of CsgA·MBP which does not binds with AgNP($Ag^{+}$). The values of parameters are the same with the first equation

Since it is impractical for us to measure precisely the changes of the concentrations of silver ions and CsgA·MBP/MT(In other words, we cannot acquire the values of $k_{on1}$ and $k_{off1}$ directly). Instead, we can use binding affinity to alternative the process.

In this part, we replace $\pmb{k_{on1}}$ and $\pmb{k_{off1}}$ with $\pmb{K_{D1}}$ which is calculated by $\pmb{k_{off1}}$ dividing $\pmb{k_{on1}}$^[11]. According to the equation: $\pmb{\frac{d[AgNP·absorbed]}{dt} = n_2(k_{on1}[CsgA·MBP][Ag^{+}]^{n_2} - k_{off1}[CsgA·MBP·n_2 AgNP])}$, when $\frac{d[AgNP·absorbed]}{dt}$ becomes zero (which means the whole system reaches steady state and cannot absorb AgNp($\pmb{Ag^+}$) any more), we can obtain that $\pmb{[AgNP·absorbed] = \frac{n_2[CsgA·MBP][Ag^+]^{n_2}}{K_{D1} + [Ag^+]^{n_2}}}$ which represents the maximal concentration of silver ions absorption. Here, $[Ag^+]$ represents the concentration of silver ions which is unbound. This equation is used to fit with experiment data to obtain the value of the binding affinity and hill coefficient.

Fig.2.9 The equation for the maximal absorption of silver ions

According to the results from experiment, we use matlab to fit the data by this model:

Fig.2.10 The fitting of experimental data

According to the fitting, we can obtian the significant values of binding affinity and hill coefficient in the absorption of silver ions.

These are the parameters we used and values acquired from fitting:

Parameters	Description	Source
$K_{D1} = 6.655(\mu M)$	Binding affinity between CsgA·MBP and AgNP($Ag^+$)	Estimated from experiment
$n_2 = 0.5086$	Hill coefficient between CsgA·MBP and AgNP(silver ions)	Estimated from experiment

The absorption of $Cu^{2+}$ and $Cd^{2+}$ by CsgA·MT

In this part, we build a model about the absorption of copper and cadmium ions and verify this model according to the fitting of the experimental data. The values of key parameters acquired from fitting lay a foundation for further development in simulation of water-purified device.

The mechanism of the absorption by MT is that it chelates the metal ions($Cu^{2+},Cd^{2+}$) and the process is reversible^[12].

Fig.2.9 The biological pathway for the absorption of $Cu^{2+}$

Fig.2.10 The biological pathway for the absorption of $Cd^{2+}$

The assumptions we made:
1, $CsgA·MT$ can either binds with $n_3$/$n_4$ molecules of $Cu^{2+}/Cd^{2+}$ to form the complex($CsgA·MT·n_3 Cu^{2+}/n_4 Cd^{2+}$) or unbound so that intermediate states where less than $n_3/n_4$ molecules are bound can be neglected^[4:2].
2, This model do not consider the process that cells themselves transport copper and cadmium ions into their bodies.

The following equations is used to describe the chelation process.

Equations	Description
$\frac{d[CsgA·MT·n_3 Cu^{2+}]}{dt} = k_{on2}[CsgA·MT][Cu^{2+}]^{n_3} - k_{off2}[CsgA·MT·n_3 Cu{2+}]$	The production of mixture of CsgA·MT and Cu^{2+}. $k_{on2}$ represents the reaction rate of association for CsgA·MT and Cu^{2+}. $k_{off2}$ stands for the reaction rate of dissociation for CsgA·MT and Cu^{2+}. $n_3$ is the hill coefficient of CsgA·MT and Cu^{2+}
$\frac{d[Cu^{2+}·absorbed]}{dt} = n_3(k_{on2}[CsgA·MT][Cu^{2+}]^{n_3} - k_{off2}[CsgA·MT·n_3 Cu^{2+}])$	The change of the concentration of the Cu^{2+} absorbed. The values of parameters are the same with the first equation for the chelation of Cu^{2+}
$\frac{d[CsgA·MT]}{dt} = -(k_{on2}[CsgA·MT][Cu^{2+}]^{n_3} - k_{off2}[CsgA·MT·n_3 Cu^{2+}])$	The change of the concentration of CsgA·MT which does not chelate with Cu^{2+}. The values of parameters are the same with the first equation for the chelation of Cu^{2+}
$\frac{d[CsgA·MT·n_4 Cd^{2+}]}{dt} = k_{on3}[CsgA·MT][Cd^{2+}]^{n_4} - k_{off3}[CsgA·MT·n_4 Cd{2+}]$	The production of mixture of CsgA·MT and Cd^{2+}. $k_{on3}$ represents the reaction rate of association for CsgA·MT and Cd^{2+}. $k_{off3}$ stands for the reaction rate of dissociation for CsgA·MT and Cd^{2+}. $n_4$ is the hill coefficient of CsgA·MT and Cd^{2+}
$\frac{d[Cd^{2+}·absorbed]}{dt} = n_4(k_{on3}[CsgA·MT][Cd^{2+}]^{n_4} - k_{off3}[CsgA·MT·n_4 Cd^{2+}])$	The change of the concentration of the Cd^{2+} absorbed. The values of parameters are the same with the first equation for the chelation of Cd^{2+}
$\frac{d[CsgA·MT]}{dt} = -(k_{on3}[CsgA·MT][Cd^{2+}]^{n_4} - k_{off3}[CsgA·MT·n_4 Cd^{2+}])$	The change of the concentration of CsgA·MT which does not chelate with Cd^{2+}. The values of parameters are the same with the first equation for the chelation of Cd^{2+}

Just like how we solve with the data about the absorption of AgNP($Ag^{+}$), we can recognize that the maximal concentration of absorbed $\pmb{Cu^{2+}}$ and $\pmb{Cd^{2+}}$ is $\pmb{[Cu^{2+}·absorbed] = \frac{n_3[CsgA·MT][Cu^{2+}]^{n_3}}{K_{D2} + [Cu^{2+}]^{n_3}}}$ and $\pmb{[Cd^{2+}·absorbed] = \frac{n_4[CsgA·MT][Cd^{2+}]^{n_4}}{K_{D3} + [Cd^{2+}]^{n_4}}}$.$[Cu^{2+}]$ and $[Cd^{2+}]$ are concentrations of unbound copper ions and cadmium ions. Combined with experiment data, we can get the values of $\pmb{K_{D1}}$, $\pmb{K_{D2}}$, $\pmb{n_3}$ and $\pmb{n_4}$.

Fig.2.11 The maximal absorption for copper ions

Fig.2.12 The maximal absorption for cadmium ions

These are the parameters we used:

Parameters	Description	Source
$K_{D2} = 0.3559(\mu M)$	Binding affinity between CsgA·MT and $Cu^{2+}$	Estimated from experiment
$K_{D3} = 0.1676(\mu M)$	Binding affinity between CsgA·MT and $Cd^{2+}$	Estimated from experiment
$n_3 = 0.4614$	Hill coefficient in the binding of between $Cu^{2+}$ and CsgA·MT	Estimated from experiment
$n_4 = 0.3182$	Hill coefficient in the binding of between $Cd^{2+}$ and CsgA·MT	Estimated from experiment

Part 3 : Population Survival Model

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Introduction

In industrial wastewater environments, metal ions have inhibitory effects on the growth of E. coli. However, the degree of tolerance to metal ions in our genetically modified E. coli still needs to be explored. Here, we first developed a model to fit the experimental data to understand the growth situation of E. coli parametrically. Then, we performed a dependence analysis of the parameters in the model and considered how these parameters change with increasing metal ions concentrations. In addition, a range of E. coli growth curves was predicted. The biological processes were parametrically evaluated in this part, leading to a more in-depth interpretation of the experimental results and further predictions to guide implementation decisions in the real world.

Model Development

Consider the functional relationship between cell number (denoted by n(t)) and time (denoted by t). In the case of unlimited resources, the growth rate r is constant, resulting in an exponential growth of E. coli. However, in wastewater environments, r depends on the cell density and varies with time. $r = r(n) = r_0$×(1- $\frac{n}{K}$), where $r_0$ is inherent growth rate.

The equation of E. coli growth hence become $\frac{dn}{dt}$ = $n(t)×r_0$×(1-$\frac{n(t)}{K})$. Therefore, the logistic growth curve of E. coli can be shown by $n(t) = \frac{K}{(1+\frac{(K-n_0)}{n_0} ×e^{-r_0 t} )}$

Since the cell number is equivalent to the cell density or approximately the $OD_{600}$ readings, the $OD_{600}$ values and the cell density follow the same $n(t)$ expression ^[13] .

To assess the extent to which the growth of E. coli was inhibited, we established the SAD model to describe the state change of E. coli after the addition of metal ions^[13:1]. It is assumed that all E. coli were knocked into a suppressed state, S, after the addition of metal ions. After that, two possible states of E. coli were available for conversion. Some E. coli regained their activity, woke up and returned to the active state A, and then continued the cycle of multiplying and growing. Some other E. coli were changed from a suppressed state to a dead state D.

Parameters	Description
$r_0$	Inherent growth rate
$n_0$	Initial cell density
$K$	Asymptotic cell density
$n_s$	Cell density in the suppressed state
$n_a$	Cell density in the active state
$n_d$	Cell density in the dead state
$\alpha$	Killing rate, from the suppressed state to the dead state
$\beta$	Weak-up rate, from the suppressed state to the active state

Then the SAD model can be quantitatively described as follows: ^[13:2]:

$\frac{dn_s}{dt} = -β×n_s - α×n_s$

$\frac{dn_a}{dt} = β×n_s+r_0×(1-\frac{n_a}{(K-n_s-n_d )}) ×n_a$

$\frac{dn_d}{dt} = α×n_s$

$n(t_0) = n_0$

Fig. 3.1 The three-state transition (SAD) model.

Results and Analysis

Here is the framework for analyzing natural biological phenomena with uncertain outcomes. The real world generates data and is evaluated by our experiments. The field of data processing uses these data to come up with a quantitative description model. Once we develop the model, we use analysis tools to explore deeper relations and evaluate them. Moreover, the results from this analysis lead to predictions and decisions about the real world.

Fig. 3.2 The relation between the real phenomena, data processing and analysis.

Conclusion and discussion

To conclude, our model provides a quantified description of the tolerance activity of E. coli to Ag ions. The toxic effect of Ag ions on E. coli fed back mainly on the variation of two parameters, the lag time and the maximum specific growth rate. Model fitting analysis of the experimental data revealed that the lag time of E. coli growth increased significantly with increasing Ag-treatment concentration. In the range of Ag ion concentrations from 0 μM to 80 μM, the E. coli could grow again, and the maximum growth rate remained basically unchanged. Furthermore, The growth situations of MG1655 pGEX-4T-1, which imported the empty plasmid, was much closer to the wild type, indicating that Ag ions mainly caused the growth inhibition of MG1655 pGEX-4T-1-csgA-MBP3, the effect caused by the operation of import plasmid was very limited.

Additionally, it is expected that more parameters characterizing the growth of E. coli can be included in our model, allowing a more detailed exploration of small changes in actual growth and the biological implications reflected. It is also important to emphasize the extensibility of our model for assessing metal ions other than Ag ions. The tolerance activity of E. coli to other heavy metal ions can be similarly obtained. By establishing a network of relationships between multiple heavy metal ion concentrations and the corresponding E. coli survival parameters, we can predict the survival of E. coli in natural wastewater environments and guide practical operations.