Math modelling

The whole system can be thought to have 3 modules:

1. Arsenic importer: Independent module
2. Gene expression: Dependent on the output from 1st module
3. Chelation: Dependent on the output from both 1st and 2nd module

Assumptions:

1. One Arsenic Species: For simplicity, let’s assume import rate, chelation rate etc. are same for both “As(III)” and “As(v)”. Therefore, we will not consider “Aox” in this model, because this only lowers the toxicity level of “As” by changing its form and thus providing a less toxic environment for both the bacteria and gut cells.This conversion of As(III) to As(V) can be modeled easily by michales menten kinetics. But here in this model we are more focused with the dynamics of chelation.
2. Phytochelatin gene under Ars promoter: T7 promoter is used just to amplifying the amount of synthetic phytochelatin expression. The system is equivalent to the operon shown here, with a higher protein production rate of phytochelatin.
   

In this section we will model those 3 modules individually.

1. Modeling Arsenic Importer

Approach 1: Using Michales Menten kinetics

Importer is the Enzyme here. Substrate is Arsenic present outside the cell membrane and the Arsenic present inside the cell can be treated as Product of the reaction. Importer binding with Arsenic outside is considered to be a bidirectional reaction. And from Importer Arsenic complex, the formation of Arsenic inside is considered as unidirectional reaction.

\(As_{out} = \) Arsenic outside
\(As_{in} = \) Arsenic inside
\(Imp = \) Importer
\(As_{out}*Imp = \) Importer and Arsenic outside complex
\(K_f = \) Forward reaction constant for the bidirectional reaction
\(K_b = \) Backward reaction constant for the bidirectional reaction
\(K_2 = \) Forward reaction constant for the unirectional reaction

Kinetic equations:

\(As_{out} + Imp \rightarrow As_{out}*Imp\)  with reaction constant \(K_f\)
\(As_{out}*Imp \rightarrow As_{out} + Imp\)  with reaction constant \(K_b\)
\(As_{out}*Imp \rightarrow As_{in} + Imp\)  with reaction constant \(K_2\)

Approach 2: Using Diffusion Equation

The flux of Arsenic through the cell membrane is considered here. As diffusion goes along a concentration gradient, it will not allow the cell to take more arsenic once Arsenic concentration inside the cell reaches to the Arsenic concentration outside the cell. That overcomes the previous drawback for Michales menten model.

Ordinary Differential Equations (ODEs) to simulate over time:

$$\frac{d[As_{out}]}{dt} = -K_f [As_{out}] [Imp] + K_b [As_{out}*Imp]$$ $$\frac{d[Imp]}{dt} = -K_f [As_{out}] [Imp] + K_b [As_{out}*Imp] + K_2 [As_{out}*Imp]$$ $$\frac{d[As_{out}*Imp]}{dt} = K_f [As_{out}] [Imp] - K_b [As_{out}*Imp] - K_2 [As_{out}*Imp]$$ $$\frac{d[As_{in}]}{dt} = K_2 [As_{out}*Imp]$$

Solutions of those ODEs are the following:

Say an initial concentration of Arsenic outside is provided, and that is 0.01 molar, and initial concentration of Arsenic inside is 0.00 molar. The concentration outside in the gut is assumed to remain as initial, over time.

Drawbacks:

Now if we consider the Arsenic concentration inside the gut to be a constant value over time, then the cells will absorb Arsenic forever. But in reality, this is not possible. Arsenic is dangerous for cell, it can’t take Arsenic forever without having any preventive mechanism against arsenic.

Flux of Arsenic through the membrane is \(J = As (Area)^{-1} (Time)^{-1} = D \cdot \frac{As_{out}-As_{in}}{d}\)
where \(D = \) diffusion coefficient, \(d = \) width of cell membrane and \(A = \) area covered by Importers on Cell Membrane.

By simplifying the Diffusion Equation, we get: $$\frac{dAs}{dt} = D \cdot \frac{As_{out}-As_{in}}{d} \cdot A$$ $$\frac{dAs}{dt} = R_{in} \cdot (As_{out}-As_{in})$$
where, \(R_{in} = \frac{D \cdot A}{d}\)

Consider that \(R_{in}\) is the Arsenic intake rate, through importer. Further assume that \(As_{max}\) is the maximum Arsenic concentration one cell can have, if Arsenic level goes beyond that, the cell dies.
So, the modified equation would be: $$\frac{dAs}{dt} = R_{in} \cdot (As_{max}-As_{in})$$

2. Modeling Phytochelatin Expression

Assumptions:

1. We will assume that the phytochelatin production directly depends on the Arsenic concentration inside the cell. As Phytochelatin production is restricted by ArsR ( which represses expression of Ars operon ), we will set a threshold of Arsenic under which Phytochelatin expression would be zero. Once the cell overcomes the threshold Phytochelatin production linearly depends on increasing Arsenic concentration inside the cell. It will contribute to the Gain Term of the equation.
2. Considering the degradation of Phytochelatin, we will need to incorporate a loss term as well to get the exact dynamics. For now, assume there is no chelation. Once we will model the Chelation, we will incorporate another loss term for that.
   

Say the threshold has been set to \(As_{thr} = 0.1 molar\)
Phytochelatin production rate \( = \gamma\)
Degradation rate of phytochelatin \( = \alpha\)
As concentration inside the cell \( = [As]\)

$$\frac{d[PC]}{dt} = 0 - \alpha[PC], \quad\quad if [As] \lt As_{thr}$$ $$\frac{d[PC]}{dt} = \gamma ([As]-As_{thr}) - \alpha[PC], \quad\quad if [As] \gt As_{thr}$$

3. Modeling the Chelation

Say one Phytochelatin chelates one Arsenic.

\(As = \) Arsenic
\(PC = \) Phytochelatin
\(PC*As = \) Phytochelatin and Arsenic complex
\(K_f = \) binding constant for forward reaction
\(K_b = \) binding constant for backward reaction

Kinetic equation:

\(As + PC \rightarrow PC*As\)  with binding constant \(K_f\)
\PC*As \rightarrow As + PC\)  with binding constant \(K_b\)

ODEs to solve through time:

$$\frac{d[As]}{dt} = -K_f [As] [PC] + K_b [PC*As]$$ $$\frac{d[PC]}{dt} = -K_f [As] [PC] + K_b [PC*As]$$ $$\frac{d[PC*As]}{dt} = K_f [As] [PC] - K_b [PC*As]$$