# Modeling

## Goals of model

The goal of the Nygone project is to grow modified E. coli cells on filters that will be put in water treatment plants to remove microplastics. However, due to time constraints, we are unable to engineer the final product. Hence a mathematical model was built in order to:

1. Give predictions on the amount of bacteria required to produce a sufficient amount of MnP enzyme to degrade micro-particles of nylon in water
2. Simulate the distribution of MnP enzyme produced by bacteria
3. Simulate the degradation of nylon micro-particles by MnP particles in water
4. Analyze the sensitivity of MnP production and nylon degradation to various factors including concentration of bacteria, production rate of MnP by unit bacteria, volume, and flow rate of waste water.

## General model

The model was separated into two parts:

1. Kinetics of MnP enzyme in water
Rate of change of MnP concentration in water = rate of diffusion + rate of production - rate of elimination
2. c is the current concentration of MnP, dp/dt is the rate of production of MnP, and J is the flow rate.
3. Reaction of MnP particles with nylon
Michaelis-Menten Equation

E is the MnP enzyme, S is the nylon substrate. kf and kr are forward and reverse reaction constants of enzyme-substrate complex formation. kcat is the reaction coefficient for oxidation of nylon substrate, and the turnover number, defined as the maximum number of substrate molecules converted to product per enzyme molecule per unit time.

## Simplification:

After a visit to the local water treatment plant, we obtained a numerical value for the following parameters:
•           Total volume of chamber where filter will be inserted = 0.879 m3
•           Average flow rate = 0.19 m3/min
•           Dimensions of water chamber: 0.46 m (length), 0.91 m (width), 2.1 m (height)

• The diffusivity of MnP was estimated as follows: *
•           Diffusivity of MnP in water = 0.6e3 m2/s
•           Pe = velocity * linear dimension / mass diffusivity = 74.33>>1
•           Since Pe is much larger than 1, the diffusion term is taken out from the equation.
•           The kinetic model is simplified as:
• *The diffusivity of MnP was not found, this estimation was based on the diffusivity table of salt) Using the above values, the Peclet number was calculated

## Assumptions:

1. Monolayer of bacteria growth on the surface of the filter inserts (i.e. MnP produced will not be trapped inside the biofilms of E.coli ).
2. In this model, the total number of bacteria is assumed to be constant.
3. Rate of production of MnP enzyme is assumed to be linearly related to the total amount of bacteria. However, this relationship can be easily modified in the model.
4. MnP and nylon micro-particles are distributed evenly in the water (i.e. minimal time is taken for MnP and nylon particles to diffuse out evenly inside the chamber).
5. The size (width and height only) of the chamber is small.

## Limits and constraints:

1. The model is only applicable to a small container where diffusion across space takes little time to occur.
2. When the number of bacteria is large, multiple layers of bacteria will be built up. This may hinder the diffusion of MnP particles as they would be trapped within the biofilm. As a result, the linear relationship between bacteria number and MnP production rate would no longer be valid.

## Result and Interpretation:

Since some of our parameters are not yet finalized, this main purpose of this section is a demonstration of the capability of the model and the ability to predict an expected outcome. The shape of the graphs may change if a different set of parameters is used.

###### Figure 1.1

Figure 1.1 shows the production of MnP and degradation of nylon over time. MnP reaches a constant level of 3000 units after 30 hours while nylonconcentration decreases to zero within 10 hours. MnP is in excess and therefore a smaller amount of starting bacteria can be used.

###### Figure 2.2

The concentration of MnP and nylon after 60 hours with respect to different starting concentrations of bacteria. Figure 2.1: MnP increase exponentially with exponential increase in starting concentration of bacteria. Figure 2.2: Given the current set of parameters, all nylon will be degraded after 60 hours for a small number of starting bacteria.

###### Figure 2.2

Response of MnP concentration to a range of parameters: Figure 2.1 shows the concentration of MnP over time for different values of p, which is the rate of production of MnP per unit of bacteria. Figure 2.2 is the concentration of MnP after 60 hours with respect to different values of p.

###### Figure 3.2

Sensitivity analysis: Figure 3.1 gives the sensitivity of MnP concentration when there is a 1 percent increase in a given set of parameters. The parameters from left to right are: rate of production of MnP, starting amount of bacteria, volume of chamber, flow rate, forward reaction constant of MnP-nylon complex formation, reverse reaction constant of MnP-nylon complex formation, forward reaction constant of oxidation of nylon particles (degradation). The sensitivity will change if a different percentage increase is used. Figure 3.2 shows the perturbation on the level of MnP concentration, bacterial level, Nylon concentration and Nylon-MnP complex concentration with respect to one percent change in the set of parameters.