## Modeling

### A Mathematical Explaination

## Overview

### Parameters and Fink's Second Law

#### For our project, diffusion plays an important role in how GFP is expressed in the biofilm. Due to the complicated structure of mucosal biofilms, our model simplified the structure into a slab. This allows for diffusion to be measured consistently throughout the biofilm. Our parameters for the diffusion model were length, diffusivity coefficient, and initial concentration. In order to model the change in concentration, we used Fick’s Second law. Fick’s Second Law pertains to changes in concentration with time and distance. The change in concentration at location x is given by the difference in flux into and out of an element of a certain width.

#### The plot shown is a percentage concentration of cation for a given depth of 10 x 10^6 nano-meters. Each line on the plot represents the concentration at a different final time.

### Parameters and Fink's Second Law

#### For our project, diffusion plays an important role in how GFP is expressed in the biofilm. Due to the complicated structure of mucosal biofilms, our model simplified the structure into a slab. This allows for diffusion to be measured consistently throughout the biofilm. Our parameters for the diffusion model were length, diffusivity coefficient, and initial concentration. In order to model the change in concentration, we used Fick’s Second law. Fick’s Second Law pertains to changes in concentration with time and distance. The change in concentration at location x is given by the difference in flux into and out of an element of a certain width.

#### The plot shown is a percentage concentration of cation for a given depth of 10 x 10^6 nano-meters. Each line on the plot represents the concentration at a different final time.

#### The second plot shows the number of binded repressor in the cell. It is important to note that as number of binded repressor increases, the amount of repressor decreases. Because we assumed that the concentration of GFP is directly proportional to the concentration of binded repressor, the GFP expression will be exactly the same as the plot shown. Repressor = Cation Responsive Protein.

#### The next objectives in terms of modeling is to increase the accuracy of our sodium biosensor model, combine the two models together, and use base splines in order to create a function using the data points received for fluorescence of GFP. Determine how much we need to fine tune the sensor in order to know when there is an overwhelming influx of sodium and whether the cell is performing its regular operations.

×