# Adhesion Model

## Overview

The locomotion and adhesion of E.coli in real human body were too complex to describe and to detect, whereas the adhesion to detachment ratio was essential for our engineered E.coli’s specific binding performance to lesions. Therefore, we want to simulate the those process through in silico modeling.

The major two questions we want to ask are:

- 1. How would microbe move in the mucus layer.
- 2. To which extent engineered E.coli could specifically bind to CRC lesions.

## Initial Assumptions

- 1. The fluid environment of colon was laminar flow with 1 atm Pa. .
- 2. The elastic properties of E.coli was mainly defined by peptidoglycan layer.
- 3. The force field in colon was in macro scale.

## Theory

The major equations used in locomotion model were:

$$\rho \frac{\partial u_{fluid} }{\partial t}+\rho (u_{fluid}\cdot \triangledown )=\triangledown \cdot [-pI+\mu (\triangledown u_{fluid}+(\triangledown u_{fluid})^{T}]+F+\rho g (1) $$

$$\rho \frac{\partial ^{2}u_{solid}}{\partial t^{2}}= \triangledown \cdot \left ( F S \right )^{t}+F_{v}, F=I+\triangledown u_{solid} (2) $$

$$S=S_{ad}+\jmath _{i}F^{-T}_{ineI}\left ( C:\epsilon _{eI} \right )F^{-1}_{ineI}, \epsilon _{eI}=\frac{1}{2}\left ( F^{T_{eI}}F_{eI}-I \right ), F_{eI}=FF^{-1}_{ineI} (3)$$

$$ S_{ad}=S_{0}+S_{ext}+S_{q} (4) $$

$$ \epsilon =\frac{1}{2}[(\triangledown u_{solid})^{T}+\triangledown u_{solid}+(\triangledown u_{solid})^{T}\triangledown u_{solid}] (5) $$

$$ C=C(E,\vartheta ) (6)$$

## Results

The velocity field of particles was in Gaussian distribution，of which the inner particles move quiker than outer ones (Fig.1).

Fig 1.The locomotion of particle in laminar flow.

## Adjusted model

We then considered some biological fators that may influence the binding of E.coli to epithelium. Considering the concentrations of IgA and mucin in colorectal microenvironment may influence the specific binding property demonstrated in Kirstie McLoughlin's work **(1)**, we ajust the Young's module based on those parameters.

$$E_{g}=\frac{\sigma _{g}}{\epsilon _{g}}\left ( 1-v_{pg}v_{g\rho } \right ) (7) $$

$$E_{p}=\frac{\sigma _{p}}{\epsilon _{p}}\left ( 1-v_{pg}v_{g\rho } \right ) (8) $$

The Young’s moduli of glycan strands Eg and peptide cross-links Ep were calculated (7-8). Where dimensionless Poisson's ratios,$v_{pg} $and $v_{gp} $ relate the spontaneous strain arising in one direction, stresses $\epsilon _{g} $ and $ \epsilon _{p} $ are the derivatives of the free energy with initial set of randomized number.

The simulation result showed a liquid-solid coupling phenomenon, in which the binding of microbe onto epithelium would change the flow direction of around liquid. Thus the velocity of small molecules' movement would slow down, suggesting a higher binding ability of E.coli in the enviroment that is rich of IgA. This conclusion could explain the slight increase in binding ability of non engineered E.coli to CRC related colorectum, compared to that to the normal colon.

Fig 2. Coupling reaction of microbe and liquid .

## Reference

[1] McLoughlin, K., Schluter, J., Rakoff-Nahoum, S., Smith, A. L., & Foster, K. R. (2016). Host selection of microbiota via differential adhesion. Cell Host & Microbe, 19(4), 550-559.

[2] Gumbart, J. C., Beeby, M., Jensen, G. J., & Roux, B. (2014). Escherichia coli peptidoglycan structure and mechanics as predicted by atomic-scale simulations. PLoS computational biology, 10(2), e1003475.