# Model

## Overview

We aim to develop mathematical models that act as a bridge between the theoretical and the physical realization of our biological work, to further optimize our experiments and understand our results.

We demonstrated that our *E. coli* can convert tyrosine - the precursor of *p*-Cresol, into *p*-Coumaric acid, a beneficial byproduct by diverting the original fermentation pathway. For that, we simulated the interactions between *E. coli* Nissle and *Clostridium difficile* (*C. difficile*) under **co-culture conditions**.

Our Goals:

Better understand and improve our

**CreSolve**therapeutic drug’s effects on reducing*p*-Cresol production.Explore the effectiveness of bacteriocin on reducing

*Clostridium difficile*population.

## Co-culture Model

**Oh My Gut’s CreSolve** is engineered as a living therapeutic and so, it is vital to simulate our therapeutic drug’s efficiency in reducing excess tyrosine. However, due to time constraints, we were not able to perform a co-culture experiment to further validate our engineered

*E. coli*Nissle. Therefore, to prove the feasibility of

**CreSolve**, we built a series of co-culture model under two different conditions:

*C. difficile*and**CreSolve**are cultured under the same initial OD value of 0.06 (1 : 1).**CreSolve**cultured in a*C. difficile*-dominant environment (1:400).

The following equations are used in the co-culture models:

Equations | Description |
---|---|

$\frac{\mathit{d}\left[\mathit{TAL}\right]}{\mathit{d}t}={k}_{1}\⁢\left[\mathit{Nissle}\right]$ | We assumed that the rate of change of TAL (tyrosine ammonia-lyase) concentration is proportional to E. coli Nissle cell density. k_{1} stands for a coefficient. |

$\frac{\mathit{d}{\left[\mathit{Tyr}\right]}_{\mathit{outside}}}{\mathit{d}t}=-\frac{{k}_{\mathit{cat1}}\⁢{\left[\mathit{TryP}\right]}_{\mathit{outside}}\⁢{\left[\mathit{Tyr}\right]}_{\mathit{outside}}}{{K}_{\mathit{m1}}+{\left[\mathit{Tyr}\right]}_{\mathit{outside}}}$ | Transportation of tyrosine into the cell through tyrosine transporter according to Michaelis-Menten equation. [TryP] stands for the tyrosine transporter concentration; _{outside}k_{cat}_{1} is the estimated turnover number of tyrosine transporter; K_{m}_{1} is the tyrosine concentration at which tyrosine transportation rate is at half-maximum. |

$\frac{\mathit{d}{\left[\mathit{Tyr}\right]}_{\mathit{Nissle}}}{\mathit{d}t}=\frac{{k}_{\mathit{cat1}}\⁢{\left[\mathit{TryP}\right]}_{\mathit{cell}}\⁢{\left[\mathit{Tyr}\right]}_{\mathit{outside}}}{{K}_{\mathit{m1}}+{\left[\mathit{Tyr}\right]}_{\mathit{outside}}}-\frac{{k}_{\mathit{cat2}}\⁢{\left[\mathit{TAL}\right]}_{\mathit{Nissle}}\⁢{\left[\mathit{Tyr}\right]}_{\mathit{Nissle}}}{{K}_{\mathit{m2}}+{\left[\mathit{Tyr}\right]}_{\mathit{Nissle}}}$ | Change of tyrosine concentration in E. coli Nissle over time based on the transportation of tyrosine into the cell and the conversion of tyrosine into p-Coumaric acid by TAL. Both tyrosine transportation and p-Coumaric production are based on Michaelis-Menten equation. k_{cat}_{2} is the estimated turnover number of TAL; K_{m}_{2} is the tyrosine concentration at which tyrosine fermentation rate by TAL is at half-maximum. |

$\frac{\mathit{d}{\left[\mathit{PCA}\right]}_{\mathit{tot}}}{\mathit{d}t}=\frac{{k}_{\mathit{cat2}}\⁢{\left[\mathit{TAL}\right]}_{\mathit{tot}}\⁢{\left[\mathit{Tyr}\right]}_{\mathit{Nissle}}}{{K}_{\mathit{m2}}+{\left[\mathit{Tyr}\right]}_{\mathit{Nissle}}}$ | p-Coumaric production rate after tyrosine is imported through tyrosine transporter and converted by TAL. Based on Michaelis-Menten equation. |

$\frac{\mathit{d}\left[\mathit{Nissle}\right]}{\mathit{d}t}={\mu}_{1}\cdot \left[\mathit{Nissle}\right]\frac{1-\left(\left[\mathit{Nissle}\right]+{\alpha}_{1}\left[\mathit{CD}\right]\right)}{{K}_{1}}$ | Rate of E. coli Nissle cell density increase based on the competitive Lotka-Volterra equation^{[1]}. μ_{1} stands for the specific growth rate of E. coli Nissle, and K_{1} stands for carrying capacity^{[2]}. α_{1} represents the effect C. difficile has on E. coli Nissle. |

$\frac{\mathit{d}\left[\mathit{PHPAD}\right]}{\mathit{d}t}={k}_{2}\⁢\left[\mathit{CD}\right]$ | We assumed that the rate of change of p-HPA decarboxylase (p-Hydroxyphenylacetate decarboxylase) concentration is proportional to C. difficile cell density. k_{2} stands for a coefficient. |

$\frac{\mathit{d}{\left[\mathit{PHPAD}\right]}_{\mathit{CD}}}{\mathit{d}t}=\frac{{k}_{\mathit{cat1}}\⁢{\left[\mathit{TryP}\right]}_{\mathit{cell}}\⁢{\left[\mathit{Tyr}\right]}_{\mathit{outside}}}{{K}_{\mathit{m1}}+{\left[\mathit{Tyr}\right]}_{\mathit{outside}}}-\frac{{k}_{\mathit{cat3}}\⁢{\left[\mathit{PHPAD}\right]}_{\mathit{CD}}\⁢{\left[\mathit{Tyr}\right]}_{\mathit{CD}}}{{K}_{\mathit{m3}}+{\left[\mathit{Tyr}\right]}_{\mathit{CD}}}$ | Change of tyrosine concentration in C. difficile over time based on the transportation of tyrosine into the cell and the conversion of tyrosine into p-Cresol by p-HPA decarboxylase. Both tyrosine transportation and p-Cresol production are based on Michaelis-Menten equation. k_{cat}_{3} is the estimated turnover number of p-HPA decarboxylase; K_{m}_{3} is the tyrosine concentration at which tyrosine fermentation rate by p-HPA decarboxylase is at half-maximum^{[3]}. |

$\frac{\mathit{d}{\left[\mathit{PC}\right]}_{\mathit{tot}}}{\mathit{d}t}=\frac{{k}_{\mathit{cat3}}\⁢{\left[\mathit{PHPAD}\right]}_{\mathit{tot}}\⁢{\left[\mathit{Tyr}\right]}_{\mathit{CD}}}{{K}_{\mathit{m3}}+{\left[\mathit{Tyr}\right]}_{\mathit{CD}}}$ | p-Cresol production rate after tyrosine is imported through tyrosine transporter and converted by p-HPA decarboxylase. Based on Michaelis-Menten equation. |

$\frac{\mathit{d}\left[\mathit{CD}\right]}{\mathit{d}t}={\mu}_{2}\cdot \left[\mathit{CD}\right]\frac{1-\left(\left[\mathit{CD}\right]+{\alpha}_{2}\left[\mathit{Nissle}\right]\right)}{{K}_{2}}$ | Rate of C. difficile cell density increase based on the competitive Lotka-Volterra equation. μ_{2} stands for the specific growth rate of C. difficile, and K_{2} stands for carrying capacity^{[4]}. α_{2} represents the effect E. coli Nissle has on C. difficile. |

The equations we used in the co-culture models are mostly based on Michaelis-Menten equation and competitive Lotka-Volterra equation. The Michaelis-Menten equation is commonly used in biochemistry to model enzyme kinetics. Under the assumption that enzyme concentration is much less than the substrate concentration, the rate of product formation is shown in the equation below:

$$v=\frac{\mathit{d}\left[P\right]}{\mathit{d}t}={V}_{\mathit{max}}\⁢\frac{\left[S\right]}{{K}_{M}+\left[S\right]}={k}_{\mathit{cat}}\⁢{\left[E\right]}_{0}\frac{\left[S\right]}{{K}_{M}+\left[S\right]}$$

$v$ is the rate of enzymatic reactions (rate of product formation, $\frac{\mathit{d}\left[\mathit{P}\right]}{\mathit{d}t}$). [*S*] is the concentration of a substrate. ${V}_{\mathrm{max}}$ represents the maximum reaction rate achieved by the system. The value of Michaelis constant ${K}_{m}$ is the substrate concentration at which the reaction rate reaches 1/2 ${V}_{\mathrm{max}}$. ${k}_{\mathrm{cat}}$, the turnover number, is the maximum number of substrate molecules converted to product per enzyme molecule per second.

Though TryP is not an enzyme, the kinetics of membrane transport can also be described by Michaelis-Menten equation.

The competitive Lotka-Volterra equation is similar to the logistic equations, which is a commonly used exponential population model, but with an additional term to account for the species' interactions:

$$\frac{\mathit{d}{x}_{1}}{\mathit{d}t}={r}_{1}\⁢{x}_{1}\left(1-\left(\frac{{x}_{1}+{\alpha}_{12}\⁢{x}_{2}}{{K}_{1}}\right)\right)$$

$$\frac{\mathit{d}{x}_{2}}{\mathit{d}t}={r}_{2}\⁢{x}_{2}\left(1-\left(\frac{{x}_{2}+{\alpha}_{21}\⁢{x}_{1}}{{K}_{2}}\right)\right)$$

$x$ is the population size of the two species. $r$ is the specific growth rate. $K$ is the carrying capacity, representing the maximum population size of the species that the environment can sustain indefinitely. The interaction between the two species is described by α_{12} and α_{21}, which stands for the effect species 2 has on the population of species 1 and the effect species 1 has on the population of species 2, respectively.

These are the parameters we used:

Parameters | Description |
---|---|

k_{1}= 0.01 |
Coefficient for the proportional relationship between the changing rate of TAL concentration and E. coli Nissle cell density. |

k_{2}= 0.01 |
Coefficient for the proportional relationship between the changing rate of p-HPA decarboxylase concentration and C. difficile cell density. |

[TryP]_{outside}=6.64*10 ^{-4} |
TyrP transport protein accessible to growth medium. |

[TryP]_{cell}=0.151 |
TyrP transport protein expression in a cell. |

k_{cat1}=50 (s^{-1}) |
The estimated turnover number of tyrosine transporter. |

k_{cat2}=0.015 (s^{-1}) |
The estimated turnover number of TAL^{[5]}. |

k_{cat3}=2.7 (s^{-1}) |
The estimated turnover number of p-HPA decarboxylase^{[3]}. |

K_{m1}=0.57 (μM) |
The tyrosine concentration at which tyrosine transportation rate is at half-maximum. |

K_{m2}=15.5 (μM) |
The tyrosine concentration at which tyrosine fermentation rate by TAL is at half-maximum. |

K_{m3}=2800 (μM) |
The tyrosine concentration at which tyrosine fermentation rate by p-HPA decarboxylase is at half-maximum^{[3]}. |

α_{1}=0.1 |
The effect C. difficile has on E. coli Nissle. |

α_{2}=0.1 |
The effect E. coli Nissle has on C. difficile. |

μ_{1}=0.82 |
The specific growth rate of E. coli Nissle^{[2]}. |

μ_{2}=0.13 |
The specific growth rate of C. difficile^{[4]}. |

K_{1}=0.62 |
The carrying capacity of E. coli Nissle^{[2]}. |

K_{2}=1.75 |
The carrying capacity of C. difficile^{[4]}. |

[Tyr]=29 (μM)_{Nissle} |
The initial tyrosine concentration in a growing E. coli cell^{[6]}. |

[Tyr]=50 (μM)_{CD} |
The initial tyrosine concentration in a growing C. difficile cell. We estimated the number to be larger than [Tyr]_{Nissle}. |

[TAL]_{cell} |
The TAL concentration in an E. coli Nissle (CreSolve) cell. |

[TAL]=0.0044_{tot}*[ TAL]_{cell} |
The total TAL concentration both inside and outside E. coli Nissle (CreSolve). V_{cell}/V_{tot}=0.0044. |

[PHPAD]_{cell} |
The p-HPA decarboxylase concentration in a C. difficile cell. |

[PHPAD]=0.0044_{tot}*[ PHPAD]_{cell} |
The total p-HPA decarboxylase concentration both inside and outside a C. difficile cell. V_{cell}/V_{tot}=0.0044. |

[Tyr]=1000(μM)_{outside} |
The initial tyrosine concentration in the environment^{[6]}. |

The assumptions we made:

- The models are used to simulate the
*in vitro*co-culture experiment. - The expression of TAL is directly proportional to
*E. coli*biomass. - Tyrosine can only be fermented to either
*p*-Cresol or*p*-Coumaric acid. *p*-Cresol can only be produced by*C. difficile*from tyrosine;*p*-Coumaric acid can only be produced by*E. coli*Nissle (**CreSolve**).- The tyrosine transporter in
*C. difficile*and*E. coli*Nissle are the same (same function, same concentration in each cell, same transportation efficiency). - The tyrosine concentration in
*C. difficile*([*Tyr]*) is larger than_{CD}*E. coli*Nissle ([*Tyr]*)._{Nissle} - In the tyrosine fermentation pathway of
*C. difficile*^{[7]}(Fig. 2), the rate-determining stage is the conversation of*p-*HPA into*p*-Cresol. The*k*_{m}_{3}and*k*_{cat}_{3}values in equations 7 and 8 are determined by the enzyme*p-*HPA decarboxylase^{[3]}. *C. difficile*and*E. coli*Nissle are grown under anaerobic conditions in BHI medium^{[8]}.- The
*p*-Cresol level is not high enough to inhibit the growth of*E. coli*Nissle^{[8]}. - The effects of
*C. difficile*and*E. coli*Nissle on each other (α_{1}, α_{2}) are the same. - CFU counting by OD value: 10
^{9}CFU/mL/OD600 nm.

### Co-culture model with the same OD value (1 : 1)

Under the first condition, *C. difficile* and *E. coli* Nissle are cultured under the same initial OD value of 0.06 (1 : 1), where the same initial amount of *C. difficile* and engineered *E. coli* Nissle (**CreSolve**) are used to simulate growth, tyrosine fermentation, *p*-Coumaric acid and *p*-Cresol productions. We started this co-culture model with an initial OD ratio of 1:1, to observe the interaction between *C. difficile* and engineered *E. coli* Nissle without additional factors. The growth model of both **CreSolve** and *C. difficile* co-culture, and *C. difficile* monoculture are then generated (Fig. 3). The results indicated that the addition of **CreSolve** slowed the growth of *C. difficile*. In addition, during co-culture, the overall tyrosine fermentation rate will decrease at a faster rate over time when compared with *C. difficile* monoculture alone (Fig. 4). From Fig. 5, the presence of **CreSolve** indeed lowered the production of *p*-Cresol by *C. difficile*. The tyrosine concentration decreases as expected (Fig. 4), and the decrease of *p*-Cresol is shown (Fig. 5). With the modeling results, we then proceed with a disparity population ratio between the two species to further explore **CreSolve**’s effectiveness.

### Co-culture model with **CreSolve** cultured in a *C. difficile*-dominant environment (1:400)

In the second condition, we aim to know whether **CreSolve** is effective in a *C. difficile*-dominant environment, we assumed the ratio of 400 for *C. difficile* and 1 for the engineered *E. coli* Nissle (**CreSolve**). From Fig. 6, we can see that even though the initial **CreSolve** population is far less than *C. difficile*, it still has a significant effect on *C. difficile* growth. The tyrosine concentration decreases faster as expected (Fig. 7), and significant decrease of *p*-Cresol is shown (Fig. 8). With this model, we are more certain that **CreSolve** is effective in decreasing the *p*-Cresol level of CKD patients.

## Bacteriocin Model

Since the *Clostridium* cluster is one of the major *p*-Cresol producers, besides changing the tyrosine fermentation pathway, we also aim to reduce the population of *Clostridium* spp. in the future. To show the killing efficiency of bacteriocin CBM-B, we modeled the relationship between *C. difficile* survival rate and CBM-B concentration.

As our model of the net growth rate of *C. difficile* population $\psi $, when exposed to CBM-B concentration $a$, we assumed the following relationship:

$$\psi \left(a\right)={\psi}_{\mathit{max}}-\mu \left(a\right)$$

${\psi}_{\mathit{max}}$ is the growth rate of the *C. difficile* population in the absence of CBM-B, and $\mu \left(a\right)$ is the survival rate of the *C. difficile* population exposed to CBM-B concentration $a$, which we assume to be a Hill function:

$$\mu \left(a\right)={E}_{\mathit{max}}\⁢\frac{{\left(a/{\mathit{EC}}_{\mathit{50}}\right)}^{\kappa}}{1+{\left(a/{\mathit{EC}}_{\mathit{50}}\right)}^{\kappa}}$$

The reason why we choose the Hill equation for this model is that according to our research, applications of bacteriocins are being tested to assess their application as a narrow-spectrum antibiotic^{[9]}, and pharmacodynamic models of antibiotics are often based on the Hill equation^{[10]}. Therefore, we consider it is suitable to use the Hill equation for our CBM-B effectivity model.

The Hill equation is commonly used in biochemistry, which refers to the binding of ligands to macromolecules. In pharmacology, it is extensively used to describe the drug concentration–effect relationship. In the equation above, ${E}_{\mathrm{max}}$ designates the maximum CBM-B-mediated survival rate, ${\mathrm{EC}}_{50}$is the CBM-B concentration at which the survival rate is at half of its maximum, ${E}_{\mathrm{max}}/2$, and $\kappa $ denotes the Hill coefficient, which determines the steepness of the sigmoid relationship between $\mu $ and $a$.

Hill coefficient describes the cooperativity of ligand binding in the following way:

$\kappa >0$ (Positively cooperative binding): Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules increases.

$\kappa <0$ (Negatively cooperative binding): Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules decreases

$\kappa =0$ (Noncooperative binding): The affinity of the enzyme for a ligand molecule is not dependent on whether or not other ligand molecules are already bound.

CBM-B’s mechanism of reducing *C. difficile* population is far more complicated than just a single ligand-receptor interaction. Therefore, the κ in this model can be seen as a coefficient reflecting the extent of cooperativity among multiple ligand binding sites, which eventually causes the reduction of *C. difficile* and inhibits the colony’s growth. This model explored how bacteriocin will definitely increase our **CreSolve**’s values.

### Bacteriocin dynamics

From the graph below, when ${\mathrm{EC}}_{50}$ value is fixed, the slope becomes steeper as the $\kappa $ value increases (Fig. 10). Similar trend can be found with fixed $\kappa $ value and varied ${\mathrm{EC}}_{50}$ value (Fig. 11). Larger $\kappa $ value and larger ${\mathrm{EC}}_{50}$ value both indicate the increasing interaction strength of bacteriocin on *C. difficile*.

After fitting with experimental data provided by one of our PIs, Professor Huang, we obtained a $\kappa $ value around 11, which shows that bacteriocin CBM-B reduces *C. difficile* efficiently. It also proves that CBM-B is a good choice for us when engineering a *p*-Cresol-reducing *E. coli* Nissle.

$\kappa $ = 11 is then used to simulate the time and concentration of bacteriocin required to perform effectively (Fig. 12). It shows that when bacteriocin concentration is higher than 1 $\mu g/\mathrm{ml}$, *C. difficile* survival rate reaches a plateau everytime, which indicates that no matter how high the concentration is, it still takes time for bacteriocin to work. After 6 hours, the *C. difficile* survival rate nearly reaches 1. Therefore, we speculate that the time bacteriocin CBM-M needs to fully kill *C. difficile* is 6 hours.

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