Overview

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

1. Better understand & improve our CreSolve therapeutic drug’s effects on reducing p-Cresol production.

2. Explore bacteriocin effectiveness on reducing Clostridium difficile population.

Co-culture Model

Oh My Gut’s CreSolve is engineered as a living therapeutic, 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:

1. C. difficile and CreSolve are cultured under the same initial OD value of 0.06 (1 : 1).

2. CreSolve cultured in a C. difficile-dominant environment (1:400)

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

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. $V_{\max }$ 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 $V_{\max }$. $k_{\mathrm{cat}}$, 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 α₁₂ and α₂₁, 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:

The assumptions we made:

1. The models are used to simulate in vitro co-culture experiment.
2. Expression of TAL is directly proportional to E. coli biomass.
3. Tyrosine can only be fermented to either p-Cresol or p-Coumaric acid.
4. p-Cresol can only be produced by C. difficile from tyrosine; p-Coumaric acid can only be produced by E. coli Nissle (CreSolve).
5. The tyrosine transporter in C. difficile and E. coli Nissle are the same (same function, same concentration in each cell, same transportation efficiency).
6. The tyrosine concentration in C. difficile ([Tyr]_CD) is larger than E. coli Nissle ([Tyr]_Nissle).
7. In tyrosine fermentation pathway of C. difficile^[7] (Fig. 2), the rate-determining stage is the conversation of p-HPA into p-Cresol. The k_m3 and k_cat3 values in equation 7 and 8 are determined by the enzyme p-HPA decarboxylase^[3].
8. C. difficile and E. coli Nissle are grown under anaerobic conditions in BHI^[8].
9. The p-Cresol level is not high enough to inhibit the growth of E. coli Nissle^[8].
10. The effects of C. difficile and E. coli Nissle on each other (α₁ α₂) are the same.
11. CFU counting by OD value: 10⁹ 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 E. coli Nissle are used to simulate growth, tyrosine fermentation, p-Coumaric acid and p-Cresol productions. 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, 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 decrease of p-Cresol is shown (Fig. 5). With the modeling results, we then proceed with a more realistic population ratio 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. From Fig. 6, we can see that even though 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, $E_{\max }$ 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, $E_{\max }/2$, 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 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 value increases (Fig B1). Similar trend can be found with fixed value and varied ${\mathrm{EC}}_{50}$ value (Fig B2). Larger 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 our PI - Dr. I-Hsiu 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 Nissle.

$\kappa$ = 11 is then used to simulate the time and concentration of bacteriocin required to perform effectively (Fig B3). 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. At 6 hr, 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.