Team:NCKU Tainan/Model

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• Co-culture Model
• Bacteriocin Model

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 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.

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Co-culture Model

Fig. 1. Co-culture model overview.

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. Native C. difficile and E. coli population ratio in the human gut, and with added CreSolve (1 : 0.05 : 0.0025).

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

$\frac{d\left[\mathrm{TAL}\right]}{dt}=k_1\⁢\left[\mathrm{Nissle}\right]$

$\frac{d{\left[\mathrm{Tyr}\right]}_{\mathrm{outside}}}{dt}=-\frac{k_{\mathrm{cat1}}\⁢{\left[\mathrm{TryP}\right]}_{\mathrm{outside}}\⁢{\left[\mathrm{Tyr}\right]}_{\mathrm{outside}}}{K_{\mathrm{m1}}+{\left[\mathrm{Tyr}\right]}_{\mathrm{outside}}}$

$\frac{d{\left[\mathrm{TAL}\right]}_{\mathrm{cell}}}{dt}=\frac{k_{\mathrm{cat1}}\⁢{\left[\mathrm{TryP}\right]}_{\mathrm{cell}}\⁢{\left[\mathrm{Tyr}\right]}_{\mathrm{outside}}}{K_{\mathrm{m1}}+{\left[\mathrm{Tyr}\right]}_{\mathrm{outside}}}-\frac{k_{\mathrm{cat2}}\⁢{\left[\mathrm{TAL}\right]}_{\mathrm{cell}}\⁢{\left[\mathrm{Tyr}\right]}_{\mathrm{cell}}}{K_{\mathrm{m2}}+{\left[\mathrm{Tyr}\right]}_{\mathrm{cell}}}$

$\frac{d{\left[\mathrm{PCA}\right]}_{\mathrm{tot}}}{dt}=\frac{k_{\mathrm{cat2}}\⁢{\left[\mathrm{TAL}\right]}_{\mathrm{tot}}\⁢{\left[\mathrm{Tyr}\right]}_{\mathrm{cell}}}{K_{\mathrm{m2}}+{\left[\mathrm{Tyr}\right]}_{\mathrm{cell}}}$

$\frac{d\left[\mathrm{Nissle}\right]}{dt}={\mu }_1\cdot \left[\mathrm{Nissle}\right]\frac{1-\left(\left[\mathrm{Nissle}\right]+{\alpha }_1\left[\mathrm{CD}\right]\right)}{K_1}$

$\frac{d\left[\mathrm{p-HPA}\right]}{dt}=k_2\⁢\left[\mathrm{CD}\right]$

$\frac{d{\left[\mathrm{PHPAD}\right]}_{\mathrm{cell}}}{dt}=\frac{k_{\mathrm{cat1}}\⁢{\left[\mathrm{TryP}\right]}_{\mathrm{cell}}\⁢{\left[\mathrm{Tyr}\right]}_{\mathrm{outside}}}{K_{\mathrm{m1}}+{\left[\mathrm{Tyr}\right]}_{\mathrm{outside}}}-\frac{k_{\mathrm{cat3}}\⁢{\left[\mathrm{PHPAD}\right]}_{\mathrm{cell}}\⁢{\left[\mathrm{Tyr}\right]}_{\mathrm{cell}}}{K_{\mathrm{m3}}+{\left[\mathrm{Tyr}\right]}_{\mathrm{cell}}}$

$\frac{d{\left[\mathrm{PC}\right]}_{\mathrm{tot}}}{dt}=\frac{k_{\mathrm{cat3}}\⁢{\left[\mathrm{PHPAD}\right]}_{\mathrm{tot}}\⁢{\left[\mathrm{Tyr}\right]}_{\mathrm{cell}}}{K_{\mathrm{m3}}+{\left[\mathrm{Tyr}\right]}_{\mathrm{cell}}}$

$\frac{d{\left[\mathrm{CD}\right]}_{\mathrm{tot}}}{dt}={\mu }_2\cdot \left[\mathrm{CD}\right]\frac{1-\left(\left[\mathrm{CD}\right]+{\alpha }_2\left[\mathrm{Nissle}\right]\right)}{K_2}$

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Bacteriocin Model

Fig B1. Bacteriocin model overview

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:

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

${\psi }_{\max }$ 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:

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

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^[1], and pharmacodynamic models of antibiotics are often based on the Hill equation^[2]. 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.

Fig B2. Bacteriocin efficiency with varying Hill coefficient. $\kappa$.

Fig B3. Bacteriocin efficiency at $\kappa$ = 11 with varying ${\mathrm{EC}}_{50}$ values.

Fig B4. Bacteriocin efficiency at $\kappa$ = 11 with varying interaction times.

References

1. Akdemir, F. E., Albayrak, M., Çalik, M., Bayir, Y., & Gülçin, I. (2017). The Protective Effects of p-Coumaric Acid on Acute Liver and Kidney Damages Induced by Cisplatin. Biomedicines, 5(4), 18. doi: 10.3390/biomedicines5020018
2. Cotter, P. D., Ross, R. P., & Hill, C. (2012). Bacteriocins — a viable alternative to antibiotics? Nature Reviews Microbiology, 11(2), 95–105. doi: 10.1038/nrmicro2937
3. Regoes, R. R., Wiuff, C., Zappala, R. M., Garner, K. N., Baquero, F., & Levin, B. R. (2004). Pharmacodynamic Functions: a Multiparameter Approach to the Design of Antibiotic Treatment Regimens. Antimicrobial Agents and Chemotherapy, 48(10), 3670–3676. doi: 10.1128/aac.48.10.3670-3676.2004

e-mail : igem.ncku.tainan@gmail.com
phone : 06-2757575