Model Overview
The 2023 iGem team has incorporated mathematical modeling to streamline and enhance the design process. Through the application of these models, the team can visualize and understand intricate bioloigcal phenomena in straightforward terms, allowing for a deeper understanding of the project to be fostered across the entire team. Most importantly, the models have helped by showing which enzymes the bacteria should express and by setting a baseline for the minimum amount of probiotic bacteria needed to reduce TMA levels in the blood.
In this year's project, two models were created to assist with different aspects of the team's research. The first model is the Bacterial E. Coli Model, which is used to predict the Build Team's in-vitro experiment and the second is the Physiologically Based Pharmacokinetic Model (PBPK), which predicts the biological reactions taking place in the human body. Both of these models are considered "compartmental" models, as they condense complex processes into manageable compartments, which can be represented via a set of equations. These models were constructed using equations from experimentally proven sources and with data found in literature.
Assumptions
Like any mathematical model, some assumptions must be made either for simplicity's sake, to increase the feasibility in the creation of the model, or to set expectations of how the model's results could look like in an optimal environment.
- The dual membrane of E. Coli is permeable to small molecules such as TMA, choline (Davis et al., 2014).
- No other reactions aside from the ones discussed (absorption, oxidation) have an effect on TMA concentration.
- The outer membrane, periplasm, and plasma membrane are considered to be one membrane with their respective widths summed up into one width.
- The E. Coli produces enzyme linearly at the average rates for transcription and translation.
- There is only one plasmid encoding for the enzyme within the cell.
- There is enough oxygen in the lumen of the small intestine to facilitate the oxidation process (Singhal & Shah, 2020).
- E. Coli ingested via the probiotic is evenly dispersed across the entire intestine, therefore implying enzymes are also evenly dispersed.
- The majority, if not all, of TMA production occurs in the small intestine (Stremmel et al., 2017).
- Consumed food travels through the lumen at a constant rate.
- All precursors to TMA are readily available to bacteria to convert into TMA. This assumption was made due to the fact that modeling food breakdown proved to be overly complicated.
- Choline converts to TMA at a 1:1 ratio (Miller et al., 2014).
- Vital dynamics of E. Coli does not significantly affect the rate TMA can be oxidized in the lumen, since the probiotic is meant to be taken frequently to constantly replenish bacteria populations and E. Coli's generation time in the human body is extremely slow (Todar, 2020).
- Any TMA remaining in the intestinal system upon exiting the small intestine is excreted via fecal matter (Yu & Amidon, 1999)
- The lumen is a perfect cylinder with uniform surface area (taking into account vilii). This assumption was made to simplify the highly irregular physical structure of the small intestine.
- No choline is absorbed in the stomach (Yu & Amidon, 1999).
Data
The tabels below contains all the data relevant to the project. These data points will be incorporated into the equations as needed to construct the model.
Enzyme Kinetics
| Enzyme | Vmax (µg*min-1*mg-1) | Km (µM) | Source |
|---|---|---|---|
| Roseovarius sp. 217 TMM | 67.0 ± 3.5 | 21.6 ± 1.9 | (Chen et al., 2011) |
| Ruegeria pomeroyi DSS-3 TMM | 15.8 ± 3.1 | 20.8 ± 2.9 | (Chen et al., 2011) |
| Pelagibacter ubique HTCC1002 TMM | 4.2 ± 0.5 | 27.5 ± 4.2 | (Chen et al., 2011) |
| Pelagibacter ubique HTCC7211 TMM | 4.0 ± 0.2 | 28.5 ± 4.4 | (Chen et al., 2011) |
| Human FMO3 | 2.5 ± 0.2 | 27.8 ± 1.7 | (Lang et al., 1998) |
| M. Silvestris TMM | 1.7 ± 0.2 | 9.4 ± 2.1 | (Chen et al., 2011) |
Molecular Properties
| Molecule | Papp Value(cm/s) | Diffusion Coefficient (cm2/s) | Source |
|---|---|---|---|
| Choline | 11.11 ± .33 * 10-6 | .00043 | (Crowe et al., 2002) |
| TMA | 3.3 ± .1 * 10-6 | .00013 | (Kamiya et al., 2020) |
Bacterial Properties
| Property | Value | Source |
|---|---|---|
| Surface Area (µm2) | 6 | (Gilbert, 2009) |
| Cytoplasm Volume (µm3) | .67 | (Neidhardt, 1996) |
| Outer Membrane Width (nm) | 7.5 | (DiRienzo et al., 1978) |
| Periplasm Width (nm) | 25 | (De Geyter et al., 2016) |
| Plasma Membrane Width (nm) | 6 | (De Geyter et al., 2016) |
| Generation time (minutes) | 20 | (Tuttle et al., 2021) |
| Transcription Rate (bases/s) | 45 | (Yu et al., 2006) |
| Translation Rate (amino acids/s | 17 | (Zhu & Dai, 2019) |
Human Body Properties
| Property | Value | Source |
|---|---|---|
| Small Intestine Length (m) | 7 | (Libretexts, 2023) |
| Surface Area (m2) | 30 | (Helander & Fändriks, 2014) |
| Intestinal Transit Speed (minutes) | 260.5 | (Worsøe et al., 2011) |
| Vc (L) | 3.5 | (Usansky & Sinko, 2005) |
Experimental Parameters
| Parameter | Value |
|---|---|
| Amount of initial TMA (µg) | 100 |
| Amount of Bacteria (cfu) | 4 * 109 |
Michaelis-Menten Equation
The Michaelis-Menten equation models the change in velocity of a chemical reaction as the subsrate saturates the enzyme. When put into an ODE form, the equation can show how much of a subsrate is consumed by the chemical reaction. In this equation, Vmax is the maximum velocity of the reaction, E is enzyme amount , S is the subsrate amount, and Km is the Michaelis-Menten constant (Seabury & Stork, 2023).
Fick's Law
The equation models the rate of diffusion across a permeable layer as the concentration gradient changes. In this equaution, D is the diffusion coefficient, A is the surface area of the layer, C1 and C2 are concentrations of the molcule on opposite sides of the layer, and W is the width of the layer (Libretexts, 2022).
Absorption Rate Constant
This equation calculates the rate absorption given the apparent permeability: Papp, the absorptive surface area A, and Vc, which is the volume of the central compartment. The central compartment is considered to be all the blood vessels and central tissues highly perfused by blood (Yim et al., 2020).
E. Coli Bacterial Model
In order to test which enzymes would be most suitable for our probiotic, we analyzed a set of 6 enzymes that were documented in various rsearch articles. Parameters regarding enzyme kinetics were collected and input into the ODE form of the Michaelis-Menten equaton. By plotting the results of each enzyme, we were able to find that the TMMs from Roseovarius sp. 217 and Ruegeria pomeroyi DSS-3 were the most viable. Thus, these two were selected along with Human FMO3, for benchmark purposes. The results of this enzyme comparison are displayed below:
Once we selected which enzymes we wanted to test in-vitro, the modeling team put together a model in an attempt to predict the results of the in-vitro lab experiment. The model takes into account the rate that TMA enters an individual cell and the rate it is oxidized by the enzymes produced by the bacteria. The equations are as follows:
The rate at which TMA exits from the space outside the cell into the cell.
The rate at which TMA enters the cell and gets consumed by the Enzyme.
The provided equations constitute a set of ordinary differential equations (ODEs). When these equations are input into an ODE solver, they enable the plotting of the remaining Trimethylamine (TMA) within the system, starting with an initial TMA quantity of 100 µg.
Furthermore, the variable 'E' is determined by a linear function, representing the rate of enzyme production per minute by each enzyme. These rates have been calculated based on transcription and translation rates, as well as the unique base pair counts of individual plasmids.
The resulting outcomes of this model are displayed below. It's anticipated that the results of the in-vitro laboratory experiment will closely align with the depicted curves. The shaded areas in the plot denote the uncertainty, which is indicated by the standard deviations for each employed parameter.
PBPK Model
To extend our previous findings to the human body, we developed a Physiologically-Based Pharmacokinetic (PBPK) model. The PBPK model enables us to observe the variations in molecule concentrations across different regions of the body. Creating this PBPK model was essential, as it played a crucial role in determining the required quantity of bacteria to include in our probiotic.
By multiplying the linear function 'E' by different amounts of bacteria, we can calculate the necessary bacterial load within the system to achieve the desired enzyme concentration. In this model, the threshold of TMA in the bloodstream for the treatment to be considered a success was set at 3.94 mg. This value was determined by comparing the ratio of TMAO to TMA levels in a healthy individual (Stremmel et al.,2017).
For the PBPK model, we selected an initial condition of 1000 mg of choline, which represents a worst-case scenario meal with an unusually high choline content. This meal scenario corresponds to consuming approximately 9-10 eggs, each containing around 1,000 mg of choline (Miller et al., 2014). The mathematical equations governing this model are provided below:
This equation models the change in choline (C), based on its absorption rate (Kc) and its conversion rate to TMA by gut bacteria (r).
The change in TMA is dependant on the rate that choline is converted to TMA, (r*C), the rate that TMA is absorbed, (Kt*TMA), and the rate that it is converted to TMAO, given by the Michaelis-Menten equation.
The rate that TMA enter the blood is given by (Kt*TMA), which leads to an increase in TMAb.
As stated before, the change in TMAO is given by the Michaelis-Menten Equation. When all these equations are solved and plotted, we can visually see the rate at which TMA increases in the blood.
The graph presented above illustrates the expected absorption of choline and TMA in a control environment, which does not involve the introduction of a probiotic. At the 260.5-minute mark, the average time for a meal to traverse the small intestine, a total of 78.711 mg of TMA has been absorbed into the bloodstream. In individuals without TMAU, this TMA undergoes a process of oxidation in the liver, with approximately 95% of it transforming into TMAO (Stremmel et al., 2017). This results in approximately 3.94 mg of TMA remaining in the bloodstream in individuals without TMAU.
For E. Esperance to be effective, the probiotic must possess the capability to rapidly oxidize TMA, aiming to replicate the amount of TMA that persists in individuals without TMAU.
The graph above shows the amount of TMA absorbed once the probiotic is delivered. To achieve TMA levels in the bloodstream of a person affected by Trimethylaminuria (TMAU) equivalent to that of an individual without TMAU after liver oxidation (3.94 mg), a population of 18.3 × 1012 colony-forming units (CFUs) is required to colonize the small intestine. To deliver this substantial quantity of bacteria with a potential survival rate of up to 100%, E. Esperance should be suspended in a suitable medium, such as liquid or gel capsules (Wang et al., 2022).
The following chart illustrates E. Esperance's ability to convert TMA into TMAO. This same bacterial population is anticipated to produce a total of 100.3 mg of TMAO.
Conclusions
Mathematical models are inherently limited in their accuracy, particularly when applied to biological systems where variables exhibit significant variability. In addition, modeling biological processes at the macroscopic level introduces additional layers of complexity. The effectiveness of a model heavily relies on its underlying assumptions and its capacity to account for irregular variables. Therefore, this model was constructed to anticipate a "worst-case scenario," which encompasses factors such as unusually high choline levels, slow rates of enzyme expression, and rapid TMA absorption into the bloodstream.
Real-life scenarios that weren't incorporated into the model must also be considered. For example, gastric emptying, which reduces the choline content in the intestine, will allow for a slower choline to TMA production rate, which should allow more TMA to be oxidized. Another noteworthy factor is the overexpression of our plasmids, leading to higher enzyme production rates than what the model accounted for. This, too, should enhance the efficiency of our probiotic and reduce the required quantity of bacteria. If the model can confirm the feasibility of a solution under harsher circumstances, it suggests that a solution for less extreme scenarios feasible.
Despite the substantial variability inherent in biological systems, this model plays a crucial role. It provides a visual representation of complex biological processes, rendering them more understandable. It also delineates the multitude of reactions that influence TMA production, absorption, and oxidation. One consideration for a future model improvement involves incorporating diffusion convection equations to more accurately depict the spread of TMA in the lumen.
References
Davis, T. D., Gerry, C. J., & Tan, D. S. (2014). General platform for systematic quantitative evaluation of small-molecule permeability in bacteria. ACS Chemical Biology, 9(11), 2535–2544. https://doi.org/10.1021/cb5003015
Singhal, R., & Shah, Y. M. (2020). Oxygen battle in the gut: Hypoxia and hypoxia-inducible factors in metabolic and inflammatory responses in the intestine. Journal of Biological Chemistry, 295(30), 10493–10505. https://doi.org/10.1074/jbc.rev120.011188
Stremmel, W., Schmidt, K. V., Schuhmann, V., Kratzer, F., Garbade, S. F., Langhans, C.-D., Fricker, G., & Okun, J. G. (2017b). Blood trimethylamine-N-oxide originates from microbiota mediated breakdown of phosphatidylcholine and absorption from small intestine. PLOS ONE, 12(1). https://doi.org/10.1371/journal.pone.0170742
Miller, C. A., Corbin, K. D., da Costa, K.-A., Zhang, S., Zhao, X., Galanko, J. A., Blevins, T., Bennett, B. J., O’Connor, A., & Zeisel, S. H. (2014b). Effect of egg ingestion on trimethylamine-N-oxide production in humans: A randomized, controlled, dose-response study , , ,. The American Journal of Clinical Nutrition, 100(3), 778–786. https://doi.org/10.3945/ajcn.114.087692
Todar, K. (n.d.). The Growth of Bacterial Populations. Online Textbook of Bacteriology. https://textbookofbacteriology.net/growth_3.html#:~:text=Generation%20times%20for%20bacteria%20vary,
to%20be%2012%2D24%20hours
Yu, L. X., & Amidon, G. L. (1999). A compartmental absorption and transit model for estimating oral drug absorption. International Journal of Pharmaceutics, 186(2), 119–125. https://doi.org/10.1016/s0378-5173(99)00147-7
Day-Walsh, P., Shehata, E., Saha, S., Savva, G. M., Nemeckova, B., Speranza, J., Kellingray, L., Narbad, A., & Kroon, P. A. (2021). The use of an in-vitro batch fermentation (human colon) model for investigating mechanisms of TMA production from choline, l-carnitine and related precursors by the human gut microbiota. European Journal of Nutrition, 60(7), 3987–3999. https://doi.org/10.1007/s00394-021-02572-6
Chen, Y., Patel, N. A., Crombie, A., Scrivens, J. H., & Murrell, J. C. (2011). Bacterial flavin-containing monooxygenase is trimethylamine monooxygenase. Proceedings of the National Academy of Sciences, 108(43), 17791–17796. https://doi.org/10.1073/pnas.1112928108
Lang, D., Yeung, C., Peter, R., Ibarra, C., Gasser, R., Itagaki, K., Philpot, R., & Rettie, A. (1998). Isoform specificity of trimethylamine n-oxygenation by human flavin-containing monooxygenase (FMO) and P450 enzymes. Biochemical Pharmacology, 56(8), 1005–1012. https://doi.org/10.1016/s0006-2952(98)00218-4
Crowe, A. P., Lockman, P. R., Abbruscato, T. J., & Allen, D. D. (2002). Novel choline transport characteristics in Caco-2 cells. Drug Development and Industrial Pharmacy, 28(7), 773–781. https://doi.org/10.1081/ddc-120005623
Kamiya, Y., Takaku, H., Yamada, R., Akase, C., Abe, Y., Sekiguchi, Y., Murayama, N., Shimizu, M., Kitajima, M., Shono, F., Funatsu, K., & Yamazaki, H. (2020). Determination and prediction of permeability across intestinal epithelial cell monolayer of a diverse range of industrial chemicals/drugs for estimation of oral absorption as a putative marker of hepatotoxicity. Toxicology Reports, 7, 149–154. https://doi.org/10.1016/j.toxrep.2020.01.004
Gilbert, R. (2009). Physical Biology of the cell, by Rob Phillips, Jane Kondev and Julie Theriot. Crystallography Reviews, 15(4), 285–288. https://doi.org/10.1080/08893110903104081
Escherichia coli and salmonella : Cellular and molecular biology. in SearchWorks catalog. (n.d.). https://searchworks.stanford.edu/view/3213374
DiRienzo, J. M., Nakamura, K., & Inouye, M. (1978). The outer membrane proteins of gram-negative bacteria: Biosynthesis, assembly, and functions. Annual Review of Biochemistry, 47(1), 481–532. https://doi.org/10.1146/annurev.bi.47.070178.002405
De Geyter, J., Tsirigotaki, A., Orfanoudaki, G., Zorzini, V., Economou, A., & Karamanou, S. (2016). Protein folding in the cell envelope of escherichia coli. Nature Microbiology, 1(8). https://doi.org/10.1038/nmicrobiol.2016.107
Tuttle, A. R., Trahan, N. D., & Son, M. S. (2021). Growth and maintenance of escherichia coli laboratory strains. Current Protocols, 1(1). https://doi.org/10.1002/cpz1.20
Yu, J., Xiao, J., Ren, X., Lao, K., & Xie, X. S. (2006). Probing gene expression in live cells, one protein molecule at a time. Science, 311(5767), 1600–1603. https://doi.org/10.1126/science.1119623
Zhu, M., & Dai, X. (2019). Maintenance of translational elongation rate underlies the survival of escherichia coli during oxidative stress. Nucleic Acids Research, 47(14), 7592–7604. https://doi.org/10.1093/nar/gkz467
Libretexts. (2023, January 17). 22.10a: Anatomy of the small intestine. Medicine LibreTexts. https://med.libretexts.org/Bookshelves/Anatomy_and_Physiology/Anatomy_and_Physiology_(Boundless)/22%3
A_Digestive_System/22.10%3A_The_Small_Intestine/22.10A%3A_Anatomy_of_the_Small_Intestine
Helander, H. F., & Fändriks, L. (2014). Surface area of the digestive tract – revisited. Scandinavian Journal of Gastroenterology, 49(6), 681–689. https://doi.org/10.3109/00365521.2014.898326
Worsøe, J., Fynne, L., Gregersen, T., Schlageter, V., Christensen, L. A., Dahlerup, J. F., Rijkhoff, N. J., Laurberg, S., & Krogh, K. (2011). Gastric transit and small intestinal transit time and motility assessed by a magnet tracking system. BMC Gastroenterology, 11(1). https://doi.org/10.1186/1471-230x-11-145
Elmeliegy, M., Lowe, P., & Krzyzanski, W. (2014). Simplification of complex physiologically based pharmacokinetic models of monoclonal antibodies. The AAPS Journal, 16(4), 810–842. https://doi.org/10.1208/s12248-014-9591-x
Seabury, R. W., & Stork, C. M. (2023). Pharmacokinetic and toxicokinetic modeling. Reference Module in Biomedical Sciences. https://doi.org/10.1016/b978-0-12-824315-2.00797-1
Team:TAS Taipei/Software. (n.d.). https://2018.igem.org/Team:TAS_Taipei/Software
Libretexts. (2022, April 23). 7.2: Fick’s Law of diffusion. Medicine LibreTexts. https://med.libretexts.org/Under_Construction/VirginiaTech/Book%3A_Pulmonary_Physiology_(Binks)/07%3A_
Fundamentals_of_Gas_Exchange/7.02%3A_Fick%27s_law_of_diffusion
Yim, D.-S., Choi, S., & Bae, S. H. (2020). Predicting human pharmacokinetics from preclinical data: Absorption. Translational and Clinical Pharmacology, 28(3), 126. https://doi.org/10.12793/tcp.2020.28.e14
Stremmel, W., Schmidt, K. V., Schuhmann, V., Kratzer, F., Garbade, S. F., Langhans, C.-D., Fricker, G., & Okun, J. G. (2017a). Blood trimethylamine-N-oxide originates from microbiota mediated breakdown of phosphatidylcholine and absorption from small intestine. PLOS ONE, 12(1). https://doi.org/10.1371/journal.pone.0170742
Miller, C. A., Corbin, K. D., da Costa, K.-A., Zhang, S., Zhao, X., Galanko, J. A., Blevins, T., Bennett, B. J., O’Connor, A., & Zeisel, S. H. (2014a). Effect of egg ingestion on trimethylamine-N-oxide production in humans: A randomized, controlled, dose-response study , , ,. The American Journal of Clinical Nutrition, 100(3), 778–786. https://doi.org/10.3945/ajcn.114.087692
Wang, G., Chen, Y., Xia, Y., Song, X., & Ai, L. (2022). Characteristics of probiotic preparations and their applications. Foods, 11(16), 2472. https://doi.org/10.3390/foods11162472
Usansky, H. H., & Sinko, P. J. (2005). Estimating human drug oral absorption kinetics from CACO-2 permeability using an absorption-disposition model: Model development and evaluation and derivation of analytical solutions for ka and fa. Journal of Pharmacology and Experimental Therapeutics, 314(1), 391–399. https://doi.org/10.1124/jpet.104.076182