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.


     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.

  1. The dual membrane of E. Coli is permeable to small molecules such as TMA, choline (Davis et al., 2014).

  2. No other reactions aside from the ones discussed (absorption, oxidation) have an effect on TMA concentration.

  3. The outer membrane, periplasm, and plasma membrane are considered to be one membrane with their respective widths summed up into one width.

  4. The E. Coli produces enzyme linearly at the average rates for transcription and translation.

  5. There is only one plasmid encoding for the enzyme within the cell.

  6. There is enough oxygen in the lumen of the small intestine to facilitate the oxidation process (Singhal & Shah, 2020).

  7. E. Coli ingested via the probiotic is evenly dispersed across the entire intestine, therefore implying enzymes are also evenly dispersed.

  8. The majority, if not all, of TMA production occurs in the small intestine (Stremmel et al., 2017).

  9. Consumed food travels through the lumen at a constant rate.

  10. 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.

  11. Choline converts to TMA at a 1:1 ratio (Miller et al., 2014).

  12. 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).

  13. Any TMA remaining in the intestinal system upon exiting the small intestine is excreted via fecal matter (Yu & Amidon, 1999)

  14. 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.

  15. No choline is absorbed in the stomach (Yu & Amidon, 1999).


      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.


     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.


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