Note: All MATLAB scripts corresponding for each of the models can be found at the following repository: https://github.com/cornelligem/2020WetLabModelingCode.
The following three differential equations were used to model our holin-antiholin kill switch. We assumed the following: the expression rate of a constitutive promoter is equal to that of the lactate-inducible promoter in an infinite-lactate environment; the protein products degrade according to a first order reaction; the holin-antiholin dimer forms according to a second order reaction; the lactate concentration in the tumor environment is constant. For the kill switch and all other protein production modeling, maturation time and efficiency were neglected.
The first equation represents the rate at which the antitoxin, antiholin, is being produced. The terms respectively represent: expression from the lactate promoter, removal due to degradation, removal due to the binding to holin (to form the inactivated holin-antiholin dimer), and addition due to dissociation of the holin-antiholin dimer. A strong promoter was used to encourage the therapeutic bacteria to survive in high lactate levels.
The second equation represents the rate at which the toxin, holin, is being produced. The terms respectively represent: the expression from the constitutive promoter, removal due to degradation, removal due to binding to antiholin (to form the inactivated holin-antiholin dimer), and addition due to disassociation of the holin from the holin-antiholin dimer. To better distinguish expression at low and high lactate levels, a weak promoter was used.
The third equation represents the net rate at which the holin-antiholin dimer is formed. The terms respectively represent: the reaction rate at which holin and antiholin bind, the rate at which the dimer disassociates, and the rate at which the dimer degrades.
| Variable | Parameter | Value | Source |
|---|---|---|---|
| [Antitoxin] | Conc. of antiholin | - | - |
| [Toxin] | Conc. of holin | - | - |
| [Dimer] | Conc. of toxin-antitoxin dimer *inactivated toxin | - | - |
| f | Fluorescence scaling factor | = const. exp / (basA + a) = (23.22 uM/min) / (7790 uM/min) = 2.98 * 10-3 | [1][2] |
| basA | Basal exp of lactate promoter | 479.3 [uM/min] | [1] |
| KdA | Dissociation constant of lactate from lactate promoter | 1075 [uM] | [1] |
| n | Exponent | 1.326 | [1] |
| a | Induced expression scaling factor | 7311 [uM/min] | [1] |
| dA | Rate const, degradation of antitoxin | 0.13 [min-1] | [3] |
| dT | Rate const, degradation of toxin | 0.0348 [min-1] | [4] |
| [Lac] | Lactate concentration | 1 to 30 [mM] | [5] |
| kf | Rate const, forward rxn: T + A ↔ TA | 1.2 * 10-4 [uM-1min-1] | [4] |
| kb | Rate const, reverse rxn: T + A ↔ TA | 0.3 * 10-4 [min-1] | [4] |
| basT | Basal exp of weak constitutive promoter | 23.22 uM/min for strong promoter J23100, scale by relative strength of J23108 * (1303 / 2547)= 11.87 [uM/min] | [3], [6] |
| dD | Rate const, degradation of toxin-antitoxin dimer | 0.0348 [min-1] | Estimated value from [4] |
To predict the maximum Trichotherapy colonization for trichosanthin and mCardinal modeling, the typical bacterial density was approximated using existing literature. According to Duong et. al, bacteria can easily reproduce within the
tumor environment, reaching a concentration of 1 × 1010 CFU/g after three days [7]. According to Monte, there are approximately 1 × 108 cancer cells in every gram of tumor; these two values gives us an approximation of 100 E. coli cells
per tumor cells (assuming one CFU equals one cell) [8].
The logistic growth model from Zweitering et al. was modified to develop a function to give growth rate as a function of cell density[9]:
The logistic growth equation was rearranged to get the effective time teff as a function of cell density y, where teff was used to find the instantaneous growth rate.
The following equation models the diffusion of our bacteria inside and outside a tumor environment using Fick’s Second Law [9]. Compared to bacteria growing outside of the tumor, bacteria inside the tumor would have a greater growth rate.
It was assumed that over the span of several weeks, the tumor size would not change significantly.
The boundary conditions were set according to the relative amount of vasculature in breast tissue, where the nearest blood vessel to the tumor was assumed to be 25 + 1/√30, since there are an average of 30 vessels/ mm2, then 1/√3 mm on
average per vessel [10]. Based on the results, we found that our treatment can potentially be effective in the long-term; one time administration of the treatment could allow the bacteria to constantly produce therapeutic until the tumor
shrinks and eventually disappears.
The initial conditions were set according to the injection, assuming that for any tumor, a total of 2e6 CFU (this was the maximum dosage before seeing some side effects [8]) were delivered evenly to the tumor site within a 0.2 mm radius
of the small tumor and 5mm for the large tumor, and all other parts of the tumor had zero initial concentration.
| Variable | Parameter | Value | Source |
|---|---|---|---|
| Concentration** | c | Boundary and initial condition for injected bacteria concentration: 2e6/(r3) CFU/mm3 | -- |
| Concentration of cells | [Cells] | -- | -- |
| Radius | r | 50000, 1000, 20000, (graph, inject, tumor) μm | -- |
| Diffusion Coefficient | Diff | 0.579 μm2/sec | [8] |
| Growth Rate In Tumor | 𝜇I | 1.736 * 10-4 sec-1 | [8] |
| Growth Rate Outside Tumor | 𝜇O | 1.736 * 10-9 sec-1 | Estimated from [7] |
| Maximum Density (used to find teff) | A | 107 CFU/mm3 | [7] |
The following equation represents the production of trichosanthin protein within the bacteria. The first term represents the rate of translation, while the second represents protein degradation.
| Variable | Parameter | Value | Source |
|---|---|---|---|
| [Trich] | Concentration of trichosanthin | -- | -- |
| basR | Basal expression of constitutive promoter | 0.387 μM/sec-1 | [2] |
| dr | Degradation rate of trichosanthin | 0.000385 sec-1 | [1] |
In MATLAB, we modeled the output of the mCardinal protein using the following differential equation, and we solved it using MATLAB’s ode45 function. The equation represents the production of mCardinal protein. The first term represents the basal expression under the constitutive promoter (BBa_J23100), while the second represents protein degradation.
| Variable | Parameter | Value | Source |
|---|---|---|---|
| [mCard] | Concentration of mCardinal | -- | -- |
| basT | Basal transcription rate | 23.22 μM/min | [2] |
| dC | Degradation rate of trichosanthin | 0.0575 min-1 | [12] |
Modeling related to Trichoscan can be found under Hardware (Link to Modeling section under Hardware).
[1] Fomitcheva, A. (2015). BBa_K1847009. Retrieved October 14, 2020, from http://parts.igem.org/Part:BBa_K1847009
[2] Yildirim, N., & Mackey, M. C. (2003). Feedback Regulation in the Lactose Operon: A Mathematical Modeling Study and Comparison with Experimental Data. Biophysical Journal, 84(5), 2841-2851. doi:10.1016/s0006-3495(03)70013-7
[3] Guan, R. (2013). BBa_K1051208. Retrieved October 14, 2020, from http://parts.igem.org/Part:BBa_K1051208
[4] Kill Switch. (2013). Retrieved October 14, 2020, from https://2013.igem.org/KillSwitch
[5] Cruz-López, K. G., Castro-Muñoz, L. J., Reyes-Hernández, D. O., García-Carrancá, A., & Manzo-Merino, J. (2019). Lactate in the Regulation of Tumor Microenvironment and Therapeutic Approaches. Frontiers in Oncology, 9.
doi:10.3389/fonc.2019.01143
[6] Anderson, J. (2006). BBa_J23117. Retrieved October 14, 2020, from http://parts.igem.org/Part:BBa_J23117
[7] Duong, M.T., Qin, Y., You, S. et al. Bacteria-cancer interactions: bacteria-based cancer therapy. Exp Mol Med 51, 1–15 (2019). https://doi.org/10.1038/s12276-019-0297-0
[8] Toley, B. J., & Forbes, N. S. (2011). Motility is critical for effective distribution and accumulation of bacteria in tumor tissue. Integrative Biology, 4(2), 165-176. doi:10.1039/c2ib00091a
[9] Zwietering, M. H., Jongenburger, I., Rombouts, F. M., & Van't Riet, K. (1990). Modeling of the Bacterial Growth Curve. Applied and Environmental Microbiology, 56(6), 1875-1881.
[10] Forbes, N. S., Munn, L. L., Fukumura, D., & Jain, R. K. (n.d.). Sparse initial entrapment of systemically injected Salmonella typhimurium leads to heterogeneous accumulation within tumors. Cancer Research, 63(17), 5188-5193.
[11] Rust, A., Partridge, L., Davletov, B., & Hautbergue, G. (2017). The Use of Plant-Derived Ribosome Inactivating Proteins in Immunotoxin Development: Past, Present and Future Generations. Toxins, 9(11), 344. doi:10.3390/toxins9110344
[12] Lambert, T. (n.d.). MCardinal at FPbase. Retrieved October 14, 2020, from https://www.fpbase.org/protein/mcardinal/