## Navigating to Model

# Model

## Microcystin Degradation Model

In accordance with the objective of the project, a mathematical model was made to predict the relationship and overall efficiency of microcystinase (MlrA) in its degradation of the microcystin (MC-LR) toxin. Initially, we hypothesized that the rate of microcystin degradation would be proportional to the concentration of *E. coli* with microcystinase expression and the concentration of the microcystin toxin. A series of differential equations was developed to mathematically predict our hypothesized relationship.

**Figure 1.**Differential Equations for Microcystin (MC-LR) degradation Model.

Through searching academic literature, constant values in common circumstances for the model were specified and are as follows:

- Initial
*E. coli*concentration = 1e12 𝜇molecule/L - Rate at which microcystinase was expressed (per minute, per
*E. coli*) = 1e5 - Initial microcystin concentration = 500 μg/L

Through the defined differential system and adopted constants, a deterministic model of the rate of microcystin degradation based on given *E. coli* and microcystin concentrations was developed. The model predicts that the rate of microcystin degradation is 0.05 μmolLper minute given our constant values.

**Figure 2.**MC-LR concentration over time in one liter of water and 109E-9

*E. coli*cells per mL.

## PP1 Assay Model

To experimentally obtain the results of microcystin degradation and verify the validity of our wet-lab solution, a protein phosphatase assay (PP1) was used to detect microcystin degradation over time. In this assay, PP1 interacts with p-nitrophenyl phosphate (pNPP) to form p-nitrophenol (pNP). In basic conditions, pNP is ionized and becomes p-nitrophenolate, a molecule that emits a detectable wavelength and discernable yellow color. However, if microcystin (MC-LR) is introduced, it will inhibit PP1 activity, leading to a solution that is less yellow (decreased absorbance). In theory, if the enzyme microcystinase (MlrA) is introduced, microcystin will be degraded and thus, there will be a greater amount of PP1 that can convert pNPP into colored p-nitrophenolate.

**Figure 3.**Mechanism reactions for the formation of pNP with a yellow wavelength.

The model, Figure 4, is a prediction model of absorbance over time in the presence of MlrA. To verify the experimental results and discern theoretical values for the absorbance of pNPP in the presence of microcystin, a model was developed and introduced that takes into account the output of the absorbance over time for a PP1 - microcystin system when MC-LR was added. The relevant constants are in the linked Python notebook for reference purposes. According to the model, it is expected that the absorbance (of non-yellow light) to increase by around 0.01 every 100 seconds.

**Figure 4.**Predicted absorbance over time graph (top) and MC-LR concentration over time (bottom). This shows an increase in PP1 activity due to MC-LR degradation in the presence of MlrA.

## References

Dziga, D., Wladyka, B., Zielińska, G., Meriluoto, J., & Wasylewski, M. (2012). Heterologous expression and characterisation of microcystinase. Toxicon, 59(5), 578–586. https://doi.org/10.1016/j.toxicon.2012.01.001

Falconer, I., Bartram, J., Chorus, I., Kuiper-Goodman, T., Utkilen, H., Burch, M., & Codd, G. (n.d.). Chapter 5. SAFE LEVELS AND SAFE PRACTICES. Retrieved October 19, 2021, from https://www.who.int/water_sanitation_health/resourcesquality/toxcyanchap5.pdf

Milo, R. (2012). What is the concentration of bacterial cells in a saturated culture? Bionumbers.org.