## Protein Degradation Model

Considering the fact that our diagnosis is protein-based, we need to consider the degradation issue (here defined as loss of enzyme activity) when our method is used in real world. In laboratory, the protein is stored in -80 ℃ refrigerator to prevent degradation. However, this is unrealistic for most families because they don’t have -80 ℃ refrigerator. We therefore want to test the protein stability under different conditions, namely room temperature, 4 ℃ (simulate the refrigerator compartment for most household fridge), and -20 ℃(simulate the freezer compartment).

We know that the more concentrate the protein is in a solution, the less stable it is. We therefore assume that the degradation speed v is in proportion with the effective protein concentration (namely the concentration of active protein) c:

where k is a constant that can be determined by experiments.

Note that by definition, the degradation speed v equals to the decrease rate of effective protein concentration c:

where t is the time. This is an ordinary differential equation (ODE). The solution for this ODE is:

where c

_{0}is the initial effective protein concentration (the c when t=0), e is the Euler number. Equation 1.3 depicts an exponential decrease of effective protein concentration over time, which has been proved by degradation assay of other proteins by previous literature^{[1]}.The parameter k can be determined experimentally. Since our protein is a fusion protein that have fluorescence, we can measure the effective protein concentration by both Coomassie blue staining and fluorescence intensity. In this way, by measuring the effective protein concentration on different time points (say, day 1,4,7,10…), we can determine the parameter k.

It is also possible that the protein degradation kinetic does not follow equation 1.2. Instead, it could be:

where alpha does not equal to 1 (if alpha equals to 1 then equation 1.4 is the same as equation 1.2). In this case, we can still determine the parameter k,alpha experimentally as stated above.

# Protein Degradation

# Reference

[1] Ovissipour, M., Rasco, B., Tang, J. et al. Kinetics of Protein Degradation and Physical Changes in Thermally Processed Atlantic Salmon (Salmo salar). Food Bioprocess Technol 10, 1865–1882 (2017). https://doi.org/10.1007/s11947-017-1958-4.

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