The basic method for visualizing the results in our project is plate assay. According to this method, agar medium containing calcium phytate is poured onto Petri dishes. Then, a few wells are made in the medium and filled with yeast culture homogenate containing phytase. Phytase diffuses in the medium and breaks down calcium phytate. Agar then gets turbid as produced calcium phosphate precipitates (similar to enzyme immunoassay method). The results of such plate assay make it possible to assess protein’s enzymatic activity. But this method is viewed as not accurate enough though very demonstrative.

We decided to develop a model making this method of assessment more accurate. We believe that our model will make it possible to calculate the exact time required for the protein to diffuse in gel for a certain distance, thus eliminating the errors associated with the ‘overstaying’ of the plate and bias in assay results.

The developing of a detailed model including the analysis of Michaelis-Menten equation for enzymatic reaction is very complicated and requires additional elaboration; for this reason, for the purpose of our trial model we used approximation where concentration of calcium phytate in the medium is considered to be rather low and concentration of phytase is high enough, and therefore protein diffusion is accompanied by simultaneous substrate breakdown.

In our calculations, we used data from article (DOI: 10.1016/j.jconrel.2006.08.006). Besides, we need the hydrodynamic radius value of phytase. There is no such value for Obesumbacterium proteus phytase available in literature; therefore, we used the radius value for fungal phytase which is known (DOI: 10.1016/j.bbrc.2004.12.111). As these enzymes are similar in their structure, it can be supposed that such approximation will not influence the results in a significant way.

The hydrodynamic radius of active phytase equals to 4 nm.

Knowing this, we can calculate the diffusion coefficient for the protein in gel. For this purpose, we will first calculate its diffusion coefficient in water according to the following equation:

- where k_b is Boltzmann's constant, 1.38*10-23 J/K;

- Т is assay temperature (in our case its room temperature), 298 К;

- η is water viscosity coefficient, 0.6947 × 10−3 Pa*s;

- r_s is hydrodynamic radius of the protein, 4 nm.

The calculated diffusion coefficient D₀ for our protein is 7.85*10^-7 cm²/s. The coefficient of diffusion in gel Dg relates to D₀ according to the following equation:

- where r_s is hydrodynamic radius of the protein;

- r_f is radius of agarose fibers (according to DOI: 10.1016/j.jconrel.2006.08.006);

- and φ and α can be calculated using the following equations:

The values of Ρ and ω are used as in article (DOI: 10.1016/j.jconrel.2006.08.006):

Ρ = 1.64 g/ml, ω = 0.625;

С = 0.018 g/ml;

then φ = 0.0176

- In our case α = 0.170.

- hen Dg/D0 = 0.66. Dg = 0.66 * 7.85*10^-7 = 5.22*10^-7 cm²/c.

- The squared radius of diffusion is calculated as:

Team:Moscow RF/Model

Modeling