The mathematical model proposed by the iGEM TecCEM team has the purpose of predicting the behavior of leptin after being released by chitosan nanoparticles, the amount of leptin that is absorbed by the cells and the amount of the mentioned protein that remains in the medium.

A simple system of differential equations was used to model the above.

Where

• k1= Degradation leptin
• k2= Leptin adsorption fibroblast
• x2= Concentration of leptin in chitosan
• x1= Concentration of leptin inside the cell

The employed equations were created based on the physical behavior of leptin's dispersion, as shown next:

• Equation 1 represents the variation of concentration of leptin in time; the liberation of leptin towards the medium at a constant velocity "k1" considering the amount of leptin left on the medium and the leptin adsorbed by cells.
• Equation 2 represents the amount of leptin adsorbed by the cells at a constant velocity "k2" with an overall performance Yxs used to harmonize units.

In order to solve the equation system with the form y'= f(t,y) ODE Matlab Solver was employed using the proposed algorithm and establishing initial conditions:



The study of drug release dynamic aims to understand the process of drug characteristics in the human body.

The graph presents the kinetics of leptin, representing the rate of reduction at a time x while simulating at the same time the increase of leptin in the medium and the rate of adsorption by the cells.



Additionally, we sought to have the relationship between a measurable product to have a direct relationship with cell proliferation while in the presence and absence of leptin. For this, a general stoichiometric analysis was carried out for the cell growth reaction. This method was used because it is possible to follow the assertion of all the carbon, hydrogen, oxygen, nitrogen atoms and all the other elements that are consumed during the cell growth process, and then these atoms are incorporated again for the formation of new cells and products, that is to say, taking into account the matter and energy balances.

If one considers the components that are present in greater quantities and there is a presence of unique products formed, the following aerobic cell growth reaction can be written:

• Without leptin



• With leptin



Glucose was considered as the main source of carbon and glutamine as the main source of nitrogen. On the other hand, lactic acid and ammonia as unique products, in addition to carbon dioxide and water.

For the resolution of this system, the following parameters were considered:

• Yps = 0.1973 g glu/g glut
• Yxs = 0.17 g glu/g lactic acid

Analogously, the respiratory rate of the fibroblasts was considered, with the ratio of moles of CO2 produced per moles of O2 consumed.

• RQ=0.97=d/a

Given all the above, a 7x7 matrix was made to obtain the stoichiometric coefficients.



Giving the following results:

• 1 mol Glucose
• a = 0.2432 mol Glutamine
• b = 4.6974 mol Oxygene
• c = 1.3533 mol Cells
• d = 0.3400 mol Lactic acid
• e = 0.1616 mol Ammoniaque
• f = 4.8427 mol Carbon Dioxide
• g = 4. 7559 mol Water

Obtaining a ratio of 0.34 moles of lactic acid per 1.2533 moles of cells.

By adding leptin to the medium, the increase in cell proliferation can be observed, giving the initial conditions of grams of leptin per grams of glucose in the medium.



Giving the following results:

• 1 mol Glucose
• a = 0.2432 mol Glutamine
• b = 0.97 mol Oxygene
• c = 0.0010 mol Leptine
• d = 2.5112 mol Cells
• e = 0.3400 mol Lactic Acid
• f = 0.0927 mol Ammoniaque
• g = 4. 4348 mol Carbon Dioxide
• h = 4.4400 mol Water

Resulting in a ratio of 0.0927 moles of lactic acid per 2.5112 moles of cells.