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Modeling

One of the ways we propose to increase the radioprotective properties of the human body is to populate modified bacteria producing radioprotective proteins and RNA into the biofilm of the human intestine.
This method of delivering therapeutic drugs is rarely used in medical practice due to significant difficulties with predicting the behavior of such a system as a whole. To facilitate the task of predicting the behavior of the system and more accurately planning experiments, we have developed a mathematical model that will help us choose the optimal ratios of all components of the system, significantly reducing the number of experiments that are planned to be conducted on animals. At the current stage, most of the parameters that we took for the mathematical model are taken from literary sources, but in the future, we plan to make refinements to the mathematical model based on the data that we receive in our laboratory.

Let x(t) and y(t) be the number of cells and protein molecules, respectively, at time. We make the following assumptions:

1. The number of cells generated per unit time is directly proportional to the number of cells at a given time t with a certain proportionality coefficient a₁ (constant of the cell reproduction rate). For a certain period of time Δt a child is born a₁xΔt.
2. The number of cells killed per unit time is directly proportional to the number of cells at a given time t with a proportionality coefficient a₂(constant of the rate of cell death). Over a period of time Δt a₂yΔt, cells die.
3. The amount of protein degraded per unit time is proportional to the amount of protein at a time t point with a proportionality coefficient B₂ (the rate constant of protein degradation). Over a period of time Δt B₂yΔt protein breaks down.

Also, we know the amount of protein B produced by one cell per unit time, and the proportion of protein A If a protein enters the bloodstream, we will have that Δt protein is produced and enters the bloodstream over a period of time. AB₁xΔt

Thus, the number of cells per time Δt changes by a₁xΔt-a₂xΔt>, the amount of protein in the blood by AB₁xΔt-B₂yΔt. Based on this, we will draw up balanced ratios for the amounts of cells and protein:

Let's rewrite them in the following form:

Assuming the dynamics of the appearance and death of cells and protein is continuous in time, we can move to the limit at Δt->0. Using the mathematical definition of the derivative, we obtain a system of two differential equations:

The solution of this system will depend on an arbitrary constant, the value of which will be determined by the initial state of the system. Therefore, we will supplement the system of equations with the values of the amounts of cells and protein at the initial time (t=0 if the time is counted from the beginning of the experiment). Then we get the following Cauchy problem:

where x₀ – number of cells in the biofilm at the initial time, y₀ – the amount of protein at the initial time (it is taken as the basic level of protein in the blood without bacteria)

Let us assume that the process establishes a dynamic equilibrium between the processes of cell birth and cell death, i.e.. a₁≈a₂ Then for the cells:

For protein, we solve the following Cauchy problem:

Let's solve it by separating variables. Divide both sides of the equation by everything on the right side and multiply by dt:

Let's integrate both sides of this equation:

After integration, we get:

where C – an arbitrary integration constant.

Multiply by -B₂ and potentiate:

Thus, we obtain a general solution of the considered differential equation:

We find the integration constant from the initial condition:

Finally, we obtain a solution to the Cauchy problem of protein quantity dynamics:

For modeling, we fix two sets of parameters corresponding to two types of proteins:

(B1 x0we will leave the parameters and unchanged, since they are associated with cells and, as expected, are not determined by the difference in proteins)

The protein degradation rate constant is calculated by the formula B2 = In(2)/T, where T is the half-life of the protein. This formula is valid if we consider separately the process of decomposition without protein reproduction, which is described by the following Cauchy problem:

Its solution:

Half-life – the time it takes for half of the molecules to be lost:

Reducing both parts of this equation by y₀ and logarithm, we get the desired formula. In this case, the half-life is set for protein1 T=2 days, for protein 2 T=7 days.

Here are graphs of changes in the number of proteins:

From the graphs, it can be seen that at first there is a sharp increase in the number of proteins in the blood (the "sharpness" of the growth depends on the initial amount of protein, the fraction entering the blood, the rate of decay). Then, at some point in time, the transition to a stationary level is gradually carried out – the processes of protein production and decay balance each other.

In accordance with the obtained mathematical model, we can select in advance the initial dosage of a bacterial preparation that allows us to bring the protein concentration to a plateau corresponding to the optimal level of radioprotection in outer space (the required protein concentration is extrapolated based on data on the concentration of proteins in the blood of people inhabiting regions of the planet with a high background level of ionizing radiation). A more precise adjustment of this model will be carried out in experiments on rats, however, thanks to the presence of a mathematical model, we can minimize the number of experiments on animals.