УДК 577.352

MATHEMATICAL MODEL OF THE INFLUENCE OF TEMPERATURE AND CHEMICAL FACTORS ON THE FUNCTIONING OF CELLULAR RECEPTORS.

Sukhorukova Irina V., Doctor of Economics, Professor of the Mathematics Department of the Russian Plekhanov University of Economics, 117997, Russian Federation, Moscow, Stremyanniy per., 36,

e-mail:suhorukovaira@yandex.ru

Vybornova I.I., associate professor of the Mathematics Department of the Russian Plekhanov University of Economics, 117997, Russian Federation, Moscow, Stremyanniy per., 36,

e-mail: inezvyb@rambler.ru

Vybornov A.N., Ph.D., associate professor, Moscow State University of Information Technologies, Radio Engineering and Electronics, 119454 Moscow, Vernadsky prospect, house 78,
e-mail : a_vybornov@rambler.ru

Shershnev Vladimir G., Professor of the Mathematics Department of the Russian Plekhanov University of Economics,

117997, Russian Federation, Moscow, Stremyanniy per., 36,

e-mail: shershnev_vg@yandex.ru

By methods of mathematical simulation study of the effect of thermal and chemical factors on cell receptors and the dynamics of their functioning. Selected two processes that are most sensitive to these impacts.

Biological membranes and their various components, such as ion channels, pumps, membrane enzymes and receptors, showed a high sensitivity to external factors. Ambient temperatures, chemical agents, рН affect both himself silipigni layer and membrane components, causing the cooperative response of the cell membrane. This work is devoted to the theoretical study of some mechanisms of the influence of temperature and chemical factors on the functioning of biological membranes: the influence of external effects on the functioning of the receptors of different classes.
The functioning of the body cells is largely due to the ability to specifically bind ligands (contained outside the cell substance) surface receptors. The processes of ligand binding and transporting it to the internal compartments of the living cell are membrane receptors. Experimental data indicate changes in the ability of receptors to bind ligand and internalional it in response to changes in temperature, concentration of cations and рН. Ionic requirements for the binding of receptors with ligands are important condition for the implementation of an adequate response of the cell to the ligand. An equally important aspect of the ability of functioning receptors look temperature conditions. The interaction of receptors with ligands leads to cooperative effects of changing the ultimate state of the membrane.

Cells of a multicellular organism are constantly changing microenvironment. They are able to maintain your metabolism and the constancy of the internal environment due to the property selectively (specifically) "learn" with a complex of proteins (glycoproteins) that are embedded in the cell membrane (receptors) that are contained outside the cell substances (ligands).

It is known [1,2] that a common feature of cellular receptors, which determines the realization of their biological functions is the ability the ability of internalization, that is, the displacement from the cell surface to the interior of cells.

J. Kaplan [3] receptors subdivided into two classes:
Receptors, transport molecules into the cell for further metabolic process --- class II (for example, low-density lipoproteins (LDL), transferrin IgG).

Receptors conditioners directly alter cellular behavior or metabolism after binding of the ligand --- class I. The class I receptors can be divided into receptors capable of internalizing its ligand (class IA), and receptors that are not internalized ligand (class IB).

To the class IA include insulin and epidermal growth factor (EGF), to the class IB --- IgG on basophils, acetylcholine.

The class II receptors reutilized many times in the course of his life. Most receptors of class IA upon binding with ligand are in the lysosomal compartment. This process is accompanied in a significant reduction of receptors on the cell surface (“down—regulation”), which leads the cell to loss of sensitivity to class IA molecules. The class IB receptor is not regulated (e.g., acetylcholine receptor, there is a low density and a very fast dissociation of the complexes the ligand-receptor).

In order to examine the changes in the functioning of cellular receptors, their ability internalional ligand depending on the ionic characteristics, pH conditions and temperature was constructed mathematical model of the dynamics of internalization and recycling of cell receptors, class I and class II.

Receptor-induced endocytosis occurs primarily through bordered pits (pores ). They exist in almost all animal cells and in approximately 2 % of the cell surface.

Receptors of class I to the initially randomly distributed on the membrane and require the binding of a ligand to movements in the pores. The majority of class II receptors reclustered in bordered pits. Bound to the receptors of most ligands are clustered in the pores and leaves the surface of the cells for 10-15 minutes.
The first stage of internalization is associated with the invagination of the cell membrane (within 1-2 minutes) and subsequent itsoriginal her dressed vesicles (i.e. vesicles, covered latinboy shell). Then dressed vesicle is transformed into receptosomes (smooth vesicles), losing latinova coat, which bordered returns to the pits at the cell surface.

The second stage is the fusion of receptosomes in the cytoplasm to lysosomes. In a weakly acidic environment of receptosomes due to the conformational changes in the receptor involves dissociation of the complex ligand-receptor full (class II) or partial (class IA). The ligand in the lysosome breaks down, the receptor or avoids crushing or subjected to limited cleavage.
The foregoing has given the basis to build a functional scheme of the process of interaction of the ligand with the receptor, including internalization, recycling and interaction with the pore.
Mathematical model of process of interaction of the ligand with the receptor is as follows:

где , , — the concentration of ligand-free receptors, ligand-receptor complexes, pore complexes in the pore, complexes in the vesicle, internalized ligand, internalized receptors, respectively;

— the speed of synthesis and catabolism of ligand and receptor, respectively;

— the baud rate and dissociation of ligand and receptor; complexes ligand-receptor with at times, respectively;

, — the rate of internalization and recycling, respectively;

, — the rate of destruction in the lysosomes of internalized ligand and receptor, respectively;

m --- the coefficient characterizing the percentage of receptors, pregesterone to pore.

;

— time intussusception;

The model reflects the process of invagination of the membrane and threshold character falling through the pores.

The model identified receptors for low-density lipoprotein , insulin, and epidermal growth factor. The mathematical model shows the periodic process of falling then, the internalization and recycling of receptors; "down-regulation" for insulin and EGF.

For LDL receptors in normal human fibroblasts with 50-70% of LDL receptors, reclustering in bordered pits, complexes leave the surface of the cells for 10 minutes . For defective fibroblasts and carcinoma cells A-431— 3-4 % LDL receptors reclustered in bordered pits, here the process of internalization of the complexes lasts much longer.

The model demonstrates for the LDL receptor, the importance of the number of receptors, reclustering in the pores. In defective condition (3 %) time of internalization increases significantly.
Receptors of class I and II are characterized by different speeds of lysis of internalized receptors. For class II —.

For class I receptors in the lysis within a few seconds-minutes, the concentration of free receptors decreases rapidly, there is "down-regulation".

In a very large factor , characterizing the rate of dissociation of the complexes ligand-receptor, the concentration of the complexes are close to zero. This corresponds to the dynamics of binding the receptors of class IB, for the realization of biological effects which are not necessary in the internalization of ligand.

The mathematical model adequately reflects the dynamics of the process of binding the ligand with all other classes of receptors, class II, class IA, class IB.
As noted in [4], for class II receptors unlike receptor class I binding of ligand to the receptor is typically sensitive to the concentration of cations and pH. The majority of class II receptors shows a lower affinity to the ligand at low pH (with the exception of transferrin).

The ability to bind the ligand for the receptor, accumulating external ligand without the depletion of receptors on the cell surface, or depends on the concentration divalent cations or pH. Receptors, transporting ligand to various intracellular compartments, such as mannose-6-phosphate receptor, sensitive to low pH, but not low concentration divalent cations. Receptors low-density lipoprotein and receptor -macroglobulin protease is sensitive to low concentrations of cations and reduced pH. At pH=5,0 binding receptors with ligand does not occur.

Thus it is clear that the concentration of cations and pH affect the rate of binding of the ligand to the receptor . In the study of the mathematical model was obtained with reducing the four times is observed for increasing absorption of the ligand is almost twice the number of complexes decreases approximately in 1.6 times. When you decrease an order of magnitude the concentration of complexes of the Lg R is reduced about three times. When increasing the order of the response time of the cell to the ligand decreases 1.8 times, while lower order of magnitude response time is increased four times. Thus the reduction value greater effect than an increase.

Experimental studies indicate the sensitivity of membrane Receptors not only reduce pH and concentration of cations, but also to temperature change. In [9] provides data about the lack of internalization of LDL within 4-5 hours at a temperature of C. In [10] it is noted that at a temperature of C for three hours internalizes only 20% EGF, at a temperature of C for thirty minutes - 73%, at a temperature C — 77% EGF. There are also data on the temperature dependence of the functioning of insulin receptors.

All this allows to conclude that the temperature affects the rate of internalization of the complexes of ligand receptors and(or) on the rate of binding of the complexes with sometimes . . In the study model, it was found that when reducing the order time absorption of the ligand in the cell increases 2.5 times, two orders of magnitude - more than five times and the concentration of free receptors on the cell surface tends to zero. For receptor class IA (e.g., insulin receptor), an abrupt increase in the rate of internalization leads to "down-regulation" and the rapid loss of cell sensitivity to class IA molecules. The increase in the mathematical model of the rate of internalization of ten times leads to the reduction of drilling down then three times and a quick, "down-regulation".

Because the class II receptors for the performance of its functions is to transport molecules into the cell, they need to form a complex of the ligand-receptor contact-lined pit. Change the speed of binding of the receptor and its complexes with sometimes certainly will affect the performance of the transport functions of the receptor. In the study model, it was found that the decrease in the rate of binding of the ligand-receptor bonds with sometimes twice leads to an increase of 1.6 times the absorption of the ligand to the cell and higher concentrations of unbound complexes sometimes several times. When reduced in 4 times the absorption of the ligand takes longer more than 3 times, the membrane is a large number of complexes unrelated to sometimes. When you decrease on the order of almost all complexes the ligand-receptor located on the membrane surface without contacting bordered pits. Eternalization ligand, effect of small amounts of complexes in the pore, hardly occurs.

In the study model, it was found that the decreased rate of binding of the ligand with the receptor in two times the model demonstrates the increasing absorption of the ligand about 1.3 times and reduce the concentration of complexes Lg-R about one and a half times.

On the basis of experimental data (e.g. [11]) we can assume that under the influence of temperature and pH may change the real number of cellular receptors located on the membrane. Given the function of receptors of class I, sostoyanie receptors with membrane becomes a barrier to perform their metabolic functions. In the study model identified receptors for class I, the following results were obtained. The initial decrease in the number of receptors on the membrane by 30% leads to a decrease in ligand-receptor complexes on the cell surface approximately in half. While "down-regulation" (DR), i.e. the depletion of receptors on the membrane, occurs five times faster than normal, resulting in the number of ligand, a total of proteinopathies with receptors decreases more than three times. When reducing the initial number of receptors by 50 % the number of ligand-receptor complexes is reduced more than twice, DR comes in 10 times faster, which leads to a reduction of the captured ligand 4 times. Reducing the number of receptors on the membrane on the right leads to the fact that they are virtually unable to perform its functions. Rapid depletion of receptors, a low concentration of ligand-receptor complexes and the unbound ligand, remaining almost the same amount outside the cell, characterize this situation.

Thus, when considering the processes of interaction of the ligand with the receptor and internalization of the ligand-receptor complexes as a stage that is most sensitive to changes in ionic state of the membrane and pH, we can distinguish, first of all, the stage of ligand binding to class II receptors on the cell membrane. The sensitivity can be achieved by changing the ability of receptors to bind. The latter may occur due to changes in the ionic properties of the membrane in the vicinity of the receptor.

Temperature change affects the stage of binding of the receptor with the ligand and to the stage of internalization of free receptors and the resulting ligand-receptor complexes. Sensitivity may occur here due to the conformational changes to the receptor and (or) by changing the viscosity of the bordered pits.

This is consistent with the results of numerical experiments conducted with the presented mathematical model.

Thus, considering the influence of the ionic state of the membrane and pH and temperature on cell receptors and the dynamics of their functioning, we can distinguish two processes that are most sensitive to this effect. First, the binding of ligand to receptors on the cell membrane. The sensitivity in this case may be provided the actual decrease in the number of receptors on the cell surface (their sostoyanie) or(and) change the ability of receptors to bind. The latter may occur either through conformational changes in the receptors themselves, or by changes in the ionic properties of the membrane in the vicinity of the receptor. Secondly, the process of internalization of free receptors and the resulting ligand-receptor complexes. Sensitivity may occur here due to the change in viscosity of bordered pits.

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2. Savin A.V., Gendelman O.V. Mechanical control of heat conductivity in molecular chains Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. 2014. Т. 89. № 1. С. 012134..

3. Kaplan J. Polypeptide-binding membrane receptors: Analysis and classification //Science. 1981. V. 212. P. 14.

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