Biological membranes are characterized not only by complex molecular structure, but also by the presence of an electric field that is generated electrostatically by fixed charges and electrodynamically the flow of ions across the cell membrane. The transmembrane potential (TMP) is a marker value, which indicates the physiological state of cells, affects feedbacks within it, and it is an integral part of monitoring of life. The work complements the earlier conducted research of the mathematical models that describe the dynamics of the cells physiological parameters. The flows of Na+, K+ and Cl– ions into and out of the cell via ion transporters are the powerful signals that regulate ionic balance of cells, they maintain water balance and TMP, indirectly regulate cell proliferation, differentiation and migration in regeneration and embryonic morphogenesis.

Thus, self-organization system of a living cell can initiate changing of its parameters; consequently, that affects the value of TMP. The mathematical modeling of membrane processes in the germ cells is based on the constant-field theory of Goldman. The reactions that occur in the cell are extremely complex and cover many intermediates. The mathematical model consists of two parts: the first part described by the plasma membrane, ion transporters: sodium, potassium, and chloride channels, sodium pump, NKCC1; and TMP. The second part describes changes of Na+, K+, and Cl– concentrations in and out of the cell.

In the mathematical model we suppose that: at the time all ion transport systems are uniformly distributed in the plasma membrane, the plasma membrane permeability to ions and conductivity are constant in time, the germ cell plasma membrane is homogeneous and neutral. The mathematical model is described by system of 12 nonlinear differential and algebraic equations. TMP is calculated by the Goldman-Hodgkin-Katz equation. According to the Fick's first law and regulation mechanisms of intracellular and extracellular ion concentrations of the value of the resulting ion flows are the algebraic sum of the ion flows created by the ion transporters for each ion type. Passive flow of ions via an ion channel is described by the Goldman equation. Active flow of ions via sodium pump is dependent on the concentration of Na+ and K+ in and out of the cell. The ions flow through NKCC is dependent on K+, Na+, and Cl– concentrations in and out of the cell and the ratio of ions are pumped via the protein.

The dynamic model has the ability to simulate: ion transport across the plasma membrane and their distribution between the intracellular and extracellular side of the plasma membrane; the changes in the ion flows for an ion transporter and the total ion flow; TMP. The results of modeling are given graphically by the time-dependence of the ion concentrations, ion flows, TMP; dependence the ion flow on ion concentration, and ion flow on TMP. The results of the mathematical modeling showed that TMP changed from –16 to –90 mV during 10 hr of growth of the early loach embryo. The intracellular potassium concentration increased, the sodium and chloride concentration in the cell decreased. Dependence of the ion flow across sodium pump on intracellular sodium and extracellular potassium concentrations was shown. This dependence had a nonlinear character. The dependences of the ion flows via sodium and chloride channels on TMP had nonlinear character. Another situation was found for the potassium flux through the potassium channel. The simulation results are in qualitative agreement with experimental observations for the loach germ cell.

Keywords: flow of ions, ion transporters, ions, membrane potential, mathematical modeling

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