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The membrane permeability coefficient for sodium and potassium ions in unicellular organisms can be calculated using the data for cell volume, surface and mean generation time during growth and dividing of cells by binary. Accordingly theory of proposed method, the membrane permeability coefficients for passed trough outer cell membrane sodium and potassium ions, is equal to the volume of unicellular organism divided to product between cell surface and mean generation time of cells. The calculated by this way diapason of values overlaps with experimentally measured diapason of values of permeability coefficient for sodium and potassium ions. The deviation between the theoretically calculated and experimentally measured values of permeability coefficient does not exceed one order of magnitude.

The ion permeability of the plasma membrane has been recognized as an important problem in cell function. The main ions in cell physiology are protons, sodium, potassium and chloride ions, because of their participation in the building of transmembrane ions gradients, the support of cell potentials and energetics mechanisms in cells [1-4]. Permeability may be defined phenomenological as the amount of substance transported across a unit area in unit time as the result of a unit force [5,6]. Permeability of given solutes is characterized by the permeability coefficient, measured by the amount of solute passing in unit time through unit area of membrane under the influence of unit concentration gradient [^{2}, cell volume in m^{3 }and concentration of solutes is given in mol per liter [

A guiding principle in the estimation of the permeability coefficient is that the cell boundary acted as a lipidlike phase through which the penetrating solute diffused. A simple hypothesis based on Fick’s laws of diffusion is that the instantaneous rate of uptake of the solutes (ds/dt) through the semi-permeable bounding membrane, into a vesicle (liposome or cell) of volume (V), is represented as the product of the surface area (S) of vesicle, the difference in solute concentration between the interior (Cin) and exterior (Cex) compartments (Cex − Cin), and the permeability coefficient (P) that connect the diffusion coefficient of solutes, thickness of membrane and relative solubility of substances in water-oil phases [

The relevant equations are:

and

where Cex and Cin are function of the time.

When cell surface area S, cell volume V and exterior concentration Cex (because of big exterior volume) are constant, these two equations simplify to:

After integration we receive for concentration Cin(t) in vesicle after time t :

where Cin(0) is the concentration of solute in vesicle at the moment t = 0.

The equation (PSt)/V = kt represents the connection between permeability coefficient P(m/s) and the firstorder rate constant k(t^{−1}) :

The initial concentration of solute in cell, at the moment t = 0 is assumed to be zero, and for C(0) = 0 the Equation (4) receives the form:

For the natural logarithm of Equation (6) we’ll receive for permeability coefficient:

The Equations (4) and (7) keep validity for measuring the permeability coefficient in vesicle and liposome (in vitro) and cell (in vivo).

For vesicles and liposome, the gradient of solute concentration ∆C = Cex − Cin dissipates up to equilibration of concentration in the interior and exterior compartments i.e. up to Cex = Cin. In this case the Equations (4) and (7) describe the establishment of the equilibrium concentration in the internal volume of the vesicle or liposome, following an exponential law equation. In this case the permeability coefficient is calculated from Equation (5) after experimental measuring of first-order rate constant “k” [

By empiric way, we established that the Equation (7) may be applied for calculation of permeability coefficient of ions passing trough outer cell membrane in growing cells. In all growing cells the transmembrane concentration of sodium and potassium ions maintained on constant level by the active transport of cell. For these ions the ratio Cin(t)/Cex is a constant and don’t depend on the time. For time-period equal to generation (doubling) time (T_{gt}) the mother cell grows and divided by binary. Similarly to the mother cell, a daughter cell has same cellular characteristics, including transmembrane distribution of sodium (Na^{+}) and potassium (K^{+}) ions. For time t = T_{gt} the Equation (7) received the form:

where P_{gt} presents the membrane permeability of cells. The volume V_{m}, surface S_{m} and genera tion time T_{gt} presented the maximum values of volume, surface and generation (doubling) time of cell. The Cin(T_{gt})/Cex ratio is given to be the constant (Const).

Because of every unicellular organism grows up to nearly the same maximum volume and surface, in the moment of end of cell cycle the V_{m}/S_{m} ratio of daughter cell is equal to the volume to surface ratio of mother cell. Thus, for given unicellular organism the volume to surface ratio has nearly a constant value in the end of cell cycle, independently of duration of generation time. For small Prokaryotes to big Eukaryotes (from bacteria to amoebas) the volume to surface ratio changes about 4 orders of magnitude [11-14]. The generation time from bacteria to amoeba changes about 4 orders of magnitude too ([11,15-19]. In unicellular organisms the transmembrane concentration of Na^{+} and K^{+} ions (ion gradient Cin/Cex) change maximum 10^{3} - 10^{4} folds^{ }[1,2,20]. Because of that, the term ln[Cin(T_{gt})/Cex] and consequently ln[1/(1 − Cin(T_{gt})/Cex)] changes maximum 3 - 4 fold. Thus, the permeability coefficient P_{gt} in Equation (8) depends firstly, on volume to surface ratio and generation time (changing 10^{4 }folds), and secondly from ln[1/(1 − Cin(T_{gt})/Cex)] (changing 3 - 4 fold).

This gives us the reason to simplify the Equation (8) as:

where P_{gt} shows the value of permeability coefficient in end of cell cycle, in the moment t = T_{gt}.

On

Accordingly Formula (9), the permeability coefficient (P_{gt}) characterizes permeability of ions, passing trough the area of outer cell membrane (S_{m}), which limits the total cell volume (V_{m}) for time-period, equal to the duration of the cell cycle (T_{gt}). The ions acting in the area of outer cell membrane could be sodium, potassium and chloride ions. In outer cell membrane are localizes the Na^{+} − K^{+} ion pumps of active transport, which creates and maintain the constant transmembrane gradients of these ions [

The permeability coefficient of K^{+} and Na^{+} ions (including CI^{−} ions), falls in diapason of 1.0 × 10^{−14} - 1.0 × 10^{−10 m/s [6,7,22]. Scaling of permeability coefficients of}

solutes passing trough outer cell membrane has showed that the sodium, potassium and chloride ions have the lowest permeability in comparison to other solutes (divalent ions, protons, phosphate, glucose, water and metabolites) [^{+}, K^{+} and CI^{−} ions don’t differ more than one order of magnitude^{ }[7,14]. The permeability coefficient of sodium ions is lower than coefficient for potassium and chloride ions, whereas the permeability coefficient of potassium and chloride ions is lower than those of protons [3,23,24].

Because of the protons act in the mitochondrial membrane (inside in the cell volume) and life cycle of mitochondria differ from life cycle of cells, the relatively high permeability coefficient for protons range in diapason of 1.0 × 10^{−10} − 1.0 × 10^{−8 m/s and can be connected rather with mitochondrial processes than the processes in outer cell membrane.}

The data for body mass (M, kg), temperature of growth (t, ˚C), minimum, maximum and mean generation (doubling) time T_{gt} (min, h, d) of cells and calculated data for volume to surface ratio (V/S, m) for 14 unicellular organisms (Prokaryotes and Eukaryotes) are presented on

The data for temperature of growth and generation time are given from the same sources. The mean generation time of cells is calculated as:

Calculation of the volume to surface ratio in unicellular organisms was made accordingly source [

The value for body density (r) of unicellular organisms is given to be 1100 kg/m^{3} and coefficient (k) is given to be 0.0542, accordingly data by Schmidt-Nielsen [

Calculation of the membrane permeability coefficient P_{gt}(m/s)_{ }equal to volume/(surface × generation time) ratio was made, accordingly theory of method by formula:

In Formula (12) the body mass (M) is in “kg”, the volume to surface ratio (V_{m}/S_{m}) is in “m” and generation (doubling) time (T_{gt)} is in “s”. The mean membrane permeability can be calculated as:

or

For calculation of P_{max} and P_{min}, the minimum (T_{min}) and maximum (T_{ max}) generation times are used.

The experimental data for sodium and potassium permeability coefficients (P_{Na+ }and P_{K+}) are received by three fashions. Firstly, from published in scientific literature data for sodium and potassium permeability coefficient. Secondly, by calculation of membrane permeability coefficient, accordingly Fick’s law dividing mass flux of ions J_{i} (mol/m^{2}·s) passing trough external cell membrane to ion concentration C_{in} (mol/m^{3}) in cells [

The data for mass flux and internal concentration of sodium and potassium ions were given from published in scientific literature data.

Thirty, by calculation of sodium and potassium permeability coefficients as a product between first-order rate constant k (in s^{−1}) and volume to surface ratio of cell V/S (in meter), accordingly Stein [

The data for first-order rate coefficient were given from scientific literature sours.

Statistical software “STATISTICA” was used for calculation of means ± standard deviation (SD), and correlation coefficients.

In given on ^{−16 kg in Psychrobacter immobilis to 1.32 × 10−4 kg in Hydrodyction africanum. The volume to surface ratio changes 4 orders of magnitude, from Psychrobacter immobilis (}1.26 × 10^{−7 m) to Hydrodyction africanum (}5.24 × 10^{−3 m). The generation time changes nearly 4 orders of magnitude too, from 20 min in Escherichia coli to 150 days in Hydrodyction africanum. }

In _{min}), maximum (P_{max}) and mean (P_{mean} ± SD) permeability coefficients accordingly formulas (12 - 14), and experimental measured data for sodium and potassium membrane permeability coefficient (P_{Na+},_{ K+}) in 7 Prokaryotes and 7 Eukaryotes.

_{min}), maximum (P_{max}), mean permeability coefficient (P_{mean} ± SD) and experimental data for permeability coefficient of sodium (P_{Na+}), and potassium (P_{K+}) ions.

\

In given on ^{−12 m/s in Methanosarcina barkeri to 1.34 × 10−10 m/s in Thermotoga maritima) and overlaps with diapason of experimental data for permeability coefficients for sodium (P}_{Na+}) and potassium (P_{K+}) ion (from 9.6 × 10^{−12 m/s in Methanosarcina barkeri to 1.55 × 10−10 m/s in Halobacterium salinarium).}

In given on ^{−11m/s in Chlorella sorociniana to 5.78 × 10−10 m/s in Hydrodictyon africanum) and overlaps with diapason of experimental data for permeability coefficient for sodium and potassium ions (from 1.0 × 10−11 m/s }in Chlorella to 9.23 × 10^{−10 m/s in Tetrahymena). The summarizing diapa}sons of data for mean permeability coefficients in Prokaryotes and Eukaryotes, accordingly

On _{mean} coefficient and the experimental data for P_{Na+}, _{K+ }coefficient differs less than one order of magnitude. From 14 studied organisms, in 12 organisms (P. immobilis, E. coli, M. barkeri, S. acidocaldarius, T. maritime, H. salinarium, Ch. Sorociniana, T. viridis, O. luteus, Tetrahymena, Ch. chaos, H. africanum) the ratio between calculated and experimental values of permeability coefficients (P_{mean}/P_{Na+}, _{K+}) changes from 1 to 3 folds. In two organisms only (Bacillus stearothermophilus and Saccharomyces cereviseae) this ratio is 4 - 5 folds.

On _{gt }= P_{mean}) and experimental values of coefficient (P_{Na+}, _{K+}), where two values are situated on the horizontal and vertical axis on coordinate system in the same scale. The position of the points with coordinates (P_{gt}, P_{Na+},_{K+}) is compared with the position of the points with equal coordinates on equilibration line of P_{gt }= P_{Na+}, _{K+}. The position of intersection coordinate (P_{gt,} P_{Na+, K+}) on _{gt} = P_{Na+}, _{K+}. This shows that the relation from type P_{Na+}, _{K+} ≈ V_{m}/(S_{m}·T_{mean}) is valid within an order of magnitude. The correlation coefficient between P_{gt} and P_{Na+}, _{K+} values is 0.845 which shows that this relation is not random.

For minimum T_{min} and maximum T_{max} generation time, the corresponding calculated permeability coefficients are P_{max} = V_{m}/(S_{m}·T_{min}), and respectively P_{min} = V_{m}/ (S_{m}·T_{max}). Because of the validity of the ratio T_{min} < T_{mean} < T_{max}, and corresponding ratio P_{min} < P_{mean} < P_{max}, the interval of values between P_{min} and P_{max} contained membrane permeability coefficients for sodium and potassium ions i.e. the relation P_{min} < P_{Na+}, _{K+} < P_{max} is valid.

In summary, the proposed method can solve two main tasks. In the first task, knowledge of the minimum and maximum generation time of cells dives possibility to determinate the interval, which contained permeability coefficients for sodium and potassium ions. In the second task, knowledge of the mean generation time of the cells gives possibility to determinate, with accuracy of an order of magnitude, permeability coefficient for sodium and potassium ions. The accuracy of the method is determined by the two terms of Equation (8):

Ignoring the term ln[1/(1 − Cin(T_{gt})/Cex)] brings error 3 - 4 times, and giving in the mind that the term (P_{mean}/ P_{Na+}, _{K+}) brings error 3 - 4 times, the product between two errors gives 9 - 12 times summarized error. For two organisms only (Bacillus stearothermophilus and Saccharomyces cereviseae), where (P_{mean}/P_{Na+}, _{K+}) ratio is 4 - 5 times, the summarized error is 12 - 15 times. This shows that the maximum error of the methods is about one order of magnitude. For better evaluation of permeability coefficient is needed to calculate the term ln[1/(1 − Cin(T_{gt})/Cex)] in Equation (8), which will reduce the error 3 - 4 times and make at least one order of magnitude. The proposed method can be improved for specific groups of microorganisms with similar characteristics.

A new method requires measure the mass or volume, surface and minimum and maximum generation time of the unicellular organism at given growing conditions. At different growth conditions (temperature of growth, substrates, air pressure and others) the generation time can change inordinately, while cell mass, volume and surface of organisms keep on relatively constant level, because the cells grow always up to gain the maximum values, before cell division. Minimum and maximum generation time can be measured by the change of the experimental conditions or can be taken from microbiology scientific sources. Recently, in this direction are accumulated many data for every cell line.

In some previous works Atanasov [

However, the mechanism of maintenance of lowest permeability of sodium and potassium ions in cells is not clear. The independence of their permeability coefficient to membrane composition [

The established theoretic equation between membrane permeability, volume, surface and generation time in unicellular organisms allows to calculate sodium and potassium ion permeability in these organisms. It is possible the equation to express cell mechanism for control the permeability of sodium and potassium ions.