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КЛІТИННА БІОФІЗИКА CELL BIOPHYSICS
Physics of the Alive, Vol. 17, No1, 2009. P.56-64. Physios of the Alive
© Dyka M.V., Humetsky R. Y. www.pa.srienra-ceiiter.net
UDC 573.2:577.95:577.3:597.554.3
THE MODEL OF DYNAMICS OF MEMBRANE-RELATED BIOELECTRICAL PROCESSES IN EARLY FISH EMBRYOGENESIS
Dyka M.V., Humetsky R.Y.
Ivan Franko National University of Lviv,
Hrushevskogo St, 4, Lviv 79005, Ukraine, e-mail: [email protected]
Was receved 30.04.2009
The research of dynamic changes in electrophysiological characteristics of loach cells embryo which pass after fertilization during synchronous cell divisions, were carried out on individual models (for dynamics of aperiodic trend and for oscillatory dynamics of membrane conductance) and also on the basis of the complex model for bioelectrical processes dynamics. The last one reflects regularity of membrane transport and the distribution of potassium and sodium ions in connection with the changes of membrane bioelectrical parameters.
Key words: membrane, bioelectrical parameters, temporary dynamics, mathematical model.
INTRODUCTION
Oscillatory changes in different physico-chemical and metabolic processes are probably one of the characteristic features of the early period of embryogenesis. Thus, it was found that characteristic oscillations of electrophysiological parameters of membranes occur synchronously with cell cycle division of embryo blastomer. In particular, such regularity is characteristic for fishes, frogs [2, 3, 5]. Periodic changes are characteristic for transmembrane potential (TMP), integral ion current and conductivity, input cell resistance. Oscillations start after fertilization of oocyte and extend to the stage of morula (as it was found for loach (Misgurnus fossilis L.)), after that the oscillations are desynchronized.
The aim of the present study is to provide mathematical description of continuous trends and oscillatory changes of transmembrane potential and electrophysiological parameters in the course of the early cell divisions. This description should account for experimental data obtained earlier for loach embryo [5] and be based on known correlations between bioelectric and ion concentration parameters. Such description can be provided by computer modeling.
Analysis of TMP, ionic currents and conductivities reveals the presence of at least two additive components in the studied processes. The first of them reflects the general monotonous trend of time-dependent parameters, and the other is related to their
oscillations. Thus, dynamics of the early development can be described by the aperiodic changes of the membrane polarization level, ionic currents and conductivities. The oscillatory changes synchronous with cell cycle division are superimposed on this monotonous trend. We can assume that periodic oscillations and trend of their general level are caused by different mechanisms. Therefore, it may be possible to suggest separate models for their description and to consider them as subsystems of complete model that describes the variations of TMP, ionic currents, voltage-dependent conductivities for ions Na+ and K+ and their concentrations. In the present study we propose such complete model of bioelectrical processes dynamics that allows to describe the time-dependent changes of TMP, voltage-dependent conductivity, ionic currents, intracellular concentrations of Na+, K+ ions for embryo cells during synchronous divisions of blastomers.
Starting with the assumption that the general model of electrogenic and electrokinetic processes can be composed using two partial models discussed above we concentrated on the solution of the following problems:
1. To model the temporal trend of bioelectric characteristics based on the common for modern electrophysiology general description of the electrogenesis process (for excited cells, in particular), and also on experimental data obtained for loach embryos. Solution to this problem can be performed in two steps. At the first step, on mathematical description
of the studied process to determine the dynamics of concentrations ions, ionic currents and conductivities as the dependence on TMP level that can be measured during the loach embryos development for complementing the model with the unknown equations of voltage-dependence of conductivities. At the second step, to determine according to this model the dynamics of the resulting TMP level and of other bioelectric parameters using the given values of potassium and sodium concentrations for the initial state and to compare the results of this modeling with the data obtained in vivo.
2. To develop a partial mathematical model for oscillatory dynamics of K+and Na+ these conductivities and other electrophysiological parameters dependent on them. To take into account the interdependence between changes of these conductivities (which must be characteristic for both excitable and non-excitable cells).
3. To unite these partial models in a consistent system, which may allow reproducing of both aperiodic and periodic components in membrane-related processes dynamics. These are the changes of the general level and the oscillatory rhythmic of basic bioelectric parameters, which is synchronous with the cycles of cell division on the initial stage of embryo development.
THE MODEL OF BIOELECTRICAL PROCESSES
The model that is considered below accounts for rather complete set of functional interrelations between characteristics of studied biosystem. It includes the dependencies between such parameters as TMP [4], equilibrium potentials EK and ENa, currents of active and passive ion transport and current of ion pump (IKact, INaact, IK, INa, IP), speed of changes of Na+ and K+ intracellular concentration, membrane conductivities for these ions (gK, gNa). The scheme of cause-consequence relations between the variations of all parameters considered by the present model is shown in Fig. 1. All correlations that were accounted in this model are presented as quantitative correlations. This model accounts for regularities of ion transport and distribution of potassium and sodium ions in connection with the dynamics of electrophysiological parameters of the cell, TMP in particular, with the account of voltage-dependence and interdependence of sodium and potassium membrane conductivity. In our view these relations allow description of bioelectric and concentrational parameters with sufficient completeness. The value of TMP depends directly on the inter- and intracellular ion concentrations [K+]o, [Na+]o, [K+]i, [Na+]i and average charge z of intracellular non-permeable anions. The ion current of the pump is determined by [Na+]i and [K+]o. We assume that parameter z does not change and its influence is stationary. Another parameters change and the correlations with them are dynamic.
parameters, which are taken into account in the model of membrane-association processes (continuous lines show dynamic dependences, interrupted lines show steady state influences).
In this scheme several contours of direct and reverse relationship can be distinguished. In particular, one can follow the connection between conductivity and transmembrane potential. The change of conductivity for K+ and Na+ leads to the changes of correspondent ion currents. The latter change the intracellular concentration of ions K+ and Na+, which determines (as one of the factors) the TMP magnitude. Dynamics of ion currents of passive transport depends on the difference between TMP and correspondent equilibrium potential, which is determined by concentration of certain ion inside and outside the cell. Dynamics of ion current of active transport also depends on [Na+]i, [K+]o. Thus, the membrane conductivity influences the membrane potential, but not directly. This occurs through the changes of ion currents and, consequently, the concentrations of electrogenic ions. In its turn, there exists the reverse connection between conductivities and TMP, which is realized as their voltage-dependence. The contour of interrelations between potassium and sodium conductivities allows to model the generation of autooscillations in a complete dynamical phenomenological way.
DERIVATION OF THE EQUATIONS
The analytic description of cells bioelectrical properties takes into account electrochemical
asymmetry of ion concentration: passive and active transported cations of K+ and Na+, free permeable ion Cl- and non-permeable intracellular anion Az-(with average charge value z). Three conditions are commonly postulated to be observed:
1) the condition of electroneutrality of intracellular and extra cellular media.
[K+]0 + [Na+]0 = [Cl~]0, (1)
[K+] +[Na+], = [Cr]t + z[A~z]i, (2)
2) the condition of water-osmotic balance for all considered ions
[K+]i+[Na+]i+[Cl~]i+[A~z]i=[K+]o+ [Na+]o+[Cl~]0, (3)
3) and also the condition of electrochemical equilibrium of ion Cl-. that is equality of Nernst’s potential (ECl) to transmembrane potential (U).
Eci = RT/F ln([Cl-]i /[Cl-]o) = U, (4)
It was accounted that the electrochemical asymmetry was created and maintained (on certain dynamics particularly, on stationary level) by different ionic flows of active and passive transport. In the regime of compensation of passive flow by active flow the stationary state is maintained. The stationary state is dynamic equilibrium of membrane processes i.e. constancy of TMP value and ion concentrations of cells. In dynamic regime the disbalance of these flows explains the changes of all bioelectrical parameters and intracellular concentrations of cations and anions. In view of that the general description of membrane processes dynamics (i.e. temporal trend of correspondent parameters) was based on the following starting statements and correspondent quantities correlation.
Transmembrane potential. Membrane potential difference is caused by electric polarization of the membrane (by transmembrane difference of ion concentration). In this case the anion content in the cell is directly related to the contents of K+ and Na+ cations. Based on the stated above conditions for concentrational and electrochemical equilibrium we have derived the equation that is consistent for modeling (5).
u=RT lni
F
z+1 z-1
-X
[K+l +[Na+l.
[K+]o +[Na ]o
+
(5),
The absolute TMP value is connected by direct monotonous dependence with total content of sodium and potassium cations, as equations (5) for z^1.
Active transport and the correspondent ion flows are determined by the intensity of the work of ion pump [7]. Since the ratio of transported Na+ and K+ is 3:2, than transitting from ion flows to currents, we obtain:
1Na = 3 X 1P
K = 2 x 1p
(6)
(7)
Where Ip is total current of active transport , which is generated by electrogenic ion pump.
r ______ r act
1p = 1 Na
+1
K
(8)
Usually the activation of ATPase by three sodium ions from inner part of the membrane and by two potassium ions from out part is necessary for hydrolysis ATP molecule.
The actual level of activation of ion pump Ip from maximal level is determined by the multiplication of simultaneous interactions probabilities with ATPase of single particles. Hence the equation of activity (intensity operation) of ion pump can be rewritten as:
1p = PNa X PK X PATP X 1 max (9)
Imax - maximal pump current, which is determined by structural characteristics of particular type of membrane.
Taking into account dependences of probability coefficient pNa and pK on concentrations of correspondent compounds of ATPase, according to equation presented above we obtain:
PATP X1 max
1P =■
(10)
1 +
K
Na
[Na +]
X
1 +
K
K
where kNa, kK are parameters of activation of ion pump; pATP is the coefficient, which takes into account the activation of ion pump as function of substrate activation of ATP by complex of Mg-ATP.
Passive transport and the correspondent ion flows (currents IK, INa) generated by electrochemical gradient are proportional to deviation TMP from correspondent equilibrium potentials:
1K = gK (U EK )’
(11)
K
1 Na = gNa (U - ENa ), (12)
where gK and gNa are electrical conductivities
E
Na
RT
F
ln
Z7 RT 1
EK =----------ln
F
Na~
K +\
K+
(13)
(14)
In steady state, if concentration of K+ and Na+ don’t change, active and passive currents are equilibrated.
Time-dependent changes of ion Na+ and K+ concentrations are the results of the unbalance of their passive and active flows. The speed of these changes
act
2
1
1
depend on the cell size, since the total transferred mass is proportional to the area of membrane, and the resulting concentration changes are reversely proportional to the cell volume.
d [Na + ]
dt vF d [K +] _
(j + j aKm
Na Na -
(15)
(16)
( + laKm ) dt vF K Voltage-dependent Na+ and K+ conductivities for the flows of passive ion transport can be phenomenologically determined by the ratio of synchronous values of correspondent current and voltage:
_ l K
gK _
U - E
S Na
K
1 Na U-ENa '
Correspondence of conductivity dynamics of to TMP changes allows to establish dependence of meanings of each conductivities from potential. As known, that voltage-dependence of conductivity is characterized by similar functions, which can be displayed by logistic curve.
(U)_
G„
1 + eXp[[ Pm - U)]
Sk (U)_
G„
(17)
(18)
1 + exp[ (Uk - U)] where GNa,GK are the maximal meanings of ion conductivities, which depend on the structural characteristics of the concrete type of membrane; kNa, kK are coefficients, that characterize steepness of voltage-dependence of each conductivity; UNa ,UK are meanings of membrane potentials (constants), which correspond to half of GNa, GK value.
FUNCTIONAL MODEL OF DYNAMICS OF CONDUCTIVITIES
At the next step we addressed the problem of the construction and testing of phenomenological model of conductivities dynamics for non-excitable embryo membrane. This functional model is proposed as the subsystem of more general system of membrane-related bioelectrical processes.
Proposed model takes into account that changes of K+ and Na+ conductivities are interrelated [1]. Besides they are limited by structural features of membrane and also depend on value of TMP. All this can be described mathematically in the following way:
ddir _0SNa - 2?Sk -a1U •
(19)
dgNa if _0 g
K - 25SNa +a2U •
(20)
where o1 ,o2 are coefficients that determined
interdependence between conductivities; y and S are coefficients that reflect the structural limits; a1 and a2 are coefficients that take into account voltage dependence of conductivities.
This system of equations can be solved relatively to any conductivity (i = K+, Na+) and by introducing certain symbol (f£) can be reduced to the differential equation of the second order:
dY dS.
----+ 2S-^ + o2g _PU
dt2 0 f f
(21)
Thus we obtain equation that corresponds to the well-known equation of harmonious oscillation.
Integration of this equation (21) gives a solution:
( \ P-
g (t)_ A exp(-St)cosl ot+p 1 + —l-U • (22)
i i V lJ o2
o
0
where o2 = o20 - S2 is cyclic frequency of oscillation; Al, q\ are constants of integration, which are determined by the initial conditions.
This solution is mathematical description of the membrane conductivities dynamics, the properties of which are reflected in parameters of the correspondent model.
It should be noted that the obtained equation characterizes the time-dependent changes of any conductivity (i= [K+, Na+] and in particular it describes the harmonious oscillation. Coefficient Al determines the amplitude of oscillation conductivity, and q\ is the initial phase of oscillations. Coefficient P is different to the given equation solution in relation to the given ion conductivity.
Pf
o1a2 + 2Sa1 o2a1 +2ya2
for K for Na
Essentially that oscillatory dynamics can be possible only on the conditions of S«o, a».
In analogy to the well-known equations of undumped harmonious oscillations the coefficients o1 and o2 characterize the cycle frequency, which is close to frequency of free undumped oscillations. Here coefficient o1 determines the sensitivity to Na+ conductivity. The coefficient of o2 characterizes the reverse dependence.
Coefficient S is the decrement of dumping, which characterizes the rate of amplitude decrease. At its very low values the frequency of undumped oscillation approaches the frequency of dumping oscillation. This
coefficient also characterizes the structural limitation of sodium conductivity as coefficient y for potassium conductivity. According to the experimental data the change of K+ conductivity in contrast to Na+ conductivity depends on the structural parameters of membrane very little. Therefore, it was assumed in modeling assume that y = 0.
Coefficients a1, a2 and P reflect the sensitivity of membrane conductivity change depending on TMP.
Thus in phenomenological model that is described by two differential equations (19-20) system the basis quantity properties of oscillation dynamics of conductivities are imposed. The model parameters characterize qualitatively interdependence of sodium and potassium conductivity changes and also their dependence on other parameters. This interrelation reflects the presence of negative reverse connection in biosystem, which provides the possibility for the appearance of the oscillation in this system.
STEADY STATE AND DYNAMIC REGIME OF BIOELECTRICAL PROCESSES
The equations presented above make up rather complete system, which is sufficient for modeling. In this system equation for TMP (5) and equations (11-12, 17-18) that determine the passive currents and conductivities are included. They allow to describe the steady state under certain conditions of their realization, determined by the equations (13-14, 6-7, 10) at constant values of concentrations characteristics and other parameters.
In the steady state the total flows of ions from the membrane is zero.
That is
lNa +lNaaCt+ h + 1^= 0.
(23)
Introducing in this equation the express passive currents (11, 12) and the total active current, we obtain the equation for steady state:
gNa(U - ENa) + gK(U - EK) + jp = 0
Its solution relative to U gives equation for the resting potential Us :
Us _
gKEK + g Na E Na 1 p
gK + gNa
(24)
The flows of ion pump make contribution to resting potential. Since Ip (represents the total cations flow from the cell) is positive, thus the pump induces additional hypopolarization of membrane.
The analysis of quantitative relations put into the background of the modeling shows that non equilibrium currents of passive and active transport for each type of ion are still not sufficient for the change of transmembrane potential. The disbalance of total
sodium and total potassium currents is necessary for that. It provides of the resulting currents 1d:
T -1- jaKm JK + 1K
(25)
■ Na Na d ’
which causes the changes of total intracellular concentration of cations and TMP.
By taking into account (11-12), we obtain:
gK (U - EK )+ gNa (U - ENa )+ JP - jd _ 0
Hence
U _ gKEK + gNaENa - 1 p + jd gK + gNa
that can be presented as
U = Uc + Up + Ud,,
(26)
(27)
where U _ gKEK + gNaENa (28)
gK + gNa
- is concentration compound of TMP, which is
caused by concentration gradients;
Up _-
L
(29)
gK + gNa
- is electrogenic compound of TMP , which is caused by work of ion pump;
1d
Ud _
(30)
gK + gNa
- is additional dynamic compound relative to total nonequilibrium of all currents through membrane.
Thus, in general TMP consists of three compounds. These are concentrational and electrogenic compounds which exist in the steady state (TMP is determined as Us = Uc + UP) and additional dynamic compound, which reveals in cell only during the transitional processes. The latest one is characterized by the absence of stationary level of both ions concentration and the total TMP. In the case of balanced oppositely directed changes of Na+ and K+ concentration the additional component is absent. And thus the TMP changes are absent.
THE MODEL AND ITS COMPUTER REALIZATION
The system of the equations (5-7, 10, 11-18) is the model of temporary trend of bioelectrical parameters, and system of the equations (19-20) is the model of interdependence of sodium and potassium conductivities. Their association has permitted to describe general changes and oscillation of electrochemical and concentration parameters. The model of temporary trends of bioelectrical parameters and model of interdependence of sodium and potassium conductivities supply one another and describe different aspects of the process of embryo cell functioning: characteristic changes of the general level
of electrophysiological parameters and their periodic fluctuations, synchronous with cycles of cell division.
The numerical methods of solution of the specified systems of the differential equations were applied to modeling. Realized in such a way complete computer model permits to describe and study membrane-associated bioelectrical processes from the moment of fertilization of oocytes and during all period of embryo cells division. The transition process of temporary changes of researched parameters begins from the moment of oocyte fertilization and is over 6-8 hours [5] later.
For definition of voltage dependence of conductivities for the initial data there were accepted TMP sizes from the moment fertilization (-15 mV) and up to the end of a stage morula (-50 mb), if TMP remains at a rather constant level after 7 hours of developing [5]. Concentration of sodium and potassium ions in environment was constant and equal accordingly to - [Na+]i = 110 mM, [K+]i = 1,4 mM. Thus, the concentration of ions K+ in cell embryo changes in the range of 70-110 mM, and the concentration of ions Na+ accordingly falls down [2]. Solving total concentration of Na+ and K+ with different meanings of TMP and taking into account the specified experimental data, the concentration of each type of ions was determined. The iterative algorithm for any of consecutive discrete steps of time was used for calculation of the dynamic compound potential
(Ud).
In computer realization of conductivity dynamics model the integration of the differential equations was carried out using system MathCAD 7.0. The differential equations in the model were solved numerically using a fourth-order Runge-Kutta method with the fixed step of integration. For conductivity dynamics modeling at the certain changes of potential more simple method of Euler, programmed on BASIC was applied. In computer realization of complete model, where model of conductivity dynamics was introduced as subsystem in trend model of membrane-associated processes the program for the solution of the differential equations system on BASIC was applied, in which the equations of conductivities interdependence were programmed.
SIMULATION RESULTS
The aperiodic dynamics changes of general level of the electrophysiological characteristics of embryo cell, which occur after fertilization during their synchronous division, was researched. It was carried out on the basis of account of temporary trend for all membrane-related bioelectrical and concentration parameters during 7 hours period of initial division [6].
Temporary changes of all components of TMP, namely concentrational, electrogenic and additional dynamic components of membrane potential are shown on Fig 2.
The concentrational component, which is determined by the chemical gradients makes the most essential contribution to the level of TMP. After 7 hours of development the dynamics of all components and TMP accordingly comes to the certain constant level. Additional dynamic component decreases to zero. It specifies that transition process has come to the end and system has entered a new stationary level.
For estimation of the model adequacy the obtained results of modeling were compared with experimental data of the dynamics of bioelectrical characteristics (TMP, conductivity, entrance resistance). Dynamics of the total level of TMP for the initial stage of development of loach embryo is received by calculation according to the given initial concentration of intracellular ion K+ and Na+. It is shown on Fig 3 and 4 for the initial stage of loach embryo development. Comparison of 7 hours trend determined in model (Fig. 4) with real dynamics of TMP determined in vivo specifies the correspondence model of membrane-related processes that occur in embryo cell at the period of initial division [5].
The oscillation of membrane conductivity both at fixed potential and at change of the TMP level with help of partial model of dynamics of conductivity (1920) was investigated. The model describes periodic changes of sodium and potassium conductivities, which are synchronous with the cell cycles and have identical frequency and are shifted on a phase.
Proposed model of conductivity dynamics, which is described by two differential equations and takes into account interrelation of conductivities, allows to suppose that it is the theoretical generalization of Antomonov's model [1]for excitable membrane and can be applied to the membranes of fish blastomeres.
The total model, which takes into account all relations between bioelectrical parameters including sodium and potassium conductivities as the appropriate system of the equations (5-7, 10-18, 19-20), permits to restore aperiodic and periodic dynamic changes of the electrophysiological characteristics of embryo cell, namely the fluctuation on the background trend of their level. For estimation of abilities of the total model the results of modeling were compared to available experimental data for dynamics of the bioelectrical characteristics of loach embryo [5].
The quantitative characteristic of dynamics of concentration of ions K+ and Na+ in matrix medium received on the basis of model, is in complete conformity with laws of their change during early development of a loach embryo known from the literature [2] .It should be noted, that in result of
modeling, and in experimental data practically there are no varying changes of concentration, synchronous with cell cycles of division, because relative change of concentration integrated parameter concerning currents is very insignificant.
The dynamics of membrane potential of embryo cells at the initial stage of their development reproduced by the described model (Fig.5) corresponds to the experimental data of 6-hour registration of TMP loach embryo (Fig.6).
The results of modeling of total ion current dynamics and conductivities also correspond to their experimental changes [6].
Fig. 2. Dynamics of TMP (U), diffusion (Uc), electrogenic (Up) and additional dynamic (Ud) components.
Fig.3. Dynamics of level TMP (Gojda, 1993) in early embryogenesis of loach (1) and trend (2).
Fig.4. Trend of TMP determined in model with initial value [Na+]i, [K+]i: 1- 40, 90; 2- 40, 100; 3- 40, 110 mM.
Fig.5. Dynamics of TMP determined in model with initial Fig.6. Experimentally determined dynamics of TMP of
value [Na+]i, [K+]i : 1- 20, 90mM; 2 - 30, 90 mM, loach embryo (Gojda, 1993).
respectively.
DISCUSSION
In this paper we present the model of bioelectrical processes dynamics for loach embryo. Our model takes into account only ions of sodium, potassium, free permeable ions of chlorine and non permeable anions
++
z-
A , and also work of Na ,K -pump. However, we did
2+
not take into account ions Ca , which influence the dynamics of TMP a little. The equation for membrane potential is deduced on the basis of physical principles and submitted by logarithmic expression. Whereas, in the majority of models for definition of TMP the
differential equations are used, except for the model Endreson [9] .In our model intracellular concentrations
of ions Na+ and K+ are dynamic variables and are determined by the unbalances of passive and active currents for certain ion. Such approach and the similar equations are characteristic of other models [8, 10, 11,
13, 14].
The system of equations, which takes into account TMP (5) and equation (6, 7, 10, 11-14, 19, 20) permits to describe a stationary condition, at which the intracellular concentration of ions and TMP remains at the constant level. Taking into account the dynamics at
the time of concentration of ions Na+ and K+, it is possible to receive the dynamics of trend for all electrophysiological characteristics. Besides, the unbalance of currents through membrane predetermines the occurrence of the resulting current Id, and it has permitted to deduce additional dynamic compound of transmembrane potential, which is inherent only for the transition period.
This model is reasonably realistic for modeling of electrophysiological processes. But for more simplicity didn’t take into account ion (Ca2+) with little influence on dynamics of membrane potential. On the other hand we try to present the model using the equations derived from basic physical-chemical principles and conservation laws. Our model involves basic assumption for most models that describe electrical activity of cells and include passive and active transport of cation K+ and Na+, free permeable anions Cl-, non-permeable ion Az-.
We describe ion currents of passive transport by known equation of Hodgkin-Katz [12]. We determined ion current of active transport by intensity of work of ion pump (Na,K-ATPase). Results of disbalances of passive and active ion flows (and correspondent currents) are time changes of concentration of Na+ and K+. This assumption is usually expressed as differential equation. Equations of voltage-dependence of conductivities are derived. By using basic conditions such as electroneutrality of intra and extra cellular medium, condition of osmotic balance for ions, and condition of electrochemical equilibrium of ion Cl-, which are taken into account in Jacobson’s model and others[4, 13] equation for TMP is obtained.
The system of equations allows to describe the dynamics of interrelations of membrane potential and other bioelectrical parameters with the changes of concentrational parameters. In general case the modeling of dynamic regime of cell processes unbalances of ion flows predetermines changes of all bioelectrical parameters, first of all, of cell processes -intracellular contents of cations and anions. Just such regime is peculiar for the period of division of oocytes of fishes, in particular loach, was also the subject of our research.
The reproduction of oscillation component of bioelectrical and concentration parameters is possible on the basis of modeling a certain endogen oscillator. Phenomenologically it is possible to consider such source of fluctuations of periodic changes of
conductivities if it is assumed that there is an
interrelation between sodium and potassium conductivities. The last follows from the dynamic
properties of Antomonov’s model [1], which takes into account interdependence of gk and gNa for the
description of conductivity dynamics and modeling of potential of calmer action axon. The changes everyone
with conductivity is proportional to value of another, depending the membrane potential and structural parameters of a membrane. The association of models of general changes and periodic component reconstructs a complete picture of ТМР dynamics and other parameters.
As it is known in the majority of physiological systems, which are capable to support the important parameters meaning in the certain range, the principle of a feedback is realized which permits to control the initial characteristics of system providing its ability to autoregulation at different levels of organization of organism. With the help of a feedback many other features of physiological systems, namely, the presence of internal oscillators are explained as well. The put forward hypothesis about interdependence between Na+ and K+ conductivities provides an opportunity of occurrence of fluctuations in system. However, the model of dynamics of conductivities is phenomenological, and parameters are formal. In general, in total model, which consists of 15 equations and contains more than 20 parameters, there are only 9 formal factors (kNa, kK, UNa, UK, ю1, ю2, 5, a1, a2, ).
CONCLUSION
In conclusion, we would like to emphasize, that the definition of formal parameters of the model is based on experimental data [5], on the basis of which we have determined voltage-dependence of Na+, K+ conductivities, necessary for trend modeling, and their interdependence for oscillations modeling. Therefore, reproduction of the dynamics of membrane potential and Na+, K+ concentration is in complete conformity with experimental data. The modeling of ion conductivities and currents separately for each kind of ions (including active and passive transport) has given new quantitative results, since in direct experimental measurements on intact cells only the dynamics of their total value is registered [5]. The information about the dynamics of activity of Na+, K+-pump its current and its contribution to ТМР can also be considered to be new. The results of modeling in this aspect are in a good conformity with experimental data.
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12. Hodgkln A.L., Katz B. The effect of sodium ions on the electrical activity of the giant axon of the squid// J. Physiol.- 1949. -Vol.108. - P. 37-77.
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14. Plonsey R., Barr R. C. Bioelectricity. A quantative approach. -Plenum Press, New York - London, 1990.360 p.
МОДЕЛЬ ДИНАМІКИ МЕМБРАНОПОВ’ЯЗАНИХ БІОЕЛЕКТРИЧНИХ ПРОЦЕСІВ У РАННЬОМУ ЕМБРІОГЕНЕЗІ В’ЮНА
Дика М.В., Гумецький Р.Я.
Дослідження динамічних змін електрофізіологічних характеристик зародкових клітин в’юна, що відбуваються після запліднення в період їх синхронних дроблень, проводились на основі часткових моделей (динаміки їх аперіодичного тренду та коливної динаміки провідностей), а також на основі комплексної (об’єднаної) моделі динаміки біоелектричних процесів. Остання відтворює закономірності мембранного транспорту та розподілу іонів калію та натрію у їх зв'язку з динамікою біоелектричних показників мембран.
Ключові слова: мембрана, біоелектричні параметри, часова динаміка,математична модель.
МОДЕЛЬ ДИНАМИКИ МЕМБРАНОСВЯЗАННЫХ БИОЭЛЕКТРИЧЕСКИХ ПРОЦЕССОВ В РАННЕМ ЭМБРИОГЕНЕЗЕ ВЬЮНА
Дыка М.В., Гумецкий Р.Я.
Исследования динамических изменений электрофизиологических характеристик зародышевых клеток проводились на частных моделях (динамики их апериодического тренда и колебательной динамики мембранной проводимости), а также на базе комплексной (объединенной) модели динамики биоэлектрических процессов. Последняя отражает закономерности мембранного транспорта и распределения ионов калия и натрия в их связи с динамикой биоэлектрических показателей мембран, в частности, трансмембранного потенциала (ТМП).
Ключевые слова: мембрана, биоэлектрические параметры, временная динамика, математическая модель.
б5
Фізика живого, Т. 16, No2, 2008. С.65-71. © Окань А.М, Санагурський Д.І.
УДК 573.2:577.152.361
УТОЧНЕННЯ МОДЕЛІ КОЛИВНОЇ ДИНАМІКИ ТРАНСМЕМБРАННОГО ПОТЕНЦІАЛУ В РАННЬОМУ ЕМБРІОГЕНЕЗІ В’ЮНА
Окань А.М, Санагурський Д.І.
Кафедра біофізики та біоінформатики Львівський національний університет імені Івана Франка, вул. Грушевського, 4, 79005 Львів E-mail: [email protected]
Надійшла до редакції 20.05.2009
Проведено математичне моделювання безперервної динаміки концентрацій потенціалгенеруючих іонів протягом 1-7 годин ембріонального розвитку зародків в’юна. Математично описано динаміку кінетичних коефіцієнтів, які відображають в моделі активність іонтранспортних систем. Досліджено амплітуду коливань концентрацій іонів та трансмембранного потенціалу, що породжуються автоколивною системою до складу якої входять незалежні системи пасивного транспорту іонів калію та натрію і система спряженого активного транспорту цих іонів. Розглянуто можливості №+/Са +-обмінника впливати на коливання концентрацій іонів та трансмембранного потенціалу.
Ключові слова: трансмембранний потенціал, концентрація іонів калію, концентрацію іонів натрію, коливання концентрацій іонів.
ВСТУП
Характерною особливістю початкових стадій розвитку зародка є строго синхронний ритмічний поділ бластомерів, який для кожного виду тварин і умов інкубації відбувається з певною швидкістю [1]. Ряд показників клітинного метаболізму, зокрема трансмембранний потенціал, здійснює коливання синхронні з ритмами клітинних поділів, що показано на зародках ряду тварин [1, 2, 3], а також на культурах клітин [4]. В той же час дроблення клітин не є причиною коливання метаболічних процесів, оскільки інгібування його не припиняє коливання метаболічних процесів [1].
Важливим є питання: які компоненти іонного транспорту здійснюють свій вклад в динаміку трансмембранного потенціалу. Показано, що синхронно зі змінами ТМП коливається вхідний опір мембрани. Коливний характер іонної провідності мембрани показаний на ряді об’єктів [3,5,6].
Важливу роль в коливаннях ТМП відіграє робота № ,К -помпи. Так при інгібуванні помпи оуабаїном відбувається не тільки деполяризація мембрани зародків в’ юна порівняно з контролем, але й затухання коливань ТМП [7]. Цікаво, що зміни рівня ТМП в ранньому розвитку в’юна дуже нагадують характер динаміки активності № ,К -помпи [8].
В літературі представлено ряд підходів, до моделювання автоколивних процесів, що мають місце в клітині [9, 10].
В попередній роботі [11] показано можливість системи, в якій відбувається незалежний пасивний транспорт іонів калію і натрію та спряжений активний транспорт цих іонів, забезпечувати коливання їх концентрацій і відповідно коливання трансмембранного потенціалу. Кінетична поведінка даної транспортної системи була описана за допомогою системи рівнянь, відомих як система Вольтера-Лотка.
^ = к,[Х ] - к2[ X ][7]; (1)
Ш
= -к3[Г ] + к4[X ][Г ].
аґ
Де [X] позначає відношення
внутрішньоклітинної концентрації іонів натрію до позаклітинної, [У] - відношення
внутрішньоклітинної концентрації іонів калію до позаклітинної їх концентрації.
В роботі [11] для з’ясування динаміки
концентрацій іонів використовувався аналітичний розв’язок системи рівнянь, який містить
невизначений параметр (сталу інтегрування) від якого залежить амплітуда коливань концентрацій іонів. Кінетичні коефіцієнти, які відображають активність роботи відповідних транспортних
систем клітини, постійно змінюються відповідно до потреб клітин, що розвиваються. В попередній
