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Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers



Parvaiz Ahmad Naik and Kamal Raj Pardasani
 
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ABSTRACT

Calcium, a second messenger for signal transduction in cells plays an important role in almost every cell of our human body. In oocytes calcium plays a significant role in oocyte maturation. In the process of reproduction calcium concentration is regulated at high levels in oocytes through various mechanisms so as to meet the requirements of oocytes maturation. Thus modelling of calcium distribution in oocytes can help us in understanding this mechanism in a better way. Here an attempt has been made to develop a finite element model to study calcium distribution in oocytes. The model incorporates the parameters like diffusion coefficient, Na+/K+ pump, Na+/Ca2+ exchanger and buffers like BAPTA and EGTA. The proposed model is solved numerically using appropriate initial and boundary conditions. The program has been developed in MATLAB 7.10 for the entire problem and simulated on a 64 bit machine to compute the numerical results.

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Parvaiz Ahmad Naik and Kamal Raj Pardasani, 2014. Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers. Asian Journal of Mathematics & Statistics, 7: 21-28.

DOI: 10.3923/ajms.2014.21.28

URL: https://scialert.net/abstract/?doi=ajms.2014.21.28
 
Received: October 13, 2013; Accepted: January 14, 2014; Published: March 29, 2014



INTRODUCTION

Panday and Pardasani (2013a) studied the effect of buffers, leak, advection and Ca2+ exchanger on calcium distribution in oocytes. Jha et al. (2013) studied the effect of voltage gated calcium channels in astrocytes. Tewari and Pardasani (2008) studied the effects of Na+ influx on cytosolic Ca2+ diffusion in neurons. Here an attempt has been made to study the spatio-temporal behaviour of Na+/K+ pump and Na+/Ca2+ exchanger on calcium distribution in oocytes in presence of buffers. The model is formed by a set of partial differential equations with appropriate initial and boundary conditions. Finite element method is used to solve the proposed mathematical model (Rao, 2004). The model incorporates the parameters like diffusion coefficient, Na+/K+ Pump, Na+/Ca2+ exchanger and buffers like BAPTA and EGTA. The effect of Na+/K+ pump and Na+/Ca2+ exchanger on calcium distribution in oocytes is studied with the help of numerical results. The main aim of this study is to study the changes in intracellular calcium concentration in oocyte during the process of fertilization in presence and absence of these parameters as calcium acts as a switch for oocyte maturation.

MATHEMATICAL FORMULATION

Calcium kinetics in oocytes is governed by a set of reaction-diffusion equations which can be framed assuming the following bimolecular reaction between Ca2+ and buffer species (Eq. 1) (Smith, 1996; Sherman et al., 2001):

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(1)

where, (Ca2+), (Bj) and (CaBj) represent the cytosolic Ca2+ concentration, free buffer concentration and calcium bound buffer concentration, respectively and ‘j’ is an index over buffer species, k+j and k-j are on and off rates for jth buffer, respectively. Using Fickian diffusion, the buffer reaction diffusion system in one dimension is expressed as (Eq. 2-4) Neher (1973) and Smith (1996):

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(2)

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(3)

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(4)

where, reaction term Rj is given by Eq. 5:

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(5)

DCa, DBj, DCaBj are diffusion coefficients of free calcium, free buffer and Ca2+ bound buffer, respectively and σCa is net influx of Ca2+ from the source. Let Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers be the total buffer concentration of jth buffer and the diffusion coefficient of buffer is not affected by the binding of calcium i.e., DBj = DCaBj. Then, Eq. 5 can be written as (Eq. 6) (Tewari, 2009):

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(6)

We assume that buffer concentration is present in excess inside the cytosol so that the concentration of free buffer is constant in space and time, i.e., [Bj] ≅ [Bj]. Under this assumption Eq. 6 is approximated by (Eq. 7) (Sherman et al., 2001):

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(7)

Where:

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers

is the background buffer concentration. Thus, for single mobile buffer species Eq. 2 can be written as (Eq. 8) (Tewari, 2009; Sherman et al., 2001):

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(8)

Here, [Ca2+] is background calcium concentration and δ(r) is the Dirac Delta function that is placed at source position. We assume a single point source of Ca2+, σCa at r = 0, there are no sources for buffers and buffer concentration is in equilibrium with Ca2+ far from the source. From GHK current equation (Neher, 1973; Keener and Sneyd, 1998), we have (Eq. 9):

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(9)

where, Ica is the current due to calcium gradient, PCa is the calcium permeability, zCa is the valency of calcium ion (i.e., +2), Vm is the membrane potential, F is the Faraday’s constant, R is the gas constant, T is the absolute temperature, Cai and Cao are the intracellular and extracellular calcium concentration, respectively. The net influx, σCa of Ca2+ ions flowing (Eq. 10) per second at the origin is Jha et al. (2011):

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(10)

where, Vcyt is the volume of the cytosol in oocytes. In Eq. 10 there is a negative sign before Ica because inward current is taken to be negative. The expression for the Na+/K+ pump is given by (Eq. 11) (Tewari, 2012):

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(11)

Here, Ipump is the scaling factor of Na+/K+ current (in μA cm-2), kf (in ms) is the forward (deocclusion) rate constant, kb (in ms) is the backward (deocclusion) rate constant, k0.5(0) is half activation (Na+)O concentration at 0 mV, HNaK is the Hill’s coefficient for half activation Na+/K+ current, λ is the fraction of electrical field droped along the access channel and τ (in ms) is some constant.

NCLX is an essential component of mitochondrial Na+/Ca2+ exchange (Palty et al., 2010). It helps in the extrusion of cytosolic calcium in oocytes and hence regulates the process of fertilization. In our model we have taken an exchange ratio of 3:1 with respect to sodium and calcium ions respectively. The net transport of Ca2+ ions through Na+/Ca2+ exchanger is given by (Eq. 12, 13) (Tewari, 2009; Panday and Pardasani, 2013b):

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(12)

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(13)

where, Cai and Cao are the intracellular and extracellular Ca2+ concentration, Nai and Nao are the intracellular and extracellular Na+ concentration. We have incorporated a deactivation function in the Na+/Ca2+ exchanger protein equation which deactivates the Na+/Ca2+ exchanger protein once the (Ca2+) reaches some value (15130 μM) concentration. Therefore, the (Ca2+) increases and increase in (Na+) decreases. Since, the change in cytosolic (Na+)i given by the Eq. 14:

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(14)

Also diminishes. Thus, as (Ca2+)i increases rate of change in (Na+)i decreases.

Now summing all the above equation we get the final model as Eq. 15:

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(15)

Along with the initial and boundary conditions (Tewari and Pardasani, 2008).

Initial condition (Eq. 16):

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(16)

Boundary conditions (Eq.17, 18):

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(17)

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
(18)

Our aim is to solve the Eq. 15 along with (16-18). We solve this model in one dimensional unsteady state by finite element method assuming the oocyte to be of circular in shape. The radius of the cell is taken as r = 5 μm and the number of elements taken for simulation are e = 1, 2, 3,..., 50. In this model, we first take the effect of Na+/Ca2+ exchanger which removes the cytosolic calcium by influx of Na+ causes the decrease in intracellular calcium after that we consider the effect of Na+/K+ pump which blocks the influx of Na+ causes the balance in cytosolic calcium. The numerical values of biophysical parameters used in the model are stated in the Table 1 (Jha et al., 2013; Panday and Pardasani, 2013a; Naik and Pardasani, 2013; Tewari, 2012).

RESULT AND DISCUSSION

The numerical results for calcium profile against different biophysical parameters have been obtained using numerical values of parameters given in Table 1 unless stated along with figures. Figure 1 shows that the calcium concentration near the source is higher and as we move away from the source the calcium concentration decreases slowly and finally tend to its initial value of 0.1 μM. The calcium concentration is higher from 0 to 1 μm and then decrease gradually upto 1.5 μm and finally reaches the initial value of 0.1 μm.

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
Fig. 1: Spatial variation of calcium concentration for the source amplitude σ = 1 pA

Table 1: Values of biophysical parameters
Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
M: Mole, Sec: Second

Figure 2 shows the calcium concentration for different concentrations of EGTA buffer. It is clear from the figure that the calcium concentration is higher for lower concentration of buffer. The calcium concentration is higher from 0 to 0.5 μm after then decreases slowly and finally tends to the initial value of 0.1 μM. The reason for lower calcium concentration in response of higher value of buffer is that the higher concentration of buffer binds more calcium in oocytes thus lowers the calcium concentration.

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
Fig. 2: Spatial variation of calcium concentration for different concentration of buffer

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
Fig. 3:Spatial variation of calcium concentration for the effect of Na+/Ca2+ exchanger and Na+/K+ pump. Na0 = 100 mM and Nai = 60 mM

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
Fig. 4:Temporal variation of calcium concentration for the effect of Na+/Ca2+ exchanger and Na+/K+ pump. Na0 = 100 mM and Nai = 60 mM

Figure 3 and 4 gives the spatial and temporal variation of calcium concentration in presence and absence of Na+/Ca2+ exchanger and Na+/K+ pump, respectively.

Image for - Finite Element Model to Study Effect of Na+/K+ Pump and Na+/Ca2+ Exchanger on Calcium Distribution in Oocytes in Presence of Buffers
Fig. 5:Temporal variation of calcium concentration for different radial positions i.e., for r = 0, 2 and 3 μm and for Buffer = 50 μM, σ = 1 pA

The effect of these parameters is clearly shown in figures. The Na+/Ca2+ exchanger works as it removes one Ca2+ ions from the cell in response of entering three Na+ ions into the cell causing the increase in Na+ inside the cell i.e., in the ratio of Ca2+:3Na+ and Na+/K+ pump works by removing three ions of Na+ from the cell by entering two ions of K+ into the cell i.e., in the ratio of 3Na+:2K+. Thus the mechanism of these two parameters is Ca2+:3Na+:2K+. From figures we see that the calcium concentration is lower in presence of Na+/Ca2+ exchanger this is because the exchanger removes Ca2+ ions from the cytosol thus causing lower concentration of calcium in the cell. Also the figures shows that in presence of Na+/K+ pump the calcium concentration is higher than in presence of Na+/Ca2+ the reason for this is that the presence of Na+/K+ pump blocks the efflux of calcium by blocking the function of Na+/Ca2+ exchanger thus increases the calcium in the cytosol. Figure 3 shows that the calcium concentration is higher from r = 0-1.5 μm after then tends to the steady state case while Fig. 4 shows that the calcium concentration is higher from t = 0 to t = 200 ms and then onwards remains in the steady state case. The presence of Na+/K+ pump thus controls the efflux of calcium from the cell by stopping the function of Na+/Ca2+ exchanger makes the balance of calcium in the cytosol as is clearly visible from the above Fig. 3 and 4.

Figure 5 gives the intracellular temporal Ca2+ distribution at different radical positions for the buffer with respect to time. As time increase the Ca2+ concentration rises sharply and after some time it achieves the steady state. With increase in distance from the source calcium concentration decreases and takes less time to reach steady state. In the figure the green curve corresponds to r = 0 μm, the red curve corresponds to r = 2 μm and the blue curve corresponds to r = 3 μm. The Ca2+ concentration is higher from t = 0-200 msec and after then tends to the steady state. The concentration of buffer in the study is taken as B = 50 μM for the oocyte.

CONCLUSION

The mathematical models developed give us interesting results regarding relationships among various parameters like calcium concentration, diffusion coefficient, radius, influx, buffers, Na+/Ca2+ exchanger and Na+/K+ Pump etc. The finite element method used is quite flexible to study relationship among these parameters and gives better relationship between them. Such models can be developed to generate information for better insights and understanding for the calcium signaling in Oocytes. The results obtained are very helpful for the Biomedical scientists in understanding the mechanisms of oocyte cell growth, maturation of oocyte and reproduction. The results obtained in this study are in close agreement with the experimental studies (Clarke and Kane, 2007; Palty et al., 2010; Lee et al., 2002; Morris, 2011) and the results obtained by Panday and Pardasani (2013b) and Tewari and Pardasani (2008).

ACKNOWLEDGMENT

The authors are highly thankful to University Grants Commission (UGC), New Delhi, India for providing financial support to carry out this work.

REFERENCES

1:  Morris, C.E., 2011. Voltage-gated channel mechanosensitivity: Fact or friction? Front. Physiol., Vol. 2.
CrossRef  |  

2:  Clarke, R.J. and D.J. Kane, 2007. Two gears of pumping by the sodium pump. Biophys. J., 93: 4187-4196.
CrossRef  |  

3:  Neher, E., 1973. Concentration profile of intracellular Ca2+ in the presence of cells presumed to be GLIA in cerebral cortex of cat. J. Neurophysiol., 36: 855-868.

4:  Jha, B.K., N. Adlakha and M.N. Mehta, 2013. Finite element model to study calcium diffusion in astrocytes. Int. J. Pure Applied Math., 78: 945-955.
Direct Link  |  

5:  Jha, B.K., N. Adlakha and M.N. Mehta, 2011. Finite volume model to study the effect of voltage gated Ca2+ channels on cytosolic calcium advection diffusion. World Acad. Sci. Eng. Technol., 5: 73-77.

6:  Keener, J. and J. Sneyd, 1998. Mathematical Physiology. Vol. 8, Springer, New York, pp: 53-56

7:  Panday, S. and K.R. Pardasani, 2013. Finite element model to study effect of buffers along with leak from ER on cytosolic Ca2+ distribution in oocytes. IOSR, 4: 1-8.

8:  Panday, S. and K.R. Pardasani, 2013. Finite element model to study effect of advection diffusion and Na+/Ca2+ exchanger on Ca2+ distribution in oocytes. J. Med. Imag. Health Informat., 3: 374-379.
CrossRef  |  

9:  Naik, P.A. and K.R. Pardasani, 2013. Finite element model to study effect of buffers in presence of voltage gated Ca2+ channels on calcium distribution in oocytes for one dimensional unsteady state case. Int. J. Modern Biol. Med., 4: 190-203.
Direct Link  |  

10:  Palty, R., W.F. Silverman, M. Hershfinkel, T. Caporale and S.L. Sensi et al., 2010. NCLX is an essential component of mitochondrial Na+/Ca2+ exchange. Proc. Natl. Acad. Sci., 107: 436-441.
CrossRef  |  PubMed  |  

11:  Lee, S.H., M.H. Kim, K.H. Park, Y.E. Earm and W.K. Ho, 2002. K+-dependent Na+/Ca2+ exchanger is a major Ca2+ clearance mechanism in axon terminals of rat neurohypophysis. J. Neurosci., 22: 6891-6899.
Direct Link  |  

12:  Rao, S.S., 2004. Finite Element Method in Engineering. Elsevier Science and Technology, USA

13:  Sherman, A., G.D. Smith, L. Dai and R.M. Miura, 2001. Asymptotic analysis of buffered calcium diffusion near a point source. SIAM J. Applied Math., 61: 1816-1838.
CrossRef  |  

14:  Smith, G.D., 1996. Analytical steady-state solution to the rapid buffering approximation near an open Ca2+ channel. Biophys. J., 71: 3064-3072.
CrossRef  |  

15:  Tewari, S.G., 2009. A variational-ritz approach to study cytosolic calcium diffusion in neuron cells for a one dimensional unsteady state case. J. Math. Math. Biosci., 2: 1-10.

16:  Tewari, S.G., 2012. The sodium pump controls the frequency of action-potential-induced calcium oscillations. Comput. Applied Math., 31: 283-304.
Direct Link  |  

17:  Tewari, S.G. and K.R. Pardasani, 2008. Finite difference model to study the effects of Na+ influx on cytosolic Ca2+ diffusion. Int. J. Biol. Med. Sci., 4: 205-209.

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