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Finite element model to study calcium diffusion with excess buffer approximation in fibroblasts cell.

Introduction

Intracellular calcium diffusion are very common and have been reported in a large variety of cell types such as: cardiac Myocytes [1], neuron [2,14],astrocytes [3], hepatocytes [16], pancreatic acinar cells [10], Fibroblasts cell [13] etc.. In these cell types, the cytosolic calcium distribution has been studied for various physiological parameters to reproduce the properties of electrical membrane model and intracellular calcium dynamics model in cell. The intracellular calcium dynamics model [10] contains two compartments: the cytocol and the ER. The plasma membrane contains calcium ATPase (PMCA) pump, L-type calcium channels, Cl-ca channel, a leak and inward rectifier potassium channels. The ER membrane contains a srca (endo) plasmic reticulum calcium ATPase (SERCA) pump, a calcium leak and an IP3 receptor channel.

Fibroblasts cell has long been considered as Non-excitable cells [5],but the presence of L-type calcium channel has been reported by experiments [4], and accounted to reproduce to behave like an excitable media in particular growth stages [12]. In the previous literature on fibroblasts cell, a mathematical model of NRK fibroblasts capturing, the basic characteristics [7,11] based on single cell data had been discussed. But the intracellular diffusion of calcium does not included in the above studies. The role of buffer in cytosol of fibroblasts cell with calcium diffusion has been discussed by M. Kotwani et.al [13] for one dimension. Here, a mathematical model is presented for the study of cytosolic calcium diffusion in two dimensions involving excess buffer approximation. Any change in calcium profile due to buffering has been studied by using finite element method.

Mathematical Formulation

Calcium dynamics in fibroblasts cell is governed by a set of reaction diffusion equation given by [9,11] :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [[B.sub.j]] and [Ca[B.sub.j]] are free buffer and bound buffer, and 'j' is an index over buffer species. So the Calcium dynamics can be in the form of following equations [6,8,14]:

[partial derivative][[Ca.sup.2+]]/[partial derivative]t = [D.sub.Ca] [[nabla].sup.2+][[Ca.sup.2+]] + [summation over (j)] [R.sub.j] + [delta][sigma](r) (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Where

[R.sub.j] = -[k.sup.+.sub.j] [[B.sub.j]][[Ca.sup.2+]] + [k.sup.-.sub.j] [Ca[B.sub.j]]

[D.sub.Ca], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are diffusion coefficients of free calcium, free buffer, and [Ca.sup.2+] bound buffer, respectively; [k.sup.+.sub.j] and [k.sup.-.sub.j] are association and dissociation rate constants for buffer j respectively. For stationary immobile buffers or fixed buffers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first term on right hand side is due to Fick's law of diffusion, the second term [R.sub.j] is known as the reaction diffusion term and the third term is source amplitude due to calcium channel. Here single mobile buffer species, i.e. [[B].sub.j] = [ B] has been taken.

The Excess Buffer approximation (EBA) [17] is incorporated in the model. It is assumed that the buffer concentration is present in excess and buffer is constant in space and time, i.e. [B] = [CaB] = const.

In the view of above, the equation (2) becomes:

[partial derivative][[Ca.sup.2+]]/[partial derivative]t = [D.sub.Ca] [[nabla].sup.2+][[Ca.sup.2+]]-[k.sub.m]B[infinity] ([[Ca.sup.2+]][infinity]) + [sigma][delta](r) (5)

It is assumed that cytosol of the cell is of circular shape with radius 5 um. Thus, diffusion equations (5) in two dimensional polar coordinates for a steady state case is given by [11,14];

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [B][infinity] and [Ca][infinity] are the buffer concentration and calcium concentration respectively.

Considering the cytosol of circular shaped fibroblasts, the region is divided into 80 coaxial circular elements [fig. 1]. The centre of circle is at ( r = 0,[theta] = 0) and radius of circle is r =5 um. It is assumed that the point source is situated at ( r = - 5,[theta] = n) and as we move away from source the calcium concentration in cytosol achieves its background value i.e. 0.1 [micro]M. So the boundary condition takes the following form [8,9]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

For other boundary condition, it is assumed that [[Ca.sup.2+]] attains its steady state concentration 0.1 [micro]M as we move far away from the source;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[FIGURE 1 OMITTED]

The discretized variational form of equations (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Here, we have used 'u' in lieu of [[Ca.sup.2+]] for our convenience, e=1, 2....80. [lambda] is the characteristic length which is equal to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Also the second term ([mu] (e)=1) for e=40, 50 and (u (e) =0) for rest of the elements. The following bilinear shape function for the calcium concentration within each element has been taken as [15]:

[u.sup.(e)] = [c.sub.1.sup.(e)] + [c.sub.2.sup.(e)]r + [c.sub.3.sup.(e)][theta] + [c.sub.4.sup.(e)]r[theta] (10)

[u.sup.(e)] = [P.sup.T][c.sup.(e)] (11)

where, [p.sup.T] = [1 r [theta] r[theta]] and [([c.sup.(e)]).sup.T] = [[c.sub.1.sup.(e)] [c.sub.2.sup.(e)] [c.sub.3.sup.(e)] [c.sub.4.sup.(e)]] from equation (10) & (11) we get,

[[bar.u.sup.(e)]] = [P.sup.(e)][c.sup.(e)] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From equation (12) we have,

[c.sup.(e)] = [R.sup.(e)] [[bar.u.sup.(e)]] (13)

where,

[R.sup.(e)] = [p.sup.(e)-1]

Substituting [c.sup.(e)] from equation (13) in equation (11) we get,

[u.sup.(e)] = [P.sup.T][R.sup.(e)][[bar.u.sup.(e)]] (14)

Now, the integral [I.sup.(e)] can be in the form

[I.sup.(e)] = [I.sup.(e).sub.k] + [I.sup.(e).sub.m]-[I.sup.(e).sub.s]- [I.sup.(e).sub.z] (15)

Where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

du{e) du(e) du(e) du{e) du{e)

On substituting values of equations, we get,

dI/d[bar.u] = [N.summation over (e=1)] [[bar.M].sup.(e)] d[I.sup.(e)]/[du.sup.(e)] [[bar.M].sup.(e)T] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

The integral I is extremized with respect to each nodal calcium concentration ui (i=1, 2.....88). This is given in the form of differential equation for steady state in terms of calcium concentration. The Gaussian Elimination Method has been used to obtain the solution.

[[[bar.X]].sub.88x88] [[[bar.u]].sub.88x1] = [[[bar.Y]].sub.88x1] (19)

Here, [bar.u] = [u.sub.1][u.sub.2]..............[u.sub.88] X are the system matrices, and [bar.Y] is system vector. A computer program in MATLAB 7.10 is developed to find numerical solution to the entire problem.

Numerical Results and Discussion

This section shows the numerical results in the form of figures for calcium profile against different biophysical parameters. The biophysical parameters used in proposed model are stated in Table (1I). The proposed model illustrates the variation of cytosolic calcium concentration for the exogenous buffers.

The one well known exogenous buffer is EGTA. Here, the calcium diffusion kinetics can be studied in fibroblasts cell with the assumption that calcium is buffered using 20[micro]M EGTA.

[FIGURE 2 OMITTED]

It is observed from fig (2) that, calcium concentration in cytosol decreases exponentially with increasing distance from the point source. It shows that as total buffer concentration in cytosol increases, the highest concentration of calcium in cytosol near the point source decreases for higher buffer concentration.

Fig [3] shows the variation of calcium concentration with respect to angular direction for different buffer concentration in cytosol, initially calcium concentration from [theta] = 0 to [theta] = [pi] increases and thereafter concentration of calcium in cytosol decreases till [theta] = 2[pi]. As total buffer concentration increases, the concentration of calcium in cytosol decreases in angular direction. At point source [theta] = [pi] concentration achieves its higher value in cytosol.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

A large amount of calcium bound to buffers and only small proportion remain free to serve as a second messenger. Thus, the buffer binding affinity plays an important role to perform [Ca.sup.2+] dependent reaction in cytosol. It is observed from fig (4) and fig (5), as buffer binding affinity of calcium in fibroblasts cell increases, the free calcium ion concentration in cytosol decreases. As proposed, near the point source, cytosolic [Ca.sup.2+] concentration is higher and as we move away from source concentration of calcium achieves background concentration i.e. 0.1 [micro]M. Thus, buffer binding affinity and calcium concentration are inversely proportional to each other.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Fig [6] shows the effect of source amplitude on the calcium profile in the presence of calcium buffering mechanisms when [k.sub.m]=0.32 and [T.sub.B] = 20 [micro]M. As stated in the problem calcium concentration is higher near the source. As source amplitude increases, the concentration of free calcium ion increases in cytosolic. Here, calcium buffer (which are already present in cytosol with concentration = 10 [micro]M) plays in important role in maintain the calcium concentration its background value and also protect against [Ca.sup.2+] toxicity. The same fact has been observed in fig (6) with respect to radius and in fig (7) with respect to angle. In both figure (6) and (7) it is observed that, calcium concentration in cytosol near source is almost proportional to the source amplitude. In this article all the simulation has been taken for [sigma]=0.1 pA

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Fig (8) shows the calcium concentration in angular direction at centre and at circumference of circle. Here, it is assumed that fibroblasts cell is of circular shape. Thus, at centre r = 0[micro]m there is no significant change in [Ca.sup.2+] profile in angular direction, but at circumference r=5 [micro]m, the calcium concentration initially increases till [theta]=[pi] and thereafter decreases till [theta]=2[pi].

References

[1] A. Michailova, F.D. Principe, M. Eggerand, E. Niggli, "Spatiotemporal features of [Ca.sup.2+] buffering and diffusion in artial cardiac Myocytes with Inhibited Sacroplasmic Reticulum", Biophysical J. 83 (2002), pp. 3134-3151

[2] A. Tripathi and N. Adlakha, "Two dimensional coaxial circular elements in FEM to study calcium diffusion in neuron cells", Applied Mathematical Sciences 6(10) (2012), pp. 455-466

[3] B.K.JHA, N. Adlakha, M.N. Mehta, "Solution of advection diffusion equation arising in cytosolic calcium concentration", Internatinal Journal of Applied .Math and Mech. 7(6) (2011), pp. 72-79

[4] De AD Ross, PH Willems, PH Peters, EJ VanZoelen and AP Theuvenet, "Synchronized Calcium Spiking resulting from Spontaneous Calcium Action Potentials in Monolayers of NRK Fibroblasts", Cell Calcium. 22 (1997b), pp. 195-200

[5] De AD Ross, PH Willems, Van EJ Zoelen, and AP Theuvenet, "Synchronized [Ca.sup.2+] signaling mediated by intercellular propagation of [Ca.sup.2+] action potential in monolayer of NRK Fibroblasts", Am J Physiol. 273 (1997c), pp. C1900-1907

[6] E. Neher, "Concentration profiles of intracellular [Ca.sup.2+] in the presence of diffusible chelator", Exp. Brain Res. Ser 14 (1986), pp. 80-96.

[7] EG Harks, JJ Torres, LN Cornelisse, DL Ypey, and AP Theuvenet "Ionic Basis for Excitability in Normal Rat Kidney (NRK) Fibroblasts" AM J Physiol, Cell Physiol. 196 (2003), pp. 493-503

[8] G.D. Smith, "Analytical Steady-State solution to the rapid buffering approximation near an open [Ca.sup.2+] channel", Biophys. J. 71 (1996), pp. 30643072

[9] G. D. Smith, L. Dai, R. M. Miura and A. Sherman, "Asymptotic analysis of buffered [Ca.sup.2+] diffusion near a point source", SIAM J. of Applied of Math 61 (2000), pp.1816-1838

[10] J. Sneyd, K..Tsaneva-Atanasova, J.I. Bruce, S.V. Straub, D.R.Giovannucci and D.I.Yule. "A model of calcium waves in pancreatic and parotid acinar cells", Biophys. J. 85(3) (2003), pp.1392-1405

[11] JJ Torres, LN Cornelisse, EGA Harks, DL Ypey, and AP Theuvenet, "Modelling action potential generation and propagation in NRK Fibroblasts", AM J Physiol, Cell Physiol. 287 (2004), pp. 851-865

[12] J M A M Kusters, M M Dernison, Van W P M Meerwijk, D L Ypey, A P Theuvenet and CCAM Gielen, "Stabilizing Role of Calcium Storedependent Plasma Membrane Calcium Channels in Action Potential Firing and Intracellular Calcium Oscillations",Bio. Phys 89 (2005), pp. 3741-3756

[13] M.Kotwani, N.Adlakha, M.N.Mehta, " Numerical model to study calcium diffusion in fibroblasts cell for one dimensional unsteady state case", Applied Mathematical Sciences paper accepted.

[14] S. Tewari and K.R. Pardasani, "Finite element model to study two dimensional unsteady state cytosolic calcium diffusion in presence of excess buffers", IAENG International Journal of Applied Mathematics 40(3) (2010) ,pp. 40_3_01

[15] S. S.Rao, "The Finite Element Method in Engineering", Elsevier Science &Technology Books, 2004

[16] T. Hofer, A. Politi & R. Hinirich, "Intercellular [Ca.sup.2+] wave propagation through gap junctional [Ca.sup.2+] diffusion: a theoretical study", Biophysical J. 80(1) (2001) , pp. 75-87

[17] V.Matveev, R S Zucker R.S. and A Sherman, "A Facilitation through Buffer Saturation: Constraints on Endogenous Buffering Properties", Biophs. J. 86 (2004), pp. 2691-2709

Mansha Kotwani (+), Neeru Adlakha and M.N. Mehta

Department of Applied Mathematics and Humanities

S. V. National Institute of Technology, Surat

Gujarat-395007, India
Table-I

E    Ie   Je   Ke   Le   E    Ie   Je   Ke   Le

1    1    2    12   13   41   45   46   56   57
2    2    3    13   14   42   46   47   57   58
3    3    4    14   15   43   47   48   58   59
4    4    5    15   16   44   48   49   59   60
5    5    6    16   17   45   49   50   60   61
6    6    7    17   18   46   50   51   61   62
7    7    8    18   19   47   51   52   62   63
8    8    9    19   20   48   52   53   63   64
9    9    10   20   21   49   53   54   64   65
10   10   11   21   22   50   54   55   65   66
11   12   13   23   24   51   56   57   67   68
12   13   14   24   25   52   57   58   68   69
13   14   15   25   26   53   58   59   69   70
14   15   16   26   27   54   59   60   70   71
15   16   17   27   28   55   60   61   71   72
16   17   18   28   29   56   61   62   72   73
17   18   19   29   30   57   62   63   73   74
18   19   20   30   31   58   63   64   74   75
19   20   21   31   32   59   64   65   75   76
20   21   22   32   33   60   65   66   76   77
21   23   24   34   35   61   67   68   78   79
22   24   25   35   36   62   68   69   79   80
23   25   26   36   37   63   69   70   80   81
24   26   27   37   38   64   70   71   81   82
25   27   28   38   39   65   71   72   82   83
26   28   29   39   40   66   72   73   83   84
27   29   30   40   41   67   73   74   84   85
28   30   31   41   42   68   74   75   85   86
29   31   32   42   43   69   75   76   86   87
30   32   33   43   44   70   76   77   87   88
31   34   35   45   46   71   78   79   1    2
32   35   36   46   47   72   79   80   2    3
33   36   37   47   48   73   80   81   3    4
34   37   38   48   49   74   81   82   4    5
35   38   39   49   50   75   82   83   5    6
36   39   40   50   51   76   83   84   6    7
37   40   41   51   52   77   84   85   7    8
38   41   42   52   53   78   85   86   8    9
39   42   43   53   54   79   86   87   9    10
40   43   44   54   55   80   87   88   10   11

Table II: List of physiological parameters used for numerical results

Symbol                      Parameter                  Value

[D.sub.Ca]                  Diffusion                   250
                            Coefficient       [micro][m.sup.2]/second
[k.sup.+] (EGTA)        Buffer association    0.32 [micro][M.sup.-1]
                          rate (Exogenous         [second.sup.-1]
                              buffer)
[[B.sub.m]][infinity]         Buffer                20[micro]M
                           Concentration
[[Ca.sup.2+]]               Background             0.1 [micro]M
 [infinity]                 [Ca.sup.2+]
                           Concentration
[sigma]                  Source amplitude              0.1pA
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Author:Kotwani, Mansha; Adlakha, Neeru; Mehta, M.N.
Publication:International Journal of Computational and Applied Mathematics
Article Type:Report
Geographic Code:9INDI
Date:Jul 1, 2012
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