CLOSED FORM SOLUTION TO PROBLEM OF ADVECTION DIFFUSION OF Ca2+ IN ASTROCYTES Brajesh Kumar Jha1, N. Adlakha2, and M. N. Mehta2 1
Department of Mathematics and Computer Science School of Technology, Pandit Deendayal Petroleum University Raisan, Gandhinagar, Gujarat-382007 2 Department of Applied Mathematics and Humanities S. V. National Institute of Technology, Surat Email:
[email protected] Received 11 December 2013; accepted 6 April 2014
ABSTRACT Astrocytes actively participate in calcium signalling in nervous system. Calcium signalling depends on cytosolic calcium concentration. Calcium itself called second messenger. Calcium ions diffuse into the cell due to concentration difference between synapse and cytosol. This process occurs in almost all type of nerve cells. Due to presence of aqua medium and verity of proteins the cross flow of calcium ion takes place using ficks law. The effect of advection diffusion (cross flow) has been studied in this paper. Other parameters like buffer, endoplasmic reticulum, etc has not been taken in account. Analytic solution of the advection diffusion equation is obtained using Laplace transform. The effect of advection is shown graphically with the help of Matlab. Keywords: Calcium Ion, Advection Diffusion, Analytic Solution
1 INTRODUCTION The mathematical formulation of calcium diffusion in astrocytes yields an initial boundary value problem. Various analytical and numerical techniques can be employed to solve initial boundary value problem. The closed form solutions to these problems are always preferable as they satisfy the continuity conditions of natural phenomena in the same natural form. These closed form solutions are mathematically sound and are useful in visualizing realistic variations from the mathematical point of view. The main objective of this paper is to obtain closed form solution to the initial boundary value problem of calcium distribution in astrocytes and study the effect of advection diffusion on cytosolic calcium concentration in nerve cell like neuron, astrocytes etc. Advection is the process in which diffusion takes place in the cross flow direction. Due to aques region it is considered that the advection diffusion occurs and calcium ion flow in cross flow direction also. A good number of investigations are reported in the literature (Jha B K, Adlakha N, Mehta M N (2011), Smith G D, Wanger J, and Keizer J (1996), Berridge M J (1997), Blaustein M P and Lederer W J (1999), Conn P M (2008)) for the study of calcium diffusion in nerve cells like neuron, astrocytes etc. Smith et. al has shown the nonlinear Int. J. of Appl. Math and Mech. 10 (10): 74-81, 2014.
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advection diffusion in the presence of repid buffers in neuron. Here an attempt has been made to study the effect of adcection diffusion on cytosolic calcium concentration in nerve cells considering the absence of all internal and external process like buffer voltage gated calcium channels etc. Some attempts are also reported in the literature (Barres A, Chun L, and Corey D P (1988), Adler E M, Augustine G J, Duffy S N, and Milton P C (1991), Fischer H B (1979), Brose N, Petrenko A G, Sudhof T C, and Jahn R (1992), Bertram R, Smith G D, and Sherman A (1999)) for the study of calcium diffusion in astrocytes and neurons. Some numerical solution of advection and diffusion equations are obtained using finite volume method (Jha B K, Adlakha N, Mehta M N (2011)). However very few attempts are reported in the literature for study of advection- diffusion of calcium in nerve cells like neuron, astrocytes, etc. In view of above advection diffusion of calcium in cytosol of astrocytes has been investigated for a one dimensional case. The motive of this paper is identify the effect of advection and diffusion on Cytosolic calcium profile in absence of internal force. The difference in calcium profile has been compared with the diffusion of free calcium ion at different speed of flow. The mathematical formulation and analytical solution of the mathematical model is given in next section.
2 MATHEMATICAL FORMULATION Here we consider a chemical species (free calcium ion Ca2+) C whose concentration C(x, t) varies in time and space, the spatial variation is considered in one spatial variable x only (Keener J P (2002). Conn P M (2008)). This situation is shown in Figure 1, where the chemical species C is contained in a long, thin tube with constant cross-sectional area A. The conservation of C can be expressed in words as (Keener J P (2002).): Rate of change of the total amount of C within R w. r. t time = rate at which C flows in to R − rate at which C flows out of R + rate at which C is generated within R − rate at which C is consumed within R.
Figure 1: Mass transfers in domain
We consider the cytosol of nerve cells of bar shape to investigate one dimensional advection diffusion of calcium. Also it is assumed that all the internal and external process like buffering, voltage gated channels, internal source flux etc are absent. The advection diffusion equation for one dimensional case in cartesian coordinate is given by: ∂C ∂ (uC ) ∂ 2C + = DCa ∂t ∂x ∂x 2
C= ( x, t ) [Ca 2+ ] − [Ca 2+ ]∞ Initial and boundary conditions are given by: Int. J. of Appl. Math and Mech. 10 (10): 74-81, 2014.
(1)
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d [C ] lim DCa = σ Ca x →0 dx
for t > 0
(2)
lim [C ] = 0
for t > 0
(3)
x →∞
C ] t =o [=
0
for 0 ≤ x ≤ 5
(4)
We employ similarity transformation in equation (1). Let us consider the coordinate transforms given by (Crank J (1975)):
η =x − ( x0 + ut )
(5)
and
t =t
(6)
Where η is moving reference frame spatial coordinate For the sake of convenience, we assume origin as a mouth of calcium channel in cytosol (plasmamembrane) i.e. x0 =0 is the injector point of tracer. u is mean velocity of flow of Ca2+ and ut is distance travelled by the centre of mass of cloud at time t. This coordinate transform (5) and (6) can be substituted in (1) using chain rule as follows: ∂C ∂t ∂C ∂η ∂ ∂t ∂C ∂t ∂C ∂η ∂ ∂η ∂C ∂t ∂C ∂η + + u + + + = DCa ∂t ∂t ∂η ∂t ∂t ∂x ∂η ∂x ∂t ∂x ∂η ∂x ∂t ∂x ∂η ∂x
(7)
Which reduces to
∂C ∂ 2C = DCa ∂τ ∂η 2
(8)
Which is one dimensional diffusion equation in the coordinates of η and τ . For co n v ience Ca 2+ = C
Applying transformations on initial and boundary conditions (2) we have: dC lim DCa= σ Ca τ > 0 η →0 dη
= lim C C∞ η →∞
τ >0
and Int. J. of Appl. Math and Mech. 10 (10): 74-81, 2014.
(9)
(10)
Closed Form Solution To Problem Of Advection Diffusion Of Ca2+ In Astrocytes
0 ≤η ≤ ∞
= Cτ o C∞ =
77
(11)
For the instantaneous point source of calcium concentration the solution of (8) is obtained using Laplace transforms (Keener J P (2002), Berridge M J (1997)). Applying Laplace transform on (8) and using initial condition (11) d 2C s − C= 0 2 dη Dca
(12)
Applying Laplace transform along boundary conditions we get: dC s (0, s ) = dn DCa s
(13)
C lim (n, s) = 0
(14)
n →∞
The solution of equation (12) is given by:
C (η , s ) c1e =
s
DCa
η
+ c2 e
−
s
DCa
η
(15)
The c1 and c2 are obtained by using boundary conditions (13) & (14) and are as given below:
s DCa s
= c1 0= and c2
(16)
Substituting expressions (16) in expression (15) we get: C (η , s ) =
− s e DCa s
s
DCa
η
(17)
Taking inverse Laplace transform of equation (17), we get: = C (η , t )
1 4p DCat
η 2 − exp 4 DCat
(18)
Using transformations (5) and (6) in equation (18), we get: = C ( x, t )
( x − ut )2 exp − A 4p DCat 4 DCat M
(19)
where A is constant of integration, x0 = 0 , M is mass which the product of concentration C and region ∆x . Int. J. of Appl. Math and Mech. 10 (10): 74-81, 2014.
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3 RESULTS AND DISCUSSION The numerical values of biophysical parameters used for computation of results are given below in Table 1.
Symbol DCa U
Table 1: List of Physiological Parameter Parameters Values Symbol 2 Diffusion 200-300 µm /s DCa Coefficient Speed of flux 10-40 µm/s U of Ca2+ ion
Figure 2 & 3 shows the calcium distribution along X- direction at different points of time step 1ms and 10 ms in the presence and absence of advection diffusion. In Figure (2) calcium concentration [Ca2+] decrease sharply up to x = 1µm after that it achieves the background concentration 0.1 µM. Initially at t = 1 ms the behavior of calcium profile is almost same but in Figure (3) the effect of advection diffusion can be seen at t = 10 ms. Calcium concentration decrease rapidly up to x= 2.5 µm and then attains the background concentration beyond x=2.5 µm. In case of advection diffusion calcium concentration first increases near the mouth of the channel up to 0.5µm then decrease rapidly up to 2.5 µm and thereafter that attains the background concentration. The change in calcium profile occurs due to the diffusion of calcium ion in cross flow direction. Thus the effect of advection diffusion is significant which is evident from Figure (3). The Figure 4 shows the variation in calcium concentration at source x=0 with respect to time in presence of advection. It is observed that the calcium concentration is high initially at the mouth of the channel then it falls sharply from 4.723 µM to 0.423 µM during t=0 to t=0.08 seconds and thereafter it achieves steady state. Figure 5 shows variation in calcium concentration along X direction at different points of time t=1 ms, 5 ms, 10 ms and 15 ms. It is observed that the peak calcium concentration is at source decrease with increase in time and the calcium concentration also falls down along X direction. This is due to the fact that with passage of time reduces due to advection diffusion along X direction. The results obtained here are in agreement with the biological facts.
4 CONCLUSION The closed form solution to problems of advection diffusion of calcium in astrocytes was obtained for a one dimensional unsteady state case. These models gave us the fair idea of effect of advection in absence of internal process. Such models can be developed further to study the relationships among various biophysical parameters like buffers, pumps, leaks, gates, source influx, diffusion coefficients etc and the effect of advection diffusion on calcium distribution in presence of internal process.
Int. J. of Appl. Math and Mech. 10 (10): 74-81, 2014.
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5 FIGURES t = 1 ms 5 Diffusion Advection Diffusion
4.5
Ca- Concentration (µM)
4 3.5 3 2.5 2 1.5 1 0.5 0
0
0.5
1
1.5
3 2.5 2 X - direction (µm)
3.5
4
4.5
5
Figure 2: Calcium distribution in cytosol along x direction at different points of time step in presence and absence of advection on time t = 1ms
t = 10 ms 1.6 Diffusion Advection diffusion
1.4
Ca-Concentration (µM)
1.2 1 0.8 0.6 0.4 0.2 0
0
0.5
1
1.5
3 2.5 2 X direction (µm)
3.5
4
4.5
5
Figure 3: Calcium distribution in cytosol along x direction at different points of time step in presence and absence of advection on time t = 1ms Int. J. of Appl. Math and Mech. 10 (10): 74-81, 2014.
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near the source 5 4.5
Ca- concentration (µM)
4 3.5 3 2.5 2 1.5 1 0.5 0
0
0.01
0.02
0.03
0.04
0.05 0.06 time (s)
0.07
0.08
0.09
0.1
Figure 4: Calcium distribution in cytosol with respect to time at the source, x=0
5 t=1 ms t=5 ms t=10 ms t=15 ms
4.5
Ca-concentration (µM)
4 3.5 3 2.5 2 1.5 1 0.5 0
0
0.5
1
1.5
3 2.5 2 X-distance in (µm)
3.5
4
4.5
5
Figure 5: Calcium distribution in cytosol along X direction for different values of time step in presence of advection Int. J. of Appl. Math and Mech. 10 (10): 74-81, 2014.
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