FINITE VOLUME SIMULATION OF TWO DIMENSIONAL CALCIUM DYNAMICS IN A HEPATOCYTE CELL INVOLVING BUFFERS AND FLUXES

The liver plays vital role in various activities like digestion, metabolism etc. in human body. The proper structure and functions of liver relies on its calcium homeostasis. The hepatocyte cell is a building block of liver. The calcium homeostasis in liver depends on intracellular calcium dynamics in a hepatocyte cell. The specific calcium levels are maintained for various activities in the hepatocyte cell. This calcium level depends on the processes of various influxes, effluxes and buffering mechanism. Any disturbances in these processes may lead to impaired function of the cell and cause disorder in function of liver. In this paper the mathematical model is proposed to study the effect of buffer and various fluxes like channel flux, leak and pump in a hepatocyte cell. The finite volume method for unsteady state case is implemented to obtain the solution of proposed problem in two dimensions. The effect of various physiological factors like concentration of exogenous and endogenous buffers, leak flux constant, channel conductance, pump rate, diffusion coefficient of medium on calcium concentration is demonstrated.


Introduction
Largest internal gland in human body is liver which is made up of parenchymal hepatocyte cells.Varieties of proteins necessary for initiation, sustentation and termination of basic cellular activities are secreted by hepatocyte cell.The functioning of liver is controlled by calcium signaling in hepatocyte cells.The concentration of free calcium ions plays crucial role in normal functioning of cell.Cell needs to keep calcium level inside the reference range from 0.1µM to 1µM to maintain normal calcium homeostasis [1].The main source of calcium in hepatocyte cell is endoplasmic reticulum (ER).The calcium releases either from the gate of calcium channels embedded on ER membrane or small leakage through ER membrane.The calcium channels are abundant near apical surface of cell [2].The released calcium from gate of calcium channel diffuses in cytosol of cell in presence of buffers to attain equilibrium calcium concentration towards basal surface of cell [3,4].The cyclic movement of calcium from ER to cytosol and viceversa is shown in Figure 1.The buffer binds with free calcium to minimize FIGURE 1.The calcium fluxes in a hepatocyte cell the excess amount of free calcium.The SERCA pumps also flushes free calcium ions back into ER.Thus to maintain balanced level of calcium inside cell, there must be fine co-ordination in many physiological factors like quantity and type of buffers, channel fluxes, leakage, pumping rate, diffusion coefficient etc. [5].

Mathematical Model
The partial differential equation describing calcium dynamics in a hepatocyte cell in presence of excess buffer incorporating calcium fluxes is given by [22], The numerical values of biophysical constants are summarized in Table 1 [5,22].

Initial condition
The background calcium concentration in hepatocyte cell before opening calcium channel gate is 0.1µM [1].With this assumption initial condition can be framed as, (2) C t=0 = 0.1µM

Boundary condition
The calcium releasing channels are concentrated near apical surface of hepatocyte cell [2].
Therefore it is considered that calcium is released from a gate of calcium channel kept at node 3, situated near midway of apical surface at point (0, 10).Therefore the first boundary condition can be set by using Ficks law of diffusion as [5]; (3) Where, σ C is influx of calcium from calcium channel.
The calcium concentration attains its equilibrium concentration 0.1µM far away from calcium channel i.e near basal surface of the hepatocyte cell.The hepatocyte cell is cubical in shape having length of approximately 20µm [1].Therefore to set second boundary condition it is assumed that the distance required to attain equilibrium calcium concentration is length of hepatocyte cell. (4)

Solution by finite volume method
In the first step hepatocyte cell is discretized using uniform grid having 25 nodal points as shown in Figure 2.

FIGURE 2. Two dimensional discretization of hepatocyte cell
For simplification we convert nonlinear partial differential Eq.( 1) into linear form.For this we consider following two cases [20].

General equation for Case-I and Case-II
The Eq. ( 6) and Eq. ( 8) can be written in general form for case I and case II as, Where, a = a 1 , b = b 1 for case-I and a = a 2 , b = b 2 for case-II.
Integrating Eq.( 9) with respect to time and space over a control volume gives [23], Solving space integration we get, Now for solving time integral we use weighted parameter θ which lies between 0 to 1.
Where the values of C at time t are super scripted with 0. Rearranging Eq.( 12) gives, To apply Crank Nicolson method we put θ = 1/2 in Eq.( 13), For all internal nodes, Eq.( 14) can be put in the form, (15) a Where, Where, Incorporating second boundary condition at node 1 by setting C ∞ = 0.1 and a W = a S = 0 for cutting link with boundary we get, Where, Similarly boundary condition can be incorporated at nodes 5, 21 and 25.The system of linear algebraic equations obtained in Eq.( 15) to Eq. ( 17) at each time step can be put in matrix form as follows; (18) Where, [A] is system matrix, [C] is concentration vector and [B] is constant vector.The Eq.( 18) is solved at each iteration to obtain solution vector [C] 25×1 .The numerical simulations has been done in MATLAB 2014a, to obtain the solution.

Results and Discussion
The effect of change in EGTA buffer concentration i.e.
[EGTA] on calcium concentration is shown in Figure 3 for case-I and Figure 4 for case-II.In both cases it is observed that with increase in value [EGTA] from 50µM to 150µM the concentration of free calcium decreases simultaneously.In case-I where K Pump >> C concentration of free calcium is less in comparison with that for case-II where K Pump << C.This is because of larger value of K Pump sequesters maximum amount of calcium back into ER.The equilibrium calcium concentration is slightly higher in case-II.The profile of calcium concentration in presence of endogenous buffer is as shown in Figure 7 for case-I and case-II respectively for steady state.The calcium concentration in presence of endogenous buffer is much less than that in case of EGTA buffer.This is because of high binding capacity of endogenous buffer.It captures free calcium ions instantly as compared to EGTA.Therefore it binds with calcium ions as soon as they released from channel gate.This leads to decrease in value of nodal calcium concentration.that it increases gradually and becomes almost constant.This is due to the equilibrium among the processes which have been achieved by the cell after 100msec.It is also observed that the difference is maximum in case-I than in case-II.This is because in case-I K pump is much higher than that in case II.

Conclusion
A finite volume model is proposed and successfully employed to study effect of excess buffers like EGTA buffer, endogenous buffer, BAPTA buffer, channel flux, leak and pump on variation of calcium concentration in a hepatocyte cell for two dimensional case.On the basis of results obtained, it is concluded that, the BAPTA buffer has more significant effect than EGTA and endogenous buffer in reducing the calcium concentration in a hepatocyte cell.Thus quantity and type of buffers play an vital role in reducing calcium concentration under various condition in which the calcium concentration becomes high in the cell during particular activity.Also the SERCA pump helps to collect calcium ions back into ER to maintain balanced range of calcium.The high levels of calcium concentration in the cell for longer periods can cause cell death.Thus these buffers and SERCA pumps protect the cell in such conditions by reducing the calcium concentration.The results obtained here are in agreement with biophysical facts, however no such experimental results are available for comparison.The finite volume method has proved to be quite versatile in incorporating the parameters and obtaining interesting results.
The information of spatiotemporal calcium profiles under various conditions can be generated from such models and can be useful to clinical applications in detection and treatment of diseases related to liver.
Incorporating first boundary condition at node 3 by setting, C W = σ C we get,(16)

FIGURE 3 .
FIGURE 3. Case-I.Variation of calcium in space in presence of EGTA buffer at time t=100 ms

FIGURE 7 .
FIGURE 7. Case-I and II.Variation of calcium in space at time = 100 msec in presence of [Endogenous buffer]= 50 µM

FIGURE 8 .
FIGURE 8. Case-I and II.Variation of calcium in space at time = 100 msec in presence of [BAPTA]= 50 µM

FIGURE 9 .
FIGURE 9. Case-I.Variation of calcium in space at time = 100 msec with pump and without pump

Figure 11 FIGURE 11 .
Figure 11 shows the difference in values of calcium concentration with pump and without

FIGURE 12 .
FIGURE 12. Case-I.Variation of calcium in space at time = 100 msec with increase in value of diffusion coefficient

TABLE 1 .
Numerical values of biophysical constants.