Abstract:

The hepatocyte cells are building blocks of the liver. The calcium dynamics in

a hepatocyte cell is responsible for maintaining structure and functions of the

hepatocyte cell and liver. Any disturbance in calcium dynamics in hepatocyte

cell can cause disturbance in structure and functions of the liver. The calcium

dynamics in hepatocyte cell is still not well understood. In this paper a model is proposed to study one dimensional

calcium dynamics in a hepatocyte cell. The parameters like excess buffers,

source influx, and diffusion coefficient have been incorporated in the model,

which is expressed in the form of reaction diffusion equation. Finite volume

method has been employed for the solution of the problem. The numerical results

have been computed and used to study the effect of endogenous buffer and

exogenous buffers on calcium dynamics in a hepatocyte cell.

Keywords: Buffer; Hepatocyte

cell; Calcium Dynamics; Finite volume method.

1. Introduction

The liver is

one of the sophisticated organs of human and animal body which performs many

essential functions for proper digestion, metabolism, immunity, and storage of

nutrients within the body 1. This liver is made up of hepatocyte cells. These

cells play an active role in maintaining the structure and function of liver.

The hepatocyte cell is non excitable cell and depends on chemical signaling to

achieve its functions. The Ca 2+ plays an important role in almost all activities of hepatocyte cell. The

concentration of Ca 2+ in the cell depends on various conditions and

the requirement of cell in order to initiate, sustain and terminate the

activity of cell. The concentration of Ca 2+ depends on various

parameters like buffers, influx, out flux and calcium transport in the cell 2.

The number of attempts are reported for

the study of calcium dynamics in various cells like neuron 3-8, acinar 9-11, astrocyte 12,13, myocyte 14,15, oocyte 16-19, and

fibroblast 20,21 etc. No attempt is reported in the literature to

study calcium dynamics in hepatocyte cell in presence of excess buffers. In

this paper an attempt has been made to propose a finite volume model to study

calcium dynamics in a hepatocyte cell in presence of excess buffer. The

mathematical formulation is presented in next section.

2. Mathematical Formulation

The

calcium dynamics in hepatocyte cell is governed by following set of reaction

diffusion equation. If we assume that there are n buffers present inside the

cell then calcium buffer reaction is 22-24,

(1)

Here, Ca 2+ is free calcium ion binds with ith

buffer Bi to form calcium bound buffer CaBi.

The

system of calcium buffer reaction diffusion can be written as follows by using

law of mass action and Fick’s law of diffusion 25,26,

(2)

(3)

(4)

Where,

(5)

are diffusion coefficient of free calcium, free buffer and

calcium bound buffer respectively;

are association and dissociation rate constants for given

buffer respectively. Square bracket represents concentration of species

enclosed in it. We assume that total buffer concentration remains conserved.

Therefore total concentration of ith

buffer

is given by,

(6)

Since diffusion coefficient of most of calcium binding species is not

affected by binding of Ca 2+ to it because of smaller molecular

weight of Ca 2+ in

comparison to most Ca 2+ binding species. Therefore we have,

. Using Eq. (6) in Eq. (2) and

adding Eq.(3) and Eq. (4) we get,

(7)

(8)

Where,

By assuming single buffer species

in excess and at equilibrium setting reaction term equal to zero gives

concentration of excess buffer

as,

,

Where,

.

Using

it in Eq. (7) we get,

(9)

Now, the third term on RHS of Eq.

(9) is approximated in term of equilibrium calcium concentration

as follows,

(10)

Thus Eq. (9) can be put in the form,

(11)

It is general model equation,

describing calcium diffusion in presence of excess buffer.

For one dimensional unsteady state

case Eq. (11) reduced in the form,

(12)

2.1. Initial

condition

It is assumed that initially before

opening the gate of calcium releasing channel, equilibrium concentration of

calcium is 0.1

2.

(13)

2.2. Boundary conditions

The calcium releasing channels are

located near apical region of hepatocyte cell 2,27. Therefore it is assumed

that, blip is produced from point source kept at node 1 located at x=0.

Thus first boundary condition can be framed as,

(14)

Where,

represents flux of calcium incorporated on

left boundary.

The calcium concentration near to

basal region of hepatocyte cell is assumed to attain background equilibrium

concentration 0.1

10. Thus, second boundary condition can be

framed as,

(15)

Ca 2+ tends to the

background concentration of

as

but here the domain taken is a hepatocyte cell

of length 15

. Therefore the length of hepatocyte cell is

taken as distance required for Ca 2+ to attain background

concentration.1,27,28.

3. Solution

The hepatocyte cell is discretized into

discrete control volumes, in order to apply finite volume method 29 as shown

in Fig. (1).

Fig.1. One dimensional discretization of hepatocyte cell.

The control volumes are the subintervals of

the problem interval in one dimensional problem and the nodes are centers of

those subintervals.

The space between two boundaries kept at A and B is uniformly discretized by

taking 30 nodal points separated by equal distance

.

Where, node 1 and node 32 represents the boundary nodes. The control volume is

considered around each node. Let G be a general nodal point in a control volume

and W and E are its neighboring nodes to west and east respectively. Also w and

e denotes west side face and east side face of control. The

and

are

distances between the nodes W and G, and between nodes G and E respectively.

Similarly

and

are the distances between face w and point G

and between G and face e respectively.

Eq.(12)

can be written in the form,

(16)

Where, C is taken for convenience

instead of

.

Dividing both sides of Eq. (16) by

we get,

(17)

Where,

.

Integration of Eq. (17) over the

control volume with respect to time and space gives,

(18)

For simplification we consider,

(19)

The

simplification of terms

is as follows,

(20)

Where,

and

represents the concentration of calcium at

general point G at time

and

respectively.

(21)

To evaluate concentrations at

combination of time

and

we take weighing parameter

which lies in between 0 and 1 in Eq. (21) we

get,

(22)

Also,

(23)

(24)

Using Eq.

(20-24) in Eq. (19) we get,

(25)

Now rearranging the Eq. (25) we

get,

(26)

It can be written in general form

as follows,

(27)

Where,

,

,

,

,

To apply Crank Nicolson method, putting

in Eq. (26)

and assuming

gives

following general form of equation for all internal nodes from 3 to 30.

(28)

Where,

,

,

,

,

Now we apply the boundary conditions at

nodes 2 and 31. At node 2 west control volume boundary is kept at specified

concentration

and

therefore

from Eq. 28 we get,

(29)

Where,

,

,

,

,

Similarly at

node 31, east control volume boundary is kept at specified concentration

and

, therefore from Eq. 28 we get,

(30)

Where,

,

,

,

,

Using all the above equations, (28-30),

we get a system of algebraic equations as follows,

(31)

Here

represents the calcium concentrations at

respective nodes, P is system matrix and Q is system vector.

MATLAB R2014a software is used to develop

a program to find numerical solution to whole problem. The time step taken for

simulation is

The numerical values of biophysical

parameters are given in the Table 1.

Table 1. Numerical

values of biophysical parameters 24

Symbol

Name of parameter

Value

Diffusion coefficient

200-300

Total buffer

concentration

50

for EGTA buffer

Buffer association

rate constant

1.5

for EGTA buffer

Buffer

dissociation constant

0.2

for endogenous buffer

Buffer association

rate constant

50

for

endogenous buffer

Buffer

dissociation constant

10

for BAPTA buffer

Buffer association

rate constant

600

for BAPTA

buffer

Buffer

dissociation constant

0.17

4. Results and Discussion

The numerical results have been

obtained by using the biophysical parameters given in Table 1.

Fig. 2. Spatial

variation of calcium concentration in presence of EGTA buffer

Fig. 2 shows the effect of

concentration of EGTA buffer on calcium concentration in a hepatocyte cell. It

is observed that, concentration of calcium is maximum at source kept at x=0. It decreases gradually away from

source and attains background equilibrium concentration 0.1

.

In absence of buffer the nodal calcium concentration is maximum, but it

decreases with increase in value of buffer concentration. Also equilibrium

calcium concentration is achieved in short distance in presence of high buffer

concentration. In the initial period of

time the nodal calcium concentration is observed minimum. But as time progress

it increases gradually to attain steady state condition. The steady state

condition is attained earlier in presence of higher value of buffer

concentration.

Fig. 3. Spatiotemporal

variation of calcium concentration in presence of EGTA buffer

Fig. 3 shows temporal variation

of calcium concentration at different nodal points. It is observed that, the

nodal calcium concentration is 0.1

at

sec. It increases gradually with

time to attain steady state concentration. The steady state concentration is

observed maximum at nodes in the vicinity of source and it decreases away from

source. The nodal

steady state calcium concentration decreases with increase in buffer

concentration.

Fig. 4,

Spatiotemporal variation of calcium concentration in presence of Endogenous

buffer

Fig. 4 shows the effect of

endogenous buffer on nodal calcium concentration with respect to time. The

oscillation of calcium concentration has been observed in presence of

endogenous buffer at node 1 and node 2. There is no any change in calcium

concentration for remaining nodes. This is because of high binding capacity of

endogenous buffer than the source influx. The maximum nodal concentration is

observed in presence of minimum value of endogenous buffer concentration. It

decreases with increase in value of endogenous buffer concentration. The time

required to attain steady state calcium concentration increases with increase

in endogenous buffer concentration.

Fig. 5. Spatiotemporal

variation of calcium concentration in presence of BAPTA buffer

Fig. 5 shows the effect of BAPTA

buffer on nodal calcium concentration with respect to time. The oscillation of

calcium concentration is observed in presence of BAPTA buffer at node 1 only,

for all remaining nodes calcium concentration is observed to be 0.1

. The binding rate of BAPTA buffer

is higher than all assumed buffers. It binds with calcium as soon as calcium is

released from calcium channel. Therefore, the oscillations of calcium

concentration have been observed at only a node nearer to source.

5. Conclusions

A finite volume model is proposed

and successfully employed to study effect of excess buffers like EGTA buffer,

endogenous buffer, BAPTA buffer on spatiotemporal calcium concentration profile

for one dimensional case. On the base of obtained results it is concluded that,

the BAPTA buffer have most significant effect than EGTA in reducing the calcium

concentration in a hepatocyte cell. Thus buffers play an important role in

reducing calcium concentration under various condition in which the calcium concentration

becomes high in the cell during particular activity. The high levels of calcium

concentration in the cell for longer periods can cause cell death. Thus these

buffers protect the cell in such conditions by reducing the calcium

concentration. The finite volume method has proved to be quite versatile in

incorporating the parameters and obtaining interested 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.

References

1. G. Dupont, S. Swillens, C. Clair, T. Tordjmann, L. CombettesL, Hierarchical organization of calcium signals in hepatocytes: from experiments to models, Biochimica et Biophysica Acta (BBA)-Molecular Cell Research. 1498 (2000)134-152.

2.

G.J. Barritt, Calcium: The Molecular Basis of Calcium Action in

Biology and Medicine. Springer. (2000) 73-94.

3.

A. Jha, N. Adlakha,

Two-dimensional finite element model to study unsteady state Ca 2+

diffusion in neuron involving ER LEAK and SERCA, International Journal of

Biomathematics. 8 (2015) 1550002.

4.

A. Jha, N. Adlakha, Finite element model

to study the effect of exogenous buffer on calcium dynamics in dendritic spines, International

Journal of Modeling, Simulation, and Scientific Computing. 5 (2014) 1350027.

5.

A. Jha, N. Adlakha, Analytical solution of two dimensional unsteady state

problem of calcium diffusion in a neuron cell, Journal of medical imaging and

health informatics. 4 (2014) 547-553.

6.

A. Jha, N. Adlakha, Finite element model to study effect of Na+

? Ca 2+ exchangers and source geometry on calcium dynamics in a

neuron cell,

Journal of Mechanics in Medicine and Biology 16 (2016) 1650018.

7.

S. Tewari, K.R. Pardasani, Finite difference model to study the effects of Na+

influx on cytosolic Ca 2+

diffusion, International Journal of

Biological and Medical Sciences. 1 (2009) 205-210.

8.

S. Tewari, 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 (2010) 108-112.

9.

N. Manhas, K.R. Pardasani, Mathematical model to study IP3 dynamics dependent

calcium oscillations in pancreatic acinar cells,

Journal of Medical Imaging and Health Informatics. 4 (2014) 874-880.

10. N.

Manhas, K.R. Pardasani, Modelling mechanism

of calcium oscillations in pancreatic acinar cells, Journal of

bioenergetics and biomembranes. 46 (2014) 403-420.

11. N.

Manhas, J. Sneyd, K.R. Pardasani, Modelling the

transition from simple to complex Ca2+ oscillations in pancreatic

acinar cells. Journal

of biosciences; 39 (2014) 463-484.

12. B.K.

Jha, N. Adlakha,

M.N. Mehta, Two-dimensional

finite element model to study calcium distribution in astrocytes in presence of

excess buffer, International Journal of

Biomathematics. 7 (2014) 1450031.

13. B.K.

Jha, N. Adlakha,

M.N. Mehta, Two-dimensional

finite element model to study calcium distribution in astrocytes in presence of

vgcc and excess buffer,

Int. J. Model. Simul. Sci. Comput. 4 (2012) 1250030.

14. K.

Pathak, N.

Adlakha, Finite element model to Study Two Dimensional

unstedy state calcium distribution in cardiac myocytes, Alexandria Journal of

Medicine. 52 (2016) 261-268.

15. K.

Pathak, N.

Adlakha, Finite Element Model

to Study Calcium Signaling in Cardiac Myocytes Involving Pump, Leak and Excess

Buffer, Journal

of Medical Imaging and Health Informatics. 5 (2015) 1-6.

16. P.A.

Naik, K.R. Pardasani, One Dimensional

Finite Element Model to Study Calcium Distribution in Oocytes in Presence of

VGCC, RyR and Buffers,

J. Medical Imaging Health Informatics. 5 (2015) 471-476.

17. P.A.

Naik, K.R. Pardasani, One dimensional

finite element method approach to study effect of ryanodine receptor and serca

pump on calcium distribution in oocytes, Journal of Multiscale Modelling. 5 (2013)

1350007.

18. S.

Panday, K.R. Pardasani, Finite element model

to study effect of advection diffusion and Na+/Ca2+

exchanger on Ca2+ distribution in Oocytes.

Journal of medical imaging and health informatics. 3 (2013) 374-379.

19. S.

Panday, K.R. Pardasani, Finite element model

to study the mechanics of calcium regulation in oocyte, Journal of Mechanics in Medicine and

Biology. 14 (2014) 1450022.

20. M.

Kotwani, N.

Adlakha, Modeling of endoplasmic reticulum and plasma membrane

Ca 2+ uptake and release fluxes with excess buffer approximation

(EBA) in fibroblast cell,

International Journal of Computational Materials Science and

Engineering. 6 (2017) 1750004.

21. M.

Kotwani, N.

Adlakha, M.N. Mehta, Finite element model

to study the effect of buffers, source amplitude and source geometry on

spatio-temporal calcium distribution in fibroblast cell. Journal

of Medical Imaging and Health Informatics. 4 (2014) 840-847.

22. B. Schwaller, Cytosolic Ca 2+

Buffers. Cytosolic Ca 2+ Buffers. 2 (2010): a004051.

23. M. Falcke, Buffers and

Oscillations in Intracellular Ca 2+ Dynamics, Biophysical journal. 84 (2003), 28-41.

24. J.

Keener, J. sneyd, Mathematical Physiology:I:Cellular Physiology, Springer

Science & Business media (2010).

25.

C.P. Fall, Computational Cell

Biology: Interdisciplinary Applied Mathematics, Springer-Verlag New York

Incorporated (2002).

26. A. Sherman, G.D.

Smith, L. Dai, R.M. Miura, Asymptotic

analysis of buffered calcium diffusion near a point source, SIAM Journal on

Applied Mathematics. 61 (2001) 1816-1838.

27. A.P. Thomas, D.C.

Renard, T.A. Rooney,

Spatial and temporal organization of calcium signaling in hepatocytes, Cell

Calcium. 12 (1996) 111-126.

28. D.E. Clapham, Calcium Signaling

Review, Cell 80 (1995) 259-268.

29. H.K. Versteeg and W. Malalasekera, An introduction to computational

fluid dynamics the finite volume method, Longman, Londres (1995).