Abstract: active role in maintaining the structure and function

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.

We Will Write a Custom Essay Specifically
For You For Only $13.90/page!


order now

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).