Abstract: the body. Therefore tissues of the body

Abstract:

Objective: To
study variation of calcium concentration in a hepatocyte cell in presence of
excess buffer. Methods/Analysis: The parameters like excess buffers, reaction diffusion etc. have been
incorporated in the model in the form of boundary value problem for one
dimensional steady state case. The boundary conditions have been framed using
physical conditions of the problem. The finite volume method has been employed
to obtain the solutions. The numerical results have been computed using MATLAB
2014a and used to study the effect of excess buffer on calcium concentration in
hepatocyte cell. Findings: The nodal concentration of calcium is maximum in presence
of EGTA buffer, and minimum in presence of BAPTA buffer. It decreases with
increase in buffer concentration and diffusion coefficient, and increases with
increase in source influx. Novelty: The
finite volume method is employed to study calcium dynamics in a hepatocyte cell.
It gives the desired results as found in literature.

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Keywords:
Calcium concentration, Buffer, Hepatocyte cell, Finite Volume Method.

1.     Introduction:

The liver is a largest internal gland in human body. It performs many
essential functions related to digestion, metabolism, immunity, and the storage
of nutrients within the body. Therefore tissues of the body cannot survive
without proper working of liver.
Hepatocyte cell is parenchymal cell of liver.  The proper working of liver depends upon the
fine coordination of calcium level in hepatocyte cell. Intracellular calcium
signaling regulates varieties of function performed by hepatocyte cell1.
The calcium binds with many proteins and modifies their enzymatic properties.
Thus cell need to keep calcium concentration in range from 0.1µM to at most 1µM
1,2. The source influx of calcium and buffers play an important role
in this calcium regulation in the cell. Buffers are large proteins that soak up
nearly 99 % of calcium 3,4. The buffers associate calcium ions to
reduce the calcium concentration in the cell. The regulation of calcium
concentration in a hepatocyte cell is still not well understood. In this paper
a model is proposed to study the calcium concentration distribution in a
hepatocyte cell. Such models can be developed to generate information of
calcium concentration in hepatocyte cell, which can be useful for developing
protocol for proper health care of liver.

 

        The attempts have been
made in the past to study calcium distribution in various cells like neurons 5-10,
myocyte 11,12, oocytes 13-16, astrocyte 17,18,
fibroblast 19 and acinar cells 20-22 under various
conditions. However, very little attention has been paid to the study of
calcium distribution in hepatocyte cell. In the present paper a finite volume
model is proposed to study one dimensional calcium concentration distribution
in hepatocyte cell for a steady state case. The parameters like excess buffer,
source influx and diffusion of calcium has been incorporated in the model. The
study has been carried out in presence of exogenous buffers and endogenous
buffer. The mathematical formulation and solution is presented in next section.

2.    
Mathematical Formulation:

If we assume that there are ‘n’ numbers of buffer species present
inside the cell then calcium buffer reaction is given by,

                                                            
                                                                                                              

                       

 
(1)

                                                                                                                                               

Assuming isotropy and homogeneity in the medium holds and using law of
mass action and Fick’s law of diffusion 23, system of reaction
diffusion equations which describes Ca 2+ buffering can be written
as follow 3,24,

 
(2)

 
(3)

 
(4)

Where,

(5)

                                                                                                            
                             are diffusion
coefficients of free calcium, free buffer and calcium bound buffer
respectively;are association and dissociation rate constants for ‘i th’
buffer respectively. Concentrations are represented by square brackets.
Assuming that total buffer concentration remains conserved i.e.,

 

Since Ca 2+  has
smaller molecular weight in comparison to most Ca 2+ binding
species, the diffusion constant of each mobile buffer is not affected by the
binding of Ca 2+. Therefore we have,

 

Using equation (5) in equation (2) and adding equation (3) and (4) we
get,

 
(7)

 
(8)

Where,

By assuming single buffer present in
excess i.e.,

Using it in equation (7) we get,

 
(9)

Now, the third term on RHS of equation
(9) is approximated by,

(10)

                                                                                                                                        

Thus the equation (9) can be rewritten
as,

 
(11)

For one dimensional steady state
condition equation (11) reduced in the form,

 
(12)

2.1.        
Boundary
Conditions:

Blip is produced from point source of calcium which is situated at the
node 1 located at x=0 near apical region of hepatocyte cell 25. The
source term is modeled as given in equation (13). The calcium concentration
near to basal region assumed to be background equilibrium concentration 0.1 µM
away from source 26. Thus with this assumptions boundary conditions
can be framed as,

(13)

(14)
 

represent the flux of Ca 2+
incorporated on the boundary and Ca 2+ tends to the background
concentration of 0.1 µM as x tends to ? but here the domain taken is a cell of
finite length. Therefore the distance required for Ca 2+ to attain
background concentration is taken as the length of hepatocyte cell 15µm 27,
5.

2.2.        
Solution:

In order to apply finite volume method 28 the domain is
divided into discrete control volumes as shown in Figure (1). For one
dimensional problem the control volumes are the subintervals of the problem
interval and the nodes are midpoints of that
subintervals.

Figure 1: One dimensional discretization of domain.

The space between A and B
is discretized by taking 30 nodal points separated by equal distance. Node 1 and 32 represents the boundary nodes. Each node is surrounded
by a control volume or a cell. A general nodal point is represented by G and
its neighboring nodes in one dimensional geometry, the nodes to west and east
are denoted by W and E respectively. The west side face of control volume is
referred by w and east side control volume face by e. The distances between the
nodes W and G, and between nodes G and E are identified by  and . Similarly the distances between face w and point G and
between G and face e are denoted by  and   respectively.

Equation (12) can be
written in the form,

 
(15)
 

 

 

                                                                                                                                       

Where C is taken for convenience in lieu of Ca 2+.
Rearranging equation (15) in the general form, we get,

(16)

Where, . Integration of equation (16) over the control volume
gives 34,

 
(17)
 

 For one
dimensional domain we consider. Thus equation (17) can be written as,

 
(18)

As regular structured grid
is considered we have  and therefore,

 
(19)

Rearranging equation (19)
gives,

 
(20)

 

As nodes are separated
uniformly we have, . The general form of equation for the interior nodes is given by,

(21)

Where,

 
(22)

 

Now the boundary conditions
are applied at nodes 2 and 31. At node 2 west control volume boundary is kept
at specified concentration  and thus we get,

 
(23)

Similarly applying boundary
conditions at node 31, east control volume boundary is kept at specified
concentration  and thus we get,

 

Where  and  be the specified boundary
conditions in terms of calcium concentrations at node 1 and 32 respectively.

Using all the values from
above equations we get a system of algebraic equations as follows,

(24)

                                                                                     

Here  represents
the calcium concentrations at respective nodes, P is system matrix and Q is
system vector.

A computer program in
MATLAB R2014a is developed to find numerical solution to whole problem. The
Gauss elimination method is used to solve the system of equation (24).

Table 1: Values of biophysical Parameters 4

Symbol

Parameter

Value

Diffusion coefficient

200-300

Total buffer concentration

50-150

 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

 

3.    
Results
and Discussion:

The numerical results have
been computed using the values of biophysical parameters 4 given in
Table 1.

Figure 2: Spatial calcium concentration in
presence of different buffers in hepatocyte cell.

Figure 2, shows
the spatial variation of calcium concentration in presence of EGTA, endogenous
buffer, BAPTA buffer respectively. From the figure it is clear that different
types of buffers have different effects on calcium concentration profile in the
cell. The calcium concentration is maximum at x=0 where source influx is
present. The calcium concentration falls down gradually in case of EGTA buffer,
but it falls down sharply in case of endogenous buffer and more sharply in case
of BAPTA buffer. This is due to fact that, BAPTA buffer is fast buffer having
large binding rate and EGTA buffer is slow buffer having small binding
rate.  The endogenous buffer is faster
than EGTA buffer and slower than BAPTA buffer. The calcium concentration
attains background concentration 0.1 beyond 10  from the source
influx.

 

Figure 3: Spatial
calcium concentration with different concentration of EGTA buffer in hepatocyte
cell.

Figure 3, shows the spatial
variation of calcium concentration when concentration of EGTA buffer is
50,100,150 respectively. From figure it is
clear that with increase in concentration of EGTA buffer the concentration of
free calcium decreases at each nodal point of hepatocyte cell. As EGTA binds
with free calcium to form calcium bound buffer, it reduces amount of free
calcium. This is why increase in concentration of buffer leads to decrease in
concentration of free calcium. At the mouth of source channel the concentration
of calcium is 0.8. The rate of decrease in calcium concentration is
increases with increase in buffer concentration. The calcium concentration
uniformly decreases towards basal part of the hepatocyte cell and attains
background equilibrium concentration 0.1.

Figure 4: Spatial distribution of calcium concentration with
different values of diffusion coefficient.

 

Figure 4, shows the spatial
variation of calcium concentration when the value of diffusion coefficient is
200, 250, 300. Diffusion coefficient is defined as amount of
diffusing substance transported from one part to other part of domain per unit
area per unit time. This shows that for higher value of D calcium ions moves
fast from apical to basal region of cell. As more amount of calcium is
transported for D = 300 the less amount of free calcium
accumulates in the space. Therefore concentration of calcium is decreases with
increase in value of diffusion coefficient. i.e amount of free calcium is
inversely proportional to diffusion coefficient.

Figure
5: Spatial distribution of calcium
concentration with different values of source influx

 

Figure 5, shows the spatial
variation of calcium concentration when the value of source amplitude sigma is
1, 2 pA respectively. The characteristic amplitude of current passing through a
channel has unit pico Amperes (pA). The open channel permits the passage of ions,
which is measured as current. The increase in value of source amplitude release
more amount of calcium into cytosol. Thus it leads to increase in concentration
of free calcium. From figure it is observed that the concentration of calcium
is 0.8 and 1.6 respectively for 1, 2
pA source amplitude at the mouth of point source. Then afterwards it decreases
uniformly up to 0.1  in presence of
EGTA buffer.  The appropriate
experimental results are not available for comparison; however the results
obtained by proposed model are in agreement with the biological facts.

 

4.     Conclusion:

The finite volume model
have been proposed and successfully employed to study the effect of different
types of buffers, source amplitude and different rates of diffusion coefficient
on the spatial calcium concentration in a hepatocyte cell. From the results it
is concluded that, calcium concentration decreases sharply for fast buffers
especially for endogenous and BAPTA buffer in comparison with exogenous EGTA
buffer. The calcium concentration in the cell is inversely proportional to
diffusion coefficient. The variation in calcium concentration in the cell is
directly proportional to the source influx. The finite volume method has proved
to be quite versatile in obtaining the interesting relationships of calcium
concentration in the cell with the type, quantity of buffer, influx rate and
diffusion coefficient. The results obtained can be of great use to biomedical
scientist for development of new protocols for treatment and diagnosis of liver
diseases.

 

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