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.

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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