Finite Element Model to Study Calcium Distribution in T Lymphocyte Involving Buffers and Ryanodine Receptors

Abstract

In this paper an attempt has been made to develop a mathematical model to study intracellular calcium distribution in T lymphocyte for a one dimensional unsteady state case. The model incorporates the parameters like diffusion coefficient, ryanodine receptors (RyRs), source influx and buffers. The boundary conditions have been framed using biophysical conditions of the problem. The finite element method has been employed to obtain the solution of the proposed mathematical model. The numerical results are used to study relationship between concentration and position with respect to source influx, buffers and ryanodine receptors (RyRs).

1 Introduction

Calcium (Ca²⁺) is a universal second messenger, found in almost all cell types. Its intracellular level is regulated by finely tuned machinery responsible for calcium uptake, release, and intracellular storage. T lymphocytes are no exception in this regard. There are several calcium channels and transporters that play a key role in balancing cytoplasmic calcium levels in T lymphocytes. Pathways of calcium homeostasis participate in a number of cellular processes that determine short and long-term function of T lymphocytes. Therapeutic strategies are now evolving based on the modulation of T lymphocyte calcium homeostasis in order to combat immune—mediated disorders [1]. Ca²⁺ entry across the plasma membrane is the most important Ca²⁺ stores for T lymphocyte activation [2]. The ion channels that regulate calcium influx from the extracellular space in T lymphocytes either by conducting calcium ions or by modulating the membrane potential that provides driving force for calcium influx [3, 4]. The best characterized calcium channel in T lymphocytes is the calcium released-activated calcium (CRAC) channel, which is composed of ORAI and stromal interaction molecule (STIM) proteins [1]. The ORAI proteins and most prominently ORAI1 are the molecular basis for Ca²⁺ release-activated calcium (CRAC) channels. CRAC/ORAI1 channels are well defined through their biophysical and pharmacological properties [2].

Ca²⁺ dynamics [5] is the exchange of Ca²⁺ ions between intracellular Ca²⁺ stores and the cytosol, entering and leaving ions between the cells and binding activity of calcium and calcium binding proteins. The most important calcium binding proteins are itself buffers that are located in Ca²⁺ stores. The buffers bind more calcium molecules when the calcium concentration is higher in the cell [6, 7]. In T lymphocytes, the main buffer within cytosol is calmodulin (CaM) [8] with 4 calcium binding sites per CaM. There is a diversity of measured CaM concentrations depending on cell type and organ [8]. The binding of Ca²⁺ concentration to buffers serves as an indicator of free calcium concentration in intracellular measurements. Furthermore other second messengers derived from the adenine dinucleotides, nicotinamide adenine dinucleotide (NAD), and nicotinamide adenine dinucleotide phosphate (NADP) have also been implicated in T lymphocyte calcium signaling [9]. Nicotinic acid adenine dinucleotide phosphate (NAADP) acts as a very early second messenger upon TCR/CD3 engagement, while cyclic ADP-ribose (cADPR) is mainly involved in sustained partial depletion of the endoplasmic reticulum by stimulating calcium release via ryanodine receptors (RyRs) [1].

A good number of theoretical attempts [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] are reported in the literature for the study of calcium signaling in neuron cells, astrocytes, oocytes, acinar cells, fibroblasts, etc. But very few attempts are reported [2, 8, 25] for theoretical study of calcium signaling in T lymphocytes. No attempt is reported in the literature for calcium diffusion based study of calcium signaling in T lymphocytes. In the present paper an attempt has been made to develop a model for intracellular calcium distribution in T lymphocyte involving calcium diffusion, buffers and influx of Ca²⁺ from ryanodine receptors (RyRs) channels. The model has been developed for a one dimensional unsteady state case. The finite element method has been used to obtain the solution.

2 Mathematical Formulation

Calcium kinetics in T lymphocytes is governed by a set of reaction–diffusion equations which can be framed assuming the following bimolecular reaction between Ca²⁺ and buffer species [17, 26,27,28]

$$ \left[ {{\text{Ca}}^{2 + } } \right] + \left[ {B_{j} } \right]\underset{{k_{j}^{ - } }}{\overset{{k_{j}^{ + } }}{\longleftrightarrow}}\left[ {{\text{Ca}}B_{j} } \right] $$

(1)

where [Ca²⁺], [B _j ] and [CaB _j ] represent the cytosolic Ca²⁺ concentration, free buffer concentration and calcium bound buffer concentration respectively and ‘j’is an index over buffer species, k ⁺ _j and k ⁻ _j are ‘on’ and ‘off’ rates for jth buffer respectively. Using Fickian diffusion, the buffer reaction diffusion system in one dimension is expressed as [10, 11, 17, 26, 27]

$$ \frac{{\partial \left[ {{\text{Ca}}^{2 + } } \right]}}{\partial t} = D_{\text{Ca}} \nabla^{2} \left[ {{\text{Ca}}^{2 + } } \right] + \sum {R_{j} } $$

(2)

$$ \frac{{\partial [B_{j} ]}}{\partial t} = D_{{B_{j} }} \nabla^{2} [B_{j} ] + \sum R_{j} $$

(3)

$$ \frac{{\partial [{\text{Ca}}B_{j} ]}}{\partial t} = D_{{{\text{Ca}}B_{j} }} \nabla^{2} [{\text{Ca}}B_{j} ] - \sum R_{j} $$

(4)

where reaction term R _j is given by

$$ R_{j} = - k_{j}^{ + } [{\text{Ca}}^{2 + } ][B_{j} ] + k_{j}^{ - } [{\text{Ca}}B_{j} ] $$

(5)

D _Ca, $D_{{B_{j} }}$, $D_{{{\text{Ca}}B_{j} }}$ are diffusion coefficients of free calcium, free buffer and Ca²⁺ bound buffer respectively and σ _RyR is the influx of Ca²⁺ from ryanodine receptors (RyRs). Let [B _j ] _T  = ([B _j ] + [CaB _j ]) be the total buffer concentration of jth buffer and the diffusion coefficient of buffer is not affected by the binding of calcium i.e., $D_{{B_{j} }} = D_{{{\text{Ca}}B_{j} }}$. The Eq. (5) can be written as [17, 26]

$$ R_{j} = - k_{j}^{ + } [{\text{Ca}}^{2 + } ][B_{j} ] + k_{j}^{ - } \left( {[B_{j} ]_{T} - [B_{j} ]} \right) .$$

(6)

We assume that the buffer concentration is present in excess inside the cytosol so that the concentration of free buffer is constant in space and time i.e., [B _j ] ≅ [B _j ]_∞. Under this assumption Eq. (6) is approximated by [12, 17, 20, 26, 28]

$$ k_{j}^{ + } [{\text{Ca}}^{2 + } ][B_{j} ] \approx k_{j}^{ - } \left( {[B_{j} ]_{T} - [B_{j} ]_{\infty } } \right) $$

(7)

where $[B_{j} ]_{\infty } = \frac{{k_{j}^{ - } [B_{j} ]_{T} }}{{\left( {k_{j}^{ - } + k_{j}^{ + } [{\text{Ca}}^{2 + } ]_{\infty } } \right)}}$ is the background buffer concentration. Thus for single mobile buffer species Eq. (2) can be written as [17, 18, 26]

$$ \frac{{\partial [{\text{Ca}}^{2 + } ]}}{\partial t} = D_{\text{Ca}} \nabla^{2} [{\text{Ca}}^{2 + } ] - k_{j}^{ + } [B_{j} ]_{\infty } \left( {[{\text{Ca}}^{2 + } ] - [{\text{Ca}}^{2 + } ]_{\infty } } \right) + \sigma_{RyR} + \delta \sigma (r) $$

(8)

Here [Ca²⁺] is background calcium concentration. σ _RyR is the influx of calcium through ryanodine receptor channels which is given by [12, 17, 29]

$$ \sigma_{RyR} = V_{RyR} P_{o} \left( {[{\text{Ca}}^{2 + } ]_{ER} - [{\text{Ca}}^{2 + } ]} \right) $$

(9)

We assume a single point source of Ca²⁺, σ(r) at r = 0; there are no sources for buffers and buffer concentration is in equilibrium with Ca²⁺ far from the source and ∇ is the Laplacian operator i.e., $\nabla^{2} = \frac{{\partial^{2} }}{{\partial r^{2} }} + \frac{2}{r}\frac{\partial }{\partial r}$.

Combining Eqs. (8) and (9) we get the mathematical model as given below

$$ \frac{{\partial [{\text{Ca}}^{2 + } ]}}{\partial t} = D_{\text{Ca}} \nabla^{2} [{\text{Ca}}^{2 + } ] - k_{j}^{ + } [B_{j} ]_{\infty } \left( {[{\text{Ca}}^{2 + } ] - [{\text{Ca}}^{2 + } ]_{\infty } } \right) + V_{RyR} P_{o} \left( {[{\text{Ca}}^{2 + } ]_{ER} - [{\text{Ca}}^{2 + } ]} \right) + \delta \sigma (r) $$

(10)

The T lymphocyte cell is assumed to be spherical in shape [8]. The Eq. (10) for a one dimensional unsteady state case in polar spherical coordinates is given by

$$ \frac{{\partial [{\text{Ca}}^{2 + } ]}}{\partial t} = \frac{1}{{r^{2} }}\frac{\partial }{\partial r}\left( {D_{\text{Ca}} r^{2} \frac{{\partial [{\text{Ca}}^{2 + } ]}}{\partial r}} \right) - k_{j}^{ + } [B_{j} ]_{\infty } \left( {[{\text{Ca}}^{2 + } ] - [{\text{Ca}}^{2 + } ]_{\infty } } \right) + V_{RyR} P_{o} \left( {[{\text{Ca}}^{2 + } ]_{ER} - [{\text{Ca}}^{2 + } ]} \right) + \delta \sigma (r) $$

(11)

The point source of calcium is assumed at r = 0 and as we move away from the source, the calcium concentration achieves its background value i.e. 0.1 μM. Thus the initial and boundary conditions for the above problem are [26, 27, 30].

The boundary conditions are given by

$$ \mathop {\lim }\limits_{r \to 0} \left( {4\pi D_{\text{Ca}} r^{2} \frac{{\partial [{\text{Ca}}^{2 + } ]}}{\partial r}} \right) = \sigma $$

(12)

$$ \mathop {\lim }\limits_{r \to 5} [{\text{Ca}}^{2 + } ] = 0.1\;\upmu{\text{M}} $$

(13)

The initial condition is given by

$$ [{\text{Ca}}^{2 + } ]_{t = 0} = 0.1\;\upmu{\text{M}} $$

(14)

Our problem is to solve Eq. (11) along with Eqs. (12–14). For our convenience we write’u’in lieu of [Ca²⁺]. The Eq. (11) in discretized variational form is given by

$$ I^{(e)} = \frac{1}{2}\int\limits_{{r_{i} }}^{{r_{j} }} {\left[ {r^{2} \left( {\frac{{\partial u^{(e)} }}{\partial r}} \right)^{2} + \alpha r^{2} \left( {u^{(e)} } \right)^{2} - 2\beta r^{2} u^{(e)} + \frac{{r^{2} }}{{D_{\text{Ca}} }}\frac{{\partial (u^{(e)} )^{2} }}{\partial t}} \right]} dr + \psi^{(e)} \left( {\frac{{\sigma u^{(e)} }}{{4\pi D_{\text{Ca}} }}} \right)_{r = 0} $$

(15)

where $\alpha = \frac{1}{{D_{\text{Ca}} }}\left( {k_{j}^{ + } [B_{j} ]_{\infty } + V_{RyR} P_{o} } \right),$ $\beta = \frac{1}{{D_{\text{Ca}} }}\left( {V_{RyR} P_{o} u_{ER} + k_{j}^{ + } [B_{j} ]_{\infty } u_{\infty } } \right),$ ψ ^(e) = 1 for e = 1 and ψ ^(e) = 0 for rest of the elements, where e = 1, 2, 3, … 50. The shape function of concentration variation within each element is defined as:

$$ u^{(e)} = c_{1}^{(e)} + c_{2}^{(e)} r $$

(16)

or

$$ u^{(e)} = p^{T} c^{(e)} $$

(17)

where

$$ p^{T} = \left[ {1\;r} \right] $$

(18)

and

$$ c^{{(e)^{T} }} = \left[ {c_{1}^{(e)} \;c_{2}^{(e)} } \right] $$

(19)

Substituting nodal conditions in Eq. (17), we get

$$ \bar{u}^{(e)} = P^{(e)} \times c^{(e)} $$

(20)

where

$$ \bar{u}^{(e)} = \left[ {\begin{array}{*{20}c} {u_{i} } \\ {u_{j} } \\ \end{array} } \right],\quad P^{(e)} = \left[ {\begin{array}{*{20}c} 1 & {r_{i} } \\ 1 & {r_{j} } \\ \end{array} } \right] $$

(21)

From the Eq. (20), we have

$$ c^{(e)} = R^{(e)} \times \bar{u}^{(e)} $$

(22)

where

$$ R^{(e)} = P^{{(e)^{ - 1} }} $$

(23)

Substituting c ^(e) from Eq. (22) in (17), we get

$$ u^{(e)} = p^{T} R^{(e)} \bar{u}^{(e)} $$

(24)

Now the integral I ^(e) can be written in the form

$$ I^{(e)} = I_{l}^{(e)} + I_{m}^{(e)} - I_{n}^{(e)} - I_{k}^{(e)} + I_{p}^{(e)} $$

(25)

where

$$ I_{l}^{(e)} = \frac{1}{2}\int\limits_{{r_{i} }}^{{r_{j} }} {r^{2} } \left( {\frac{{\partial u^{(e)} }}{\partial r}} \right)^{2} dr $$

(26)

$$ I_{m}^{(e)} = \frac{1}{2}\int\limits_{{r_{i} }}^{{r_{j} }} {\alpha r^{2} } \left( {u^{(e)} } \right)^{2} dr $$

(27)

$$ I_{n}^{(e)} = \int\limits_{{r_{i} }}^{{r_{j} }} {\beta r^{2} } u^{(e)} dr $$

(28)

$$ I_{k}^{(e)} = \psi^{(e)} \frac{\sigma }{{4\pi D_{\text{Ca}} }}u^{(e)} \mathop |\nolimits_{r = 0} $$

(29)

$$ I_{p}^{(e)} = \frac{1}{{D_{\text{Ca}} }}\frac{d}{dt}\int\limits_{{r_{i} }}^{{r_{j} }} {r^{2} } \left( {u^{(e)} } \right)^{2} dr $$

(30)

Now, we extremize the integral I ^(e) w.r.t. each nodal calcium concentration u _i as given below

$$ \frac{{dI^{(e)} }}{{d\bar{u}^{(e)} }} = \frac{{dI_{l}^{(e)} }}{{d\bar{u}^{(e)} }} + \frac{{dI_{m}^{(e)} }}{{d\bar{u}^{(e)} }} - \frac{{dI_{n}^{(e)} }}{{d\bar{u}^{(e)} }} - \frac{{dI_{k}^{(e)} }}{{d\bar{u}^{(e)} }} + \frac{{dI_{p}^{(e)} }}{{d\bar{u}^{(e)} }} = 0 $$

(31)

$$ \frac{dI}{{d\bar{u}}} = \sum {\bar{M}^{(e)} } \frac{{dI^{(e)} }}{{d\bar{u}}}\bar{M}^{{(e)^{T} }} = 0 $$

(32)

where $\bar{M}^{(e)} = \left[ {\begin{array}{*{20}c} 0 & 0 \\ . & . \\ 1 & 0 \\ 0 & 1 \\ . & . \\ 0 & 0 \\ \end{array} } \right],\quad \bar{u} = \left[ {\begin{array}{*{20}c} {u_{1} } \\ {u_{2} } \\ . \\ . \\ . \\ {u_{51} } \\ \end{array} } \right]$

Assembling the integrals (25), we get

$$ I^{{}} = \sum\limits_{e = 1}^{50} {I^{(e)} } $$

(33)

This leads to a following system of linear differential equations

$$ [A]_{51 \times 51} [\bar{u}]_{51 \times 1} + [C]_{51 \times 51} \left[ {\frac{\partial u}{\partial t}} \right]_{51 \times 1} = [B]_{51` \times 1} $$

(34)

Here, $\left[ {\bar{u}} \right] = [u_{1}$ u ₂ u ₃ … u ₅₁]^T, A and C are the system matrix and B is the system vector. A computer program has been developed in MATLAB 7.10 for the whole problem and executed on Intel(R) Core™ i3 CPU, 4.00 GB RAM, 2.40 GH_z processor.

3 Results and Discussion

In this section, the numerical computations were performed using a “finite difference in time, finite element in space” using linear basis functions. The simulations reported here have been designed to investigate the intracellular calcium concentration in T lymphocytes in presence and absence of the parameters. Our simulations also reproduce the relative values of intracellular calcium in the measured time courses.

The numerical values of biophysical parameters used in the model are stated in the Table 1. [8, 12, 17, 18, 26, 29].

Table 1 Values of biophysical parameters [8, 12, 17, 26, 29]

Figure 1 shows radial calcium concentration for different source amplitudes and B = 25 μM. The calcium concentration is maximum at r = 0 μM i.e., source and it decreases as we go away from the source up to r = 5 μM. It is observed that the calcium concentration is higher for higher values of source amplitudes.

Fig. 1

Radial variation of calcium concentration for different source amplitudes i.e., σ = 1 pA, 2 pA, 3 pA, B = 25 μM

Figure 2 shows radial calcium concentration distribution for σ = 1 pA and different values of BAPTA buffer concentrations B = 50, 100, 150 and 200 μM. The calcium concentration is maximum at source i.e. r = 0. The peak value of calcium decreases with the increase in the buffer concentration. The gaps among the curves in Fig. 2 indicate that buffer has significant effect on calcium concentration distribution in the cell.

Fig. 2

Radial variation of calcium concentration for σ = 1 pA for different concentration of BAPTA buffer i.e., B = 50, 100, 150, 200 μM

Figure 3 shows the radial calcium concentration distribution in presence and absence of RyR. It is observed that the presence of RyR raises the peak calcium concentration at the source r = 0. Further the gap between the curves indicates the effect of RyR on calcium concentration distribution in the cell.

Fig. 3

Radial variation of calcium concentration for σ = 1 pA in presence and absence of RyR

Figure 4 shows the radial calcium concentration distribution due to the two different types of buffers namely BAPTA and EGTA buffers. The fall in the calcium concentration profiles for BAPTA buffer is sharper as compared to that for EGTA buffer. This is because the BAPTA buffer is a fast buffer and binds the calcium ions at faster rate than the EGTA buffer.

Fig. 4

Radial variation of calcium concentration for two different types of buffers EGTA = 100 μM and BAPTA = 100 μM, σ = 1 pA

Figure 5 shows that the temporal variation of Ca²⁺ concentration in T lymphocytes for different concentrations of buffer. The effect of changing buffer concentration is clear in this figure. We observe that calcium concentration reaches steady state in less than 1000 ms. The peak value of Ca²⁺ concentration is different for different values of buffer concentration. The peak value of Ca²⁺ concentration is higher for lower concentration of buffer. It is also observed that the steady state is achieved early for higher buffer concentration. The reason for this is that the higher concentration of buffers binds more calcium and then forcing the system to reach steady state early.

Fig. 5

Temporal variation of calcium concentration in T lymphocytes for different concentrations of Buffers i.e., B = 50, 100, 150, 200 μM

Figure 6 shows temporal variation of Ca²⁺ concentration in T lymphocytes in presence and absence of Ryanodine Receptor Ca²⁺ channel. It is observed that Ca²⁺ concentration is lower in absence of Ryanodine receptor. The figure further shows that the Ca²⁺ concentration is higher in presence of the receptor as receptor releases the Ca²⁺ thus causes increase in Ca²⁺ concentration. The Ca²⁺ concentration is higher from t = 100 ms tot = 500 ms and then remains in steady state. The Ca²⁺ concentration in presence of ryanodine receptor is maximum and reaches up to 3 μM. The Ca²⁺ concentration is higher at 100 ms after that it decreases rapidly and becomes almost steady at 800 ms.

Fig. 6

Temporal variation of calcium concentration in T lymphocytes in the presence and absence of ryanodine receptor

The results obtained here are in agreement with the biological facts. But no such experimental results are available for comparison. Similar results have been observed in other cells like neuron cells, astrocytes, oocytes etc. [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] and our results are in agreement with them.

4 Conclusion

The proposed finite element model has been employed successfully to study the relationships of spatial–temporal calcium concentration distribution with buffers, source amplitude, RyR etc. in T lymphocytes. From the results it can be concluded that the source amplitude, buffers and RyR have significant impact on calcium concentration distribution patterns in T lymphocytes required for maintaining the structure and function of the cell. Such models can be developed further to get deeper insights of calcium concentration regulation mechanism in T lymphocytes and generate information which can be useful to biomedical scientists for developing protocols for diagnosis and treatment of diseases associated with T lymphocytes.