Effects of disorders in interdependent calcium and IP3 dynamics on nitric oxide production in a neuron cell

Abstract

Calcium ([Ca²⁺]), IP₃, and nitric oxide (NO) play a significant role in cell signaling to maintain various physiological functions. Calcium and IP₃ regulation has been investigated independently in a variety of cells like myocyte, hepatocyte, and neuron cells. However, very little attention has been paid to the study of interdependent calcium and IP₃ dynamics regulating nitric oxide production in neurons and other cells. Nitric oxide and its derivatives are reported to be involved in the pathogenic process leading to neurogenerative disorders like Parkinson’s disease. The production of nitric oxide depends on the calcium dynamics in a neuron cell. Therefore a model is proposed to study the regulatory and dysregulatory effects of interdependent calcium and IP₃ dynamics in a neuron cell. The system of reaction–diffusion equations for calcium and IP₃ is coupled with the production of nitric oxide in a neuron cell to formulate an initial boundary value problem. The finite element simulation is performed to obtain results for regulatory and dysregulatory conditions of interdependent calcium and IP₃ dynamics along with nitric oxide production in the cell. It is observed that disorders in mechanisms of calcium dynamics are balanced to some extent by IP3 dynamics. The dysregulation of calcium or IP3 dynamics causes an increase or decrease in nitric oxide production in the cell, which can lead to various neurodegenerative disorders. The information obtained from the present study can be used in the development of diagnostic and therapeutic measures.

Appendix: summary of the model equations

The shape function of calcium and IP₃ concentration for each element is taken as,

$$ {\text{u}}^{{\text{(e)}}} {\text{ = q}}_{{1}}^{{\text{(e)}}} {\text{ + q}}_{{2}}^{{\text{(e)}}} {\text{ x}} $$

(19)

$$ {\text{v}}^{{\text{(e)}}} {\text{ = r}}_{{1}}^{{\text{(e)}}} {\text{ + r}}_{{2}}^{{\text{(e)}}} {\text{ x}} $$

(20)

$$ {\text{u}}^{{\text{(e)}}} {\text{ = S}}^{{\text{T}}} {\text{q}}^{{\text{(e)}}} {\text{, v}}^{{\text{(e)}}} {\text{ = S}}^{{\text{T}}} {\text{r}}^{{\text{(e)}}} $$

(21)

$$ S^{{\text{T}}} { = [}\begin{array}{*{20}c} {1} & {\text{x}} \\ \end{array} {\text{], q}}^{{{\text{(e)}}^{{\text{T}}} }} { = [}\begin{array}{*{20}c} {{\text{q}}_{{1}}^{{\text{(e)}}} } & {{\text{q}}_{{2}}^{{\text{(e)}}} } \\ \end{array} {\text{], r}}^{{{\text{(e)}}^{{\text{T}}} }} { = [}\begin{array}{*{20}c} {{\text{r}}_{{1}}^{{\text{(e)}}} } & {{\text{r}}_{{2}}^{{\text{(e)}}} } \\ \end{array} {]} $$

(22)

Putting nodal conditions in Eq. 21, we obtain

$$ \overline{{\text{u}}} ^{{({\text{e}})}} = S^{{({\text{e}})}} q^{{({\text{e}})}} ,\,\,\,\overline{{\text{v}}} ^{{({\text{e}})}} = S^{{({\text{e}})}} r^{{({\text{e}})}} , $$

(23)

where

$$ {\overline{\text{u}}}^{{\text{(e)}}} { = }\left[ {\begin{array}{*{20}c} {{\text{u}}_{{\text{i}}} } \\ {{\text{u}}_{{\text{j}}} } \\ \end{array} } \right]\,\,\,\,{\overline{\text{v}}}^{{\text{(e)}}} { = }\left[ {\begin{array}{*{20}c} {{\text{v}}_{{\text{i}}} } \\ {{\text{v}}_{{\text{j}}} } \\ \end{array} } \right]\,\,\,{\text{and}}\,\,\,\,S^{{\text{(e)}}} { = }\left[ {\begin{array}{*{20}c} {1} & {{\text{x}}_{{\text{i}}} } \\ {1} & {{\text{x}}_{{\text{j}}} } \\ \end{array} } \right], $$

(24)

By the Eq. 23, we have

$$ {\text{q}}^{{\text{(e)}}} {\text{ = R}}^{{\text{(e)}}} {\overline{\text{u}}}^{{\text{(e)}}} {\text{, r}}^{{\text{(e)}}} {\text{ = R}}^{{\text{(e)}}} {\overline{\text{v}}}^{{\text{(e)}}} $$

(25)

And

$$ {\text{R}}^{{\text{(e)}}} {\text{ = S}}^{{{\text{(e)}}^{{ - 1}} }} $$

(26)

Putting ${\text{q}}^{{\text{(e)}}}$ and ${\text{r}}^{{\text{(e)}}}$ from Eq. 25 in  21, we obtain

$$ {\text{u}}^{{\text{(e)}}} {\text{ = S}}^{{\text{T}}} {\text{ R}}^{{\text{(e)}}} {\overline{\text{u}}}^{{\text{(e)}}} {\text{, v}}^{{\text{(e)}}} {\text{ = S}}^{{\text{T}}} {\text{ R}}^{{\text{(e)}}} {\overline{\text{v}}}^{{\text{(e)}}} $$

(27)

Discretized form of Eq. 1 and 7 is given by,

The integral ${\text{I}}_{{1}}^{{\text{(e)}}}$ and ${\text{I}}_{{2}}^{{\text{(e)}}}$ can be expressed in this formation

$$ {\text{I}}_{{1}}^{{\text{(e)}}} {\text{ = I}}_{{{\text{a1}}}}^{{\text{(e)}}} {\text{ - I}}_{{{\text{b1}}}}^{{\text{(e)}}} {\text{ + I}}_{{{\text{c1}}}}^{{\text{(e)}}} {\text{ - I}}_{{{\text{d1}}}}^{{\text{(e)}}} {\text{ + I}}_{{{\text{e1}}}}^{{\text{(e)}}} {\text{ - I}}_{{{\text{f1}}}}^{{\text{(e)}}} {\text{ - I}}_{{{\text{g1}}}}^{{\text{(e)}}} $$

(28)

where

$$ {\text{I}}_{{{\text{a1}}}}^{{\text{(e)}}} { = }\int\limits_{{x_{i} }}^{{x_{j} }} {\left\{ {\left( {\frac{{\partial {\text{u}}^{{\text{(e)}}} }}{{\partial {\text{x}}}}} \right)^{2} } \right\}{\text{dx}}} $$

(29)

$$ {\text{I}}_{{{\text{b1}}}}^{{\text{(e)}}} { = }\frac{{\text{d}}}{{{\text{dt}}}}\int\limits_{{{\text{x}}_{{\text{i}}} }}^{{{\text{x}}_{{\text{j}}} }} {\left[ {\frac{{{\text{(u}}^{{\text{(e)}}} {)}}}{{{\text{D}}_{{{\text{Ca}}}} }}} \right]{\text{ dx}}} $$

(30)

$$ {\text{I}}_{{{\text{c1}}}}^{{\text{(e)}}} { = }\frac{{{\text{V}}_{{{\text{IPR}}}} }}{{{\text{D}}_{{{\text{ca}}}} {\text{F}}_{{\text{c}}} }}\int\limits_{{{\text{x}}_{{\text{i}}} }}^{{{\text{x}}_{{\text{j}}} }} {\left[ {{\alpha u}^{{\text{(e)}}} + {\beta v}^{{\text{(e)}}} + {\upgamma }} \right]{\text{ dx}}} $$

(31)

$$ {\text{I}}_{{{\text{d1}}}}^{{{\text{(e)}}}} = \frac{{{\text{V}}_{{{\text{SERCA}}}} }}{{{\text{D}}_{{{\text{ca}}}} {\text{F}}_{{\text{c}}} }}\int\limits_{{{\text{x}}_{{\text{i}}} }}^{{{\text{x}}_{{\text{j}}} }} {\left[ {\kappa u^{{{\text{(e)}}}} {\text{ + }}\eta } \right]{\text{ dx}}} $$

(32)

$$ {\text{I}}_{{{\text{e1}}}}^{{\text{(e)}}} { = }\frac{{{\text{V}}_{{{\text{LEAK}}}} }}{{{\text{D}}_{{{\text{ca}}}} {\text{F}}_{{\text{c}}} }}\int\limits_{{{\text{x}}_{{\text{i}}} }}^{{{\text{x}}_{{\text{j}}} }} {\left[ {{\text{[Ca}}^{{2 + }} {]}_{{{\text{ER}}}} {\text{ - u}}^{{\text{(e)}}} } \right]{\text{ dx}}} $$

(33)

$$ {\text{I}}_{{{\text{f1}}}}^{{\text{(e)}}} { = }\frac{{{\text{K}}^{ + } }}{{{\text{D}}_{{{\text{ca}}}} }}\int\limits_{{{\text{x}}_{{\text{i}}} }}^{{{\text{x}}_{{\text{j}}} }} {\left[ {{\text{u}}^{{\text{(e)}}} {\text{ - [Ca}}^{{2 + }} ]_{\infty } } \right]{\text{ dx}}} $$

(34)

$$ {\text{I}}_{{{\text{g1}}}}^{{\text{(e)}}} {\text{ = f}}^{{\text{(e)}}} \frac{{\upsigma }}{{{\text{D}}_{{{\text{ca}}}} }}_{{\text{ x = 0}}} $$

(35)

$$ {\text{I}}_{{2}}^{{\text{(e)}}} {\text{ = I}}_{{{\text{a2}}}}^{{\text{(e)}}} {\text{ - I}}_{{{\text{b2}}}}^{{\text{(e)}}} {\text{ + I}}_{{{\text{c2}}}}^{{\text{(e)}}} {\text{ - I}}_{{{\text{d2}}}}^{{\text{(e)}}} $$

(36)

$$ {\text{I}}_{{{\text{a2}}}}^{{\text{(e)}}} { = }\int\limits_{{x_{i} }}^{{x_{j} }} {\left\{ {\left( {\frac{{\partial {\text{v}}^{{\text{(e)}}} }}{{\partial {\text{x}}}}} \right)^{2} } \right\}{\text{ dx}}} $$

(37)

$$ {\text{I}}_{{{\text{b2}}}}^{{\text{(e)}}} { = }\frac{{\text{d}}}{{{\text{dt}}}}\int\limits_{{{\text{x}}_{{\text{i}}} }}^{{{\text{x}}_{{\text{j}}} }} {\left[ {\frac{{{\text{v}}^{{\text{(e)}}} }}{{{\text{D}}_{{\text{i}}} }}} \right]{\text{ dx}}} $$

(38)

$$ {\text{I}}_{{{\text{c2}}}}^{{\text{(e)}}} { = }\frac{{{\text{V}}_{{{\text{PROD}}}} }}{{{\text{D}}_{{\text{i}}} {\text{F}}_{{\text{c}}} }}\int\limits_{{{\text{x}}_{{\text{i}}} }}^{{{\text{x}}_{{\text{j}}} }} {\left[ {{\mu u}^{{\text{(e)}}} + {\uptau }} \right]{\text{ dx}}} $$

(39)

$$ {\text{I}}_{{{\text{d2}}}}^{{{\text{(e)}}}} = \frac{\lambda }{{{\text{F}}_{{\text{c}}} {\text{D}}_{{\text{i}}} }}\int\limits_{{{\text{x}}_{{\text{i}}} }}^{{{\text{x}}_{{\text{j}}} }} {\left[ {\delta u^{{{\text{(e)}}}} {\text{ + }}\xi {\text{v}}^{{{\text{(e)}}}} {\text{ + }}\omega )} \right]{\text{ dx}}} $$

(40)

The various parameters $\alpha ,\beta _{1} ,\gamma ,\kappa ,\eta ,\mu ,\tau ,\delta ,\xi ,{\text{and}},\omega$ are obtained by the linearization of nonlinear interdependent calcium and IP3 dynamics. The equations are analyzed and boundary conditions are included to give the following system of equations.

$$ \frac{{{\text{dI}}_{1} }}{{{\text{d}}\overline{{\text{u}}} ^{{{\text{(e)}}}} }} = \sum\limits_{{{\text{e = 1}}}}^{{\text{N}}} {\overline{{\text{Q}}} ^{{{\text{(e)}}}} } \frac{{{\text{dI}}_{1}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{u}}} ^{{{\text{(e)}}}} }}\overline{{\text{Q}}} ^{{{\text{(e)}}^{{\text{T}}} }} = 0 $$

(41)

$$ \frac{{{\text{dI}}_{2} }}{{{\text{d}}\overline{{\text{v}}} ^{{{\text{(e)}}}} }} = \sum\limits_{{{\text{e = 1}}}}^{{\text{N}}} {\overline{{\text{Q}}} ^{{{\text{(e)}}}} } \frac{{{\text{dI}}_{2}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{v}}} ^{{{\text{(e)}}}} }}\overline{{\text{Q}}} ^{{{\text{(e)}}^{{\text{T}}} }} = 0 $$

(42)

where

$$ \overline{{\text{Q}}} ^{{{\text{(e)}}}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ {\begin{array}{*{20}c} . \\ {\begin{array}{*{20}c} 0 \\ 1 \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} . \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} 0 \\ {\begin{array}{*{20}c} . \\ {\begin{array}{*{20}c} 0 \\ 0 \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} . \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right]\,{\text{and}}\,\overline{{\text{u}}} {\text{ = }}\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\text{u}}_{1} } \\ {{\text{u}}_{2} } \\ \end{array} } \\ {{\text{u}}_{3} } \\ \end{array} } \\ {\begin{array}{*{20}c} . \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} . \\ {{\text{u}}_{{19}} } \\ \end{array} } \\ {{\text{u}}_{{20}} } \\ \end{array} } \\ {{\text{u}}_{{21}} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right],\;\overline{{\text{v}}} {\text{ = }}\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\text{v}}_{1} } \\ {{\text{v}}_{2} } \\ \end{array} } \\ {{\text{v}}_{3} } \\ \end{array} } \\ {\begin{array}{*{20}c} . \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} . \\ {{\text{v}}_{{19}} } \\ \end{array} } \\ {{\text{v}}_{{20}} } \\ \end{array} } \\ {{\text{v}}_{{21}} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right] $$

(43)

$$ \frac{{{\text{dI}}_{{{\text{1}}}}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{u}}} ^{{{\text{(e)}}}} }} = \frac{{{\text{dI}}_{{{\text{a1}}}}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{u}}} ^{{{\text{(e)}}}} }} + \frac{{\text{d}}}{{{\text{dt}}}}\frac{{{\text{dI}}_{{{\text{b1}}}}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{u}}} ^{{{\text{(e)}}}} }} + \frac{{{\text{dI}}_{{{\text{c1}}}}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{u}}} ^{{{\text{(e)}}}} }} + \frac{{{\text{dI}}_{{{\text{d1}}}}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{u}}} ^{{{\text{(e)}}}} }} - \frac{{{\text{dI}}_{{{\text{e1}}}}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{u}}} ^{{{\text{(e)}}}} }} - \frac{{{\text{dI}}_{{{\text{f1}}}}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{u}}}^{{{\text{(e)}}}} }} $$

(44)

$$ \frac{{{\text{dI}}_{2}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{v}}} ^{{{\text{(e)}}}} }} = \frac{{{\text{dI}}_{{{\text{a2}}}}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{v}}} ^{{{\text{(e)}}}} }} + \frac{{\text{d}}}{{{\text{dt}}}}\frac{{{\text{dI}}_{{{\text{b2}}}}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{v}}} ^{{{\text{(e)}}}} }} + \frac{{{\text{dI}}_{{{\text{c2}}}}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{v}}} ^{{{\text{(e)}}}} }} + \frac{{{\text{dI}}_{{{\text{d2}}}}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{v}}} ^{{{\text{(e)}}}} }} - \frac{{{\text{dI}}_{{{\text{e2}}}}^{{{\text{(e)}}}} }}{{{\text{d}}\overline{{\text{v}}} ^{{{\text{(e)}}}} }} $$

(45)

$$ \left[ {\text{A}} \right]_{{{\text{42}} \times {\text{42}}}} \left[ {\begin{array}{*{20}c} {\left[ {\frac{{\partial \overline{{\text{u}}} }}{{\partial {\text{t}}}}} \right]_{{{\text{21}} \times {\text{1}}}} } \\ {\left[ {\frac{{\partial \overline{{\text{v}}} }}{{\partial {\text{t}}}}} \right]_{{{\text{21}} \times {\text{1}}}} } \\ \end{array} } \right] + \left[ {\text{B}} \right]_{{{\text{42}} \times {\text{42}}}} \left[ {\begin{array}{*{20}c} {\left[ {\overline{{\text{u}}} } \right]_{{{\text{21}} \times {\text{1}}}} } \\ {\left[ {\overline{{\text{v}}} } \right]_{{{\text{21}} \times {\text{1}}}} } \\ \end{array} } \right]{\text{ = }}[{\text{F}}]_{{{\text{42}} \times {\text{1}}}} $$

(46)

Here, the system matrices are A and B with the system vectors F. The Crank–Nicolson method which is numerically stable is applied to solve the time derivate in FEM.