Regulatory disturbances in the dynamical signaling systems of $Ca^{2+}$ and NO in fibroblasts cause fibrotic disorders

Abstract

Studying the calcium dynamics within a fibroblast cell individually has provided only a restricted understanding of its functions. However, research efforts focusing on systems biology approaches for such investigations have been largely neglected by researchers until now. Fibroblast cells rely on signaling from calcium $(Ca^{2+})$ and nitric oxide (NO) to maintain their physiological functions and structural stability. Various studies have demonstrated the correlation between NO and the control of $Ca^{2+}$ dynamics in cells. However, there is currently no existing model to assess the disruptions caused by various factors in regulatory dynamics, potentially resulting in diverse fibrotic disorders. A mathematical model has been developed to investigate the effects of changes in parameters such as buffer, receptor, sarcoplasmic endoplasmic reticulum $Ca^{2+}$-ATPase (SERCA) pump, and source influx on the regulation and dysregulation of spatiotemporal calcium and NO dynamics in fibroblast cells. This model is based on a system of reaction-diffusion equations, and numerical simulations are conducted using the finite element method. Disturbances in key processes related to calcium and nitric oxide, including source influx, buffer mechanism, SERCA pump, and inositol trisphosphate $(IP_3)$ receptor, may contribute to deregulation in the calcium and NO dynamics within fibroblasts. The findings also provide new insights into the extent and severity of disorders resulting from alterations in various parameters, potentially leading to deregulation and the development of fibrotic disease.

Appendix

Shape function for each element;

$$\begin{aligned} u1^{(e)}=l_1^{(e)}+l_2^{(e)}x, \end{aligned}$$

(16)

$$\begin{aligned} v1^{(e)}=m_1^{(e)}+m_2^{(e)}x, \end{aligned}$$

(17)

$$\begin{aligned} u1^{(e)}=SS^T\times l^{(e)}, \, v1^{(e)}=SS^T\times m^{(e)}. \end{aligned}$$

(18)

where

$$\begin{aligned} SS^T=[1 \, \, \, x], \, l^{{(e)}^{T}}=[l_1^{(e)}\,l_2^{(e)}] \, and \, m^{{(e)}^T}=[m_1^{(e)}\,m_2^{(e)}]. \end{aligned}$$

(19)

From Eq. (18), we get

$$\begin{aligned} \bar{u1}^{(e)}=SS^{(e)}\times l^{(e)},\, and \, \bar{v1}^{(e)}=SS^{(e)}\times m^{(e)}. \end{aligned}$$

(20)

where

$$\begin{aligned} \bar{u1}^{(e)}= \begin{bmatrix} u1_i\\ u1_j \end{bmatrix}, \, \bar{v1}^{(e)}= \begin{bmatrix} v1_i\\ v1_j \end{bmatrix} \, and \, S^{(e)}= \begin{bmatrix} 1 &{} x_i\\ 1 &{} x_j \end{bmatrix}. \end{aligned}$$

(21)

From (20) we have

$$\begin{aligned} l^{(e)}=(M^{(e)})\times \bar{u1}^{(e)}\ \text{and}\ m^{(e)}=(M^{(e)})\times \bar{v1}^{(e)}. \end{aligned}$$

(22)

where

$$\begin{aligned} M^{(e)}=\frac{1}{SS^{(e)}}. \end{aligned}$$

(23)

From Eqs. (22) and (18), we get

$$\begin{aligned} u1^{(e)}=SS^T \times M^{(e)} \, \bar{u1}^{(e)} \, and \, v1^{(e)}=SS^T\times M^{(e)} \, \bar{v1}^{(e)}. \end{aligned}$$

(24)

From Eqs. (1) and (8) we get

$$\begin{aligned} I_1^{(e)}=I_{ao1}^{(e)}-I_{bo1}^{(e)}+I_{co1}^{(e)}-I_{do1}^{(e)}+I_{eo1}^{(e)}-I_{fo1}^{(e)}-I_{go1}^{(e)}. \end{aligned}$$

(25)

where

$$\begin{aligned} I_{ao1}^{(e)}=\int _{x_{i}}^{x_{j}}\left[ \left[ \frac{\partial u1^{(e)}}{\partial x} \right] ^2\right] dx, \end{aligned}$$

(26)

$$\begin{aligned} I_{bo1}^{(e)}=\frac{d}{dx}\int _{x_{i}}^{x_{j}}\left[ \frac{u1^{(e)}}{D_{Ca} } \right] dx, \end{aligned}$$

(27)

$$\begin{aligned} I_{co1}^{(e)}=\frac{V_{IPR}}{D_{Ca}F_C}\int _{x_{i}}^{x_{j}}\left[ \alpha u1^{(e)}+\beta v1^{(e)}+\gamma \right] dx, \end{aligned}$$

(28)

$$\begin{aligned} I_{do1}^{(e)}=\frac{1}{D_{Ca}F_C}\int _{x_{i}}^{x_{j}}\left[ V_{sp}ku1^{(e)}-\eta +V_{PMCA}k1u1^{(e)}-\eta 1\right] dx, \end{aligned}$$

(29)

$$\begin{aligned} I_{eo1}^{(e)}=\frac{V_{leak}}{D_{Ca}F_C}\int _{x_{i}}^{x_{j}}\left[ [Ca^{2+}]_{ER}-u1^{(e)} \right] dx, \end{aligned}$$

(30)

$$\begin{aligned} I_{fo1}^{(e)}=\frac{K^+}{D_{Ca}}\int _{x_{i}}^{x_{j}}\left[ u1^{(e)}-[Ca^{2+}]_{\infty } \right] dx, \end{aligned}$$

(31)

$$\begin{aligned} I_{go1}^{(e)}=f^{(e)}\left[ \frac{\sigma }{D_{Ca}}\right] _{x=0}, \end{aligned}$$

(32)

$$\begin{aligned} I_2^{(e)}=I_{ao2}^{(e)}-I_{bo2}^{(e)}+I_{co2}^{(e)}-I_{do2}^{(e)}, \end{aligned}$$

(33)

$$\begin{aligned} I_{ao2}^{(e)}=\int _{x_{i}}^{x_{j}}\left[ \left[ \frac{\partial v1^{(e)}}{\partial x} \right] ^2\right] dx, \end{aligned}$$

(34)

$$\begin{aligned} I_{bo2}^{(e)}=\frac{d}{dx}\int _{x_{i}}^{x_{j}}\left[ \frac{v1^{(e)}}{D_{NO} } \right] dx, \end{aligned}$$

(35)

$$\begin{aligned} I_{co2}^{(e)}=\frac{V_{pro}}{D_{NO}}\int _{x_{i}}^{x_{j}}\left[ \mu u1^{(e)}+\tau \right] dx, \end{aligned}$$

(36)

$$\begin{aligned} I_{do2}^{(e)}=\frac{V_{deg}}{D_{NO}}\int _{x_{i}}^{x_{j}}\left[ v1^{(e)} \right] dx. \end{aligned}$$

(37)

Linearizing $[Ca^{2+}]$ and NO dynamics yields the values of $\gamma$, k, $\eta$, $\mu$, $\alpha$, $\beta$, k1, $\eta 1$, and $\tau$.

$$\begin{aligned} \frac{dI_1}{d\bar{u1}^{(e)}}=\sum _{e=1}^N\left[ \bar{l^{(e)}}\frac{dI_1^{(e)}}{d\bar{u1}^{(e)}}\bar{L^{(e)}}^{T}\right] =0, \end{aligned}$$

(38)

$$\begin{aligned} \frac{dI_2}{d\bar{v1}^{(e)}}=\sum _{e=1}^N\left[ \bar{L^{(e)}}\frac{dI_2^{(e)}}{d\bar{v1}^{(e)}}\bar{L^{(e)}}^{T}\right] =0. \end{aligned}$$

(39)

where

$$\begin{aligned} \frac{dI_1^{(e)}}{d\bar{u1}^{(e)}}=\frac{dI_{ao1}^{(e)}}{d\bar{u1}^{(e)}}+\frac{d}{dt}\left[ \frac{dI_{bo1}^{(e)}}{d\bar{u1}^{(e)}}\right] +\frac{dI_{co1}^{(e)}}{d\bar{u1}^{(e)}}+\frac{dI_{do1}^{(e)}}{d\bar{u1}^{(e)}}-\frac{dI_{eo1}^{(e)}}{d\bar{u1}^{(e)}}-\frac{dI_{fo1}^{(e)}}{d\bar{u1}^{(e)}}, \end{aligned}$$

(40)

$$\begin{aligned} \frac{dI_2^{(e)}}{d\bar{v1}^{(e)}}=\frac{dI_{ao2}^{(e)}}{d\bar{v1}^{(e)}}+\frac{d}{dt}\left[ \frac{dI_{bo2}^{(e)}}{d\bar{v1}^{(e)}}\right] +\frac{dI_{co2}^{(e)}}{d\bar{v1}^{(e)}}+\frac{dI_{d2}^{(e)}}{d\bar{v1}^{(e)}}-\frac{dI_{eo2}^{(e)}}{d\bar{v1}^{(e)}}, \end{aligned}$$

(41)

$$\begin{aligned} \overline{M}^{(el)}= \begin{bmatrix} 0 &{} 0\\ . &{} .\\ 0 &{} 0\\ 1 &{} 0\\ 0 &{} 1\\ 0 &{} 0\\ . &{} .\\ 0 &{} 0 \end{bmatrix}, \overline{u}=\begin{bmatrix} u_{1}\\ u_{2}\\ u_{3}\\ .\\ .\\ .\\ u_{20}\\ u_{21} \end{bmatrix} and \; \overline{v}=\begin{bmatrix} v_{1}\\ v_{2}\\ v_{3}\\ .\\ .\\ .\\ v_{20}\\ v_{21} \end{bmatrix}, \end{aligned}$$

(42)

$$\begin{aligned} \begin{bmatrix} A \end{bmatrix}_{42 \times 42} \begin{bmatrix} \begin{bmatrix} \frac{\partial \bar{u}}{\partial t} \end{bmatrix}_{21 \times 1}\\ \begin{bmatrix} \frac{\partial \bar{v}}{\partial t} \end{bmatrix}_{21 \times 1} \end{bmatrix}+\begin{bmatrix} B \end{bmatrix}_{42 \times 42} \begin{bmatrix} \begin{bmatrix} \bar{u} \end{bmatrix}_{21 \times 1}\\ \begin{bmatrix} \bar{v} \end{bmatrix}_{21 \times 1} \end{bmatrix}=\begin{bmatrix} F \end{bmatrix}_{42 \times 1}. \end{aligned}$$

(43)

FEM and Crank-Nicolson methods are applied to the matrices A, B, and F. The resulting system of equations is solved by the Gaussian elimination method.