Two-dimensional interdependent \({\text{Ca}}^{2+}\) and \({\text{IP}}_{3}\) dynamics in a T lymphocyte cell

Abstract

Calcium (\({\text{Ca}}^{2+}\)) signaling is crucial for the functioning of the immune system, particularly in lymphocytes. The activation of \({\text{Ca}}^{2+}\) influx in T cells involves the participation of inositol 1,4,5-trisphosphate (\({\text{IP}}_{3}\)). Previous studies reported were focused on one-dimensional relationship between \({\text{IP}}_{3}\) formation and \({\text{Ca}}^{2+}\) mobilization in T lymphocyte cells and have resulted in limited insights due to simplifying assumptions. To examine the more realistic dynamics of \({\text{Ca}}^{2+}\) and \({\text{IP}}_{3}\) in T lymphocytes, we propose a two-dimensional mathematical model that integrates \({\text{Ca}}^{2+}\)-induced \({\text{Ca}}^{2+}\) release via \({\text{IP}}_{3}\) receptors and feedback regulation of \({\text{IP}}_{3}\) production and degradation. We utilized the Crank–Nicolson technique and the Rayleigh–Ritz finite element approach for solving time-dependent partial differential equations, successfully simulating the relative behavior of \({\text{Ca}}^{2+}\) and \({\text{IP}}_{3}\) signals. Our model emphasizes the role of \({\text{Ca}}^{2+}\)-dependent \({\text{IP}}_{3}\) modulation in creating complex \({\text{Ca}}^{2+}\) homeostasis and explores the effects of source, leak, and diffusion coefficients on the dynamics of these molecules. The proposed model provides more realistic insights for improving our understanding of \({\text{IP}}_{3}\)/Ca\(^{2+}\) signaling in T lymphocytes which could reveal their wider roles in immunity and inflammation, significantly advancing medical science. This study uses the two-dimensional finite element method for detailed modeling of cellular activities, enabling precise analysis of concentration gradients and intracellular behaviors. Rectangular elements improve the discretization of T cells, allowing simultaneous evaluation of reaction kinetics, membrane dynamics, and diffusion to highlight their effects on cell behavior.