Eur. Phys. J. Plus (2025) 140: 252
https://doi.org/10.1140/epjp/s13360-025-06185-4

Regular Article

Two-dimensional interdependent ${\text{Ca}}^{2+}$ and ${\text{IP}}_3$ dynamics in a T lymphocyte cell

¹ Department of Mathematics, Ram Lal Anand College, 5 Benito Juarez Marg, 110021, New Delhi, Delhi, India
² Department of Mathematics, Sardar Vallabhbhai National Institute of Technology, Dumas Road, 395007, Surat, Gujrat, India

^a hemant.maths@rla.du.ac.in

Received: 9 December 2024
Accepted: 27 February 2025
Published online: 25 March 2025

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.