On the dynamics of antigen receptors on the B-cell membrane through a two-dimensional stochastic process

B cells are important components of the adaptive immune system, responsible for {antibody} production and working as antigen-presenting cells. B cells display protein receptors on their membrane, {which} bind with foreign antigens and process them before presenting them to T cells. In this work, we present a stochastic process modeling the dynamics of such receptors on the B cell. The model consists of a two-dimensional birth-death process $\displaystyle \{(X(t), Y(t)), \; t \geq 0\}$ having linear transition rates, where $X(t)$ and $Y(t)$ represent the number of free and occupied receptors, respectively. After determining the partial differential equation for the probability generating function of the process, we compute the main moments of the process, including the covariance. The transient and asymptotic behavior of the means of $X(t)$ and $Y(t)$ is also studied. Throughout the paper, we provide insights into the biological significance of each parameter on the system's dynamics. In addition, we conduct a sensitivity analysis to assess how variations in the model parameters affect the first-order moments. Such analysis shows that minimal variations of the parameters representing the binding frequency of antigens and {B-cell} receptors, when happening in the initial instants of the process, result in noticeable alterations of the number of occupied receptors.