FIRST PASSAGE TIMES TO T CELL ACTIVATION.

Effective recognition of foreign antigens by the adaptive immune system relies on T cells being activated by antigen-presenting cells (APCs) in lymph nodes. Here, diffusing T cells may encounter cognate APCs that present matching antigen fragments or non-cognate ones that do not; they are also subject to degradation. We develop a stochastic model in which T cell-APCs interact via a sequence of recognition steps, represented as a multistage Markov chain. T cells are successfully activated only if the terminal state associated with a cognate APC is reached. We compute the probability of successful activation in the presence of interfering non-cognate APCs, T cell degradation, and lymph node exit, and analyze the mean first-passage time to activation. We also incorporate a kinetic proofreading mechanism that enables state resetting, and show how this enhances specificity toward cognate APCs. Relevance to Life Sciences: We present a quantitative framework to study T cell activation within the lymph node that integrates diffusion, the presence and abundance of cognate and non-cognate APCs, and T cell death and exit from the lymph node. Relevant spatio-temporal parameters, such as T cell diffusivity and residence time within the lymph node, are estimated from existing literature. Quantification of the activation probability and time to first activation provide fundamental insights into the onset of the adaptive immune response. Mathematical Content: T cell recognition is modeled as a multistage Markov process, coupled with spatial diffusion, exit, and death. All four processes are represented through a system of partial differential equations that are analyzed under Robin and Neumann boundary conditions. Using first-passage time theory, we calculate activation probabilities and mean activation times. We also show how kinetic proofreading through stochastic resetting enhances specificity.