Modulation of antigen discrimination by duration of immune contacts in a kinetic proofreading model of T cell activation with extreme statistics

Abstract

T cells form transient cell-to-cell contacts with antigen presenting cells (APCs) to facilitate surface interrogation by membrane bound T cell receptors (TCRs). Upon recognition of molecular signatures (antigen) of pathogen, T cells may initiate an adaptive immune response. The duration of the T cell/APC contact is observed to vary widely, yet it is unclear what constructive role, if any, such variations might play in immune signaling. Modeling efforts describing antigen discrimination often focus on steady-state approximations and do not account for the transient nature of cellular contacts. Within the framework of a kinetic proofreading (KP) mechanism, we develop a stochastic First Receptor Activation Model (FRAM) describing the likelihood that a productive immune signal is produced before the expiry of the contact. Through the use of extreme statistics, we characterize the probability that the first TCR triggering is induced by a rare agonist antigen and not by that of an abundant self-antigen. We show that defining positive immune outcomes as resilience to extreme statistics and sensitivity to rare events mitigates classic tradeoffs associated with KP. By choosing a sufficient number of KP steps, our model is able to yield single agonist sensitivity whilst remaining non-reactive to large populations of self antigen, even when self and agonist antigen are similar in dissociation rate to the TCR but differ largely in expression. Additionally, our model achieves high levels of accuracy even when agonist positive APCs encounters are rare. Finally, we discuss potential biological costs associated with high classification accuracy, particularly in challenging T cell environments.

Fig 1. The First Receptor Activation Model (FRAM) as a classifier of APC status ξ ∈ {0, 1}.

The T cell and APC form a cellular contact at t = t₀. An APC is agonist positive (ξ = 1) with probability $\mathbb{P}(\xi =1)={\rho }_{ag}$ and agonist negative (ξ = 0) with probability $\mathbb{P}(\xi =0)=1-{\rho }_{ag}$. If ξ = 1 then there is n_ag = 1 agonist antigen and n_self = 10⁴ self antigen. Activation is the event ($\text{min}({\text{T}}_{1,n_{self}},{\text{T}}_{1,n_{ag}})<\tau$) which results in a true positive (TP) classification. If ξ = 0 then correct classification occurs when none of the self antigen activate (${\text{T}}_{1,n_{self}}>\tau$) which results in a true negative (TN). We measure the accuracy ($\mathbb{P}\left(\text{TP}\right)+\mathbb{P}\left(\text{TN}\right)$) of the FRAM as an APC classifier with respect to the cellular contact duration (τ). Cellular contacts of too short duration result in a high false negative rate while overly long contacts return a high false positive probability, with both scenarios reducing accuracy. Decision accuracy is maximized at fixed contact duration (τ*). Published with permission under the CC BY 4.0 licence.

Fig 2. T cell accuracy as contact duration τ varies.

Kinetic proofreading parameters given in Table 1. A–B The T cell classification accuracy Γ(τ) for n_KP = 3 (A) and n_KP = 10 (B). For each value of σ, the time τ* of maximum accuracy is highlighted (gray vertical lines). C The maximum accuracy Γ(τ*) as n_KP varies for values σ = {2, 3, 4, 5}. D–F Receiver operating characteristic (ROC) for n_KP = 3 (D) and n_KP = 10 (E) at various σ values. F The ROC for σ = 5 and varied n_KP.

Fig 3. Effect of agonist positive prevalence (ρ_ag) on T cell accuracy (Γ(τ*)) and false positive rate at the optimal contact duration (6).

A–C Accuracy Γ(τ*) versus ρ_ag for n_KP = 3 (A), n_KP = 10 B, and varied n_KP with σ = 5 (C). The baseline (black dashed line) shows the worst accuracy a model can accomplish given that the optimal cellular contact duration is chosen. D–F False positive rate (6) at the maximizing cellular contact time (τ*). G–I Effect of agonist positive prevalence on the optimal cellular contact duration. Curve endpoints indicate where the FRAM reduces to baseline, i.e., τ* → 0 if ρ_ag < 0.5 or τ* → ∞ if ρ_ag > 0.5.

Fig 4. Plots showing the classification accuracy in terms of the number of energy utilizing reactions (n_e) and n_KP.

The scaled accuracy $\tilde{\Gamma }$ is such that $\tilde{\Gamma }=0$ is Γ ≤ Γ^(BL) and $\tilde{\Gamma }=(\Gamma -{\Gamma }^{\left(BL\right)})/(1-{\Gamma }^{\left(BL\right)})$. The left, center, and right column of plots show results for σ = 10, σ = 10², σ = 10³, respectively. Each top, center, and bottom row shows results for ρ_ag = 0.01, ρ_ag = 0.1, and ρ_ag = 0.5.