A mathematical framework for analyzing T cell receptor scanning of peptides

Abstract

T cells continuously search for antigenic peptides presented on major histocompatibility complexes expressed on nearly all nucleated cells. Because only a few antigenic peptides are presented in a sea of thousands of self-peptides, the T cells have a critical task in discriminating between self- and nonself-peptides. This search process for antigens must be performed with sufficient speed in order to induce a fast response against invading pathogens. This study presents a mathematical framework for analyzing the scanning process of peptides. The framework includes analytic expressions for calculating the sampling rate as well as continuous-systems- and stochastic-agent-based models. The results show that the scanning of self-peptides is a very fast process due to fast off-rates. The simulations also predict the existence of an optimal sampling rate for a certain range of on-rates based on the recently proposed confinement time model. Calculations reveal that most of the self-peptides located within a microdomain are scanned within just a few seconds, and that the T cell receptors have kinetics for self-peptides, facilitating fast scanning. The derived mathematical expressions within this study provide conceptual calculations for further investigations of how the T cell discriminates between self- and nonself-peptides.

Figure 1

(A) Sampling rate as a function of ${\overline{k}}_{\text{on}}$values calculated with one TCR and 300 pMHC molecules/μm². (Inset) Effect of different densities of pMHC molecules. (B and C) Fraction unsampled pMHC molecules within a microdomain as a function of time (M_tot = 300 molecules/μm²; ${\overline{k}}_{\text{on}}$= 0.01 μm²/s) simulated with one TCR/μm² (B) or with 100 TCRs/μm² (C). Continuous model simulations (solid lines) are compared with one stochastic simulation (dashed lines) with the corresponding parameter values. (Insets) Time at which unsampled pMHC molecules bind to a TCR from one representative stochastic simulation.

Figure 2

Fraction of sampled pMHC molecules within a microdomain after τ seconds as a function of ${\overline{k}}_{\text{on}}$values, with T_tot = 100 and M_tot = 300 molecules/μm². (A) Simulation obtained with the ODE system (Eqs. 4 And 5) with different scan time values (τ = 5–100 s) at a k_off value of 0.5/s. (B) Mean values of the corresponding stochastic simulation as in panel A, where the error bars represent the standard deviation from 50 repeats. (C) The effect of k_off values is shown with a scan time of 5 s.

Figure 3

Fraction of sampled pMHC molecules within a microdomain after 5 s (τ = 5 s) as a function of ${\overline{k}}_{\text{on}}$values, with a k_off value of 5/s is calculated with the ODE system (Eqs. 4 And 5). (A) Effect of different densities of both the interacting molecules. (B) The effect of changing only the TCR density.

Figure 4

Results obtained with the confinement time model. (A) Calculations of the sampling rate of the analytic expression (Eq. 11) as a function of ${\overline{k}}_{\text{on}}$values with different off-rates at a density of 300 pMHC molecules/μm². (Inset) Effect of different densities of pMHC molecules. (B–D) Simulations obtained with the ODE system (Eqs. 13–16) where the fraction of sampled pMHC within a microdomain as a function of ${\overline{k}}_{\text{on}}$values is shown (T_tot = 100; M_tot = 300 molecules/μm²; D_T = 0.1; D_M = 0.05 μm²/s). Effect of scan time (B) and k_off values with a scan time of 20 s (C) and 5 s (D).

Figure 5

Results obtained with the confinement time model (Eqs. 13–16). The simulations show the fraction of sampled pMHC molecules within a microdomain as a function of ${\overline{k}}_{\text{on}}$values (k_off = 5/s; D_T = 0.1; D_M = 0.05 μm²/s; τ = 5 s). (A) Effect of varying the diffusivities for TCR and pMHC molecules 10-fold where D represents the sum of D_T and D_M. (B) Results obtained by varying the densities of both molecules 10-fold. (C) The effect of changing the TCR density separately.