Phenotypic model for early T-cell activation displaying sensitivity, specificity, and antagonism

Abstract

Early T-cell activation is selected by evolution to discriminate a few foreign peptides rapidly from a vast excess of self-peptides, and it is unclear in quantitative terms how this is possible. We show that a generic proofreading cascade supplemented by a single negative feedback mediated by the Src homology 2 domain phosphatase-1 (SHP-1) accounts quantitatively for early T-cell activation, including the effects of antagonists. Modulation of the negative feedback with SHP-1 concentration explains counterintuitive experimental observations, such as the nonmonotonic behavior of receptor activity on agonist concentration, the digital vs. continuous behavior on certain parameters, and the loss of response for high SHP-1 concentration. New experiments validate predictions on the nontrivial joint dependence on binding time and concentration for the relative effect of two antagonists: We explain why strong antagonists behave as partial agonists at low concentration and predict that the relative effect of antagonists can invert as their concentrations are varied. By focusing on the phenotype, our model quantitatively fits a body of experimental data with minimal variables and parameters.

Fig. 1.

Scheme of our model for early T-cell response and its behavior. (A) Idealized sensitivity-specificity curve of the immune response as a function of ligand concentration and dissociation time. Antigen discrimination is supposed to hinge on the antigen-TCR dissociation time. (B) Our model of KPR with negative feedback. The blue box indicates the core KPR, and the red box illustrates the negative feedback mediated by phosphatase, assumed to be SHP-1. Equations are defined in SI Appendix. (C) Numerical integration for the steady-state concentration of the effector , which controls the response of the system, as a function of antigen concentration for different dissociation times. All concentrations are expressed per cell. Parameters used are as in Table 1. Threshold K on for activation of response is indicated by the dotted line. The color (from blue to orange) corresponds to values of τ =1, 2, 3, 5, 10, and 20 s. Crosses indicate the analytical derivation performed in SI Appendix assuming no receptor saturation. The discrimination range in antigen concentration between τ = 3 s and τ =10 s is indicated by the red arrow. (D) Numerical integration of the system of equations with no phosphatase (i.e., pure KPR) for comparison. The discrimination range in antigen concentration between 3 and 10 s is indicated by the red arrow. Crosses indicate analytical KPR without receptor saturation. (E) Fraction of activated SHP-1 as a function of antigen concentration for different dissociation times (same color code as in C and D). SHP-1 controls the strength of the negative feedback; it increases with ligand number but clearly does not depend much on the dissociation time. (F) Concentration of antigen-triggering response as a function of its dissociation time. Note that for intermediate dissociation time, the response can disappear for a small range of high ligand concentrations. From this curve, one can define the agonist region for τ > 3 s; the response is specific. For 3 s < τ < 10 s, ligand concentration to trigger response drops from more than 100,000 to values of order 10, spanning ligand concentrations across four orders of magnitude; the response is sensitive.

Fig. 2.

Summary of parameter dependencies and digital/analog response for the deterministic system. (A, B, and E) Blue slope symbolizes the KPR regime of the response, the red slope symbolizes the negative feedback part of the response where phosphatase is activated, and the green slope symbolizes the saturated regime for the negative feedback. The dotted line indicates the threshold of activation of response by . Blue stars indicate the position of first activation of response, and red stars indicate the position of deactivation of response. All simulations are for agonist dissociation time τ = 10 s (A) Schematic of the three regimes at low, intermediate, and high antigen concentrations. The solid yellow lines indicate the interpolation for three different values of dissociation times. (B) Qualitative effect of phosphatase increase. The KPR part of the response (blue) is unchanged, whereas the negative feedback slopes move toward the left and the saturated regime on the right disappears. As the phosphatase increases, the response gets deactivated at higher concentration. (C) Numerical integration of the model with increasing SHP-1 concentration (solid lines; blue to orange indicate, respectively, γS_T = 50%, 100%, 200%, 400%, and 800% of reference value used for integrations of Fig. 1). The oblique dashed lines indicate the theoretical asymptotic slope for the negative feedback computed in SI Appendix, showing that the behavior gets closer to it as phosphatase increases. The dashed line is the threshold of activation. (D) Phase diagram computed numerically shows the antigen concentration triggering response as a function of relative SHP-1 concentration (with Fig. 1 being the reference). The response is abolished for a fourfold increase of SHP-1. (E) Qualitative effect of phosphorylation rate increase. Both the KPR part of the response and the negative feedback part of the response increase. (F) Numerical integration of the model with increasing (solid lines; blue to red indicate, respectively, = 50%, 100%, 200%, and 400% of reference value used for integrations of Fig. 1). (G) Phase diagram computed numerically shows the antigen concentration triggering response as a function of relative phosphorylation rate (with Fig. 1 being the reference). As phosphorylation rates increase, the minimum antigen concentration for activation decreases continuously.

Fig. 3.

Scheme of our model for the response to mixtures of antigens; antagonistic effects. (A) Schematic of the interactions with two types of ligands and , with different dissociation times. Ligands are engaged in parallel at the same time in the same cell. Both activate the negative feedback symmetrically and contribute to the triggering of response. (B) Total response as a function of agonist concentration (τ=10s) in the presence of 10³ antagonists with dissociation times of 0.5, 1, 1.5, 2, and 2.5 s (light blue to orange). Dark blue indicates the response without any antagonist. Crosses indicate the analytical approximation detailed in SI Appendix showing perfect agreement. Other parameters are as in Fig. 1. (C) Same as B, but with antagonists, shows a strong dependency of antagonism on the antagonist concentration. (D) Agonist concentration triggering response as a function of antagonist concentrations and dissociation time. Dark blue, light blue, yellow, and red correspond to 1,000, 5,000, 10,000, and 50,000 ligands, respectively. As the quantity of antagonists increases, the maximum dissociation time for response decreases. It should be noted that the maximum concentration triggering response goes through a maximum shortly before its sharp collapse; this explains why antagonism is maximum for dissociation times just below the agonist regime.

Fig. 4.

Comparison between stochastic simulations, assuming quasistatic SHP-1 concentration, and experimental results. Parameters are as in Table 1. The percentage of active cells is computed over a total number of 1,000 realizations for each of the binding times and ligands. (A) Percentage of responding cells as a function of antigen number; different colors correspond to different dissociation times following the conventions of Fig. 1B. (B) Average response times (seconds) of responding cells for simulations of A show a minimum response time for ligands between 100 and 1,000 [as observed by Altan-Bonnet and Germain (14)]. (C) Percentage of responding cells as a function of increasing SHP-1 concentration. Conventions follow Fig. 2C. As SHP-1 increases, fewer and fewer cells are responding at high concentration. (D) Percentage of responding cells with increasing SHP-1 concentration, redrawn from the study by Feinerman et al. (21), shows collapse of response. In these experiments, total SHP-1 concentration varies from 0.1-fold (blue) to 2.5-fold reference (red). Calibration reveals that 10⁻² μmol corresponds roughly to 3,000–10,000 ligands per cell (21). Thus, the agreement on the position of the decrease in response at high ligand concentration between simulations and experiments is very good. The strength of collapse is weaker than in C, which could be because of either a lower SHP-1 reference concentration or response smoothening due to other forms of noise (e.g., on parameters).

Fig. 5.

Comparison between stochastic simulations and experimental results for antagonism. (A) Simulation of the percentage of responding cells for the standard parameters in Table 1 and an agonist with τ₁ = 10 s (red); then, the same agonist with 10,000 antagonistic ligands with τ₂ = 0.5 s (green) and τ₂ = 3 s (blue). We see the same trends as in Fig. 3, with more responding cells for the strong antagonist at low ligand concentration and the reverse relation at larger agonist levels. (B) Percentage of responding cells after 5 min as a function of ligand concentration for ovalbumin (OVA) ligands alone (red), OVA + 10 μmol of E1 ligands (green), and OVA +10 μmol of G4 ligands (blue). The trends are the same as in the simulation in A, and the fraction of cells responding when G4 is present follows the strong antagonist simulations. Antagonistic effects for these cells are less dramatic than in A and previously published data (notably ref. 14), suggestive of smaller SHP-1 concentration. (C) Stochastic simulation similar to A with half of the SHP-1 concentration in Table 1 and using a dissociation time of 2 s for strong antagonist gives better quantitative agreement with the experiments in B.