The geometry of pMHC‐coated nanoparticles and T‐cell receptor clusters governs the sensitivity‐specificity trade‐off in T‐cell response: a modeling investigation

Abstract

T cells must reliably discriminate between foreign‐derived antigens that require an adaptive immune response and nonspecific self‐antigens that do not. This discrimination is highly specific to the affinity of the bond between the ligand and T‐cell receptors (TCRs), and highly sensitive to the concentration of ligand. We examined the features of T‐cell‐mediated immunity in the context of multivalent ligand–receptor interactions between clusters of TCRs with peptide major histocompatibility complex‐coated nanoparticles (NPs). Using Monte Carlo simulations of NP‐T‐cell surface interactions, we compared the effect of TCR clustering on the dose–response curves of bound TCRs when various NP design parameters were altered. These simulations revealed a trade‐off between sensitivity and specificity, mediated by TCR clustering and NP geometry. Large TCR clusters enhance sensitivity to both NP valence and NP concentration at the expense of antigen specificity. This loss of specificity arises from two key effects of TCR clustering on NP binding: (1) steric hindrance caused by TCR proximity and NP size, leading to early saturation of bound TCRs; and (2) increased the avidity of multivalent low‐affinity NPs. The combination of saturated high‐affinity binding and amplified low‐affinity binding resulted in impaired affinity‐based discrimination. Finally, we demonstrated how kinetic proofreading (KPR) mechanisms mediated by TCR phosphorylation were able to recover specificity in models of T‐cell activation. Together, these results suggest that multivalent ligand–receptor interactions promote greater sensitivity at the expense of specificity, and provide mechanistic insights into early T‐cell activation that can guide the design of NPs for therapeutic applications.

Using numerical Monte Carlo simulations of nanoparticle‐T‐cell receptors binding, we show that the spatial organization of surface receptors is important for affinity‐based discrimination of peptide‐major histocompatibility complex ligands. Interaction geometry modulates the sensitivity of the T‐cell response to both ligand concentration and binding kinetics.

INTRODUCTION

T‐cell activation is initiated by the binding of surface T‐cell receptors (TCRs) to specific antigen peptides associated with major histocompatibility complex (pMHC) molecules. The challenge from the T‐cell's perspective is to recognize specific foreign‐derived antigens requiring an immune response, while tolerating nonspecific self‐derived antigens. T cells perform this task of ligand discrimination with remarkable sensitivity and specificity. Experiments have shown that as few as 1–10 foreign pMHCs are sufficient to elicit activation of the T cell, whereas that same cell will regularly encounter thousands of similar self‐pMHC without triggering a response. ¹ , ² , ³ , ⁴ Furthermore, it has been demonstrated that the affinity of the pMHC–TCR interaction is a critical feature for the ligand discrimination task, wherein small differences in affinity can result in significant differences in the T‐cell response. ⁵ , ⁶ , ⁷ High sensitivity to ligand concentration and high specificity to ligand affinity are hallmarks of T‐cell‐mediated immunity. ⁸ , ⁹ There is strong evidence that during T‐cell activation, TCRs and other surface proteins form microscale clusters that are necessary for proper signal transduction. ¹⁰ , ¹¹ , ¹² , ¹³ , ¹⁴ Other studies have shown that TCRs may also exist within nanoscale clusters even prior to encountering pMHC. ¹² , ¹⁵ , ¹⁶ It has been suggested that these TCR nanoclusters may be relevant for effective ligand discrimination by increasing the local TCR density, ¹³ promoting cooperative binding ¹⁷ and enabling serial engagement of TCRs by a single pMHC. ¹⁸ , ¹⁹ Despite the TCR–pMHC interaction being itself monovalent, ²⁰ TCR clustering could result in coordinated TCR engagement and cooperative binding kinetics characteristic of polyvalent systems. ²¹ , ²²

Current biophysical models describing the sensitivity and specificity of the T‐cell response place particular emphasis on the intracellular signals downstream of TCR engagement. The KPR mechanism is a famous example, which was originally proposed by McKeithan ²³ in 1995, and it has since led to variations incorporating more sophisticated signaling networks. ⁸ , ²⁴ , ²⁵ , ²⁶ Most of these studies investigated mechanisms of KPR in the absence of specific geometrical considerations. Those that do account for TCR clustering usually assume that the clustering is explicitly induced by the binding of high‐affinity ligands, which naturally introduces a positive feedback component, leading to enhanced discrimination. ²³ , ²⁷ In the absence of this positive feedback, it is unclear whether the presence of TCR clusters, such as those observed by Pageon et al., ¹⁶ enhances ligand discriminability, particularly in the context of multivalency.

pMHC‐coated nanoparticles (NPs) have been shown to successfully stimulate T‐cell activation. ²⁸ A growing body of experimental work is dedicated to the broad applications of NP‐based immunotherapies in the context of autoimmunity, viral infection and cancer. ²⁹ , ³⁰ NPs coated only with disease‐specific pMHCs, in the absence of any costimulation, have been used to promote the expansion of specific regulatory T‐cell populations to slow the progression of Type 1 diabetes in mice. ³¹ , ³² , ³³ , ³⁴ These NPs benefit from being able to exploit polyvalency to enhance their avidity via multiple axes of design. For example, in addition to the affinity of a chosen ligand, the size of the NPs and the number of pMHCs per NP (also known as the valence) also contribute to the avidity of the NPs for a given T‐cell surface. ²¹ , ³⁵ Quantifying this avidity essentially relies on knowing the number of TCRs and pMHC in the contact area between T cells and NPs, as well as the monovalent pMHC–TCR affinity. ²¹ The organization of TCRs into nanoclusters adds another layer of complexity to this interaction, making it unclear how it impacts ligand sensitivity and specificity.

In this study, we explore this question by investigating how stationary TCR clusters, multivalent ligands and KPR mechanisms may interact to regulate the sensitivity and specificity of the TCR to NP stimulation. We present results from stochastic simulations of the polyvalent binding kinetics between these pMHC‐coated NPs and TCR nanoclusters performed by a modified Gillespie algorithm. We examine the consequences of TCR spatial organization and NP design on steady‐state dose–response curves, providing quantitative metrics pertaining to the specificity and sensitivity of our simulated TCR. Our solutions reveal important trade‐offs along the sensitivity/specificity axis, regulated by the size and number of TCR clusters, as well as geometric properties of the NPs.

RESULTS

TCR clusters promote polyvalent binding at the expense of NP capacity

To investigate the interaction between TCR clusters and pMHC‐coated NPs, we simulated the binding of 8, 14 and 20 nm NPs, coated with 1–10 pMHC per NP, to an idealized T‐cell surface (Figure 1a–c). Design parameters of the NPs were obtained from experimental work published by Singha et al. ²⁸ The simulated T‐cell surface consists of a 2D circular region of radius 1000 nm containing 300 TCRs randomly distributed either uniformly or in variously sized clusters (Figure 1b). Individual TCRs are represented by an exclusive radius of 5 nm, ¹⁶ accounting for the space occupied by co‐receptors. ¹²

Figure 1.

Design properties of pMHC‐coated nanoparticles (NPs) and T cell receptor (TCR) landscape impose steric constraints on receptor–ligand interactions. (a) Simulated NPs are defined by their size, valence (i.e., number of pMHC coated on NPs) and affinity of the pMHC ligand binding to a TCR. (b) Three distinct TCR landscapes on a cell surface containing 300 TCRs within a 1000 nm radius. TCRs occupy a 5 nm radius and are randomly organized in clusters containing either one (left), three (center) or 20 (right) TCRs per cluster (TPC). The low‐density cluster (left) corresponds to uniformly distributed TCRs. (c) Schematic of the contact area between NPs and T‐cell surface. (d) Monte Carlo estimates of NP carrying capacity of each surface in (b) for three different NP radii, averaged over 30 simulations. Curves are color‐coded by NP radius (r) according to the legend. (e) Average number of covered TCRs per NP as a function of NP binding order. Each subpanel corresponds to a different NP size with r = 8 nm (top), r = 4 nm (middle) or r = 20 nm (bottom). Curves are color‐coded by the type of surface landscape specified in (b). (f) Examples of time‐series simulations showing the number of bound TCRs for NPs (r = 20 nm, 5 pMHC per NP, K _D  = 10⁻¹) binding to 1 TPC (black), 3 TPC (blue) and 20 (red) TPC surfaces. (g) Examples of steady‐state distributions of bound TCRs obtained from 30 independent simulations for the same surfaces specified in (b). NP properties are the same as in (f).

We defined the surfaces by the number of TCRs belonging to each cluster, maintaining the total number of TCRs fixed at 300 for all simulations. The uniform surface contained only one TCR per cluster (1 TPC), whereas the surface with the largest clusters contained 20 TPC (Figure 1b). The cluster radius for each surface was defined in such a way that the intracluster TCR density was constant for all surfaces, with the exception of the 1 TPC surface.

NPs are simulated to collide with these surfaces at a rate determined by the NP concentration (expressed in NP molecules per second). The contact area between the NP and cell surface defines the TCRs covered by a specific NP, allowing for pMHC–TCR binding and excluding other NPs from binding these TCRs (Figure 1c). NPs that do not bind any of their pMHC are assumed to diffuse quickly away from the surface and are removed from the simulation. Individual NP and TCR positions are randomly initialized and are assumed to be static to obtain steady‐state results. Consequences of modeling static TCRs are addressed in the Discussion section. These configurations of T‐cell surface landscapes will be maintained throughout this study.

We defined the carrying capacity of each surface to be the greatest number of nonoverlapping NPs that can simultaneously bind a given landscape. This capacity was estimated via Monte Carlo simulations by enforcing the condition that NPs remain attached to the surface provided they cover at least one TCR (by essentially setting the dissociation constant to K _D  = 0). Running these simulations until the probability of a new NP binding is below a certain threshold (P < 0.002), we then counted the number of bound NPs per surface (Figure 1d) as well as the number of TCRs covered by each NP (Figure 1e) as a function of the order of NP arrival. The sequence by which NPs bind to T cells influences the number of TCRs that are likely to be covered by that NP, with the earliest NPs to arrive generally binding more TCRs, on average, than later NPs. Three NP sizes were considered, choosing their radii to be r = 8, 14 and 20 nm, and results were averaged over 300 independent realizations. As expected, larger NPs covered more TCRs per NP at the cost of lower carrying capacity due to steric hindrance. This effect was amplified by the clustering of TCRs.

Interestingly, the carrying capacity appeared to decrease faster than the increase in covered TCRs. For example, from 1 TPC to 20 TPC, we observed a 3‐fold decrease in the carrying capacity for NPs of radius 20 nm. However, this did not manifest in a 3‐fold increase in covered TCRs per NP. This indicated that a significant portion of TCRs is inaccessible for binding, resulting from the inefficient packing of spheres on a plane. While this may appear to suggest that large NPs would be a poor choice to maximize TCR engagement, this is only true once the surface is saturated with NPs. Indeed, Figure 1e indicates that the first bound NPs cover more TCRs, facilitating polyvalent interactions with TCR clusters and thus enhancing their avidity at low NP concentrations.

To further quantify these effects, we simulated the time series of the number of bound TCRs for different T‐cell surfaces. The time series obtained from these simulations (Figure 1f) revealed differences in binding kinetics for each surface. The 1 TPC and 20 TPC surfaces exhibited, respectively, the slowest/fastest dynamics due to the lack/abundance of cross‐linking reactions. Simulations were run for 10⁶ s of simulated time to ensure that the time series converged to a steady state. By computing the steady‐state distributions of bound TCRs from the last 50 time points of each simulation, for 30 independent realizations (Figure 1g), we found that these distributions were affected both by the TCR landscape as well as by the properties of the NPs, presenting a significant challenge for the T cells tasked with affinity‐based ligand discrimination. This motivated the question of how TCR clustering influences the sensitivity and specificity of the dose–response profiles of bound TCRs when various NP design parameters are altered.

TCR clusters impact ligand discriminability in a K _D ‐dependent manner

To determine whether TCR clustering enhances ligand discriminability, we examined how variations in the dissociation constant K _D affect the features of the dose–response profiles. Simulations were conducted by setting the NP radius and valence to (r,v) = (20 nm,5 pMHC), ensuring multivalent binding. These simulations were run across a range of NP concentrations for three distinct T‐cell surfaces: 1 TPC (uniform), 3 TPC and 20 TPC landscapes (Figure 2a). As before, all surfaces contained the same total number of 300 TCRs distributed in a 1000 nm radius. We considered four different ligand–receptor affinities defined by the dissociation constant K _D , ranging from 10⁻³ to 10⁰ arbitrary units (a.u.). These values are obtained from the ratio K _D  = k _off /k _on by varying k _off and maintaining k _on = 0.1 s⁻¹ fixed, where k _off and k _on represent the monovalent TCR–pMHC unbinding and binding rates, respectively. For each condition, we simulated 30 independent realizations using randomized initial TCR and NP positions. Each simulation was run for 10⁶ s of simulated time to ensure steady‐state distributions of bound TCRs (Figure 2b).

Figure 2.

How the T‐cell receptor (TCR) landscape influences nanoparticle (NP) binding to various surfaces. (a) Examples of three surface configurations of TCRs per clusters (TPC), showing surfaces with 1 TPC, 3 TPC and 20 TPC. Unless otherwise specified, the results shown correspond to simulations performed using 20 nm NPs with valence 5 pMHC per NP. (b) Simulated dose–response curves showing bound TCRs as a function of pMHC concentration for three choices of ligand affinity indicated by k _off. Colors correspond to the 1 TPC (black), 3 TPC (blue) and 20 TPC (red) surfaces. Points indicate median steady‐state bound TCRs obtained from 30 randomly initialized simulations, with error bars corresponding to the range of the bound TCR distributions. The dose–response profiles are fit using Hill functions and plotted as solid lines on the same axes. (c) E _Max and EC₅₀ trends estimated from fitting the dose–responses and plotted against each other for different values of k _off, as indicated by the legend. The circle size correlates with the size of the TCR clusters, corresponding to 1, 3, 5, 10 and 20 TPC in ascending order. (d) Heatmaps of the mutual information between distributions of bound TCRs and K _D for surfaces with 1 TPC (left), 3 TPC (middle) and 20 TPC (right). Hotter regions indicate better discriminability of ligand affinity. (e) Minimum pMHC concentration required to attain 10 bound TCRs plotted against dissociation constant K _D for surfaces with 1 TPC (black), 3 TPC (blue) and 20 TPC (red). Dashed lines correspond to the average slope for the three highest affinity ligands: K _D  = {10⁻³, 10⁻², 10⁻¹}. Dotted lines correspond to the average slope for the three lowest affinity ligands: K _D  = {10⁻²,10⁻¹,10⁰}. (f, g) Discriminatory power for high‐ (f) and low‐ (g) affinity ligands, shown as a function of TCR clustering, corresponds to the slope of the dashed and dotted lines in panel e, respectively. Colors represent the type of surfaces matching the horizontal axis and facilitate comparisons with other panels. (h) Simulated dose–response curves of NPs binding to mixed surfaces containing both clustered and nonclustered TCRs for three choices of k _off. Each cluster contains 20 TCRs, identical to the 20 TPC surface, with colors representing surfaces containing 0, 5, 10 and 15 clusters per surface. Examples of the 1 TPC and mixed surfaces are shown above. The 1 TPC (black) and 20 TPC (red) surfaces are further included in the dose–response curves below for comparison. (i) E _Max versus EC₅₀ trends are obtained from the dose–response curves of mixed surfaces in (h). Circle size correlates with the number of TCR clusters, including 0, 5, 10 and 15 clusters per surface. Colors represent ligand affinity as indicated by the legend.

These simulations were then used to plot the steady‐state dose–response curves of the number of bound TCRs as a function of pMHC concentration (NP concentration multiplied by NP valence) for different ligand dissociation constants k _off (Figure 2b); this revealed that increasing the affinity of the ligand results in both a horizontal shift toward lower pMHC concentrations as well as a vertical scaling of the dose–response reflecting greater TCR engagement. These transformations result in reduced EC₅₀ and greater E _Max for higher‐affinity ligands (Figure 2b). Indeed, plotting the fitted E _Max against the EC₅₀ showed that larger clusters of TCRs can amplify the response by similarly reducing the EC₅₀ and increasing E _Max (Figure 2c). This can be understood as enhanced sensitivity due to increased amplitude response while simultaneously decreasing the necessary quantity of ligands. Intersecting trajectories for different affinities in the EC₅₀–E _Max space suggests that increasing the size of TCR clusters may not be sufficient for the cell to resolve k _off. Furthermore, large TCR clusters may reduce E _Max due to steric hindrance and competition for TCRs between NPs, particularly for high‐affinity ligands. Therefore, large TCR clusters may impair discriminability at high ligand concentrations.

Calculating the mutual information ³⁶ between the distributions of bound TCRs and K _D as heatmaps (Figure 2d) indicated that peak K _D discrimination occurs at intermediate ligand concentrations, and that TCR clusters greatly affect the size of this discriminatory regime (orange and red regions in Figure 2d). Interestingly, these heatmaps further show an expansion of the yellow regime between 3 TPC and 20 TPC in the upper right corner (Figure 2d), representing impaired discriminability due to saturation of bound TCRs.

To examine the effect of cluster size on discriminatory power, ³⁷ we defined a threshold of 10 bound TCRs and used it to subsequently calculate the minimum pMHC concentration necessary to cross this threshold for each surface and K _D value. The slope of the threshold concentration against K _D defines the discriminatory power of a particular TCR landscape (Figure 2e–g). We found that increasing the cluster size consistently lowers the minimum pMHC concentration to cross the 10‐bound TCR threshold but in a K _D ‐dependent manner. We compared the discriminatory power of the three lowest and three highest affinity ligands (Figure 2f, g), revealing that larger TCR clusters enhance the discriminatory power of low‐affinity ligands at the expense of high affinities.

Fixing the size of the clusters to 20 TPC, we then examined how varying the number of clusters affects the dose–response in mixtures of clustered and free TCRs (Figure 2h). We considered 4 surfaces containing either no clusters (equivalent to 1 TPC), 5 clusters, 10 clusters, and 15 clusters (equivalent to the original 20 TPC surface). Increasing the number of clusters primarily affects E _Max for low‐affinity ligands, but its effect progressively shifts toward EC₅₀ as ligand affinity increases. Once again, this suggests that TCR clusters primarily enhance the sensitivity of the TCR by raising E _Max and lowering EC₅₀.

Examining valence sensitivity across TCR landscapes

It has been previously demonstrated that T cell responses to NP stimulation are influenced by the valence of pMHCs displayed on NPs. ²⁸ , ³³ , ³⁸ , ³⁹ We therefore simulated the dose–response profiles of bound TCRs as a function of NP valence in the presence of 1 TPC and 20 TPC surfaces (Figure 3a, left). The valence range was taken to be between 1 and 10 pMHC per NP, ensuring that the parameter regimes considered are rate‐limited by either pMHC or the TCR number. The obtained dose–response curves of bound TCRs suggest that the clustered 20 TPC surface is more sensitive to changes in NP valence, whereas the 1 TPC surface exhibits greater sensitivity to the kinetic parameter K _D (Figure 3a, right). Increasing NP valence was found to lead to greater numbers of bound NPs for the 1 TPC surface but not for the 20 TPC surface, with the latter allowing for greater pMHC–TCR cross‐linking reactions per NP (Figure 3b).

Figure 3.

Role of valence of nanoparticles (NPs) in shaping their interaction with different T cell receptor (TCR) surface landscapes. (a) Simulated dose–response curves showing bound TCRs for both 1 TPC and 20 TPC surfaces. Colors from light to dark represent decreasing NP valence with values v = {1–5, 10} pMHC per NP. (b) Same as panel a for bound NPs instead. (c) Heatmaps depicting the mutual information between distributions of bound TCRs and NP valence for both 1 TPC and 20 TPC surfaces. Hotter regions indicate better discrimination of NP valence. (d–f) E _Max versus EC₅₀ trends for the 1 TPC (d), 3 TPC (e) and 20 TPC (f) surfaces, respectively. Colors correspond to different ligand affinities according to the legend. Circle size grows with increasing NP valence for values v = {1–5, 10}. (g) The minimum pMHC concentration required to achieve binding of at least 10 TCRs is plotted against the dissociation constant K _D for valences 2, 5 and 10 pMHC per NP. Dashed lines correspond to the average slope of the three highest affinity ligands: K _D  = {10⁻³, 10⁻², 10⁻¹}. Dotted lines correspond to the average slope of the three lowest affinity ligands: K _D  = {10⁻², 10⁻¹, 10⁰}. (h) Discriminatory power as a function of NP valence for low‐ (top) and high‐ (bottom) affinity ligands, obtained from the slopes of the respective dotted and dashed lines of panel g. Colors represent NP valence and facilitate comparison with panel g.

The enhanced valence‐sensitivity of the 20 TPC surface is further supported by the mutual information encoded by the distribution of bound TCRs relative to a uniformly sampled NP valence (Figure 3c). Interestingly, the highest mutual information score was obtained for low‐affinity pMHCs at high NP concentrations. This is likely once again due to greater cross‐linking reactions per NP associated with the 20 TPC surface.

Examining the E _Max–EC₅₀ trends revealed that NP valence modulates both the amplitude and sensitivity of the dose–response curves (Figure 3d–f). This effect of NP valence is strongly dependent on the spatial organization of TCRs. Specifically, it was found that 1 TPC surface has difficulty resolving NP valence, although trajectories in E _Max–EC₅₀ space appeared well separated based on k _off (Figure 3d). Meanwhile, the 3 TPC surface contained roughly the same number of TCRs as pMHC in the NP contact area, allowing it to prevent steric hindrance between NPs while ensuring good discrimination of both NP valence and k _off (Figure 3e). In the 20 TPC surface, a shift toward very good discrimination of NP valence at the expense of k _off discriminability was observed, as highlighted by the relative proximity of the k _off trajectories in the EC₅₀–E _Max space (Figure 3f).

To characterize the discriminatory power of the 20 TPC surface, we used the same method described earlier (Figure 2e). Once again, the pMHC concentration at which the number of bound TCRs crosses a threshold of 10 TCRs was obtained and plotted against the dissociation constant K _D (Figure 3g). Greater NP valence led to consistently reduced concentrations necessary to attain the aforementioned threshold of bound TCRs. This parallels the effect of TCR clustering, in which greater levels of multivalent binding led to enhanced sensitivity.

As previously described (see Figure 2), the slopes of the linear fits for low and high affinity ligands (Figure 3g) were used to show that discriminatory power is also affected by NP valence (Figure 3h). The peak discriminatory power for weaker ligands (K _D  = 10⁻² to 10⁰) occurred at intermediate NP valence, whereas for high‐affinity ligands, this peak occurs at lower valence. This demonstrates that weaker ligands disproportionately benefit from high‐valence NPs, possibly contributing to reduced specificity at such large numbers of pMHCs per NP.

Taken together, these analyses indicate that surfaces with clustered TCRs are more sensitive to the valence of NPs rather than the affinity of the ligand, but the converse is true for spatially uniform TCR landscapes.

Dual effects of TCR clustering and KPR on sensitivity and specificity

Upon ligand binding, TCRs undergo a number of modifications in the form of phosphorylation steps, a well‐documented process central to T‐cell activation. ⁴⁰ The KPR model for TCRs was developed ²³ (Figure 4a) and further modified ⁸ , ¹⁸ , ²³ , ²⁵ , ⁴¹ to explore the impact of these phosphorylation steps on T‐cell sensitivity and ligand discrimination. Here, we investigated how the interplay between TCR clustering and KPR affects the sensitivity and specificity of the early T‐cell activation signal.

Figure 4.

Impact of the kinetic proofreading mechanism (KPR) on the sensitivity and specificity of T cells with clustered T‐cell receptor (TCR) surface. (a) Schematic of the canonical KPR mechanism with a phosphorylation rate k _p and N phosphorylation steps. Only the final state C _N contributes to activation of the T cell. (b) Schematic of the modified KPR mechanism by Francois et al., ⁸ including negative feedback mediated by Src homology 2 domain phosphatase‐1 (SHP‐1) represented by state variable S. The concentration of activated SHP‐1 is determined by the occupancy of the C ₁ state and acts to increase the backward dephosphorylation rates, given by b _p  + S. (c) Simulated dose–response profiles of bound (top row) and phosphorylated (middle and bottom rows) TCRs for three different surfaces: 1, 3 and 20 TPC, obtained using NPs of radius 20 nm and valence 5. Colors from light to dark correspond to increasing ligand affinity for K _D  = {10⁰, 10⁻¹, 10⁻², 10⁻³}. Top row depicts dose responses without KPR, middle row with the original KPR (k _p  = 0.14, N = 5) and bottom row for modified KPR with negative feedback (k _p  = 0.14, b _p  = 0.07, N = 5, S _T  = 0.14, C _s  = 200). (d) E _Max versus EC₅₀ trends for the canonical KPR mechanism. Increasing circle radius corresponds to increasing k _p  = {10⁻³, 5 × 10⁻³, 10⁻²}, with the largest circle representing the no‐proofreading case for comparison. The number of phosphorylation steps is fixed at N = 3. (e) Discriminatory power of the canonical KPR mechanism represented by the slope of the minimum pMHC concentration needed to reach at least one fully phosphorylated TCR plotted against K _D when k _p  = 0.01. Different colors correspond to different numbers of phosphorylation steps N, as specified in the legend. (f) Heatmaps showing the activating regimes of various thresholds for the 1 TPC (top row) and 20 TPC (bottom row) surfaces. Threshold values are labeled by the level curves and correspond to either bound TCRs for the no‐proofreading case (left) or fully‐phosphorylated TCRs for the canonical (middle) and modified (right) KPR mechanisms.

Using the canonical form of the KPR mechanism (Figure 4a), it has been previously shown that the number of signaling TCRs ²³ is given by

$$C_N=B\times {\left(\frac{k_p}{k_p+k_{\mathrm{off}}}\right)}^N,$$ (1)

where B is the number of bound TCRs, N is the number of phosphorylation steps, k _p is the rate of each phosphorylation step and k _off is the off‐rate of the pMHC–TCR bond.

Francois et al. ⁸ proposed a phenomenological KPR mechanism that includes negative feedback mediated by the Src homology 2 domain phosphatase‐1 (SHP‐1) (Figure 4b). The model assumes that SHP‐1 is activated by the occupancy of the first phosphorylation state (C ₁) according to the expression $S=S_T\frac{C_1}{C_1+C_{*}}$, where S _T and C _* are free parameters, respectively, representing the total SHP‐1 concentration and the concentration of phosphorylated TCRs required to activate SHP‐1. The equilibrium concentration of fully phosphorylated TCRs (C _N ) was derived analytically, yielding an expression for the number of productively signaling TCRs, given by

$$C_N=B\times \left(1-\frac{r_{-}}{r_{+}}\right)r_{-}^N,$$ (2)

where B is the number of bound TCRs, and r _± are functions of S (Eqn. 6).

We compared the effects of both models on our dose–response curves and found that both can selectively attenuate the low‐affinity response across the full range of NP concentration (Figure 4c). By visual inspection of the dose–response profiles, KPR via TCR phosphorylation (according to both mechanisms) does not contribute significantly to the specificity of the 1 TPC surface, but becomes particularly relevant for the clustered surfaces as they achieve greater numbers of bound pMHC–TCR complexes. This suggests complementary roles for both TCR clustering and TCR phosphorylation acting in tandem to regulate the sensitivity and specificity of the early TCR.

Focusing initially on the canonical KPR mechanism, we found that it affects the dose–response profile by a vertical scaling factor, resulting in reduced E _Max without affecting the EC₅₀ concentration (Figure 4d). The phosphorylation rate k _p affects the attenuation of E _Max, with lower k _p values corresponding to lower E _Max. More significantly, k _p sets the range of K _D values most influenced by the canonical KPR mechanism, with higher values of k _p significantly reducing the E _Max of low‐affinity ligands while leaving high‐affinity ligands largely unaffected (Figure 4d). As the value of k _p decreases, the KPR mechanism begins to affect the E _Max of higher‐affinity ligands as well.

The number of phosphorylation steps, N, also dictates the magnitude of signal attenuation. In the canonical KPR mechanism, greater values of N lead to fewer TCRs reaching the final state necessary for productive signaling. Notably, only a few phosphorylation steps are required to enhance the discriminatory power (Figure 4e). Additional phosphorylation steps disproportionately reduce the phosphorylation of low‐affinity ligands, while leaving high‐affinity ligands relatively unaffected. This means that low‐affinity NPs require much higher concentrations to attain the same threshold of fully phosphorylated TCRs, which accounts for the higher discriminatory power.

Because T cell responses are often described as digital, ⁸ exhibiting an all‐or‐none response to antigen stimulation, most models of T‐cell activation incorporate a threshold ³⁷ , ⁴² for the number of phosphorylated TCRs above which the T cell is considered activated. To assess this, we compared the activation regimes of the 1 TPC and 20 TPC surfaces for various thresholds with or without a KPR mechanism (Figure 4f). In the absence of KPR, we found that TCR clusters enlarge the regimes of activation for all thresholds of bound TCRs. The shift of these regimes toward lower concentrations of pMHC reflects the enhanced sensitivity conferred by TCR clustering. We also observed that the choice of threshold can be used to discriminate low‐ from high‐affinity NPs on the 1 TPC surface without KPR. However, these same thresholds on the 20 TPC surface can be crossed by all values of K _D , provided the pMHC concentration is large enough. This further supports the notion that TCR clusters enhance sensitivity at the expense of ligand specificity.

Interestingly, specificity can be recovered via either KPR mechanism (canonical and modified), depending on the choice of threshold on the number of fully phosphorylated TCRs. In fact, the inclusion of negative feedback in the modified KPR mechanism was found to effectively prevent low‐affinity activation independent of ligand concentration, consistent with previous observations. ⁸ We propose that the combination of surface TCR clustering, combined with KPR mediated by TCR phosphorylation, is necessary and sufficient to account for both sensitivity and specificity of the early T cell response to multivalent ligand stimulation.

DISCUSSION

Our study explored the importance of TCR clustering and polyvalent interactions in mediating NP‐induced T‐cell activation. Our model of polyvalent binding is governed by three geometric features: NP size, NP valence and the spatial distribution of TCRs. These variables, together with the affinity of the pMHC–TCR complex, determine the avidity of the NP‐surface interaction. Within this framework, we found that TCR clustering enhances sensitivity to both NP concentration and valence. However, this increased sensitivity comes at the expense of NPs' carrying capacity. This reduced capacity leads to earlier saturation of bound TCRs, resulting in hampered K _D discrimination at high ligand concentrations.

Previous modeling work ²³ , ²⁷ has demonstrated that the induced clustering of TCRs enhances K _D discrimination. These models assume positive feedback between ligand binding and TCR clustering while also ignoring steric hindrance exacerbated due to clustering. By explicitly accounting for the spatial arrangement of these TCRs and by preventing TCR mobility, we demonstrated that TCR clusters do not necessarily confer specificity in multivalent interactions. Instead, we found that these clusters tend to enhance both the sensitivity and amplitude of the response. This is reflected by the trends in the EC₅₀ and E _Max of the dose–response curves, which, respectively, decrease and increase with TCR clustering. If we consider a T cell undergoing activation with a surface transitioning from 1 TPC to 20 TPC via induced clustering, our results suggest that the 1 TPC surface is more effective at restricting the initial binding to high‐affinity ligands. The subsequent transition to 20 TPC then serves to amplify the signal at low concentrations of the cognate ligand. Although our findings do not exclude the possibility that induced clustering can serve as a KPR mechanism, they indicate that positive feedback in the induced clustering model is necessary to account for enhanced specificity.

It is important to note that preclustered TCRs on the surface of T cells may sacrifice specificity in exchange for greater sensitivity to ligand concentration and amplified responses. ¹⁶ This trade‐off would benefit antigen‐experienced memory T cells by ensuring early and rapid detection at low concentrations of antigen. ¹²

T‐cell activation can enhance sensitivity by means of TCR clustering while also retaining downstream specificity via an intracellular KPR mechanism. Depending on the kinetic parameters of early TCR phosphorylation events, a small number of such events (e.g., N = 2) are sufficient to filter out low‐affinity signals, without affecting the amplitude of the high‐affinity response. TCR clusters enhance the sensitivity of the TCR by lowering the EC₅₀ and raising the E _Max of the dose–response curves. However, this tends to raise the E _Max of the low‐affinity response more than that of higher affinities. The KPR mechanism resolves this issue by attenuating the amplitude of the low‐affinity signal while retaining the elevated E _Max for high‐affinity ligands. This mechanism, combined with the cooperativity associated with clustered TCRs, allows the TCR to be highly sensitive to both ligand affinity and concentration.

Our results allowed us to postulate about certain design parameters of NP‐based immunotherapy. Notably, the choice of NP geometry, including the valence of NPs, determines the specificity of the treatment. Our model predicts that higher valence NPs are less specific to any given TCR, thereby inducing the activation of a larger portion of the T‐cell population. High‐valence NPs can provide the necessary avidity to activate noncognate T cells using weaker antigens. This would be especially relevant in the treatment of autoimmune diseases, wherein NPs coated with self‐antigens are able to promote the expansion of regulatory T cells. ³² A recent study by Ols et al. ⁴³ has demonstrated that using multivalent antigen‐coated NPs increased B‐cell clonotype diversity via the recruitment of low‐affinity B cells, in agreement with the conclusions of our model. Alternatively, low‐valence NPs may prove useful for identifying suitable TCR clonotypes for adoptive cell therapies. Our results indicated that relatively few pMHC per NP are necessary to achieve high cooperativity. Coating more pMHCs per NP would increase the chances of cross‐reactivity without much benefit to the avidity of the cognate TCR. Hence, a strategy for identifying disease‐specific TCR clonotypes may rely on using NPs with limited valence. The experimental use of NP valence as an axis to modulate TCR selectivity versus population‐level activation would represent important validation of the results presented here.

The combination of NP valence and size could also serve to target cells based on TCR organization. Since observing individual NPs bound to the cell surface is easier than observing individual TCRs, this may provide an indirect method for detecting TCR clusters without relying on superresolution imaging, similar to what has been done previously. ¹² Leveraging the known NP properties, perhaps using combinations of NP sizes, the number of bound NPs of each type can then be compared to another measure of T‐cell activation to infer meaningful information about the size and distribution of the clusters and the density of TCRs within them. This would help resolve the ongoing debate regarding the existence of preclustered TCRs. ²⁰ , ⁴⁴

It has been widely reported that monovalent pMHC–TCR interactions are sufficient for initiating T‐cell activation. ³ , ²⁰ In contrast, our simulations suggest that monovalent NPs fail to engage the number of TCRs required for robust and reliable activation. This discrepancy between our simulations and experimental observation can be partly attributed to the fact that most of these experiments rely on supported lipid bilayers coated with either monomeric TCR or pMHC ligands. This effectively traps the ligand–receptor interaction between the cellular surface and supported lipid bilayers, ensuring ample opportunities for productive TCR–pMHC encounters. In solution, T‐cell activation was observed with as few as three peptides bound to any given cell ¹ ; however, this result was achieved using a highly specific, strong agonist (K _D  = 5 pM).

Finally, the conclusions of our study challenge the conventional wisdom that greater levels of polyvalent binding always result in more specific interactions. ³⁵ Competition for TCRs, steric hindrance and the inherent stochasticity of ligand–receptor binding can place fundamental limits on the ability of T cells to resolve various features of polyvalent TCR engagement. ²¹ , ²² , ⁴⁵ , ⁴⁶ This study highlighted the critical role of TCR spatial organization in adaptive immunity and underscored the need for further experimental investigations into TCR surface arrangement. Many open questions still remain pertaining to the dynamics of TCR clustering during activation ¹⁴ , ⁴⁷ and the role of co‐receptors and the complex spatial relationships with other surface proteins. ⁴⁸ Despite this, the present analysis established a theoretical framework able to guide the design and evaluation of future such studies.

METHODS

TCR landscapes

TCR positions were generated on a circular 2D surface with a 1 μm radius. First, the number of TCRs per cluster was defined; given that the total number of TCRs was 300 for all simulations, which also determined the number of clusters on the surface relative to the cluster sizes. The radius of the TCR clusters was set to ensure a consistent intracluster TCR density across surfaces. This intracluster TCR density corresponds to 20 TCRs within a circular radius of 50 nm, consistent with experimentally detected nanocluster densities from Pageon et al. ¹⁶ Cluster center positions were then generated using a random Poisson point process within a polar coordinate system. To ensure all clusters were fully contained within the model surface, centers were placed at least one cluster radius away from the boundary. Overlapping clusters were avoided by rejecting positions within a cluster diameter of one another, with resampling performed until the required number of nonoverlapping cluster centers was achieved. Individual TCR positions within each cluster were subsequently sampled using another Poisson point process constrained within the cluster radius. Minimum spacing between TCRs was set to 10 nm, consistent with experimental measurements ¹⁶ ; this TCR diameter also accounts for space occupied by co‐receptors. For more details regarding the design of these TCR clusters, see Table 1.

Table 1.

Size and density of simulated TCR clusters.

TCRs per cluster	No. of clusters	Radius of clusters (nm)
20	15	$50$
10	30	$50/\sqrt{2}$
5	60	$25$
3	100	$50/\sqrt{6}$
1	300	Undefined

NP geometry

The geometry of the simulated NPs was modeled based on experiments performed by the Santamaria Lab (University of Calgary), as outlined by Singha et al. ²⁸ These NPs were developed to stimulate T cells via direct engagement of the TCR for the treatment of Type 1 diabetes. Notably, these NPs were coated exclusively with disease‐specific pMHC and were engineered using a variety of NP sizes and pMHC density. For our simulations, we chose to primarily simulate 20‐nm radius NPs with a pMHC density of approximately 0.4468 pMHC 100 nm⁻² in order to examine the role of multivalent binding. This choice ensures that multiple TCRs and approximately 5 pMHCs per NP are present in the contact area.

Stochastic simulations

Monte Carlo simulations of NP–cell surface interactions were run in MATLAB using a Tau‐Leaping algorithm to model the binding kinetics. ⁴⁹ This algorithm is based on the well‐known Gillespie algorithm, ⁵⁰ but uses an adaptive time‐step of length τ. Our implementation of this algorithm consisted of the following steps:

1. Defining the state variable x(t) = {X _i (t)}, where X _i  = {−1,0,1} is the state of the ith TCR (for i < 300). Each TCR could exist as either free (unbound) with X _i  = 0, covered (but still unbound) with X _i  = −1, or bound to a pMHC with X _i  = 1. The covered state corresponding to X _i  = −1 is necessary to track which TCRs are competing for binding with the same NP, and consequently, modulates the cross‐linking binding rates.

2. Updating reaction rates R _j . These rates included the propensity of TCR–pMHC binding and unbinding, phosphorylation events and NP adsorption. These reactions and their corresponding propensities are detailed in Table 2.

3. Choosing the time step τ. The maximum step size was set to τ = 500 s.

4. Generating a random number of events K _j  ∼ Poisson(R _j τ) for each event type E _j . These events correspond to the binding and unbinding of ligand–receptor complexes, TCR phosphorylation and the adsorption of new NPs to the cell surface. The step size was reduced if the number of reactions exceeded the current number of bound NPs. This was done to ensure efficient computations at high NP concentrations without compromising the accuracy of the steady‐state quantification.

5. Checking for overlap of NPs. New NP arrivals had to avoid overlap with NPs already bound to the surface. Failing this condition resulted in the removal of the new NP.

6. Updating state variables. This was done using the following equation:

$$x\left(t+\tau \right)+x\left(t\right)+{\sum }_j{K}_j{v}_{\mathit{ij}},$$ (3)

where v _ij represents the change in state X _i due to event E _j .

• 7Repeating steps 2–6, until the end of the simulation, taken to be t _end = 10⁶ s.

Table 2.

Reactions and propensities implemented in the Tau‐Leaping algorithm.

Reaction	Propensity	Description
NP adsorption	ρK 0	The arrival rate of NPs on the cell surface, determined as the product of NP concentration (ρ) with the adsorption rate (K 0)
TCR cross‐linking	k on (v − B TCR) (n t  − B TCR)	The reaction that depends on the pMHC–TCR binding rate (k on), the NP valence (v), the number of TCR in the contact area (n t ) and the current number of bound pMHC–TCR complexes (B TCR)
TCR unbinding	k off B TCR	The product of the pMHC–TCR unbinding rate (k off) with the number of bound TCRs (B TCR)
TCR phosphorylation	k p B TCR	The product of the TCR phosphorylation rate (k p ) with the number of bound TCRs (B TCR)

Every simulation run was composed of 30 independent realizations using randomized initial conditions and TCR positions. Trials were run in parallel by means of the Parallel Computing Toolbox available in MATLAB, using servers from Compute Canada. Our simulations did not include TCR mobility and did not account for the dynamic reorganization of TCRs known to occur during T‐cell activation. This was to ensure that our sensitivity and specificity metrics are obtained at steady state, while also disambiguating the effects of TCR geometry from feedback due to induced clustering by ligand–receptor binding. Furthermore, simulating uniform cluster sizes on an idealized TCR landscape enabled precise quantification of cluster parameters and their impact on ligand discriminability.

Creating and fitting the dose–response curves

The steady‐state number of bound TCRs was estimated from the average number of bound TCRs over the final 50 time points. Doing this across 30 independent realizations provided a distribution of bound TCRs for each condition generated by perturbing a specific component of the model (e.g., TCR landscape and valence). The average of this distribution was plotted against either the NP or pMHC concentration to produce the dose–response curves for bound TCRs or bound NPs.

These dose–response curves were then fit using Hill functions of the form

$$f\left(x\right)=\frac{E_{\max }{x}^n}{{\mathrm{EC}}_{50}^n+x^n},$$ (4)

where E _Max is the maximum response, EC₅₀ is the half‐maximum activation and n is the Hill coefficient. Fitting was performed using the fit function from the Curve Fitting Toolbox in MATLAB to generate the best fit estimates for E _Max, n and EC₅₀.

Estimates for the Hill coefficient n range between 0.5 and 2. We accepted noninteger estimates of n, as this reflects the average cooperativity resulting from the stochastic ligand–receptor interactions. Due to the random configurations of NP and TCR positions, identical NPs do not share the same avidity. Furthermore, restricting our fits to only integer values of n yielded poor estimates of the EC₅₀ and E _Max, which represented the primary focus of our analysis.

Discriminatory power

To quantify T‐cell discriminatory power, we adopted the measure introduced by Pettman et al. ³⁷ This measure relies on defining a threshold for activation (chosen to be 10 bound TCRs in our case) to define pMHC potency, that is, the concentration of pMHC required to cross the selected threshold. pMHC potency was then plotted against K _D on a log–log scale and the discriminatory power was estimated from the slope of the linear fit on this graph. Setting the threshold to 10 bound TCRs ensured that even weak ligands are not excluded from the analysis. Other threshold values were considered and led to similar conclusions.

Kinetic proofreading

We incorporated two KPR mechanisms to examine the consequences of TCR geometry on downstream activation. The first mechanism is the canonical KPR proposed by McKeithan ²³ with N irreversible phosphorylation steps, while the second is a modified KPR mechanism with N reversible phosphorylation steps and a negative feedback, as described by Francois et al. ⁸ In both mechanisms, the N steps, corresponding to the phosphorylation of the TCR's ITAMs on the cytoplasmic tail of the CD3 subunit and mediated by Zap70 and LcK, ⁴¹ were assumed to occur following the initial pMHC binding event (Figure 4a, b). These intermediate steps are critical for T‐cell activation and have been shown to enhance the specificity of the TCR to cognate antigen at a cost to sensitivity and speed of activation. The inclusion of a negative feedback term, as proposed by Francois et al. ⁸ was meant to represent the contribution of the SHP‐1 phosphatase, a known negative regulator of TCR phosphorylation.

Both mechanisms assume that T‐cell activation occurs once the occupancy of the final KPR step exceeds a certain threshold. Furthermore, dissociation of the pMHC–TCR complex immediately reverts the TCR to its original unphosphorylated state. These assumptions are common to other models of the KPR mechanism. ⁸ , ²³ , ⁴¹ The two mechanisms were implemented as deterministic sets of ordinary differential equations, whose steady‐state solutions can be readily determined either analytically (for the canonical KPR) or numerically (modified KPR with negative feedback). This was done in part to lighten the computational cost of the stochastic simulations, but this choice also reflects the simplifying assumption that the KPR steps are independent of the surface geometry and of TCR positions. While this assumption may not fully capture the underlying biology, in the absence of detailed biophysical information relating to downstream signaling pathways, these phenomenological models are well established and have been shown to be capable of reproducing the qualitative features of early T‐cell activation. ⁸ The resulting steady‐state solutions yielded the number of fully phosphorylated TCRs as a function of the number of bound TCRs obtained from the Monte Carlo simulations.

For the modified KPR mechanism with negative feedback, the number of fully‐phosphorylated TCRs was obtained by numerically solving Eqn. 2 alongside the following system of equations ⁸ :

$$S=S_T\frac{C_1}{C_1+C_{*}}=S_T\frac{r_{-}\left(1-r_{-}\right)}{r_{-}\left(1-r_{-}\right)+\frac{C_{*}}{B}}$$ (5)

$$r_{\pm }=\frac{k_p+b_p+S+k_{\mathrm{off}}\pm \sqrt{{\left(k_p+b_p+S+k_{\mathrm{off}}\right)}^2-4k_p\left(b_p+S\right)}}{2\left(b_p+S\right)}$$ (6)

This formalism introduces three new free parameters to the original KPR mechanism: the spontaneous dephosphorylation rate b _p , the total concentration of phosphatase S _T , and the number of phosphorylated TCRs C _* required to activate SHP‐1. The parameter values for the negative feedback KPR mechanism are presented in Table 3.

Table 3.

Parameters of the negative feedback KPR mechanism. The effective dephosphorylation rate is given by b _p  + S, which is the reason why the units of S and S _T are presented as rates. The units of C _* are expressed in the actual number of TCRs.

Symbol	Value	Description
k p	0.14 s−1	Forward phosphorylation rate
b p	0.07 s−1	Backward phosphorylation rate
S T	0.05 s−1	Maximum contribution of SHP‐1 to the dephosphorylation rate
C *	200	Concentration of C 1 required to activate SHP‐1

Software and packages

Monte Carlo simulations of the NP carrying capacity and the analysis of the dose–response curves were performed in MATLAB R2024b using functions from the Curve Fitting Toolbox, the Parallel Computing Toolbox, the Deep Learning Toolbox and the Statistics and Machine Learning Toolbox. The stochastic simulations of NP binding dynamics for generating the dose–response curves were performed using the Compute Canada servers Narval, Cedar and Graham, running MATLAB R2018b and the aforementioned packages. The codes for running the simulations and generating the figures are available on the Khadra Lab GitHub repository: https://github.com/khadralab/NP_TCR_surface_interactions (uploaded Nov 23, 2025). Simulation results can be found on the Federated Research Data Repository: https://doi.org/10.20383/103.01515. NP and KPR schematics were made using BioRender: https://BioRender.com.

AUTHOR CONTRIBUTIONS

Louis Richez: Conceptualization; investigation; writing – original draft; methodology; visualization. Anmar Khadra: Funding acquisition; writing – review and editing; supervision.

CONFLICT OF INTEREST

The authors declare no competing interests.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are openly available in NP TCR surface interactions at https://doi.org/10.20383/103.01515.