A cell-based kinetic framework enables TCR specificity prediction

Abstract

The ability to predict T-cell receptor (TCR) specificity from sequences could transform immunotherapy, vaccine development, and our understanding of immune recognition. However, progress has been shaped by “edge cases”, in which specificity appears to be captured by simplified descriptors, such as sequence motifs or correlations between binding affinity and functional activation. Although informative, these regimes are not representative of the general mode of TCR recognition. Emphasizing such cases has contributed to a drift in the field, where both experimental assay design and computational modeling increasingly rely on nonrepresentative signals, limiting generalizability across antigens and TCRs. We argue that this drift stems from a lack of a clear biophysical definition of TCR specificity and continued reliance on equilibrium binding assays that are not well suited to capture it. These limitations propagate into training datasets, constraining the performance and generalizability of predictive models. To address them, we introduce three key elements. First, we develop a cell-based assay for quantitative measurement of TCR–pMHC binding kinetics in a physiological context. Second, we introduce a mechanistic framework for interpreting these data, showing that the widely used reversible ligand–receptor model is insufficient in the general case and proposing the TCR cycle model as a minimal systems-level description. Third, we describe a strategy for generating multiplexed, high-throughput datasets and integrating mechanistic modeling with machine learning. Together, this work establishes a foundation for a mechanistically grounded and scalable approach to TCR specificity prediction.

Introduction

A few decades ago, multimer-binding technology revolutionized immunology¹ by enabling researchers to directly identify and quantify subsets of antigen-specific T cells (i.e., T cells that become activated upon binding to their target). Public databases such as IEDB (https://www.iedb.org/) and VDJdb (https://vdjdb.cdr3.net/) are largely composed of data generated during this “multimer-binding era”. While this technology remains invaluable in certain scenarios, recent insights have revealed its limitations. Over time, multimer-binding technology has biased research toward predominantly higher-affinity TCRs that are not necessarily specific. This bias is evident from two key observations: the growing number of empirical studies revealing a disconnect between multimer-binding capacity and T-cell activation (Suppl. Table 1, Fig. 1) and the realization that, while we still lack a clear biophysical definition of TCR specificity (i.e., the parameters of TCR-pMHC interaction that determine whether a T-cell becomes activated upon binding to its target), it is evident that affinity alone cannot serve as that measure.^2–4 As a result, assays that measure the binding strength of TCR–pMHC interactions at equilibrium, when decoupled from T-cell activation, cannot reliably serve as TCR specificity assays. From a machine learning perspective, training data generated from such binding assays are likely to contain both false positives and false negatives (Fig. 1), making it challenging to predict TCR specificity without explicit measurements of T-cell activation. Until a clearer biophysical definition of TCR specificity is established, it will remain necessary to rely on training data in which binding assays are coupled with T-cell functional readouts.

Fig. 1.

Current multimer-based methods for assessing TCR specificity do not support the separation of specificity and activation prediction into two independent tasks. Omitting functional readouts from multimer-binding assays is effectively equivalent to removing an essential dimension that helps differentiate between specific and nonspecific TCR populations. Figure elements created with BioRender.com

That said, while equilibrium pMHC binding, and by extension, affinity, is not a definitive proxy for specificity, pMHC multimers still provide practical value in certain contexts. For instance, using soluble pMHCs in assays where TCRs are presented on cells remains an informative entry point. These assays allow immediate evaluation of coreceptor contributions, such as CD8 and CD4, and enable pairing of binding measurements with assessments of T-cell triggering and activation. Nevertheless, existing models in the literature, including kinetic proofreading⁵ and digital signaling,⁶ underscore the importance of further investigating the kinetic parameters that govern TCR–pMHC interactions. The kinetic proofreading model proposes that productive T-cell activation depends on sustained engagement of the TCR–pMHC complex, during which a cascade of phosphorylation events occurs. This model emphasizes the complex half-life (t₁_/₂) as a key determinant of signaling. Complementing this, digital signaling models suggest that T cells convert analog inputs (e.g., dwell time, affinity) into binary outputs, reflecting a threshold-based decision-making process. These frameworks are not mutually exclusive and may represent different phases of TCR discrimination.

Nonetheless, direct experimental validation of these models has been hampered by the lack of scalable, quantitative methods for measuring TCR–pMHC binding kinetics in a cellular context. Existing cell-based assays typically rely on multimeric reagents and equilibrium binding measurements, which limit their ability to resolve individual on- and off-rates (kon and koff) or to capture the dynamics of receptor engagement. Other approaches, such as micropipette adhesion assays,⁷ offer high-resolution insights but are constrained by very low throughput, often allowing measurement from only a handful of cells per experiment. To address this gap, we developed a cell-based assay that quantitatively measures TCR–pMHC interaction kinetics using monomeric pMHC reagents. By leveraging the advantages of fast dissociation rates inherent to monomers, this assay enables precise estimation of kon and koff while preserving the physiological context of CD8 + T cells. In parallel, we assessed early signaling via CD3ζ phosphorylation to probe the relationship between binding kinetics and functional activation. As we demonstrate here, this framework not only deepens our understanding of TCR–pMHC dynamics but also establishes a foundation for predictive models that integrate mechanistic biophysical parameters, such as kinetic rates, as features for TCR specificity prediction.

In this work, we use the term “edge cases” to refer to scenarios in which TCR specificity appears to be adequately captured by simplified descriptors, such as the presence of recognizable sequence motifs or an apparent correlation between binding affinity and functional activation. Although such relationships have been observed and leveraged in prior studies, they arise in restricted regimes and do not represent the general mode of TCR recognition. Outside these special cases, affinity and sequence features alone fail to consistently predict activation, particularly across different antigens, ligands and experimental conditions. Emphasizing these edge cases can therefore obscure the underlying biophysical determinants of specificity, reinforcing the need for time-resolved, mechanistically grounded measurements of TCR–pMHC interactions that explicitly link binding kinetics to early cellular signaling.

Building on these considerations, this work follows a structured progression aimed at addressing fundamental limitations in the study and prediction of TCR specificity. We first highlight that the field lacks a clear biophysical definition of TCR specificity and remains heavily reliant on equilibrium binding assays that are not well-suited to capture it, resulting in datasets that constrain the performance and generalizability of current computational approaches. This motivates three key needs. First, there is a need for experimental assays that measure parameters more directly linked to TCR specificity; to this end, we develop a cell-based assay that enables quantitative measurement of TCR–pMHC binding kinetics in a physiological context, where TCRs are presented on cells together with their coreceptors. Second, there is a need for a mechanistic framework capable of interpreting such data; we show that the widely used reversible ligand–receptor model is insufficient in the general case and introduce the TCR cycle model as a minimal, systems-level framework that captures the observed kinetics and enables extraction of interpretable parameters. Third, there is a need to generate such measurements at scale to support predictive modeling; therefore, we outline a strategy for multiplexed, high-throughput data generation and describe how mechanistic modeling can be integrated with machine learning approaches within this framework. Together, these contributions establish a foundation for a more mechanistically grounded and scalable approach to TCR specificity prediction.

Results

Using equilibrium binding as a measure of specificity and as ground truth for predictive modeling

The initial success of equilibrium multimer-binding assays in identifying antigen-specific T cells gave rise to the hypothesis that TCRs with similar sequences or shared dominant binding motifs⁸ are likely to recognize the same pMHC. This assumption spurred the development of unsupervised machine learning models that use TCR sequence similarity as a proxy for specificity.^9,10 Recent studies¹¹ that claim these approaches are effective classifiers, despite reporting low discriminatory power, underscore the need for thorough evaluation. Here, we focus on evaluating the conceptual limits of sequence-similarity–based inference under controlled conditions rather than on optimizing model performance. While our previous study¹² is focused on supervised machine learning for TCR–pMHC specificity prediction, the present work complements these findings by examining the limitations of unsupervised approaches and introducing mechanistically grounded measurements to improve predictive modeling.

To assess the validity of this approach, we systematically reanalyzed published clustering data.¹³ (All analyses were performed under HLA-fixed conditions using the same preprocessing, distance metrics, and clustering procedures described in Materials and Methods.) Our analysis revealed that the ability of these methods to group TCRs by specificity is limited. Only a small subset of TCRs formed pure clusters enriched for a single peptide specificity (Fig. 2a), consistent with prior findings reported in Leary et al.¹³ (Fig. 4 and Supplementary Materials therein). Despite claims of classifier effectiveness,¹¹ the observed clustering patterns demonstrated low discriminatory power, highlighting the need for more robust strategies to predict TCR–pMHC interactions.

Fig. 2.

Unsupervised models group TCRs independently of their epitope specificity. a Performance of representative unsupervised machine learning models in sorting TCR sequences into antigen-specific clusters, as reported in Leary et al.¹³. The plot summarizes clustering outcomes for TCRβ chains using multiple published methods, which together span the most commonly used distance metrics and embedding approaches. Over 70% of datasets result in impure clusters, i.e., clusters that contain mixtures of TCRs specific for different peptides, demonstrating the limited specificity of current unsupervised models. Similar performance trends were observed for clustering based on TCRα or paired TCRαβ sequences (data not shown, see Leary et al.¹³). b Hierarchical clustering of 17 peptide-specific TCRβ repertoires from the IMMREP_2022 benchmark dataset¹⁴ using the tcrdist3 distance metric.^15,16 Each TCR is assigned to one of 17 clusters. The first four repertoires (indicated by the dashed line) correspond to peptides with significant CDR3β sequence motifs, as determined by sequence logo analysis. Despite motif presence, TCRs recognizing the same peptide are distributed across multiple clusters. The inset shows cluster distributions for only those TCRs that contain the dominant sequence motif for each of the four peptides, further illustrating the lack of clear separation based on motif-bearing sequences alone

To better understand the limitations underlying these observations, we conducted a targeted analysis of motif-based sequence patterns within antigen-specific repertoires. We systematically evaluated peptide-specific TCR repertoires from the IMMREP_2022 benchmark dataset¹⁴ to assess the prevalence and utility of conserved sequence motifs for specificity inference. Conserved binding motifs, as described by Culka et al.¹² were identified in only 4 out of 17 repertoires (Fig. 2b, supplementary Fig. 1), indicating limited motif occurrence across antigen-specific datasets. Even when present, TCRs containing the dominant motif were distributed across multiple clusters rather than forming a distinct group (Fig. 2b, inset), suggesting that motif presence alone does not guarantee cluster homogeneity.

To further evaluate the reliability of sequence similarity as a proxy for antigen specificity, we compared TCR sequence distances using multiple representations, including ESM2 embeddings, plain BLOSUM62 pairwise similarity, and tcrdist3 metrics.^15,16 These representations span learned embeddings, substitution-matrix similarity, and structure-informed distance metrics (see Materials and Methods). Across these metrics, we observed that TCRs targeting different peptides frequently exhibited higher sequence similarity to each other than to TCRs recognizing the same antigen (Fig. 2b, supplementary Fig. 1), reinforcing the inconsistency of sequence similarity-based grouping. These findings are consistent with recent observations by Simpson et al.¹⁷ where motifs failed to generalize beyond a few specific cases (Fig. 6, Suppl. Fig. 11 in ref. ¹⁷).

Nonetheless, in simpler peptide-specific classification tasks, distance-based clustering approaches performed on par with supervised methods,¹⁴ highlighting their limited but context-dependent utility. However, their comparable performance in narrow settings does not translate to broader applicability, especially when addressing the full diversity of TCR–pMHC interactions. As a result, the reported performance likely represents an upper bound; introducing MHC/HLA diversity would be expected to further degrade both clustering purity and classifier performance. Introducing variation in MHCs/HLAs within peptide-specific datasets would add further complexity to TCR specificity prediction. Conversely, determining the MHC/HLA type recognized by a TCR may be inherently easier than identifying the specific peptide epitope.

Unsupervised machine learning approaches based on commonly used similarity features, such as binding motifs or readouts from equilibrium multimer-binding assays, are insufficient to fully capture TCR specificity. In our recent study,¹² we also demonstrated the limitations of supervised ML methods that rely on similar data types, including equilibrium multimer-binding assays and, in some cases, functional assay readouts, particularly when applied in a general panpeptide context. The fact that neither supervised nor unsupervised ML approaches have proven broadly effective in predicting TCR specificity suggests that the underlying data may lack the conceptual resolution required to describe this complex phenomenon. This highlights the need for more advanced strategies, as discussed in the following section.

Together, these analyses indicate that limitations in current ML approaches stem not from model choice but from the nature of the data used to define specificity, motivating the need for alternative, mechanistically grounded measurements.

TCR-pMHC binding kinetics as a basis for predicting TCR specificity and enabling modeling

Several decades ago, early efforts to study TCR–pMHC binding kinetics utilized soluble monomeric pMHC molecules with photoaffinity labeling assays.¹⁸ Shortly thereafter, flow cytometry–based assays were introduced as a tool to study TCR binding to pMHC.¹⁹ These newer assays primarily focused on measuring equilibrium binding, where stable interactions are necessary. As a result, monomeric pMHCs were deemed impractical due to their rapid koff, which hindered stable binding and accurate affinity measurements. To address this, multimeric pMHC reagents became the standard, offering increased binding stability through avidity effects.

However, for kinetic measurements, a rapid koff is advantageous, as it allows direct quantification of both association (kon) and dissociation (koff) rates — key parameters for characterizing TCR–pMHC interactions. In this context, monomeric pMHCs avoid the avidity-related artifacts introduced by multimeric reagents. Building on this principle, we developed a cell-based assay (Fig. 3a) that tracks the binding and unbinding of intact CD8 + T cells to target peptide–MHC monomers using flow cytometry. In parallel, we measured CD3ζ phosphorylation to evaluate the earliest steps of T-cell activation and explore their correlation with TCR occupancy and TCR–pMHC binding kinetics.

Fig. 3.

A cell-based monomeric flow cytometry assay enables measurement of TCR–pMHC binding kinetics and recapitulates the previously observed distinctive shape of kinetic curves. a Schematic representation of the assay. Peptide–MHC (pMHC) monomers labeled with Alexa Fluor 488 are incubated with T cells to allow pMHC–TCR binding. Following fixation, cells are stained with phospho-specific monoclonal antibodies (pAb) targeting phosphorylated sites on the ζ domain of the TCR. These pAbs are directly conjugated to APC (indicated by red circles in the schematic). b Representative binding kinetics curves for OT-I T cells interacting with OVA-derived peptides (N4, T4, V4) and a “UV peptide.” Curves are shown with (red) and without (blue) anti-CD8α antibodies across varying concentrations of pMHC. The degree of labeling (DOL) for each pMHC is reported in the table. c Previously reported TCR–pMHC binding kinetics measured using microscopy³⁰ (left), photoaffinity labeling assay¹⁸ with soluble monomeric pMHC in the presence of CD8 coreceptor (middle), and in the absence of CD8α (right; inset shows T cells expressing CD8α). Figure elements created with BioRender.com

As shown here, this assay successfully recapitulated the previously observed hump-shaped TCR–pMHC–CD8 kinetic curves (Fig. 3b). The reappearance of this hump, reported by Luescher et al.¹⁸ and others (Fig. 3c), and largely overlooked since, is a fortunate observation that serves at least three key purposes: (1) it enables identification of a biokinetic model capable of describing these data by dramatically narrowing the space of possible models; (2) it indicates that a simple reversible ligand–receptor mechanism (i.e., association–dissociation) is insufficient to explain TCR–pMHC–CD8 binding kinetics; and (3) it suggests the presence of a feedback loop, whereby TCR phosphorylation triggers modifications, such as conformational changes, in the extracellular domain of the TCR that in turn affect its binding capacity to pMHC.

We leveraged the distinctive hump in the binding curve to derive a biokinetic model best suited to describe the observed kinetic patterns. Starting with the classical reversible ligand–receptor model (L + R ⇌ LR), we progressively increased model complexity and fit each version to the experimental kinetic data. The simplest model that provided the best fit (Fig. 4a) included not only the association and dissociation steps but also a transition of the receptor into an “inactive” state (Ra → Rn) following its interaction with pMHC, followed by a “relaxation” step back to the “active” state (Rn → Ra) (see Materials and Methods for a detailed description of the “TCR cycle” model).

Fig. 4.

We identified the minimal biokinetics model, referred to as the “TCR cycle”, capable of reproducing the hump-shaped binding kinetics curve. Model (a) is consistent with empirical observations (b): N4 exhibits a higher kon than T4, which in turn has a higher kon than V4. Additionally, the use of anti-CD8α antibodies has the most pronounced effect on N4 (c). k_on: Effective association rate of ligand (pMHC) with the active receptor population (R_a), averaged over TCR-CD8 (higher affinity), TCR alone (weaker affinity), and CD8 alone (likely negligible). k_off: Effective dissociation rate of ligand‒receptor complex (X), reflecting combined off-rates of TCR-pMHC and TCR-CD8-pMHC complexes. The rate of ligand-induced receptor inactivation captures all routes (e.g., ligand-triggered phosphorylation) by which a bound receptor transitions to a ligand-free inactive state. k_rel: Effective activation rate of receptors, including dephosphorylation and conformational changes enhancing pMHC-binding competency. Rate constant values (kon, koff, and krel) are presented on a log₁₀ scale and reflect data from three or more independent experiments (see supplementary Data 1). Phosphorylated TCR (pTCR; CD3ζ phosphorylation) levels are presented as the mean fluorescence intensity measured during the early phase of the binding experiment, averaged across three pMHC concentrations, and are used here as an operational readout of early TCR triggering rather than functional avidity. Representative data from one of three independent experiments are shown (see supplementary Data 2 for additional datasets). Δk_s: Differences in the corresponding rate constants measured in the absence versus the presence of anti-CD8α antibodies. Error bars represent the standard deviation of values across independent experiments. Figure elements created with BioRender.com

We further applied this mechanistic mathematical model to a series of well-characterized OT-I–OVA (N4, T4, V4) peptides and compared it to nonspecific peptide (UVpep; see Fig. 4b) binding experiments to systematically estimate the underlying TCR–pMHC binding kinetics parameters, including kon, koff, and krel (see supplementary Data 1). This approach provided quantitative insights into the kinetics of each peptide variant and enabled assessment of the impact of anti-CD8α antibodies on TCR–pMHC–CD8 binding kinetics and the resulting level of TCR phosphorylation (Fig. 4b; see supplementary Data 2 for pMHC concentration-dependent pTCR signal).

Upon treatment with anti-CD8α antibodies, both kon and koff values decreased (Fig. 4b) relative to untreated conditions. This effect could arise from two factors: first, anti-CD8α antibodies impaired CD8’s ability to participate in TCR–pMHC–CD8 interactions²⁰; second, the antibodies could introduce steric hindrance that affected both binding and unbinding kinetics. The first mechanism is supported by the peptide-dependent patterns in Δkon and Δkoff (Fig. 4c); notably, N4 showed the largest positive Δkon, consistent with the empirical observation that N4 was more heavily dependent on CD8 coreceptor engagement than T4 or V4. The second mechanism was corroborated by our steric hindrance test (supplementary Fig. 2). kon decreased due to a reduced frequency of productive collisions, as the bulky antibody obstructed proper alignment of TCR, CD8, and pMHC. koff also decreased because the steric environment restricted conformational freedom, effectively stabilizing the bound complex and making dissociation less favorable.

We further investigated the role of kinetic parameters in shaping the binding kinetics curve (Fig. 5a), emphasizing that the early phase of the binding reaction, occurring over the first few minutes, was the most informative for distinguishing pMHCs based on their specificity for a given TCR (Fig. 5b, supplementary Fig. 3a). This initial phase was primarily governed by the kon and koff parameters. We also explored the relationship between kinetic parameters and phosphorylated TCR (pTCR) values (Fig. 5c), with koff showing the strongest correlation with the pTCR signal. This observation is consistent with the TCR cycle biokinetic model, in which koff reflects the overall rate of receptor transition to a ligand-free inactive state, including ligand-triggered phosphorylation pathways. Having quantified these kinetic parameters, we next examined how they relate to functional T-cell responses.

Fig. 5.

Predictive modeling enables simulation of binding kinetics across varying pMHC concentrations and kinetic parameters. This figure integrates biophysical binding kinetics, early TCR signaling metrics, and downstream functional avidity measurements, with each panel reporting the corresponding operational readout. a Representative binding kinetics data (dots) and corresponding fits using the “TCR cycle” model (solid lines) at two pMHC concentrations (10 µg/mL, blue; 5 µg/mL, red). Key kinetic parameters, kon, koff, and krel, are annotated at the positions on the curve where they exert the greatest influence. For example, kon primarily determines the initial slope and intercept of the binding curve at the onset of the TCR–pMHC–CD8 interaction. The dotted line indicates the maximum signal amplitude, while the blue-shaded region highlights the most informative timeframe for distinguishing highly specific peptides from less specific peptides. b Maximum (i.e., peak) MFI value observed in the binding kinetics curve during the first 5 min of the TCR–pMHC interaction is presented on a log₁₀ scale. Values represent averages from three or more independent experiments. To enable comparison across different experimental days, MFI values were normalized using the formula: (peak − B)/A, where A and B represent flow cytometer settings (e.g., voltage), as described in the “TCR cycle” biokinetic model methods section. Error bars represent the standard deviation of values across independent experiments. c Relationship between biophysical kinetic parameters (kon, koff, krel) and TCR triggering, quantified by pTCR (CD3ζ phosphorylation) measured during the early binding phase. The Pearson’s ρ and corresponding p-value calculated from mean values are shown in the panel. Pearson correlations calculated from all individual matched data points (not just group means) were as follows: kon Pearson’s ρ = 0.65, p-value = 0.02, koff Pearson’s ρ = 0.49, p-value = 0.10, krel Pearson’s ρ = 0.21, p-value = 0.53. Error bars represent the standard deviation of values across independent experiments. d Functional avidity assay results for OVA peptide variants measured as cytokine response (IFNγ EC₅₀), shown in the absence (left) and presence (right) of anti-CD8α antibodies. Three negative controls (UVpep, DMSO, and VSV) are included. e Binding kinetics curves simulated using the “TCR cycle” biokinetic model at varying pMHC concentrations using kon, koff, and krel parameter values corresponding to N4 (left) and T4 (right). f Binding kinetics simulated at a fixed pMHC concentration for varying kon (left) and koff (right) values. Here, kon *1 refers to the empirically measured kon for the OTI–OVA–N4–CD8 interaction, as identified in our manuscript. Accordingly, kon*0.5 represents half that value, and so on. The same convention applies to koff (right). g Kinetic parameter space (kon, koff) showing regions that produce peak MFI values equal to or lower than the OT-I–UVpep peak (violet area) and those producing higher peaks (red area). Peak MFI is used here as an early signaling metric. Parameter values are shown relative to N4 on a log₁₀ scale. Simulations were performed at a pMHC concentration of 10 µg/mL. Positions of N4, T4, V4, and UVpep are indicated by colored stars

The Δkoff-based ranking of peptides (Fig. 4c) aligns well with the functional avidity results (Fig. 5d). In addition to reflecting the extent of CD8 involvement in peptide recognition, it suggests that if binding kinetics were measured between OT-I TCRs and the OVA-A2 or -Q4 peptides, the resulting curves would likely still exhibit humps, with peak MFI values falling between those observed for N4 and T4. More broadly, having an analytical solution enables predictive modeling of binding curve behavior in response to changes in pMHC concentration (Fig. 5e) and even allows simulation of how the curves would shift with changes in kinetic parameters (Fig. 5f, supplementary Fig. 3b–d). This framework opens the opportunity to computationally explore which combinations of biokinetic parameters yield binding curves that resemble those of highly specific TCR–pMHC interactions, such as OTI–OVA–N4 (Fig. 5g, supplementary Fig. 3b–d), and to identify the optimal pMHC concentrations for TCR–pMHC pairs with such kinetic profiles.

Consistent with these insights, early binding features (0–5 min) more accurately predict TCR functional avidity than late-time binding measurements (20–30 min), highlighting the importance of initial association kinetics for discriminating peptide variants (supplementary Fig. 4). Correlations between kon, koff, and krel with functional avidity further support the mechanistic relevance of these kinetic parameters in shaping T-cell responses. Although kon shows the strongest individual correlation with functional avidity (supplementary Fig. 4), it is clear that the observed relationship arises from a combination of kinetic parameters, including koff and krel, underscoring that T-cell responsiveness is shaped by the integrated dynamics of the TCR–pMHC interaction rather than any single rate constant.

To further assess the generality of the TCR cycle model, we applied it to previously published murine and human TCR–pMHC systems (Fig. 6). Importantly, these datasets were generated by independent laboratories and were not used in any way to develop or tune the TCR cycle model, providing an unbiased external validation. Kinetic traces were extracted from these studies using a standardized digitization workflow, and the model was fitted to association time courses. Across diverse systems and experimental conditions, the TCR cycle model accurately recapitulated the temperature dynamics of binding, including the effects of coreceptor engagement and peptide-specific variations. In particular, inferred koff values from the model strongly correlated with dissociation-only measurements, supporting the robustness of the model for capturing intrinsic receptor–ligand kinetics without functional readouts.

Fig. 6.

Validation of the TCR cycle model using previously published TCR–pMHC systems. All panels report biophysical binding kinetics, with no functional or signaling readouts included in this figure. a, b Representative association and dissociation kinetic curves for murine TCR–pMHC systems investigated in this study, averaged over three experimental replicates. The datasets were drawn from previously published studies and include measurements performed under different temperature regimes, as indicated. c Correlation between the dissociation rate constants (koff) inferred from fitting the association time courses using the TCR cycle model and the koff values obtained by fitting dissociation-only kinetic measurements with a standard single-exponential model. Strong agreement is observed (Pearson R > 0.94). Error bars represent the standard errors of the fitted parameters, derived from the Hessian matrix of the fit. d Fits of the TCR cycle model to the representative association kinetic curves shown in panel a, demonstrating the model’s ability to capture the full temperature dynamics of TCR–pMHC binding, as well as the biological effect of mutating the T1 TCR to disrupt CD8α engagement (see panel e for details). Error bars represent the standard errors of the fitted parameters, derived from the Hessian matrix of the fit. e Summary of the murine and human TCR–pMHC (HLA) systems extracted from the published literature^18,31,32 and analyzed in this manuscript, including receptor–ligand identities and experimental conditions. As raw numerical data were not publicly available, kinetic traces in (a, b) were digitized from the original publications listed in panel e using custom, open-source tools developed by the authors (see Data availability), ensuring consistent and reproducible extraction across studies

To test whether the additional ‘competence cycling’ step (krel) is mechanistically needed, we compared the TCR cycle model to a simpler two-state reversible model. Across ligands and experimental conditions, the full TCR cycle model consistently provided superior fits (higher R²) and lower AIC values, indicating that the observed binding kinetics, particularly hump-shaped curves, cannot be explained by a simple reversible interaction alone (supplementary Fig. 5a–b). Examination of the fitted parameter correlations further confirms that kon, koff, and krel are identifiable within the experimental time windows and concentration ranges used (supplementary Fig. 5c), supporting the mechanistic relevance of competence cycling in shaping TCR–pMHC dynamics. Notably, one of the systems analyzed in Fig. 6 exhibits nonmonotonic (‘hump-shaped’) association kinetics even when peptides are presented on antigen-presenting cells (Fig. 6b), supporting the physiological relevance of the kinetic behavior captured by the TCR cycle model.

Discussion

Leveraging machine learning for TCR specificity prediction is an ambitious and promising goal. However, success critically depends on asking the right biological questions, framing the problem in a way that machine learning can meaningfully address, and generating training datasets that are well aligned with that formulation. As we have shown in our prior work¹² and in this study, currently available datasets, primarily derived from equilibrium TCR-pMHC binding assays, sometimes supplemented with T-cell activation readouts, remain insufficient for enabling accurate TCR specificity prediction in the general case, using either supervised or unsupervised machine learning approaches. This insufficiency stems not only from limited dataset size (i.e., the number of characterized TCR-pMHC pairs) but also from fundamental limitations in the type of data being collected and perhaps even in the questions being asked.

Using the well-characterized OT-I/OVA system, along with other previously published examples, we demonstrate that TCR–pMHC interactions are fundamentally kinetic in nature, shaped by a complex interplay between kinetic parameters, receptor and ligand concentrations, and intracellular feedback mechanisms, which collectively manifest as an effective adaptive behavior of the TCR at the systems level. Rather than relying solely on affinity or avidity measurements at equilibrium, it may therefore be more informative to assess TCR specificity based on early binding kinetics. Importantly, competence cycling should not be interpreted as evidence for a specific extracellular conformational switch; rather, it represents a minimal systems-level description of the feedback and adaptation required to explain the observed kinetics, which could arise from multiple intracellular processes, including phosphorylation-dependent signaling, kinase-phosphatase balance or receptor–coreceptor coupling. Consistent with this interpretation, prior studies using Src family kinase inhibition (e.g., dasatinib) report selective stabilization of surface TCR–pMHC interactions,^21,22 an observation that aligns with the competence cycling framework described here. Accordingly, the OT-I–OVA system serves as a well-controlled validation platform, while the mechanistic framework itself is intended to be broadly applicable across receptors, peptides, and species. Practically, for empirical selection of specific TCRs, direct measurement of TCR–pMHC binding kinetics may not always be needed. Existing tetramer staining protocols could be adapted, for example, by reducing incubation time to 3–5 min and using lower tetramer or monomer concentrations, rather than the conventional > 20–30 min incubation. More generally, such cell-based assays provide a strong starting point for TCR selection and for generating training data, as they inherently account for coreceptor involvement and the feedback mechanisms that influence TCR discrimination.

To illustrate these principles, consider OVA peptide variants (e.g., N4, Q4, T4, V4), which differ in their MHC binding and recognition strength by the OT-I TCR. Both functional avidity assays and binding kinetics parameters independently indicate that peptide discrimination in this system relies heavily on the CD8 co-receptor. Co-engagement of CD8 significantly enhances the OT-I–OVA interaction, increasing kon and facilitating signal transduction, thereby amplifying subtle differences between peptide variants. Does this strong dependence on coreceptor involvement render prediction of TCR specificity from sequence hopeless? Not necessarily. In our dataset, the only differences between TCR–pMHC pairs were single amino acid substitutions among N4, T4, and V4, suggesting that sufficiently large and well-structured training data could reveal patterns of amino acid combinations on both TCR and pMHC that determine CD8 engagement. The challenge, however, lies in the fact that there is no simple mapping from sequence alone to definitive T-cell specificity or functional response because activation also depends on the concentrations of TCR and pMHC. Therefore, how can machine learning help in this context? Here, machine learning can play a complementary role when integrated with predictive mechanistic modeling. Our analytical framework enables simulation of binding curves under varying pMHC concentrations, but these predictions rely on empirically measured biokinetic parameters, such as kon and koff. Developing ML methods that approximate these parameters from TCR–pMHC sequences would provide critical inputs for downstream predictive modeling. A scalable path to generate the necessary training data is to apply the binding kinetics framework introduced here to diverse TCR and pMHC repertoires or even a library-on-library setup. By barcoding different incubation time points to capture stages of the TCR–pMHC interaction and sequencing the TCRs, one could reconstruct binding kinetics profiles and extract kon and koff values by fitting the mechanistic model.

From the TCR–pMHC interaction perspective, predicting kinetic parameters (kon, koff) from sequence is a more tractable and generalizable modeling approach than predicting functional avidity directly. Kinetic measurements are grounded in biophysics and can be obtained consistently across diverse receptor–ligand systems, enabling larger and more standardized training datasets. In contrast, functional avidity data are challenging to scale and standardize across multiple TCRs, antigens, and cellular contexts. Focusing on kinetics also opens opportunities for rational receptor design, allowing tunable binding dynamics that provide both predictive insight and translational utility. In this work, we outline a potential path for linking kinetic parameters to downstream functional outcomes, such as functional avidity, providing a conceptual and computational bridge between biophysical binding data and biological function. The multiplexed version of our platform will enable large-scale kon and koff measurements, which can serve as supervised training labels for machine-learning models designed to predict TCR–pHLA kinetic parameters directly from sequence. In this framework, experimentally derived kon/koff values will act as regression targets, while model inputs will include TCR and pHLA sequence embeddings, together with sequence-to-structure–derived interface features. This approach builds on recent advances in computational and machine-learning methods for predicting biomolecular binding kinetics from structural and biophysical representations, including graph-based and hybrid physics-informed models (reviewed in ref. ²³). Once trained on systematically generated kinetic data, such models could enable scalable inference of TCR–pHLA binding kinetics across diverse sequence repertoires. These predictions would then be rigorously validated using held-out experimental measurements. Looking further ahead, a multiplexed library-on-library approach (Supplementary Materials, supplementary Fig. 6) could allow pooled measurement of hundreds of TCRs against hundreds of pHLA ligands, with time-resolved barcoding reconstructing association and dissociation kinetics and enabling large-scale estimation of kon, koff, and krel.

More broadly, the cell-based TCR–pMHC kinetics approach we present here opens new opportunities to explore fundamental questions about T-cell activation mechanisms. For example, Wooldridge et al.²¹ reproduced an intriguing empirical observation initially reported by Lissina et al.²²: incubation with the protein tyrosine kinase inhibitor dasatinib paradoxically enhances the stability of surface TCR–pMHC interactions, improving staining of cognate T cells with pMHCI tetramers. Dasatinib blocks phosphorylation of the CD3ζ chain and ZAP70, early steps required for T-cell activation. These observations are consistent with the TCR cycle model, which posits that phosphorylation of intracellular TCR domains acts as a feedback mechanism promoting ligand dissociation, effectively increasing koff. Within this framework, kinase inhibition is predicted to slow the transition into the inactive state following ligand engagement, effectively prolonging the time receptors remain in the competent state. Conversely, phosphatase inhibition decreases krel, slowing recovery to the competent state and thereby prolonging receptor incompetence. While the TCR cycle model provides a systems-level description of the observed kinetics, we do not assign its effective states and transitions to specific molecular mechanisms. Mapping these onto defined biochemical processes will require further experimental work. Perturbations such as kinase and phosphatase modulation, as well as controlled variation of receptor–ligand conditions, provide a clear experimental framework to distinguish among possible molecular implementations. Revisiting these systems with such perturbations in the context of the TCR cycle model would be a valuable direction for future studies.

The TCR cycle model consistently outperforms a simple reversible binding framework across the datasets analyzed; however, we do not claim universal validity for all TCR–pMHC interactions. Rather, our aim is to demonstrate that the widely used reversible ligand–receptor paradigm is insufficient to explain key kinetic features, including the recurrent nonmonotonic (“hump-shaped”) association behavior. Importantly, the datasets analyzed here were independently generated across several laboratories and experimental conditions, indicating that the model captures generalizable features rather than being fitted to a curated set of examples. Nonetheless, systematically generated, multiplexed kinetic datasets, such as those enabled by a library-on-library framework, would allow further unbiased evaluation of the model across diverse TCR–pMHC pairs. Taken together, our results support the view that competence cycling represents a generalizable feature of TCR signaling dynamics, while its full scope and molecular implementation remain to be established through future high-throughput and prospective studies.

Materials and methods

To clarify the terminology used throughout this study, we define the following operational metrics: affinity refers to the strength of the TCR–pMHC interaction, typically quantified by the dissociation constant (Kd) or the association/dissociation rates (kon/koff); specificity refers to the ability of a TCR to selectively recognize its cognate pMHC over unrelated pMHCs; and functional avidity captures the overall potency of a T-cell’s functional response to a given pMHC, operationally measured here by readouts such as cytokine EC₅₀ (e.g., IFNγ) or EC₅₀ for cell-mediated lysis. These definitions will be used consistently throughout the Results to link measured biophysical parameters to functional outcomes.

Analysis of previously published data

For the assessment of the peptide-specific predictive power of various clustering methods (Fig. 2a), we used data from the Source Data table in Leary et al.¹³

For the assessment of peptide specificity assignment using agglomerative clustering (Fig. 2b), based on various distance metrics, we used the IMMREP_2022 benchmark dataset (Meysman et al.¹⁴ available at https://github.com/viragbioinfo/IMMREP_2022_TCRSpecificity), which contains 17 peptide-specific data buckets. For our clustering analysis, we used concatenated training set data organized epitopewise.

For Fig. 2a, we plotted a subset of an already published analysis (Leary et al.¹³). Specifically, the data can be found in fig4cd_sup_fig_1 to 7 tab of the Source Data table. We selected a subset of data points with minimal cluster size 3, without a spike of irrelevant data (spike_x = 0), tcrvalid reference labels, and clustering based on TCRβ (TRB). For the methods where it is defined (all except GLIPH and clustcr), we selected data points for the distance parameter ε = 0.5 (or the closest value available). For each method, we plotted the value in the c-CSI column (clustering Critical Success Index).

For Fig. 2b and supplementary Fig. 1, we clustered aggregated peptide-specific TCRs using agglomerative (hierarchical) clustering with the scikit-learn²⁴ Python library. We used distance matrices based on the tcrdist3^15,16 package (TCRβ only), Euclidean distance in the latent space of the ESM2 language model²⁵ (esm2_t36_3B_UR50D variant, using mean representations of CDR3β with 2560 dimensions), and negative BLOSUM62 pairwise sequence similarity (using pairwise2.align.globaldx in the BioPython package²⁶). We then divided the data by epitope and plotted the portions corresponding to each cluster within each epitope-specific data bucket. The subset of motif-containing peptides was selected based on sequence logos (created using logomaker²⁷) of CDR3β sequences, aligned based on IMGT numbering using ANARCI.²⁸

pMHC reagent generation and characterization

UV-MHCI generation and fluorophore labeling

Recombinant H2-Kb heavy chain and B2M were refolded in the presence of a high-affinity peptide with the sequence FAPGNYJAL, with residue “J” denoting a nonnatural UV cleavable amino acid as previously described.²⁹ The resultant UV-MHCI was incubated with a 2-20X molar excess of Alexa Fluor 488 NHS ester (Thermo Fisher, Cat # A20000) in phosphate buffered saline pH 7.4 for 2 hours at room temperature. Excess, unconjugated fluorophore was removed by dialysis. The sample was loaded into a 10 K molecular weight cutoff dialysis cassette (Slide-A-Lyzer 10 K MWCO cassette, Thermo Fisher) and placed in 25 mM TRIS pH 8.0, 150 mM NaCl, 4 mM EDA at a ratio of 1:2000, and sample:dialysate. Samples and dialysate were incubated at 4 °C for 8 h while continually mixing via a magnetic stir plate. The dialysate was then discarded and replaced for an additional 8 h of incubation at 4 °C while stirring. After the second round of dialysis, the sample was recovered, and the protein concentration was determined using a UV‒Vis spectrophotometer and corrected for the contribution of the fluorophore to the absorbance at 280 nm.

Determining the degree of MHCI fluorophore labeling

The degree of fluorophore labeling (DOL) was determined by reversed-phase liquid chromatography‒mass spectrometry (RP LC‒MS). A total of 2–3 µg of the MHC-fluorophore conjugation reaction were injected on an Agilent 1290 Infinity series HPLC in line with an Agilent 6230 time-of-flight electrospray ionization mass spectrometer. The reversed-phase column (Agilent PLRP-S 1000 Å, 8 µm, 50 × 2.1 mm) was exposed to a gradient of 25–45% mobile phase B in 5 min at 0.50 ml/min with the column heated to 80 °C. Mobile phase A was 0.05% TFA, and mobile phase B was 0.05% TFA in acetonitrile. The column eluent was sent to the TOF-MS for mass spectrometry data acquisition. The degree of fluorophore conjugation was determined by using the deconvoluted mass spectra of the peaks corresponding to the B2M and H2-Kb heavy chains. The average number of fluorophores per pMHCI was calculated by using the relative abundance of each mass corresponding to a different number of fluorophore additions to each protein species and determining a weighted average for the complex.

UV-mediated peptide exchange

OVA synthetic peptides (Anaspec, Elim Bio) were solubilized in ethylene glycol to a concentration of 20 mg/ml and were added to fluorophore-labeled peptide-MHCI at a 25X molar excess. The peptide exchange reaction was performed in 25 mM TRIS pH 8.0, 150 mM NaCl, and 4 mM EDTA and contained 5% ethylene glycol v/v after the addition of peptide. The final concentration of the peptide-MHCI in the exchange reaction was 2.0 mg/ml. The peptide-exchange reaction was then incubated under UV light set to 365 nm (Analytikjena UVP 3UV Lamp) for 20 min. After exposure to UV light, the exchange reaction was allowed to proceed at room temperature for a minimum of 4 hours or overnight incubation.

Determination of peptide binding to MHCI

A 2-dimensional liquid chromatography‒mass spectrometry (2D LC‒MS) method was used to characterize peptide binding to MHCI complexes. Between 2 and 3 µg of MHCI-peptide mixtures were injected into the instrument and sent to the first dimension column. The first dimension LC method employed an analytical size exclusion column (SEC) (Agilent AdvanceBio SEC 300 Å, 2.7 µm, 4.6 × 15 mm) to separate intact complex from excess peptide run at an isocratic flow of 0.7 ml/min in 25 mM TRIS pH 8.0, 150 mM NaCl for 10 min with signal acquisition at 280 nm. A sampling valve collected the entirety of the complex peak corresponding to monomeric pMHCI that eluted between 1.90 and2.13 min in a volume of 160 µl and injected it onto the second dimension reversed-phase column (Agilent PLRP-S 1000 Å, 8 µm, 50 × 2.1 mm). The second dimension column was exposed to a gradient of 5–50% mobile phase B in 4.7 min at 0.55 ml/min with the column heated to 80°C. Mobile phase A was 0.05% TFA. Mobile phase B was 0.05% TFA in acetonitrile. The column eluent was sent to an Agilent 6224 TOF LCMS for mass spectrometry data acquisition.

The MHC-I complex peak area in the first dimension and mass spectrometry detection of the peptide in the second dimension were used to determine successful peptide binding. Successful binding of a peptide into the complex after cleavage of the conditional ligand during the peptide exchange reaction stabilizes the complex and results in nearly complete recovery of the starting complex measured in the first dimension SEC analysis. The peptide that has exchanged into the complex can then be detected in the second dimension, where the complex was run under denaturing conditions with mass spectral analysis, allowing for direct detection of the peptide of interest. The first dimension SEC method also allowed for the detection of any aggregate species. Aggregation formation during the peptide exchange reaction is most often formed from partially unfolded HLA from a complex that failed to bind a peptide. The pMHCIs used in this study were >98% monomeric and are unlikely to have contained any functional higher-order complexes.

Flow cytometry-based association and dissociation kinetics assay (including pTCR)

OTI CD8 T cells were purified by negative selection (Miltenyi Biotec, 130-104-075) from the spleens and lymph nodes of OTI mice (Jackson Laboratory or Genentech in-house colony). All animal usage was conducted by following relevant ethical regulations detailed in animal use protocols approved by the Genentech Institutional Animal Care and Use Committee. Cells were stained with live/dead Aqua (Thermo Fisher Scientific, L34966) and TCRb BV711 (BioLegend, clone H57-579) prior to dispensing in a 96-well plate at 3–4 × 10⁵/well. For association assays, titrated amounts of peptide:MHC monomers were added at 4 °C, and at specified time points, 4% paraformaldehyde fixation buffer was added. For dissociation assays, titrated amounts of peptide:MHC monomers were prebound to cells at 4 °C for 1 h, washed, and then fixed at specified time points.

For CD8 blockade conditions, purified anti-CD8 (Thermo Fisher Scientific, clone MA5-17594) was prebound at 10 µg/mL and then at 50 µg/mL in the presence of the peptide:MHC monomers. Association and dissociation assays were run in MACS buffer (1x PBS + 0.5% BSA + 2 mM EDTA) at 4 °C. To measure the level of phosphorylation of the ζ domain of TCR, cells were washed with 1x permeabilization buffer (Thermo Fisher Scientific, 00-8333-56) prior to incubation with pCD247 CD3ζ Tyr142 APC (Thermo Fisher Scientific, 17-2478-42) in 1x permeabilization buffer for 30 minutes at room temperature. Cells were then washed with 1x perm buffer and resuspended in MACS buffer for subsequent flow cytometric analysis. Samples were run on a BD Symphony flow cytometer, followed by data analysis using FlowJo software.

Association kinetics assay automation

Plate washing and fixation procedures were performed using a fully integrated automation platform developed by HighRes Biosolutions operated under Cellario scheduling software. The system was configured to execute time-sensitive liquid handling and environmental control steps with high precision and reproducibility. The automation platform included the following key components: an Agilent Bravo liquid handling workstation for accurate dispensing of reagents during titration and fixation. An Agilent VSpin centrifuge was used for gentle and programmable plate spinning to ensure even distribution and cell settling. A Thermo Scientific Cytomat automated incubator was used to carry out incubation following PFA addition, ensuring consistent fixation conditions at controlled temperature. A temperature control pad is integrated with the deck layout to maintain localized cold zones. A multiposition plate hotel for temporary plate storage and routing across devices. Throughout the process, 96-well microplates were maintained at 4 °C on a temperature-controlled pad to preserve sample integrity during fixation. All steps were scheduled and coordinated through Cellario, ensuring seamless plate transfers and timing synchronization.

TCRβ/αCD8 steric hindrance test

OTI CD8 T cells were isolated by negative selection (Miltenyi Biotec, 130-104-075) from OTI Thy1.1 spleens and mesenteric lymph nodes (Genentech in-house colony). Cells were first stained with live/dead aqua (Thermo Fisher L34966). In one scenario, TCRb BV711 (BioLegend, 109243) was added first at 1:400 (corresponding to 0.5 µg/mL or 0.05 µg per test) and incubated for 30 miutes at 4 °C, washed in MACS buffer, followed by anti-CD8 (clone CT-CD8a, Thermo Fisher MA5-17594) at either 10 µg/mL or 50 µg/mL and incubated for an additional 30 min at 4 °C. In the second scenario, anti-CD8 was added first at either 10 µg/mL or 50 µg/mL and incubated for 30 min at 4 °C, washed, and then 1:400 of TCRb BV711 was added and incubated for an additional 30 min at 4 °C. Cells were washed twice in MACS buffer prior to flow cytometric analysis on a BD Symphony.

Functional avidity test

OTI CD8 T cells were isolated by negative selection (Miltenyi Biotec, 130-104-075) from OTI Thy1.1 spleens and mesenteric lymph nodes (Genentech in-house colony). A portion of the spleen and lymph node cell suspensions were used as antigen-presenting cells after T-cell depletion. T-cell depletion was performed by first staining with anti-CD3 FITC (Tonbo Biosciences, 35-0032-U100), followed by removal using anti-FITC microbeads (Miltenyi Biotec, 130-048-701). Purified OTI CD8s and T-depleted APCs were plated at a 1:1 ratio with titrated purified peptides N4, T4, V4, UV peptide or VSV peptide. Peptide concentrations started from 10 µg/mL and were serially diluted 10-fold across 12 wells. If CD8 blockade was included, anti-CD8 (clone CT-CD8a, Thermo Fisher MA5-17594) was added at a final concentration of 50 µg/mL. Cells were cultured for 4 hours with protein transport inhibitors (Thermo Fisher, 00-4970-03). Cells were then harvested and stained with live/dead aqua (Thermo Fisher L34966), TCRb BV711 (BioLegend, 109243), and Thy1.1 PE (BioLegend 202524) and incubated for ~30 minutes at 4 °C. Cells were washed in MACS buffer and then fixed in 1x fix/perm buffer (Thermo Fisher, 00-5523-00 buffer kit), followed by 1x permeabilization and staining with IFNg APC (BioLegend, 505810) for 30 min at room temperature. Cells were washed in 1x perm buffer, followed by flow cytometric analysis on a BD Symphony.

“TCR cycle” biokinetics model

Definitions of key terms:

k_on: Effective association rate of ligand (pMHC) with the active receptor population (Rₐ), representing an average over contributions from TCR–CD8 (higher affinity), TCR alone (lower affinity), and CD8 alone (likely negligible).

k_off: Effective dissociation rate of the ligand–receptor complex (X), reflecting the combined off-rates of TCR–pMHC and TCR–CD8–pMHC interactions, as well as ligand-induced receptor inactivation pathways (e.g., phosphorylation-driven transitions to a ligand-free inactive state).

k_rel: Effective reactivation (or relaxation) rate of receptors, encompassing processes such as dephosphorylation and conformational changes that restore pMHC-binding competency.

The simple two-state model cannot account for the presence of the hump in the TCR-pMHC-CD8 binding kinetic curve. To explain this feature, we extend the basic reaction scheme to a slightly more complex model, as follows:

$$\left\{\begin{array}{c}L+R_a\stackrel{k_{\mathit{on}}}{\to }X\stackrel{k_{\mathit{off}}}{\to }L+R_n\\ R_n\stackrel{k_{\mathit{rel}}}{\to }{R}_a\end{array}\right)$$ 1

Here, L is the external ligand molecule; R_a and R_n are different types of free receptors, X is the ligand‒receptor complex, and k_j are rate constants of the corresponding reactions.

The following differential equations describe the reaction scheme shown in [1]:

$$\left\{\begin{array}{c}\frac{\mathit{dX}}{\mathit{dt}}=k_{\mathit{on}}L{R}_a-k_{\mathit{off}}X\\ \frac{\mathit{dX}}{\mathit{dt}}=k_{\mathit{off}}X-k_{\mathit{rel}}{R}_n\\ \frac{d{R}_a}{\mathit{dt}}=k_{\mathit{rel}}{R}_n-k_{\mathit{on}}L{R}_a\end{array}\right)$$ 2

Equations [2] can be rewritten solely in terms of active receptors R_a and complexes X by applying the conservation of the total number of receptors R₀

$$R_0=X+R_a+R_n$$ 3

as follows:

$$\left\{\begin{array}{cc}\frac{dX}{dt}\hfill & =-k_{off}X+k_{on}L{R}_a\hfill \\ \frac{d{R}_a}{dt}\hfill & =-k_{rel}X-\left(k_{rel}+k_{on}L\right)R_a+k_{rel}{R}_0\hfill \end{array}\right)$$ 4

and solve it using the initial conditions for association kinetics:

$$\begin{array}{c}{R}_a=1\\ X=0\end{array}$$ 5

Or in matrix form:

$$\frac{\mathit{dX}}{\mathit{dt}}=\mathit{A}\mathit{X}+\mathit{b}$$ 6

where:

$$\mathit{A}=\left(\begin{array}{cc}-k_{\mathit{off}}& k_{\mathit{on}}L\\ -k_{\mathit{rel}}& -\left(k_{\mathit{rel}}+k_{\mathit{on}}L\right)\end{array}\right),$$ 7

$$\mathit{b}=\left(\begin{array}{c}0\\ k_{\mathit{rel}}{R}_0\end{array}\right),$$ 8

$$\mathit{X}=\left(\begin{array}{c}X\\ R_a\end{array}\right).$$ 9

Matrix A has the following eigenvalues:

$${\mathrm{\lambda }}_{1,2}=\frac{-\left(k_{\mathit{off}}+k_{\mathit{on}}L+k_{\mathit{rel}}\right)\pm \sqrt{{\left(k_{\mathit{off}}-k_{\mathit{on}}L-k_{\mathit{rel}}\right)}^2-4k_{\mathit{on}}L{k}_{\mathit{rel}}}}{2}$$ 10

For example, for fitting with the following optimal parameters:

k_on = 2.52e-4 mL/(µg*s);

L = 10 µg/mL;

k_off = 1.35e-2 S^-1;

k_rel = 2.13e-3 S^-1;

We have the matrix:

$$\mathit{A}=\left(\begin{array}{cc}-0.0135& 0.00252\\ -0.00213& -0.00465\end{array}\right){\mathrm{s}}^{-1}$$ 11

And the eigenvalues:

${\mathrm{\lambda }}_1=-0.005305s^{-1}$; ${\mathrm{\lambda }}_1=-0.012845s^{-1}$.

The intensity Y(t) of the MFI signal is then expressed in terms of the solution X(t) to the system of differential equations [4] as follows:

$$\mathit{Y}\left(t\right)=\mathit{B}+A\mathit{X}\left(t\right)$$ 12

Here, B represents the background signal, and A = C × R₀, where C is the instrument calibration constant (for the flow cytometer). The same notation is used for the concentrations of species as for the species themselves in the reaction scheme (e.g., X). In the fitting procedure, the fluorescence scaling factor A was intentionally held constant across conditions within each experimental batch. All OT-I–OVA measurements were acquired using identical flow cytometry settings within a batch (including PMT voltages and detector gains); accordingly, A represents a fixed experimental scaling factor rather than a condition-dependent biological variable. Allowing condition-specific values of A would violate the experimental design and introduce an unphysical degree of freedom capable of absorbing apparent kinetic differences unrelated to binding or competence cycling. For this reason, A was treated as a shared parameter by construction rather than optimized independently across conditions. In contrast, for datasets obtained from the public domain, A was fitted independently for each experimental condition, as the corresponding instrument settings were not reported and may have varied between experiments.

Reaction scheme [1] simplifies to the classic reversible receptor–ligand reaction (R + L ⇌ X) in the limit as 1/k_rel → 0. During fitting, we require the same value of A for the same batch of T cells, regardless of CD8 blocking or peptide type, since A = C × R₀ represents the total number of receptors (R₀) on the cell.

We developed a software package called TCRkin to fit binding kinetics data using the analytical solution of the reaction scheme [1]:

$$Y\left(t\right)=B+A\frac{k_{\mathit{on}}L}{\left(\left(k_{\mathit{on}}L+k_{\mathit{off}}\right)k_{\mathit{rel}}+k_{\mathit{on}}L{k}_{\mathit{off}}\right)}\left(k_{\mathit{rel}}+\frac{\left(k_{\mathit{on}}L+k_{\mathit{off}}+{\mathrm{\lambda }}_1\right)k_{\mathit{rel}}+k_{\mathit{on}}L{k}_{\mathit{off}}}{\left({\mathrm{\lambda }}_2-{\mathrm{\lambda }}_1\right)}{e}^{{\mathrm{\lambda }}_{2}t}-\frac{\left(k_{\mathit{on}}L+k_{\mathit{off}}+{\mathrm{\lambda }}_2\right)k_{\mathit{rel}}+k_{\mathit{on}}L{k}_{\mathit{off}}}{\left({\mathrm{\lambda }}_2-{\mathrm{\lambda }}_1\right)}{e}^{{\mathrm{\lambda }}_{1}t}\right)$$ 13

where λ₁ and λ₂ are the eigenvalues [10].

We emphasize that competence cycling is introduced here as a systems-level description of adaptive TCR behavior required to explain the observed kinetics. While intracellular phosphorylation-dependent regulation is a biologically plausible implementation, the model does not assume a unique molecular mechanism and is compatible with multiple biochemical realizations.

Author contributions

Conceptualization and study design were carried out by M.C., J.D., J.C., M.C.T., S.N.P., M.D., R.A.S., A.S., I.M., A.C., and D.O.; Data acquisition and curation were performed by M.C., J.C., M.C.T., S.N.P., and D.O.; Methodology development was conducted by J.C., M.C.T., S.N.P., M.D., G.S., A.C., and D.O.; Data analysis and interpretation involved M.C., J.D., J.C., M.C.T., S.N.P., R.A.S., E.T., A.S., I.M., A.C., and D.O.; Manuscript drafting and critical review/editing were performed by M.C., J.D., J.C., M.C.T., S.N.P., M.D., R.A.S., E.T., G.S., A.S., I.M., A.C., and D.O.; Supervision and funding acquisition were provided by IM and DO. All authors have read and approved the article.

Data availability

The source data table in Leary et al.¹³ is available through the journal web page: https://static-content.springer.com/esm/art%3A10.1038%2Fs41467-024-48198-0/MediaObjects/41467_2024_48198_MOESM4_ESM.xlsx. The IMMREP_2022 benchmark dataset¹⁴ is available at https://github.com/viragbioinfo/IMMREP_2022_TCRSpecificity. The OT1–OVA kinetic data newly generated as part of this study are publicly available at Zenodo: https://zenodo.org/records/17069783. Digitized kinetic datasets reconstructed from previously published TCR–pMHC(HLA) studies analyzed in this manuscript (summarized in Fig. 6e) are also publicly available at https://github.com/tabatsky/TCRkin/tree/main/digitized%20data. These datasets were generated using custom digitization tools developed by the authors. The TCRkin source code, including the digitization algorithms used to reconstruct kinetic data from published figures, is available at https://github.com/tabatsky/TCRkin.

Competing interests

M.C.T., S.N.P., M.D., and A.S. are employees of Genentech, Inc. J.D. is an employee of Servier Symphogen. I.M. is an employee of Medici Therapeutics. J.C., M.C.T., M.D., and D.O. are inventors on a patent related to cellular assays and methods to assess TCR–pMHC interactions and kinetics. All other authors declare no conflict of interest.

Supplementary information

The online version contains supplementary material available at 10.1038/s41392-026-02855-6.