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

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


Reviewer #2
Overall Comment: I think this is a very interesting paper that deserves to be published in PLOS Comp Bio.The authors address an important biological problem which has been studied with theoretical models for quite some time, and yet the authors present an important new perspective that prior "steady-state" analyses seem to miss.My main critiques are the following more technical questions which I think the authors should address.I think most of these critiques could be addressed by a clearer description of the model in the main text.
Response: Thank you for your positive remarks on our work and for your helpful comments.We have addressed your comments below and believe the presentation of the manuscript is much improved.Specific Comments: 1. Page 6 of the main text says that the formula for the accuracy is given in the supplement.Where is the formula for in the supplement?I couldn't find it.
Response: Added more clarity on this definition of and its derivation.The definition = P(T P ) + P(T N) is given in equation ( 3) with the relevant statistics given in equation ( 2).The practical calculation of requires us to determine the distribution of T 1,n for n = 1 (shown explicitly in Sec.1.1 of supplement) and for n 1 (approximated in Sec.1.2 of supplement).
2. In some instances, the authors find optimal values of ⌧ are on the order of ⌧ = 10 5 .Looking at Table 1, ⌧ is said to have units of seconds, in which case this is ⌧ ⇡ 30 hours which I assume is unphysiological.By saying that ⌧ has units of seconds, do the authors merely mean that it has units of time, and the authors have simply scaled time for that k 1 , k p , and k 1 are all unity (along the lines of section 5 in the supplement)?If so, can the authors estimate the values of these rates (from the literature) to translate their optimal values of ⌧ into real time values?Along these lines, are there available estimates for ?
Response: In some of the earlier works by Kersh et. al. (e.g. 1998 Immunity), the parameter could be approximated from the experimental data.In some cases, a 5-fold di↵erence in dissociation rate ( = 5) could result in large di↵erences in T cell activation su cient to di↵erentiate an agonist antigen from a weak antagonist.However, there is some degree of uncertainty about the actual T cell environment.For example, when our model predicts near-perfect accuracy for = 5, we have also explicitly modeled the self and agonist antigen populations.It is not clear in experiments Kersh et.al. the population of weak antagonists and agonists present in the T cell/APC contact and it would be interesting to compare our model with experimental data in which these variables are controlled.Furthermore, Pettman et.al. (2021 eLife) have recently shown that this "very good" (low ) antigen discrimination observed in early experiments may be an artefact.So it may be necessary to re-evaluate any possible values for .In terms of obtaining real-time optimal ⌧ values, this would indeed be possible if we were to use biologically relevant rates, KP steps, and estimated from the literature.However, there is a similar problem here.The optimal ⌧ ⇤ we would find in this case would be specific to some self and agonist antigen populations.Thus, we would also need, at the very least, an estimate of the distribution of these quantities for the T cell environment in question.Furthermore, the number of e↵ective KP steps is still under investigation.In the past, there were many phosphorylation sites being investigated as potential KP steps.But now only a small number of these sites appear to be relevant to the type of antigen (Voisinne, 2022 Nature Immunology).Other structures, such as microvilli on the surface of the T cell, could influence the temporal dynamics of the T cell response.Our main contribution is a method to evaluate the relevance of the cellular contact duration in T cell accuracy.A potential set of experiments could be to systematically vary cellular contact duration, in both agonist positive and agonist negative APC populations, and derive accuracy curves with respect to the cellular contact duration.This could establish if such a thing as optimal contact duration exists.It may also demonstrate that the mechanism of antigen discrimination operates, at least in part, in response to first-passage activation events.
3. Are the units of k 1 correct?Table 1 says that it is an inverse time, but looking at equation (14a) in the supplement, it seems like it is a bimolecular reaction rate and thus has units of 1/(concentration ⇤ time) assuming L T has units of concentration.It says L T is the total population of ligands, so the units are simply the raw number of ligands?Perhaps R T , L T should be added to Table 1?
Response: k 1 is indeed a bimolecular reaction rate, and would make more sense to have units 1/(concentration ⇤ time) in a continuous mass action model.This may just be confusion in perspective or semantics, but equation ( 14) in the supplement is technically not an approximation of the mass action model.Instead, it is an approximation for the extreme first passage time (equation 14e) from a discrete stochastic system.By treating R(t) and L(t) as dimensionless counts, this gives units for dA/dt to be probability/unit time, which can be interpreted as a hazard function.
4. The supplement presents a few di↵erent approximations.Which one of these approximations is used to plot in the main text figures?
Response: In the single agonist case (n = n ag = 1), we use the density given in equation ( 12) of the supplement.In the large self antigen population (n = n self 1), we use equation ( 14) of the supplement.The calculation of then follows from equations (3) and (2) in the main text.We have added a sentence in the main text clarifying this.In particular, we do not use the Weibull approximation in any of our main results -it is only included for validation in the n 1 limit.
5. I think the ODE approximation in the supplement should be clarified.In particular, I don't understand the "nonlinear binding rate K 1 (R(t), L(t))" mentioned at the top of page 5 of the supplement.
Response: This is the bi-molecular (mass-action) binding rate at which available ligands (L(t)) and receptors (R(t)) interact.We have added more clarifying text to the supplement.
6.The self-activation time is defined as the minimum of t 1 , t 2 , . . .t n self .Are these random variables independent and identically distributed (i.i.d.)?I think they are, but then I don't understand the following statement from the Supplement: "A source of numerical error in applying this result [3] to a large antigen population in the FRAM is that the first passage activation of receptors are not i.i.d., since the propensity for binding events is K 1 = k 1 RL."I don't understand this sentence.I think this sentence should be clarified and the model assumptions (namely of independence or dependence) should be clarified in the main text.
Response: They are not i.i.d..This is the reason the approximation displayed in equation ( 14) of the supplement is necessary for the extreme statistic, rather than the simpler case of finding the minimum of i.i.d.samples.The reason the receptor activation times are not i.i.d. is that the binding rate depends on both the populations of unbound receptors and the population of unbound ligands at time t.This means the binding propensity of a realization of the stochastic process at time t is K 1 = k 1 R(t)L(t), where R(t) and L(t) is the number of unbound receptors and ligands at time t.We have added more clarification to this in the supplement.
7. On page 4 of the supplement, there are some references to equation ( 13) which should be references to equation (12) (for example, it says "The integral ( 13) is readily..." Response: Fixed this.