article: Experimental work has shown that T cells of the immune system rapidly and specifically respond to antigenic molecules presented on the surface of antigen-presenting-cells and are able to discriminate between potential stimuli based on the kinetic parameters of the T cell receptor-antigen bond. These antigenic molecules are presented among thousands of chemically similar endogenous peptides, raising the question of how T cells can reliably make a decision to respond to certain antigens but not others within minutes of encountering an antigen presenting cell. In this theoretical study, we investigate the role of localized rebinding between a T cell receptor and an antigen. We show that by allowing the signaling state of individual receptors to persist during brief unbinding events, T cells are able to discriminate antigens based on both their unbinding and rebinding rates. We demonstrate that T cell receptor coreceptors, but not receptor clustering, are important in promoting localized rebinding, and show that requiring rebinding for productive signaling reduces signals from a high concentration of endogenous pMHC. In developing our main results, we use a relatively simple model based on kinetic proofreading. However, we additionally show that all our results are recapitulated when we use a detailed T cell receptor signaling model. We discuss our results in the context of existing models and recent experimental work and propose new experiments to test our findings.
  T cells of the adaptive immune system use their T cell receptors (TCR) to scan the surfaces of antigen-presenting-cells (APC) for antigen in the form of specific peptides bound to major-histocompatibility complexes (pMHC). Scanning of APCs by T cells is rapid, with estimates suggesting that an individual T cell spends only 1–5 minutes interacting with a single APC if it lacks specific pMHC [1]. Experiments have demonstrated that T cells are extremely sensitive to specific pMHC, responding to as few as 1–10 pMHC in a sea of thousands of chemically similar self (null) pMHC [2], [3], [4], [5]. It has also been demonstrated that a single amino acid substitution on a presented peptide can dramatically alter the T cell response [6]. Speed, sensitivity, and specificity have been dubbed the S3 characteristics of antigen detection by T cells [1]. The observation that T cells transiently interact with APCs that do not express specific pMHC suggests that the decision to respond occurs within seconds of an encounter. Rapid turnover of T cell-APC contacts in vivo accelerates the search for specific pMHC by allowing numerous unique T cell-APC interactions. The decision to respond gives rise to a ‘stop’ signal [7] and is commonly followed by the formation of the immune synapse [8], a stable adhesion between the T cell and APC that persists for upwards of 30 minutes and facilitates a second, sustained phase of signaling. Experiments and mathematical modeling have been extensively used to understand the efficiency of T cell activation. During the sustained signaling phase, the serial binding of many TCR by a single pMHC has been postulated to increase T cell sensitivity [9]. Serial binding is expected because the bonds formed between TCR and agonist pMHC are transient, with half-lives in the range of 1–100 s [10], [11], [12]. On the other hand, T cell specificity has been addressed by the kinetic proofreading model [13], [14]. This model postulates that a series of TCR-proximal steps, such as the binding and subsequent phosphorylation of the TCR associated immunoreceptor tyrosine-based activation motifs (ITAMs) by signaling molecules, occur upon pMHC binding, and that these signaling events require continued TCR engagement to proceed. A productive signal is transduced only after several such transformations have taken place. In this model, T cells are able to discriminate between different pMHC by imposing a threshold on the TCR-pMHC dissociation rate constant (). Combining serial binding and kinetic proofreading reveals that a balance between sensitivity and specificity gives rise to an optimal for efficient T cell activation [15], [16], [17], an effect which has been experimentally observed [18], [17], [19]. The efficiency of T cell activation in these models, and others [20], [21], is usually quantified by the number of activated TCRs (or the phosphorylation level of a downstream signaling molecule) integrated over the whole cell, and after a relatively long period of interaction (min) with an APC. Additionally, several studies have reported correlations between the TCR/pMHC bond dissociation constant (), but not, and the efficiency of T cell activation as measured after hour by cytoxicity and/or cytokine assays [12], [22], [23]. However, T cells have been observed to respond to stimulatory pMHC in less than a minute [1] and, at least for cytotoxic T cells, a stable contact interface is not required for pMHC detection [4]. Serial binding/kinetic proofreading models do not predict specificity on these short time scales, in part because signals generated by high concentrations of weakly binding self pMHC are found to be comparable to signals generated by low concentrations of high affinity agonist pMHC [16], [1]. Moreover, the early T cell response is unlikely to be determined by an equilibrium parameter, such as, as it is quite unlikely that the T cell-APC interface attains equilibrium at such short times. It is more likely that is an important determinant of the efficiency of T cell activation during the sustained phase of signaling, well after the initial decision to respond. In this paper we investigate a putative mechanism for antigen discrimination during the early phase of TCR signaling. Specifically, we examine the role of TCR/pMHC rebinding in allowing T cells to make rapid (s) and reliable decisions to respond. By explicitly accounting for TCR/pMHC rebinding within existing formulations of diffusion-limited membrane reactions, we find that rebinding has very little effect in canonical proofreading models. A simple modification that accounts for signal persistence at the TCR allows individual TCR to integrate the duration of multiple rebinding events. The consequence of this scheme is that discrimination in this ‘sum-of-binding’ model is now sensitive to both the association and dissociation rate constants of the TCR-pMHC bond. This enhanced sensitivity leads to the finding that a T cell can discriminate between a wider spectrum of antigens than would be predicted by a traditional serial binding/kinetic proofreading model. We further show that coreceptors, but not TCR clustering, are important to achieve these rapid rebinding events. In addition, we show that signal persistence at the TCR does not allow high concentrations of endogenous pMHC to generate spurious signals. Finally, we show that our general conclusions are unchanged when our cartoon kinetic proofreading model is replaced by a detailed model of TCR-proximal signaling. We propose that T cells discriminate antigen based on and via a threshold in the sum-of-binding which allows for rapid and reliable T cell responses to specific pMHC.
  We investigate the effect of rebinding between TCR and pMHC by modifying the canonical kinetic proofreading model [13] to explicitly account for the possibility of TCR/pMHC rebinding. The scheme is shown in Figure 1A, where the bound TCR goes through a series of steps (e. g. binding and phosphorylation by signaling molecules), all occuring with an identical forward rate constant (), after which it becomes activated. We describe the system using a set of ordinary differential equations (ODE) where () denotes the probability of the TCR being in the proofreading step. We assume that the pMHC is initially bound to TCR (, for all). In addition to this modified kinetic proofreading model, we establish the role of rebinding in an explicit T cell signaling model (Figure 1C, discussed later). The model explicitly accounts for a state where the pMHC is chemically dissociated but within physical proximity of the TCR () such that rebinding is possible. In this state, the pMHC can either re-bind the TCR (with a first order rate constant) or diffuse away (also with a first order rate). These first order rate constants (in units of) are related to the ensemble binding rate constant () and the diffusion-limited association rate constant at the membrane () via and, where is the local TCR concentration (see Methods). In this model, the effective lifetime of a TCR/pMHC bond is: (1) the mean lifetime of an individual bond () multiplied by the mean number of rebinding events (). The number of rebinding events is then determined by the relative values of and. We note that the effect of rebinding described here is distinct from the serial binding of many different TCR by a single pMHC, which has been the topic of previous studies [24], [15], [25] and is usually implicitly captured by continuum mathematical models. We verified the accuracy of our ODE model using spatial Monte Carlo simulations as described in the Methods. Briefly, we first established the accuracy of the simulation algorithm by comparing results to calculations from a partial-differential-equation (PDE) model that explicitly accounts for spatial effects. We next show that the spatial Monte Carlo simulations are in good agreement with the ODE model. Therefore, we conclude that this ODE model, which is computationally efficient to solve, accurately captures the effects of membrane diffusion. To address antigen discrimination at short time scales, we chose a threshold of s () for a productive signal, and computed the probability of productive signaling after 30 s () as a read-out for the T cell response. In Figure 2A we plot as a function of for a range of biologically relevant on-rates spanning two orders of magnitude in. As expected, we observe a sharp productive signaling threshold in terms of. However, we find that accounting for rebinding events has only a small effect on antigen discrimination. Figure 2B shows contours of as a function of both and in the case of 1 pMHC (as in panel A) and in Figure 2C we show results when 10 pMHC are presented by plotting contours of the probability that at least 1 out of 10 pMHC elicits a productive TCR signal in s. It is clear from the vertical contours that increasing the number of rebinding events (by increasing) has only a negligible effect on productive signals in the canonical kinetic proofreading model. We now examine a model within which TCR signaling is not disrupted by brief unbinding events, allowing a single TCR to integrate signals from multiple binding and rebinding events. This departure from a canonical proofreading model is plausible, given that a finite period of time is required to return the signaling state of the TCR to basal levels (e. g. by the action of phosphatases). We investigated this effect in our model, by introducing signal persistence states, denoted as, which represent unbound TCR from intermediate step (Figure 1B). Unbound TCR in an intermediate step return to the unmodified state with rate but kinetic proofreading resumes if the pMHC rebinds to the TCR. We chose a value of that allows for the signaling state of the TCR to persist for 0. 01 s (). If than signal persistence occurs rarely and when we recover standard kinetic proofreading. Results for several values of are discussed below in the context of endogenous pMHC. As seen in Figure 2D, allowing for signal persistence leads to a much greater sensitivity to the on-rate. For example, for a pMHC with (dashed line in Figures 2D), we observe that an order of magnitude change in the on-rate increases the probability of productive signaling by several orders of magnitude. Comparing the contours of in the absence and presence of signal persistence (Figure 2; panels B vs. E and C vs. F) further illustrates this dependence on in the latter case. With signal persistence, the T cell is able to discriminate between pMHC based on both and over a large portion of the parameter space. Examining Figure 2F, it is clear that in certain parameter regimes the T cell is still only able to discriminate pMHC based on. Discrimination is independent of when because for such small on-rates there is negligible rebinding before the pMHC diffuses away. In contrast to standard kinetic proofreading where the productive signaling threshold is determined entirely by (Figure 2A–C), in our model the threshold depends on the total number of rebinding events, (Equation 1) and is therefore determined by both and. We refer to this model as ‘sum-of-binding’ discrimination. Productive signaling can be achieved by a few binding events of long duration or by many binding events of shorter duration. We further support the notion of sum-of-binding discrimination in Text S1 by showing that this molecular model is comparable to a perfect detector that ‘samples’ and subsequently integrates the binding durations arising from a single pMHC, making the decision to respond based on a threshold in the sum duration of binding events. Next, we investigated the role of TCR clustering in enhancing localized rebinding. Sub-micron TCR clusters have been observed immediately upon T cell contact with a supported planar bilayer containing specific pMHC [26], [27], [28], suggesting that they form within seconds. This rapid clustering is consistent with a simple diffusion-trapping mechanism, e. g. by cytoskeleton binding (see Text S2). Localized within these clusters are many signaling molecules important for T cell activation [29], [26], [27], [28] and there is strong evidence that coreceptors are present [30], [26]. After this rapid initial formation phase, TCR clusters translocate to the center of the contact interface forming a large scale aggregate. Their accumulation at the center, over a period of 5–10 minutes, is a marker for the formation of the immune synapse. The spontaneous formation of a few TCR clusters has also been observed in experiments using planar bilayers containing non-stimulatory (null) pMHC [28]. The function of TCR clustering remains controversial [31]. Given the observation that TCR proximal signals are confined to rapidly forming TCR clusters, we tested the possibility that TCR clustering can enhance detection of weakly binding pMHC. We utilized spatial Monte Carlo simulations to model a single pMHC interacting with TCR, and considered two different distributions of an identical number of TCR: 1) a homogeneous distribution at a concentration of 100 and 2) a specified number of TCR confined to a cluster of radius 0. 1 at the center of the domain surrounded by the remaining TCR distributed uniformly. To maximize the effect of clustering, we initialized the simulations with the pMHC bound at the center of a TCR cluster. All simulations were terminated when s or when a productive signal was attained. We take the simulation domain to be sufficiently large to avoid any boundary effects. The Monte Carlo simulation algorithm is described in the Methods. In Figure 3A we plot the fraction of simulations that reached a productive signal in 30 s as a function of for the indicated number of TCR in the cluster. It is clear that increasing the number of TCR in the cluster has a negligible effect on productive signaling, though TCR clustering does increase the number of unique TCR bound by a pMHC (Figure 3B), and it substantially reduces the probability of escaping the cluster (Figure 3C). We conclude that receptor clustering has no impact on productive signaling, further underscoring the importance of pMHC rebinding to the same TCR versus serial binding of pMHC to different TCR. Another mechanism for increasing the number of rebinding events between a single pMHC and a single TCR is to include the T cell coreceptors CD4 or CD8 [32], [33]. Experiments have demonstrated that coreceptors bind MHC independent of TCR [34], [35]. The kinetics of TCR-coreceptor association are presently unknown, with some evidence suggesting constitutive association in resting T cells [36] and that TCR-coreceptor association increases on the time scale of minutes [37]. In our simple model, we assume that coreceptors are constitutively associated with TCR. In a following section, we use a detailed model of TCR-proximal signaling, in which coreceptors undergo reversible binding to TCR, and we show that our conclusions are unchanged. We used a heterodimerization model to capture the effect of coreceptors [38] (Figure 4A) and obtained binding parameters from the literature [34], [35], [36], [39]. The inclusion of coreceptors effectively decreases the mobility of pMHC, thus increasing the probability of rebinding TCR, and leading to a higher probability of productive signaling (Figure 4B vs. Figure 2D). We note that coreceptors increase sensitivity to weakly binding pMHC but do not enhance pMHC discrimination as such. Comparing Figure 4F to Figure 2C (no coreceptors), we conclude that, due to the presence of coreceptors, pMHC discrimination based on may occur over a wider range of parameter space. We also examine the role of coreceptors in the TCR cluster simulations described in the previous section. As expected, we find that coreceptors increase the probability of productive signaling but the clustering of TCR remains unimportant (Figure S1). A key advantage of signal persistence is that it allows the T cell to set a high threshold for productive signals (high specificity) while maintaining sensitivity to antigenic pMHC, provided they rebind. The threshold we have used is 15 s which requires pMHC to have in order to frequently achieve productive signals by a single binding event (see Figure 4C for). Thus, agonist pMHC with larger than this value will usually have to rebind (via a high) to achieve productive signals. The affinity and kinetics of endogenous pMHC binding to TCR have yet to be determined but are expected to be characterized by and endogenous pMHC are therefore unlikely to transduce productive signals in a single binding event. However, endogenous pMHC are present in large numbers and therefore sequential binding could result in activated TCR. We examine this possibility in our model by comparing the rate of signal decay at the TCR () to a conservative estimate for the rate of endogenous pMHC binding to an individual TCR (, assuming and that endogenous pMHC are present at a density of 500). Intuitively, given the separation of scales between the binding and signal decay rates, we expect that the TCR will always revert back to the unmodified state before additional endogenous pMHC can bind. We confirm this physical argument by formulating the ODE model in the case of a high concentration of pMHC (see Text S3). In Figure 5 we show the probability that at least one TCR at the T cell-APC interface has achieved a productive signal when identical pMHC are presented. We find that unless is very small (panel D), the region of parameter space corresponding to endogenous pMHC (large, small) has a low probability of productive signals (). In the previous sections, we used generic proofreading models to test the role of localized rebinding and signal persistence. This raises the important question of whether our findings are relevant to more realistic TCR signaling models. To this end we have implemented a particular realization of TCR signaling based on the work of Altan-Bonnet and Germain [40]. The model is depicted in Figure 1C and represents a coupled system of 139 ODEs which were generated using BioNetGen [41] (see Text S4 for the BioNetGen file defining the model). The model explicitly accounts for the enzymatic kinetics of Lck sequentially phosphorylating the and stabilization of each ITAM by Zap70. A productive signal is defined as a fully phosphorylated with three bound Zap70 molecules. We do not include further events, such as the phosphorylation of Zap70 which subsequently leads to the modification of cytosolic molecules involved in feedback (e. g. SHP-1, ERK). In the context of this model we are defining a productive signal, and hence a T cell response, as the generation of cytosolic molecules. As with the generic models we explicitly model rebinding and allow for signal persistence. In Figure S2 we show the probability of a productive signal as a function of both and (A) without signal persistence, (B) with signal persistence, and (C) when including both signal persistence and coreceptors. We find that all our results are in qualitative agreement with those from the generic models presented earlier (Figure 2,4), supporting the general applicability of our conclusions. Quantitative agreement is difficult to obtain because the threshold in the detailed model is determined by many individual parameters. Coreceptors are not constitutively associated with TCR in this model but undergo reversible binding. We have used a reasonable TCR-coreceptor affinity but if the TCR-coreceptor reaction on-rate is very small or the off-rate very large, coreceptors will have little impact on immobilizing pMHC.
  We model the interaction between a single pMHC and a homogeneous TCR distribution at the T cell-APC interface using a two-step binding model [60], [61], (2) where, , and are the probabilities of finding the pMHC spatially separated from TCR (no binding may take place), unbound but within binding range of TCR, and bound to the TCR, respectively and is the concentration of TCR. The diffusion-limited on-rate is given by, where is the diffusion coefficient, is the mean distance between TCR (), and is the reaction radius of the TCR. When the pMHC is in state, the rate at which it binds (or couples) TCR is and when pMHC is in state, the rate at which it fully unbinds (or uncouples) from TCR is. At equilibrium we expect that and therefore. Assuming that, it then follows that. We define to be the area occupied by a single TCR and therefore. The parameter also sets the lattice size used in the spatial Monte Carlo simulations that are used to validate this ODE model (see below). We couple the above rebinding model to the three TCR signaling models shown in Figure 1. The detailed TCR-proximal signaling model is described in the main text and the two generic proofreading models (Figure 1A, B) are modeled with the following set of ODEs, where, and represent states where the pMHC and TCR are spatially separated, unbound but within binding proximity, and bound, respectively. We consider steps in the kinetic proofreading scheme with a forward rate and denote intermediate states as. Coreceptor binding is denoted by a superscript. The pMHC may diffuse away (at rate) when it is unbound. We introduce signal persistence through the quantities which allow the TCR to remain in state when the pMHC unbinds. In this state the unbound yet modified TCR may resume kinetic proofreading if the pMHC rebinds (), may return to the the unmodified state (), or the pMHC may diffuse away (). The reaction off-rate between TCR-pMHC is and between pMHC and coreceptor is. Since we are considering the interaction between a single pMHC and a single TCR the reaction on-rates are first order (in units of s−1) and can be related to macroscopic quantities by a reaction parameter (): and. The macroscopic on-rates, and, are 2D quantities in units of and can be related to experimentally determined 3D quantities, see below. Justification for the relations between the microscopic () and macroscopic () on-rate are provided below. In all calculations the pMHC is initially bound to the TCR in state (, all other states are zero at). We set which corresponds to discrimination based on a threshold of (canonical kinetic proofreading) and a threshold in the sum-of-binding () of 15 s (kinetic proofreading with signal persistence). In both cases the probability of a productive signal, after time, is. Numerical solutions are obtained using the Matlab function ode23. We validate the ODE model described above with explicit spatial Monte Carlo simulations based on a discrete-space continuous-time model [62], [63]. We simulate a single pMHC diffusing on the APC membrane and binding to TCR on the T cell membrane. We use a lattice simulation which allows for reactions when the pMHC is in a lattice site opposite a TCR. The intrinsic binding rate is, where is the 2D (macroscopic) bimolecular reaction on-rate and is the lattice spacing. The unbinding rate is and diffusion () is captured by a first order reaction, of rate, to nearest neighbour lattice sites. We take to be the size of a TCR (nm) in our simulations, making it possible to capture both reaction-limited and diffusion-limited regimes. Depending on the state of the pMHC only certain reactions are possible and the quantity is the combined rate. In the absence of coreceptors, three states are possible: pMHC is unbound (), pMHC is unbound above TCR (), or pMHC is bound (). Based on the overall rate, , we compute the time for the next reaction, where is a uniformly distributed random number between 0 and 1. Knowing, we use a second random number, , to determine which reaction actually took place. Finally, we simulate the reaction, advance time by, recompute, and repeat. For clarity we have described the method without kinetic proofreading, signal persistence, and coreceptors, but these effects just increase the number of possible reactions which we have implemented. This Monte Carlo algorithm is used to explore the effects of TCR clustering, as described in the main text. We first tested the accuracy of our basic spatial simulations by comparing them to a deterministic partial differential equation (PDE) model. The simulations depend solely on intrinsic/microscopic quantities between individual TCR/pMHC (,) while the PDE model depends on ensemble/macroscopic quantities measured in bulk (,). Simulations were initialized with the pMHC bound to TCR at the centre of the domain and terminated once the pMHC reached a distance. This termination conditions simulates a stochastic first passage time process. We performed 1000 simulations and in Figure S3 we show the binned (and normalized) first passage time (grey line). The PDE describing this first passage time process is, with on the boundary (at), , and. Note that and are the macroscopic quantities. To obtain the probability of a first passage we numerically solve the above PDE and compute the total flux through the boundary, , as a function of time. We plot in Figure S3 (dotted black line) and find that it is in good agreement with the stochastic simulation in both the reaction- and diffusion-limited regimes. Good agreement is also observed over a wide range of parameters (not shown). Having validated the Monte Carlo model with the PDE computations, we next implemented the complete Monte Carlo simulations that includes kinetic proofreading, signal persistence, and coreceptors and compared these simulations to the ODE model described above and used to generate Figures 2 and 4. As in the ODE computations, the simulations are now terminated after s or once a productive signal is transduced. In the first column of Figure S4 we show the fraction of simulations (out of 500 per data point) that terminated with a productive signal (colored circles) for A) kinetic proofreading, B) kinetic proofreading with signal persistence, and C) with the addition of coreceptors. We find good agreement with the ODE calculations (solid lines). In the second column we show good agreement across a wide range of TCR concentrations. We conclude that the ODE model accurately captures the effect of membrane diffusion. The calculations and simulations we have used rely on several parameters, many of which have been experimentally determined. The solution reaction parameters for many TCR/pMHC have been measured using SPR [10], [11], [12], [23]. Solution on-rates have been reported in the range of 500–300000 and off-rates in the range of 0. 01–1 s−1. SPR measurements have revealed similar equilibrium binding constants between CD4 or CD8 and MHC [34], [35]. Wyer et al [34] report the solution on-rate to be and the off-rate to be. Experiments probing the interaction between CD8 and MHC in living cells report roughly similar on-rates [36] and off-rates [39]. We have used an off-rate of between MHC and coreceptors. We have simulated reactions between membrane proteins and hence the bimolecular reaction parameter, , is a 2D quantity. However, reaction measurements between TCR-pMHC and CD4/CD8-MHC using SPR provide solution or 3D on-rates. To obtain estimates of 2D quantities we multiplied the 3D on-rate (in units of) by a factor of (where is Avagadro' s number) to obtain units of m3/s. We next divide this 3D quantity by a confinement length to obtain the 2D on-rate in units of. We use a confinement length of 0. 262 nm, which can be obtained by comparing 3D and 2D dissociation constants [24], [64], [25]. This conversion indicates a range of for TCR-pMHC on-rates and for CD4/CD8-MHC on-rates. Recent experiments have revealed membrane on-rates in this range for other T cell molecules [49]. Nonetheless, the accuracy of this method is unknown and we therefore explore a larger range in the reaction on-rates, see Figure 2 for example. We have taken the diffusion coefficient of pMHC to be [51]. Calculations with or give similar results. We have assumed that the diffusion coefficient of TCR in a cluster is zero. We have assumed steps in the kinetic proofreading scheme but results are qualitatively similiar with any. The rate of each step in the proofreading scheme, , was equal and set to. In order to obtain signal persistence the value of the decay-of-signaling parameter () must be smaller than. We have taken which may represent specific phosphatases having a concentration of and effective on-rates of. Standard kinetic proofreading models implicitly assume that (i. e).
summary: T cells are essential players in the immune response to pathogens such as viruses and bacteria. They can be activated to respond when they recognize molecular signatures of infection (antigens) on the surface of antigen-presenting-cells of the immune system. The T cell response is highly specific (a particular T cell responds to only the right antigen), sensitive (a T cell will respond to as few as 1–10 antigens) and speedy (antigen binding may induce signaling within seconds). We wish to understand how the T cell, using its surface antigen receptors, is able perform this task. To do this, we developed mathematical models of antigens binding and unbinding from T cell surface receptors. Our primary finding is that T cells can discriminate antigens based on both their binding and unbinding rates from the T cell antigen receptor. We examine potential impacts of T cell antigen receptor clustering, T cell surface coreceptor molecules, and background self-antigens on this process.
keywords: biophysics/cell signaling and trafficking structures
  biophysics/theory and simulation
  immunology/antigen processing and recognition
  cell biology/cell signaling
title: A Role for Rebinding in Rapid and Reliable T Cell Responses to Antigen