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* Mathematics Institute, University of Warwick, Coventry, United Kingdom; and Sir William Dunn School of Pathology, Oxford University, Oxford, United Kingdom
Correspondence: Address reprint requests to Nigel J. Burroughs, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK. E-mail: njb{at}maths.warwick.ac.uk.
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ABSTRACT |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHEORYRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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). Conformational change models propose that the TCR undergoes a conformational change in response to agonist binding. Structural studies have largely ruled out conformational changes in TCR
ß itself as a mechanism but it remains possible that binding leads to conformational changes between TCRß and the associated CD3 complex or between two TCRßs in a preformed TCRß dimer. Recent studies have shown that TCR binding leads to a conformational change in the cytoplasmic tail of the CD3
, which enables a proline-rich motif therein to bind Nck (2,3). This change appears to be independent of tyrosine phosphorylation but it remains to be shown whether it is required for TCR activation. Segregation models propose that TCR activation involves redistribution of the TCR and other cell-membrane associated molecules. Specifically, in the kinetic segregation hypothesis (4), regions of close contact form between T-cells and other cells from which molecules with large ectodomains, including the tyrosine phosphatases CD45 and CD148, are excluded. This leads to the creation of a tyrosine kinase-rich domain (KRD) within which tyrosine phosphorylation is favored. TCRs that bind ligand within these regions will be trapped here for long periods, leading to tyrosine phosphorylation of the cytoplasmic portions of the CD3 subunits. Key to the mechanism is the dependence of the TCR/CD3 phosphorylation state on the KRD residence time.
We developed a model to determine the signaling characteristics of the kinetic segregation hypothesis. We use a patterned environment of KRDs within a predominantly phosphatase-rich environment (PRE); thus kinases such as Lck are inactivated (and primed) in the latter, whereas they autoactivate in phosphatase-excluded regions. These KRDs, or phosphatase exclusion regions, are identified with regions of close membrane-membrane proximity (separation 14–15 nm), since the predominant phosphatases have large ectodomains (25–40 nm) and therefore exclusion is energetically favored. Thus, KRDs also function to localize bound TCRs (Fig. 1). Signaling through the TCR involves the stepwise formation of a large multi-molecular complex that has been termed the signalosome (5). One of the earliest events in agonist detection is the phosphorylation of immunotyrosine-based activation motifs (ITAMs) on TCR-associated CD3 chains by Lck, which then recruit and activate the tyrosine kinase ZAP70. ZAP70 phosphorylates the transmembrane adaptor protein LAT, which acts as a scaffold to recruit adaptor proteins and effector proteins including SLP-76, Grb-2, GADs, Sos, and PLC
1, which are, in turn, further activated by tyrosine kinases including Itk and Tec. Thus, to attain a fully active complex requires a number of phosphorylation and recruitment steps (Fig. 2), a process that has a number of similarities to kinetic proofreading models of signaling and DNA motif discrimination (6–8). Such a sequence of events could therefore underpin ligand specificity within the context of differential localization of bound TCR to regions of close contact. We demonstrate that the residence time of unbound TCRs in the KRDs is very short (milliseconds), and that TCR binding to peptide-MHC significantly lengthens this residence time. For appropriate parameters, a significant number of activation steps can accumulate and thus provide the basis for ligand discrimination and detection. We note, however, that bound TCRs are not absolutely localized to KRDs since they are only localized by a finite energy barrier, and thus leave the KRDs presumably as a state where the membranes are locally distorted, the membranes being more flexible than the actual proteins. However, these excursions out of KRDs are short and do not significantly reduce levels of triggering. We demonstrate that background levels of activation from unbound TCRs and discrimination of weak versus strong agonists relies on both efficient relocation and the activation sequence, the latter functioning as a kinetic proofreading scheme. Further, we incorporate a sequestered, fully activated state that reproduces the observed increase in phosphorylated TCRs on ligand exposure. We interpret this state as a stabilized signalosome.
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,10) This large-scale segregation is slower than the small-scale segregation that we describe here, and it occurs after TCR triggering. Indeed, synapse models assume upregulation of LFA-1, which follows, and depends on, TCR triggering (11). However, there are some aspects of similarity with these models given their common dependence on differential extracellular domain sizes. The kinase-phosphatase balance as a component in driving TCR triggering has previously been modeled in a feedback context (12,13). |
THEORY |
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TOPABSTRACTINTRODUCTIONTHEORYRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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), since synapse patternation fails to form in the absence of agonist. These close-contact domains are therefore transient and formed by fluctuation of the intermembrane separation, fluctuations forming a distribution of domain sizes, probably ranging from nanometers to hundreds of nanometers. These regions will exclude the large phosphatases (CD45, CD148) and thus coincide with regions of a high kinase to phosphatase balance of activity, regions, which we call kinase rich domains. We simulate a region of the cell-cell contact interface formed between a T-cell and an antigen-presenting cell. Within this contact interface, we assume there are regions of close membrane proximity (membrane separation 
15 nm) of area fraction f (Fig. 3). We use a static pattern of KRDs, which is justified provided their lifetime is larger than the timescales of the triggering process and the TCR localization time. They will also diffuse in the surface; however, this will have negligible impact on the triggering kinetics and thus can be safely ignored.
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between a KRD and the phosphatase-rich environment for a bound TCR-pMHC complex. To estimate , we model bonds as springs (9,14), with natural bond length l and elasticity 
to give 
, where z(x) is the membrane-to-membrane distance in the contact interface at position x. Thus, the ratio of bond affinities between KRDs and the PRE is exp(/kT), where 
and 
l is the difference in membrane separations between the two regions. Since the TCR-pMHC bond does not break under application of a pN force, we assume for simplicity that the off-rate is identical in the two environments, i.e.,
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The results are in fact not strongly sensitive to this assumption. A bound TCR will experience a free energy barrier at the KRD/PRE boundary; thus, in the random walk simulation of a bound complex, a move from a KRD to the PRE is accepted with probability exp(–/kT). Both migration and reaction kinetics thus generate a relative complex density of e–/kT.
The same principles can be used to compute the depletion of CD45 in the KRDs, i.e., phosphatase activity is reduced by e–'/kT in KRDs, where ' 
is the difference between the chemical potentials of a CD45 molecule in the two environments. However, it has been proposed that CD45 has an active and an inactive state, whereas Lck activation is nonlinear (autoactivation) and requires priming by CD45, effects not determined by these assumptions. Because of this, we assume that kinase and phosphatase activities are independent in the two environments. We do not explicitly model CD45 molecules or Lck, but assume that these are at sufficiently high concentration that concentration fluctuations are negligible.
Activation dynamics
We model TCR triggering as a sequence of states, effectively a reversible kinetic proofreading scheme of length m (Fig. 2 B), where a step is either a tyrosine phosphorylation or recruitment of a component of the signalosome (5). For instance, earlier steps may correspond to ITAM phosphorylation or kinase recruitment (ZAP70, Lck), and later steps to recruitment and phosphorylation of adaptors (e.g., LAT, SLP-76, Grb2) and effector proteins (e.g., phospholipaseC1). The rates of these activation steps depend on the environment; thus, in a KRD the activation rate 
is high, whereas there is a low rate of reversion 
, and in the PRE scheme the corresponding rates are p and q with 
(Fig. 2, Table 1). This environment dependence strictly relates to phosphorylation steps, since the two environments only differ in efficiency of kinases and phosphatases; hence, we are assuming that there are a number of key rate-limiting phosphorylation steps in the construction of the signalosome. For specificity to arise from this sequence of steps they must also have similar rates, a likely consequence of the fact that the same kinase is probably responsible for subsequent steps. Thus, our model is simplified in that the rates of subsequent steps are all assumed equal (faster steps are ignored) and events are assumed independent. We also modify the activation sequence by incorporating a final "sequestered" state (Fig. 2), corresponding, for example, to the formation of an enlarged fully competent signaling complex of adaptors and kinases where cooperative binding within the signalosome decreases the rate of signalosome decay/unbinding. Other models with a gradual increase of stability of later stages in the activation sequence, or unequal rates of (in)activation are likely to give similar results. Note that the pMHC unbinding rate is not influenced by the activation state of the TCR; thus, coreceptor recruitment is not explicitly included as part of the activation sequence. We justify this either by modeling activation in the absence of coreceptors, or, when coreceptors are present, they are in excess and thus MHC is predominantly already bound to CD4/CD8.
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). For reversible kinetic proofreading schemes with m > 3 the probability of reaching the end of the sequence instead of unbinding, and thus achieving full activation, is approximately exp –ßkoff, where 
for a TCR that remains in the KRD environment (high ). Thus, under variation of the length m of the sequence the rates of the intervening steps should be rescaled by m to preserve the sensitivity, i.e., 
etc. Accordingly, the mean time to reach the end of the sequence conditioned on not unbinding is conserved, whereas the variance of this time decreases as m–1. For an agonist with a lifetime of 10 s, koff = 0.1 s–1, we fix this probability at e–1 0.367, thereby determining the scale of ß. This implies that in TCR excess, the triggering rate of a pMHC agonist is koff exp –ßkoff, and thus maximal at koff = ß–1 0.1 s–1. In the simulations we use m = 10, although all sufficiently large m give similar results (not shown). To complete determination of the model, we need to determine appropriate values for p and q. Dephosphorylation rates in the PRE must be sufficiently large to reduce levels of activation in the absence of agonist; in fact, in signal interruption studies using specific kinase inhibitors or blocking antibodies, tyrosine phosphorylation is lost within a minute, indicating that dephosphorylation is rapid (16,17). We also assume that although KRDs are predominantly kinase active regions, kinase activity is also present in the PRE, whereas phosphatases cannot be absolutely excluded from KRDs. We illustrate our model under relative kinase/phosphatase activities varying only by two orders of magnitude (Table 1).
The high stability of the final sequestered state is achieved with a low inactivation rate 
, a half-life of 5 min, and a binding rate + on a scale of seconds, both independent of the environment. The lifetime was set to match TCR down-modulation timescales (18). This is reasonable given that only triggered TCRs are likely to be downmodulated, whereas TCRs remain marked for downmodulation after dissociation from pMHC (18,19). If all triggered TCRs are not downregulated, i.e., if there is a probability of inactivation (20), the sequestered state lifetime will be <5 min.
Self-peptides
We do not explicitly model self-pMHC molecules, since they are at high density. We used a uniform fixed density of unbound self-pMHC, Mself; thus, unbound TCRs have a constant rate konMself of binding to self. In the simulations, all self-pMHC have the same off rate (3 s–1 or 5 s–1), i.e., we only model the most stable peptide component (longest half-life) for simplicity. We use three self-profiles for illustration: self, with koff = 3 s–1 at density 50 µm–2; low-activity self, with koff = 5 s–1 at density 300 µm–2; and high-density self, with koff = 3 s–1 at density 300 µm–2. The first two are likely to be representative of typical antigen presenting cells, whereas the latter is used here to allow trends to be observed (i.e., triggering levels are high enough to be visible) and thus compared to agonist triggering rates.
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RESULTS |
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30%, (koff = 0.1 s–1), the average rate of triggering of a single pMHC is at most two per minute, thereby determining the timescale. As fully activated TCRs are assumed to inactivate (or be downregulated) with a half-life of 5 min (
), the number of activated TCRs saturates at a level of 10–15 per pMHC on achieving equilibrium, a small fraction of the total TCR pool (30,000 per cell (18)). Without the stabilization of the final state, fully activated TCRs have a very short lifetime when released from the MHC, since they diffuse into the PRE where phosphatase activity is high. In these circumstances, although TCR triggering is high under agonist exposure, the density of fully activated TCRs is negligible; in fact, in the absence of stabilization of the final state we observed a fully activated TCR fraction of only 0.25% at an agonist density of 1 µm–2, compared to up to 15% with a stabilized final state.
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, D decrease, or q increases.
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= backgnd + self + agonist. For efficient agonist detection, the signal from a good agonist agonist must be clearly distinguishable from background signaling backgnd and from self self. To illustrate the determinants of self-triggering we will use an unrealistically active self-peptide profile, specifically high-density self, with koff = 3 s–1 at a density of 300 µm–2. This would mean that 100,000 MHC, or 10–50% using 200,000 to 1,000,000 MHC per cell, are loaded with peptides with koff = 3 s–1. More realistic profiles are self, with koff = 3 s–1 at a density of 50 µm–2, and low-activity self, with koff = 5 s–1 at a density of 300 µm–2. For the parameters chosen in Table 1, the model clearly demonstrates that a single agonist pMHC has a triggering rate above background levels and populations of self (neutral) peptides (Figs. 4 and 6). The shift of occupancy to higher activity states in the presence of peptides (self or agonist) is shown in Fig. 6. The two realistic self profiles, self and low-activity self, show a similar distribution among the partially active states, demonstrating that both peptide quality and density play a role in self, whereas the fully active state is only occupied with more stable self-peptides (high-density self) (Fig. 6). These self and background triggering rates have to be scaled by the appropriate contact area to obtain the triggering for a contact. Typical areas of contact are on a scale of 10 µm2; thus, a single peptide agonist would still give significant triggering relative to realistic self-peptide profiles (multiplying self or low-activity self by 10 relative to Fig. 6), although self is still likely to generate fully triggered TCRs with our parameters. Background triggering on the free surface of the T-cell will be negligible, since KRDs are absent.
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(except for the final sequestered state) (Fig. 6). This is similar to that observed in kinetic proofreading schemes and follows from the assumed independence of the kinase and phosphatase kinetic rates with position k in the activation sequence of Fig. 2. In the case of background triggering, the fraction of time spent in KRDs is in fact f, the KRD area fraction, because there is no preference for either environment. The ratio is then the ratio of phosphorylation to dephosphorylation rates averaged over the environment, 
and 
respectively. Thus, despite a large population of N 30000 TCRs per cell, the background signal is easily controlled; both an increase in the length of the kinetic proofreading scheme m, or a decrease of , by either increasing the dephosphorylation rate in the PRE or reducing the fraction of KRDs, are effective in reducing the background signal. Note that should be small for a low level of activation in the absence of agonists, and thus the constraint 
follows, as f cannot be negligible, i.e., the dephosphorylation rate in the PRE is faster than tyrosine phosphorylation in KRDs. The most effective reduction, without compromising agonist detection, is through increasing the dephosphorylation rate q, e.g., high CD45 activity, and increasing the length m. With the parameters of Table 1, the background triggering rate is <10–14 µm–2 s–1 with = 0.024.
A second key attribute of T-cell activation is the ability to discriminate peptides, essentially discriminating between agonists with different off-rates koff (21,22). Although feedback (13,23) and signal integration (16) may play a role in differentiating agonists, there must be differences in the signal at its source, i.e., in the rates of triggering. We find good levels of discrimination between agonists with off rates 0.01, 0.1, and 1.0 s–1, whereas agonists with koff 0.1 s–1 are optimal (Fig. 7). Longer half-lives decrease activation, eventually plateauing at one triggered TCR per pMHC for half-lives of 1000 s (Fig. 7). This reflects the importance of serial triggering (18) to T-cell activation; in this case, the pMHC fails to exchange TCRs, remaining bound to the same (activated) TCR. Self, being comprised of a high number of poor binding peptides is effectively controlled because the residence time of bound complexes in KRDs remains small because of rapid TCR exchange. In contrast to background signals from unbound TCRs, triggering by self has the same dependencies on model parameters as agonist signaling; thus, in particular, it cannot be reduced by continually increasing the sequence length m (unless sensitivity is reduced (24)).
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), the triggering rate for an agonist agonist is given by 
where [T] is the free TCR density and , with a corresponding queue equilibrium agonist/–. Here we assume the number of agonists is small so that we can ignore TCR competition; the free TCR density is therefore approximately equal to the TCR surface density. We also ignore the time to enter the final sequestered state. These expressions clearly demonstrate that the fraction f must not be negligible for agonist detection; specifically, fractions f as low as koff/kon[T] 20% will cause a reduction in signaling. For finite , a time-averaged kinase and phosphatase activity can be used to derive an appropriate approximation. These approximations give reasonable fits to the data (not shown), and capture the essential qualitative features.
Environment heterogenity also affects signaling characteristics; both the strength of the trapping potential and the size of the KRDs affect signaling. If the trapping potential is insufficiently high, bound TCRs spend sufficient time in the PRE to become inactivated and the triggering rate falls (Fig. 8 A), whereas at high potentials triggering becomes independent of . Similarly, as KRD size increases, triggering increases because bound TCRs remain longer in a KRD (Fig. 8 B). However, self and background signals also increase, as shown in Fig. 8 with high-density self, reducing the ability to filter nonspecific (relative) signals. KRD size only affects triggering above a certain threshold that depends on the trapping potential and activation rates. This can be explained from a consideration of the relative timescales. We find that localization of the bound complex, although significant compared to unbound TCRs, is far from absolute, and TCR-pMHC complexes escape from the KRDs on timescales significantly shorter than 10 s (optimal agonist complex lifetime). For instance, KRDs with localization potentials = 1, 2, 3, 5, 7, and 10kT have mean residence times of 0.005, 0.015, 0.04, 0.3, 2.1, and 45 s, respectively, whereas for = 15kT, the residence time in excess of 1000 s. In contrast, the majority of excursions from KRDs are very short, mean 4.5 ms, and thus only when the excursion frequency is sufficiently high is signaling reduced. This compares to phosphorylation in KRDs and dephosphorylation in the PRE with timescales of 0.5 s and 0.02 s, respectively. Therefore, for < 5, bound TCRs experience an average kinase-phosphatase environment, and isolated events of localization to a KRD are of insufficient duration to induce a single activation step. However, for = 7 and above, localization to a KRD will on average increase activation levels by more than one activation step. The KRD localization duration relative to the activation timescale provides an explanation for the enhanced triggering rate observed on larger domains (Fig. 8).
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for efficient ligand detection are interdependent. This follows from an analysis of the residence time of unbound TCRs and bound complexes in KRDs (Fig. 9). Since the system is stochastic, there is a distribution of residence times that should be compared to the exponential distribution for the lifetime of the TCR-pMHC complex Exp(koff) to assess probable event sequences. Escape from a localized potential can be explained by a rough argument as follows. At each attempt of the TCR-pMHC complex to cross the KRD boundary, there is a probability of success of e– (setting kT = 1). Diffusion gives a space-time dependence of 
(root mean-square distance); thus, the frequency of attempts to cross the boundary has a dependence cD/r2 (where c is a constant), which can clearly be very high for small domains. In time t, there are therefore cDt/r2 attempts; provided this is large, we obtain the probability of no escape by time t as
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/(cD) with c 50 from numerics; thus, smaller domains require higher trapping potentials to achieve the same level of localization. Using this relation, we find that for a TCR-agonist pMHC complex to reside in the KRD for its lifetime (10 s) requires a KRD size of order 
, or 700 nm for = 5kT, whereas to remain in a KRD for 0.5 s (kinase phosphorylation timescale) requires a KRD size of 150 nm. Thus, KRD size-enhanced triggering will occur for KRDs with radii >75 nm (Fig. 8 B). Finally, we examine the requirements on the area fraction f. Since TCR-pMHC complexes form most efficiently in KRDs, ligand detection decreases as f decreases; however, since the PRE damps activation in the absence of agonists, a large area fraction f also results in poor discrimination since the self contribution is large (Fig. 8 C). This can be compensated by increasing the phosphatase efficiency in the PRE, i.e., there is no particular area fraction selected by the dynamics except that it cannot be negligible or agonist triggering will be negligible, and must be <100%.
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DISCUSSION |
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). This timescale is determined by the triggering rate, two activated TCRs per agonist per minute, and the inactivation rate, 0.2 per minute, whereas the timing of cell responses will depend on where thresholds are set and on what variables (15). Our model thus provides a basis for ligand detection distinct from the traditional mechanisms of conformational change or dimerization, a mechanism that isn't limited to the immunological context discussed here. For efficient agonist detection, we identify three crucial properties:
). In fact, only phosphatase exclusion is necessary, i.e., kinase activity can be uniform 
(not shown). ).
We demonstrated this mechanism with a 240-fold difference in dephosphorylation rates between the PRE and KRDs, of the same order as the segregation energy potential, e' = 240 with ' = 5.5kT, as required for consistency (see discussion in Theory). There are, however, other cytoplasmic phosphatases implicated in TCR triggering that would not be excluded from regions of close contact. Our exclusion criterion refers to phosphatase activity targeting certain key substrates critical for TCR triggering. It is plausible that this specific phosphatase activity is decreased by such an amount for the following reasons. First, CD45 is exceptionally abundant, comprising >10% of the T-cell surface (26) and >90% of the membrane-associated tyrosine phosphatase activity (27). Second, tyrosine phosphatases show exquisite substrate specificity when expressed at normal levels in their normal environment (28), making it unlikely that other phosphatases can efficiently substitute for CD45. Clear evidence for this is that T-cells deficient in CD45 show profound defects, demonstrating that other tyrosine phosphatases are not able to substitute for CD45 (29). Finally, we have not optimized this model; thus, the limits on the parameters for functionality are not at present known and a phosphatase activity difference <240 may well be possible. However, to validate an analysis of this type, the specificity and sensitivity properties for early TCR triggering need to be determined, specificity most likely being function-dependent and thus differing between cytokine secretion and early signals (15). Our analysis also indicated a dependence on KRD size above a certain threshold; for our parameter values, this large domain enhancement occurred for domains >100 nm in radius. However, for particular parameter regimes, triggering could be solely dependent on this enhancement, i.e., small domains could produce negligible triggering. Otherwise, the system is robust to a reasonable level of variability between cells, but extreme variations in area fraction or kinase or phosphatase densities disrupt ligand detection.
A number of T-cell activation dependencies can be explained by our model. First, truncation of the ectodomain of tyrosine phosphatases CD45 and CD148 inhibits TCR triggering (30,31), exclusion and segregation being less effective in this case. Second, activation has a dependence on the KRD patternation. Significantly, as the kinase-rich domains increase in size—for instance, as they coalesce in the immunological synapse—activation signals would increase and the ability to filter out signals from null peptides decreases. Contributions to the signals from null peptides has been observed in the synapse (32); our model suggests that this contribution only arises from large KRDs. Third, the coreceptors CD4 and CD8, by stabilizing the TCR-pMHC complex and localizing the kinase Lck with the TCR, are likely to increase the residence time of bound TCRs to KRDs and also increase the rate of early signalosome activation steps and thus increase signaling. Fourth, it has recently been shown that progressively increasing the size of the pMHC abrogates TCR triggering (33). It was proposed, with some supporting evidence, that this was the result of an increase in intermembrane distance in the region of TCR engagement of pMHC, which resulted in less effective exclusion of CD45 from this region. The model presented here suggests an additional mechanism, namely, less effective trapping of elongated TCR-pMHC within the KRD. The model predicts that the potential energy barrier preventing diffusion of engaged TCR-pMHC complexes will be strongly dependent on changes in TCR-pMHC dimensions.
The essential requirement for achieving differential signaling between unbound and bound TCRs is the segregation of bound TCRs from phosphatases. Our model uses the ingredients of the kinetic segregation hypothesis (4). However, the key mechanism of CD45 exclusion from regions of close contact through its large extracellular domain size also suggests that CD45 access to bound TCRs may be reduced compared to access to unbound TCRs or, more specifically, that a bound TCR can generate its own kinase-rich shell environment (in analogy to lipid shells (34)). Here, the distinction is between local membrane distortion around the single TCR-pMHC complex compared to a region of close contact on a larger scale (>10 nm) and stabilized by other molecules, e.g., coreceptors such as CD2-CD58. This suggests that the event of TCR binding alone is sufficient to exclude CD45 from the local bond proximity, reducing phosphatase access and thus reducing TCR dephosphorylation. Lck may also autoactivate within the shell, leading to TCR tyrosine phosphorylation. This model would display triggering characteristics similar to those described here (not shown), but without the requirement for spatial segregation/patternation if we assume that bound TCRs generate a local kinase-dominant environment similar to that of larger KRDs. In contrast to the kinetic segregation relocation mechanism, there is a direct correlation between binding and phosphorylation in this case. However, there are a number of arguments against this assumption. First, local membrane distortion around a single TCR-pMHC bond produces graded levels of exclusion; thus, CD45 exclusion will be less efficient as the signalosome grows in size, and will never be as effective as a larger KRD where cooperative effects mean that the TCR is highly protected within the body of the KRD. Kinase activity may also be reduced in kinase-rich shells relative to larger KRDs. Second, the cell membranes apply a pulling force to the bond, which in larger KRDs is shared among any bonds in the domain. This is likely to reduce the half-life of the bond and thus reduce the triggering, which is very dependent on complex stability. In our simulations, we assumed that koff was not affected by the intermembrane separation (Eq. 1) or the application of a pulling force on the TCR-pMHC complex. For KRDs, this assumption could be relaxed. A 50:50 split of the effects of the potential between the on- and off-rates in the KRD segegration model, (kon(PRE) = kon(KRD) exp(–/2kT), koff(KRD) = koff(PRE) exp(–/2kT)), was found to have a negligible effect on the agonist and self signals. This is because excursions of bound TCRs are short in the PRE (milliseconds). We conclude that signaling from TCRs in the absence of KRDs is likely to be inefficient because of these effects. We suspect that formation of a kinase-rich shell by bound TCRs in the PRE may enhance the early steps in the activation cascade, whereas as the developing signalosome increases in size phosphatase access increases, giving a much stronger requirement for finite-size KRDs for later steps in the activation sequence. This suggests that the early phosphorylation of CD3 may not be as dependent on kinase-rich domains as the construction of a fully competent signalosome.
A number of our assumptions do not affect our results. We used a simple reversible activation sequence that possessed kinetic proofreading characteristics. The exact structure of the activation sequence is not important and other models will give similar results provided it retains specificity; in fact, we argue that sequences with m > 3 all give similar results. Activation steps can be unequal in importance (the slowest steps determining the specificity properties, i.e., m is the number of rate-limiting steps), for instance, the first couple of tyrosine phosphorylations may be easily acquired, which would raise the average phosphorylation state of the background. In fact, background phosphorylation is observed in the absence of agonist (35); this, however, is not an argument against kinetic proofreading. In addition, dephosphorylation steps can be grouped (multiple dephosphorylation events), which will allow the dephosphorylation rate in our model to be reduced. In contrast, activation events that require protein recruitment cannot be so grouped. We also used the total number of fully activated TCRs as a measure of system (cell) activation, the accumulation of fully activated TCRs functioning as a signal integrator. Other outputs could easily be used, for instance, sustained internal calcium levels retain transcription factors such as NFAT and NFB in the nucleus and thereby integrate TCR triggerings. We note that the stability of the sequestered state was essential for signal integration since the inactivation rate – directly determined the equilibrium level of fully activated TCRs, and a less stable complex would result in an increased level of noise in the output. Thus, our output has two sources of noise, first the signal noise, i.e., from the source of TCR triggerings, and second from the accumulation of fully activated TCRs. The latter is in fact very noisy, since the equilibrium load is small, 10–25 (Fig. 4), with a relative error of 20–30%, and thus the signaling characteristics can in fact be superior to that indicated here. Other components of the signaling cascade, e.g., the calcium signal, may therefore have higher specificity under appropriate signal integration, for instance, the sequestering of NFB in the nucleus.
Finally, we note that mathematical models of the immunological synapse (9) indicate that segregation domains have a minimum possible size. This means that very small KRDs are unstable, and thus can only be transient. During the development of the synapse early signals are essential, since in the absence of agonist, receptor segregation does not occur and the immunological synapse does not form (except on dendritic cells), suggesting that patternation is not thermodynamically favored in the absence of agonist. We suggest that thermal fluctuations generate regions of close contact in the contact interface over a range of sizes from nanometers to hundreds of nanometers, such regions having gross kinase activity, and thus lead to TCR activation in the presence of ligand. For the parameters used here, arbitrarily small domains contributed to triggering, size-enhanced signaling occurring only on domains >100 nm in size. These signals alter the thermodynamic properties of the interface, e.g., upregulation of the adhesion receptor LFA1 affinity, which induces stable receptor segregation as predicted by the mathematical models (9,10). Provided KRDs of sufficient size are seeded during this early stage, patternation will be observed. Thus the size stability threshold is not inconsistent with initial triggering being predominantly through small KRDs.
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ACKNOWLEDGEMENTS |
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Submitted on January 4, 2006; accepted for publication May 1, 2006.
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