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Copyright © 2008 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 94, Issue 9, 3447-3460, 1 May 2008

doi:10.1529/biophysj.107.116897

Biophysical Theory and Modeling

Analysis of Serial Engagement and Peptide-MHC Transport in T Cell Receptor Microclusters

Omer Dushek and Daniel Coombs^,  

Department of Mathematics and Institute of Applied Mathematics, University of British Columbia, Vancouver, Canada

Address reprint requests to Daniel Coombs.

T cells play a central role in adaptive immunity by regulating immune responses and performing targeted killing of infected cells. For T cells to carry out these functions they must be stimulated by antigen-presenting cells (APC) bearing cognate antigen. Stimulation is mediated by interactions between T cell receptors (TCR) and specific antigen presented in the form of peptide-major-histocompatibility complexes (pMHC), in a tight adhesion region between the T cell and APC 1,2. In this region, known as the immunological synapse (IS), fewer than 10 agonist pMHC molecules have been shown to transduce sufficient intracellular signaling to cause measurable stimulation of T cells 3,4,5. Further, the TCR-pMHC bond is weak, with solution K_D usually in the μM range 6.

A partial explanation of the sensitivity of T cells to such weak stimuli was proposed by Valitutti et al. 7. TCR downregulation was measured over the course of several hours of T cell interaction with APC carrying a known number of antigenic pMHC. The ratio of downregulated TCR to number of pMHC was found to be as high as 200. Assuming that every internalized TCR has previously bound pMHC, this suggests that pMHC sequentially bind hundreds of TCR in the IS. This is known as the serial-engagement hypothesis. Similar results were found by Itoh et al. 8 but the notion of serial engagement was weakened by findings that TCR that have never bound pMHC can be internalized in a pMHC-dependent manner 9. Further investigation revealed that when few pMHC are present on the APC, TCR are downregulated so rapidly that a strict serial engagement model could not fit the data 10,11. Rather, a model allowing for downregulation of nearby, unstimulated TCR, in a TCR-density-dependent manner, provided a more plausible explanation 11. The importance of serial engagement of TCR for T cell activation has been further challenged by data of Holler and Kranz 12, which did not show a decline in T cell activation by high-affinity pMHC. We reanalyze parts of this data here.

We see that the rate of engagements (i.e., the hitting rate) between TCR and pMHC is of great importance. Using a simple mathematical model, Wofsy et al. 13 calculated the hitting rate for a single pMHC moving diffusively on the APC surface. Using measured parameters for the TCR-pMHC bond, it was found, for instance, that a single agonist pMHC can engage 5–35 TCR during one sojourn in the IS. The highest rates of serial engagement were achieved by those pMHC that bind TCR very transiently. On the other hand, fast-dissociating pMHC would lead to only weak signaling of individual TCR. These considerations led to a model of overall pMHC signaling efficacy based on the lifetime of the TCR-pMHC bond, where the optimal pMHC has a lifetime that is high enough for reliable signaling of individual TCR, but low enough for serial engagement to proceed efficiently 7,14,15. These verbal predictions were confirmed experimentally and theoretically in a series of experimental and theoretical articles 16,17,18.

In the calculations of Wofsy et al. 13, it was assumed that TCR are distributed homogeneously within the IS. It has since been shown that TCR aggregate into half-micron-sized clusters, often referred to as TCR microclusters, in T cell synapses formed with both APC 19 and suspended planar bilayers 19,20,21 bearing agonist pMHC. These TCR clusters form in the periphery of the IS, a region known as the peripheral supramolecular activation cluster (pSMAC), and migrate toward the center of the IS (the central-SMAC (cSMAC)) 19. Various molecules in the signaling cascade that begins with a pMHC interacting with a TCR have been shown to localize in pSMAC clusters and not in the cSMAC 19,20. TCR clusters are observed to form when bilayer pMHC densities are as low as 0.2μm⁻²21.

Weak TCR-pMHC interactions can also be stabilized by more complex binding reactions. For instance, the cell-surface coreceptors CD4 and CD8 22 are thought to bind the nonpolymorphic regions of the pMHC molecule 23, strengthening the TCR-pMHC interaction. The cytoplasmic tails of CD8 and CD4 are also associated with Lck, a molecule known to interact with the cytoplasmic tail of engaged TCR, perhaps further stabilizing the complex 24. In a similar vein, pMHC are thought to form multimeric complexes that may allow for stabilization of bonds formed by the individual pMHC (further discussion below).

In this article, we revisit the calculations of Wofsy et al. 13 in the context of moving TCR clusters and stabilization of the TCR-pMHC bond by coreceptors and pMHC dimerization. We show that serial engagement, in the context of mobile clusters, is negligible unless the bond is stabilized, in which case substantial serial engagement of TCR by a single pMHC molecule will occur. We also show that stabilization is necessary for pMHC transport to the cSMAC within a single cluster and that pMHC transport to the cSMAC is well correlated with T cell stimulation. Finally, we consider a field of immobile clusters, and examine how many TCR a pMHC can expect to engage during one sojourn in the IS in such a situation.

We derive a model for the escape time of a pMHC molecule diffusing within the immunological synapse and potentially binding and unbinding from TCR. We do not resolve individual TCR in this model, nor do we consider TCR signaling after pMHC engagement of TCR. Principally, we are concerned with understanding how physical parameters of the TCR-pMHC interaction are related to pMHC motion within the IS and serial engagement of TCR by pMHC. TCR signaling and more complex effects within this model could be addressed using an agent-based approach 25.

We want to calculate the length of time that a pMHC is expected to remain within a TCR cluster before escaping. We model a particle (i.e., a pMHC molecule) advecting and diffusing in a two-dimensional membrane. The particle transitions between four states (i=A, B, C, and D) by a Markov process with first-order reactions between the states, as shown in Fig. 1We define the escape time from some domain, for a particle in state i at position (x, y), to be t_i(x, y).

Display large version of this figure

Figure 1

The most general reaction scheme we consider. Transitions between the four states occur with first-order reactions, with transition rates given by λ_±i. Detailed balance (microscopic reversibility) requires that λ₁λ₂λ₃λ₄=λ_–1λ_–2λ_–3λ_–4. Specific reaction schemes are illustrated in Fig. 2.

This is calculated for a diffusing particle (without reactions) by solving the Poisson equation on the region of interest using Dirichlet (zero) boundary conditions (see Appendix A of Goldstein et al. 26 for a heuristic derivation based on a two-dimensional random walk). Following the procedure in the literature 13,26, we derive a system of equations governing the t_i(x, y):

(1)

Although we will apply the general reaction scheme illustrated in Fig. 1 to several scenarios, we will always identify state i=A with a free pMHC on the APC or supported planar bilayer and states i=B, C, D with pMHC bound to molecule(s) on the T cell (e.g., TCR, CD4, CD8). Since the pMHC molecule undergoes directed motion toward the cSMAC only when bound to molecules within a cluster, we can make the equations simpler by working in the cluster reference frame (i.e., and ), and picking the cluster velocity vector along a coordinate axis We further assume that, when bound, the pMHC has a negligible rate of diffusion (i.e., D_B=D_C=D_D=0), to find

(2)

(3)

principle of detailed balance (microscopic reversibility) which requires that Λ₁Λ₂Λ₃Λ₄=1. For simplicity we will assume that the cluster is a square of side b.

We formulate the solution to Eq. (3) as t_A(x, y)=β Φ(x, y), with

(4)

(5)

where We note that the escape time is only dependent on transition affinities, through the parameter β, and not on the individual forward and backward reaction rates. We obtain the mean escape time by averaging over all possible starting positions of the pMHC,

(6)

We now derive expressions for the number of transitions between each state in Fig. 1 during a sojourn in a TCR cluster. We begin by calculating the probability, P_i, of finding the particle in state i by considering the transition matrix associated with Fig. 1 augmented by the conservation equation P_A+P_B+P_C+P_D=1,

The solution of this system is (P_A, P_B, P_C, P_D)=(1/β)(1, Λ₁, Λ₁Λ₂, Λ₁Λ₂Λ₃).

We can now use the P_i to approximate the mean number of transitions between each state. As an example, consider the mean number of transitions between state i=A and state i=B. The total amount of time the particle spends in state A is 〈t〉 P_A. The particle transitions out of state A into state B or D with a total rate λ₁+λ_–4 and therefore the number of transitions is 〈t〉 P_A(λ₁+λ_–4). We can calculate how many of these transitions were strictly to state B by noting that the probability of transitioning, given that a transition occurred, from A to B is λ₁/(λ₁+λ_–4). Therefore, the number of transitions from A to B is A similar calculation shows that hits (B → A)=hits (A → B). In this way we can estimate the mean number (#) of transitions along any arrow in the reaction network, while the particle (pMHC) stays in the cluster:

(7a)

(7b)

(7c)

(7d)

We begin our study of the TCR/pMHC dynamics by focusing on a single TCR cluster that forms in the pSMAC and travels toward the cSMAC. We first calculate an upper bound on the number of TCR a pMHC can hit (bind to) during one visit to that cluster. Yokosuka et al. 19 experimentally measured the mean velocity of 81 TCR clusters as a function of their initial formation location. They found a mean velocity of V=0.0249μm/s for pSMAC clusters traveling toward the cSMAC and found that these clusters migrated a maximum of L_max=4.5μm toward the cSMAC (Fig. S1 in their work). Therefore the maximum journey time is t_max ≈ 181s. If, during this time, a pMHC molecule can continually engage TCR (i.e., the pMHC molecule cannot escape the cluster) and the mean time for a free pMHC to bind a TCR is negligible (i.e., the reaction on-rate is very large), then the number of TCR engagements, or hits, can be approximated by dividing t_max by the mean TCR-pMHC bond lifetime, 1/k_off. Typical off-rates for agonist pMHC are 0.03–0.3s⁻¹1, and therefore the maximum number of possible hits, if the pMHC remains in the cluster for 181s is in the range of ∼5–54.

This rough calculation suggests that, in principle, a single pMHC molecule can engage a substantial number of TCR in a cluster before arriving at the cSMAC. However, the number of engagements can be significantly reduced by considering the effects of pMHC diffusion, TCR cluster composition and mobility, and the finite reaction on-rate. In the next section we will use our escape time formulation to quantify the decrease in the number of hits when these effects are considered.

To investigate the behavior of a single pMHC within a mobile TCR cluster, we consider the pMHC to be in one of two states; unbound (i=A) or bound (i=B) to a TCR; see Fig. 2(basic model). The reaction scheme is with a forward rate λ₁, backward rate λ_–1, and transition affinity Λ₁=λ₁/λ_–1.

Display large version of this figure

Figure 2

Reactions schemes we consider. (a) Two-state reaction scheme between single pMHC and TCR. Inclusion of coreceptors (b) or dimeric pMHC molecules (c) requires a four-state reaction scheme. Ag=agonist pMHC; En=endogenous (nonstimulatory) pMHC.

We define the number of hits for this simple model to be the number of times (on average) the pMHC binds a TCR during its time in the cluster. We compute the mean escape time, the pMHC transport distance, and the number of hits by reducing the four-state model described earlier to a two-state model. This reduction is achieved by setting Λ₂=Λ₃=Λ₄=0 in Eqs. (6). The number of hits in this two-state model is then

(8)

Display large version of this figure

Figure 3

Residence time in a TCR cluster as a function of initial position. We plot t_A(x, y) as a function of x for y=b/2 for several values of the microcluster velocity V. We use reaction parameters for the MCC88-103 peptide (Table 2): b=0.59μm, T_mc=286μm⁻², D_P=0.03μm²/s, and k_off=0.057s⁻¹.

Before we can determine the number of TCR engagements by a pMHC molecule in a cluster we need to estimate parameters. Several recent studies have characterized TCR cluster composition and mobility 19,20,21. By comparing background and cluster anti-TCR-fab fluorescence intensity, Campi et al. 20 determined that 140 TCR are contained in a single cluster. Consistent with these findings, Yokosuka et al. 19 reported 40–150 CD3ζs per cluster. Cluster area was observed to be 0.35–0.5μm²20. The concentration range of TCR in a cluster is then T_mc=80–430μm⁻². We also take the diffusion coefficient of pMHC to be D_P=0.03μm²/s 13. These parameters are summarized in Table 1.

Table 1 Parameter definitions and estimations

	Parameter	Description	Reported parameter ranges
	Amc	TCR microcluster area	0.35–0.5μm220
	b	TCR microcluster dimension	Amc=b2, b=0.59 – 0.71μm or Amc=πb2, b=0.33 – 0.40μm
	R	Synapse radius	5–6μm
	NT	TCR number in a microcluster	40–150 19,20
	Tmc	TCR concentration in microclusters	80–430μm−2
	T0	TCR concentration outside microclusters	50μm−2
	Cmc	Coreceptor concentration in microclusters	Not known
	KD	TCR/pMHC dissociation constant	0.005–1500μM (see Table 2)
		Coreceptor/pMHC dissociation constant	∼200μM 32,36
	V	TCR microcluster velocity	0.0249μm/s 19
	DP	pMHC diffusion coefficient	0.03μm2/s 13,49

We summarize experimentally determined reaction rates between various TCR and pMHC in Table 2. The transition rates in our model are related to these using the relations

(9)

Table 2 Estimates of TCR hits and pMHC transport by a mobile TCR cluster

							〈t〉 (s)		〈L〉μm		Hits
	TCR	pMHC	KD(μM)	kon(μM−1s−1)	kon(μm2s−1)	koff(s−1)	CR–	CR+	CR-	CR+	CR-	CR+
	Data from Grakoui et al. (1)
	2B4	MCC88-103	60.2	900	0.0057	0.057	12	121	0.03	3.0	0.67	6.7
	2B4	T102S	238	1500	0.0095	0.36	3.5	35	0.087	0.87	1.1	11
	2B4	T102G	1520	3400	0.022	5.0	0.91	9.2	0.023	0.23	2.5	25
	3.L2	Hb64-76	12.1	5557	0.035	0.064	65	180	1.6	4.5	4.1	12
	3.L2	N72T	9.90	15,374	0.097	0.136	84	181	2.1	4.5	11	25
	3.L2	N72I	14.9	16,600	0.11	0.248	50	181	1.2	4.5	12	45
	Data from Krogsgaard et al. 50
	2B4	102S	90	2240	0.014	0.20	8.7	87	0.22	2.2	1.7	17
	2B4	PCC	32	1080	0.0068	0.035	23	181	0.58	4.5	0.80	6.2
	2B4	MCC95-103	8.7	2200	0.014	0.019	86	181	2.1	4.5	1.6	3.4
	2B4	K2	8.7	6670	0.042	0.058	86	181	2.1	4.5	4.9	10
	2B4	K3	33	2120	0.013	0.071	23	181	0.56	4.5	1.6	12.6
	2B4	K5	2.9	4900	0.031	0.014	181	181	4.5	4.5	2.5	2.5
	Data from Garcia et al. 51
	172.10	MBP1-11	5.9	37,200	0.24	0.219	126	181	3.1	4.5	28	39
	1934.4	MBP1-11	31	5130	0.032	0.160	24	181	0.60	4.5	3.8	28
	Data from Willcox et al. 52
	JM22z	HLFA-A2	17	69,000	0.44	1.2	43	181	1.1	4.5	51	215

We calculate 〈t〉, 〈L〉, and hits as described in the main text in the absence (CR–) and presence (CR+) of coreceptors. The three-dimensional molar values are converted to two-dimensional values using a confinement length of 0.262mm (13). Parameters: b=0.59μm, Tmc=100 TCR/(0.59μm)2=286μm2, Cmc=Tmc, =200μM, V=0.0249μm/s, and DP=0.03μm2/s.

We can now compute the mean escape time 〈t〉 (Eq. (6), β=1+ Λ), pMHC transport distance 〈L〉, and total hits (using Eq. (8)). We summarize these three results for specific TCR and pMHC in the CR^– columns of Table 2. We can draw two main conclusions:

However, exceptions to these conclusions exist. For example, clusters comprised of 172.10 TCR transport MBP1-11 to the cSMAC and MBP1-11 can engage 40 TCR. There are also examples of substantial TCR engagement even though the pMHC is not transported to the cSMAC (e.g., 172.10/MBP1-11, JM22z/HLFA-A2).

In constructing Table 2 we have imposed the total journey time, t_max=181s, as an upper limit to 〈L〉, which subse-quently imposes an upper bound on 〈L〉 and hits. We impose this upper bound because once in the cSMAC, TCR begin to be internalized 21, an effect not accounted for in our model. We also focus on the cluster journey because experiments have shown that pMHC-dependent signaling through the TCR occurs during the journey to the cSMAC and not in the cSMAC itself 19,20,21,29.

We can also determine the required TCR/pMHC affinity to achieve pMHC transport to the cSMAC by a single mobile cluster from Eq. (6). Using the parameters given above, we calculate 〈Φ〉=0.41s (Eq. (4)). Setting 〈t〉=t_max, we find that, on average, the pMHC remains in the microcluster for tmax provided β>. 442.2. For a TCR concentration of T_mc=286mm^–2 this corresponds to KD,0:65mm^–2: The maximum three-dimensional dissociation constant that permits transport to the cSMAC, based on the TCR-pMHC interaction alone is, therefore,

We note that estimates of TCR numbers in clusters and estimates of cluster size, discussed above, rely on optical fluorescence microscopy 19,20. Since clusters cannot be resolved by optical microscopy, the latter estimate is probably an upper bound. We find that variations in cluster size do not have a significant effect on 〈t〉 and hits, provided we fix the number of TCR per cluster, because a decrease (increase) in cluster size is proportionally balanced by a larger (smaller) reaction on-rate.

There is a growing body of evidence suggesting that T cell activation by pMHC molecules is dependent upon coreceptors CD4 1,5,30,31 or CD8 12,22,32,33. In cases where the pMHC concentration is low 1,5,35 or the pMHC exhibits small affinity to TCR 12, T cells lacking coreceptors have been observed to be less likely to form an immune synapse 1,5, flux calcium 5,35, proliferate 1, or secrete IL-2 12. How coreceptors facilitate the activation of T cells remains largely unknown, in part because the experimentally determined affinity between CD4 or CD8 and MHC is very weak (KD ∼ 200mM) 32,36 (reviewed in 37). CD4 and CD8 are known to associate with TCR via the signaling molecules Lck and ZAP-70 38,39, possibly providing additional stabilization to the TCR-pMHC-coreceptor complex.

In this section we will show that coreceptors, although having weak binding to MHC, sufficiently augment the TCR/pMHC interactions such that substantial TCR engagement and pMHC transport to the cSMAC are achieved.

To determine the degree to which coreceptors augment the TCR/pMHC interaction we will need to consider the full four-state escape time problem. In this reaction scheme, the pMHC molecule can exist in four states: unbound from both TCR and coreceptor (i=A), bound to TCR (i=B), bound to TCR and coreceptor (i=C), and bound to a coreceptor (i=D), see Fig. 2 (coreceptor model). There are two fundamental reactions in this scheme: TCR/pMHC (transition rates: λ₁,λ–₁) and coreceptor/pMHC (transition rates: λ₂, l_–2). We assume the reactions occur independently and therefore identify the transitions between states C and D with those of A and B (i.e., λ₃=1, λ3=λ_–1) and transitions between states D and A with those of B and C (i.e., λ_–4=λ₂, λ₄=l_–2).

As before, the mean escape time is given by Eq. (6), with β=(1+Λ₁)(1+Λ₂) in this case. We also keep track of the total number of times the pMHC binds TCR or “hits” (in this case determined by summing Eqs. (7a). The Pi in Eqs. (7a) can be simplified to the form

(10)

The only additional parameter we introduce by including coreceptors is their transition affinity for the MHC molecule, L2. This parameter is related to the equilibrium dissociation constant as follows,

where C_mc is the coreceptor concentration in the cluster and is the two-dimensional dissociation constant between pMHC and coreceptors. The three-dimensional dissociation constant has been reported to be ∼200mM between class I MHC and CD8αα (32) and ∼199mMbetween class II MHC and CD4 36. As before, we convert three-dimensional values to two-dimensional values using a confinement length (see Table 2).

The results when coreceptors are incorporated into the TCR/pMHC model are summarized in the CR⁺ columns of Table 2. We find that in most cases, the addition of coreceptors maximizes the mean escape time resulting in pMHC transport to the cSMAC. Furthermore, pMHCmolecules that exhibit fewTCR engagements in the absence of coreceptors are able to engage a substantial number of TCR in the presence of coreceptors. These increases are observed despite the large dissociation constant for the coreceptor-MHC interaction.

The importance of coreceptors to T cell stimulation was highlighted in a study by Holler and Kranz (12). Briefly, they measured IL-2 release after interactions between APC bearing a particular pMHC and CD8^– or CD8⁺ T cells. In separate experiments, they determined the binding constants for some of the TCR-pMHC pairs in their experiments. They found that coreceptors were necessary for T cell stimulation only when the dissociation constant for the TCR/pMHCbond roughly exceeded 5μM. This is comparable to our previous estimate for the minimal affinity required to retain Pmhc within a TCR cluster for travel to the cSMAC (described above) of 4.1μM.

In Table 3 we summarize the reaction rates for TCR/ pMHC combinations used in their study and indicate the particular experiments where coreceptors were required for T cell stimulation. In this table we also compute 〈t〉, 〈L〉, and total hits in the absence (CR^–) and presence (CR⁺) of coreceptors. We find a striking correlation between the experimental determination of CD8 dependence and our theoretical determination of whether the pMHC can be expected to travel to the cSMAC. We also find that, if T cell stimulation is independent of CD8, the predicted number of hits is also independent of the presence of coreceptors, but the opposite is true when CD8 is required for T cell stimulation.

We can write a simple formula to estimate the maximum pMHC-TCR dissociation constant, that gives pMHC transport to the cSMAC. Rearranging β=(1+Λ₁)(1+Λ₂) and substituting the physical parameters for Λ₁ and Λ₂ we obtain

(12)

We obtain μm⁻² (using parameters β=442.2, μm⁻², and C_mc=T_mc=286μm⁻²). The three-dimensional value is then μM, an order-of-magnitude larger than in the case without coreceptors. Examining

In our analysis of the effects of coreceptors on pMHC transport and TCR hits we have assumed equal concentrations of coreceptors and TCR. Lower concentration of coreceptors would decrease In Fig. 4 we plot K_D versus using Eq. (12) for We find that at lower coreceptor concentration, a dissociation constant of ∼200μM is too large to achieve pMHC transport to cSMAC. We conclude that if coreceptors facilitate pMHC transport to the cSMAC, they must be present in clusters at concentrations comparable to that of TCR.

Display large version of this figure

Figure 4

Maximum dissociation constant () of coreceptors required to achieve transport of agonist pMHC to cSMAC. The dissociation constant between the agonist pMHC and TCR is given on the x axis (K_D). Results are shown for different ratios of coreceptor/TCR in the microcluster.

Biochemical assays have provided evidence that MHC class II molecules form dimers 40,41 and it is reasonable to suppose that they may form dimers in experiments using supported bilayers or APC. pMHC dimers have also been shown to be the minimal unit required for T cell activation in an assay where soluble multimeric pMHC complexes were used to stimulate T cells 42, although it is not clear to what extent this result informs the physiological situation where binding occurs at a cell-cell interface. Furthermore the coreceptor and the TCR that it associates with may bind different pMHC complexes 31,43. This forms a “pseudodimer” model of TCR triggering, proposed in part to explain the observation that a single agonist pMHC complex can lead to TCR triggering. It was suggested that, when a TCR with an associated coreceptor binds to an agonist-pMHC, the coreceptor binds a distinct self/null-pMHC complex. A TCR pseudodimer is formed when a second TCR binds to this self-pMHC complex.

We will show that pMHC dimers can boost the effective affinity of their constituent pMHC for the cluster and thus allow enhanced pMHC transport to the cSMAC. Therefore, higher-order complexes such as pseudo-dimers will also allow enhanced transport.

To study the effects of pMHC dimers we use the full four-state escape time formulation; see Fig. 2 (dimer model). This is exactly analogous to the coreceptor theory, with Λ₂ in this case related to the second pMHC molecule in the pMHC dimer,

(13)

where is the dissociation constant for the TCR-second pMHC bond. In the case of homodimers (identical presented peptide and MHC molecule), we ignore any cooperative effects and set In the case of heterodimers, will be different from K_D.

As in the previous cases, we find a simple relationship to determine the maximum dissociation constant, K_D*, required for pMHC transport to the cSMAC. Using the definition of β we find

(14)

The observation that endogenous peptides accumulate in the IS 35 and that heterodimers of agonist/endogenous pMHC stimulate T cells 43 suggest that endogenous peptides may play a role in pMHC transport to the cSMAC and in serial triggering. Endogenous peptides generally have an undetectable affinity for TCR and therefore have a dissociation constant >200μM. Equation (14) (shown in Fig. 4, 1:1 case)) indicates that endogenous peptides are able to transport the heterodimer only when the dissociation constant of the agonist peptide is stronger than ∼40μM. In cases when the agonist pMHC dissociation constant is >40μM, we predict that the addition of endogenous peptides alone will not sufficiently augment the interaction to result in pMHC transport to the cSMAC and substantial serial engagement of TCR.

In the previous sections we have shown that agonist pMHC can be expected to escape from TCR clusters during the journey to the cSMAC unless the reaction is augmented by coreceptors or pMHC dimers. Substantial TCR engagements and transport to the cSMAC could be possible in the absence of these factors if a single pMHC molecule visits multiple TCR clusters whose collective motion sieves the pMHC molecules into the cSMAC.

We leave a complete examination of pMHC dynamics in a field of moving clusters for future work, focusing here on the question of how many engagements (hits) a pMHC would experience during a single sojourn in an IS that holds a number of immobile clusters enriched in TCR as well as a background concentration of nonclustered TCR. Varma et al. 21 generated such synapses by treating T cells with latrunculin-A, a cytoskeleton poison, shortly after synapse formation. The result, shown in Fig. 6, F–J, of their work, is an immobile and stable field of TCR clusters. In Figure 3CD, of their work, they also illustrate the rapid (∼60s) reduction in calcium signaling upon administrating latrunculin-A to T cells forming an IS. To form a clear picture of the potential effects of multiple TCR clusters, in what follows we will not include coreceptors.

We reduce system 1 by setting λ_±i=0, for all i ≠ 1, and removing the advective field by setting for all i. We take the synapse to be a disk of radius R containing N disk-shaped clusters centered at the points r₁, r₂, …r_N and having a radius of b. Each cluster contains N_T TCR. These simplifications allow us to write a single equation governing the escape time t^syn,

(15)

(16)

where the superscript, syn, indicates the escape time is from the entire synapse. The boundary condition is t^syn=0 when |r|=R. To account for the difference between background and clustered TCR concentrations, we have split the reaction term into two parts using the indicator function I, which is zero everywhere except within clusters where it is equal to one. Therefore the term with λ₁/λ₋₁ captures reactions within clusters having a forward transition rate λ₁. The term with λ₁*/λ₋₁ captures reactions outside of clusters with a forward rate λ₁*. The transition rates are related to the physical parameters by Eq. (9) and the relation where T₀ is the TCR concentration outside clusters, which we take to be 50μm⁻²19. The TCR concentration within clusters is N_T/(πb²).

As discussed earlier, there is uncertainty in the amount of area covered by TCR in clusters. Consequently, in the analysis that follows we fix the number of TCR per cluster, N_T, and vary the cluster size, b. We can decompose the solution to Eq. (15) into three parts, t^syn=t₁+t₂+t₃, satisfying

(17a)

(17b)

(17c)

All the t_i=0 on the synapse boundary (i.e., t_i(r=R)=0). The expression t₁(r)=(R² – r²)/(4D) is the escape time from a synapse without any TCR, leading to the average time of escape 〈t₁〉=R²/8D_P (averaged over all possible starting positions). The value t₂ is the time spent bound to clustered TCR, and t₃ is the time spent bound to nonclustered TCR. By linearity, Substituting in for the physical parameters, we obtain This shows that as b → 0, in agreement with the theory for a uniform distribution of TCR presented in Wofsy et al. 13. We can obtain the mean number of transitions in this case by dividing the total time spent bound to TCR, t₂+t₃, by the mean time per binding event, 1/λ_–1 (i.e., hits=λ_–1 〈t₂+t₃〉).

If b/R≪1 then we can use matched asymptotics to obtain t₂ as a power series in the small parameter ϵ=b/R (see Appendix). Averaging over all starting positions of the pMHC, we find, to first order in ϵ,

(18)

(19)

We see that, to first order in ϵ, 〈t^syn〉 depends only on physical parameters and the locations of the clusters relative to the synapse boundary, and that centrally located clusters have the biggest impact on the mean time a pMHC spends in the synapse.

We can also consider the case where we assume that the pMHC starts within a cluster. In this case, we average over all possible starting positions within clusters to obtain the mean time spent bound to clustered TCR,

(20)

We begin by randomly placing TCR clusters within a synapse of radius 5.5μm. In Figure 5a we show the locations, r_j, of N=5 (black), N=25 (black+dark gray), and N=50 (all disks) TCR clusters. In Figure 5b we plot the total hits to clustered TCR (solid line) and to nonclustered TCR (dashed line) as a function of b for the three values of N. We obtain 〈t₂〉, and hence the total hits, by using a central difference scheme for the Laplacian in Eq. (17a) when b>0.07μm. When b<0.1μm we use the asymptotic solution, Eq. (18), which agrees with the numerical solution on the overlapping range of b. We take reaction rates for a typical TCR-pMHC interaction: =0.05μm²/s, k_off=0.05s⁻¹.

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Figure 5

Number of pMHC engagements to clustered TCR is almost independent of cluster size. We compute the number of binding events (hits) in two scenarios. In panel a, we use an idealized disk synapse of radius 5.5μm containing N randomly distributed TCR microclusters each of radius b and containing N_T=100 TCR. The positions of N=5 (black), N=25 (black+dark gray), and N=50 (all disks) TCR clusters is shown. In panel b, we use the TCR microcluster distribution from panel a to compute the total hits to clustered (solid line) and nonclustered (dashed line) TCR for the three values of N as a function of the microcluster size, panel b. In panel c, we use experimental data (Fig. 6 H in 21) to obtain a physiological microcluster distribution, see main text for details. Also shown in panel c are three steps along the multiple erosion operations performed on the microcluster stencil. Using these microcluster distributions we compute the total hits (d) as a function of the total microcluster area. Parameters: k_off=0.05s⁻¹, and D_P=0.03μm²/s.

We find that the total hits to clustered TCR increases with N but is independent of the cluster size. Indeed, the asymptotic solution, Eq. (18), is independent of b. As b → 0, for all values of N, the total hits to TCR external to clusters asymptotes to the value it would have if there were no clusters, as it should. For larger values of b, the total hits to nonclustered TCR decreases as N increases. The small changes in the number of hits at large occur because, in the simulation, parts of the microclusters end up outside the idealized synapse region.

As discussed earlier, Varma et al. 21 created synapses containing immobile TCR clusters. To obtain a physiological TCR microcluster distribution, we performed simple image analysis on Fig. 6 H of their work. We acquired a high resolution version of their Fig. 6 H and performed thresholding to convert the grayscale total internal reflection fluorescence image into a binary (black/white) image. Morphological open and close operations were performed to remove isolated pixels. The resulting TCR microcluster stencil, shown in Figure 5c, was used as the indicator function for a numerical solution of Eq. (17a). The synapse boundary, where t_i=0, was obtained by alternate thresholding and multiple morphological open operations on the total internal reflection fluorescence image (see black outline in Figure 5c).

To examine the effect of cluster size we use a morphological operation that removes pixels that are not surrounded. Applying this morphological operation multiple times on the TCR cluster stencil progressively decreases the total TCR cluster area. We show the TCR cluster stencil at three instances in Figure 5c. We keep the total number of TCR in clusters fixed at 5000, equivalent to 100 TCR distributed across 50 clusters (N=50 in the idealized synapse). We take k_off, and D_P as in the idealized synapse calculation. In Figure 5d we plot the total hits as a function of the microcluster to synapse area ratio (synapse area is fixed at 85μm²). The total hits to clustered TCR remains almost unchanged.

The main trend observed in Fig. 5, that the number of hits to clustered TCR is only weakly dependent on cluster size, is a result of keeping the number of TCR per cluster constant. Decreases in cluster area are balanced by increases in the reaction on-rate within clusters leaving the number of hits unchanged. The number of hits that a single pMHC makes with clustered and nonclustered TCR during a single sojourn in the IS is substantial, even without the aid of bond stabilization by coreceptors or dimerization.

Experimental observations show pMHC-dependent signaling in TCR clusters and the accumulation of pMHC in the cSMAC. Using a series of mathematical models we have analyzed serial engagement of TCR and transport of pMHC by mobile TCR clusters. We have shown that the TCR-pMHC interaction alone does not support substantial serial engagement of TCR or pMHC transport to the cSMAC by a single cluster but that if the TCR-pMHC bond can be stabilized (for instance, by coreceptor molecules such as CD4/8 or by dimerized pMHC, or by combined effects thereof), transport to the cSMAC and serial engagement within a single cluster can be expected. We have calculated minimum affinities of the TCR-pMHC bond that allow pMHC transport to the cSMAC to proceed efficiently in each scenario. We found evidence that coreceptors CD4/8 must be present in concentrations comparable to that of clustered TCR for pMHC transport in a coreceptor-dependent manner. Using experimental data 12 we were able to correlate predicted pMHC transport to the cSMAC with T cell stimulation as measured by IL-2 production. We also analyzed the role of multiple clusters in trapping pMHC in the synapse and boosting serial engagement. Our conclusions are based on a number of modeling assumptions and suggest future directions of experimental and theoretical enquiry. We discuss these in turn.

Our results underline the importance of measuring the kinetic parameters for TCR-pMHC bonds. The parameters which have the largest uncertainty in our model are probably the two-dimensional dissociation constant, and two-dimensional on-rate, We obtained two-dimensional values by converting their respective experimentally measured three-dimensional values using a constant factor (confinement length) determined for a specific TCR/pMHC interaction. In a review, Davis et al. 6 discuss the importance of directly determining two-dimensional rates, as they can be substantially different from their respective three-dimensional values. Recently, we have proposed a method to directly determine two-dimensional affinities and on-rates using live cells based on fluorescence recovery after photobleaching 53.

In this article we used the macroscopic or long-range diffusion coefficient for pMHC which can be substantially smaller than the microscopic diffusion coefficient 44. Increases in D_P would decrease our predictions of 〈t〉, pMHC transport distance, and hits. We also remark that we have assumed that pMHC binding to TCR creates TCR-pMHC complexes that cannot diffuse. Diffusion of these complexes within the cluster would also decrease 〈t〉. In the absence of other uncertainties concerning the binding parameters, these considerations would mean that the values of 〈t〉 in Table 2,Table 3 would be upper bounds. Fluorescence recovery after photobleaching experiments could indicate the extent to which diffusion constants change after TCR-pMHC binding.

Our c