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Edited by: Bruno Laugel, Cardiff University School of Medicine, UK

Reviewed by: Brian M. Baker, University of Notre Dame, USA; Yoram Louzoun, Bar-Ilan University, Israel

This article was submitted to T Cell Biology, a section of the journal Frontiers in Immunology.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The interaction between T-cell receptors (TCRs) and peptide epitopes is highly degenerate: a TCR is capable of interacting productively with a wide range of different peptide ligands, involving not only cross-reactivity proper (similar epitopes elicit strong responses), but also polyspecificity (ligands with distinct physicochemical properties are capable of interacting with the TCR). Degeneracy does not gainsay the fact that TCR recognition is fundamentally specific: for the vast majority of ligands, the functional sensitivity of a given TCR is virtually null whereas this TCR has an appreciable functional sensitivity only for a minute fraction of all possible ligands. Degeneracy can be described mathematically as the probability that the functional sensitivity, of a given TCR to a randomly selected ligand, exceeds a set value. Variation of this value generates a statistical distribution that characterizes TCR degeneracy. This distribution can be modeled on the basis of a Gaussian distribution for the TCR/ligand dissociation energy. The kinetics of the TCR and the MHCI molecule can be used to transform this underlying Gaussian distribution into the observed distribution of functional sensitivity values. In the present paper, the model is extended by accounting explicitly for the kinetics of the interaction between the co-receptor and the MHCI molecule. We show that T-cells can modulate the level of degeneracy by varying the density of co-receptors on the cell surface. This could allow for an analog of avidity maturation during incipient T-cell responses.

Thymus-derived lymphocytes (T-cells) recognize peptide antigens via antigen-specific receptors (TCRs); in particular, CD8^{+} cytotoxic T lymphocytes (CTLs) recognize short peptides presented by major histocompatibility complex (MHC) class I molecules (^{8} different antigen receptors in the naïve T-cell pool (^{15} distinct pMHCs (

The interaction between TCR and pMHCI ligand can be modulated by the co-receptor CD8 in several ways: (i) promoting the association of TCR and pMHCI; (ii) stabilizing the TCR/pMHCI interaction; and (iii) enhancing the rate at which the TCR/CD3 complex attains signaling status by association of TCR/CD3 with protein tyrosine kinases such as p56^{lck} and adaptor molecules such as LAT and LIME (

These findings suggest that CD8 not only controls degeneracy, but also differentially regulates functional sensitivity, that is, the T-cell can increase its sensitivity for one ligand, while reducing it for others. By varying the level of CD8 expression, the T-cell can increase its sensitivity to the disease-associated antigen, while at the same time decreasing its sensitivity to antigens associated with healthy conditions. This novel mode of co-receptor action could be critical in ensuring that the TCR repertoire retains the ability to respond to antigenic challenges, while avoiding autoimmunity.

T-cell antigen recognition can be expressed in terms of its

The aim of the present study is to explore the kinetic basis of the role of CD8 in regulating degeneracy and functional sensitivity. Our model generalizes the classical kinetic proofreading model, as proposed by McKeithan (

The statistics of TCR degeneracy is modeled by relating the TCR/pMHCI mean interaction time with the dissociation energy according to Arrhenius theory. We show that by varying the total co-receptor density and key kinetic parameters, the T-cell can modulate the level of degeneracy. Furthermore, we compare our results with experimental data for HLA A^{∗}0201 mutants with altered binding affinity for CD8.

We assume (i) that the TCR/CD3 complex on the T-cell surface becomes triggered (achieves signalosome status) during an interaction with a pMHCI ligand if it undergoes ^{lck}, which phosphorylates tyrosine residues within the ITAMs.

To determine the kinetics of the TCR/pMHCI/CD8 interactions, consider the four binding states of a pMHCI molecule, labeled (I–IV) in Figure _{R}_{R}_{R}_{R}

_{i}_{−}_{i}_{−1} or λ_{−4}) to the unbound states (I) or (III), respectively, can occur. Transitions between the sequences occur at rates λ_{2} and λ_{−2}. Left-to-right transitions within the sequences correspond to ITAM phosphorylation by kinases such as p56^{lck} and ZAP-70, at rate _{R}

The model describes the kinetics of the interactions between the TCR, pMHCI molecules, and co-receptor CD8 in the contact area between a T-cell and an APC. This area is occupied by TCRs and CD8s on the T-cell side and by pMHCIs on the APC side, whereby CD8 binds pMHCI at a distinct site of the TCR/pMHCI complex. ITAMs are located on the ζ-chains of the CD3 complex associated with the TCR and are represented schematically in Figure

Key quantities in the model (summarized in Table _{i}_{1} − Λ_{4} are the two-dimensional association rates (cm^{2}s^{−1}) for TCR/pMHCI or CD8/pMHCI binding. Two-dimensional dissociation constants (cm^{−2}) are defined as follows:

Free pMHCI density | |

_{R} |
TCR/pMHCI density without CD8 bound |

_{X} |
pMHCI/CD8 density without TCR bound |

_{XR} |
TCR/pMHCI/CD8 density |

Free CD8 density | |

Free TCR density | |

_{T} |
Total pMHCI density |

_{T} |
Total CD8 density |

_{T} |
Total TCR density |

In order for a system of reactions to be in thermal equilibrium, each individual reaction must be at equilibrium (the principle of detailed balance):
_{1}_{2} = _{3}_{4}. Combining this with the conservation laws with we obtain:

These results can also be expressed in terms of standard affinity constants:

The co-receptor CD8 modulates the rate of TCR triggering. Three major modulatory functions of the co-receptor have been documented: modulation of TCR/pMHCI on-rate, TCR/pMHCI off-rate, and of the ITAM phosphorylation rate. These effects can be represented by dimensionless multipliers.

enhanced TCR/pMHCI on-rate:

reduced TCR/pMHCI off-rate:

increased phosphorylation rate, which is equivalent to a reduced TCR triggering threshold

It is sometimes convenient to combine the on-rate and off-rate effects into a single coefficient, as follows:

From equation (_{kin} ≤ γ_{off}. We rewrite _{R}_{X}_{XR}_{kin}:

To non-dimensionalize this system, we introduce the following dimensionless quantities:

It follows from equation (_{T}_{3} = κ_{T}

This system is readily solved numerically for _{kin}, κ, _{T}_{T}

The functional sensitivity of the TCR is represented in the present model as the rate at which TCR/CD3 complexes attain signalosome status. To calculate this TCR triggering rate, which we shall denote by _{R}_{R}

In reality, signalosome formation involves several other types of event besides ITAM phosphorylation, such as binding of ZAP-70, engagement of LAT, and so on. To avoid cumbersome notation we shall formulate the model as if ITAM phosphorylation were the only type of event; the essential theory is not materially affected by this simplification. We do not assume an equilibrium state for the Markov chain: the complex starts at zero phosphorylations at the beginning of every encounter with a pMHCI ligand and proceeds forward stochastically.

The pMHCI/TCR/CD3 complex may not attain the _{−1} when the co-receptor is not bound and at rate λ_{−4} ≤ λ_{−1} when the co-receptor is engaged. We assume that upon TCR/pMHCI dissociation the CD3 complex reverts to the basal state of zero ITAM phosphorylations sufficiently rapidly that the TCR/CD3 complex will be in this completely unphosphorylated state when the next encounter with a pMHCI molecule occurs. Essentially, this means that the CD3 complex is more susceptible to the action of phosphorylases and/or less susceptible to the action of kinases when the TCR is not engaged. A mechanistic explanation underpinning this assumption lies outside the scope of the present model.

The probability that the TCR/CD3 complex will undergo another ITAM phosphorylation is given by

The law of total probability yields the following system of coupled difference equations:

This boundary condition expresses the basic assumption that triggering is attained when the sequence has been completed. To render the equations dimensionless, we introduce the following parameters:

The scaled (dimensionless) TCR triggering rate is then given by the following expression:

The scaled TCR triggering rate

_{T} |
Scaled total pMHCI density |

_{T} |
Scaled total CD8 density |

_{T} |
Scaled total TCR density |

α | Scaled TCR/pMHCI off-rate without CD8 bound |

δ | Scaled pMHCI/CD8 off-rate with TCR bound |

ν | Scaled kinetic effect of pMHCI/CD8 interactions with and without TCR |

κ | Ratio of dissociation constants _{1} and _{3} |

γ_{off} |
Factor by which CD8 modulates TCR/pMHCI off-rate |

γ_{kin} |
Factor by which CD8 modulates the TCR/pMHCI affinity |

γ_{R} |
Factor by which CD8 modulates the TCR triggering threshold |

To express TCR degeneracy mathematically, we consider the distribution of the triggering rate over the set of peptide ligands. This is just the set of _{ij}_{ij}_{ij}_{−1} (we have thus far suppressed subscripts for clone _{ij}_{0} is the frequency factor and Δ_{ij}_{ij}_{ij}_{ij}_{R}/T_{0}} we have _{ij}^{2}), where μ > 0 and σ are the underlying parameters of the normal distribution

We investigated the effect of variations of the total CD8 density on the functional sensitivity of hypothetical ligands with various TCR/pMHCI off-rates. All variables and parameters are dimensionless (scaled) in the model simulations. The scaled parameters are summarized in Table

Figure _{T}_{T}_{T}

_{T}_{T}_{T}_{T}^{−8} and ℙ(^{−5}. Parameter values: δ = 300, ν = 0.5, _{kin} = 0.5, γ_{off} = 0.5, γ_{R}_{T}_{T}

It can be observed that a ligand with α = 2.5 (solid line) and a ligand with α = 0.5 (dotted line) show opposing changes in the scaled functional sensitivity: a ligand that is less potent at low CD8 becomes more potent at high CD8 and

The corresponding degeneracy curves ℙ(_{T}_{T}_{T}

A high degree of degeneracy can increase the risk autoimmune disease. On the other hand, too low a degree of degeneracy could compromise the immune system’s ability to mount a timely and efficient response. To analyze these risks, suppose that the T-cell is activated if its integrated TCR triggering rate exceeds a certain value, termed _{j}_{I}_{act} is the activation threshold. For a given _{j}_{I}_{ij}_{ij}^{−8} to 10^{−5}. Figure

Figures

_{T}_{R}_{kin} = 0.5, γ_{off} = 0.5, γ_{R}_{T}_{T}

_{T}_{off} expressing the modulatory effect of CD8 on the TCR/pMHCI off-rate. Parameter values in _{kin} = 0.05, γ_{off} = 0.2, γ_{R}_{T}_{T}

Figure _{R}_{R}

The effect of CD8 levels on functional sensitivity is shown in Figure _{off}, the functional sensitivity remains at near-optimal levels when CD8 levels are increased. The value of γ_{off} may be expected to be different for different TCR/ligand combinations. In particular, when CD8 makes a substantial contribution to the binding energy, the multiplier γ_{off} will be low, and the co-receptor role in governing ligand optimality will be more pronounced.

Wooldridge et al. (

In keeping with (_{D}_{D}_{D}^{b} with enhanced affinity (_{D}

The two-dimensional dissociation constant for pMHCI/CD8 interaction without TCR bound, _{3}, appears in the scaled parameters κ, _{T}_{3} to be proportional to _{D}

^{∗}0201 mutants with altered binding affinity for CD8: A245V (dotted line), wild-type (semi-dashed line), Q115E (dashed line), and A2/α3k^{b} (solid line). The three regions represent the overall pattern of CD8^{+} T-cell antigen specificity and the arrow indicates the strength of pMHCI/CD8 interaction. ^{∗}0201 mutants with altered binding affinity for CD8. The parameter values are the same as in A except for ν = 10. Parameter values in _{kin} = 0.5, γ_{off} = 0.5, γ_{R}_{T}_{T}_{T}

The co-receptor CD8 can modulate the specificity of antigen recognition, as shown in Figure ^{b}) is the most degenerate, with the largest antigen sensitivity. The three regions are a schematic representation of the overall pattern of CD8^{+} T-cell antigen specificity, as defined by Cole et al. (^{+} T-cell antigen degeneracy becomes wider. Enhancing the kinetic effect of pMHCI/CD8 interactions (setting ν ≫ 1) results in the reversed pattern, as shown in Figure

Whereas in the MHC-limited kinetic regime, the behavior is as shown in Figure

The co-receptors CD4 and CD8 are glycoproteins that modulate the interactions of the TCR with pMHCI and pMHCII molecules, by binding to invariant sites on these molecules (

Co-receptor-directed ligand focusing may allow the T-cell response to an antigen challenge to undergo an adaptive evolution that would be functionally analogous to affinity maturation in B-cell immunity. Moreover, CD8 modulation could allow for an elevated degeneracy among the earliest responding clones. This would ensure that at least one or more responding clones are activated sufficiently early in the course of the infection. Moreover, a gradual restriction of the degeneracy, coupled with an increase in functional sensitivity to the salient epitope, would then reduce the degeneracy of the response, which would gradually evolve from oligoclonal to one that is dominated by an optimally tuned single clone.

Disrupting the pMHCI/CD8 interaction impairs the ability of T-cells to recognize antigens. In particular, T-cell activation can be abrogated if the pMHCI/CD8 interaction is blocked (^{+} T-cell degeneracy using combinatorial peptide libraries and APCs expressing mutant HLA A^{∗}0201 molecules with altered pMHCI/CD8 affinity has shown that the co-receptor enhances T-cell degeneracy by increasing the range of agonist ligands that can elicit T-cell activation (^{∗}0201 by at least an order of magnitude resulted in the loss of cognate antigen specificity (^{+} T-cell antigen sensitivity, but reduces CD8^{+} T-cell antigen specificity (

A cornerstone of the present theory is that a certain amount of degeneracy is unavoidable, in view of the vast universe of possible peptides and the relatively modest number of TCR clonotypes that even a large mammal might be able to maintain in its standing repertoire. Moreover, salient epitopes, those associated with a disease state, and non-salient ones, such as self-peptides for which immune tolerance is required, will of necessity be “finely interleaved” subsets of the peptide universe (a mathematician would say that one subset is “dense” in the other), lest the tolerant subset forms a target for the rapidly evolving pathogens: the system cannot work if molecular mimicry is readily attained. From these two premises, it follows that a TCR must be degenerate, and also that this degeneracy must be susceptible to exquisite modulation. Against this line of reasoning a case could be made that the size of the ligand universe is effectively much smaller. For instance, if one considers ^{m}^{4} × 35 = 5.6 × 10^{6} functionally distinct ligands, which is of the same order as the TCR repertoire size. Whereas there may be some merit to this argument, its underlying image, essentially of CDR3 as a tape recorder head that interacts with only

The present model indicates that intermediate levels of CD8 are associated with the lowest functional sensitivity. This suggests the following mechanism to maintain quiescent (naïve) T-cells in a relatively unresponsive state. When the T-cell receives the appropriate stimuli, it either up-regulates or down-regulates the co-receptor and a specific subset of its potential agonists “comes into focus.” Such signals are known to be transmitted via cytokine profiles in the T-cell’s surroundings (

Whilst the model includes the key components of TCR triggering, many important aspects have been omitted. In particular, we have neglected the spatial dynamics of TCR, CD8, and pMHCI within the immunological synapse, where the relative concentrations of p56^{lck} and CD45 will determine how quickly partially phosphorylated TCR/CD3 will reset to the basic state (

In addition to kinetics of the interaction between TCR, MHCI, and CD8, we have only considered ITAM phosphorylation steps. It is well known that phosphorylated ITAMs orchestrate the activation of the Src-related protein tyrosine kinases which initiate TCR signaling. These kinases induce tyrosine phosphorylation of several polypeptides, including the transmembrane adaptors. Protein tyrosine phosphorylation subsequently leads to the activation of multiple pathways such as ERK, NF-κB, and NFAT (

In summary, the present findings suggest that the co-receptor CD8 can differentially modulate functional sensitivity to its potential agonists, thereby modulating TCR degeneracy in a tunable fashion. The ligand focusing mechanism would allow each T-cell to have a wide range of potential agonists, even while only one of these would be a ligand of high functional sensitivity at any particular moment in time.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was supported by the Biotechnology and Biological Sciences Research Council (grant BB/H001085/1) and the Wellcome Trust (Grants WT079848MA, WT096454AIA, and WT099067AIA). The authors are indebted to the reviewers for their constructive comments.

^{+}T cell activation is governed by TCR-peptide/MHC affinity, not dissociation rate

^{−}T cell transfectants that express a high affinity T cell receptor exhibit enhanced peptide-dependent activation