Molina-Par?s C., Lythe G. (editors) Mathematical Models and Immune Cell Biology

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2 T Cell Receptor Triggering and Antigen Discrimination 43

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Chapter 3

Dynamic Tuning of T Cell Receptor Speciﬁcity

by Co-Receptors and Costimulation

Hugo A. van den Berg and Andrew K. Sewell

Abstract Mounting evidence that each clonotype of T cell antigen receptor can

productively interact with hundreds or even thousands of peptide antigens would

appear to conﬂict,

prima facie

, with the immune system’s primary task of guarding

against auto-immunity and singling out harmful “non-self” epitopes against a back-

ground of “self” epitopes. This paradox dissolves somewhat once it is appreciated

that, at any one time, a TCR will only have high functional sensitivity to a small

subset of all its potential agonists, that is, when presented at low copy numbers,

only this small subset will be able to activate the T cell. In this light, the self-nonself

problem becomes a matter of keeping the TCR trained on the appropriate subset of

salient epitopes. We review evidence and models that show how the co-receptors

CD4 and CD8, as well as the “signal 2” costimulatory system, act to keep the TCR

focussed on the appropriate agonist subset. On the theory of dynamic tuning of

TCR speciﬁcity, both immune tolerance and speciﬁc reactivity against salient epi-

topes rely on continual regulation by other components of the immune system. We

present a model of “avidity maturation” during the early phase of a T cell response.

The TCR repertoire, while vast, is thought to be too small by at least one order

of magnitude to encompass a speciﬁc TCR against any pathogen-derived peptide

the host may ever encounter [1, 2]. Several lines of evidence suggest that a given

TCR may recognise a wide variety of peptides, a phenomenon known as

TCR de-

generacy

[3, 4]. This raises the questions of how the immune system ensures that

such degeneracy does not engender autoimmunity, and if any of these mechanisms

may also play a role in TCR maturation. This chapter reviews models of T cell ac-

tivation that propose that the degeneracy of a TCR is not merely a static property

determined by the molecular identity of its CDRs, but instead a highly malleable

property of the TCR/pMHC/co-receptor complex, subject to dynamic modulation

that can restrict or expand the set of ligands for which the TCR has high functional

H.A. van den Berg (



)

Warwick Systems Biology Centre, University of Warwick, Coventry CV47AL, UK

e-mail: hugo@maths.warwick.ac.uk

C. Molina-Par´ıs and G. Lythe (eds.), Mathematical Models and Immune Cell Biology,

DOI 10.1007/978-1-4419-7725-0

 Springer Science+Business Media, LLC 2011

48 H.A. van den Berg and A.K. Sewell

sensitivity, and which, moreover, can act to focus the TCR on one particular ligand,

suggestive of an avidity maturation mechanism analogous to afﬁnity maturation in

humoral immunity.

A key variable in the analysis is

functional sensitivity

, a term coined by Wool-

dridge et al. [5]. This quantity corresponds to what has been termed

TCR avidity

by some immunologists (e.g. [6]), although use of this latter term may invite confu-

sion, since avidity already has an established biochemical meaning. Quantiﬁcation

of TCR signalling strength has been controversial, which is perhaps not surpris-

ing since a read-out at the cellular/population level must somehow be related to

biomolecular parameters. The importance of both the TCR/pMHC off-rate (equiv-

alently, the mean TCR/pMHC interaction time) and the afﬁnity (off-rate divided

by on-rate) has been emphasised, as well as that of TCR and pMHC densities

on the surfaces of, respectively, the T cell and the APC. The TCR triggering rate

model of functional sensitivity due to Rand and co-workers [7–9] is perhaps the

simplest mathematical model that combines all these parameters, and is brieﬂy

reviewed below.

When one considers TCR degeneracy in statistical terms, each TCR is viewed

as functionally sensitive to every possible peptide ligand, and characterised by its

distribution over these ligands. Almost all of the probability mass is concentrated

about zero sensitivity; these

null peptides

trivially include those that cannot bind

to the appropriate MHC isoform. On the order of 10

5

of the probability mass is

concentrated over the non-null range of functional sensitivities, with  10

6

–10

7

found at the maximum functional sensitivity end of the spectrum.

Looking at TCR degeneracy in statistical terms is necessary because a T cell can

actively alter its functional sensitivity distribution, and hence its TCR degeneracy.

That is to say that a T cell can tune itself to become more broad-spectrum promis-

cuous, or sensitive only to a very small set of peptide ligands.

Moreover, it seems

possible that the T cell can select this ligand from among a set of potential strong

agonists. These abilities can be understood, in mechanistic terms, on the basis of the

TCR triggering rate model, as will be explained below.

Dynamic degeneracy would vastly amplify the ﬂexibility of the T cell response

and allow it to mount a speciﬁc response to virtually every antigen even with a lim-

ited repertoire. However, dynamic degeneracymight equally well allow a substantial

portion of T cells to become autoreactive – even those who have been exposed to

negative selection. This underlines the importance of costimulation in regulating the

T cell’s functional sensitivity, in both health and disease. The role of costimulation

and threshold adaptation in the maintenance of the na¨ıve T cell repertoire has been

analysed elsewhere [10, 11]; we consider the role of costimulation in adjusting the

functional sensitivity spectrum associated with an immune response.

A residual level of promiscuity is inherent in the physico-chemical properties of the TCR/pMHC

contact; one might say that a TCR tuned to minimal degeneracy is sensitive to a single molecular

ﬁngerprint.

3 Tuning of TCR Speciﬁcity by Co-Receptors 49

The Triggering Rate Model for Functional Sensitivity

The T cell integrates, at the whole-cell level, intracellular signals emanating from in-

dividual TCR/CD3 complexes that have been triggered by a peptide/MHC molecule

(the triggered TCR corresponds to a signalosome, a complex involving several ki-

nases and adapter linkers). The philosophy of the triggering model, due to Rand and

co-workers, is that any pMHC can trigger a given TCR, but will do so with widely

varying likelihood. In particular, the rate at which TCRs are triggered on the surface

of a given T cell is given by a simple formula:

trig

.i; j /

where the sum ranges over all pMHC species j ; ˝

denotes the rate at which cou-

plings occur between the TCR of clonotype i and a ligand of species j , whereas

trig

.i; j / denotes the probability that a given interaction between a TCR of type i

and a pMHC of type j results in the TCR being triggered.

The kinetic pre-factor ˝

generally depends on several factors: the density of

free (unbound) TCR molecules on the T cell surface; the density of free pMHC

molecules on the APC surface; and the afﬁnity constant of the TCR/pMHC interac-

tion. A detailed derivation of these dependencies is given by van den Berg et al. [8].

The kinetic pre-factor depends only on the free TCR density when the latter is suf-

ﬁciently low relative to the afﬁnity constant; in this

TCR-limited triggering regime

the expression levels of ligand are immaterial. This regime may prevail during pos-

itive selection in the thymus, and is perhaps also important in CTLs. On the other

hand, interactions between T cells and APCs in secondary lymphoid tissues may be

expected to take place in the

MHC-limited triggering regime

, where the rate factor

depends only on the pMHC levels:

(3.1)

where Z

is the copy number of pMHC species j engaged by the T cell and T

denotes the mean interaction time of the TCR/pMHC interaction (that is, 1=T

the off-rate). Although the afﬁnity constant does not appear explicitly in this ex-

pression, it plays an important role in governing the change-over between TCR- and

MHC-limited triggering; a theoretical prediction that was borne out by experimental

observations [12]. The theory postulates that the T cell response is a functional of

the triggering rate

trig

.i; j / (cf. [13–15]); e.g. the T cell is activated if the

integrated rate

trig

.i; j / exceeds a certain value, called the

cellular activa-

tion threshold

, during the T cell:APC conjugation [16].

Equation (3.1) forms a link between theory and experiment. Write the TCR trig-

gering rate due to pMHC species j as W

and assume MHC-limited triggering. Then

D Z

trig

.i; j /

: (3.2)

50 H.A. van den Berg and A.K. Sewell

Also assume that the T cell responds when W

act

where W

act

>0denotes the

cellular activation threshold. The functional sensitivity is determined experimentally

by exposing T cells to APCs with various ligand levels and determining the midpoint

of the dose-response curve.

Let Z

crit

denote this midpoint level. Then the reciprocal

is taken as a measure of functional sensitivity: the higher the pMHC copy number

required, the lower the functional sensitivity of the TCR for that ligand. In formula:

crit

act



trig

.i; j /

(3.3)

which shows that functional sensitivity is proportional to the MHC-speciﬁc TCR

triggering rate

trig

.i; j /=T

. Moreover, (3.3) shows how functional sensitivity

might be modulated: through changes of P

trig

.i; j / and T

at the receptor level and

through changes of W

act

at the cellular level. We discuss modulation of P

trig

.i; j /

and T

via the co-receptor and then modulation of W

act

via the costimulatory ligands.

Co-Receptor Tuning of Functional Sensitivity

T cells vary considerably with respect to the amount of stimulation that is re-

quired to elicit a response: na¨ıve, nearly quiescent T cells require an interaction

with a professional APC that may last for hours, whereas fully differentiated CTLs

may be triggered by very few ligand molecules [17, 18]. These differences re-

ﬂect differences in the expression of the TCR, various kinases, phosphorylases,

including CD45 which occurs in isoforms that correlate with the cellular activa-

tion threshold of the T cell, as well as different forms of costimulation which

modulate the strength of the cognate TCR/pMHC interaction. The T cell sur-

face glycoproteins CD4 and CD8 interact with invariable regions of the MHC

molecule, independently of the cognate contact [19], and hence termed

co-receptors

(see Fig. 3.1). This section focuses on the effects exerted by these co-receptors on

TCR triggering.

The width of the dose-response curve is attributable to natural variability in Z

among the APCs

and W

act

among the T cells; this variation comes into play because a population-level response is

measured.

In fact, an additional step intervenes between experiment and theory: the immunologist does

not control the presentation level Z directly, but rather the concentration at which the APC is

incubated with peptide ligand. The latter may be assumed to increase with Z and will often be

simply proportional.

3 Tuning of TCR Speciﬁcity by Co-Receptors 51

pMHCI pMHCII CD80 CD86

TCRCD8 CD4 CD28 CD152

antigen-presenting cell

T cell

Fig. 3.1 Receptors and co-receptors at the T cell:APC interface

The TCR Triggering Probability

The probability that the TCR/CD3 complex achieves signalosome status (i.e. starts

to contribute to the activation signal that is conveyed through intracellular signalling

pathways) following an interaction with a pMHC molecule is given by the following

formula

trig

.t/f

.t/dt (3.4)

where f

.t/ is the probability density function of the TCR/pMHC residence time,

that is, the probability that the TCR remains bound to the pMHC molecule for at

least an amount of time t equals

.u/du,andP

.t/ is the probability that the

TCR/CD3 complex is phosphorylated at all n sites at time t. If the dissociation rate

is not inﬂuenced by the age of the ternary complex, the TCR/pMHC interaction

time follows the exponential distribution:

.t/ D

exp



t=T



(3.5)

where T

denotes the average docking time of a TCR/pMHC complex constituting

a TCR molecule of clonotype i and a pMHC molecule of species j ; the half-life of

the interaction is log.2/T

If there are a large number of individual reaction steps that all have to occur in

order for the TCR to be triggered, the waiting time for completion of all steps may

be expected to be well-approximated by a narrow Gaussian, in view of the Central

Limit Theorem. Such steps include phosphorylations of ITAMs, binding of kinases

For the sake of simplicity, the dependence on i and j will be suppressed if it is less important in

the immediate context.

52 H.A. van den Berg and A.K. Sewell

and linker proteins, which themselves may need to undergo phosphorylations for

further steps to happen [20, 21]. The narrow Gaussian may be expected to be

obtained if the steps are strictly ordered in a linear succession. For more complex

transition graphs, the narrow Gaussian remains a good approximation if all steps

are forward and all completion paths through the graph contain many steps; in this

case the shortest path dominates. The T cell can modulate which path is shortest by

diverting certain routes through the transition graph by means of ITIM phosphory-

lation. If backward steps, such as dephosphorylations, are allowed, the waiting time

distribution widens.

In the limit for large n, the narrow Gaussian becomes degenerate. The func-

tion P

.t/ is then well-approximated by the Heaviside step function:

.t/ D



0 for t  T

1 for t>T

(3.6)

where T

denotes the receptor threshold duration: whenever the TCR remains lig-

ated with a pMHC molecule for longer than this threshold time T

, the TCR/CD3

complex is triggered. Equations (3.4)–(3.6) together give:

trig

exp.t=T

dt D exp.T

/: (3.7)

Recall from (3.2) that the MHC-limited kinetic co-factor is inversely propor-

tional to T

. The TCR triggering rate is then proportional to exp.T

/=T

which implies that the TCR triggering rate depends non-monotonically on T

, with

a maximum at T

. This is the

serial triggering

effect ﬁrst proposed by Valitutti &

Lanzacecchia [22] and experimentally conﬁrmed by Kalergis et al. [23]. Clearly, the

serial triggering effect will only be found under MHC-limited conditions.

The Classic Proof-Reading Model

One may question how robust approximation (3.7) is, given the simplifying assump-

tions. Some support is lent by the classic kinetic proof-reading model [24], which

approaches the narrow Gaussian very neatly via a sequence of Gamma distributions.

On this model, the TCR/CD3 complex has to undergo n ITAM phosphorylations

before it acquires the ability to mediate intracellular signal transduction, leading to

various cellular responses (notably gene activation in the na¨ıve T cell) [25, 26]. It is

assumed that the ith phosphorylation can take place only after phosphorylations 1

through i  1 have taken place. Let the ITAM phosphorylation rate  be nil if the

TCR is not bound to a peptide/MHC (pMHC) complex, and take >0during

TCR/pMHC ligation. The probability P

.t/ that there are i phosphorylated ITAMs

at time t then obeys the following differential equations during ligation: