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

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3 Tuning of TCR Speciﬁcity by Co-Receptors 63

0 0.2 0.4 0.6 0.8

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

proportion of CD86 (s)

Fig. 3.3 Differential afﬁnity signalling. Left: CD28 sequestering (%

CD28

); middle: CD152 seques-

tering (%

CD152

); right: strength of stimulation (%

CD28

=.%

CD152

.1  //) all as a function of .

D 0:05, A

D 0:005,  D 0:9 (top panels), 0:5 (lower panels);  D 1. Curves

for (top to bottom in the sequestering graphs)  D

;

;1;2;4;8(lowest curves for lowest

CD80(86) densities). The costimulatory balance (in favour of T cell activation) increases with 

when  is large (i.e. when CD80(86) densities are low)

Evolution of the Responder Functional Sensitivity Spectrum

Clonal expansion involves antigen-driven selection of TCR clonotypes that have the

highest functional sensitivity for the pathogen-derived epitopes presented on profes-

sional APCs [88]. Costimulation through CD28 and CD152 is thought to inﬂuence

the shape of the TCR avidity spectrum among the clones responding to a given

pMHC antigen. In some instances, the most efﬁcacious T cell response may be com-

posed of a few (or one) dominating clonotypes of very high functional sensitivity

(a narrow spectrum response), whereas in other cases the immune response needs

to maintain T cell diversity, with a broader spectrum [62]. The professional antigen

presenting cells involved in priming na¨ıve T cells may play an active role in guiding

the evolution of the ‘avidity spectrum’ during the primary expansion, employing the

differential afﬁnity properties of the CD80/CD86–CD28/CD152 system. This sec-

tion describes a model for the dynamics of the ‘avidity spectrum’ which elucidates

the possible roles of APCs and co-stimulation.

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

Let w

denote the functional sensitivity of the interaction between a TCR of

clonotype i and a pMHC ligand of species j :

def

trig

.i; j /

(the quantities on the right are those for the prevailing levels of co-receptor, not the

‘unboosted’ values; a slight abuse of notation for the sake of simplicity). The aim is

to describe the statistical distribution of w

-values among the TCR clones that are

activated by a given pMHC ligand j . The probability that a T cell is activated during

a given encounter with a professional APC is calculated as follows:

P.Z

act

/ D P



APC

act

T cell



(3.31)

(with (3.22)). For the random variable Z

=.m

APC

act

/, it is convenient and rea-

sonable to choose a log-logistic distribution. With location parameter $ and scale

parameter # one then has:



APC

act

T cell





1 C .m

T cell

$=w



1

Responding T cells will experience a number of encounters with pAPCs, each of

which may activate the T cell to undergo further proliferation or terminal differen-

tiation. It is believed that a T cell responds to such an activation event by increasing

its activation threshold W

act

through up-regulation of CD152 [79]. Consider a T cell

with functional sensitivity w

which has thus far experienced k activating encoun-

ters. The probability that this T cell will be activated in the next encounter is

given by:

act

.k C1Iw

/ D



=.m

APC

act

/>m=w



.mIw

/ (3.32)

which is obtained by conditioning the activation probability, (3.31), on the distribu-

tion F

, deﬁned as follows:

.mIw

def

D P.m

T cell

 m after k encounters, with functional speciﬁcity w

If the encounter is successful, the T cell’s threshold multiplier changes as follows:

T cell

.k C 1Iw

/ D m

T cell

.kIw



act

.kIw



Egen et al. [62] proposed that the increment of the activation threshold depends on

the strength of the TCR signal; this effect is represented in the above equation by

3 Tuning of TCR Speciﬁcity by Co-Receptors 65

the non-negative parameter  (for  D 0, the Egen effect is absent and the threshold

increase is always by a factor f

>1).

Equation (3.32) gives the activation probability for a T cell with a given func-

tional sensitivity w

; the probability that any of the T cells that were activated in the

k previous rounds is activated during the next encounter is found by conditioning

on the distribution of functional sensitivities among T cells that have been activated

k times. Denote this distribution as G

.w/. Then:

act

.k C1/ D

act

.k C 1Iw/dG

.w/ (3.33)

where

.!/

def

act

.k/

act

.kIw/dG

k1

.w/: (3.34)

The initial condition for this recurrence relation is the distribution G

.w/ of func-

tional sensitivities among antigen-inexperienced T cells

; this distribution has been

described in detail by van den Berg & Rand [89]. To complete the mathematical

description, a recurrence relationship for F

must be found. Straightforward proba-

bility considerations give the following expression:

kC1

.mIw

/ D 1 

m=f

APC

act





=f





1=

.Iw



m=f



APC

act





.Iw

act

.kIw

(3.35)

The initial condition F

.mIw

/ is the degenerate distribution at m D 1 for all val-

ues of w

. Equations (3.32), (3.33)–(3.35) constitute a system of integro-difference

equations which can be solved numerically.

By the avidity principle, a T cell whose TCR has comparatively low functional

sensitivity w

for the relevant ligand can still be activated if the presentation level Z

is sufﬁciently high. Professional APCs (mature dendritic cells) that activate T cells

from the na¨ıve repertoire can, in principle, prevent T cells from exploiting this avid-

ity advantage, by correlating high presentation levels with high multiplier (m

APC

)

values,

whereas a negative correlation would emphasise the avidity advantage. In

the present model this correlation is expressed by the parameter #. The variance of

It is reasonable to assume that this function does not depend on the ligand at hand. This na¨ıve

distribution G

has the following features: most of its probability mass is concentrated near zero

avidity (w

 0 for almost all clonotypes i , as the vast majority of na¨ıve T cells have no or

very little sensitivity for any given ligand j ), while a minute fraction of the probability mass is

concentrated near the maximum avidity; these latter cells constitute likely responders.

Such correlations require a connection between the antigen presentation pathways and co-

receptor expression; such a correlation may be induced by DC maturation, which sees an increase

in presentation levels as well as a shift from CD80 to CD86 expression [90, 91], which appears to

depend on stimulation of CD80 [92].

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

the random variable Z

=.m

APC

act

/, which represents the modulated inﬂuence of

the APC, is minimised (i.e. # is large) when high m

APC

correlate positively with Z

so that a high threshold tends to cancel out the higher number of ligand molecules

presented to the T cell. By contrast, a negative correlation tends to increase this

variance (i.e. to lower #). The effect of this parameter is shown in Fig. 3.4.Thetop

panel shows that large variability of Z

=.m

APC

act

/ gives rise to a broad spectrum

centred on

act

During successive rounds of activation, this spectrum shifts toward the right,

that is, becomes enriched with TCRs with greater functional sensitivity.

When

the variance is low (Fig. 3.4, bottom panel) the spectrum is just the na¨ıve spectrum,

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

scaled functional sensitivity

Fig. 3.4 Functional sensitivity spectra among responding clonotypes: the effect of APC variability

alone. $ D 0:25; f

D 1;  D 0; # D 10, 50, 500 (top to bottom); abscissa is scaled functional

sensitivity T

; ordinate is probability density

Generally, the avidity spectrum among the responders at any given moment in time is a weighted

sum of the spectra for k D 1; 2; 3; : : : . However, if the number of divisions between activation

3 Tuning of TCR Speciﬁcity by Co-Receptors 67

left-truncated at

act

. It is interesting that successive rounds do not lead to an enrich-

ment of strong TCRs in this case; this is because clones just above

act

are likely to

be re-activated in later rounds when the APC-related variance is low, whereas only

those with high functional sensitivity are secure when APC-related variance is high.

Thus, selection for high functional sensitivity can be driven by APC variability

alone, in the absence of threshold increases on the part of the T cell. Figure 3.5

shows the effect of changing f

from 1 to 1:05. As one would expect, this improves

the drive to select stronger clonotypes. At low APC variability the threshold eventu-

ally becomes too stringent for any of the responding T cells to receive stimulation,

whereas high APC variability ensures that a left tail of intermediate-strength TCRs

is retained.

0.05 0.1 0.15 0.2 0.25 0.3 0.35

scaled functional sensitivity

Fig. 3.5 Functional sensitivity spectra among responding clonotypes: the effect of activation-

dependent threshold increments. $ D 0:25; f

D 1:05;  D 0; # D 10, 50, 500 (top to bottom);

abscissa is scaled functional sensitivity T

; ordinate is probability density

events is sufﬁciently large, the actual spectrum will be dominated by that corresponding to the

latest activation round.

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

The effect of stimulation-dependent threshold increments is demonstrated in

Fig. 3.6, for various positive values of .AsEgenetal.[62] predicted, the effect

of ‘penalising’ high-strength TCRs is to make the spectrum more uniform, so that

it acquires the appearance of a plateau. Moreover, the spectrum converges quickly

after 10 activation rounds.

The spectrum G

is continuous probability density function, effectively describ-

ing an ensemble of identically prepared immune systems; in any one of these

systems, the spectrum will be a histogram describing a relatively small

number

of responding clones. This number is Poisson distributed

with parameter

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

scaled functional sensitivity

Fig. 3.6 Functional sensitivity spectra among responding clonotypes: the effect of stimulation-

dependent threshold increments. $ D 0:25; f

D 1;  D 0:1, 0:5, 0:9 (top to bottom); # D 50;

abscissa is scaled functional sensitivity T

; ordinate is probability density

The detectable expansion consists of on the order of 10 clones [93]; clonotypical heterogeneity

typically has a “quasi-species” structure with only one or a few clones constituting the bulk [94];

the number of clones initially activated may be up to an order of magnitude larger.

As may easily be shown using a technique called compounding.

3 Tuning of TCR Speciﬁcity by Co-Receptors 69



D N

`D1

act

.`/

after k rounds; here, N

denotes the number of TCR clonotypes in the na¨ıve reper-

toire. The number of clones whose functional sensitivity exceeds a given value !

is similarly Poisson distributed, with parameter 

.!/ D .1  G

.!//

after k

rounds (clearly 

 

.0/). An open problem is to specify the parameters

act

, ,and# so that 

.!/ is maximised, given !.

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