MHC ligands

ARTICLE IN PRESS

Journal of Theoretical Biology 231 (2004) 535–548 .www.elsevier.com/locate/yjtbi

Foreignness as a matter of degree: the relative immunogenicity of peptide/MHC ligands Hugo A. van den Berg, David A. Rand Interdisciplinary Programme for Cellular Regulation, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK Received 29 March 2004; received in revised form 7 July 2004; accepted 12 July 2004 Available online 26 August 2004

Abstract The ability of T lymphocytes (T cells) to recognize and attack foreign invaders while leaving healthy cells unharmed is often analysed as a discrete self/non-self dichotomy, with each peptide/MHC ligand classified as either self or non-self. We argue that the ligand immunogenicity is more naturally treated as a continuous quantity, and show how to define and quantitate relative ligand immunogenicity. In our theory, self-tolerance is acquired through reduction of the relative immunogenicity of autoantigens, whereas xenoantigens, typically not presented during induction of deletional tolerance, retain a high degree of relative immunogenicity. Autoantigens that are not prominently presented in deletional tolerance likewise retain a high relative immunogenicity and remain essentially foreign. According to our analysis, any given autoantigen can attain a high level of relative immunogenicity, provided it is presented at sufficiently high levels. Our theory provides a quantitative tool to analyse the immunogenicity of tumour-associated neoantigens and the ætiology of autoimmune disease. r 2004 Elsevier Ltd. All rights reserved. Keywords: T cell tolerance; T cell ignorance; Negative selection; Immunogenicity; Autoimmunity; Large deviations theory

1. Introduction An immune response is set in motion by an intricate network of concerted interactions between many cell types, through a host of intercellular signalling molecules (Gallucci and Matzinger, 2001; Granucci et al., 2003; Guermonprez et al., 2002). The immune system reacts to the invasion and proliferation of pathogens: its response is triggered by recognition of foreign (non-self) elements, whereas the immune system has learned to be non-responsive to host (self) elements (Parham, 2000). However, some non-self-antigens derive from a harmless source, and, obversely, some self-antigens are associated with disease, for instance tumour-associated antigens (Pardoll, 2003; Strober et al., 2002). The view that immunoresponsiveness is predicated on the self/non-self Corresponding author. Tel.: +44-24-76-52-3698; fax: +44-24-7652-4182. E-mail address: [email protected] (H.A. van den Berg).

0022-5193/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2004.07.008

distinction is therefore overly simplistic inasmuch as the notion of self versus non-self unduly emphasizes the proteomic provenance of the antigens rather than characteristics that matters, which is the immunogenicity of the antigen regardless of provenance (cf. Matzinger, 2002). Accordingly, we focus on the concept of relative immunogenicity, which compares the immunogenicity of any given ligand, whether self or foreign, to that of a ligand to which the immune system has never been exposed before. Our aim is to show that relative immunogenicity can be experimentally quantitated, and thus allows a realistic classification of both xenoantigens and autoantigens. Rather than a stark self/non-self dichotomy, we associate with each antigen a relative immunogenicity function from which two key quantities (critical presentation level and minimum relative immunogenicity) are derived. Our analysis is based on a quantitative model of autoantigen presentation in the thymus, which takes into account how often a

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H.A. van den Berg, D.A. Rand / Journal of Theoretical Biology 231 (2004) 535–548

thymocyte encounters an autoantigen on a negatively selecting cell, as well as the presentation level of the antigen. Section 2 is aimed primarily at the immunologist, comprising a self-contained exposition of the main results and a discussion of their immunological significance. Section 3 is more technical: the main results are derived and a possible experimental approach to relative immunogenicity is outlined. Mathematical notation is summarized in Table 1.

2. Relative immunogenicity We propose that the immunogenicity of a ligand derived from the host proteome (an autoantigen) is a matter of degree; see Fig. 1. To explain this quantitative concept of immunogenicity, we explain how autoantigen immunogenicity is determined by exposure of the ligand in the thymus as well as critical presentation level in the secondary lymphoid tissues. In this section, we state our two main results and discuss their immunological significance. 2.1. A simple model of T cell activation Thymus-derived lymphocytes (T cells) are activated through specific recognition mediated by the T cell antigen receptor (TCR), located on the surface of the T cell (Parham, 2000). The ligand of the TCR is a peptide, non-covalently bound to a glycoprotein encoded by the major histocompatibility complex (MHC). Inside the antigen-presenting cell (APC), the peptide ligand is derived from a protein and loaded onto the MHC molecule through a dedicated loading pathway (Chicz et al., 1993; Yewdell and Bennink, 2001). The peptide/ MHC (pMHC) ligands subsequently appear on the surface of the APC. By interacting with the TCR, the pMHC ligands may activate the T cell; this TCR/ pMHC interaction can only take place when T cells and APC conjugate to form a region of close contact (Dustin and Shaw, 1999; Grakoui et al., 1999). In contrast to the TCR molecules on the T cell, which all have the same molecular identity, a multitude of different peptides is presented on the MHC molecules. Moreover, a ligand that is capable of cognate interaction with the T cell may be present at widely varying copy numbers (Chicz et al., 1993; Hunt et al., 1992; Rudensky et al., 1991). Our first objective is to model the activation of T cells through recognition of pMHC ligands which takes place when a T cell of clonotype i forms a conjugate with an APC (indexed by q). To this end, we consider the rate at which TCR molecules are being triggered by interactions with pMHC. We assume that the TCR triggering rate W iq can be written as a sum of contributions from

Table 1 Notation Symbol

Interpretation

i j k q W iq

index for TCR clonotype index for pMHC ligand species index for focus pMHC ligand species index for APC TCR triggering rate registered by T cell of clonotype i in contact with APC q thymic selection threshold (thymocyte) cellular activation threshold (mature naı¨ ve T cell) presentation level of ligand j on APC q minimal detectable presentation level of ligand j on APC q minimal detectable presentation level of potent agonist on APC q presentation level of ligand j during negative selection critical presentation level of ligand j Total number of MHC molecules antigen presentation profile on APC q ubiquity of ligand j in the thymus

W thy W act Zjq Zjq % Z


ZjðthyÞ Z% j MT Zq uðthyÞ ; uk k ; rk rðthyÞ k Ij wij

presentation propensity of ligand j in the thymus indicator variable for ligand j TCR triggering rate induced in T cell of clonotype i per molecule of species j maximum TCR triggering rate induced in clonotype i per molecule of species j truncation point number of thymocyte contacts with negatively selecting cells thymic exposure of ligand j mean interaction time of TCR of clonotype i and pMHC of species j minimal productive interaction time of TCR of clonotype i and pMHC of species j incipient cumulative distribution function of TCR triggering rate cumulative distribution function of TCR triggering rate for ligand j post selection incipient cumulative distribution function of mean TCR/pMHC interaction time likelihood ratio (selection kernel) relative immunogenicity of ligand j post selection parameters for F T ‘th moment of F W ‘th cumulant of F W cumulant generating function of TCR triggering rate registered by thymocyte number of ligand species presented to thymocyte diversity of thymic antigen presentation profile likelihood ratio (selection kernel) tolerance factor

w^ ek w m ej m T ij TR FW F W ;j FT Lk wj n; T 0 m‘ k‘ L n ndiv Lk t

TCR: T cell antigen receptor; pMHC: peptide/major histocompatibility complex molecule; APC: antigen-presenting cell. Not included is notation used and defined only locally.

the various pMHC species j present on the surface of the APC, within the T cell:APC contact area, as follows: X W iq ¼ Z jq wij ; ð1Þ j

ARTICLE IN PRESS H.A. van den Berg, D.A. Rand / Journal of Theoretical Biology 231 (2004) 535–548

HOST PROTEOME (autoantigens)

harmless self subcritical self LOW

potentially harmful self “essentially foreign”

RELATIVE IMMUNOGENICITY

harmless nonself “essentially self”

HIGH

pathogenic nonself

FOREIGN PROTEOMES (xenoantigens)

Fig. 1. Relative immunogenicity. Peptide/MHC ligands are assigned a position on a continuous scale of relative immunogenicity, whether they derive from the host genome or other genomes; a ligand scores low on the relative immunogenicity scale if it is prominently presented during the induction of deletional tolerance. For autoligands, this tolerization takes place primarily in the thymus (the main focus of this paper), while xenoligands (as well as certain autoligands) partake in extra-thymic tolerization, which depends on the appropriate accessory stimulation to distinguish tolerizing from immunoinductive contexts.

where Z iq denotes the number of molecules of pMHC species j present in the T cell:APC contact area (the presentation level of pMHC species j); and wij denotes the contribution per MHC molecule to the TCR triggering rate. The latter quantity depends on TCR clonotype i and pMHC species j since TCR triggering depends on molecular recognition between the TCR and the pMHC complex. We assume that a given cellular response ensues if W iq exceeds a cellular threshold value associated with that particular response (Valitutti and Lanzavecchia, 1997; Viola and Lanzavecchia, 1996). For each of the cellular responses of which the T cell is capable, there is a corresponding threshold value. In the case of a thymocyte undergoing negative selection in the thymus, the cellular response is apoptosis, and we denote the cellular threshold by W thy ; the thymic selection threshold. A T cell is deleted from the repertoire whenever the TCR triggering rate W iq exceeds the selection threshold W thy during an interaction with a negatively selecting cell. Similarly, a naı¨ ve mature T cell is assumed to be activated when W iq exceeds the activation threshold W act : Each T cell continually adjusts W act under tolerizing co-stimulatory conditions (i.e. absence of ‘‘danger’’ signals) on the basis of the autostimulation it registers (Grossman and Paul, 1992; Nicholson et al., 2000; Wong et al., 2001). The statistics of these background fluctuations determine a safe lower bound on W act : Tuning of the cellular activation threshold W act is a prime component of extra-thymic tolerance. Other components are functional removal of from the reper-

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toire via anergy, differentiation into regulatory T cells (Roncarolo and Levings, 2000), and possibly TCR signalling-induced apoptotic death (analogous to thymic selection). While our theory is readily extended to accommodate these phenomena, we prefer to focus here on thymic deletion and extra-thymic cellular activation threshold tuning as the two major components of tolerance. Our aim is to characterise the relative immunogenicity of a given autoantigen (a pMHC species derived from the host proteome). The relative immunogenicity of a ligand compares the ligand’s immunogenicity to that of a reference ligand which is chosen at random, but previously ‘unseen’ by the TCR repertoire. The relative immunogenicity at presentation level Z is calculated as the probability that a T cell will be activated by that ligand, divided by the probability of activation by a random reference ligand that has not previously been presented to any T cell in the repertoire, when both the ligand of interest and the reference ligand are presented at Z copies in the T cell:APC interface. The relative immunogenicity of a pathogen-derived ligand is virtually always maximal (i.e. equal to 1); many autoantigens, by contrast, are less immunogenic. Both autoimmune disease and anti-tumour surveillance rely on T cell responses against autoantigens. Autorecognition is a double-edged sword; sometimes it is clinically desirable to decrease the immunogenicity of autoantigens, while for others, such as tumour-associated antigens, an increase of immunogenicity is required. 2.2. The critical presentation level and thymic exposure Relative immunogenicity has two main determinants, as is indicated schematically in Fig. 2, and described by the following two main results. 2.2.1. Result 1 Critical presentation level: The relative immunogenicity of an autoantigen j approaches the maximum value 1 when its presentation level Z jq exceeds a critical value defined by def

Z %j ¼ Z ðthyÞ j

W act ; W thy

ð2Þ

where W act is the activation threshold of the T cell to which autoantigen j is being presented, W thy is the thymic selection threshold of the T cell when it was a thymocyte undergoing negative selection, and Z ðthyÞ is j the level at which autoantigen j was presented during negative selection. The claim is that any autoantigen can become maximally immunogenic, if presented at sufficiently high levels. An important caveat is that the level required (indicated by Z % ) may not always be

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removal of T cells from the naı¨ ve repertoire: here xenoantigens derived from the mucosal microflora are essentially ‘self’ (Strober et al., 2002). Foreignness is a matter of degree, rather than a stark ‘self/non-self’ dichotomy. All autoantigens are to some extent foreign, and more prone to evoke autoimmunity when their critical presentation level Z % and/or thymic e are low. exposure m

relative immunogenicity

1 0.8 0.6 0.4 0.2

2.3. Predictions and comparison to experimental results

0 0

0.2

0.4

0.6

0.8

1

fraction of MHC molecules presenting autoantigen Fig. 2. Immunogenicity of an autoantigen following negative selection. Relative immunogenicity wj as a function of the relative presentation level Zj =M T : The vertical dashed line indicates the =M T : The horizontal critical presentation level, ðW act =W thy ÞZðthyÞ j dashed line indicates the approximate lowest immunogenicity, e j g: Parameter values: ZjðthyÞ =M T ¼ 0:05; W thy =ðM T T R Þ ¼ expfm 0:018; W act =W thy ¼ 5; m ¼ 100; uj ¼ 0:02; n ¼ 1; T 0 =T R ¼ 0:0005; see the appendix for details.

physiologically attainable. The critical level Z %j depends on both the ligand species j, via the thymic presentation level Zthy j ; and on the T cell, as different T cells generally have different values of W act =W thy : The notion of a critical presentation level above which an autoantigen becomes immunogenic is consistent with the finding that tumour-associated antigens are autoantigens that are expressed in the peripheral tissues, albeit normally at very low levels (Guermonprez et al., 2002; Pardoll, 2003). 2.2.2. Result 2 Thymic exposure: At presentation levels well below the critical level, the relative immunogenicity of pMHC species j is approximately equal to e j g; expfm e j denotes the thymic exposure, defined as the where m average number of interactions with negatively selecting cells such that the thymocyte is presented with pMHC species j (that is, the number of negatively selecting conjugations in which j is present on the APC). Ligands derived from pathogens are not usually presented during thymic selection; such a xenoligand has the maximum relative immunogenicity of 1 (irrespective of its presentation level). Similarly, a ligand derived from the host proteome, but not presented e ¼ 0), will also have a relative during tolerization (i.e. m immunogenicity of 1 at any presentation level. Such a ligand is just as ‘foreign’ as is a pathogen-derived ligand; it is prior exposure to a ligand that determines its immunogenicity, rather than whether or not it is derived from the host proteome. By the same token, the obverse applies to mucosal tolerance, insofar as it relies on

The present theory predicts that autoantigens presented at low levels in the thymus should essentially behave as foreign, while, at the other extreme, those with sufficiently high thymic presentation levels (and thymic ubiquity) should be virtually non-immunogenic. Autoantigens presented during negative selection in the thymus at intermediate presentation levels should exhibit a truncated avidity distribution, depleted in high-avidity clonotypes. This implies that fewer clones will be induced to expand in comparison to a response to a foreign ligand, and those that do will be of moderate avidity at best. These predictions are borne out by the experimental findings of Gross et al. (2004), who studied the immune response to autoligands presented at low thymic presentation levels, when these ligands are presented at high levels during induction of the immune response (to achieve this, they employed mutant peptides with increased affinity for the MHC molecule, but unaltered TCR recognition characteristics). The cytotoxic T lymphocyte (CTL) responses against these ligands exhibit the high avidities typical of responses against foreign-derived epitopes, as well as comparable CTL frequencies, confirming that such ligands are essentially foreign. Furthermore, autoligands presented at high thymic presentation levels were non-immunogenic, whereas responses against autoligands presented at intermediate thymic levels were less vigorous, with fewer CLTs and a wide range of intermediate avidities well below that of a response against a foreign ligand. Thus, thymic presentation level was found to affect the T cell repertoire in accordance with our predictions. We argue that vigorous responses against normally ‘cryptic’ autoligands will ensue if these are presented at sufficiently high levels (and this notion is buttressed by the findings of Gross et al., 2004). This suggests the idea of a tolerance factor, which is developed next. 2.4. Tolerance factors and the efficacy of self-directed responses For physical reasons, the MHC-specific TCR triggering rate can never exceed a certain maximum value, which we denote by w^ (thus wij pw^ for all i and j; see Berg van den et al., 2002, for a detailed explanation).

ARTICLE IN PRESS H.A. van den Berg, D.A. Rand / Journal of Theoretical Biology 231 (2004) 535–548

Whenever wij   w^ for some antigen j  and TCR clonotype i, the antigen is said to be a potent agonist for that clonotype. The T cell activation threshold W act must allow low levels of potent salient agonists (xenoantigens or autoantigens associated with disease) % def ^ to a good to elicit T cell activation. Let Z
¼ W act =w; % approximation, Z
represents the minimum detectable presentation level of a salient potent agonist. Clearly, the lower the cellular activation threshold W act ; the % lower the detection threshold Z
: However, with a decrease in W act comes an increase of the risk that ‘harmless self’ autoantigens evoke the horror autotoxicus (the term ‘harmless self’ is due to Matzinger, 2002). The lowest safe value that W act can assume depends on the TCR clonotype. Berg van den and Rand (2004) discuss how activation threshold tuning can achieve a uniformly % low detection threshold Z
across the TCR repertoire and across the secondary lymphoid tissues. A consequence of thymic selection is that the presentation levels of ‘harmless self’ autoantigens are allowed to be markedly higher than those of detectable salient ligands and yet fail to evoke an autoimmune response. In particular, each ligand may be characterised in terms of its tolerance factor (t factor): a ligand is said to have a tolerance factor t if it can safely occur at levels that are at most t times higher than a detectable hitherto unseen ligand. It should be noted that the tolerance factor is invariant under cellular activation threshold tuning. For a ligand to have tolerance tX1; its thymic presentation level must satisfy W thy ð3Þ w^ % (this follows from t ¼ Z %j =Z
). For autoantigens with e j ; which are essentially foreign autoantigens, low m Eq. (3) is vacuous: little protection is gained by ensuring that Z jq is lower than Z %j : Since they are essentially foreign, they are only of reduced immunogenicity if their % presentation level satisfies Zjq oZ
: It follows that their presentation level must remain below that of the immunologically relevant ligands; the tolerance factor of essentially foreign autoantigens equals 1. Conversely, xenoantigens may become ‘essentially self’: when presented in the appropriate (‘‘no danger’’) co-stimulatory context, they induce deletional tolerance, and can acquire t-factors significantly larger than 1 (this tolerization is extra-thymic, and ZðthyÞ in Eq. (3) should j be replaced by the tolerizing presentation level; we will continue to use ‘thy’ in our symbols, leaving the extension to extra-thymic deletional tolerance understood). The tolerance factor makes the notions of tolerance and ignorance more precise: an autoantigen with t ¼ 1 % must remain undetectable (below Z
) lest it elicit autoimmunity; thus, the immune system relies entirely Z ðthyÞ ¼t j

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on ignorance of the ligand. On the other hand, t-factors much larger than 1 indicate a high degree of tolerance: the autoantigen may be presented at ‘‘detectable’’ levels, yet fail to evoke an autoimmune response. The cryptic repertoire may be identified with the autoantigens that have low t-factors; the harmless self with autoantigens that have high t-factors. The tolerance factor associated with an autoantigen will influence the nature of the autoimmune response elicited by that autoantigen. The essentially foreign autoantigens have the lowest possible t-factor, t ¼ 1: The immune response they evoke may be expected to share many of the characteristics with responses mounted against xenoimmunogens: a high-avidity repertoire is expanded which will efficiently target and remove those cells presenting the autoantigen at elevated levels. An autoimmune response elicited by an autoantigen with a high t-factor will generally be of low avidity (by definition, as high presentation levels on the professional APCs are required to activate the responding clones). Therefore, the response will generally be less efficient, and may be associated with cytolytic activity that fails to be narrowly focussed on the cells presenting the autoantigen. We speculate that efficient tumourassociated neoantigens can be characterised and recognized by their low tolerance factors, and that the morbidity in autoimmune disease correlates positively with the t-factor.

3. Derivation of the main results from thymic selection parameters We characterise relative immunogenicity more precisely, outline our model of negative selection, and discuss how the (relative) immunogenicity of an autoantigen can be determined experimentally. 3.1. Definition of relative immunogenicity Prior to negative selection, the MHC-specific TCR triggering rate wij elicited in a randomly chosen clonotype i by a fixed pMHC ligand species j follows the same incipient distribution for all autoantigens: Pfwij po j before deletional toleranceg ¼ F W ðoÞ:

ð4Þ

The distribution F W is relative to the set of ligands presented on the MHC isoform with which the TCR is capable of productive interaction. Thus, F W is conditional on survival of positive selection, another process of thymic selection which eliminates all thymocytes incapable of productive interaction with any of the MHC isoforms expressed by the host (Sebza et al., 1999; Berg van den et al., 2002, discuss positive selection in more detail; the interactions between positive and negative selection are discussed in Berg van den and

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1 0.01 0.0001 1e-06 1e-08 1e-10 1e-12 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Fig. 3. Incipient distribution function of the TCR triggering rate. The probability that the TCR triggering rate wij for an arbitrary clonotype i and a foreign ligand j exceeds the abscissa. Graphs depict 1  F W ðoÞ; the complement of the cumulative distribution function (or ‘‘survivor’’). Prior to deletional tolerization, all autoligands follow the same distribution F W ; thus these ligands all start out as essentially foreign. of the TCR triggering rate. Curves are shown for various parameter values, as indicated; see Appendix A for computational details.

Rand, 2003). Fig. 3 shows examples of F W for the model derived in Appendix A. The effect of deletional tolerance (such as thymic negative selection) can be expressed as Pfwij po j after deletional tolerancegXF W ðoÞ

Pfwij  pog ¼ F W ðoÞ barring cross-reactivity with autoantigens or crosspresentation in the thymus. Let F W ;j ð Þ denote the distribution function of the MHC-specific TCR triggering rate wij elicited in a randomly chosen clonotype i by a fixed pMHC ligand species j after negative selection. The relative immunogenicity of ligand j is defined as follows: def

Pfwij 4wg 1  F W ;j ðwÞ ¼ ; Pfwij  4wg 1  F W ðwÞ

Pfwik 4wg Pfwij  4wg R1 R1 Lk ðoÞ dF W ðoÞ w dF W ;k ðoÞ ¼ R1 ¼ w R1 ; w dF W ðoÞ w dF W ðoÞ

wk ðwÞ ¼

ð5Þ

for any ligand j that has been presented during deletional tolerization, with a stronger inequality at the low values of o: By contrast, a pathogen-derived ligand j  virtually always follows the incipient distribution:

wj ðwÞ ¼

We have

ð6Þ

where ligand j  follows the incipient distribution F W : Thus, ligands derived from a foreign proteome typically satisfy wðwÞ  1: The immunogenicity function wj expresses the relative foreignness of pMHC species j. When a xenoantigen and an autoantigen j are both presented at level Z, the quantity 1 wj ðW act =ZÞ indicates how much more likely the T cell is to be activated by the xenoantigen as compared to the autoantigen j (a more accurate formula, taking into account the contribution made Rby other ligands, which follows the distribution F bg ; is dF bg ðoÞ=wj ððW act  oÞ=ZÞ). Our objective is to calculate the relative immunogenicity function for a given ligand k (the focus ligand).

ð7Þ

where def

Lk ðwÞ ¼

dF W ;k ðwÞ : dF W ðwÞ

ð8Þ

This is a likelihood ratio which compares the probability that the TCR triggering rate evoked by the focus epitope in a randomly selected TCR clonotype is in the interval ðw; w þ dwÞ to the corresponding probability for a xenoantigen. The function Lk ðwÞ serves as a selection kernel: the incipient probability density function of the TCR triggering rate at W ¼ w; multiplied by Lk ðwÞ; gives the probability density function after negative selection. Similarly, by multiplying the probability density function of the mean TCR/pMHC residence time at T ¼ t (the derivative of F T ) with Lk ðexpfT R =tgT R =tÞ (Eq. (A.4)) we obtain the postselection density function of the mean residence time (Fig. 4). 3.2. Thymic presentation statistics To evaluate the likelihood ratio, we need to characterise the distribution of the focus ligand k after m rounds of negative selection, that is, after m exposures to negatively selecting cells. The residence time of thymocytes in the thymus may allow for dozens or perhaps even hundreds of rounds of negative selection (Scollay and Godfrey, 1995); however, m may well be much smaller (Merkenschlager, 1996; Merkenschlager et al., 1994). Each of those selecting cells presents a number of autoantigens at various levels: collected together, these constitute the antigen presentation profile (APP) of the

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1

0.01

0.01

0.0001

0.0001

1e-06

1e-06

1e-08

1e-08

1e-10

1e-10

1e-12

1e-12 0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

Fig. 4. Post-selection probability density of an autoantigen’s mean TCR/pMHC residence time T ij over the TCR repertoire. The density function prior to negative selection is indicated by the top curve. Left panel: varying thymic ubiquity: uj ¼ 0:0475; 0:05; 0:06; 0:07 at thymic presentation level ZjðthyÞ =M T ¼ 0:05: Right panel: varying thymic presentation level: ZðthyÞ =M T ¼ 0:005; 0:01; 0:05; 0:1 at thymic ubiquity uj ¼ 0:05: Parameter values: j W thy =ðM T T R Þ ¼ 0:018; W act =W thy ¼ 5; m ¼ 100; n ¼ 1; T 0 =T R ¼ 0:0005; see the appendix for details.

APC. We expect that, as a result of negative selection, Lk ðwÞ41 for low w-values and Lk ðwÞo1 for w near the ^ The intermediate w-value such that the maximum w: likelihood ratio equals 12 represents the truncation point. The focus ligand will not be able to induce deletional tolerance unless it is presented. We consider therefore the probability that the focus ligand k will be presented during an encounter with a negatively selecting cell, and define this to be its thymic ubiquity: def

uðthyÞ ¼ PfI kq ¼ 1g; k

ð9Þ

where the indicator variable I kq equals 1 when pMHC species k is presented on APC q, and 0 otherwise; since we will not be treating peripheral ubiquities, we will usually write uk for uðthyÞ : The expected number of k presentation rounds in which the focus ligand is presented is the thymic exposure of this ligand: def

e k ¼ uk m: m

ð10Þ

Another important selection parameter is the thymic presentation level of the focus ligand. The presentation level of a peptide depends on numerous factors, such as the level at which it is expressed, whether the peptide is expressed in the APC (Jardetzky et al., 1991), as well as various factors associated with the protein processing and MHC loading pathway, such as degradation in the proteasome; transport to the intracellular compartment containing the MHC molecules; and the peptide’s affinity for the MHC-binding cleft (Lennon-Dume´nil et al., 2002; Stevanovic´ and Schild, 1999). The propensity model (Berg van den et al., 2001; Berg van den and Rand, 2003) lumps all these factors into a single, aggregate propensity rX0: The ratio between presentation levels of every pair of expressed peptides j and j 0 is given by the ratio of their propensities, Z j =Z j0 ¼ rj =rj0 ; where rj and rj 0 are the propensities of the two peptides. The unit of propensity is arbitrary; we choose the average propensity, so that hri ¼ 1 by convention.

According to the propensity model, the thymic presentation level is given by ¼P Z ðthyÞ k

rðthyÞ k j

I jq rjðthyÞ

MT ;

ð11Þ

where q indicates a negatively selecting APC, M T denotes the number of MHC-molecules (of the appropriate isoform) on such an APC, and rðthyÞ is the ligand’s k thymic presentation propensity. 3.3. Negative selection On our model of negative selection, a thymocyte is deleted as soon as the TCR triggering rate exceeds the selection threshold W thy of that thymocyte (different thymocytes may have different values of the selection threshold; we assume that we have fixed a thymocyte/T cell throughout). We obtain: P ðthyÞ ½Pf N wj oW thy j wk ¼ wgm j¼1 I j Z j Lk ðwÞ ¼ ; ð12Þ Psurv where Psurv denotes the probability that a thymocyte survives negative selection: )# m Z " (X N ðthyÞ Psurv ¼ P I j Z j wj oW thy j wk ¼ w dF W ðwÞ: j¼1

ð13Þ The probability in these formulae may be rewritten as follows: ( ) N X P I j Z ðthyÞ wj oW thy jwk ¼ w j j¼1

(

¼ uk EAPPjI k ¼1 P

X j2Znk

) Z ðthyÞ wj oW thy j (

þ ð1  uk ÞEAPPjI k ¼0 P

X j2Z



ZðthyÞ w k )

Z ðthyÞ wj oW thy j

:

ð14Þ

ARTICLE IN PRESS H.A. van den Berg, D.A. Rand / Journal of Theoretical Biology 231 (2004) 535–548

Here, EAPP j I k ¼1 denotes the expectation over APPs in which the focus peptide k occurs, while EAPPjI k ¼0 denotes the expectation over APPs in which the focus peptide k does not occur. The APP is denoted by Z: def

Z ¼ fj j I j ¼ 1g

ð15Þ

and Znk indicates the APP except the focus ligand k. If we assume, for simplicity, that the focus peptide k is always presented at a fixed presentation level Z ðthyÞ ; k whenever I k ¼ 1 during negative selection, we can pull the probabilities out of the expectation operators in Eq. (14). We assume that the probability of not being deleted by central tolerance is close to 1 (Laufer et al., 1999; Surh and Sprent, 1994). The likelihood ratio then reduces to " ( ) #m X ðthyÞ ðthyÞ Lk ðwÞ ¼ 1  uk P Zj wj 4W thy  Z k w ; j2Znk

ð16Þ which combines the four thymic characteristics W thy ; m, Z ðthyÞ ; and uk : The parameters W thy and Z ðthyÞ determine k k the truncation point:

 W thy  ðM T  Z ðthyÞ ÞmW W thy 1 def k ek ¼ L1 w  ðthyÞ ;  k ðthyÞ 2 Zk Zk ð17Þ where mW is the expected value ofR the incipient TCR def triggering rate distribution (mW ¼ o dF W ðoÞ). When uk is small, the likelihood ratio is well approximated by ( ( )) X ðthyÞ ðthyÞ ekP Lk ðwÞ  exp m Zj wj 4W thy  Z k w j2Znk

ð18Þ e k defined by Eq. (10). with the thymic exposure m 3.4. The main results The probability of the focus ligand k activating a naı¨ ve T cell (chosen at random from the mature repertoire) is approximately PfZ kq wik 4W act g ¼ Pfwik 4W act =Zkq g ¼ wk ðW act =Z kq ÞPfwij  4W act =Z kq g; where the ligand is presented at level Z kq ; the T cell has activation threshold W act ; and we neglect the contribution from all other autoantigens. The probability Pfwij  4W act =Zkq g is the probability of activation by a foreign ligand presented at the same level as the focus ligand k. Observe that the relative immunogenicity wk ðwÞ ek and of the focus ligand k remains near 1 for wow e k g for w4e subsequently decreases to expfm wk (Fig. 5).

1

relative immunogenicity

542

0.1

0.01 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

MHC-specific scaled TCR triggering rate Fig. 5. The relative immunogenicity function w: Relative immunogenicity curves are depicted for various values of the thymic presentation level: Z ðthyÞ =M T ¼ 0:005; 0:01; 0:02; 0:05; 0:1 (correspondj ej ¼ 0:4; 0:2; 0:1; 0:04; 0:02). Top set of curves: ing to truncation point w thymic ubiquity uj ¼ 0:005 (corresponding to thymic exposure e j ¼ 0:5); bottom set of curves: uj ¼ 0:05 (corresponding to thymic m e j ¼ 5). Parameter values: W thy =ðM T T R Þ ¼ exposure thymic exposure m 0:002; m ¼ 100; n ¼ 1; T 0 =T R ¼ 0:0005:

Thus, to ensure that the focus ligand is less immunogenic than a foreign ligand presented at the same level, wk ; its presentation level Z kq must remain below W act =e which by Eq. (17) is approximately equal to : ðW act =W thy ÞZ ðthyÞ k This latter quantity has been defined in Eq. (2) to be the critical presentation level. We have thus established our first main result. Our second main result, on the role of thymic exposure, follows immediately from Eq. (18): e k g: Lk ðwÞX expfm

ð19Þ

The finding that the relative immunogenicity at subek can be undercritical presentation levels equals em stood as follows: to activate a naı¨ ve T cell at low presentation levels, a sufficiently high MHC-specific ^ TCR triggering rate is required (wik must be close to w); however, thymocytes that register such a high TCR triggering rate for ligand k are virtually assured to be deleted whenever they encounter ligand k; thus, among this class of thymocytes, only those remain that escape deletion by never encountering the autoantigen; and the ek : Poisson probability of that eventuality is just em 3.5. Experimental determination of relative autoimmunogenicity To quantify the relative degree of autoimmunogenicity of a given autoantigen j, immunologists must be able to determine the following key quantities experimentally: the incipient (pre-selection) TCR triggering rate distribution F W ; the autoantigen’s immunogenicity e j and function wj ; as well as its thymic exposure m ej : A possible protocol follows. truncation point w

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Consider the following experimental procedure. Mature naı¨ ve T cells are sampled from the peripheral blood and are exposed to APCs which have been pulsed with varying concentrations of foreign peptide; the experimenter determines the fraction of the population that is activated (see Hamilton and Harty, 2002; Oelke et al., 2003, for possible experimental techniques). A dose– response curve is thus obtained, which we denote cðC j Þ; where C j  is the foreign peptide incubation concentration and c denotes the fraction of responders. Ignoring the self-background contribution, and assuming that the foreign peptide presentation level is related to the incubation concentration by the standard Langmuir binding model, we have F W ðwÞ ¼ 1  cðk=ðw  w0 ÞÞ;

ð20Þ

where k and w0 are parameters arising from the Langmuir equation. Appendix A below presents an explicit two-parameter model for F W ; if this model is used, we have four parameters in total, which can be estimated from the repertoire-level dose–response data by means of the maximum-likelihood method (the underlying assumption that the T cells all share the same value for the activation threshold W act can be relaxed). The same experimental procedure can also be carried out with an autoantigen. The post-selection TCR triggering rate distribution of this autoantigen j may then be plotted by transforming data according to the formula 1  cj ðk=ðw  w0 ÞÞ; using the previous estimates for k and w0 : Similarly, the immunogenicity curve of autoantigen j may be plotted using the transformation cj ðk=ðw  w0 ÞÞ=cðk=ðw  w0 ÞÞ: Thymic exposure and truncation point may then be gleaned from this curve; the critical presentation level is inversely proportional to the truncation point as shown in Fig. 5. Knowledge of an autoantigen’s critical presentation level, in relationship to that of a xenoantigen, is of manifest importance in avoiding autoimmunity, for instance in dendritic cell-based immunotherapy (Guermonprez et al., 2002). An alternative, more direct, approach is based on the distribution of TCR/pMHC off-rates (koff  1=T ij ) over the repertoire for a given ligand j. The theory (see the appendix) predicts how this distribution is modified by negative selection (Fig. 4; the progressive truncation of an intermediate range of values as shown in this figure is due to the non-monotone dependence of TCR triggering rate on mean TCR/pMHC residence time, Eq. (A.4); see Appendix B). A direct comparison with the data is possible in principle. However, the determination of millions of off-rates for a single ligand, by means of BIAcore for example, would appear to be prohibitive. However, approaches based on tetramer-staining may provide a suitable experimental proxy.

543

4. Discussion The classic notion of self/non-self discrimination has been challenged by Matzinger and co-workers (Gallucci and Matzinger, 2001; Matzinger, 2001, 2002), who claim that foreignness of a pathogen or toxin is not the relevant trigger for an immune response. We concur: were the immune response to be predicated on intrinsic chemical differences, pathogens would quickly evolve proteomic mimicry. We argue, rather, that the immune system distinguishes ligands on the basis of varying relative immunogenicities, without regard to the proteome from which any given ligand derives: the immune system can only recognize foreignness insofar as it correlates with high relative immunogenicity. The immune system induces a correlation between foreignness and immunogenicity by deleting thymocytes which experience a high relative immunogenicity relative to the autoantigens they encounter on the selecting APCs (Kappler et al., 1987; Kisielow et al., 1988; Surh and Sprent, 1994). It follows directly from the principle that foreignness is not chemically intrinsic that a ligand derived from the host proteome, but never presented during induction of deletional tolerance, has exactly the same recognition statistics over the TCR repertoire as a hitherto unseen xenoantigen. Thus, a peptide ligand may be said to be completely foreign if it has not been implicated in deletional tolerance, and less foreign to the extent that it has played a more prominent role in deletional tolerance. We argue that foreignness is a matter of degree and that immunogenicity can be quantified on a continuous scale. A corollary is that every autoantigen can potentially be recognized as ‘dangerous self’; it needs to exceed its characteristic critical presentation level to become immunologically relevant. For many common autoantigens, the critical presentation level will be unreasonably high, in which case the ligand has a very high tolerance (t) factor; for others the critical level may be dangerously close to the level at which pathogenic ligands are detected, in which case the ligand has a low t-factor, very close to 1. One might argue that the immune system could effectively recover a discrete self/non-self dichotomy by ensuring that thymic ubiquity and presentation levels are amply sufficient to ensure tolerance factors well in excess of unity for all autoantigens. However, there are intrinsic bounds on thymic presentation statistics, imposed by the thymocyte’s average sojourn time; Mu¨ller and Bonhoeffer (2003) review the relevant data. It is now generally acknowledged that danger stimuli emanating from diseased or injured tissue are important in eliciting immunity, and that professional APCs play a key role in orchestrating both the innate and adaptive immune responses via cytokines and co-stimulatory receptors (Egen et al., 2002; Granucci et al., 2003;

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Guermonprez et al., 2002; Riley et al., 2002). TCR specificity must nonetheless have a role to play; otherwise it would be difficult to understand why there is a repertoire, and why only a minute fraction of this repertoire is allowed to expand in response to any given antigen. According to our theory, TCR specificity helps the T cells to discriminate between ligands of high and low relative immunogenicity. We may contrast the principle of discrimination based on relative immunogenicity to the notion of self/non-self discrimination: on the one hand, these two concepts agree inasmuch as non-self (xenoantigens) will almost invariably have a high relative immunogenicity, whereas self (autoantigens) has varying degrees of relative immunogenicity, dependent on the ubiquity and presentation propensity during deletional tolerization. On the other hand, the correspondence breaks down for essentially foreign autoligands and essentially selfxenoligands. Deletional tolerance is not an exclusively thymic affair: otherwise the immune system could not acquire tolerance for ‘harmless foreign’ xenoantigens or selfneoautoantigens first expressed later in life. The presentation of hitherto unseen ligands in the absence of ‘danger’ stimuli or costimulation may lead to nonresponsiveness in various ways. The stimulated T cell may be anergized, which is a form of peripheral deletional tolerance: as the T cell is removed from the naı¨ ve repertoire, the ligand’s avidity distribution over the remaining TCR repertoire is modified (Eq. (5)). Instead of being anergized, the T cell may differentiate into a regulatory T cell; the anergetic stage may be an intermediary state between mature naı¨ ve T cell and regulatory cell (Roncarolo and Levings, 2000). Finally, the ligand may induce the T cell to increase its cellular activation threshold W act : Extra-thymic deletional tolerance with respect to a given autoantigen may be relatively inefficient: first, the T cell (not the clone) is the unit of selection and second, systemic ubiquity of topically expressed autoantigens will be low; for both these reasons complete eradication of a clonotype with high avidity for the given autoantigen may take a long time to be accomplished. However, matters may well be different for subsets of T cells restricted to local recirculation in specific organs or organ systems. Thus, foreignness would ultimately acquire a spatial dimension. If autoimmune activation is viewed as the cellular activation threshold W act straying too far below the safe lower bound given above, one might view autoimmune disease as a failure of the mechanisms employed by the T cell to adjust the threshold. Alternatively, the lowering of the cellular activation threshold to levels that allow certain autoantigens to become immunogenic may be regarded as arising from the need to detect illnessrelated epitopes. As distress-signals from the affected

tissue become more and more prominent, pressure mounts on the cellular immune system to activate responding clones. It may be possible that the balance between the need to detect harm-related antigens and to avoid responding to non-harm-related antigens is itself dynamic, and subject to constant adjustment, with major directions coming from the innate immune system and, in particular, the professional APC system. We have focused on the ‘deletion’ or ‘apoptosis’ threshold W thy in thymocytes. In fact, a thymocyte may be capable of various adaptive responses besides apoptosis; for instance, there may be a threshold W reg such that the T cell differentiates into a regulatory T cell whenever W iq exceeds W reg during an interaction in the thymus (Jordan et al., 2001). Also, there might be a ‘reverse’ threshold W rec such that the TCR gene undergoes further recombination if the TCR triggering rate W iq fails to exceed W rec (Buch et al., 2002; Fink and McMahan, 2000). As with all cellular responses, such thresholds are tied to the specific developmental stage during which the T cell is capable of exhibiting the corresponding responses. For instance, cytotoxic T cells (CTLs) feature a lytic threshold W lyt as well as a threshold associated with cytokine secretion (W sec ); the former appears to be significantly lower than the latter (Faroudi et al., 2003). It is tempting to suppose that the post-selection distribution functions F W ;j can be used to model the statistical fluctuations of the TCR signal arising from the self-background: these fluctuations are important in understanding activation threshold tuning and TCR repertoire maintenance, for example (Goldrath and Bevan, 1999). However, this is impossible because the distributions of TCR triggering rates from self-epitopes are correlated within each particular clonotype: deletional tolerance induces a statistical dependence between the distributions F W ;j across the autoantigens. A technique to tackle is difficult as described by Berg van den and Molina-Parı´ s (2003). The same difficulty underlies our assumption that the TCRs of the thymocytes follow the incipient (pre-selection) triggering rate distribution F W for all autoantigens. Strictly speaking, this is only correct for the very first round of negative selection. The error associated with this approximation depends on m and the mean thymic ubiquity of the autoantigens. For instance, if all autoantigens have ubiquity 1, the thymocyte will register the same triggering rate W iq on all negatively selecting cells (if we ignore fluctuations associated with MHC loading and TCR signalling). Further rounds therefore have no effect: the effective number of rounds is 1, e j  1 for this regardless of the actual number (and m extreme case). In an earlier paper (Berg van den et al., 2001), the essentially foreign autoantigens were denoted as the ‘variable’ component of self-presentation, and we

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observed that relevant epitopes must be presented at higher presentation levels than those belonging to the variable component, while this restriction did not apply for the other ‘constant’ component. Scherer et al. (2004) erroneously claim that our theory relies on different presentation levels to explain self/non-self discrimination; this criticism, due perhaps to a failure to appreciate the nature of essentially foreign autoantigens, is explicitly refuted by Eq. (3) above. While we focus on ligand presentation levels, we note that the TCR signal registered by the T cell can be independent of the presentation level when the TCR surface density is so low that the triggering rate becomes TCR-limited (see Berg van den et al., 2002 for more details). Since thymocytes have low TCR counts (Nakayama et al., 1990; Shortman et al., 1996), at least part of negative selection may be TCR-limited, inducing tolerance against autoantigens, virtually regardless of their thymic presentation level; Scherer et al. (2004) analyse the statistics of the repertoire structure for this case, on the implicit assumption that all self-epitopes are constitutive (i.e. have ubiquity 1).

Acknowledgements HAB was supported by the Wellcome Trust. The authors are grateful to Peter Cock for his comments on an earlier draft, and indebted to two anonymous reviewers for helpful criticisms and valuable insights.

Appendix A. Model for the TCR triggering rate distribution The incipient TCR triggering rate distribution F W ; that is, the distribution of W ij prior to negative selection, may be derived from the underlying molecular kinetics of the interactions between TCR and pMHC molecules. The Arrhenius equation represents the average duration of a TCR/pMHC interaction as T ij ¼ expfU ij g=f 0 ;

ðA:1Þ

where U ij is the dissociation energy barrier (in Boltzmann units, i.e. dimensionless) for the binding of a TCR molecule of clonotype i to a pMHC molecule of species j, and f 0 is the frequency factor, which we assume to be independent of i or j. The dissociation energy U ij depends on a large number of interactions between the atoms at the TCR/pMHC interface; if these interactions contribute additively to the energy barrier, and if their number is large enough, the central limit theorem states that U ij is normally distributed; approximating this Gaussian with a logistic distribution, we have PfU ij  E½U ij pug ¼ 1=ð1 þ expfnugÞ;

ðA:2Þ

545

where E½U ij  is the expected value of U ij (i.e. the average over the TCR clonotypes and the pMHC ligands) and n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi def is a dimensionless scale parameter; n ¼ p= 3V½U ij  where V½U ij  is the variance of U ij : On this model, the mean dwell time T ij follows a two-parameter quasiPareto distribution: def

F T ðtÞ ¼ PðT ij ptÞ ¼

1 ; 1 þ ðT 0 =tÞn

ðA:3Þ

def

where T 0 ¼ expfE½U ij g=f 0 : The rate at which a pMHC molecule of species j triggers a TCR of clonotype i, which we have denoted as wij ; is dependent on the mean dwell time T ij ; and it appears that the dependence is non-monotonic: the rate is maximal at an intermediate value of T ij (Kalergis et al., 2001; Lanzavecchia et al., 1999), believed to be the minimum duration required to enable the intracellular signal transduction capabilities of the TCR/CD3 complex (Lyons et al., 1996; Matsui et al., 1994). This nonmonotonic dependence is represented as follows: wij ¼ expfT R =T ij g=T ij ;

ðA:4Þ

where T R is the TCR triggering threshold, the minimum time required to trigger the TCR (see Chan et al., 2003, for a detailed discussion of this formula). At T ij ¼ T R ; the TCR triggering rate wij attains its maximum w^ ¼ e1 =T R : The statistics of the TCR triggering rate wij (as given by Eq. (A.4)) is found from the cumulative distribution function F T of the mean dwell time T ij :





 r1=ð1rÞ ln r r1 r1 P wij 4  FT ¼ FT T R ðr  1Þ r ln r ln r ðA:5Þ with F T according to Eq. (A.3) and where r is a parameter ranging from 0 to 1. Let ro denote the unique solution of the following equation: oT R ¼

ln ro 1=ð1ro Þ r : ro  1 o

ðA:6Þ

We then obtain the following expression for the cumulative distribution function of the TCR-triggering rate: def

F W ðoÞ ¼ Pðwij poÞ



 1 T 0 ro ln ro n ¼1 1þ T R ro  1



 1 T 0 ln ro n þ 1þ ; T R ro  1

ðA:7Þ

where we have used the dwell time distribution as specified by Eq. (A.3). The complement of the cumulative distribution function, 1  F W is shown in Fig. 3, for various choices of the parameters n and T 0 =T R : as the shape parameter n decreases, the distribution approaches the Bernoulli distribution with probability mass concen^ trated at o ¼ 0 and w:

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546

The TCR triggering rate distribution F W may be characterised in terms of its cumulants, which we denote k1 ; k2 ; . . . ; k‘ : ln E expfWtg ¼

1 X ‘¼1

k‘

t‘ : ‘!

ðA:8Þ

The cumulants can be obtained recursively from the moments, which we write m1 ; m2 ; . . . ; m‘ ; defined as Z w^ def o‘ dF W ðoÞ; ðA:9Þ m‘ ¼ 0

where mW ¼ m1 : The recursive formula is (Smith, 1995)  ‘1

X ‘1 k‘ ¼ m‘  ðA:10Þ k‘1 mi i i¼1 with initial condition k1 ¼ m1 :

Appendix B. Estimating the probability of surviving a round of negative selection ThePTCR triggering rate registered at the cellular level, Z ðthyÞ wj ; has the following cumulant generating j function: ( ) X ½n;Z 1 def LðtÞ ¼ lim ln E exp nt Z j wj ðB:1Þ n!1 n j2Znk (we assume that all thymocytes follow the incipient distribution, and accordingly suppress the dependence on i; we also leave the indication ‘thy’ understood). The parameter n denotes the number of autoantigens present at non-zero presentation levels. We seek approximations for the probability appearing in Eqs. (12) and (13) based on the large n limit. The relative presentation level is represented as explicitly dependent on n and the APP Z: rj Z ½n;Z ¼P ðM T  I k Zk Þ: j j2Z rj We are of course aware that the number of presented pMHC species is finite; our assumption is rather that the number is large enough for the approximations to be valid; in the simulations presented in this paper we used n ¼ 3000 (Hunt et al., 1992; Jardetzky et al., 1991). The Ga¨rtner–Ellis theorem (Dembo and Zeitouni, 1998) furnishes a large deviations estimate for the probability that the TCR triggering rate, induced by the autoantigens other than the focus epitope, exceeds o; we have the ersatz ( ) X P Z j wj 4o  expfnðoWo  LðWo ÞÞg; ðB:2Þ

The rate function used in this ersatz is oWo  LðWo Þ;

which can be evaluated by expressing L in terms of the cumulants of F W and the presentation propensity structure of the thymic APP, as we will show presently. The cumulant generating function Lð Þ is well defined only if the limit in Eq. (B.1) exists, which requires a systematic way of increasing n, that is, of adding autoantigens to the antigen presentation peptide ad infinitum. We assign propensities to the new additions according to an underlying propensity distribution, the moments of which are P ‘ j2Znk rj lim ¼ hr‘ i for ‘ ¼ 1; 2; ::: ðB:5Þ n!1 n by the law of large numbers (a propensity unit was chosen by setting hri ¼ 1). The cumulant generating function can now be evaluated as follows: ( ) X ½n;Z 1 LðtÞ ¼ lim ln E exp nt Z j wj n!1 n j2Znk Y 1 E expfnZ ½n;Z wj tg ¼ lim ln j n!1 n j2Znk 1 X ¼ lim ln E expfnZ ½n;Z wj tg j n!1 n j2Znk ¼ lim

n!1

¼

1 X

1 1 X X k‘ ðtnZ ½n;Z Þ‘ j n j2Znk ‘¼1 ‘!

‘¼1

 lim

n!1

¼

1 X

o ¼ L0 ðWo Þ:

t‘ ‘! !‘ P

k‘ ðM T  I k Z k Þ‘ P

n

‘ j2Znk rj

j2Znk rj

k‘ ðM T  I k Z k Þ‘ hr‘ i

‘¼1

n t‘ : ‘!

ðB:6Þ

Thus, the cumulants of the autoantigen presentation background (which excludes the focus epitope) are just the cumulants of the underlying distribution F W multiplied by the ‘presentation moment’ ðM T  I k Z k Þ‘ hr‘ i: In the simulations presented in this paper, autoantigen presentation propensity was taken to be a Gamma distribution with moment-generating function E expfrtg ¼ ð1  t=2Þ2

ðB:7Þ

which corresponds to the probability density function 4r expf2rg for rX0: Combining Eqs. (B.6) and (B.4) we obtain a series expansion for the large deviations rate function:

j2Znk

oWo  LðWo Þ ¼

where o4ð1  Z k ÞmW and Wo satisfy

ðB:4Þ

1 X

k‘ ðM T  I k Zk Þ‘ hr‘ ið‘  1ÞW‘o =‘!;

‘¼2

ðB:3Þ

ðB:8Þ

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which was truncated at order 11 for the calculations presented in this paper. In the case where ooðM T  Z k ÞmW ; the same method can be used to estimate the probability that the TCR triggering rate is smaller than o: For moderate deviations, we have ln P1 / ndiv (Dembo and Zeitouni, 1998) where the presentation diversity is defined by def

ndiv ¼

n 1 ¼P 2 hr2 i j2Z ðZ j =M T Þ

ðB:9Þ

(the second equality shows that the presentation diversity is the reciprocal of Simpson’s diversity). The presentation diversity ndiv can be regarded as the effective size of the APP; it satisfies 1pndiv pn and the maximum n is attained when all autoantigens are presented at the same level. On the other hand, ndiv is small when the APP is dominated by one or a few highpropensity ligands. We refine the erzatz (B.2) as follows: ( ) X P Z j wj 4o j2Znk

expfnðoWo  LðWo ÞÞg  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : pW2o nL00 ðWo Þ=2 þ 4 þ pW2o nL00 ðWo Þ=2

ðB:10Þ

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