Polly Matzinger & Michel Oldstone

Anderson et al J Immunol 2001 Graft male tissue onto female mouse lacking an immune system o graft is rejected when immune system reconstitutes o graft is accepted when female receives a male thymus

Ohashi et al Cell 1991, Oldstone et al Cell 1991: Induction of diabetes by viral infection of transgenic mice expressing LCMV protein in pancreas o mice are normal, no evidence for tolerance o infect mice with LCMV: virus is rejected → but later mice develop diabetes A simple mathematical model A ﬁrst model (DeBoer.prsl93)

S number of self epitopes evoking tolerance (105) R0 potential repertoire (before tolerance) R “functional”A repertoire ﬁrst model after(DeBoer.prsl93) tolerance (> 108) p lymphocyte recognition probability (precursor frequency)

S number of self epitopes evoking tolerance (105) Probability of responding to foreign epitope: A simple mathematical model R0 potential repertoire (before tolerance) S number of self epitopes evoking toleranceR (105) [Burroughs.i04]8 R “functional” repertoireR0 potentialP =1 repertoire after (before(1 tolerance tolerance)p) (huge: 10 (15>) 10 ) The “optimal”i speciﬁcity R “functional” repertoire− after tolerance− (> 108) [Arstila.s99] wherep lymphocyte recognitionp recognition probability probability (precursor freq. 10 (precursor-5) [Blattman.jem02] frequency) Size of functional repertoire: We had S R = R0(1 p) S Probability of responding toR foreign0(1 p) epitope: Pi =1 (1 p) − − Probability of −responding− to foreign epitope: R Pi =1 (1 p) AccommodatesThis probability two of problems:responding,−Pi, recognition has− its maximum and at deletion. Taking the derivative of Pi to p gives the optimum where 1 5 13 p = 10− S " R = R (1 p)S 0 − Thus speciﬁcity seems determined by number of self epitopes (DeBoer.prsl93,Whitaker.jtb93,Nemazee.it96,Borghans.ji99). Accommodates two problems: recognition and deletion.

14 13 A simple mathematical model 1 Too Too much Pi specific deletion 0.8 1/S

0.6

0.4

0.2

0 -14 -12 -10 -8 -6 -4 -2 0 SPECIFIC DEGENERATE Cross-reactivity (log(p))

Wide range of cross-reactivities for which functional repertoire is complete. True optimum at p =1/S.

15 LymphocytesLymphocytes are are specific speciﬁc AND AND recognize recognize many many peptides peptides

13 10 peptides

5 > 10 peptides

Repertoire: 8 10 clones

Because there are many more peptides (2010 ≃ 1013) than clones we need Because there aresufficient many cross-reactivity more peptides (Mason.it98). (2010 1013) than ! clones we need suﬃcient cross-reactivity (Mason.it98). −5 5 But a peptideBut each peptide recognized is recognized by very by very few few clones: clones: p p≃ 1010 − ! 16 1 1 1 1 1 1

1 1 1 0.8 0.8 0.8 0.8 0.8 0.8

0.8 1 0.8 1 0.8 1 0.6 0.6 0.6 0.6 0.6 0.6 Pi Pi Pi 0.6 Pi 0.8 0.6 Pi0.8 0.6 0.8 Pi

Pi 0.4 Pi 0.4 Pi 0.4 0.4 0.4 0.4

0.4 0.6 0.4 0.6 0.4 0.6

Pi Pi Pi 0.2 0.2 0.2 0.2 0.2 0.2

0.2 0.4 0.2 0.4 0.2 0.4

0 –10 –8 –6 –4 –2 0 –10 –8 –6 –4 –2 0 –10 –8 –6 –4 –2 0 0 0 0.2 –10 –8 –6 –4 –2 0.2 –10 –8 –6 –4 –20.2 –10 –8 –6 –4 –2 0 0 0 –10 –8 –6 –4 –2 log[p] –10 –8 –6 –4 –2 –10log[p] –8 –6 –4 –2 log[p] log[p] log[p] log[p] log[p] log[p] log[p]

0 –10 –8 –6 –4 –2 0 –10 –8 –6 –4 –2 0 –10 –8 –6 –4 –2 Figure 1:FigureThe probability 1: The probability of mounting of an mounting immune an response immuneP responsefrom Eq.Pi (2)from as a Eq. function (2) as of a function the specificity of thep specificityof the p of the Figure 1: The probability of mountinglog[p] an immune response5 Pi from Eq.log[p] (2)9 asi a function of the specificity plog[p]of the lymphocytes. Parameters5 S = 10 and9 R = 10 . Panel (b) depicts the effect of decreasing the initial repertoire size lymphocytes. Parameters5 S = 109 and R0 = 10 . Panel0 (b) depicts the effect of decreasing the initial repertoire size lymphocytes. Parameters S = 10 and9 R0 = 10 .8 Panel (b) depicts7 the effect of decreasing the initial repertoire size 9 from8 9 R0 = 10 78, R0 = 10 , to7 R0 = 10 . Panel (c) depicts the effect of incomplete tolerance induction, i.e., f = 1 and from R0 = 10from, RR0 0== 10 10, to, RR00 == 10 10 ., Panel to R0 (c)= 10depicts. Panel the e (c)ffect depicts of incomplete the effect tolerance of incomplete induction, tolerance i.e., f = 1 induction, and i.e., f = 1 and Figuref 1:=0The.8 in probability Eq. (6). of mounting an immune response Pi from Eq. (2) as a function of the specificity p of the f =0.8 in Eq.f =0 (6)..8 in Eq. (6). 5 9 lymphocytes. Parameters S = 10 and R0 = 10 . Panel (b) depicts the effect of decreasing the initial repertoire size 9 8 7 from R0 = 10 , R0 = 10 , to R0 = 10 . Panel (c) depicts the effect of incomplete tolerance induction, i.e., f = 1 and ∂ P of Eq. (3)f and=0.8 solving in Eq.∂ (6).P = 0 to find that that the maximum is at P =1/S. This optimum suggests p i ∂pPi of Eq.∂pP (3)i of andp Eq.i solving (3) and∂p solvingPi = 0∂ topP findi = that 0 to that find thatthe maximumi that the maximum is at Pi =1 is/S at.P Thisi =1 optimum/S. This suggests optimum suggests that the lymphocytethat the lymphocytethat specificity the lymphocyte is specificity largely determined specificity is largely by is determined the largely number determined of by self the epitopes number by the the of number immune self epitopes system of self the has epitopes immune the system immune has system has to be tolerant to. Thus, the specificity is not determined by the recognition! of pathogens, but by the demand to be∂pP toleranti ofto Eq. be to. (3) tolerant and Thus, solving to. the Thus, specificity∂pPi = the 0 to specificity is find not that determined that is not the determined maximum by the recognition is by at P thei =1 recognition! of/S. pathogens, This optimum! of pathogens, but suggests by the but demand by the demand to remain tolerant to a large number of self epitopes. Once lymphocytes are specific the repertoire has to be to remainthat theto tolerant lymphocyte remain to tolerant a specificity large to number a is large largely of number self determined epitopes. of self by epitopes. Once the number lymphocytes Once of self lymphocytes epitopes are specific the immune are the specific repertoire system the has has repertoire to be has to be sufficiently diverse to guarantee recognition of foreign epitopes (Fig. 1b). suffitociently be tolerantsu diversefficiently to. to Thus, diverse guarantee the to specificity guarantee recognition is not recognition of determined foreign of epitopes by foreign the recognition (Fig. epitopes 1b).! of (Fig. pathogens, 1b). but by the demand to remain tolerant to a large number of self epitopes. Once lymphocytes are specific the repertoire has to be sufficiently diverse to guarantee recognition of foreign epitopes (Fig. 1b). 1.1 Incomplete tolerance 1.1 Incomplete1.1 Incomplete tolerance tolerance Although there1.1 is promiscuous Incomplete expression tolerance of self antigens in the thymus, it remains unlikely that self tolerance is complete.Although HealthyAlthough there individuals is promiscuous there do harbor is promiscuous lymphocytes expression expression of that self can antigens recognize of self in antigens the self thymus, epitopes. in the it To thymus, remains study how it unlikely remains the that unlikely self tolerance that self tolerance results changeis complete. whenAlthough toleranceis complete. there Healthy is is promiscuous incomplete individuals Healthy we expression individuals definedo harbor a of new self do lymphocytes parameter harborantigens lymphocytes inf thefor that thymus, the can fraction recognize that it remains of can self self recognize unlikely epitopes epitopes. that that self self To epitopes. tolerance study how To study the how the manage toresults induceis complete. tolerance. change when Healthy For f tolerance= individuals 1 the newis incomplete do model harbor should lymphocytes we bedefine identical a that new to can parameter the recognize previousf self one.for epitopes. the For fraction a foreign To study of self how epitopes the that epitope we now requireresults that change it is recognized, when tolerance but that is none incomplete of the clonotypes we define recognizing a new parameter the foreignf epitopefor the fraction of self epitopes that manageresults to change induce when tolerance. tolerance For is incompletef = 1 the we new define model a new should parameter be identicalf for the to fraction the previous of self epitopes one. For that a foreign also recognize onemanage ofmanage the to (1 inducef to)S tolerance. induceself epitopes tolerance. For f that= 1 fail For the to newf induce= model 1 the tolerance. should newIncomplete model be Otherwise identical should tolerance the to be the clone identical previous will be one. to held the For previous a foreign one. For a foreign − responsibleepitope forepitope auto-immunity. weepitope we now now require we require Following now that that require Borghans it it is is that recognized,et it al. is[4] recognized, but we but thatlet thatα nonebe none but the of fraction thethat of the clonotypes none clonotypesof clonotypes of the recognizing clonotypes recognizing recognizing the foreign recognizing the foreign epitope the epitope foreign epitope also recognize one of the (1 f)S self epitopesfS number that of self fail epitopes to induce evoking tolerance. tolerance Otherwise the clone will be held at least one ignoredalso recognize selfalso epitope, recognize one i.e., of the one (1 off the)S self (1 epitopesf)R0S potentialself that epitopes fail repertoire, to induce that R “functional” fail tolerance. to induce repertoire, Otherwise tolerance. p precursor the clone Otherwise freq. will be held the clone will be held −− − (1 f)S responsibleresponsibleresponsible for for auto-immunity. auto-immunity. for auto-immunity.α Following Following=1 (1 Borghans Following Borghansp) − et. Borghanset al. al.[4][4] we we letetα let al.beα[4] thebe we fraction the let fractionα ofbe clonotypes the of fraction clonotypes(4) recognizing of clonotypes recognizing recognizing at leastat least one one ignored ignored self self epitope, epitope, i.e., i.e.,− − Fraction of clones recognizing at least one ignored self epitope: The chance that theat system least remains one ignored tolerant self when epitope, stimulated i.e., with a foreign(1 f(1)S f epitope)S is the probability that αα=1=1(1(1 p) p−) − . . (1 f)S (4) (4) none of the clones in the functional repertoire R will respond (with− − chanceα =1p) and(1 isp) potentially− . auto-reactive (4) − − − − (with chanceTheα),The chance i.e., chance that that the the system system remains remains tolerant tolerantSize when of when functional stimulated stimulated repertoire: with with a foreign a foreign epitope epitope is the probabilityis the probability that that The chance that the systemR remains tolerantfS when stimulated with a foreign epitope is the probability that nonenone of the of the clones clones inP in thet the= (1 functional functionalpα) repertoirewhere repertoireRR=RwillRwill0(1 respond respondp) (with. (with chance chancep) andp) is and potentially is potentially(5) auto-reactive auto-reactive (with chancenone ofα), the i.e., clones− in the functional repertoire− R will respond (with chance p) and is potentially auto-reactive Now the chance(with of chance a “successfulα), i.e., immune response” is theProbability probabilityR to remain that tolerant: the systemfS remains tolerant and (with chance α), i.e., Pt = (1 pα) R where R = R0(1 p) . fS (5) responds to the foreign epitope, which is the chancePt = to (1 remain− pα) tolerantwhere minusR R the= R chance−0(1 top) not. respondfS at all: (5) −Pt = (1 pα) where −R = R0(1 p) . (5) Now the chance of a “successful immune response”− is the probability that the system− remains tolerant and Now the chance of a “successful immune response”R is the probability that the system remains tolerant and respondsNow to the the foreign chance epitope, of aP “successful whichs = Pt is the(1Probability chance immunep) , to of remain response”a successful tolerant immune is the minus response: probability the chance tothat not the(6) respond system at all: remains tolerant and responds to the foreign epitope, which is− the− chance to remain tolerant minus the chance to not respond at all: responds to the foreign epitope, which is the chanceR to remain tolerant minus the chance to not respond at all: where the functional repertoire R is given in Eq. (5). To studyPs = howPt incomplete(1 p) , tolerance affects the results we (6) − − R plot Eq. (6) for f =0.8 and f = 1 in Fig. 1c. Ps = Pt (1 p) , R (6) where the functional repertoire R is given in Eq. (5). To− studyP −= howP incomplete(1 p) tolerance, affects the results we (6) s t − − Fig. 1c demonstrateswhereplot the Eq. that functional (6) the for efffect=0 repertoire. of8 and incompletef =R 1is in tolerance given Fig. 1c. in is Eq. enormous. (5). To The study region how of incomplete specificity values tolerance where affects the results we where the functional repertoire R is given in Eq. (5). To study how incomplete tolerance affects the results we the chanceplot of a Eq. successful (6) for responsef =0.8P ands approachesf = 1 in one Fig. is 1c. much narrower. Moreover the optimum has shifted leftwards, i.e.,Fig. towards 1cplot demonstrates a specificity Eq. (6) for that muchf the=0 smaller e.ff8ect and of than incompletef =p 1=1 in/S Fig. tolerance. Thus 1c. the is enormous.p =1/S estimate The region [8] of is specificity an upper values where bound for the lymphocytethe chance of crossreactivity: a successful response when thePs approaches initial repertoire one is is much sufficiently narrower. large Moreover the immune the optimum system has shifted Fig.leftwards, 1c demonstrates i.e., towards that a the specificity effect of much incomplete smaller than tolerancep =1/S is. enormous. Thus the p The=1/S regionestimate of specificity [8] is an upper values where operates even betterFig. when 1c lymphocytes demonstrates are that more the specific effect [4]. of The incomplete conclusion tolerance remains is that enormous. lymphocytes The are region of specificity values where thebound chance for of the a successful lymphocyte response crossreactivity:Ps approaches when the initialone is repertoire much narrower. is sufficiently Moreover large the the immune optimum system has shifted specific to avoid auto-immunity, and not to recognize many pathogens. leftwards,operatesthe i.e., even chance towards better of when a a specificity successful lymphocytes much response are smaller morePs specificapproaches than p [4].=1 The one/S. conclusion is Thus much the narrower. remainsp =1/S that Moreoverestimate lymphocytes [8] the is are optimum an upper has shifted boundspecific forleftwards, theto avoid lymphocyte auto-immunity, i.e., towards crossreactivity: and a specificity not to recognize when much the many smaller initial pathogens. repertoire than p =1 is/S su.ffi Thusciently the largep =1 the/S immuneestimate system [8] is an upper operates evenbound better for the when lymphocyte lymphocytes crossreactivity: are more specific when [4]. the initialThe conclusion repertoire remains is sufficiently that lymphocytes large the immune are system 2 specific tooperates avoid auto-immunity, even better when and not lymphocytes to recognize are many more pathogens. specific [4]. The conclusion remains that lymphocytes are specific to avoid auto-immunity, and not to2 recognize many pathogens.

2 2 0.3

R/R0

0.2

0.1

0 1 2 3 4 MHC diversity (log M )

Figure 2: Positive and negative selection according to the avidity model [13]. The curve in (a) depicts the distribution of thymocyte avidities for self peptide–MHC complexes. In our model, the chance p to be positively selected by a single MHC type is the chance that the avidity between the thymocyte T cell receptor and any of the self peptide– MHC complexes exceeds threshold T1. Thymocytes with avidities for self peptide–MHC complexes exceeding the upper threshold T2 are negatively selected (with chance n per MHC type). Panel (b) depicts the size of the T cell repertoire as a function of MHC diversity. The number of clones in the functional repertoire R is plotted as a fraction of the total initial lymphocyte repertoire R0. Parameters are: p =0.01, and n =0.005.

2 MHC diversity within the individual

Since individual MHC diversity increases the presentation of pathogens to the immune system, one may wonder why the number of MHC genes is not much higher than it is. The argument that is mostly invoked is that more MHC diversity within the individual would lead to T cell repertoire depletion during self tolerance induction. This argument is incomplete, however, because more MHC diversity could also increase the number of clones in the T cell repertoire through positive selection. In order to be rescued in the thymus, lymphocytes need to recognize MHC–self peptide complexes with suﬃcient avidity. A high MHC diversity thus increases both the number of lymphocyte clones that are positively selected and the number of clones that are negatively selected. To calculate the net eﬀect of these two opposing processes we develop a simple mathematical model [5].

Consider an individual with M diﬀerent MHC molecules and an initial T lymphocyte repertoire consisting of

R0 diﬀerent clones. Let p and n denote theMHC (unconditional) diversity within an individual chances that a clone is positively selected by a single MHC type, because its avidity is higher than a threshold T1, or negatively selected because its avidity

p0.3 probability of positive selection exceeds a higher threshold T , respectively (see Fig.R/R 2).0 By this deﬁnition, thymocytes can only be negatively 2 n probability of positive selection selected by MHC molecules by which they are also positivelyNote0.2 p > n selected, i.e. n < p. Since T cell clones need to M number of MHC molecules per host be positively selected by at least one of the MHC molecules,0.1 and avoid negative selection by all of the MHC

0 1 2 3 4 molecules, the number of clones in the functional repertoireMHC diversity (logR M ) can be expressed as ClonesFigure 2: Positiveshould and negative not selection be negatively according to the avidity selected model [13]. The curveon in any (a) depicts MHC the distribution but should be of thymocyte avidities for self peptide–MHC complexes. In our model, the chance p to be positively selected by a positivelysingle MHC type selected is the chance that theby avidity at betweenleast the one thymocyte MHC: T cell receptor and any of the self peptide– MHC complexes exceeds threshold T1. Thymocytes with avidities for self peptide–MHC complexes exceeding the upper threshold T2 are negatively selected (with chance n per MHC type). Panel (b) depicts the size of the T cell repertoire as a function of MHC diversity. The number of clones in the functionalM repertoire R is plotted as a fraction of the total M Rinitial lymphocyte= repertoireR0R0. Parameters(1 are: p =0.01, andnn =0).005. (1 p) , (7) In2 mice MHC about diversity 3% within of the the −thymocytes individual survive− and −about half of the

positivelySince individual MHC selected diversity increases cells the presentation become of pathogens negatively to the immune selected system, one may wonder why the number of MHC genes is! not much higher than it is. The argument that is mostly invoked is that more " [5]. The functional repertoire R thus[Van containsMHC diversity Meerwijk within the individual et al all wouldJ Exp lead to T Med T cell repertoire cell1997] depletion clones during self tolerance induction. that fail to be negatively selected, minus the This argument is incomplete, however, because more MHC diversity could also increase the number of clones in the T cell repertoire through positive selection. In order to be rescued in the thymus, lymphocytes need to recognize MHC–self peptide complexes with sufficient avidity. A high MHC diversity thus increases both the ones that also fail to be positively selectednumber of lymphocyte by clones that any are positively of selected and the the number ofM clones that aredi negativelyfferent selected. MHC molecules of the host. To calculate the net effect of these two opposing processes we develop a simple mathematical model [5].

Consider an individual with M different MHC molecules and an initial T lymphocyte repertoire consisting of R0 different clones. Let p and n denote the (unconditional) chances that a clone is positively selected by a single MHC type, because its avidity is higher than a threshold T1, or negatively selected because its avidity exceeds a higher threshold T2, respectively (see Fig. 2). By this definition, thymocytes can only be negatively selected by MHC molecules by which they are also positively selected, i.e. n < p. Since T cell clones need to Experimental estimates for the parametersbe positively selected of by at least this one of the MHC model molecules, and avoid negative have selection by all of recently the MHC become available. In mice, around 3% molecules, the number of clones in the functional repertoire R can be expressed as

R = R (1 n)M (1 p)M , (7) 0 − − − of the T cells produced in the thymus end[5]. The functional up repertoire inR thus contains the! all T cell mature clones that" fail to be negatively T selected, cell minus the repertoire, and at least 50% of all positively ones that also fail to be positively selected by any of the M diﬀerent MHC molecules of the host.

Experimental estimates for the parameters of this model have recently become available. In mice, around 3% selected T cells have been shown to undergoof the T cells produced in thenegative thymus end up in the mature T cell selection repertoire, and at least 50% of all positively in the thymus [17]. Thus, 94% of all thymic T selected T cells have been shown to undergo negative selection in the thymus [17]. Thus, 94% of all thymic T cells fail to be positively selected by any of the MHC molecules in the host [17]. These estimates can be used to calculate the chances p and n of a T cell clone to be positively or negatively selected by a single type of MHC cells fail to be positively selected by anymolecule. of Taking the into account that MHC inbred mice are homozygous molecules and therefore express 3 types of class in I MHC the host [17]. These estimates can be used to and 3 types of class II MHC molecules, p and n follow from: (1 p)6 =0.94 and n = p/2. This yields p =0.01 and n =0.005. − calculate the chances p and n of a T cellUsing these clone experimental estimates, to the number be of clones positively in the functional T cell repertoire R increases or with negatively selected by a single type of MHC molecule. Taking into account that inbred mice are3 homozygous and therefore express 3 types of class I MHC and 3 types of class II MHC molecules, p and n follow from: (1 p)6 =0.94 and n = p/2. This yields p =0.01 and n =0.005. −

Using these experimental estimates, the number of clones in the functional T cell repertoire R increases with

3 0.3

R/R0

0.2

0.1

0 1 2 3 4 MHC diversity (log M )

2 MHC diversity within the individual

Consider an individual with M different MHC molecules and an initial T lymphocyte repertoire consisting of R0 different clones. Let p and n denote the (unconditional) chances that a clone is positively selected by a single MHC type, because its avidity is higher than a threshold T1, or negatively selected because its avidity exceeds a higher threshold T2, respectively (see Fig. 2). By this definition, thymocytes can only be negatively selected by MHC molecules by which they are also positively selected, i.e. n < p. Since T cell clones need to be positively selected by at least one of the MHC molecules, and avoid negative selection by all of the MHC molecules, the number of clones in the functional repertoire R can be expressed as

R = R (1 n)M (1 p)M , (7) 0 − − − 0.3 ! " [5]. The functional repertoire R thus containsR/R0 all T cell clones that fail to be negatively selected, minus the ones that also fail to be positively selected0.2 by any of the M diﬀerent MHC molecules of the host.

Experimental estimates for the parameters0.1 of this model have recently become available. In mice, around 3% of the T cells produced in the thymus end up in the mature T cell repertoire, and at least 50% of all positively 0 1 2 3 4 selected T cells have been shown to undergo negativeMHC diversity selection (log M ) in the thymus [17]. Thus, 94% of all thymic T cells fail to be positively selected by any of the MHC molecules in the host [17]. These estimates can be used to Figure 2: Positive and negative selection according toParameters the avidity model p=0.01 [13]. and The n=p/2=0.005 curve in (a) [Borghans depicts the Eur distribution J Imm 2003] of thymocyte avidities for self peptide–MHC complexes. In our model, the chance p to be positively selected by a calculate the chancessingle MHC typep isand the chancen thatof the a avidity T between cell the clone thymocyte to T cell be receptor positively and any of the self peptide– or negatively selected by a single type of MHC MHC complexes exceeds threshold T1. Thymocytes with avidities for self peptide–MHC complexes exceeding the upper molecule. Takingthreshold intoT2 are accountnegatively selected (with that chance inbredn per MHC type). mice Panel (b) are depicts homozygous the size of the T cell repertoire and therefore express 3 types of class I MHC as a function of MHC diversity. The number of clones in the functional repertoire R is plotted as a fraction of the total 6 and 3 types of classinitial lymphocyte II MHC repertoire R0 molecules,. Parameters are: p =0.01,p andandn =0.005.n follow from: (1 p) =0.94 and n = p/2. This yields p =0.01 − and n =0.005. 2 MHC diversity within the individual

Since individual MHC diversity increases the presentation of pathogens to the immune system, one may wonder why the number of MHC genes is not much higher than it is. The argument that is mostly invoked is that more Using these experimentalMHC diversity within the estimates, individual would lead the to T cell number repertoire depletion of clones during self tolerance in the induction. functional T cell repertoire R increases with This argument is incomplete, however, because more MHC diversity could also increase the number of clones in the T cell repertoire through positive selection. In order to be rescued in the thymus, lymphocytes need to recognize MHC–self peptide complexes with suﬃcient avidity. A high MHC diversity thus increases both the number of lymphocyte clones that are positively selected and the number of clones that are negatively selected. To calculate the net eﬀect of these two opposing processes we develop a simple mathematical3 model [5].

Consider an individual with M different MHC molecules and an initial T lymphocyte repertoire consisting of R0 different clones. Let p and n denote the (unconditional) chances that a clone is positively selected by a single MHC type, because its avidity is higher than a threshold T1, or negatively selected because its avidity exceeds a higher threshold T2, respectively (see Fig. 2). By this definition, thymocytes can only be negatively selected by MHC molecules by which they are also positively selected, i.e. n < p. Since T cell clones need to be positively selected by at least one of the MHC molecules, and avoid negative selection by all of the MHC molecules, the number of clones in the functional repertoire R can be expressed as

R = R (1 n)M (1 p)M , (7) 0 − − − [5]. The functional repertoire R thus contains! all T cell clones that" fail to be negatively selected, minus the ones that also fail to be positively selected by any of the M diﬀerent MHC molecules of the host.

Experimental estimates for the parameters of this model have recently become available. In mice, around 3% of the T cells produced in the thymus end up in the mature T cell repertoire, and at least 50% of all positively selected T cells have been shown to undergo negative selection in the thymus [17]. Thus, 94% of all thymic T cells fail to be positively selected by any of the MHC molecules in the host [17]. These estimates can be used to calculate the chances p and n of a T cell clone to be positively or negatively selected by a single type of MHC molecule. Taking into account that inbred mice are homozygous and therefore express 3 types of class I MHC and 3 types of class II MHC molecules, p and n follow from: (1 p)6 =0.94 and n = p/2. This yields p =0.01 and n =0.005. −

Using these experimental estimates, the number of clones in the functional T cell repertoire R increases with

3 A simulation model: Borghans, Int. Immunol. 2002

bacterial peptides

viral peptides ....

toxin peptides

self peptides Repertoire

Set of peptides for each antigen group. Clones will recognize peptides from all sets with probability p.

21 Play an immune system game

1. Make a pathogen by drawing e epitopes from its set 2. Determine phenotype of all clones responding 3. If all naive: switch them to memory of corresponding mode 4. If memory available: switch all to that mode • determine succes • 5. Goto 1.

22 A simple example

Clone numbers: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 ... Ro

Initial types: 0 0 0 0 0 0 1 0 0 0 0 0 1 0 ... 0

Antigen 1, type 7: 7 0 0 7 0 0 1 7 0 0 0 0 1 0 ... 0 Zero score

Antigen 2, type 5: 7 5 0 7 5 0 1 7 0 0 0 5 1 0 ... 0 Zero score

Antigen 3, type 5: 7 5 5 7 5 0 1 7 0 0 0 5 1 0 ... 0 Positive score

Antigen 4, type 9: 7 5 5 7 5 5 1 7 0 0 5 5 1 0 ... 0 Negative score

23 10e3 epitopes per set, 10e6 clones

1 response positive negative 0.8 autoimmunity

0.6

0.4 Fraction immune responses 0.2

0 10!8 10!6 10!4 10!2 100 Lymphocyte cross!reactivity (p)

24 528 Diversity and speci®city in the immune system

Fig. 3. The performance of a diverse immune system. The fractions Fig. 4. The performance of a less diverse immune system. The 4 of different immune responses have been plotted for different repertoire consists of R0 = 10 clonotypes; for other parameters and degrees of cross-reactivity (p), after challenge with 103 different the interpretation of the different curves, see the legend of Fig. 3. pathogens. The thick curve denotes the fraction of challenges that This repertoire has the largest chance of making an immune induce any immune response at all. The fraction of challenges in response at p ~ 10±4. With such a high cross-reactivity, however, which pre-existing effector/memory clones induce the correct type of many mistakes are made. immune response is denoted by the thin curve. The fraction of challenges in which pre-existing effector/memory clones induce an inappropriate immune response is denoted by the dashed curve. example of a small simulation is given in Fig. 2. The above The fraction of challenges leading to autoimmunity, caused by gives a full description of our model simulations. The C-code of ignorant self-speci®c clones that are triggered by foreign antigens, our program is available upon request. is denoted by the dash-dotted curve. This repertoire is most functional at a cross-reactivity of p ~ 10±6, because the chance of immunity is then close to 1, and the net contribution of pre-existing effector/memory cells is high. There are e = 6 different epitopes per Results pathogen, pathogens come from m ± 1 = 8 different groups, each consisting of N = 103 different epitopes, a fraction f = 0.8 of all S = Somatic learning requires speci®city 4 6 10 self epitopes induces tolerance and there are R0 = 10 clonotypes. We have studied how the performance of an immune system that is challenged with 103 different pathogens depends on the cross-reactivity p of its lymphocytes (Fig. 3). The probability of immunity (i.e. the probability that all 103 pathogens with the corresponding effector mechanism. Even if an are recognized by at least one non-tolerant clone, see the thick inappropriate type of response is induced, naive lymphocytes line in Fig. 3) is close to 1 in a wide range of intermediate cross- acquire the corresponding (incorrect) effector mechanism. reactivities. A too cross-reactive immune repertoire fails to Memory clonotypes involved in a response to an antigen do respond to foreign antigens because the majority of lympho- not switch effector type (15,61). Pathogens never kill their cytes has been rendered non-functional during self-tolerance hosts, i.e. the simulations are continued even if an inappro- induction (53,54,62). If lymphocytes are too speci®c, on the priate response is induced. other hand, the immune repertoire frequently fails to recognize The performance of the model immune system is followed a foreign antigen. Thus, the simulations con®rm that suf®cient by counting the fractions of infections in which the presence of cross-reactivity is required to ensure an immune response effector/memory clones helps or hinders the induction of an against any pathogen (63). appropriate immune response. In the default situation, the Within the range of cross-reactivities yielding a high chance pathogen is only recognized by naive clonotypes, and the of immunity, effector/memory clones only tend to make correct effector type of the responding clonotypes is determined by decisions if they are suf®ciently speci®c (see the thin line in the innate immune system and the immunological context of Fig. 3). At a relatively high cross-reactivity, i.e. p ~ 10±3, the the pathogen. All cases in which pre-existing effector/memory positive contribution of effector/memory clones is largely clones establish the correct type of response against a coincidental. Even if there is no structural relationship between pathogen (without being responsive to any self antigens) the pathogens, such a protective effect is observed because contribute to the protection of the individual. The cases in of random cross-reactions (not shown). Since there are eight which effector/memory clones establish an incorrect type of different responsive modes in our simulations, the probability response against a pathogen are detrimental. We also count with which effector/memory clones coincidentally induce the the number of autoimmune responses, which are induced right type of immune response is 1/8. If lymphocytes are when naive clonotypes that are ignorant of their self epitopes suf®ciently speci®c, this randomness disappears and the are triggered into one of the responsive modes (49±52). An fraction of immune responses in which the presence of pre-