Deriving Quantitative Constraints on T Cell Selection fromData on the Mature T Cell Repertoire1
Vincent Detours,*†‡ Ramit Mehr, § and Alan S. Perelson2*
The T cell repertoire is shaped in the thymus through positive and negative selection. Thus, data about the mature repertoire maybe used to infer information on how TCR generation and selection operate. Assuming that T cell selection is affinity driven, wederive the quantitative constraints that the parameters driving these processes must fulfill to account for the experimentallyobserved levels of alloreactivity, self MHC restriction and the frequency of cells recognizing a given foreign Ag. We find thataffinity-driven selection is compatible with experimental estimates of these latter quantities only if 1) TCRs see more peptideresidues than MHC polymorphic residues, 2) the majority of positively selected clones are deleted by negative selection, 3) between1 and 3.6 clonal divisions occur on average in the thymus after completion of TCR rearrangement, and 4) selection is driven by103–105 self peptides. The Journal of Immunology,2000, 164: 121–128.
T he T cell repertoire is shaped in the thymus by three pro-cesses. First, TCR V-region coding genes are generatedby randomly rearranging V(D)J segments and inserting
random nucleotides between them. This mechanism creates thediverse repertoire of TCRs needed for the immune system to copewith unpredictable pathogens. Second, positive selection (1, 2)promotes the differentiation to a further developmental stage ofcells bearing receptors with a sufficiently large affinity for peptidespresented on molecules of the MHC expressed in the thymus. Thisconfers to T cells the property of self MHC restriction: they rec-ognize peptides presented in the groove of host MHC molecules,but ignore them when presented on foreign MHC (3–7). Third,negative selection (8, 9) deletes cells whose TCRs bind thymicMHC-peptide complexes with very high affinity, thus preventingthe emergence of self reactive T cells. Overall, only about 3% ofthymocytes have the intermediate affinity needed to fully mature (10).
In addition to self MHC restriction, the mature repertoire is alsocharacterized by a high alloreactivity. Typically, 1–24% of T cellsreact against the product of a given foreign MHC allele (11, 12).This high response frequency is hard to reconcile with the fact thatonly one T cell in 104–106 of the naive repertoire recognizes agiven pathogen (13, 14).
What should the quantitative properties of the processes drivingTCR generation and selection be to produce the experimentallyobserved levels of self restriction, alloreactivity, and Ag response?Previously, we developed a mathematical model relating these lat-ter quantities to the parameters driving affinity-based selection (15)and showed that this model gives a reasonable quantitative accountof self MHC restriction, alloreactivity, and Ag response frequency(16). In particular, we found that the difference between alloreac-tivity and Ag response frequencies is satisfactorily explained by
the affinity model. We go one step further in the present paper bydetermining what ranges of parameters driving repertoire genera-tion are implied by the observed properties of the maturerepertoire.
The parameters of our model fall into three categories. TCR/MHC-peptide interaction is quantitatively controlled by the num-bers of peptide and of MHC residues involved in binding to TCR.At the level of sets of molecules, peptides and MHCs are charac-terized by their respective diversity. Finally, the stringencies ofpositive and negative selection are expressed as affinity selectionthresholds. The stringency of positive selection can be inferredfrom data on the overall stringency of selection and on the amountof thymic clonal expansion following TCR rearrangement (16).Our analysis consists of calculating the levels of self MHC restric-tion, alloreactivity, and foreign Ag response frequencies for all thecombinations of parameter values that could be inferred from ex-perimental data. Although a significant portion of the parameterspace thus defined is consistent with the generation of a repertoirewith realistic properties through affinity-based selection, somemeasurements reported in the literature are incompatible with it.
Quantitative Model of T Cell SelectionWe give a concise verbal description of the concepts underlyingour model of T cell selection. A rigorous mathematical definitionof this model can be found in Refs. 15 and 16. A computer im-plementation is also available (a software package in C languageimplementing the model and related simulations can be down-loaded from ftp://ftp-t10.lanl.gov/pub/detours/abs-lab-1.1.tar.gz).
Protein shapes and binding affinities
The features of two proteins that determine their binding can bedescribed with a relatively small number of parameters, such astheir geometric shape, charges, and hydrophobicity. All these pa-rameters combine to form the protein’s “generalized shape” asdefined in Ref. 17. As in previous simulation studies (reviewed inRef. (18), we model the generalized shape of a protein as a stringof digits, from an alphabet of up to 255 digits. (The size of thealphabet does not affect the results, as long as it is large enough(15).) The strength of binding of two proteins is then defined as thedegree of complementarity between the digits representing theirgeneralized shapes (Fig. 1). Only the interface between TCRs andMHC-peptide complexes (framed region in upper diagram in Fig.
*Theoretical Biology and Biophysics, and†Center for Nonlinear Studies, Los AlamosNational Laboratory, Los Alamos NM 87545;‡Santa Fe Institute, Santa Fe NM87501; and§Department of Molecular Biology, Princeton University, Princeton, NJ08544
Received for publication May 18, 1999. Accepted for publication October 18, 1999.1 Portions of this work were performed under the auspices of the U.S. Department ofEnergy. This work was supported by National Institutes of Health (NIH) GrantsRR06555 and AI28433 (A.S.P.), NIH Grant GM20964-25 for the study of geneticsand regulation of autoimmunity, and NIH Grant AI10227-01 (R.M.).2 Address correspondence and reprint requests to Dr. Alan S. Perelsun, TheoreticalBiology and Biophysics, MS K710, Los Alamos National Laboratory, Los Alamos,NM 87545. E-mail address: [email protected].
1) is represented in the model. We define the affinity,K, betweentwo digit string proteins, as the sum of their individual digitinteractions.
Our model describes residues at the interface between TCRs andMHC-peptide complexes, not the full structure of these molecules.MHC and peptide are random strings,lm and lp digits long, re-spectively. In computing binding interactions the strings arealigned so that the central digits of a TCR always contact a peptide,and the digits at the extremities contact MHC. This modelingchoice follows from studies according to which TCRs bind MHC-peptide complexes with a common orientation (19). The model isindependent of the linear arrangement of digits. In particular, it isequivalent to a model with digits arranged in a two dimensionalarray mimicking the solvent accessible surface of proteins (as pro-posed in Refs. 20 and 21).
TCRs, MHCs, and peptides
In our model, it is assumed thatlm and lp are the same for allMHCs and peptides. This assumption is reasonable because werestrict our analysis to class I MHC, which present peptides offixed length. The number of MHC alleles expressed in an individ-ual is nm. A given MHC allele can present a panel ofnp distinctself peptides. We also assume that because of allele-specific bind-ing motifs, MHC molecules of different haplotypes present differ-ent subsets of self peptides (22). This is mathematically equivalentto presenting the same peptides in different conformations (15, 16),a property that we used in our previous modeling (16). Thus, aTCR is selected by a self environment composed ofnm 3 np
MHC-peptide complexes.Our goal is to measure self restriction and alloreactivity, which
depend, by definition, on MHC polymorphism and on the speci-ficity of TCRs. Therefore, MHC polymorphism-independent ef-fects do not need to be part of the model. Hence we make thefollowing simplifications. The effect of T cell coreceptors is omit-ted. Conserved MHC residues are not represented, i.e., thenm
MHC segments are interpreted as the polymorphic parts of MHCmolecules accessible to TCRs. To our knowledge, there is no ev-idence for a germline-encoded bias of TCRs toward recognition ofsome particular peptides, and bias toward recognition of MHCmost likely results from interaction with MHC conserved residues(23–25), which are not taken into account here. Thus, assumingthat preselection TCRs are random is justified in the context of themodel (see Ref. 16 for a more extensive discussion of this issue).The peptide does not influence the MHC in the model. Thus, weexclude from our scope of investigation the altered self hypothesis
(6, 26) according to which the TCR senses peptide-induced struc-tural features of the MHC rather that the peptide itself.
Positive and negative selection
Selection is implemented by introducing two affinity thresholds forpositive and negative selection,KP and KN (KP , KN). Clonesbinding at least one self MHC-peptide complex with affinityKlarger thanKP survive positive selection. Negative selection de-letes clones binding one or more self MHC-peptide complexeswith affinity K larger thanKN. The values ofKP andKN are derivedfrom experimental data by considering the fractions of clones sur-viving the different stages of selection. Thus, a clone will becomepart of the peripheral repertoire if its affinityK falls betweenKP
andKN. The fractionf of clones allowed to reach the periphery is:
f 5 fP z fN (1)
where fP is the fraction of clones surviving positive selection (asimilar parameter is used in Ref. 27), andfN is the fraction ofpositively selected clones that survive negative selection (27–30).The values off and fN can be inferred from recent experimentaldata (see below).
T cell activation and self-tolerance
The activation of selected T cells has to be defined in our model inorder to study alloreactivity and Ag response frequency. A clone isconsidered activated by a set of MHC-peptide complexes if theaffinity of binding between its TCR and at least one MHC-peptidecomplex in this set is greater thanKN. The repertoire in the modelis thus self tolerant by construction, because no clone having anaffinity larger thanKN to a self MHC-peptide complex can survivenegative selection.
Parameter Ranges for Affinity-Driven SelectionSeven parameters (Table I) determine mature repertoire properties(see Table II) in the model. All can be estimated from biologicaldata, except one, the number of possible digits,dmax, used in thestring representation of TCR, MHC, and peptides. This latter pa-rameter controls the discretization of the model and has no effecton it as long as it is chosen large enough (15).
Diversities of self MHC products and self peptides
Essentially all progress in the identification and characterization ofself peptides in alloreactivity has involved MHC class I systems(31). Thus, we focus on class I MHC in this paper. There are threeclass I loci in mice (32). Alloreactivity and self restriction exper-iments use inbred mouse strains (33), thus only one allele ispresent at each locus. Therefore, we setnm 5 3. (The analysis forclass II can be done in an analogous way.) Because the number ofclass I loci is known with absolute certitude and is not subject tovariation among inbred animals,nm is not varied.
About 103–104 different peptides can be eluted from moleculesof a given MHC allele (34–37). Accordingly, Bevan (38) suggeststhat 103–104 self peptides drive selection. Thus, unless specifiedotherwise, the number of self peptides presented by a given selfMHC string is set tonp 5 104. However, the effect of varyingnp
is also explored. Because there are about 105 genes in mammals,assuming that all of them are expressed and encode proteins ofaverage length 300 aa, leads to the conclusion thatnp , 3 3 107.Thus, 108 is a theoretical upper bound fornp, and consequently westudynp in the range 102–108.
FIGURE 1. Digit-string representation of MHC-peptide and TCR inter-action. MHC-peptide complexes are constructed by inserting a peptidestring of lengthlp digits in an MHC string of lengthlm digits. TCRs aresequences oflm 1 lp digits chosen randomly. The interaction strength,I,between two facing digits in the two aligned strings, is a measure of theircomplementarity (see text). Affinity,K, is the sum of interaction strengthsof contacting digits in the two aligned strings.
122 QUANTITATIVE CONSTRAINTS ON T CELL REPERTOIRE SELECTION
Contribution of peptide and MHC polymorphic residues tobinding
The number of peptide residues and MHC polymorphic residues incontact with TCRs,lp and lm, respectively, are derived from crys-tallographic data. The structure of TCR/MHC-peptide complexA6/HLA-A2-Tax (39) reveals 7 peptide and 5 MHC polymorphicresidues in contact with TCR A6, which giveslp 5 7 andlm 5 5digits. Performing a similar measurement for B7/HLA-A2-Tax(40), 2C/H2-Kb-dEV8 (41), and 2C/H2-Lb-QL9 (42), gives an av-erage of 5.75 peptides and 3.5 MHC residues in contact with TCR.Becauselp andlm must be integers, we setlp 5 6 andlm 5 4. A6,B7, and 2C are all known to be positively selected when expressedin the relevant MHC background. Consequently, the above esti-mate might not reflect a property of the preselection repertoire.Counting solvent-accessible peptides and MHC polymorphic res-idues in a class I MHC-peptide crystal structure giveslp 5 11 andlm 5 5 (43). This approach is independent of any selection-inducedbias, but has its own caveat because only part of the solvent-ac-cessible surface of the MHC-peptide complex is covered by theTCR (39–41, 44). In the absence of conclusive data, (lp, lm) 5 (6,4) will be used as default value, butlp will be varied in the range4–11, andlm in 2–8.
Stringencies of selection processes
About 20–50% of positively selected thymocytes survive negativeselection (23, 24, 45–48). Interestingly, earlier probabilistic mod-els of clonal deletion based on the hypothesis that evolution opti-mizes the size of the repertoire predicted a very compatible value,fN 5 37% (28–30). (These three models do not consider positive
selection, but assume implicitly that the repertoire submitted tonegative selection is constituted exclusively of functional Ag re-ceptors, i.e., of receptors that have been positively selected. There-fore it is legitimate to establish a connection between the fractionof deleted clones in these models andfN.) Alternatively, Laufer etal. (49) estimated thatfN , 95% by counting the number of apo-ptotic cells in the thymic medulla. The lower boundfN 5 20%(46), the intermediate valuefN 5 37% (28–30), and the upperboundfN 5 95% (49) will be investigated here.
Three percent of T cells produced in the thymus reach the pe-riphery (50). However, the fraction of clones, which our modeldeals with, and the fraction of cells differ because a significantportion of mature T cells divide before emigrating to the periphery(51–53). Scollay and Godfrey (52) suggest that one division occursbefore emigration to the periphery. Division also occurs earlier inclonal development, with the fraction of CD41CD81TCR1 cellsthat proliferate estimated as being 1.5- to 2-fold larger than thefraction of dividing mature thymocytes (10, 51, 54, 55). Overallthese data suggest that TCR1 cells go through one to three divi-sions in the thymus. Thus, if 3% of thymocytes survive selection,the fraction of clones reaching the periphery lies betweenf 5 3%andf 5 1⁄8 3 3% ' 0.33%. In the absence of more precise infor-mation, we assume that two divisions occur on average and henceeach clone consists, on average, of four cells. Here,f 5 0.75% willbe used as the default setting, but the effect of this parameter willbe explored in the range 0.19–3%. Hare et al. (56) report that upto six clonal divisions could take place in the thymus (i.e.,f 5 3%4 26 5 0.047%). This upper estimate will also be discussed.
ResultsCalculations were done for all the 5880 parameter combinationsthat can be obtained from Table I. The likely operating parameterregime of affinity-driven selection is determined by fitting data onAg response frequency, alloreactivity, and possibly self restriction(experimental ranges of these quantities are given Table II).
High alloreactivity implies stringent negative selection and amoderate number of thymic clonal divisions
Alloreactivity, a, is the fraction of clones in the selected repertoirethat are activated by a complex of self peptides associated with anyof thenm MHCs of a foreign haplotype. Any peptide driving rep-ertoire selection is considered as part of “self” in our model. Weshowed elsewhere (15, 16) that under the hypotheses stated in the“Model” section,a 5 f z (1/fN 2 1).
Alloreactivity is given in Table III as a function off and fN.Increasing the positive selection threshold,KP, either by reducing
Table I. Parametersa
Name Definition Default Explorationb
nm Number of MHC class I loci 3 3np Number of self peptides 104 {102, 103, . . . , 108}
lm Number of MHC digits 4 {2, 3, . . . , 8}lp Number of peptide digits 6 {4, 5, . . . , 11}
f Fraction of selected clones 0.75% {0.19, 0.37, 0.75, 1.5, 3}%fN Fraction of positively selected clones that
survive negative selection37% {20, 37, 95}%
dmax Largest digit 255 255
a Default values are used unless specified otherwise. All the parameter combinations that can be generated using the valuesin the “Exploration” column are investigated in this paper.
b The parameternm is known with absolute certitude and is not subject to variation in the congenic mice used in alloreactivityand self restriction experiments.dmax has no effect on the model’s results if chosen large enough (15).
Table II. Quantities to be fitted with the modela
Name Definition Range Ref.
R Frequency of clonesresponding to aforeign peptidepresented on selfMHC
1026–1024 13, 14
a Frequency of clonesresponding toMHC moleculesof a given foreignhaplotype
1–24% 11, 12
r Restriction ratio .1b 13
a See Ref. 15 for a mathematical derivation of these quantities.b Conflicting experimental estimates (see text).
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f or by increasingfN, lowers alloreactivity. The alloreactivity is inthe experimental range (1–24%) only whenfN # 37%. Thus themodel is not compatible with the upper boundfN 5 95% proposedby Laufer et al. (49).
In addition, realistic alloreactivity levels could be obtained onlywhen f $ 0.25%. Thus, the average clone size should be at most12 cells, and the average number of thymic clonal divisions at most3.6. The estimate of Hare et al. (56) that up to six clonal divisionsoccurs in the thymus, i.e.,f 5 3% 4 26 5 0.047%, impliesa #0.19% if fN 5 20%,a # 0.08% if fN 5 37%, anda # 0.0025% iffN 5 95%. Therefore, we conclude that the lower boundf #0.047% (56) is incompatible with high alloreactivity in the contextof affinity-driven selection. According to Hare et al. (56) the av-erage clone size of single positive thymocytes should be 9–12cells, which is compatible with the upper bound of 12 predicted byour model. (We obtained this range by averaging data from figures2c and 2d in Ref. 56.)
Response frequency to foreign peptides is inversely proportionalto self peptide diversity
The response frequency to foreign peptides,R, is defined in ourmodel as the frequency of TCRs in the mature repertoire recog-
nizing a given foreign peptide. We find that the impact off, fN, lm,and lp on R is weak within the explored region of the parameterspace. Varyingf within the range given in Table I never leads toa variation ofR greater than 1.1log10, whatever the values offN,np, lm, andlp. Similarly, the difference inRbetweenfN 5 20% andfN 5 95% never exceed 1.6log10. Finally, varyinglm or lp changesR at most by 1.6 and 1.7log10 respectively.
By contrast,R varied over at least 4.6log10 when np waschanged. We conclude that, within the explored region of the pa-rameter space,R is affected mostly bynp. This is apparent in Fig.2, whereR is plotted as a function off, fN, lm, andnp. Changinglponly translates the plots along thelm axis (data not shown), abehavior consistent with our claim that relative, rather than abso-lute contributions of MHC and peptide to TCR binding influencethe model (15).
Using the default parameters in Table I, we previously foundthatR is inversely proportional tonp (16). To check the generalityof this result, we measured for each value off, fN, lm, andlp howwell R could be approximated by linear regression onnp. The fitswere always excellent (error,0.5%). Although the parameters ofthe regression line depend onf, fN, lm, andlp, the slope remains ina narrow range (21.2 to20.8).
Data on alloreactivity and foreign peptides response frequencyimplies that 103–105 self peptides drive selection
Response frequencies in the experimental range 1026–1024 (13,14) are possible only ifnp , 105. Low self peptide diversity,np 5102, implies R in the biological range only iffN 5 95%, a valueincompatible with an alloreactivitya $ 1% (see above). IffN #37%, thennp must be greater or equal to 103 for R to be in therange 1026–1024. Thus, considering together the constraints onaandR leads to the conclusion that the number of self peptides that
Table III. Alloreactivity,a, for different values off and fNa
a All numbers are expressed in percent. Italic font indicates thata is in the phys-iological range 1–24%.
FIGURE 2. The logarithm of foreign peptide response frequency, log10(R) is visualized using gray scales. Light shades represent high values of log10(R)and dark shades low values. Different scales are used for each value offN investigated (bars at the right hand side of the diagram). Big dots indicateparameter sets that implyR in the experimental range 1026–1024. Parameterslm and np are varied within each square diagram, and each diagramcorresponds to a particular setting off and fN. For example, the value of log10(R) resulting fromf 5 3%, fN 5 37%,np 5 104, and lm 5 5 is shown bythe dot of coordinates [5, 104] in the first diagram of the second row.lp is set to 6 in all diagrams. Calculations were performed for the other values oflpgiven Table I and forf 5 0.19%, but are not plotted here.
124 QUANTITATIVE CONSTRAINTS ON T CELL REPERTOIRE SELECTION
drive thymic selection must be in the range 103 # np # 105. Thisrange is compatible with the estimate 103–104 proposed by Bevan(38) and corroborates earlier elution studies (34–37).
Self MHC restriction increases with the number of thymic clonaldivisions but decreases with the stringency of negative selection
Experiments based on the comparison between allogeneic and syn-geneic immune responses demonstrated strong restriction in someinstances (4, 5, 57–59), but weak or absent restriction in others (58,60–64). Thus, it is difficult to draw any conclusion on the averagelevel of self restriction from experimental data. Here, we make theminimal assumption that T cells are better activated, by anyamount, by peptides presented on self MHC than by peptides pre-sented on foreign MHC. The extent of self MHC restriction hastypically been estimated by comparing the effector activity againstforeign peptides presented by self MHC with the activity againstforeign peptides presented by foreign MHC. There are no effectorfunctions in our model, but it is reasonable to assume that responseintensity is proportional to the number of responding clones, whichis measurable in the model. Stockinger et al. (13) assessed selfrestriction levels by comparing the frequency of precursors againstAgs presented in the context of self and foreign MHCs (also seeRef. 6). This protocol is closely related to our approach becausethe frequency of responding precursors is expected to relateclosely to the frequency of responding clones.
We defineRa as the response frequency to a foreign peptidepresented on foreign MHC, and the self MHC restriction ratio,r,asR/Ra. A restriction ratio greater than one indicates that on av-erage a larger number of clones in the selected repertoire recognizea foreign peptide if it is presented on self MHC than if it is pre-sented on a foreign MHC. Stockinger et al. (13) givesr in therange 6–10.
The relationf 5 fP z fN (see “Model” section) implies that eitherdecreasingf or increasingfN result in a decrease infP, i.e., makingpositive selection more stringent. The result is a higher restriction
ratio, r. This effect is clearly apparent in Fig. 3. The relative con-tribution of peptide to TCR binding,cp 5 lp/(lm 1 lp), and selfpeptide diversity,np, also have a major impact onr. The repertoirecannot be restricted ifnp is large andcp small, or ifcp is large andnp small.
Compatibility with precursor frequency based estimate of selfMHC restriction implies at least one clonal division in thethymus
We showed above thata $ 1% could be obtained only iff $0.25% (i.e., 3.6 or less clonal divisions in the thymus on average).Further, a value ofr $ 6 could not be obtained withf 5 3%. Thusif r $ 6 reflects biological conditions, as suggested by Stockingeret al. (13), thenf should lie in the range 0.25–1.5%; i.e., between1 and 3.6 clonal divisions should occur on average in the thymus.Complying further with Stockinger et al. (13) by imposingr # 10does not imply more stringent constraints on parameter ranges.
Combined peripheral repertoire data imply that TCRs contactmore peptide residues than MHC polymorphic residues
It is possible to obtain physiological values for alloreactivity, i.e.,1% # a # 24%, the restriction ratio, i.e.,r . 1, and the foreignpeptide response frequency, i.e., 1026 # R # 1024, for any of theexplored values oflm provided that other parameters are set prop-erly. The same is true forlp. However, the relative contribution toTCR binding by peptide,cp 5 l/(lm 1 lp), is completely deter-mined by the above constraints: we find that the model meets theabove experimental requirements only ifcp is in the range 36–85%. Experimental estimates of the level of self restriction rangefrom weak to absolute restriction (63). The most minimal require-ment for self restriction isr . 1. Requiringr to exceed 6, the lowerrange reported by Stockinger et al. (13), narrows the admissibleinterval forcp to 50–80%. Thus our model suggests that if T cell
FIGURE 3. Restriction ratior. See Fig. 2 for explanations. Large dots indicate that repertoire is self MHC restricted (r $ 1). lp is set to 6. Calculationswere performed for other values oflp given Table I and forf 5 0.19%, but are not plotted here.
125The Journal of Immunology
selection is affinity-driven, then the contribution of peptide resi-dues to TCR binding should be greater than that of MHC poly-morphic residues.
DiscussionWe proposed estimates of the parameters driving TCR generationand selection (Table I) from the data on the mature T cell reper-toire (Table II). The model underlying this analysis is based on thecentral hypothesis that thymocyte selection is affinity driven.Therefore, the results discussed in the following are predicated onthe correctness of the affinity model. Although it plays a major rolein current immunological thinking, the view that affinity controls Tcells behavior remains controversial (25, 65).
Our analysis provides information on the average behavior ofthe T cell repertoire; it does not give a full account of the diversityof biological situations.
Alloreactivity frequency is, by definition, a global property ofthe T cell repertoire, not the property of any particular TCR/MHC-peptide ternary complex. Our model is a first step in bridging thegap between microscopic molecular events controlling T cell se-lection and activation and their effect on macroscopic properties ofthe repertoire. We show that if alloreactivity lies in the range1–24% (11, 12), one T cell in 104–106 recognizes a given pathogen(13, 14), and if the repertoire is self MHC restricted (by anyamount), then the relative contribution of peptide (as opposed toMHC polymorphic) residues to TCR binding, on average, shouldbe in the range 36–85%. In other words, if a TCR contacts onaverage 4 peptide residues, as suggested by several MHC-peptidecrystal structures (66–69), then it should also contact an averageof 0.7–6.5 MHC polymorphic residues. Assuming that the numberof T cells activated by a foreign peptide presented on self MHC ison average at least 6-fold larger than the number of T cells acti-vated by a foreign peptide presented on foreign MHC (as sug-gested in Ref. 13) implies a relative peptide contribution of 50–80%. That is, if four peptides residues mediate TCR binding, thenone to four MHC polymorphic residues should also be involved inTCR binding. Thus our analysis suggests that more peptide resi-dues than MHC polymorphic residues should interact on averagewith TCR. The contribution of conserved MHC residues is notaddressed by the current version of our model.
We established elsewhere (15) that alloreactivity both prior toand after selection is given byf z (1/fN 2 1), wherefN is the fractionof positively selected clones also surviving negative selection, andf is the fraction of clones surviving the overall selection process.Examining this relation provides insights into the stringency ofnegative selection. The fractionfN has been estimated using threedifferent approaches. The first consists of counting the number ofapoptotic cells in the thymic medulla, where negative selection issupposedly taking place. Doing so, Laufer et al. (49) found that aminimum of 5% of positively selected cells undergo negative se-lection (i.e.,fN , 95%). The second approach consists in gener-ating T cells in an environment in which only one peptide se-quence is covalently attached to all MHCs. Measuring the numberof cells produced in this single peptide environment that also re-spond to APCs harboring a normal diverse array of self peptides,provides an estimate for the number of cells that would have beendeleted by negative selection in a normal animal. This protocolleads tofN in the range 20–50% (45–48). Rescuing thymocyteswith anti-CD3 (23), or anti-TCRab (24) Abs instead of MHC mol-ecule with a covalently bound peptide, gives similar estimates.Finally, theoretical models based on the assumption that evolutionminimizes the size of the preselection repertoire, suggestfN ' 37%(27–30). Our analysis supports these latter estimates by showing
that a repertoire in which more than 1% of the clones are alloreac-tive can be obtained only if a large fraction of the cells is deletedby negative selection. Low negative selection,fN 5 95% (49) cou-pled with 3% survival of thymocytes, implies high affinity thresh-olds for positive selection, which in turn implies high self MHCrestriction, and therefore low alloreactivity.
The advent of highly active antiretroviral therapy raised consid-erable interest about immune system regeneration capabilities. Thethymus plays an important role in this process (70), especially forrestoring the diversity of the T cell repertoire, because peripheralproliferation in not believed to be associated with TCR rearrange-ment. Our analysis makes it possible to estimate the extent ofclonal proliferation occurring in the murine thymus, which is cru-cial to the understanding of how the number of cells produced inthe thymus relates to the number of new clonal specificities gen-erated, as well as to determining the fraction of recent emigrantsthat carry T cell receptor excision circles (70).
Recent evidence shows that the TCRa-chain (71–74) gene re-arrangement may not stop after a productive receptor gene hasbeen formed and expressed. Rearrangement appears to continueuntil the cell is either positively selected or dies (75, 76). The resultmay sometimes be the maturation of a T cell with two productivelyrearranged TCRa genes, both of which may be expressed (77, 78),and multiple TCR excision circles. These results have no directbearing on the study presented here, because we are dealing withthe repertoire of TCRs rather than of actual T cells. If a cell makesa secondary rearrangement, it is equivalent, from the point of viewof our model, to the deletion of its previous TCR and the intro-duction of a new TCR. However, any future model of T cell se-lection that attempts to deal with the population dynamics of T cellclones, will have to take secondary rearrangements into account.
The fraction of clones surviving the overall selection,f, is equalto the fraction of cells surviving selection, which is well estab-lished (;3%; Ref. 50), divided by the average size of a T cellclone in the thymus. Estimating thymic clonal expansion from theincorporation of 5-bromo-29-deoxyuridine (BrdU) leads to theconclusion that thymocytes undergo one to three divisions aftercompletion of TCR rearrangement, which corresponds to a clonesize of two to eight cells in the naive repertoire. Alternatively, onecan measure the dilution of 5- (and 6-) carboxyfluorescein diac-etate succinimidyl ester (CFSE), a membrane binding dye, on thesurface of dividing thymocytes. Using this technique, Hare et al.(56) found that the clone size of mature thymocytes could be ashigh 64 cells, and an additional analysis of their data show that itaveraged 9–12 cells. According to our model, an alloreactivity$1% implies that, on average, no more than 3.6 clonal divisionsoccur in the thymus, which means that average clone size shouldbe#12 cells. Also, we find that at least one division is required toget a restriction ratio$6. Thus, thymocytes should undergo 1–3.6clonal divisions on average, and the average clone size should be2–12 cells. This prediction of the model is consistent with esti-mates based on BrdU and CFSE experiments. Thus, the upperbound of six thymic divisions (56) should hold only for a minorityof thymocytes.
We found that if affinity drives selection, then the frequency ofclones responding to a foreign peptide presented on self MHC isinversely proportional to the number of distinct self peptides con-trolling thymocyte development. The model predicts that a re-sponse frequency in the experimental range 1026–1024 (13, 14) ispossible only if any given self MHC presents 103–105 self pep-tides. This range is supported by elution studies, in which selfpeptides were eluted from the surface of APC and separated bymass spectroscopy or HPLC (34–37). Given that as many as 108
distinct self peptides could in principle be extracted from all self
126 QUANTITATIVE CONSTRAINTS ON T CELL REPERTOIRE SELECTION
proteins (see section on parameters), the range 103–105 impliesthat the overall process of protein cleavage, and peptide transportand presentation is extremely selective. Only one peptide in 103–105 would be presented on a given MHC. These latter numbers aredisturbingly small considering that only;3 3 103 different pep-tides can be extracted from a virus of;104 base pairs, such asHIV-1. (For example, the 9 genes of HIV-1 strain HXB2 (Gen-Bank accession number txid11706) encode 9 proteins with a totalof 3034 aa. For each protein nonamers can be generated ending atposition 9, 10, . . . . Thus, 30342 (9 3 8) 5 2962 nonamers canbe extracted from HXB2.)
T cell development has been investigated from a variety of per-spectives encompassing, on the one hand, molecular events under-lying T cell selection and activation, and on the other hand, dy-namics of thymic cell populations. However, the two levels ofobservation are not independent: molecular processes control pop-ulation dynamics, and both determine T cell repertoire properties.A thorough understanding of T cell selection cannot be achievedwithout revealing the quantitative relationships between the thesetwo levels of description of thymic selection. Our study is a firststep toward such a unification. It reveals how quantitative param-eters inferred from data on cellular and molecular level experi-ments contribute to the shaping of the mature T cell repertoire.
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