We previously developed a simple mathematical model for the probability, P_i , that an immune response to a foreign antigen is mounted from a functional repertoire of R antigen receptors (15, 16). In these models, p is the probability that an antigen receptor binds a pMHC with an avidity exceeding the threshold for a cell to become activated and mount an immune response. We call p the ‘binding probability’ and we will use ‘epitope’ to refer to a particular pMHC. Specific T-cell receptors have a low value of p and crossreactive TCRs have a high value of p. Because this probability, p, directly defines the ‘precursor frequency’ of clonotypes responding to a foreign epitope, we know that 10⁻⁶ ≤ p ≤ 10⁻⁵ would be a reasonable range (2–7). In the models, R ₀ is the diversity of the pre-selection repertoire, i.e., the total number of unique antigen receptors made by V(D)J recombination, and S is the number of self-epitopes that require tolerance by clonal deletion, anergy or the formation of Tregs. The diversity of the post-selection (or functional) repertoire, R, is then determined by the probability, P_s (for P _survival), that a clonotype fails to recognize all self-epitopes S,

According to the simplest model based upon complete self tolerance (15), the probability that a functional repertoire of R TCRs fails to respond to a foreign epitope is the probability that none of its clonotypes recognize the epitope, P_e = (1 − p) ^R , where the e stands for ‘escape’. Expressing one minus this chance of escape, as the probability of mounting an immune response to a foreign epitope, we obtain

Since ( 1 − x ) n ≃ e − x n when x is small, we can approximate P_s and P_i by

The value of p that maximizes P_i is computed by taking the derivative ∂_pP_i and solving ∂_pP_i = 0. One finds that the maximum is at p = 1/S (15). Evolution is therefore expected to select for individuals with lymphocyte binding probabilities around p = 1/S. Because S ≃ 10 5 (10), this prediction was strikingly confirmed by the 1:10⁵ estimates for the T-cell precursor frequency (15).

Taking the previously estimated S = 10⁵ self-epitopes (10) as an example, the probability of mounting an immune response, P_i , is depicted in Figure 2A for pre-selection repertoires of R ₀ = 10⁵ to 10⁹ clonotypes (as the diversity of the pre-selection repertoire is expected to differ markedly between small and large animals). Large vertebrates like Homo sapiens have post-selection repertoires exceeding R = 10⁹ different T-cell clonotypes (1), and given that only 5% of the T cells maturing in the thymus survive positive and negative selection (25), should have pre-selection repertoires well exceeding R ₀ = 10¹⁰ different T-cell clonotypes. One of the smallest vertebrates is the fish species Paedocypris which is known to have about R = 37000 T cells (and about 12000 self-proteins) (26). Such a small species is not expected to be able to generate more than say R ₀ = 10⁶ different T-cell clonotypes. The P_i curves in Figure 2A indeed have their optimum at p = 1/S = 10⁻⁵ (as indicated by the vertical dotted line). The dashed sigmoid in Figure 2A depicts the probability, P_s , with which a clone with binding probability p survives negative selection, which illustrates that when p = 1/S, this probability becomes P s = e − 1 ≃ 0.37 (as indicated by the horizontal dotted line). Reassuringly, the predicted fraction of clonotypes surviving negative selection is higher than the estimated 5% T cells surviving both positive and negative selection (25).

In this simple model all cells were considered to have the same probability, p, of recognizing a random epitope. The models and the data reviewed in the Introduction suggest that the post-selection repertoire also becomes specific because thymic selection weeds out the most crossreactive clonotypes from the pre-selection repertoire (see Figure 1D ). We therefore extend the model by allowing for a range of binding probabilities defined by a log normal distribution, with a mean µ and a standard deviation σ,

where D ₀(x) is a probability density, and x = log₁₀ p defines a log specificity, meaning that p = 10 ^x . Note that the log of the probability, p, obeys a normal distribution, that p = 1 when x = 0, that D ₀(x) is only defined for −∞< x ≤ 0, and that we are using a log₁₀ for the specificity, instead of the conventional natural logarithm that is typical for a log normal distribution, because specificity levels are usually expressed as order of magnitudes (e.g., 10⁻⁶< p< 10⁻⁵). Since D ₀(x) is a probability density function it has an area under the curve of one. To define the total number of clonotypes, we therefore still need to multiply D ₀(x) with R ₀ (i.e., R ₀(x) = R ₀ D ₀(x)). The probability density function of Equation (4) is depicted in Figure 2B for various values of µ and for σ = 1/2. Note that for σ = 1/2 each repertoire contains a wide variation of antigen receptors, differing by several orders of magnitude in their specificity.

In the same panel of Figure 2B we also depict the survival probability, P s ( x ) = e − p S = e − 10 x S from Equation (3) (see the dashed sigmoid lines representing S = 10³,10⁴, 10⁵ and 10⁶ self-epitopes). A 10-fold increase of S shifts the P_s curve an order of magnitude to the left (as more clonotypes will be lost by tolerance induction). ¹ Since the solid D ₀(x) and the dashed P_s (x) curves are independent, as the pre-selection repertoire D ₀ does not depend on S, it would still be possible to evolve an average specificity, µ, such that the D ₀(x) curve intersects the P_s (x) curve at the same height in species with different numbers of self-epitopes, S. We therefore color both curves red, blue, green or orange, when log₁₀[1/S] or µ equals −6,−5,−4 or −3, respectively (throughout the paper). This visualization reveals in Figure 2B that a similar level of survival during negative selection is expected when the average binding probability were decreased 10-fold for any 10-fold increase in S, which is similar to our previous results (14–16).

We study this further by explicitly defining the remaining density of receptors in the post-selection repertoire as D(x) = P_s (x)D ₀(x),

As an example, the probability density D(x) of the post-selection repertoire is depicted in Figure 2C for the previously estimated S = 10⁵ self-epitopes (10), and for various values of the mean, µ, of the lognormal distribution. Pre-selection repertoires composed of specific receptors, e.g., µ = −6, are hardly affected by tolerance induction to S = 10⁵ self-epitopes, whereas in repertoires composed of crossreactive receptors only a small fraction of the clonotypes survive tolerance induction to these S = 10⁵ self-epitopes (compare the height of the red µ = −6 curve with that of the green µ = −4 curve in Figure 2C , and compare the pre- and post-selection curves between Figures 2B, C ). The fraction of clonotypes surviving tolerance induction can be quantified by plotting the area under the curve, ²

as a function of the pre-selection average log specificity, µ (depicted for various values of S in Figure 2D ). These curves reveal that whenever p ≫ 1 / S ( or μ > lo g 10 [ 1 / S ] ) , only a small fraction of the clonotypes survive tolerance induction. Since the combined survival of positive and negative selection was estimated to be 5% (25), the dashed horizontal line at P_s = 0.05 depicts an experimental lower bound: negative selection by itself should not be lower than P_s = 0.05. Together the curves in Figure 2D suggest that the average binding probability of the preselection repertoire cannot be larger than µ = −4 for S = 10⁵, µ = −3 for S = 10⁴ and µ = −2 for S = 10³; otherwise less than 5% of the clonotypes in the pre-selection repertoire survive negative selection. Because specific receptors preferentially survive, the post-selection distributions of the more crossreactive repertoires, e.g., µ ≥−5 (for S = 10⁵), are skewed to the left in Figure 2C (compare the location of the peaks with the color-matching vertical dotted lines at µ = −6,−5 and µ = −4). Thus, this model confirms that negative selection makes a crossreactive pre-selection repertoire more specific (18, 21).

Figure 2D confirms that one obtains the same fraction of clonotypes surviving (the same AUC), whenever a 10-fold increase in S is compensated by a 10-fold decrease in the average pre-selection binding probability, µ. This was already suggested by the color-matching curves in Figure 2B . Since the same fraction of clonotypes survive when an increase in S is perfectly compensated for by a decrease in the average binding probability, µ, the results remain similar to the optimum, p = 1/S, obtained with our earlier model (15), which did not consider a distribution of receptor binding probabilities. According to both models, species with more self-epitopes should thus have a more specific pre-selection repertoire to achieve a similar completeness of the functional post-selection repertoire.

Because negative selection skews the binding probabilities, we explicitly compute the average binding probability of the post-selection repertoire of conventional T cells (by using the general definition of an average),

Figure 3A reveals that the average post-selection binding probability, µ _Tconv, is always lower than the preselection binding probability, µ (observe that all curves are located below the diagonal). Moreover, the skewing of µ _Tconv increases when there are more self-epitopes, and when the pre-selection repertoire is more crossreactive (observe that the distance to the diagonal increases with µ and S). Despite this skewing, µ _Tconv increases monotonically with the average binding probability, µ, of the pre-selection repertoire.

Although the loss through negative selection increases with the crossreactivity of the pre-selection repertoire ( Figure 2D ), the Tconv clonotypes surviving selection in a crossreactive pre-selection repertoire, do have a relatively high probability to respond to a foreign antigen ( Figures 2C , 3A ). To quantify the immune response we computed the expected breadth of an immune response. For functional clonotypes with a binding probability p = 10 ^x , we define the contribution, C(x), to the immune response to a foreign epitope as C(x) = pR(x) = pR ₀ D(x),

which is depicted in Figure 3B for each specificity, x, and for various values of µ. This reveals that for S = 10⁵, the largest contribution is expected from cells with a binding probability of p ≃ 10 − 5 = 1 / S . The total number of clonotypes in an immune response, i.e., the breadth of the response, is then defined as the integral B = ∫ − ∞ 0 C ( x ) d x , which is depicted in Figure 3C for various values of S. This confirms that a pre-selection repertoire centered around µ = −5, is expected to mount the most diverse immune response to a foreign antigen (compare the location of the peaks with the color-matching vertical dotted lines). Note that this breadth, B, replaces the probability of an immune response, P_i , of the previous model. Because due to its continuous nature there is always an immune response in the extended model (although it can become extremely narrow).

These results suggest that the binding probabilities of the functional post-selection repertoire are indeed determined by negative selection because the most crossreactive clonotypes in a pre-selection repertoire have the highest chance of becoming deleted. In Figures 2C , 3A we saw that in crossreactive preselection repertoires, negative selection skews the distribution of binding probabilities to more specific clonotypes. Hence the previous observation (14–16) that the evolutionary optimum, p = 1/S, coincides with the typical precursor frequency (2–7), can also be explained by negative selection. Such a specific binding probability of the post-selection repertoire naturally follows from strong negative selection within a crossreactive pre-selection repertoire with many self-epitopes, e.g., for µ = −3 and S = 10⁵, see Figure 3A . Nevertheless, our extended model also confirms the previous results, as the optimal binding probability of the pre-selection repertoire remains to be centered around p = 1/S (see the color-matching vertical dotted lines in Figure 3C ), because pre-selection repertoires composed of too specific receptors have a low probability to respond to foreign epitopes ( Figure 3B ), whereas repertoires composed of too crossreactive receptors suffer too much from clonal deletion ( Figure 2D ). Thus, evolution is still expected to select for immune systems with pre-selection lymphocyte binding probabilities centered around p = 1 / S ≃ 10 − 5 .

In addition to maximizing the probability of mounting an immune response by optimizing the recognition probability, we previously (15) also computed the size of the pre-selection repertoire required for having a sufficiently complete (8) functional repertoire for any given value of p. The probability that a foreign epitope is not recognized by any of the clonotypes in the functional repertoire was defined as P_e = (1−p) ^R [see Equation (2) and (15)]. Solving R ₀ for a particular probability of escape, P_e , corresponds to

Since most pathogens express several epitopes, picking P_e ≤ 0.1 would allow most pathogens to be recognized. ³ Plotting R ₀ as a function of p reveals that this function has a minimum (see Figure 4A ), and solving ∂_pR ₀ = 0 shows that this minimum is again located at p = 1/S. At this minimum R ₀ = −ln[P_e ]eS, which is proportional to the number of self epitopes, S, and only depends logarithmically on the probability of escape, P_e . This confirms that the immune system needs to be specific largely because there are so many self epitopes. ⁴ For p values larger than 1/S, the required R ₀ rapidly becomes prohibitively large (see Figure 4A ).

In the extended model a foreign epitope never completely escapes recognition, but the breadth of its immune response, B, can become extremely narrow. We can perform a similar analysis by increasing R ₀ to compensate for the loss of clonotypes due to negative selection. Thus, rescaling the area under the curve of the post-selection repertoire to one for every value of µ, we define D_N (x) = D(x)/AUC, where the subscript N stands for ‘normalized’. The required compensation in the size of the pre-selection repertoire, 1/AUC, is depicted in Figure 4B . We again observe that this compensation becomes prohibitively large for repertoires that are considerably more crossreactive than p = 1/S (or µ = log₁₀ 1/S). We conclude that both models agree on the fact that an unrealistically large pre-selection repertoire is required whenever the pre-selection repertoire is too crossreactive ( Figures 4A, B ). Evolution should therefore select for pre-selection binding probabilities in a medium range that does not exceed p = 1/S too much.

In the extended model, negative selection selects for more specific receptors in the functional repertoire ( Figures 2C , 3A ). The receptors that become negatively selected should therefore be less specific, i.e., more crossreactive. This prediction by Chao et al. (20) and Kosmrlj et al. (21) was recently confirmed by Lagattuta et al. (24), who demonstrated that the more ‘sticky’ hydrophobic amino acids, such as phenylalanine, leucine, tryptophan and tyrosine, are enriched in Treg cells, while the more weakly interacting amino acids, such as aspartic acid and glutamic acid, are enriched in Tconv cells. The repertoire of receptors that are negatively selected in our model is defined as

Loosely calling this the ‘Treg’ repertoire, D _Treg(x) is depicted in Figure 5A (for various values of µ and for S = 10⁵). The area under the curve, and the average log specificity, are defined as

Negative selection increases the crossreactivity of Tregs, especially when there are few self-epitopes and when the pre-selection repertoire is specific (see Figure 5B ), because the most crossreactive receptors have the highest probability [1 − P_s (x)] to become a Treg. The average binding probability of the Treg repertoire is hardly affected when there are many self-epitopes and the pre-selection repertoire is crossreactive because most receptors then become tolerized.