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Models for Antigen Receptor Gene Rearrangement. II. Multiple Rearrangement in the TCR: Allelic Exclusion or Inclusion?¹

Hannah Piper^*, Samuel Litwin^2, and Ramit Mehr^*

^* Department of Molecular Biology, Princeton University, Princeton, NJ 08544; and Fox Chase Cancer Center, Philadelphia, PA 19111

	Abstract

Top Abstract Introduction Review of Previous Models... A Simulation of TCR... Results Discussion References

 
This series of papers addresses the effects of continuous Ag receptor gene rearrangement in lymphocytes on allelic exclusion. The previous paper discussed light chain gene rearrangement and receptor editing in B cells, and showed that these processes are ordered on three different levels. This order, combined with the constraints imposed by a strong negative selection, was shown to lead to effective allelic exclusion. In the present paper, we discuss rearrangement of TCR genes. In the TCR -chain, allelic inclusion may be the rule rather than the exception. Several previous models, which attempted to explain experimental observations, such as the fractions of cells containing two productive TCR rearrangements, did not sufficiently account for TCR gene organization, which limits secondary rearrangement, and for the effects of subsequent thymic selection. We present here a detailed, comprehensive computer simulation of TCR gene rearrangement, incorporating the interaction of this process with other aspects of lymphocyte development, including cell division, selection, cell death, and maturation. Our model shows how the observed fraction of T cells containing productive TCR rearrangements on both alleles can be explained by the parameters of thymic selection imposed over a random rearrangement process.

	Introduction

Top Abstract Introduction Review of Previous Models... A Simulation of TCR... Results Discussion References

 
Multiple Rearrangements and Allelic Exclusion: A Contradiction?

Recent evidence shows that in the B cell receptor (BCR)³ light chain (1, 2), or the TCR -chain, rearrangement may not stop after a productive receptor gene has been formed and expressed. This raises the question: how is allelic exclusion maintained, if at all, in the face of continued rearrangement? The first paper in this series (41) showed, using computer simulation of BCR gene rearrangement, how continued light chain gene rearrangement can be reconciled with allelic exclusion. For ß T cells, the situation is more complex. As with the BCR heavy chain, allelic exclusion seems to be quite complete in the TCR ß-chain (3, 4, 5, 6, 7, 8, 9, 10). The expression of a functional TCR ß-chain (in conjunction with a surrogate TCR -chain (6)) triggers several successive cell divisions, which contributes to the shutdown of TCRß gene rearrangement (7), followed by further differentiation (8) and the rearrangement of TCR genes (9). Rearrangement and expression of TCR -chain genes, on the other hand, does not stop after the expression of the first rearranged -chain (3, 4, 5, 11, 12, 13, 14). Rearrangement appears to continue until the cell is either positively selected, or dies (15, 16).

Due to the lack of allelic exclusion in TCR, a T cell may not only contain two productively rearranged TCR alleles, but also simultaneously express the two resulting TCRs. This is an alarming concept, because an ß T cell that matures in the thymus expressing two different TCRs may be positively selected on one of them, while the other TCR may be autoreactive (17). The frequency of T cells simultaneously expressing two different V genes was found in one study to vary between 10^-3 and 10^-4. Only the cell surface expression of V2, V12, and V24 was monitored, which means that the frequency of T cells expressing any pair of V genes may be orders of magnitude higher (18). Independently, Malissen et al. (13) found that 26% of various T cell clones contained two productive V-J rearrangements (19).

The observations of allelic inclusion in TCR raise the following questions. Can allelic inclusion be fully accounted for by multiple rearrangements alone? Do these rearrangements occur completely at random, or is there some underlying order? What is the role of positive and negative selection in driving, or limiting, the process of TCR gene rearrangement? Several models (reviewed below) were suggested in an attempt to answer the first question, but have not sufficiently addressed the issues of order in rearrangement and the role of selection. Here, we develop a model of the TCR gene rearrangement process, and use it to examine competing explanations for TCR allelic inclusion. We aim to elucidate the mechanisms of allelic exclusion (or inclusion), and, in particular, to examine the degree of order in TCR gene rearrangement. Since the questions we study are probabilistic in nature, we use stochastic computer simulation of gene rearrangement and thymocyte selection. We perform simulations of our model under various parameter sets, and derive the constraints under which rearrangement and selection must operate (such as the average number of rearrangements performed per allele). Our results, briefly summarized, are: the ß: ratio can largely be explained based on the number of cell divisions after ß selection and thymic selection, but cannot be accounted for by rearrangement mechanisms alone. This is in contrast to the : ratio in B cells, which can be explained without invoking preferential expansion of B cells. The percent of TCR "double-productive" T cells, on the other hand, is mainly determined by the probabilities of positive and negative thymic selection, and the probabilities of resulting cell death. Death probabilities due to selection are smaller than death probabilities of developing B cells, which allow, on average, only two or three rearrangement attempts per cell, thus accounting for the apparent allelic exclusion in BCR chains. The presence of residual rearrangements in ß T cells (20, 21, 22) can be used to further delimit selection parameters. The fraction of productive residual rearrangements out of all residual rearrangements is found by all models, including ours, to be around 20% for a -first rearrangement pathway, in agreement with the experimental observations. This agreement supports the suggestion that TCR and rearrangement precedes and ß rearrangement.

In the following sections, we review previous models of TCR gene rearrangement, present our model and the results of computer simulations of this model, and conclude with a comparison of our findings for B and T lymphocyte gene rearrangement and development.

	Review of Previous Models of TCR Rearrangement

Top Abstract Introduction Review of Previous Models... A Simulation of TCR... Results Discussion References

 
TCR rearrangements on both alleles

In this section, we review previous models of TCR rearrangement, on which our computer simulations rely. The value of 26% TCR "double-expressors" found by Malissen et al. (19) was considered close to the value (20%) that one would expect if rearrangement of alleles proceeded on both alleles, allowing only one rearrangement per chromosome (Fig. 1). However, Malissen’s calculation did not take into account the possibility of multiple rearrangements on a single allele.

View larger version (11K): [in this window] [in a new window]   FIGURE 1. Malissen’s model of the generation of "double-expressors" (adapted from Ref. 13 ). The probability of a rearrangement being productive is at most one-third (this is the probability of joining in the correct reading frame), or slightly less if we take into account the existence of pseudo-genes. Thus, out of every nine cells, three cells would succeed in the first attempt to productively rearrange an -chain gene, and, out of those, one will also productively rearrange the second allele; out of the remaining six cells, two will productively rearrange the second allele. (We denote the genotype of cells as follows: "0" denotes the unrearranged, germline configuration, a "+" denotes a productive rearrangement, and a "-" denotes a nonproductive rearrangement.) Three different genotypes will result: one cell will be +/+ ("double-productive"), (2 + 2) cells will be +/-, and four cells will be -/-. The latter four will not survive selection, and hence the fraction of "double-productives" among the cells that did survive thymic selection will be 1/5, or 20%.

Our aim is to use a similar model to promote understanding of the mechanisms of rearrangement, addressing issues such as the average number of rearrangements performed per TCR allele and the order (if any) in which they are performed. These factors could not be directly obtained from Mason’s model, because it does not take into account the following two opposing constraints. First, Mason’s model assumes rearrangement can continue ad infinitum, while, in reality, the numbers of V and J gene segments, though large, are not infinite. TCR V-J rearrangements delete all gene segments between the two segments being joined (22), and hence, after several rearrangements, either the V or the J gene segment pool would be exhausted on that allele. Second, a mechanism that may partially compensate for gene segment pool exhaustion is order in gene rearrangement. This order refers to the apparent preference to rearrange first those J segments that are closer to the 5' region of the J locus (10, 22, 24, 25). Additionally, we wanted to address the possibility of preference to rearrange the allele that was rearranged last, as suggested by studies on B cells (the first paper in this series). In T cells, TCR rearrangement seems to go on simultaneously on both alleles (10, 13). However, weak preference for the most recently rearranged allele may still exist. In this study, we evaluate which of the two potentially opposing forces, the limited number of gene segments or the order in rearrangement, is more important in limiting TCR rearrangement.

Mason’s model lumps together the two nonpositive possible outcomes of the selection process: negative selection, or no selection (when the cell does not bind any self-MHC successfully, or the signals it receives are too weak for positive selection). These have to be addressed separately, due to their different effects on the probabilities of differentiation and death. The dependence of the outcome on the number, strength, and duration of signals the cell receives through its TCR is not yet fully known (26). This issue becomes more complicated when we consider that, if a cell expresses more than one receptor, the two receptors may be expressed with different cell surface densities (3). Our models do not directly address receptor expression; we assume that any productively rearranged gene is expressed at the maximum possible level and that the cell is selected according to the last rearrangement performed. However, our models deal with thymic selection through modifying the probabilities of the cell’s death, maturation, cell division, or further rearrangement. A cell expressing an autoreactive TCR may receive strong negative selection signals, and, hence, survive for a shorter time (and thus be allowed fewer rearrangement attempts) than a cell expressing a receptor that does not bind any thymic MHC-peptide complexes. Hence, our models take into account the probability of intrathymic cell death as a function of the quality of the cell’s TCR. The interplay between the strength of selection signals, and the potential for secondary TCR gene rearrangement, will determine a cell’s fate.

T cells may mature out of the thymus expressing a potentially autoreactive TCR, and the chance of this happening is probably higher for "double-positive" T cells. There exists no experimental data indicating how many of the "double-positive" T cells contain an autoreactive TCR, in addition to the TCR on which these cells were positively selected and allowed to mature. In the simulations presented below, it is easy to determine this value because we record the fate of every TCR rearrangement.

The ß: ratio and ß cells containing rearrangements

Any model of the TCR rearrangement process should also be able to account for the observed ratio of ß to T cells, observed to be 20:1 or larger, depending on the tissue being studied (27, 28). Not much is known about T cells, their function (29), development (30, 31, 32), or TCR - and -chain gene rearrangement. Most thymocytes try first to rearrange the -chain genes (22, 27), but this is not a rule (33); expression of TCRß does not preclude differentiation into T cells (20, 34), and TCRß and - transcripts can be detected simultaneously in the same cells (21).

Hayday and colleagues (20) suggested a model in which rearrangement is assumed to be strictly sequential (first , then , then ß, and then ), and each allele can only be rearranged once. According to this model, 56 out of every 81 cells would end up in the ß lineage, after failing to productively rearrange a TCR -or -chain (Fig. 2). This results in an ß: ratio of, at most, 2:1, 10-fold smaller than the observed ratio. The observed ratio must hence be explained as a result of subsequent cell proliferation in the ß T cell lineage. However, modifying this model to allow ß rearrangements at any stage (before, during, or after or rearrangement), and multiple TCR rearrangements, would reduce the final number of T cells produced. Thus, one question our models can be used to answer is: can the high ratio of ß to T cells be accounted for by assuming that rearrangement is not strictly sequential? Or do we have to also invoke multiple rearrangements on both alleles to account for this high ratio?

View larger version (17K): [in this window] [in a new window]   FIGURE 2. Hayday’s model of the factors determining the ß: T cell ratio. This model is strictly sequential and assumes no editing. Assuming strictly sequential rearrangement, cells that have failed to productively rearrange (4/9 of total) or (4/9 of the 5/9 that succeeded in rearrangement) proceed to the ß pathway (a total of 56/81 of the cells). Even if all cells that turned to the ß pathway had matured, the resulting ratio is 2:1, an order of magnitude lower than the observed value; the difference was attributed to cell divisions in the ß lineage.

View larger version (23K): [in this window] [in a new window]   FIGURE 3. Our extension of Hayday’s model. We calculate the number of productive and nonproductive rearrangements in ß T cells. Calculations follow the same rules demonstrated in the previous figures; however, they were extended to trace the fate of TCR alleles in ß cells. Out of the 56/81 cells that proceed to the ß lineage, 20/81 or 25% of the total number of cells will contain productive rearrangements, but rearrangement of the locus may later excise the rearranged gene fragments. Fig. 3 thus shows that there are three starting points for cells going to the ß lineage: TCR+/0-/-, TCR-/+-/-, and TCR-/-. The figure shows the calculation for each of these starting points. As we are only interested in percentages among mature ß cells, we do not follow ß rearrangement in this calculation because it does not affect the status of alleles. We only present the results of rearrangement. For the 12/56 cells starting with a TCR+/0 genotype (top left), there is a probability of 1/3 that rearrangement of the first allele will be productive. Out of these cells, 50% will have rearranged the allele that contained the productively rearranged allele, and 50% will have rearranged the allele that contained the nonrearranged allele. Hence, half of the cells with an +/0 phenotype will contain a nonrearranged allele, and the other half will be left with a productively rearranged allele. Those cells that have failed to rearrange the first allele but have succeeded with the second allele, ending up with an -/+ phenotype, will have erased both alleles. Similar calculations were done for the 8/56 cells starting as TCR-/+-/- (top right) and the 36/56 cells starting as TCR-/- (bottom right). When one adds up all the above combinations, the result is that 53.6% of all ß cells retain a rearranged allele, and 20% of the retained rearrangements are productive. The above calculation does not include the possibility of multiple TCR rearrangements. We can, however, derive an upper bound for the fraction of ß cells that retain a rearranged allele, for the case of strict allele preference. This is done by replacing the probability of a rearrangement being productive (1/3) in the previous calculation, by P1, the probability that the cell has reached a successful rearrangement on one allele only (possibly after a number of attempts), without rearranging the other allele. (The probability of failure, 2/3, is accordingly replaced by (1 - P1)). Then, out of all cells going to the ß pathway, the fraction of ß cells with no allele surviving will be P1(1 - P1); the fraction of ß cells with an unrearranged allele surviving will be 6P1/56; the fraction of ß cells with a nonproductively rearranged allele surviving will be 40P1/56; and the fraction of ß cells with a productively rearranged allele surviving will be 10P1/56. The remaining (1 - P1)2 of the cells will die. Summing these numbers, the fraction of surviving ß cells that contain rearrangements will now be 50/56(2 - P1), which can be at most 89% (if P1 1). It will be smaller if P1 < 1, or if allele preference is not absolute, as shown in the simulations presented below. The fraction of productive (out of total) rearrangements is independent of the editing process and again equals 20%.

	A Simulation of TCR Gene Rearrangement

Top Abstract Introduction Review of Previous Models... A Simulation of TCR... Results Discussion References

 
We constructed a stochastic simulation of TCR gene rearrangement.⁴ A cell is "born" into the simulation and followed throughout its life in the thymus as it undergoes TCR gene rearrangement, cell divisions, and selection. Each cell of the final progeny is either allowed to mature or else dies intrathymically. This process is repeated for a large number of clones. The program is constructed of a number of modules, which correspond to the various processes the simulated cell undergoes, as follows.

1) Cell birth: a new cell is born; its TCR genes are all assigned the germline configuration.

2) Cell death: the cell is deleted from the simulation. Since thymocytes are thought to spend only 3 wk in the thymus, cells that have not matured, but survived in the thymus up to the age of 20 days, die anyway.

3) Cell maturation: the cell’s features are added to the accumulated statistics of T cells produced in the simulation, and it is deleted from the simulation.

4) Cell division: an additional copy of the current cell is produced (without changing the probabilities associated with the cell). The current cell’s development is followed first, and the other daughter cell’s development is followed next. This is a recursive process.

5) ß-selection: after rearranging a productive TCR ß-chain, the cell undergoes ß-selection, that is, selection for the expression of a functional TCR ß-chain; if the cell passes this obligatory step (with a probability P_ßsel), it proceeds to rearrange the TCR genes, possibly performing a few cell divisions first (depending on the probability P_divß).

6) Thymic selection: operates on cells that have productively rearranged both TCRß and TCR. There are three possible outcomes: positive selection (with probability P_+sel), negative selection (P_-sel), or no selection (P_0sel). Positively selected cells may perform a few additional cell divisions before maturing (depending on the probability P_divß). Cells that were not positively selected may be allowed to perform additional attempts at rearrangement of TCR, but death may occur before each attempt. Negatively selected cells are assigned a high probability of death.

7) TCR gene rearrangement: one of the TCR alleles is chosen and rearranged. If productive, the cell proceeds to rearrange a TCR gene. If not, it proceeds to rearrange the other TCR allele. If both failed, it proceeds to TCR ß rearrangement (if TCR ß alleles are still unrearranged).

8) TCR gene rearrangement: if productive, the cell matures as a T cell. If not, it proceeds to rearrange the other TCR allele; if both failed, it proceeds to TCR ß rearrangement. Secondary rearrangement, if allowed, occurs only once per allele.

9) TCRß gene rearrangement: if it is productive, the cell proceeds to ß-selection. If not, it proceeds to rearrange the other allele. If both failed, then the cell either goes back to (if this pathway was not tried earlier), or dies.

10) TCR gene rearrangement: if productive, the cell proceeds to thymic selection. If not, it proceeds to rearrange the other allele, or rearrange the same allele when this is allowed. If both failed, the cell dies.

The simulation of the rearrangement process is similar to our model of BCR gene rearrangement (41), but is more elaborate, taking into account the rearrangement of all TCR chains, and all segments in each chain. One segment from each library (V, J, and, if applicable, D) is chosen at random. The probabilities for further rearrangements of the other segments are then renormalized, to account for deletion of intermediate segments. For example, if, at a given TCR rearrangement, the simulation used V₃₀ and J₁₀, then the probabilities of rearrangements for all V segments downstream of V₃₀ and J segments upstream of J₁₀ are set to zero, since we assume all rearrangements are deletional (22). The probabilities of choosing each of the remaining segments are assumed to be equal, unless we apply specific biases (see below). Rearrangement is deemed productive with probability P_product.

Parameters used in the simulations

Parameters for the simulations (shown in Table I) are interactively determined for each run. The values shown in Table I are those that we used as a "baseline," because they give a reasonable fit to most experimental observations (see below). P_product was taken to be 0.33 in all simulations. The probabilities of cell division are discussed below. The probabilities of cell death at various stages are unknown and were varied in the simulations, as were the probabilities of rearrangement of vs ß.

	Results

Top Abstract Introduction Review of Previous Models... A Simulation of TCR... Results Discussion References

 
Interclonal variability and cell division

Preliminary simulations indicated that even 10,000 individual cells per simulation are insufficient, as demonstrated by the high variability between simulations that was observed (data not shown); with high cell division probabilities, 10,000 individual cells may all belong to a small number of clones. Thus, it was necessary to include a large number of independent clones (each possibly containing many cells) in each simulation, for parameters such as the ß: ratio to stabilize. The number of clones was considered to be sufficient when both inter- and intrasimulation variabilities were small (<10% of the initial variability). This has been achieved for all quantities measured by generating 4000 independent clones in each simulation (data on variability not shown).

The above variability criterion helped to identify the proper division probabilities (probability of division following ß-selection and following positive selection of ß T cells) to be used in the simulations. To understand what "division probabilities" mean in the model, we discuss what the simulated cell can do in each simulation step: it can either rearrange one of its TCR genes, go through a selection process, divide, or die, depending on the outcome of the previous step. Each of these operations takes a few hours in the real thymus, hence we think of our simulation steps as representing a time period of 6 h (the minimum time required for cell division). Data shows that thymocytes do not usually perform more than one division per day on average (38). Specific measurements of cell divisions following ß-selection suggest that cells passing this checkpoint perform about eight divisions in the course of 4 days (39). Later, cells that are positively selected may perform one or two additional divisions before leaving the thymus (40). Hence, in our simulations, values of P_divß around 0.5, and lower values for P_divß, are reasonable.

Allowing cell divisions only following ß-selection (up to a rate of P_divß = 0.5) had a very small effect on the ß: ratio, because the cells still have to rearrange TCR and pass thymic selection. On the other hand, simulation results are very sensitive to increases of the probability of division after positive selection. With P_divß = 0.5 and P_divß = 0.25, the ß: ratio is only 2.5. However, the ratio increases quickly when we increase P_divß; the ß: ratio is 4.7 when P_divß = 0.4, 27.0 when P_divß = 0.5, and as high as 200 when P_divß = 0.6 or higher (data not shown). Thus, values of P_divß, P_divß 0.5 result in nonrealistic proliferation of positively selected cells. A simulation of 10,000 cells is largely taken over by one clone (even though we do not allow cells to remain in the simulated thymus for more than 80 simulation steps, or 20 days). The variability between simulations also becomes very large under these conditions, since individual clones may grow to high numbers so that each simulation represents a smaller number of thymocyte clones (data not shown). It is believed that positively selected cells do not perform more than a few divisions before maturing (40). Hence, in the following simulations the values of P_divß = 0.5, P_divß = 0.4 were used.

As long as proliferation is kept within reasonable limits, the total numbers of cells maturing from the thymus in the simulations remain small. Between 90 and 96% of the cells die intrathymically, as observed (38), which confirms our choice of selection probabilities (P(-sel) = 0.67 and P(0sel) = 0.30).

Preferential expansion must be invoked to account for the ß: ratio

We proceeded to use the ß: ratio as a way to identify the regions of parameter space that would give biologically reasonable results. We studied the dependence of this ratio on division probabilities, the probability P_R of starting with rearrangement, selection parameters, and death probabilities. The following results were obtained. First, as noted above, without cell divisions, or with small cell division probabilities, the ratio of ß to T cells remains small; it increases with division probabilities in the ß pathway. We did not consider the possibility of extensive cell death in the T cell lineage, because there is no evidence for such extensive death. Second, as expected, the highest ratio was obtained when it was mandatory to start with ß rearrangement (P_Rß1 + P_Rß2 + P_d = 1, P_R1 = P_R2 = 0), so that the cell rearranges and only if it failed to productively rearrange a ß-chain gene and express a functional ß-chain. This case does not reflect the real dynamics and was used only to demonstrate the extreme limit. Third, the ß: ratio is always higher when secondary rearrangement is allowed, than in its absence. Obviously, if we allow secondary rearrangements in TCR as well, the ß: ratio decreases (Fig. 4). However, multiple rearrangements in TCR did not significantly affect most results (data not shown), because there are only two J segments, so secondary rearrangements can only occur once per allele. The most important insight was gained from simulations combining the above parameter variations: secondary rearrangements alone cannot give an ß: ratio higher than 15, even in the extreme unrealistic case of a mandatory start with ß rearrangement (Fig. 4). Cell divisions in the ß pathway thus have to be responsible for a large part of the ß: ratio.

View larger version (24K): [in this window] [in a new window]   FIGURE 4. The ß: T cell ratio. Ratios obtained in simulations as function of the probability of starting with rearrangement of a TCR allele. Since there are two such alleles, PR = 0.495 means that the probability of starting with a TCR rearrangement is 0.99 = 1 - Pd, that is, starting with rearrangement is obligatory. "Biased editing" means we used Pallele = P5' = 1.

Random TCR gene rearrangement is compatible with TCR allelic inclusion

Our main goal was to understand TCR rearrangement; hence we studied the fate of productively rearranged TCR alleles. We asked whether multiple rearrangements are necessary for reconstructing the experimental observations such as the fraction of "double positives" and the fractions of ß T cells containing residual rearrangements. To answer this, we studied the effects of changes in simulation parameters, especially the degree of order in rearrangement and the death probabilities, on results such as the fraction of "double positives".

In each simulation, we recorded the number of ß T cells that have matured with both alleles productively rearranged. In one series of simulations, P_d was maintained at 0.1, while P_das was varied between 0.1 and 0.9. In another series of simulations, P_das was maintained at 0.7, while P_d was increased from 0.1 to 0.9. In each series, simulations were performed without secondary rearrangements or with biased or unbiased multiple rearrangements. The latter terms can be explained as follows. There are two types of bias that can be applied to rearrangements. The first type of bias is a preference to rearrange the same allele that was rearranged last. In our simulations, we used a parameter called P_allele, which took the value 0.5 if there was no preference, the value 1 if there was absolute preference to rearrange the last-rearranged allele (unless it was impossible), and the value 0 if there was absolute preference to rearrange the other allele. Intermediate values would mean partial preference; but as the effects of allele bias were not usually very large, we only used the values 0.5 or 1, that is, allele-unbiased or same allele-biased rearrangement. The second type of bias is a preference to use the 5' J segments first. We used a parameter called P_5', which took the value 0 if the choice of J segments was completely random, and 1 if the probabilities were biased. We only used a modest bias: when P_5' = 1, the probability of choosing the most 5' J segment available is twice the average, and the probability of choosing the most 3' J segment available is 0, while the probabilities of choosing intermediate segments changes linearly between the latter values. This still leaves some randomness in the choice of segments, and does not force a completely ordered rearrangement. Again, only the extreme cases of P_5' = 0 and P_5' = 1 were simulated, as the results fall between these extremes when intermediate values are used. When we refer to "biased rearrangement" or "biased receptor editing" without giving details, we mean that we used both types of bias, i.e., P_allele = P_5' = 1.

Not surprisingly, the percent of "double productives" decreased with the increase of P_das, the death probability of "autoreactive" cells (Fig. 5). The effect on the percent of "double productives" was not very strong, but it was consistent. In simulations performed with biased rearrangements, the fraction of "double productives" was lower than the observed (and no higher than that obtained with no multiple rearrangements at all): it did not exceed 15% even for low values of P_das and/or P_d (even when they were both set to 0; data not shown). Only when we simulated unbiased multiple rearrangements did we get higher fractions of "double productives" (up to 25%). The conclusion from this result is that the experimental observations imply that rearrangements are not biased, at least not as ordered as receptor editing in B cells seems to be. Furthermore, multiple unbiased rearrangements must be combined with a relatively weak negative selection to explain the observed 26% of "double productives."

View larger version (31K): [in this window] [in a new window]   FIGURE 5. Effects of death probabilities. A, The fraction of "double-productives" vs Pdas. B, The fraction of "double-productives" vs Pd. C, The fraction of "potentially autoreactive" cells within the "double-productives," vs Pdas. D, The fraction of "potentially autoreactive" cells within the "double-productives," vs Pd. NE, no editing; UBE, unbiased editing; BE, biased editing (including both allele preference and sequentiality); SEQ, sequentiality only; AlB, allele bias only.

Is order in TCR rearrangement masked by a large number of rearrangements per allele?

Our result on unbiased TCR rearrangement agrees with the experimental measurement of "double expressors," implying that there is no order in TCR rearrangement. However, we cannot completely exclude the possibility that some small degree of order does exist in the biological system, and is masked by a large number of rearrangements per cell. Even if P_allele, the probability for staying on a previously rearranged allele during a single rearrangement attempt is relatively high, the probability for staying on a previously rearranged allele after multiple rearrangements still decreases as the number of rearrangement attempts increases. To demonstrate this point, we show in Fig. 6 the distribution of the number of rearrangements per allele obtained in our simulations for the "default" parameter set (Table I) and unbiased rearrangement. This number was usually 3 or less, but in some cases was as high as 6, giving up to 12 rearrangements per cell. Even with only six rearrangements per cell, and with P_allele as high as 0.9, the probability of staying on a single allele throughout six rearrangements will be (P_allele)⁵, which is only 0.59 for P_allele = 0.9, largely masking the inherent order.

View larger version (33K): [in this window] [in a new window]   FIGURE 6. Rearrangements per allele. The distribution of number of rearrangements per TCR allele in: A) a representative generic simulation with Pdas = 0.7, and B) a simulation with Pdas = 0.2, with all other parameter values remaining at default values (Table I).

Prediction: a potentially high fraction of TCR double-expressors will carry an autoreactive receptor

One of the advantages of modeling is that it enables us to make predictions on quantities not previously measured in experiments. In the simulations described above, we have also counted the number of "double positives" in which the second allele (that which did not result in positive selection and maturation of the cell) encoded an -chain resulting in an autoreactive (vs nonselected) TCR. This is a worst-case estimate only, because it is obtained under the assumption that a cell is selected only according to its last rearrangement.

Theoretically, the fraction of "autoreactive double-positives" should only depend on the last rearrangements on both alleles. The last of the two rearrangements will be the one that the cell was positively selected upon, but the previous one, being productive, must be either an anti-self rearrangement or a "neglected" one. Thus, we expect the fraction of cells expressing a "potentially autoreactive" TCR allele to be at most P(-sel)/[P(-sel) + P(0sel)], where P(-sel) is the probability that a cell is negatively selected because its second to last rearrangement resulted in an autoreactive receptor, and P(0sel) is the probability that a cell is neither negatively nor positively selected after its second to last (productive) rearrangement. With P(-sel) = 0.67 and P(0sel) = 0.30, the maximum fraction of "autoreactive" "double-positives" will be 0.69. The effective value obviously decreases if we assign a larger death probability to negatively selected cells.

In the simulations, the fraction of "potentially autoreactives" decreased from the >60% predicted above for low P_das, to <30% with high P_das or P_d, which clearly shows the strong dependence on negative selection (Fig. 5). This result was independent of rearrangement order, as predicted by our analytical considerations above. It would be interesting to compare this value to experimental measurements, if and when these become available.

Residual TCR rearrangements in ß cells

We next considered the rearrangement status of TCR alleles in ß T cells. There can be at most one rearranged allele within the locus on the chromosome in a mature ß T cell. Our analysis (Fig. 3) shows that, in a model of strictly ordered rearrangement (, , ß, ), without multiple rearrangements, the fraction of ß cells containing a surviving rearranged allele within the locus would be at most 53.6%. With multiple rearrangements, this value be as high as 89% (see legend to Fig. 3). However, this is only an upper bound, derived in the case of strict allele bias, and when P₁ = 1, P₁ being the probability that a cell will succeed in rearranging a nonautoreactive TCR -chain gene on one allele only. Hence, we again turned to the simulation, recording the status of alleles, wherever there are undeleted alleles, in mature ß T cells. (When the rearranged allele is productive, the cell has become an ß T cell because no productive rearrangement of TCR was achieved on either allele.) In these simulations, values above 60% are only observed when there is order, and not with random rearrangement (Fig. 7). Thus, high fractions of ß T cells with chromosomal rearrangements imply some degree of order in TCR rearrangement, although masked by the large number of rearrangements per cell. Since we have concluded above that high fractions of "double productives" require a low degree of rearrangement order, it is unlikely that there are high percentages of ß T cells with chromosomal rearrangements. As the currently published experimental measurements (22) do not clearly distinguish between chromosomal and extrachromosomal residual rearrangements, future measurements of residual rearrangements on the chromosomes will reveal further information on the degree of order in TCR gene rearrangement.

View larger version (27K): [in this window] [in a new window]   FIGURE 7. Residual TCR rearrangements in ß T cells. The fraction of ß cells that have a rearranged allele vs PR (A), vs Pdas (B), and vs Pd (C). The percent of productively rearranged alleles, out of the rearranged alleles found in ß cells, vs PR (D), vs Pdas (E), and vs Pd (F). Each point is an average of at least three simulations; error bars were not indicated, to make the figures clearer, but the typical variability was similar to that in Fig. 5.

As an additional test of our simulations, we studied the fraction of productive rearrangements within the rearranged alleles. All analytical models predict that this fraction would converge to a value of 20%, which is within the range of experimentally observed values. Our simulations are also compatible with this observation, the value of 19 ± 3% is obtained. Even when the rearrangement process is not strictly sequential, we get similar results (Fig. 7, C–E).

	Discussion

Top Abstract Introduction Review of Previous Models... A Simulation of TCR... Results Discussion References

 
This paper is one of two studies that address the relationship between the allelic exclusion paradigm and the rapidly accumulating evidence for multiple rearrangements of Ag receptor genes in lymphocytes. The first study showed how allelic exclusion in B cells is maintained by the combination of ordered rearrangements and strong negative selection. Here, we extended our computer simulation of Ag receptor gene rearrangement to study rearrangement of TCR genes. We examined random vs ordered models of TCR gene rearrangement, and the interplay between this process and thymic selection. Our simulation takes into account stochastic rearrangement of the TCR variable region genes from their corresponding V(D)J gene segment libraries, several selection steps (ß-selection; positive and negative selection of TCR-ß-expressing cells), cell division, and cell death. The simulation follows each TCR clone from the start of rearrangement and records the fate of all daughter cells. We studied the properties of the emerging T cell repertoires under varying assumptions concerning the degree of order in the process of rearrangement. The main conclusions of the present study are the following. 1) High values of the ß: ratio cannot be obtained with multiple rearrangements alone; cell division must also be taken into account. 2) Multiple rearrangements of TCR genes are most likely random, rather than ordered. 3) A high fraction of TCR "double-productives" may express an autoreactive receptor. 4) The fraction of residual rearrangements in ß T cells that are productive is 20%, in agreement with experimental observations, thus confirming the accuracy of the analytical models. These conclusions are discussed in detail below.

The ß: ratio

One of the measurable quantities that has received much attention in the literature is the ß: ratio in thymocytes and mature T cells, which can be 20:1 or even higher, depending on the tissue studied. Theoretical predictions based on models that do not include multiple rearrangements fall around 2:1, which is far from the experimentally observed range of values. The difference was attributed to cell division. We set out to check to what extent multiple rearrangements may serve as an alternative explanation.

In our simulations, values compatible with the experimental observations were obtained only in the presence of multiple TCR rearrangements. This is similar to our finding in B cells that the : ratio cannot be explained without multiple rearrangements. In contrast to the : ratio in B cells, however, high values of the ß: ratio cannot be obtained without also taking cell division into account. The ratio increases with the number of cell divisions after ß selection and after thymic selection. Conversely, the ratio decreases when we increase the death probabilities of cells that fail ß selection or ß selection. Additionally, the ß: ratio increases with the probability that ß rearrangement will precede rearrangement.

The fraction of TCR "double-productives"

The measure for the extent of allelic exclusion, or rather allelic inclusion, in T cells, is the fraction of TCR "double-productives," T cells that carry productive TCR rearrangements on both alleles. Theoretical models predict that secondary rearrangements are necessary to explain the experimentally observed fraction of up to 26% TCR "double-productives." Our simulations examined the dependence of this quantity on the degree of order in TCR rearrangement and on selection probabilities. Fractions of TCR double-productives higher than 20%, as observed experimentally, are obtained in our simulations only when we allow multiple TCR rearrangements but assume they are unbiased, as in Mason’s model. Thus, the results of these simulations cannot exclude the hypothesis that multiple rearrangements in T cells are random, rather than ordered as was found for the B cell light chain.

The fraction of autoreactive TCR "double-productives"

A novel quantity defined in this study, for which no observations exist, is the fraction of cells with an autoreactive receptor among the TCR double-productives. This fraction is independent of multiple rearrangements, because it depends only on the last rearrangements on the two alleles. However, we found that the fraction of autoreactive double-productives is highly sensitive to the death rate of autoreactive thymocytes. If this rate is low, as it must be to get 26% "double-productives," then the fraction of autoreactive double-productives can be as high as 70%. This value is only an upper bound, since it was obtained for the case in which the cell is selected only according to its last rearrangement. Otherwise, this number will be lower, and will also depend on the relative expression levels of the two receptors, which are not addressed by the current model. More experimental data would be beneficial for settling this issue, which may help elucidate instances of escape from central tolerance in T cells.

Residual rearrangements in ß T cells.

The fraction of residual rearranged alleles in ß T cells may also be helpful in revealing the details of the rearrangement process, due to the nesting of the locus within the locus. The amount of residual TCR DNA in ß T cells was observed experimentally to be as high as 80%; however, most of these rearranged alleles probably exist on extrachromosomal excised DNA circles (22). Our analysis of chromosomal rearrangements predicts that this value will very between 45 and 89%, depending on the parameters of TCR editing. Our simulation confirms this prediction. Thus, these simulations can be used, in conjunction with future experimental measurements of the fraction of ß T cells containing TCR rearrangements on chromosomes, to estimate currently unknown parameters, such as the death probability of unselected cells or the probability of rearranging before ß. In our simulations, the fraction of rearranged alleles in ß T cells increases when we increase the death probabilities of unselected or negatively selected ß-expressing thymocytes, because an increase in a death probability reduces the probability that the cell would rearrange both alleles before maturing. The effect of the death probability of unselected cells is much more pronounced.

Among ß T cells that contain rearranged alleles, the fraction of these rearrangements that are productive was shown by all models to be independent of the number and order of rearrangements and predicted to be around 20%, which is within the range of experimental observations. Our simulations are consistent with this prediction under all conditions studied, confirming the accuracy of the analytical models.

TCR vs BCR gene rearrangement.

The first paper in this series (41) discussed rearrangement of the BCR light chain, for which isotypic exclusion and allelic exclusion have been established experimentally (1) in spite of observations on multiple rearrangement (reviewed in Ref. 2). Our computer simulation of BCR gene rearrangement enables us to reconcile multiple rearrangement with allelic exclusion. We provided evidence that, in B cells, 1) secondary rearrangements are negative-selection driven, in the sense that a cell has a limited time window in which it can edit its receptor and be rescued from deletion; and that 2) light chain rearrangement is an ordered process, on three levels: a preference for rearranging rather than light chain genes; a preference to make secondary rearrangements on the allele that has already been rearranged, rather than choosing the location of the next rearrangement at random; and, moreover, a sequentiality of rearrangement within each allele, such that J1,2 are preferentially used before J4,5. This order, combined with the stringency of negative selection, was shown to lead to effective allelic exclusion: the likelihood of a cell producing two productive rearrangements on two light chain alleles, within a limited time window and under the constraints of ordered rearrangement, becomes extremely small.

In spite of the strong similarities revealed in our studies between the way rearrangement seems to operate in B and T cells, it is worthwhile to note a crucial difference between the development of T cells to that of B cells. While BCR rearrangement seems to be limited by negative selection only, T cell development, on the other hand, seems to be limited by positive, rather than by negative, selection: developing T cells in the thymus are allowed a much more generous time window for continued TCR rearrangement, so that multiple rearrangements on both alleles become the rule rather than the exception. Furthermore, while B cells generally exhibit allelic and isotypic exclusion, due in part to ordered rearrangement, in T cells, receptor gene rearrangement is far less ordered. As a result, the probability that a cell will contain more than one productive rearrangement of the TCR -chain, and even express two TCR -chains simultaneously, is far from negligible. Moreover, positive selection may rescue the cell from death, and allow it to mature, based on the virtues of only one of its expressed receptors, as long as the other receptor is not so extremely autoreactive as to cause immediate deletion of the cell. This is a potentially dangerous situation, because the second receptor may still be weakly autoreactive, or, worse, may be specific, with high affinity, to a self-peptide that is not presented in the thymus. In spite of the existence of peripheral mechanisms of self-tolerance, which safeguard against improper activation of T cell, such improper activation does sometimes happen. Thus, understanding TCR gene rearrangement, selection, and editing is key to understanding autoimmunity.

	Acknowledgments

 
We thank Drs. Martin Weigert and Philip Seiden for useful discussions, and Michele Shannon and Dr. Lee Segel for reviewing the manuscript.

	Footnotes

 
¹ This study was supported by National Institutes of Health Grants GM20964-25 for the study of genetics and regulation of autoimmunity, R01 AI34882 (to S.L.), and AI10227-01 (to R.M.).

² Address correspondence and reprint requests to Dr. Samuel Litwin, Fox Chase Cancer Center, Institute for Cancer Research, Biostatistics Department, 7701 Burholme Avenue, Philadelphia, PA 19111. E-mail address:

³ Abbreviation used in this paper: BCR, B cell receptor.

⁴ The simulation program, and a program manual, containing a detailed description of the algorithm, are available from the authors upon request.

Received for publication December 17, 1998. Accepted for publication June 1, 1999.

	References

Top Abstract Introduction Review of Previous Models... A Simulation of TCR... Results Discussion References

 

1. Klein, J.. 1990. Immunology Blackwell Scientific Publications, Cambridge.
2. Radic, M. Z., M. Zouali. 1996. Receptor editing, immune diversification, and self-tolerance. Immunity 5:505.[Medline]
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