J Recombination Based on Allele Allelic Exclusion in Dynamical Models Β

TCRβ Allelic Exclusion in Dynamical Models of V(D)J Recombination Based on Allele Independence

This information is current as Etienne Farcot, Marie Bonnet, Sébastien Jaeger, Salvatore of October 1, 2021. Spicuglia, Bastien Fernandez and Pierre Ferrier J Immunol 2010; 185:1622-1632; Prepublished online 28 June 2010; doi: 10.4049/jimmunol.0904182 http://www.jimmunol.org/content/185/3/1622 Downloaded from

Supplementary http://www.jimmunol.org/content/suppl/2010/06/28/jimmunol.090418 Material 2.DC1 http://www.jimmunol.org/ References This article cites 63 articles, 20 of which you can access for free at: http://www.jimmunol.org/content/185/3/1622.full#ref-list-1

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The Journal of Immunology is published twice each month by The American Association of Immunologists, Inc., 1451 Rockville Pike, Suite 650, Rockville, MD 20852 Copyright © 2010 by The American Association of Immunologists, Inc. All rights reserved. Print ISSN: 0022-1767 Online ISSN: 1550-6606. The Journal of Immunology

TCRb Allelic Exclusion in Dynamical Models of V(D)J Recombination Based on Allele Independence

Etienne Farcot,*,1 Marie Bonnet,† Se´bastien Jaeger,† Salvatore Spicuglia,† Bastien Fernandez,* and Pierre Ferrier†

Allelic exclusion represents a major aspect of TCRb gene assembly by V(D)J recombination in developing T lymphocytes. Despite recent progress, its comprehension remains problematic when confronted with experimental data. Existing models fall short in terms of incorporating into a unique distribution all the cell subsets emerging from the TCRb assembly process. To revise this issue, we propose dynamical, continuous-time Markov chain-based modeling whereby essential steps in the biological procedure (D-J and V-DJ rearrangements and feedback inhibition) evolve independently on the two TCRb alleles in every single cell while displaying random modes of initiation and duration. By selecting parameters via ﬁtting procedures, we demonstrate the capacity of the model to offer accurate fractions of all distinct TCRb genotypes observed in studies using developing and mature T cells Downloaded from from wild-type or mutant mice. Selected parameters in turn afford relative duration for each given step, hence updating TCRb recombination distinctive timings. Overall, our dynamical modeling integrating allele independence and noise in recombination and feedback-inhibition events illustrates how the combination of these ingredients alone may enforce allelic exclusion at the TCRb locus. The Journal of Immunology, 2010, 185: 1622–1632.

n developing T and B lymphocytes, allelic exclusion typically allelic exclusion continues to puzzle immunologists. For instance, restricts the assembly by V(D)J recombination of a pro- the mere fact that a small group (#5%) of allelically included http://www.jimmunol.org/ I ductively rearranged (in-frame) variable exon to only one al- abT cells carrying two productively rearranged (VDJ+/VDJ+) lele of, respectively, TCR and Ig genes. The resulting allelically TCRb alleles eventually develop alongside the overwhelming mass excluded (e.g., abT cells) commonly display a TCRb genotype ($95%) of allelically excluded cells still eludes a comprehensive made of a combination of one in-frame assembled allele (hence- explanation (2). + forth denoted VDJ ) and either one unrearranged germline (GL) Two types of modeling theories have prevailed in an attempt to 2 or partially (DJ)-rearranged allele or one out-of-frame (VDJ ) tackle this conundrum (3). The so-called stochastic models com- rearranged allele (1). Consequently, at the phenotypic level, the monly considered a low probability p for an in-frame joint to occur b TCR proteins expressed by these cells are encoded at a single due to inefficiency in either the onset or achievement of recombi- by guest on October 1, 2021 chromosome, a feature that may contribute in preserving the nation (4–6). With the probability for allelic inclusion given by the working of an adaptive immune system founded on clonal-cell square p2, the models were able to account for the rare appearance selection procedures. Despite years of efforts, the phenomenon of of allelically included cells. Soon afterward, however, these sim- plistic views appeared incoherent with a mass of experimental *Centre de Physique Theórique, Centre National de la Recherche Scientifique Unite´ findings, including those of a vast majority of T cells carrying Mixte de Recherche 6207, Universite´ de la Me´diterraneé-Universite´ de Provence- b Universite´ Sud Toulon Var, Centre National de la Recherche Scientifique Luminy dually rearranged TCR alleles, and B cells carrying dually rear- + Case 907; and †Centre d’Immunologie de Marseille-Luminy, Centre National de la ranged Ig H chain alleles as well, to a ratio of ∼60% VDJ /DJ to Recherche Scientifique Unite´ Mixte de Recherche 6102-Institut National de la Sante´ 40% VDJ+/VDJ2 cells (not mentioning the few VDJ+/VDJ+- et de la Recherche Me´dicale U631-Universite´ de la Me´diterraneé, Campus de Luminy Case 906, Marseille Cedex 9, France equipped cells; see Refs. 1, 2, 7). Thus, purely stochastic models 1Current address: Institut National de Recherche en Informatique et en Automatique, were found to insufficiently explain allelic exclusion. Virtual Plants Team, Cooperation Centre for Agronomic Research in Development/ Feedback inhibition is the hallmark of the current regulated Unite´ Mixte de Recherche De´veloppement et Ame´lioration des Plantes, Montpellier models of allelic exclusion. These models support the notion that Cedex 5, France. V(D)J recombination initiates at one allele at a time due to Received for publication December 29, 2009. Accepted for publication May 7, 2010. a specific yet ill-defined molecular control(s) (8–10) or as the result This work was supported by the Agence Nationale de la Recherche program BioSys number 06-135161. Work in P.F.’s laboratory is also supported by Institut National de of a stochastic, low-probability onset (11, 12). Whatever the la Sante´ et de la Recherche Me´dicale, Centre National de la Recherche Scientifique, cause, a VDJ+ outcome, one in every three (e.g., Vb-to-DJb) the Association pour la Recherche sur le Cancer, the Institut National du Cancer, the joints on average (13), eventually leads to the prohibition of Fondation Princesse Grace de Monaco, and the Commission of the European Com- munities. E.F. was supported by a fellowship from Agence Nationale de la Recherche further rearrangement via a signal conveyed from the immature BioSys, number 06-135161. M.B. was supported by fellowships from the Marseille- receptor (the so-called pre-TCR) built from the newly synthesized Nice Genopole and Association pour la Recherche sur le Cancer. TCRb polypeptide (14–17). This concept agrees with the 60:40 Address correspondence and reprint requests to Dr. Pierre Ferrier, Centre d’Immuno- logie Marseille-Luminy, Marseille, Parc Scientifique de Luminy, Case 906, Marseille ratio mentioned above. However, it remains unclear as to how Cedex 09, 13288 Paris, France (P.F.), or Bastien Fernandez, Centre de Physique The´- recombination proceeds to the opposite allele in the relatively orique, CNRS Luminy, Case 907, 13288 Marseille Cedex 09, France (B.F). E-mail frequent cases of an out-of-frame VDJ2 initial assembly. More- addresses: [email protected] (P.F.) or [email protected] (B.F.). over, these models again fall short in terms of properly depicting The online version of this article contains supplemental material. the production of VDJ+/VDJ+ cells, unless we assume that a Abbreviations used in this paper: DN, double-negative; GL, germline; WT, wild-type. loosened control sporadically tolerates a synchronization of re- Copyright Ó 2010 by The American Association of Immunologists, Inc. 0022-1767/10/$16.00 combination at the two homologous TCRb alleles. These concerns www.jimmunol.org/cgi/doi/10.4049/jimmunol.0904182 The Journal of Immunology 1623 led us to consider an alternative scenario that is not tied into possible statuses as depicted in Fig. 1), at time t in a differentiating T cell a strictly sequential mode of interallelic activation for V(D)J re- population, the recombination time window of which started at t0. The combination. Markov transition matrix Q = Q(tDJ, tVDJ, tf) (Table I), composed of the probabilities 1/tDJ, 1/3tVDJ, 2/3tVDJ, and 1/tf for transitions through the In this study, we aimed to use a dynamical approach to model corresponding cell states (for a definition of the genomic statuses, re- TCRb gene recombination in an effort to comprehend the sto- combination time window, and transition rates, see Results, Formulation of chastic and regulated premises of allelic exclusion within a de- the model: overview and Formulation of the model: basic features) is upper velopmental scheme that would integrate all the observed cell triangular due to the feed-forward structure of the transition graph. The Markov chain was exploited using the relation subsets at once. Principally, successive D-to-J and V-to-DJ rear- t0 t0 rangements at TCRb alleles of individual T cells and ensuing x t ¼ x t0 exp t2t0 Q ;t t0; ð1Þ feedback inhibition are seen throughout as independent, possibly t t0 14 t in which x 0 ðtÞ represents the vector ðxi ðtÞÞ and, likewise, x 0 ðt0Þ¼ t 14 i¼1 concurrent and mostly not stringently simultaneous biochemical ðx 0 ðt ÞÞ i 0 i¼1 denotes the initial distribution at the window origin. transactions, with fluctuating modes of initiation and duration. In this study, we were first concerned with data from TCRb+ T cells Indeed, evidence that developing T cells exhibit uniformed, bial- (in the form of hybridomas) that corresponded to final states in the model + lelic transcription of a Vb gene before recombination (18) makes (denoted sVDJ /…; see below). Based on relation (1), we made explicit allele autonomy in the conduct of TCRb rearrangements a plausible assumptions as follows: the distribution remains constant from the end of the time window onwards [meaning in particular homogeneous hypothesis. Besides, the variety of rearranging sequences (19) and proliferation of cells harboring the distinct (sVDJ+/…) statuses] t0 t0 multiplicity of molecular factors/mechanisms involved in TCRb (i.e., x ðtÞ¼x ðt0 þ 1Þ if t $ t0 + 1) and in line with biological evidence gene recombination and feedback signaling (16, 20) strongly favor that thymus seeding by T cell progenitors (of status denoted GL/GL in the a widespread noise in, and randomness of, all these events. Fur- model) proceeds via gated intervals of receptiveness followed by longer refractory periods (26), we postulated that such cells enter the process on Downloaded from thermore, because none of the shaping events leading up to a sig- a cyclic basis (instead of a constant flow), with every single cell com- naling launch (from TCRb gene transcription to protein synthesis pleting its window within the seeding cycle (i.e., for all cells, t0 +1# end and formation of the pre-TCR) nor the resulting feedback block in of the refractory periods). Accordingly, the distribution resulting from any rearrangement are likely to take place instantly, the notion of feed- such cycle does not depend formally on t0. It reads back control implies a deferred outcome(s) with the possible pro- xð1Þ¼xð0ÞexpðQÞ; ð2Þ longation of recombination after a VDJ+ has been made. Hence, 14

xð Þ¼ðx ð ÞÞ http://www.jimmunol.org/ dynamical aspects are expected to impinge in a general sense on with the initial distribution 0 i 0 i¼1 containing GL/GL cells only (i.e., xGL/GL(0) = 1 and xi(0) = 0 for all i GL/GL). Steady-state TCRb gene recombination and, potentially, allelic exclusion. recombination takes place during consecutive seeding cycles. Thus, the + With this framework in mind and building on a recent study by fractions xi(1) for all four final states in the model i 2{sVDJ /GL, Sepulveda et al. (21), which illustrated stochastic modeling of sVDJ+/DJ, sVDJ+/VDJ2, sVDJ+/VDJ+} (see Results, Formulation of the model: basic features) actually proffer a formal expression for the distri- TCRg1/g4 gene rearrangement, we depicted TCRb gene re- + combination as a continuous-time Markov process contingent on bution of TCRb T lymphocytes. Given the triangular structure of the matrix Q, analytic formulas for the variables x (1), which, via Q, explicitly adjustable parameters. Using available data from wild-type (WT) i depend on the parameters (tDJ, tVDJ, tf), may be obtained by solving in and mutant mice, we obtained quantitative values for these param- a recursive procedure their corresponding linear differential equations. In eters by standard fitting procedures. In these settings, stochastic practice, for simplicity, we used in this study the matrix exponential tool of by guest on October 1, 2021 dynamics combined with feedback control thoroughly accounted Mathematica (Wolfram Research, Champaign, IL) to directly deduce these variables from expression (2). for the generation, in experimentally compatible ratios, of cell From relation (1), we also computed cell fractions from preselected DN cohorts comprised of a large proportion of allelically excluded data in studies by Aifantis et al. (24, 25) using another series of specific cells along with small subsets of VDJ+/VDJ+ cells and those har- statements. In this case, TCRb+ DN thymocytes prior to b-selection were + boring a VDJ+/GL genotype. The parameter values further prof- identifiedwithtransientstates(VDJ/…) in the model. Practically, we fered estimates for the mean length of the joining and inhibiting further presumed that cells ending their recombination time window and not in the (sVDJ+/. . .) form, including (VDJ+/. . .), have a limited intervals, thus providing a scale of relative timings tailored to lifetime before committing to apoptosis or diverging toward another TCRb gene recombination. Our modeling reveals concepts award- lineage. Thus, differing from above where the computations relied on ing a discrete chromosomal system with the property to display cell fractions integrated over a stationary regimen comprised of multiple robust allelic exclusion at minimal regulatory cost while keeping cell-seeding cycles, we then had to consider statistical numbers cumulated over a limited interval. As an integration period, we chose the the opportunity to maximize genetic diversity via dual allele usage. nonstationary section of the model corresponding to the timing window T = 1. Therefore, the sampled fractions (supposedly collected at an in- Materials and Methods stant t independent of the cell-differentiation course) had to be fitted by the following expression: Source of TCRb+ experimental distributions ðt + to Experimental distributions of mouse TCRb T cells used were obtained XiðtÞ¼ xi ðtÞdt0; ð3Þ from published studies and compiled in Tables II and III. Numbers of t21 b-selected T lymphocytes well-defined in terms of TCRb genotype (Table which integrates the cell populations over the time window that began after II) were derived from data sets available in studies by Khor and Sleckman t21. A simpler expression was achieved by further assuming that experimen- (22) and Senoo et al. (23), which used peripheral T cell-derived, cloned tal sampling is effected once the flood of cells initiating TCRb recombination hybridomas from WT (22, 23) and engineered mutant bLD/LD (23) mice, has reached a steady-state course (i.e., with the proportions of GL/GL precur- respectively. Distributed numbers of early-developing double-negative sors entering the recombination window being constant in time). (DN) thymocytes prior to b-selection (Table III) were derived from data t0 t21 sets in studies by Aifantis et al. (24, 25), which used FACS-sorted intra- Explicitly: x ðt0Þ¼x ðt21Þ for t21 t0 t; cytoplasmic TCRb+, cell-surface CD25+ small DN cells from WT and ð 2 2 2 2 1 ð4Þ a a / / t21 pT - or SPL-76-deficient animals (pT and SPL-76 , respectively). implying XiðtÞ¼ xi ðt21 þ t0Þdt0; 0 Simulation of TCRb+ T cell distributions and analytic xt21ðt2 þ t Þ¼xt21ðt2 Þ ðt QÞ expressions in which i 1 0 1 exp 0 with, as before, the initial distribution xt21(t21) = x(0) containing only GL/GL cells. Again, we To modelize the random dynamics of TCRb gene recombination in early used the matrix exponential and formal integration tools in Mathematica developing T lymphocytes, we generated a Markov chain based on the (Wolfram Research) to obtain an explicit expression of Xi(t)=Xi, which t0 transition graph shown in Fig. 1, in which xi ðtÞ2½0; 1 symbolizes the likewiseÐ depends on (tDJ, tVDJ, tf), by using the t-independent relations X ¼ xð Þ 1 ðt QÞdt X ¼ðX Þ14 fraction of individual cells harboring the same genomic status i (among 14 0 0exp 0 0 and i i¼1. 1624 DYNAMICAL MODELING OF TCRb ALLELIC EXCLUSION

Least-squares fits and parameter estimation addition, to ascertain that solutions were method-independent, we also verified that the same curves of parameter triples were obtained when To fit the analytic predictions of our statistical model to the experimental applying the maximum likelihood method, which determines the most data sets, we primary used the least-square method, a standard technique in likely parameters by maximizing the sum numerical analysis (27). The method is based on an algorithm designed to tune parameters so as to minimize the Euclidean distance between the Ni xið1Þ predicted distributions and the experimental distributions, relying on the + ðusing data sets for selected T lymphocytesÞ fact that the distance vanishes if (and only if) the two distributions co- i2S +x + incide. In this study, given a collection of TCRb cell numbers {Ni}i2S (in or which S denotes any subset of cells harboring one potential TCRb+ genomic status as depicted in Fig. 1 and in Tables II and III (e.g., sVDJ+/DJ; + 2 + + + 2 + + Ni sVDJ /VDJ ; sVDJ /GL; or VDJ /DJ; VDJ /VDJ ; VDJ /VDJ ), the Xi + ðusing data sets for preselected DN thymocytesÞ: quantities to minimize would be the distances i2S +X N x ð Þ 2 + i 2 i 1 Least-squares fitting procedure and parameter estimation: + + i2S N x worked example ðin fitting data sets for selected T lymphocytesÞ Using data from WT mice in the study by Senoo et al. (23), the distance to or minimize was 2 N þ x þ ð Þ N X 2 sV DJ =GL sV DJ =GL 1 + i 2 i 2 + + 76 Sx i2S N X 2 N þ x þ ð Þ

sV DJ =DJ sV DJ =DJ 1 Downloaded from ðin fitting data sets for preselected DN thymocytesÞ þ 2 76 Sx 2 NsV DJþ=V DJ2 xsV DJþ=V DJ2 ð1Þ Ni þ 2 ; between the experimental proportions + (in which +N ¼ + Ni is N i 2 S i2S 76 Sx a normalization factor) retrieved from Tables II or III, respectively, and the in which Sx ¼ xsV DJþ=GLð1Þ þ xsV DJþ=DJ ð1Þ þ xsV DJþ=V DJ2 ð1Þ. xið1Þ To perform minimization using standard fitting algorithms [including corresponding relative fractions + (in which, likewise, x i 2 S when applying the widely used Nealder-Mead algorithm (28) for parameter http://www.jimmunol.org/ Xi validations via the routine “optim” in R] always firstly requires assigning +x ¼ + xið1Þ), or + , computed as described in the previous i2S X i2 S arbitrary values to all the parameters and iteratively traversing the param- paragraph. eter space through a succession of small displacements along each param- In practice, numerical calculations consisted of applying built-in proce- eter axis. dures and functions (especially function FindMinimum) in Mathematica In practice, to explore the predicted continuum of parameters for the distribution ðNsV DJþ=GL; NsV DJþ=DJ ; NsV DJþ=V DJ2 Þ = (1, 39, 36), we (Wolfram Research) that, starting from arbitrarily chosen values of (tDJ, therefore initially ran the least-square algorithm for 500 parameter triples tVDJ, tf), searched for minimizing the above mentioned distances and, from there, directly yielded the matching, resultant parameter triples. Of equidistributed on a cubic grid. In applying the validating criterion that note, due to finite computation time, a perfectly vanishing distance could we adopted in this study [closest integers for the computed cell numbers + not be reached by numerical means. Therefore, to validate the parameter x ð Þ N i by guest on October 1, 2021 1 + coinciding with experimental figures (1, 39, 36)], the pro- values, we adopted the following criterion: we only retained parameter x i 2S triples for which the closest integers to the predicted cell numbers cedure returned .450 solutions (parameter triples) that aligned on

+N +N a smooth curve (i.e., the red dots depicted in Fig. 2; also see Supplemental xið1Þ or Xi all coincided with the experimental figures + + Table I). x i2 S X i2 S {Ni}i2S. Continuum of fitting parameters Results Formulation of the model: overview As described above, the least-squares approach was a comprehensive and practical way to numerically solve the equations We implemented a stochastic modeling of TCRb gene recombination based on the Markov process formalism (29). Thus, xið1Þ Ni Xið1Þ ¼ ;i2S ðor TCRb dual-allele rearrangement status in a collection of individ- Sx SN SX Ni ual single cells is represented via a time-varying random variable ¼ ; i2SÞ for the parameters ðtDJ; tVDJ; tf Þ: SN distributed into a finite set of states. Holding to the Markov property, this variable evolves with time from one state to the next at Incorporating in this scheme as many equations as the number s of analyzed states in S, there are indeed only s21 independent conditions to a rate specific to the considered transition, independently of all comply with, because one equation is always ascertained by the normal- past statuses. Permitted transitions and associated rates are x ð Þ i 1 ¼ Ni ¼ encoded into the so-called transition matrix (30), the structure of ization + + + + 1. i2S x i2S N which is determined by: 1) biological constraints on TCRb gene For all data sets in Tables II and III, s = 3 (i.e., states sVDJ+/DJ; + 2 + + + 2 + + recombination (D-J occurring first, prior to V-DJ; no direct V-J); sVDJ /VDJ ; sVDJ /GL; and VDJ /DJ; VDJ /VDJ ; VDJ /VDJ , re- + spectively). Accordingly, in each case, there were only two independent and 2) feedback inhibition following a VDJ assembly (see Ref. 1, equations to resolve the three parameters tDJ, tVDJ, and tf. Therefore, Fig. 1, Table I). Transition rates represent adjustable parameters precisely defining those parameters was possible aside from one degree of expressed in terms of the mean duration of the given rearrange- freedom, and, consequently, the fitting procedure, when feasible, yielded ment step and average time lapse to achieve inhibition (covering a continuum of triple solutions in the parameter space. the whole period from TCRb gene expression and protein synthesis to trans-allelic block in recombination), respectively. Collec- Additional validation of parameter triples tively, the distribution of cell subpopulations harboring distinct To further access the effectiveness of parameter evaluations, we sought to TCRb gene configurations (b-genomic statuses) evolves in a de- confirm the above calculations using additional tools and methods. We thus terministic way along the linear flow generated by the transition applied the routine optim in R, another minimizing procedure that relies on the well-known Nelder-Mead (downhill simplex) algorithm (28). To be matrix. Provided an initial distribution is known for t = 0, the agreed, the proposed estimations had to yield indistinguishable or, if not, fraction of cells included in each subpopulation at any one time similar (according to the above-defined criterion) triple solutions. In t . 0 can be calculated. The Journal of Immunology 1625

VDJ+/DJ are seen as one identical status). In this way, the model only features general aspects regarding the structural organization and use of gene segments at the TCRb locus, without dwelling on which particular cluster(s) of gene segments is being used for recombination. Upgrading prospects (e.g., by taking into account the regular usage of separate Db–Jb clusters) are considered in the Discussion. TCRb gene recombination in jawed vertebrates takes place in minor subpopulations of thymic cells at the very initial stages of T cell development between thymus seeding by lymphoid progenitors and pre-TCR–driven selection of thymocytes committed into the abT cell lineage (also known as b-selection). In the mouse, FIGURE 1. Transition graph of the Markov process modeling TCRb these cell subsets, delineated via cell-surface expression of discrete gene recombination and feedback inhibition. The graph also depicts or- molecular markers, are commonly named DN1–3 cells, with DN2 dered rearrangement events at the TCRb locus (Db-to-Jb joining and DN3 displaying predominant DJb- and VDJb-rearranged occurring first, prior to Vb gene recombination); arrows represent products, respectively. Allelic exclusion/feedback inhibition is authorized transitions in TCRb gene rearrangement/cell genomic status. tightly coupled to b-selection prior to the DN3–DN4 stage transi- Transition rates (in red) are computed from those at a single allele (e.g., the tion, with cells not passing this checkpoint doomed to apoptosis or, → first GL DJ transition occurs at rate 2/tDJ as a Db-to-Jb rearrangement possibly, diverted toward a distinct gdT cell lineage (31). Hetero- may occur at each individual allele). Notice that, at the TCRb locus, there Downloaded from geneity in DN1–3 cell distribution, despite evidence of cycled is only one potential open reading frame through the various Jb gene segments and Cb exons and no stop codon within the Db gene segments seeding by T cell precursors (26), suggests an uncoordinated mode (Ref 33; S. Jaeger and P. Ferrier, unpublished observations). Accordingly, of activation and course of DN cell differentiation and/or recom- in a situation in which there is no evidence that DJb joints encode bination programs. To account for all these features in our model- functionally relevant, truncated TCR b-chains, it is thus the final recom- ing (notice that developmental stages are not explicitly represented bination outcome (the Vb-to-DJb joining event) that is assumed to impinge as separate variables in this study), we assumed that, for each and on in-frame/out-of-frame readability [i.e., at this later transition, the pos- every single cell of initial GL/GL status, all requirements for b http://www.jimmunol.org/ sible occurrence of a rearrangement-generated premature stop codon(s) sequential rearrangements (a nonexhaustive list including notably along the unique VbDJbCb open reading frame actually becomes relevant availability of discrete transcription/coactivation factors, recombi- in terms of productivity]. nase expression and activity, and changes in chromatin structure and in chromosomal organization/positioning; see Ref. 32) could be met during a time window of length T (without loss of general- Formulation of the model: basic features ity, we take T = 1). The window begins at variable time points for The model features a standardized compartmentalization of TCRb each individual cell depending on both intrinsic (e.g., gene/ gene recombination such that, at the single-cell level, each TCRb protein expression landscapes, all items in the list mentioned by guest on October 1, 2021 allele displays one of the following configurations: 1) GL; 2) above) and extrinsic (e.g., thymic environment) fluctuation fea- partially D-J rearranged (DJ); or 3) completely V-DJ rearranged, tures. Within the window, successive D–J and V–DJ rearrange- either productively (VDJ+) or not (VDJ2). The genomic status at ments proceed concurrently on opposite alleles such that the b this locus is defined by the configuration of the two TCRb alleles, status evolves according to the Markov process based on the tran- regardless of how the allelic pairs are ordered (e.g., DJ/VDJ+ and sition graph featured in Fig. 1. At individual alleles, the GL→DJ

Table I. Transition matrix Q =(qrs)r,s = {1,…,14} associated with the Markov model 0 1 3 2 000000000000 B tDJ C B 0 3 2 1 1 00 000 0 000C B 3tVDJ tDJ 3tVDJ C B 00 3 0001 0000000C B tDJ C B 00 0 3 004 2 000000C B 3tVDJ 3tVDJ C B 00003 1 0 1 000000C B tf tDJ C B 00000000000000C B C B 0000003 002 1 000C B 3tVDJ 3tVDJ C B 0000000 3 1 0 2 1 00C B tf 3tVDJ 3tVDJ C B C B 00000000000000C B C B 00000000000000C B 0000000 000 3 0 1 0 C B tf C B 3 1 C B 0000000 000 0 0 tf C @ 00000000000000A 00000000000000

The order of rows and columns follows the numbering in transition graph shown in Fig. 1. In each row, the symbol 3 14 denotes the opposite of the sum of row entries, namely the quantity 2+ qrs. s¼1 1626 DYNAMICAL MODELING OF TCRb ALLELIC EXCLUSION

2 and subsequent DJ→VDJ+ or DJ→VDJ transitions occur as in- Table II. Experimental numbers of cells harboring the sVDJ+/GL, + + 2 dependent Poisson processes, with rates equating to, respectively, sVDJ /DJ, or sVDJ /VDJ genomic status 1/tDJ, 1/3tVDJ, and 2/3 tVDJ, in which tDJ and tVDJ represent the T Cell Type (Data Source) N VDJþ =GL NsV DJþ =DJ NsV DJþ =V DJ2 average lengths of time of the corresponding transitions, and the V- s to-DJ transition alone bears susceptibility to the 1/3 rule of in- WT (22) 0–2a 118–120a 92b frame recombination, as it is thought to be the case at the TCRb WT (23) 0–2a 38–40a 36 LD/LD locus (1, 33) (Fig. 1; the figure shows the actual rates for TCRb Mutant b (23) 14 32 36 dual allele usage). Should a VDJ+ be completed and the resulting The distinct genomic statuses are indicated and were obtained from data in Khor and Sleckman (22) and Senoo et al. (23). pre-TCR produced and made operational, the given cell can then a + Numerical variability results from formatting the experimental data so as to fit switch to one final state in the model (symbolized sVDJ /GL, the genomic statuses in the model [especially the status of TCRb alleles carrying an 2 sVDJ+/DJ, sVDJ+/VDJ , or sVDJ+/VDJ+). The later transitions unrearranged Db1 gene segment is uncertain (i.e., could be GL or DJ depending on the configuration of the cis-linked Db2–Jb2 cluster)]. all arise at rate 1/tf, in which, similar to above, the parameter tf bIn this study, the productive versus nonproductive outcome in cells harboring + denotes the mean time interval between completion of a VDJ and two complete VDJ rearrangements was specified for only 44 out of 92 hybridomas, 2 effectiveness of pre-TCR–mediated allelic exclusion. As time runs and all 44 displayed a VDJ+/VDJ genotype. Accordingly, in the corresponding N þ ¼ N þ 2 þ fitting procedure, we assumed that 92 = sV DJ =V DJ s VDJ =V DJ out, the end of the recombination window eventually puts a stop to N þ þ s VDJ =V DJ . We note that, in all cases of fitted parameters, the model did predict any further change in the status of lingering cells [strictly speaking, at least 44 out of 92 sVDJ+/VDJ cells that harbor 1 VDJ+ only. the latter statement only pertains to the b genomic status; a more elaborate model would be needed to also integrate the outcome— death, commitment to a distinct lineage, etc.—of those cells that specifying the model so as to account for the given data set). Even did not reach a (sVDJ+/. . .) closing stage during their recombination though parameter solutions are depicted for a specific T cell dis- Downloaded from window; see Ref. 34]. Given all of the above statements, we relied tribution in the particular study (Fig. 2, see legend), fits using on standard results from the Markov process theory and on different distributions within the limits of those indicated in Table biologically relevant formulations to explicitly compute parameter- II led to similar solutions with identical properties (data not dependent estimations of TCRb+ cell distributions; notably, expres- shown). Notably, the solution curves obtained for WT mice in sions (2) and (4) proffered analytic predictions on statistical values of the two studies investigated in this paper displayed sound consis- developing T cells harboring the distinct b genomic statuses (for tency (Fig. 2, blue and red dots; slight deviation between the two http://www.jimmunol.org/ details, see Materials and Methods). curves likely reflects statistical variations inherent to relatively small-sized biological samples). Overall, these results thus imply Fitting statistics from b-selected T lymphocytes that, provided a proper calibration of the parameters has been To calibrate parameters in the model and, ultimately, assess the performed, a Markov process simulating the basic steps of TCRb statistical relevance of modeling predictions, we relied on available locus assembly in developing T cells reproduces in silico the data from experimental analyses of TCRb+ cell distributions and the application of standard fitting procedures. Briefly, we applied least-squares minimizing methods to fractions ascertained from by guest on October 1, 2021 published numbers of T cell-derived, cloned hybridomas well characterized in terms of TCRb genotype (for details on the biological approach, see, for example, Ref. 22). The mathematical procedures were implemented using Mathematica (Wolfram Re- search), function FindMinimum. The parameter triples (tDJ, tVDJ, tf) obtained in this way were further validated using the maximum likelihood method (see Materials and Methods, Least-squares fits and parameter estimation and Additional validation of parameter triples; all methods also described in Ref. 27). In practice, studies in the literature yielded quantitative information on cell populations displaying only three out of the four potential TCRb+ final states outlined above, namely sVDJ+/GL, sVDJ+/DJ, and sVDJ+/VDJ2 (22, 23) (Table II). Hence, resolving the three parameters from these fragmentary statistics was possible aside from one degree of freedom (see Materials and Methods, Continuum of fitting parameters). Consequently, for each data set in Table II, the fitting process, if doable, was predicted to yield a continuum of triple solutions situated on a curve in the parameter space. To draft the curves, we ran the minimizing procedures for 500 initial parameter triples, equidistributed along each parameter axis. The validated solutions (full range of numerical values) are displayed in Supplemental Table I and the definitive FIGURE 2. T lymphocyte data fitting. Plots of parameter triples curves shown in Fig. 2 (for a practical application of the global ap- obtained in fitting the experimental data sets summarized in Table II proach, also see Materials and Methods, Least-squares fitting pro- [three-dimensional drawing and two-dimensional projections (A–C) are shown]. Blue dots: fitting the distribution (NsV DJþ=GL;N VDJþ=DJ ; cedure and parameter estimation: worked example). Thus, fitting s N þ 2 þ N þ þ sV DJ =V DJ sVDJ =V DJ ) = (2, 118, 92) chosen from the experimen- the model to quantitative data from hybridomas generated in two tal data provided for WT T cells in Khor and Sleckman (22); red dots: studies using WT mice (22, 23) and, in one case, also mice car- fitting the distribution ðNsV DJþ=GL;NsV DJþ=DJ ;NsV DJþ=V DJ2Þ = (1, 39, rying a modified TCRb locus impinging on some aspects of gene 36) chosen from data provided for WT T cells in Senoo et al. (23); green recombination (23, 35) yielded parameter triples that aligned onto dots: fitting the distribution ðNsV DJþ=GL;NsV DJþ=DJ ;NsV DJþ=V DJ2Þ = (14, a smooth curve in all cases (Fig. 2; each individual triple 32, 36) provided for the bLD/LD mutant T cells in Senoo et al. (23). The Journal of Immunology 1627 statistical data on TCRb genotype/cell-subset distribution delin- the control of the initiation phase of allelic exclusion (allelic asyn- eated experimentally in independent analyzes of mature T lym- chronism) at the TCRb locus impinges on the Vb-to-DJb phocytes (as discussed in detail below). rearrangement step (1, 2, 20, 42). In addition, the longer GL→DJ and, conversely, slightly shorter DJ→VDJ transitions observed in Parameter estimates and model predictions parallel between bLD/LD mutant and WT T cells represent Examination of the solution curves in Fig. 2 afforded important provocative results as the bLD mutation: 1) specifically enforces information concerning parameter calibration and related pre- the sole usage of the D2–J2 cluster, the segment assembly of dictions relevant to the specific experiments. First and markedly, which is suspected to proceed less efficiently than that involving + + in those situations in which the number of VDJ /VDJ cells was the D1–J1 cluster; and 2) greatly reduces the numbers of 59 Vb ignored, the explicit solutions could in all cases be obtained for gene segments and the intervening distance from the D2–J2 cluster, → ‘ arbitrary values of tf (in fact, unlimited from 0 + ) with, con- possibly increasing their recombination rate (23, 39, 43). + versely, tDJ and tVDJ showing variations of small amplitude. In N xið Þ Finally, in using the numerical expressions 1 + (see Mate- other words, the parameter tf entirely epitomized the anticipated x degree of freedom. Clearly, this parameter (and corresponding rials and Methods, Simulation of TCRb+ T cell distributions and time delays) cannot be inferred from such data sets neglecting the analytic expressions; expressions (1) and (2) and parameter sol- VDJ+/VDJ+ cells. In this context, however, we noticed that the utions from Fig. 2 and Supplemental Table I, we made predictions + parameter tVDJ in general, as well as the parameter tDJ for the concerning the distribution of all TCRb cell populations and, more + + mutant mice (Fig. 2, green dots), displayed either substantial specifically, of the fraction of sVDJ /VDJ T cells not supplied + + 2 variations or saturation for, respectively, small (,0.1) and larger by the given data sets. Strikingly, all sVDJ /DJ, sVDJ /VDJ ,

+ + + Downloaded from (.0.2) tf values (Fig. 2, most evident in A and C, respectively), sVDJ /GL, and sVDJ /VDJ entities were correctly predicted by pointing to more robust estimates of recombination periods using the model, with the first two approximating the 60:40 ratio (pre- , , this later interval. An interval of 0.1 , tf , 0.2 would imply an cisely, for 0.1 tf 0.2, 56 and 39%, respectively; in agreement approximate 0.3–0.5 d (7–12 h) period to achieve pre-TCR syn- with exact measuring; e.g., see Ref. 2) and the latter ones restricted thesis and feedback inhibition assuming that the recombination to a minority (∼5%), as expected (Fig. 3A; similar distributions window is comprised within the 2.5–3 d needed by T cell precur- were obtained using parameters from WT data in Ref. 23; E. Farcot sors to transit through the DN2–DN3 compartments (36, 37). In- and B. Fernandez, unpublished observations). Specifically, the http://www.jimmunol.org/ + + deed, within the 0.1 , tf , 0.2 range, the curves exhibited limited fraction of sVDJ /VDJ cells turned out to be fairly low in all three variations enabling a precise determination of tDJ values [0.036 6 cases, varying monotonically with tf (Fig. 3B). Depending on the + 0.006 and 0.038 6 0.007 for the WT T cells in Khor and Sleck- data set, solutions ranged from 3.6–6.4% of total TCRb cells. man (22) and Senoo et al. (23), respectively; 0.205 6 0.020 for the These estimates are in accordance with assessments on TCRb mutant T cells in Senoo et al. (23)] and fairly consistent estimates allelically included cells based on genomic sequencing of a number of tVDJ values (WT T cells: 0.55 6 0.06 and 0.4 6 0.05; mutant of VbDJb joints (3–10%; even though, at the phenotypic level, dual T cells: 0.33 6 0.03); as already mentioned, slight variations TCRb+ cells represent a smaller percentage due to postrearrange- between numerical figures from WT data sets may reflect statis- ment events, such as competition between Vb gene promoters, tical fluctuations linked to limited-size sampling. Accordingly, in TCR b- and a-chain pairing, etc.) (24, 25, 44, 45). In addition, in by guest on October 1, 2021 LD/LD + WT T cells, tDJ would display smaller values (by one order of the case of the b mice, the predicted proportions of sVDJ / + magnitude) compared with those of tVDJ, implying, in this sce- VDJ cells are similar to those in WT animals and thus corroborate nario, a prompt completion of D–J assembly and a time window experimental observations arguing that the specific mutation does dominated by the V–DJ interval. These results are reminiscent of not impact on TCRb allelic exclusion (23). Overall, the ability of a handful of experimental data implying an increased intricacy of the model to integrate predominant and rare subpopulations and to Vb-to-DJb rearrangement as compared with that of Db-to-Jb, satisfactorily predict their statistical behaviors is testimony to the requiring at least: 1) the partitioning of distant genomic regions power of the approach. via developmentally regulated chromosomal looping (38); and, perhaps, 2) the breaking of Vb-privileged connection to nuclear Fitting statistics from preselected DN thymocytes repressive compartments (12, 38); 3) overriding the relatively in- Besides integrating statistical data on TCRb allelic status in effective Vb recombination signal sequences; and/or 4) supplant- differentiated T cells, potentially accessing such quantitative ing of inter-Vb antagonisms for productive coupling with a DJb figures in early developing DN thymocytes prior to b-selection unit (39–41). In this regard, the tVDJ .. tDJ picture accom- would be an additional advantage. So far, very few experimental modates the assumption generally endorsed in the literature that studies have addressed this issue. Notably, Aifantis et al. (24, 25)

FIGURE 3. Predictions on cell distributions. Pro- ductively rearranged cells related to the parameter tf were computed via the model in using the solution curves shown in Fig. 2. A, Estimations of the four cell populations of the form (sVDJ+/. . .) using parameter triples derived from data in Khor and Sleckman (22). B, Estimations of allelically included cells using parameter triples derived from data sets summarized in Table II. The color code is identical to the one used in Fig. 2. 1628 DYNAMICAL MODELING OF TCRb ALLELIC EXCLUSION

used single-cell PCR in comparative analyses of TCRb rear- X 2 + N 2N i rangements in purified (FACS-sorted) intracytoplasmic TCRb , Si i X preselected DN cells from WT mice versus mutant animals in which pre-TCR setting and/or signaling was perturbed. We thus to infer optimal parameter pairs tDJ and tVDJ.However,in used their results to further challenge our model by fitting the these cases, the model no longer accurately matched the reported reported numbers (Table III) with those computed in formal figures, with best-predicted scores ðNVDJþ=DJ ;NVDJþ=V DJ2 ; + 2/2 states (VDJ /. . .), starting with data from WT animals. Differing NVDJþ=V DJþ Þ = (17, 15, 4) [instead of (16, 13, 7); pTa from above, where the fits relied on cell fractions integrated T cells] and ðNVDJþ=DJ ;NVDJþ=V DJ2 ;NVDJþ=V DJþ Þ = (19, 21, over a stationary regime comprised of multiple cell-seeding 5) [instead of (18, 19, 8); SPL-762/2 T cells] (minimized cycles, we now had to consider cells in a transient state(s) of distances .0.25, of ∼29 in both cases). We can think of two limited length(s) and thereby statistical numbers cumulated over reasons to account for this unique setback. Firstly, we cannot shorter intervals. In this framework, we thus designed the least- formally eliminate the possibility that the model in its present squares parameter-fitting algorithm so as to minimize the corres- form does not fit data sets from pre-TCR–deficient T cells. For ponding quantity example, current predictions may underestimate the length of survival of pre-TCR–deficient cells. That such cells may behave N X 2 + i 2 i oddly is suggested by their unusual high level of expression + + i2S N X (compared with DN thymocytes in WT mice) of a surface [mathematical details in Materials and Methods, Simulation of marker (CD25) associated with DN differentiation (24). + However, we favor the alternative explanation that the model TCRb(b) T cell distributions and analytic expressions; ex- Downloaded from indeed matches the biological reality better than the given pressions (3) and (4)]. experimental data. As mentioned in the original articles, the Because experimental distributions were again related to only + + + biological numbers/cell distributions obtained experimentally three b genotypes (in this study including VDJ /VDJ , VDJ / 2/2 2/2 2 from the pTa and SPL-76 situations may be ques- VDJ , and VDJ+/DJ cells, but not VDJ+/GL cells), a continuum tionable, notably being prone to overestimate of NVDJþ=V DJþ , of triplet solutions was similarly predicted. In fact, the solution possibly linked to a sorting procedure-induced bias that would

curve for the WT cells now adopted a distinctive shape, with http://www.jimmunol.org/ result in preferential sampling of cells harboring a VDJ+ on the nonmonotonous variations of tVDJ and tf (the latter limited to 2 first rearranging allele (and so inevitably increasing the proportion a remarkably small amplitude ,5 3 10 3) and the degree of of VDJ+/VDJ+ cells; see discussion in Refs. 24, 25). In support freedom effecting t (Fig. 4, Supplemental Table II). Although DJ of the latter scenario, we found that when the VDJ+/VDJ+ frac- the curve provided evidence for arbitrary solutions of t (note DJ tions were ignored, each distribution of the remaining cells that the mean length of these rearrangements matters little when ðNVDJþ=DJ ;NVDJþ=V DJ2 Þ was then precisely predicted using VDJ+/GL cells are ignored), it also proffered optimal values for a continuum of parameter pairs (t , t ) (Supplemental Fig. t and t that were in good agreement with those reported DJ VDJ DJ VDJ 1). Moreover, for each continuum, the fitted parameters (t , t ) above using data sets from hybridomas [e.g., the triplet (t , t , DJ VDJ DJ VDJ could be chosen arbitrarily close to some of those determined t ) = (0.04, 0.52, 0.046) fitted the curve]. As for t , the latter value by guest on October 1, 2021 f f from WT data sets in Fig. 4. Finally, expected scores for VDJ+/ (0.046) denotes a mean delay for feedback inhibition shorter than VDJ+ cells were then lower (of three and five cells for the pTa2/2 that deduced earlier (0.1–0.2). However, the discrepancy may be and SPL-762/2 situations, respectively) but still of a larger moderated by the fact that this parameter appeared fairly sensitive fraction compared with that generally found in pre-TCR– to the fraction of the VDJ+/VDJ+ population. For instance, the proficient animals (i.e., again validating a disruption of allelic figures 0.04, 0.67, and 0.13 were found, respectively, for t , t , DJ VDJ exclusion). These findings corroborate the biological belief that and t in fitting a slightly different distribution ðNVDJþ=DJ ; f pTa or SLP-76 deletion prevents feedback signaling without af- NVDJþ=V DJ2 ;NVDJþ=V DJþ Þ = (34, 19, 2), workable within the fecting prior rearrangement features including the 1/3 in-frame data range (Table III) (24). versus 2/3 out-of-frame Vb recombination outcomes. Indeed, they Knockout deletion of pTa (the invariant protein that pairs with better fit the expectation of 20% (at most) of TCRb allelically the TCR b-chain to form the pre-TCR) or SLP-76 (a scaffolding included cells among those having completed rearrangements on factor that plays a critical role in pre-TCR signaling) were each both alleles in this situation (versus 35% as per the data in Refs. reported to result in the suppression of pre-TCR–mediated 24, 25) and could then emphasize the projecting power of this feedback inhibition, hence in a disruption of allelic exclusion 2 2 2 2 modeling method. (24, 25). Therefore, to fit data from pTa / and SPL-76 / T cells (Table III, second and third rows), we simply set up tf = Alternative model of feedback inhibition targeting the V-DJ +‘, which comes to eliminate the ultimate states (sVDJ+/. . .) in recombination step the model and used a least-squares algorithm with In the above-described first version of our model (henceforth called M1), we simulated the inhibitory feedback to impact both the Table III. Experimental numbers of cells harboring the VDJ+/DJ, VDJ+/ D-J and V-DJ rearrangement steps (Fig. 1). In this way, our esti- 2 VDJ ,orVDJ+/VDJ+ genomic status mates implied that the initial step may be substantially shorter than the average time delay for feedback inhibition (tDJ ∼ 0.04 T Cell Type (Data Source) NVDJþ =DJ NVDJþ =V DJ2 NVDJþ =V DJþ ,, tf ∼ 0.1–0.2). In practice, this would suggest that WT (24) 34 18–20a 1–3a most D-J joining has already been completed long before inhibi- Mutant pTa2/2 (24) 16 13 7 tion can take place, according to current understanding (42). How- 2 2 Mutant SLP-76 / (25) 18 18–19b 8–9b ever, to our knowledge, no study has yet been published in which The distinct genomic statuses are indicated and were obtained from data in the time course of Db–Jb recombination in the very early hours of Aifantis et al. (24, 25). TCRb gene assembly has been investigated. Moreover, in a gen- Productive versus nonproductive outcomes were not provided for, respectively, two cellsa and one cellb harboring two complete VDJ rearrangements (for details, see eral scheme whereby immature thymocytes employ distinct pre- Table 4 and Table I in Refs. 24 and 25, respectively). TCR downstream pathways to signal for allelic exclusion versus The Journal of Immunology 1629 Downloaded from

FIGURE 4. Thymocyte data fitting. Legend as in Fig. 2 [three- FIGURE 5. T cell data fitting. Plots of parameter triples obtained in dimensional drawing and two-dimensional projections (A–C) are shown], fitting the distributions previously used in Fig. 2 [three-dimensional draw- except that parameter triples were obtained in fitting the distribution ing and two-dimensional projections (A–C) are shown], applying the ðNVDJþ=GL;NsV DJþ=V DJ2 ;NsV DJþ=V DJþÞ = (34, 20, 1) chosen within the

model M2 (instead of M1; see text for details). http://www.jimmunol.org/ range of experimental data obtained for WT T cells in Aifantis et al. (24) (Table III). (sVDJ+/. . .) cells that were indistinguishable in this situation from cell expansion and differentiation (46, 47), feedback inhibition those already depicted in Fig. 3. However, one main difference may well depend on a series of non-mutually exclusive processes related to the mean length tDJ, which remained at a (roughly) con- impinging on the activity of the recombination apparatus and on stant value throughout the curve’s evolution and corresponded to a the epigenetic status of the Vb versus DJb chromosomal tem- significantly higher grade than previously observed (tDJ = 0.208 6 plates (38, 48–50). Additional molecular mechanism(s), most of 0.02 and tDJ = 0.2201, WT T cells in Refs. 22, 23, respectively; which remain incompletely elucidated, likely also act beyond tDJ = 0.5086, mutant T cells in Ref. 23). We conclude that the by guest on October 1, 2021 chromosomal accessibility to impair recombination between sites experimental data sets are faithfully reproduced by either the M1 eluding epigenetic silencing at the TCRb locus (51–53). In such or M2 version of the model with, in the latter case, an extended an intricate context, another relatively unexplored question duration of D-J rearrangement. Importantly, both versions can ac- remains of whether feedback signaling primarily (or initially) tar- commodate the minute fraction of VDJ+/GL cells displayed in gets the V-to-DJ rearrangement step. Our modeling framework Fig. 3A and basic experimental findings that expression of a rear- might help in appraising this issue by revising the behavior of ranged TCRb transgene, although inhibiting Vb–DJb recombina- sVDJ+/GL cells in the transition matrix; in practical terms, in tion, generally does not block Db-Jb rearrangement (1). Hence, + + permitting a sVDJ /GL→sVDJ /DJ switch at rate 1/tDJ, similar within the framework of our model, the hypothesis that allelic to the other final transitions during the recombination window exclusion might differentially impinge on the course of Vb-DJb (model M2; notice that this does not amend the clause that all versus Db–Jb rearrangements represents an authentic option, even types of recombination, including D-J, are arrested as the window if this has received little attention so far. ends). Using the same numerical procedure as used above (Materials Discussion and Methods, Simulation of TCRb+ T cell distributions and ana- We used Markov processes to model TCRb gene recombination, lytic expressions and Least-squares fits and parameter estimation, considering cis-rearrangement and trans-inhibition intervals at op- simulations of b-selected T lymphocytes, simulations of b-selected posite alleles in individual cells from a dynamical point of view, T lymphocytes), we found that such an M2 alternative also fitted with all relevant events obeying a probabilistic rule. From frac- data sets from Khor and Sleckman (22) and Senoo et al. (23), to the tional data, we proceeded by systematically determining parameter same level of accuracy as M1 did (see the solution curves in Fig. 5 values, firstly by analytical calculations and then by numerical and Supplemental Table III; importantly, because the M2 formula- optimization. Remarkably, we thereby reproduced experimental tion does not affect the VDJ+/DJ, VDJ+/VDJ2, and VDJ+/VDJ+ observations from different strains of mice and could proffer exact subpopulations, this new version of the model then obviously also predictions on the distribution of cells harboring the various TCRb fits the DN thymic cell data from Refs. 24, 25). Hence, M2 appears genomic statuses arising during T cell development. While keeping to represent a plausible hypothesis. Notably, the curves in Fig. 5 allele cross talk at a minimum, our model thus comprehensively exhibited all the main characteristics previously found for model satisfies the allelic exclusion landmark, suggesting that its basic M1 (with domain ranges of the parameter tVDJ almost unaffected; precepts truly capture the nature of the underlying controls at the [e.g., compare A in Figs. 2 and 5]), implying that similar conclu- TCRb locus. This work, together with previous studies in which sions could be drawn in the M2 situation (i.e., indetermination of tf; Markov modeling (using Monte Carlo simulations and empirical large versus small variations of tVDJ depending on tf intensities). predictions or, as in the current study, explicit calculations and This similarly applied to predictions on the final distribution of analytical predictions) was shown to fit the distribution of subsets 1630 DYNAMICAL MODELING OF TCRb ALLELIC EXCLUSION of TCR/BCR-rearranged T and B lymphocytes (21, 34, 54, 55), TCRb allelic expression and rearrangement, as both processes strengthens evidence for the effectiveness of stochastic-based for- may be perturbed by gene dosage and/or deficiency of transcrip- malisms at simulating developmental milestones in lymphoid cells. tion factors, such as the helix-loop-helix protein E47 (58). How- One hallmark of the modeling design used in this study is to ever, recent investigations questioning the validity of the GFP award total independence to the onset and course of GL→DJ and system in supplying evidence for probabilistic gene expression DJ→VDJ stepwise transitions at the two opposite TCRb alleles, at Igk alleles (59, 60) cast doubt on these numerical data. In fact, prior to the completion of feedback inhibition. Although an instant as with the Vb locus in pro-T cells (18), GL Igk transcription shift between two consecutive states is, in principle, conceivable seems to occur biallelically in single pre-B cells (59–61). Sepa- with the selected formalism, such an outcome rarely occurs. In- rately, using immunofluorescence in situ hybridization, TCRb stead, the transition typically occurs on a biological timeline in a alleles were found to associate at a high frequency and presumably stochastic manner in individual cells, and thus entails, on average, stochastically with repressive nuclear compartments (i.e., the peri- positive time slots tDJ or tVDJ. As the independence premise also centric heterochromatin and nuclear lamina), such that only 5% of entails positive time in events committed to autonomous Poisson nuclei showed two alleles free of both compartments against processes (30), allelic asynchrony generally ensues. In most cases, ∼60% still showing potentially repressive dual interactions (12). the time lag separating VDJ assembly at opposite sites will there- Although evidence has been provided suggesting that Vb-DJb fore be significantly larger than that of feedback inhibition, a pre- rearrangement occurs more effectively on nonassociated alleles, condition to allelic exclusion. Occasionally, however, the two values the analysis did not permit the determination of the extent to may be comparable, thereby permitting the emergence of a few which recombination may be inhibited by the interaction, how VDJ+/VDJ+ cells, an integral part of this developmental process. frequently associated alleles separate from the repressive A possible criticism to the present analysis is that the model is only compartments, or whether dually nonassociated alleles undergo Downloaded from as good as the data it uses for testing and refinement, and, arguably, concomitant rearrangement, all figures that would help to a weakness of the approach could be its reliance on data sets that, due validate a probabilistic factor in the initiation of TCRb allelic to the technical difficulties, do not have high enough numbers to exclusion. On the subject, and contrary to common notion, we make definitive statements. However, we note that, in fitting sta- emphasize in this study that a stochastic-based view of allelic tistics from the small data sets recorded in comparative analyses of exclusion does not necessarily hinge on poorly efficient, monoal- TCRb rearrangements in preselected DN thymocytes from WT lelic (and strictly sequential) recombination, as long as the inhib- http://www.jimmunol.org/ mice versus pTa2/2 or SPL-762/2 animals (55, 36, and 45 records, iting control becomes effective during the time lag otherwise respectively; Table III) (24, 25), we not only observed, for the WT separating VDJ assembly at opposite sites. situation, parameter values that were in general agreement with In their modeling of Vg4Jg1/Vg1Jg4 recombination, Sepulveda those assessed using larger data sets from hybridomas (e.g., 210 et al. (21) opted for a Markov chain whereby recombination rates, records; Table II) (22), but also, for the mutant situations, numerical feedback inhibition, and differential chromosomal accessibility all figures that, compared with the biological data, better fitted the contribute to isotypic and allelic exclusion at these loci. Regulated outcomes predicted for a disruption of allelic exclusion. modulations of the chromatin template are known to commonly Consistency and predictive ability are two criterions that back up orchestrate gene expression and recombination in developing by guest on October 1, 2021 the strength of our modeling approach. thymocytes (62). In this context, it is initially surprising that The property of equal autonomy allocated to each individual TCRb gene recombination can be modeled using only the first two allele differs from proposals intended to explain the control of criteria, with maximum fitting to the experimental data at hand. allelic exclusion (especially of its initiation phase) based on de- One obvious reason for this discrepancy may relate to the re- terministic models. These proposals argue that opposite alleles are spective numbers of genotypic/isotypic states under consideration nonequivalent substrates for the VDJ recombinase and that this (12 in the TCRg study—see Table IV in Ref. 21—versus at most regulated choice directs the order, on a strictly sequential mode, of three in the present study), as an additional parameter, delineating, subsequent allelic rearrangements in that particular cell (10, 38, 56) in the former study, the basic epigenetic changes that might sus- (even though reassessment of the data initially presented in Ref. tain a switch from a densely packed to decondensed chromatin at 56 led to a less definite view in this case; see addendum in Ref. TCRg1–g4 alleles, compulsory to VgJg recombination, clearly 57). However, our calculations showing that experimental cell aids the fitting procedures. In any case, this practical consideration distributions can be predicted by combining allele autonomy and does not preclude the potential contribution of an epigenetic- stochasticity in rearrangement and feedback inhibition events related process(es) in instigating, still on a stochastic basis, allele better accommodate scenarios whereby a global decrease in re- asynchrony in TCRb gene assembly. For example, the extended combination efficiency reduces the likelihood of the two alleles value of tVDJ may relate to an epigenetic-based recombination initiating rearrangement simultaneously (11, 12). Overall, they hindrance that would bestow on the two alleles differential capa- reinforce the plausibility that stochastic events cause asynchro- bilities for Vb-to-DJb assembly. In this context, it is notable that nous rearrangements, at least at the TCRb locus. Yet, the model the unrearranged Vb domains/alleles were found to be preserves deterministic facets (ordered rearrangements, feedback overmethylated and to frequently associate with repressive control) displaying an oriented and constrained evolution as stated compartments in DN nuclei, even though the Vb region is by the triangular form of the transition matrix. biallelically expressed at this stage (12, 18, 63). Of note, as long The molecular mechanism(s) that might be involved in adapting as the key issue rests on a mechanism capable of generating a time the dynamics of TCRb gene assembly and instigating allelic ex- lag in the assembly of homologous alleles, a two-step recombina- clusion remains largely unknown. Based on an experimental tion process should mechanically help to this outcome, not only in system consisting of the introduction of the GFP marker into the mathematical simulation but also in reality at the given locus. Igk locus, it has been argued that limiting transcription factors Then, one intriguing possibility would be that the mechanisms randomly and infrequently activates this locus (,5% of contributing to allelic asynchrony are not necessarily identical at pre-B cells apparently expressed the marker, a value claimed to distinct loci, with those assembled from V, D, and J gene segments provide a first statistical insight into this framework; see Ref. 11). (such as TCRb) relying less on epigenetic cues compared with Such an effect has also been implied as potentially impacting on those (such as TCRg) utilizing V and J gene segments only. The Journal of Immunology 1631

Parameter values deduced from data obtained from WT T cells in August 2007). We thank Dr. J. Carneiro (IGC, Lisbon, Portugal) and Dr. R. separate studies and further validated using both mature and de- Lima (CPT, Marseille, France) for helpful discussions and advice during veloping T cell subpopulations, implied that Vb–DJb recombination the early stages of this project. occurs on average over a longer time frame compared with Db–Jb rearrangement [0.5 versus 0.04 time window units (i.e., 30–36 h Disclosures versus 2.4–3 h according to an estimated 2.5–3-d period of DN2–DN3 The authors have no financial conflicts of interest. cell transit) or 0.5 versus 0.2 time window units in the M2 version (i.e., 30–36 h versus 12–14 h)]. As already mentioned, extended tVDJ References intervals may be justified by the existence of several structural and 1. Khor, B., and B. P. Sleckman. 2002. Allelic exclusion at the TCRbeta locus. biological features particular to the Vb gene segments/genomic Curr. Opin. Immunol. 14: 230–234. 2. Mostoslavsky, R., F. W. Alt, and K. Rajewsky. 2004. The lingering enigma of the regions, and relatively short tDJ timing (as is especially true for the allelic exclusion mechanism. Cell 118: 539–544. M1 option) is usually accepted. Concerning this shorter interval, the 3. Gorman, J. R., and F. W. Alt. 1998. Regulation of immunoglobulin light chain fact that our formalism allows any given cell to proceed with Db–Jb isotype expression. Adv. Immunol. 69: 113–181. assembly instantly (albeit rarely) following activation of the re- 4. Claverie, J. M., and R. Langman. 1984. Models for the rearrangements of immunoglobulin genes: a computer view. Trends Biochem. Sci. 9: 293–296. combination window suggests it merely spans the period required 5. Coleclough, C. 1983. Chance, necessity and antibody gene dynamics. Nature for these rearrangement events. Even so, when related to the esti- 303: 23–26. mated length of the window (a few days), these particular values 6. Coleclough, C., R. P. Perry, K. Karjalainen, and M. Weigert. 1981. Aberrant rearrangements contribute significantly to the allelic exclusion of immuno- may seem exceedingly long (a few hours) with regards to the re- globulin gene expression. Nature 290: 372–378. combination reaction itself. This being the case, the length of the 7. Malissen, M., J. Trucy, E. Jouvin-Marche, P. A. Cazenave, R. Scollay, and window might have been overestimated. Alternatively, the values B. Malissen. 1992. Regulation of TCR a and b gene allelic exclusion Downloaded from during T-cell development. Immunol. Today 13: 315–322. may cover additional events including the local epigenetic changes 8. Bergman, Y., and H. Cedar. 2004. A stepwise epigenetic process controls im- that in effect trigger this first rearrangement step (64, 65); prompt munoglobulin allelic exclusion. Nat. Rev. Immunol. 4: 753–761. 9. Mostoslavsky, R., N. Singh, A. Kirillov, R. Pelanda, H. Cedar, A. Chess, and Db–Jb transitions would have no biological reality (with little im- Y. Bergman. 1998. Kappa chain monoallelic demethylation and the establish- pact on our data because this option only bears on tiny cell per- ment of allelic exclusion. Genes Dev. 12: 1801–1811. centages). In the future, thorough examination of Db–Jb recom- 10. Mostoslavsky, R., N. Singh, T. Tenzen, M. Goldmit, C. Gabay, S. Elizur, P. Qi,

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