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Predicting the spectrum of TCR repertoire sharing with a data-driven model of recombination

Yuval Elhanati, Zachary Sethna, Curtis G. Callan Jr., Thierry Mora, Aleksandra M. Walczak

arXiv:1803.01056v1q-bio.GN

TL;DR

The paper asks how recombination and selection shape TCR sharing across individuals. It models sharing with quantitative V(D)J recombination models and a thymic-selection factor, finding that predictions agree well with human and mouse data. It also finds that publicness depends on cohort and sampling context as well as sequence generation probability.

  • Problem

    Although public TCRs had been linked to generation biases, the relative contributions of convergent recombination, V(D)J bias, and thymic and later selection remained to be quantified.

    Source: §I · p. 1

  • Method

    The paper uses inferred stochastic V(D)J recombination models augmented by a simple thymic-selection model to predict sharing across individuals and sampling depths.

    Source: §I · p. 1, C. Using TCR recombination models to… · p. 3

  • Results

    The model’s predictions of TCR sharing show excellent quantitative agreement with data from human and mouse repertoire studies.

    Source: §I · p. 1

  • Takeaways & Limitations

    Publicness depends on cohort size and sequencing depth as well as generation probability, and the PUBLIC score uses generation probability to predict sharing status with very high accuracy.

    Source: §I · p. 2, §VI · p. 9

Abstract

from arXiv · show

Despite the extreme diversity of T cell repertoires, many identical T-cell receptor (TCR) sequences are found in a large number of individual mice and humans. These widely-shared sequences, often referred to as `public', have been suggested to be over-represented due to their potential immune functionality or their ease of generation by V(D)J recombination. Here we show that even for large cohorts the observed degree of sharing of TCR sequences between individuals is well predicted by a model accounting for by the known quantitative statistical biases in the generation process, together with a simple model of thymic selection. Whether a sequence is shared by many individuals is predicted to depend on the number of queried individuals and the sampling depth, as well as on the sequence itself, in agreement with the data. We introduce the degree of publicness conditional on the queried cohort size and the size of the sampled repertoires. Based on these observations we propose a public/private sequence classifier, `PUBLIC' (Public Universal Binary Likelihood Inference Classifier), based on the generation probability, which performs very well even for small cohort sizes.

I. INTRODUCTION

The paper examines why TCR sequences are shared across individuals and models sharing using recombination biases and convergent recombination. It treats publicness as a spectrum shaped by both sequence generation probability and cohort and sampling conditions.

I. INTRODUCTION

  • Sharing across individuals has been attributed to biased V(D)J generation and convergent recombination, which can make some TCR sequences easier to generate than others.The relative contributions of recombination, thymic selection, and later selection had remained to be quantified.Source: §I · p. 1
  • The degree of sharing depends on the sequence, cohort size, and repertoire sampling depth, so publicness is not a fixed public/private dichotomy.The paper proposes PUBLIC, a classifier based on the sequence’s recombination probability.Source: §I · p. 2

A. Spectrum of sharing numbers

  • A sequence’s sharing number is the number of samples containing its CDR3 amino-acid sequence, ranging from one sample to all sampled individuals.The resulting distribution counts sequences at each sharing number; defining sharing by full nucleotide sequence is possible but yields reduced statistics.Source: A. Spectrum of sharing numbers · p. 2
  • A simpler model based on fewer N insertions predicted pairwise sharing only within the correct order of magnitude and missed detailed recombination biases that influence the sharing spectrum.Uniform generation had underestimated observed human TCR sharing by several orders of magnitude.Source: A. Spectrum of sharing numbers · p. 2

B. TCR generation bias

  • Nonuniform gene usage, deletion and insertion patterns, and convergent recombination create unequal generation probabilities across TCR sequences.Different recombination events can yield the same nucleotide sequence, and different nucleotide sequences can yield the same amino-acid sequence.Source: B. TCR generation bias · p. 2

C. Using TCR recombination models to predict sharing

The recombination model with a simple thymic-selection factor closely predicts observed sharing in human and mouse cohorts. The results show both a broad spectrum of sharing and a long-tailed distribution in the large human cohort.

C. Using TCR recombination models to predict sharing

  • The analysis compared observed CDR3 amino-acid sharing in 14 mice and 658 human donors with simulations and analytical predictions from recombination models, incorporating a species-dependent thymic-selection probability.Human and mouse models were inferred from unproductive rearrangements; these were used to infer generation models, not in the sharing analysis.Source: C. Using TCR recombination models to… · p. 3

D. Model predicts many degrees of publicness in the data

  • q = 0.15 for mice and q = 0.037 for humans yielded excellent agreement between model predictions and observed sharing, with selection improving mouse agreement despite slight overestimation at high sharing numbers.The selection factor was estimated independently from the number of unique amino-acid CDR3 sequences relative to unique nucleotide sequences.Source: D. Model predicts many degrees of… · p. 3
  • More than 90% of TCRs in the 658-donor human dataset appeared in only one repertoire, while a very small fraction appeared in all donors; the model reproduced the long-tailed sharing distribution.The paper attributes this range of sharing behaviors to variation in TCR generation probabilities captured by the model.Source: D. Model predicts many degrees of… · pp. 3–4

III. FROM SAMPLES TO FULL REPERTOIRES

Observed sharing depends on both sampling depth and cohort size, so publicness is conditional on study design. Model-based extrapolations indicate that cumulative TCR diversity continues to grow far beyond sampled cohorts, with an estimated 10–20% of sequences truly public under the paper’s definition.

A. Sampling depth affects sharing

B. Cumulative diversity and extrapolation to full repertoires

  • Cumulative unique TCRβ sequences grow almost linearly with pooled samples, and model extrapolation predicts no saturation even beyond the number of humans who have ever lived.The extrapolation to very large cohorts uses mathematical expressions rather than simulating billions of repertoires.Source: B. Cumulative diversity and… · p. 4
  • With public TCRs defined by generation probability above 1/N, the predicted public fraction is 10–20% in both humans and mice, depending on repertoire-size estimates.For a sequence above that threshold, the paper estimates a 63% chance of occurrence in an individual’s repertoire; the species have different repertoire sizes and TCRβ diversities.Source: B. Cumulative diversity and… · pp. 4–5

IV. PREDICTING PUBLICNESS

The generation model predicts sharing across individuals, while generation probabilities vary widely across TCRβ CDR3 sequences. Sequences with greater sharing have higher generation probabilities, although the model predicts narrower distributions than observed and does not account for sequence-dependent selection.

A. Sharing and TCR generation probability

  • The sequence-generation model correctly predicts the amount of sharing across individuals.The reported generation-probability distributions of sequences shared across multiple individuals are narrower and shifted toward higher probabilities than the distribution across all sequences.Source: A. Sharing and TCR generation… · pp. 5–6
  • In pooled repertoires, each added individual contributes about 200,000 new CDR3 sequences, and the theoretical cumulative-diversity curve agrees well with the data.The observed growth rate showed little dependence on the order of adding individuals across 30 random orderings.Source: Figure 4 · p. 6
  • Generation probabilities span many orders of magnitude, and distributions shift toward higher probabilities as the minimum sharing number increases.The model captures this trend but predicts narrower distributions than the data.Source: A. Sharing and TCR generation… · p. 6, Figure 5 · p. 7
  • Computing an amino-acid CDR3 generation probability sums probabilities across nucleotide sequences and their possible recombination events, using dynamic programming to make the calculation tractable.The model refines the relationship between generation probability and sharing by accounting quantitatively for recombination biases and convergent recombination.Source: A. Sharing and TCR generation… · p. 6
  • The model’s distributional shift reflects selection not represented in its sharing calculations, which select TCRs randomly with probability q regardless of sequence.The model was inferred from nonproductive sequences; sequence-dependent selection factors could improve distributional predictions but may vary with individuals’ HLA types.Source: A. Sharing and TCR generation… · p. 6

B. PUBLIC: Classifier of public vs. private TCRs based on generation probability

PUBLIC uses generation probability to classify TCR sequences as public or private, with accuracy improving under more restrictive sharing definitions. The paper also compares bladder-cancer and healthy blood repertoires, finding no statistically significant excess sharing in the cancer cohort.

B. PUBLIC: Classifier of public vs. private TCRs based on generation probability

  • PUBLIC classifies a TCR as public when its generation probability exceeds a threshold θ, with publicness defined as presence in at least m of n repertoires.The classifier uses a recombination model trained without public/private labels; its generation-probability score is proposed as a proxy for publicness.Source: B. PUBLIC: Classifier of public vs.… · p. 7
  • PUBLIC performs well even at small sharing thresholds, and classification accuracy improves as the minimum sharing number m increases.Highly public human TCRs are predicted almost perfectly; the authors attribute lower mouse performance to the small cohort, which makes publicness less reliably defined.Source: B. PUBLIC: Classifier of public vs.… · p. 7
  • Higher AUROC with increasing m reflects an easier task: public sequences become rarer and more extreme in generation probability as m approaches cohort size n.AUROC is the probability that a randomly chosen public sequence is ranked above a randomly chosen private sequence.Source: B. PUBLIC: Classifier of public vs.… · p. 7

V. PUBLIC SPECIFIC RESPONSE

  • The sharing distribution was nearly identical in 30 bladder-cancer patients and 30 sample-size-matched healthy individuals, with no statistically significant over-representation of blood TCRs.The comparison uses healthy repertoires downsampled to match cancer-patient sample sizes.Source: §V · pp. 8–9, Figure 8 · p. 9
  • Larger bladder-cancer tumor cohorts are needed to distinguish explanations for the blood result, which the authors limit to bladder cancer.Other tumor types with greater infiltration into blood may be more likely to show detectable signatures there.Source: §V · p. 9

VI. DISCUSSION

TCR publicness spans a continuum and depends on both sequence generation probability and cohort sampling conditions. The authors find that generation probability supports accurate sharing predictions, while noting important limits and extensions of their model.

VI. DISCUSSION

  • Characterizing TCR sharing may help identify universal immune-response features relevant to vaccine design and candidate T-cell clones for immunotherapy.The authors identify these as potential uses of the knowledge, not demonstrated outcomes.Source: §VI · p. 9
  • Publicness depends on sequence generation probability, cohort size, sequencing depth, and the threshold used to define sharing; the authors also propose an absolute repertoire-wide definition.Under that definition, a TCR is public when its occurrence probability exceeds the inverse of the number of unique TCRs in the full repertoire.Source: §VI · p. 9
  • A wide spectrum of publicness follows the broad distribution of TCR generation probabilities, with high-probability sequences common and low-probability sequences making up most of the repertoire.The functional role of high-probability sequences remains inconclusive.Source: §VI · p. 9
  • Read abundance can be confounded by peripheral proliferation, PCR amplification noise, and expression variability, so the model uses unique sequences and ignores abundance.These effects are expected to dilute the correlation between abundance and publicness.Source: §VI · pp. 9–10
  • The unsupervised PUBLIC score predicts sharing with high accuracy using generation probabilities, without abundance or sharing-status information.The authors suggest that supervised use of sharing status and abundance could further refine the classifier.Source: §VI · p. 10
  • The sharing model needs an overall sequence-independent selection factor, while selection may also vary among individuals and include universal amino-acid features.The authors suggest that incorporating universal selection features could improve sharing predictions and the assessment of sequence publicness.Source: §VI · p. 10
  • The analysis focuses on TCRβ but can extend to other recombined chains; paired αβ sharing remains for future study because paired repertoires are currently too small or unavailable.The authors also suggest BCR analysis, while noting that somatic hypermutation is expected to reduce sharing and its predictability.Source: §VI · p. 10

VII. METHODS

The methods assign generation probabilities to TCR sequences using a probabilistic recombination model. The model is inferred from non-functional reads, and sequence probabilities are obtained by summing over compatible recombination scenarios.

VII. METHODS

  • The recombination model parameterizes V, D, and J gene choices, gene deletions, and nucleotide insertions.The probability of a recombination scenario is calculated from these event probabilities.Source: A. The probability of generating a TCR… · p. 10
  • A TCR sequence’s generation probability is the sum of the probabilities of all recombination scenarios that produce it.This applies to nucleotide or amino-acid sequences, including the full sequence or CDR3 alone.Source: A. The probability of generating a TCR… · p. 10
  • IGoR infers model parameters by maximizing the likelihood of non-functional recombination reads, and the study trains models on the corresponding human and mouse datasets.The reads are non-functional because they are out of frame or contain stop codons.Source: A. The probability of generating a TCR… · p. 10
  • An efficient dynamic-programming algorithm calculates CDR3 amino-acid generation probabilities without brute-force summation over all scenarios.Source: A. The probability of generating a TCR… · p. 10

B. Evaluating the number of shared sequences using simulations

The simulation procedure generates productive CDR3β sequences from an inferred recombination model and applies a random-fraction selection step. Mathematical expressions use generation probabilities and sample sizes to predict unique and shared sequence counts.

B. Evaluating the number of shared sequences using simulations

  • Simulated samples draw recombination events, filter for productive sequences, translate CDR3β, and retain a random fraction q to model thymic selection.The selection rule is consistent across simulated individuals, and samples can match real cohort sizes and sequencing depths.Source: B. Evaluating the number of shared… · p. 11

1. Predicting sharing numbers from the distribution of generation probabilities

  • For N independent draws from a model assigning sequence probabilities p(s), the expected number of observed unique sequences is M0.Source: §1 · p. 11
  • For n individuals with sample sizes {Ni}, the model gives the expected number of sequences found in exactly m individuals, using a Poisson approximation for small p(s).Source: §1 · p. 11
  • The expected sharing counts can be computed through a generating polynomial whose coefficients are the Mms, obtained by expansion and summation over sequences.The expression uses Jm, the set of all combinations of m individuals.Source: §1 · p. 11

2. Density of states approximation

Because summing over all possible sequences is intractable, the method approximates sequence sums with an integral over the density of states. It estimates that density from Monte Carlo samples of the generative model.

2. Density of states approximation

  • The exact sequence sum is replaced by an integral over energy E(s) = −ln p(s), with the density of states counting sequences in each energy interval.This transforms a sum of a function of sequence probability into an integral over energy.Source: §2 · p. 11
  • The density of states is estimated from Monte Carlo samples—10^7 for humans and 10^6 for mice—by histogramming E(s) and normalizing the resulting probability density P(E).Source: §2 · p. 11
  • The sharing-count equations are then rewritten as integrals using the estimated density of states.Source: §2 · p. 11

3. Sharing modified by selection

The model renormalizes generation probabilities over productive sequences that pass selection. This adjustment uses the functional-sequence fraction and the fraction of productive sequences that pass selection.

3. Sharing modified by selection

  • The sequence model is renormalized to the reduced space of productive, selected sequences rather than all possible recombination events.Many possible recombination events yield nonfunctional or negatively selected sequences.Source: §3 · pp. 11–12
  • The renormalization uses f, the functional-sequence fraction, and q, the fraction of productive sequences that pass selection.The model computes f from the generative model and infers q.Source: §3 · p. 12
  • Renormalizing pgen(s) to p(s) shifts the energies by ln f + ln q and reduces the density of states by a factor of f × q.ggen(E) is computed from pgen(s), while g(E) is derived from p(s).Source: §3 · p. 12

4. Inferring the selection factor q

The selection factor q is inferred by fitting model predictions to the observed M0(N) curve using least-squares regression.

4. Inferring the selection factor q

  • q is inferred by least-squares regression of model predictions for the M0(N) curve.The curve relates unique amino acid CDR3 sequences observed to productive, selected recombinations generated.Source: §4 · p. 12
  • M0(N) relates the number of unique amino acid CDR3 sequences observed to the number of productive, selected recombinations generated.Source: §4 · p. 12
  • To determine the curve from data, the number of productive selected recombinations must be determined for each sample.The passage notes that unique productive nucleotide reads are used, but its description of this step is incomplete.Source: §4 · p. 12

5. Analytic computation of public fraction of a repertoire

A sequence is operationally classified as public when its probability meets the 1/N threshold, and the public fraction is computed from the probability of finding sequences in a repertoire of size N.

5. Analytic computation of public fraction of a repertoire

  • A sequence s in a repertoire of size N is defined as public when p(s) ≥ 1/N.Source: §5 · p. 12
  • The calculation concerns the fraction of a repertoire comprised of sequences meeting the publicness criterion.Source: §5 · p. 12
  • The public fraction is computed by evaluating the probability that a sequence with probability e−E is found in a repertoire of size N.Source: §5 · p. 12
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