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arxiv: 2205.08750 · v2 · submitted 2022-05-18 · 🧬 q-bio.QM · cond-mat.stat-mech· q-bio.PE

Mathematical Characterization of Private and Public Immune Repertoire Sequences

Lucas B\"ottcher , Sascha Wald , Tom Chou This is my paper

Pith reviewed 2026-05-24 12:05 UTC · model grok-4.3

classification 🧬 q-bio.QM cond-mat.stat-mechq-bio.PE
keywords immune repertoiresclone richnessrepertoire overlappublic sequencesprivate sequencesprobabilistic modelTCR sequencessampling protocols
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The pith

A general probabilistic model for clone abundances yields exact formulas for the mean and variance of immune receptor richness and overlap across individuals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a probabilistic framework that defines public and private receptor sequences and supplies closed-form expressions for the expected number of distinct clones in a sample plus the expected number of sequences shared between individuals. These expressions also include the corresponding variances and hold for any abundance distribution that supports independent draws from different people. The formulas make it possible to predict how sampling depth influences observed richness and commonality without first specifying the detailed recombination process that generates the sequences. A sympathetic reader would use the results to interpret sequencing data from multiple donors and to design sampling protocols that reliably detect shared clones.

Core claim

Using a general probabilistic model for T/B cell receptor clone abundances to define publicness or privateness and information-theoretic measures for comparing the frequency of sampled sequences observed across different individuals, we derive mathematical formulae to quantify the mean and the variances of clone richness and overlap. Our results can be used to evaluate the effect of different sampling protocols on abundances of clones within an individual as well as the commonality of clones across individuals. Using synthetic and empirical TCR amino acid sequence data, we perform simulations to study expected clonal commonalities across multiple individuals and compare them with the analyt­

What carries the argument

A general probabilistic model for clone abundances that permits independent sampling across individuals, allowing overlap statistics to be derived without specifying the exact VDJ recombination mechanism.

If this is right

  • The formulas quantify how different sampling protocols change the observed abundances of clones within and across individuals.
  • Explicit closed-form expressions for richness and its uncertainty are available when clone abundances follow a single-parameter truncated power-law distribution.
  • The information loss incurred by grouping receptor sequences together, as in spectratyping, can be calculated directly from the model.
  • Simulations with synthetic and real TCR data confirm that the analytical predictions match observed clonal commonalities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separation of abundance statistics from the generation mechanism could let researchers test whether observed public clones exceed what is expected from sampling alone.
  • Variance formulas supply a direct route to statistical tests for whether sharing between patient groups differs from healthy controls.
  • The same expressions could be reused to compare repertoires collected before and after vaccination once an abundance model is fitted.

Load-bearing premise

Clone abundances follow a probabilistic distribution that permits independent sampling across individuals.

What would settle it

Fit an abundance distribution to TCR data from many individuals, then check whether the measured mean and variance of sequence overlap in a new cohort fall within the predicted ranges given the fitted distribution.

read the original abstract

Diverse T and B cell repertoires play an important role in mounting effective immune responses against a wide range of pathogens and malignant cells. The number of unique T and B cell clones is characterized by T and B cell receptors (TCRs and BCRs), respectively. Although receptor sequences are generated probabilistically by recombination processes, clinical studies found a high degree of sharing of TCRs and BCRs among different individuals. In this work, we use a general probabilistic model for T/B cell receptor clone abundances to define "publicness" or "privateness" and information-theoretic measures for comparing the frequency of sampled sequences observed across different individuals. We derive mathematical formulae to quantify the mean and the variances of clone richness and overlap. Our results can be used to evaluate the effect of different sampling protocols on abundances of clones within an individual as well as the commonality of clones across individuals. Using synthetic and empirical TCR amino acid sequence data, we perform simulations to study expected clonal commonalities across multiple individuals. Based on our formulae, we compare these simulated results with the analytically predicted mean and variances of the repertoire overlap. Complementing the results on simulated repertoires, we derive explicit expressions for the richness and its uncertainty for specific, single-parameter truncated power-law probability distributions. Finally, the information loss associated with grouping together certain receptor sequences, as is done in spectratyping, is also evaluated. Our approach can be, in principle, applied under more general and mechanistically realistic clone generation models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper introduces a general probabilistic model for T/B cell receptor clone abundances that treats sampling across individuals as independent. It derives closed-form expressions for the mean and variance of clone richness and overlap using linearity of expectation and indicator variables, validates the formulas via Monte Carlo simulations on both synthetic draws from the model and empirical TCR frequency data, specializes the results to single-parameter truncated power-law distributions, and quantifies information loss from sequence grouping as in spectratyping.

Significance. If the derivations hold, the work supplies analytical, parameter-light tools for quantifying public versus private repertoire sequences and the effects of sampling depth on observed richness and overlap. These expressions can directly inform experimental design in immunology without requiring a full mechanistic model of VDJ recombination, and the simulation checks provide a concrete falsifiability route for the formulas.

minor comments (2)
  1. [Model definition] The model definition paragraph states that sampling across individuals is independent, but the text does not explicitly list the precise independence assumptions used when deriving the overlap variance (e.g., whether clone abundances are drawn once per individual or re-sampled). Adding a short enumerated list of the independence statements would remove any ambiguity for readers applying the formulas.
  2. [Truncated power-law specialization] In the section presenting explicit expressions for the truncated power-law case, the normalization constant for the truncated distribution is left implicit; writing the closed form for the normalizing factor (even if standard) would make the richness and variance formulas fully self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation for minor revision. No specific major comments were provided in the report, so we have no individual points requiring point-by-point rebuttal or clarification. The manuscript stands as submitted, and we are prepared to address any minor editorial suggestions that may arise.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central derivations apply linearity of expectation and variance formulas for indicator variables to a stated general probabilistic model of clone abundances that treats sampling across individuals as independent; the resulting closed-form expressions for mean and variance of richness and overlap are compared to independent Monte Carlo simulations on synthetic draws and empirical frequencies, with no reduction of the reported quantities to fitted parameters or self-citation chains by construction. The model is defined explicitly without embedding the target statistics, and explicit expressions are also given for specific truncated power-law distributions without circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on a domain-level probabilistic model of clone abundances whose parameters are not enumerated in the abstract; no new physical entities are postulated.

free parameters (1)
  • parameter of the truncated power-law distribution
    Single free parameter used to obtain explicit expressions for richness and its uncertainty.
axioms (2)
  • domain assumption T/B cell receptor clone abundances follow a general probabilistic model that can be sampled across individuals
    Invoked to define publicness/privateness and to derive overlap statistics.
  • standard math Standard rules of probability for computing expectations and variances apply to the clone abundance distribution
    Required for all mean and variance derivations.

pith-pipeline@v0.9.0 · 5808 in / 1297 out tokens · 31062 ms · 2026-05-24T12:05:17.939444+00:00 · methodology

discussion (0)

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Reference graph

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