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Estimating Clonality

Yi Liu, Andrew Z. Fire, Scott Boyd, Richard A. Olshen

arXiv:1408.1149v1stat.MEq-bio.PE

TL;DR

Estimating population structure matters across immune, microbial, and tumor research, and this paper focuses on clonality when clonal origins can be identified from sequences. It combines empirical Bayes estimation with jackknifing to estimate clonality from replicate data; simulations report a large reduction in mean squared error, while the authors conclude the method performs well across frequency distributions.

  • Problem

    The paper asks how to estimate clonality in mixed biological populations whose clonal origins can be identified pairwise, a problem relevant to immune, microbial, and tumor populations.

    Source: §1 · p. 2

  • Method

    The authors combine empirical Bayes estimation and jackknifing to estimate clonality, a quadratic functional of population frequencies, from replicate data.

    Source: §1 · p. 2, §7 · p. 16

  • Results

    More than 98% lower mean squared error for estimated clonality than another recent estimator was reported in simulations, and the approach performed well across frequency distributions.

    Source: §1 · p. 3, §7 · p. 16

  • Takeaways & Limitations

    A functional such as clonality can be estimated well without knowing the full vector of relative frequencies equally well.

    Source: §1 · p. 3

    The model assumes rearrangements absent from the circulation are measured without error, an assumption the authors say may not apply to some technologies or experiments.

    Source: §2.2 · p. 6

Abstract

from arXiv · show

Challenges of assessing complexity and clonality in populations of mixed species arise in diverse areas of modern biology, including estimating diversity and clonality in microbiome populations, measuring patterns of T and B cell clonality, and determining the underlying tumor cell population structure in cancer. Here we address the problem of quantifying populations, with our analysis directed toward systems for which previously defined algorithms allow the sequence-based identification of clonal subpopulations. Data come from replicate sequencing libraries generated from a sample, potentially with very different depths. While certain properties of the underlying clonal distribution (most notably the total number of clones) are difficult to estimate accurately from data representing a small fraction of the total population, the population-level "clonality" metric that is the sum of squared probabilities of the respective species can be calculated. (This is the sum of squared entries of a high-dimensional vector $p$ of relative frequencies.) The clonality score is the probability of a clonal relationship between two randomly chosen members of the population of interest. A principal takeaway message is that knowing a functional of $p$ well may not depend on knowing $p$ itself very well. Our work has led to software, which we call {\it lymphclon}; it has been deposited in the CRAN library.

1 Introduction

This section introduces clonality as the probability that two independently chosen rearrangements belong to the same clone. It proposes estimating this functional of population frequencies even when the frequencies themselves are not well known.

1 Introduction

  • Clonality is the Gini-Simpson index of relative frequencies and equals the probability that two independently chosen rearrangements belong to the same clone.The method uses empirical Bayes estimation and jackknifing, including for bias and variance estimation.Source: §1 · p. 2
  • Simulations report more than 98% lower mean squared error for estimated clonality than another recent estimator, perhaps the best then available in immunology.The cited comparison is with Parawesmaran et al. (2013).Source: §1 · p. 3
  • Because clonality is quadratic in relative frequencies, its mean squared error involves fourth moments, which the authors compute under an estimation model.A special case is estimating the squared length of a Euclidean multinomial vector from its frequencies.Source: §1 · p. 3
  • Clonality can be estimated well without knowing the full vector of species’ relative frequencies equally well.The problem arises in multiple contexts, including ecology.Source: §1 · p. 3
  • The paper uses B-cell receptor rearrangements as an example, with replicate PCR-amplified sequencing libraries sampling blood-derived B cells.The setup assumes a correspondence between clonal subpopulation frequencies and the representation of their characteristic sequences across replicate amplifications.Source: §1 · pp. 3–4

2 The Model, Assumptions and Implications

The model treats each replicate as a noisy estimate of a population frequency vector and defines clonality as the squared length of that vector. Inference depends on assumptions about replicate errors, and the straightforward average of replicate squared lengths is typically biased.

2.1 Basic assumptions

  • The population is represented by a frequency vector p, and clonality θ is its squared length; richness C is not studied explicitly.Replicates have frequency vectors pi with conditional expectation p, and each replicate observes fewer rearrangements than the full population.Source: §2.1 · p. 4
  • The direct average of replicate squared lengths is typically extremely biased for clonality because squaring is convex.Jensen’s inequality gives the stated bias.Source: §2.1 · p. 5

2.2 Assumptions regarding errors

  • Inference on θ relies on assumptions about replicate errors, including conditional independence across coordinates and replicates.The errors have conditional mean zero and additional moment constraints, but those constraints do not determine their exact distributions.Source: §2.2 · p. 5, §2.1 · p. 5
  • The error assumption implies that rearrangements absent from the population are measured without error, a condition the authors say may not apply to some technologies or experiments.The authors assign minimizing fake clones from sequencing errors to experimentalists and first-stage analysis.Source: §2.2 · p. 6

3 Covariances of Pairs of {θ(k,l)}

This section examines how pairwise replicate-overlap quantities covary under the model’s assumptions. Nonoverlapping replicate pairs have zero conditional covariance, while shared indices yield covariance terms that can be analyzed using moment calculations and concentration arguments.

3 Covariances of Pairs of {θ(k,l)}

  • Pairwise replicate-overlap quantities have zero conditional covariance when their two pairs contain four distinct replicate indices.The calculation then turns to pairs sharing one index.Source: §3 · p. 7
  • For pairs sharing indices, the conditional covariance is expressed through expectations of products of pairwise overlaps minus the product of their expectations.The section also considers the case where the two pairs are identical.Source: §3 · pp. 7, 9
  • Only two of the 16 expanded terms in one product-of-inner-products calculation are nonzero.This reduces the covariance calculation to a smaller set of terms.Source: §3 · p. 7
  • The section applies large-deviation results to averages of independent, identically distributed variables to bound deviations from their expectations.The constants in the bounds depend on the calculation of Kullback-Leibler information for the assumed distribution.Source: §3 · p. 8

4 Empirical Bayes Estimation and Jackknifing.

The authors develop an empirical Bayes and jackknife approach to combine clonality estimators using replicate-based covariance estimates. They use covariance regularization because the unregularized structure can have a poor condition number, while noting that some arguments remain heuristic.

4 Empirical Bayes Estimation and Jackknifing.

  • The proposed jackknife-based estimators are close to unbiased given p and {Ci} and have far smaller variance than estimator (6).The approach estimates a covariance matrix for fixed coordinates across replicates and combines five estimators, including one exactly unbiased estimator.Source: §4 · p. 10
  • The method constructs a C × n matrix X from the observed replicate vectors and jackknifes estimates of θ using covariance estimates for its rows.The jackknife is intended to eliminate the leading 1/n term in the bias of an estimator.Source: §4 · p. 10
  • The authors use hard thresholding, soft thresholding, and cor.shrink to regularize covariance estimates after identifying a poor condition number in the approximated covariance structure.The resulting methods are available through the R command corpcor.Source: §4 · p. 12

5 Combining Estimators

The section combines five clonality estimators into final estimates using their estimated covariance matrix. The set includes an estimator that is close to unbiased and another that is exactly unbiased.

5 Combining Estimators

  • The estimator from leaving estimator (13) unregularized is not exactly unbiased, although it is close; another estimator, θ̂∗, is exactly unbiased.The nearly unbiased estimator is one of the five combined estimators.Source: §5 · p. 13
  • The five estimators are combined using an estimated 5 × 5 covariance matrix, with jackknifing across replicates to estimate that matrix.The resulting combination yields the final estimates of clonality.Source: §5 · p. 13
  • The final estimator is returned by lymphclon’s “inter.clonality” function within the “mixture.clonality” component.Source: §5 · p. 14

6 Simulations

The authors use simulations to assess their complex final clonality estimators, whose development includes heuristic arguments. The reported simulation describes striking results, while its scope excludes read errors and PCR amplification bias.

6 Simulations

  • The simulation uses eight replicates, with six containing 1,000 cells and two containing 10,000; C is set to 200,000.The number of replicates is user-specified, and C can be changed in lymphclon.Source: §6 · p. 14
  • Replicate cell counts are generated from a truncated Zipf distribution, and cell assignments to rearrangements use a Pareto-Poisson model.For each replicate, a Pareto random value with location and shape parameters 1 sets the means of the Poisson assignment distributions.Source: §6 · pp. 14–15
  • Replicates have identical distributions unconditionally, but not conditional on replicate cell counts and rearrangement assignments.Source: §6 · p. 15
  • The authors compare their estimators with the algorithm implied by (6) alone and describe their configurations as an attempt to circumvent amplification preferences and PCR jackpots.The simulations do not model read errors or PCR amplification bias.Source: §6 · p. 15
  • The one reported simulation demonstrates “remarkable results” for the final clonality estimators.The authors say simulation is their route to evaluating estimators developed partly with heuristic arguments.Source: §6 · pp. 14–15

7 Summary

The paper combines new and existing approaches to estimate functionals of V(D)J rearrangement frequencies, focusing on clonality. It reports that performance holds across rearrangements with very different observed frequencies.

7 Summary

  • For quadratic functionals such as clonality, pairs of quadratic functions of the data can estimate accurately, with jackknifing addressing bias and empirical Bayes reducing variance.The empirical Bayes arguments enable estimation of key parameters.Source: §7 · p. 16
  • The goal is accurate estimation of functionals from V(D)J rearrangement frequency data using new approaches combined with existing methodologies.Source: §7 · p. 16
  • The approach performs well across V(D)J rearrangement frequencies, from rearrangements seen once within a replicate to those seen many times.Rare rearrangements vary by replicate, while some are observed repeatedly.Source: §7 · p. 16
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