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Models for transcript quantification from RNA-Seq

Lior Pachter

arXiv:1104.3889v2q-bio.GNstat.ME

TL;DR

RNA-Seq quantification seeks accurate relative transcript abundance estimates from sequenced reads, which are important for transcriptome analysis and downstream comparisons. This review synthesizes published stochastic models and inference approaches in a common framework, showing that many apparently different formulations produce identical estimates and that one general model captures many existing methods. It also examines model use in differential analysis and practical model-selection considerations.

  • Problem

    Accurately estimating relative transcript abundances from RNA-Seq reads is a central challenge because RNA-Seq supports transcript-level analyses and downstream comparisons.

  • Method

    The review organizes published stochastic RNA-Seq models and inference approaches within a common framework, including a general model that specializes to many existing methods.

  • Results

    Many existing models yield identical relative transcript abundance estimates despite differing formulations, and a single general model captures many of their elements.

  • Takeaways & Limitations

    Model relationships and equivalences clarify how published RNA-Seq quantification methods relate, while accurate abundance estimates support downstream differential analyses.

  • Takeaways & Limitations

    Read-mapping errors beyond a mapper's threshold create missing data that the presented error model does not address, potentially affecting expression estimates.

Abstract

from arXiv · show

RNA-Seq is rapidly becoming the standard technology for transcriptome analysis. Fundamental to many of the applications of RNA-Seq is the quantification problem, which is the accurate measurement of relative transcript abundances from the sequenced reads. We focus on this problem, and review many recently published models that are used to estimate the relative abundances. In addition to describing the models and the different approaches to inference, we also explain how methods are related to each other. A key result is that we show how inference with many of the models results in identical estimates of relative abundances, even though model formulations can be very different. In fact, we are able to show how a single general model captures many of the elements of previously published methods. We also review the applications of RNA-Seq models to differential analysis, and explain why accurate relative transcript abundance estimates are crucial for downstream analyses.

1. Introduction

RNA-Seq supports diverse transcriptome applications, but its central analytical challenge here is accurately quantifying relative transcript abundances. The review focuses on this problem and clarifies that RNA-Seq measures relative RNA transcript amounts rather than absolute abundance or gene expression itself.

  • RNA-Seq enables applications including genome annotation, fusion identification, novel isoform discovery, genome assembly, and single-cell analysis.
  • Accurate relative transcript abundance quantification is identified as a singularly important RNA-Seq analysis challenge.
  • RNA-Seq measures relative RNA transcript amounts, not technically gene expression or the abundance of translated protein products.
  • RNA-Seq cannot measure absolute transcript abundances because molecules are sampled proportionately.
  • The review focuses on relative transcript abundance quantification and organizes existing methods within a common framework.

2. The RNA-Seq model hierarchy

The review organizes RNA-Seq models into a hierarchy that represents their nesting, complexity, modeled features, and mathematical relationships. It also shows that count-based estimates arise from an underlying multinomial model, while mathematically equivalent models may still differ in implementation and performance.

  • Count based models: Count-based RNA-Seq methods use normalized mapped-read counts as proxies for relative transcript abundances.These normalized counts can be interpreted as maximum likelihood estimates under a multinomial model.
  • Model hierarchy: The model hierarchy represents published methods as a partially ordered set in which descending paths indicate special-case relationships.Model complexity corresponds to the number of nested feature ellipses.
  • Multi-reads (isoform resolution): Multi-read models address reads that may map ambiguously to transcripts, including isoform-resolution settings with multiple isoforms.
  • Mathematical equivalence: Models grouped in the same box are mathematically identical for quantification results, although this equivalence need not extend to differential analysis.
  • Implementation: Equivalent model formulations can be implemented by programs with drastically different performance and usability.Implementation details such as data types and protocol-tailored read processing affect quantification results.

3. Count based models

Count-based models estimate relative transcript abundances from uniquely mapped single-end reads under a generative sampling model. Their likelihood connects normalized read counts to maximum-likelihood abundance estimates, while extensions address multi-isoform ambiguity and normalization concerns.

  • Model setup: Count-based models assume single-end reads map uniquely to transcripts and are generated by uniformly selecting transcriptome positions.The simplest model contains no bias modeling.
  • Model setup: Relative transcript abundances ρ are normalized probabilities, and effective length is the number of valid read-start positions, ˜l_t = l_t − m + 1.The read set is partitioned by transcript, with X_t denoting the number of reads mapped to transcript t.
  • Inference: The count-based likelihood can be reparameterized as a log-linear model, with transcript-selection probabilities α transformable to relative abundances ρ.The paper states that the maximum-likelihood solution for α can be transformed into the unique maximum-likelihood solution for ρ.
  • Inference: RPKM is proportional to the maximum-likelihood estimate of relative abundance, and its derivation supports using effective rather than transcript length.RPKM values measure relative, not absolute, abundance.
  • Normalization: Additional normalization can improve robustness because extensive transcription of one gene can strongly affect RPKM values.Quantile normalization was proposed as one such additional step.
  • Extensions and limitations: Single-isoform models are limited for alternatively spliced genes, motivating projective normalization or uniquely mappable-read approaches for multi-isoform genes.Projective normalization can underestimate relative gene abundances, whereas maximum-likelihood transcript estimation has empirical support for improved accuracy; uniquely mappable-read restriction can omit valuable data.

4. Models for multi-reads: estimating isoform abundances

This section examines models for estimating relative transcript abundances when reads map ambiguously to multiple isoforms. It relates multinomial and Poisson formulations, showing that their maximum-likelihood estimates can coincide while length normalization remains an important modeling choice.

  • Multi-read models: Ambiguously mapped reads require extending transcript-abundance models beyond uniquely mappable reads.Multi-reads arise from overlapping transcript regions and duplicated genomic segments in gene families.
  • Model representation: The reviewed models use compatibility matrices to represent which transcripts can explain each read or fragment.For the Poisson formulation, compatibility is defined by whether a transcript appears in an equivalence class of compatible positions.
  • Length normalization: The omission of isoform-length normalization is harmless when all isoform lengths are equal but is an error for RNA-Seq models when lengths differ.Equal lengths change the likelihood only by a scalar factor, leaving maximum-likelihood estimates unchanged.
  • Parameter interpretation: In the Poisson model, transcript-specific parameters represent expected read counts, while normalized parameters represent probabilities for choosing reads from transcripts.The Poisson parameters κk are expected read counts, and the β parameters have the same interpretation as the multinomial α parameters.
  • Model equivalence: Poisson and multinomial likelihood formulations can have different generative interpretations yet yield exactly the same relative transcript abundance estimates.The equivalence follows by reparameterizing the Poisson model and maximizing its unconstrained normalization constant at Ẑ = 1.

5. A general model for RNA-Seq

The review presents a general RNA-Seq generative model that incorporates transcript abundance, fragment-length distributions, positional and site-specific biases, and read errors. This model contains previously discussed models as special cases, supports likelihood-based inference, and has an equivalent Poisson formulation.

  • Model specification: The model assumes each fragment has a well-defined transcript-relative length based on a unique 5′ alignment for a given 3′ alignment.The assumption simplifies notation but is described as slightly restrictive.
  • Generative process: The generative process selects a transcript and 3′ site, then models fragment length, sequence context, positional effects, and errors.The conditional fragment probabilities and likelihood are derived from these components.
  • Model specification: The general model includes transcript abundances, fragment-length distributions, site-specific and positional biases, and read-error probabilities.Its parameters specify how fragments are generated and how observed sequences relate to transcripts.
  • Inference: For fixed bias and error parameters, the model is a concave linear model whose maximum-likelihood parameters can be translated into relative abundances.Reported abundance estimates are commonly expressed in FPKM.
  • Caveats: Different paired-end generative formulations alter the likelihood denominator slightly, and the best formulation for abundance estimation remains unclear.The alternatives differ in whether the 3′ site, 5′ site, or fragment length is selected first.
  • Caveats: Read errors can interact with mapping thresholds, creating missing data that may affect expression estimates, especially for allele-specific estimation.The review notes that fully general compatibility matrices can be impractical and recommends correcting errors before mapping.
  • Model relationships: The general multinomial model can also be formulated as an equivalent Poisson model.The review states that the equivalence is proved in an appendix.

6. Inference

Inference methods estimate relative transcript abundances from models whose mathematical properties vary with read-mapping ambiguity. Closed-form solutions are available for simple log-linear models, whereas ambiguous mappings generally require numerical optimization such as EM.

  • Inference properties: Simple single-read, single-isoform models are log-linear and admit closed-form abundance estimates.Ambiguous read mappings remove log-linearity but can preserve concavity under known bias effects.
  • Inference properties: Ambiguous-mapping models are concave when bias effects are known, allowing numerical algorithms to find a unique global maximum when the model is identifiable.Identifiability is tested through the rank of the compatibility matrix.
  • Example: The three-isoform example is identifiable because its compatibility matrix has rank 3.The maximum-likelihood problem is constrained by abundances summing to one.
  • EM algorithm: The EM algorithm alternates between proportionally assigning compatible reads and recalculating abundances from expected counts.Its likelihood increases at every iteration.
  • Related approaches: The multi-read rescue method is equivalent to one EM iteration.This connects a published heuristic to the general iterative optimization procedure.
  • Alternative inference: Least-squares estimates correspond to maximum likelihood under normal-count assumptions, with variance structures varying across methods.Multivariate normal approximations can be suitable when counts are high.
  • Scope of comparison: A published L1-minimization method cannot be directly compared with other methods because its details are insufficiently explained.The review therefore omits it from the model-relationship figure.

7. Differential analysis

The review distinguishes models that are equivalent for quantification from their behavior in differential analysis. Multinomial and Poisson models remain equivalent, whereas observed replicate counts often require alternatives that account for dispersion.

  • Accurate relative transcript abundance estimates support power to detect differentially transcribed RNAs.
  • Observed technical and biological replicate counts often violate multinomial behavior through over- or under-dispersion.
  • Multinomial and Poisson models are equivalent for both quantification and differential analysis.
  • Negative binomial models match multinomial quantification but produce more conservative differential-analysis results.
  • Optimized differential analysis also depends on appropriate experimental design alongside model selection and robust parameter estimation.

8. Discussion and future directions

The discussion emphasizes that RNA-Seq modeling remains important as technologies and applications evolve. Future priorities include choosing models according to read characteristics, modeling bias, validating accuracy, and accounting for transcriptome assembly.

  • RNA-Seq modeling must continue improving to make best use of current and foreseeable sequencing technologies.
  • Multi-read models are essential for ambiguous mappings, especially with short reads, while longer reads make effective-length corrections important.
  • Using transcript length instead of effective length can change relative abundance estimates by up to 30% for 200-base average fragments.
  • Observed fragment-location bias makes general bias modeling desirable, with recent results showing better agreement with qRT-PCR estimates when bias is modeled.
  • The exact accuracy of reviewed abundance estimates remains unknown because systematic benchmarks with known abundances are still needed.
  • Incomplete transcriptomes can bias abundance estimates, making transcriptome assembly important for accurate quantification.

Appendix I: Equivalence between the paired-end Poisson and multinomial models for RNA-Seq

This appendix establishes that the general paired-end RNA-Seq model is equivalent to a Poisson model with the same bias model. It defines the combinatorial notation used to derive the likelihood and shows that the resulting likelihoods share the same maximizer.

  • The general model in Section 5 is equivalent to a paired-end Poisson model when both use the same bias model.
  • A weak composition is an ordered tuple of non-negative integers whose parts sum to a specified total.
  • A fragment-assignment function induces a weak composition by counting assignments to each transcript-position element.
  • In the example, four fragments assigned among three positions produce composition (2, 0, 2), with 3^4 possible assignment functions.
  • Factoring the polynomial reduces evaluation from mn monomials with n terms each to O(nm) operations.
  • The paired-end likelihood generalizes an earlier likelihood directly, and its likelihood expression matches the earlier form after replacing α_t with β_t.
  • The two likelihood functions are maximized at the same estimates, ˆα = ˆβ.

Appendix II: Notation

The notation appendix organizes the paper’s symbols by their relationship to experiments and model calculations. It separates transcript structures, experimental fragments and alignments, model parameters, and derived helper variables.

  • The notation table divides variables into transcript structures, experimental fragments and alignments, model parameters, and helper variables.
  • The appendix includes Table 1, titled “Notation.”
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