[Paper Review] Estimating intrinsic and extrinsic noise from single-cell gene expression measurements

Audrey Qiuyan Fu, Lior Pachter|arXiv (Cornell University)|Jan 13, 2016

Gene Regulatory Network Analysis3 references3 citations

TL;DR

This paper provides a rigorous statistical framework for estimating intrinsic and extrinsic noise in single-cell gene expression using two-reporter assays. By modeling gene expression through a hierarchical random-effects model and applying the law of total variance, the authors derive corrected estimators that minimize mean squared error, especially for small sample sizes, and justify quantile normalization and correlation-based extrinsic noise estimation.

ABSTRACT

Gene expression is stochastic and displays variation ("noise") both within and between cells. Intracellular (intrinsic) variance can be distinguished from extracellular (extrinsic) variance by applying the law of total variance to data from two-reporter assays that probe expression of identical gene pairs in single-cells. We examine established formulas for the estimation of intrinsic and extrinsic noise and provide interpretations of them in terms of a hierarchical model. This allows us to derive corrections that minimize the mean squared error, an objective that may be important when sample sizes are small. The statistical framework also highlights the need for quantile normalization, and provides justification for the use of the sample correlation between the two reporter expression levels to estimate the percent contribution of extrinsic noise to the total noise. Finally, we provide a geometric interpretation of these results that clarifies the current interpretation.

Motivation & Objective

• To address the statistical limitations of existing estimators for intrinsic and extrinsic noise in single-cell gene expression data.
• To improve estimation accuracy, particularly for small sample sizes, by minimizing mean squared error.
• To provide a formal hierarchical model that justifies the use of the law of total variance in noise decomposition.
• To clarify the role of quantile normalization and the use of sample correlation in estimating extrinsic noise contribution.
• To offer a geometric and probabilistic interpretation of noise decomposition formulas in two-reporter experiments.

Proposed method

• Formalizing the two-reporter assay as a hierarchical random-effects model where reporter expression is conditionally i.i.d. given cellular state.
• Applying the law of total variance to decompose total variance into intrinsic (within-cell) and extrinsic (between-cell) components.
• Deriving corrected estimators for intrinsic and extrinsic noise that minimize mean squared error under the hierarchical model.
• Using moment-based estimation to derive explicit formulas for noise components from observed expression pairs (c_i, y_i).
• Justifying the use of sample correlation between reporter pairs as a measure of extrinsic noise contribution.
• Providing geometric interpretations of the noise decomposition using vector space projections and covariance structure.

Experimental results

Research questions

• RQ1 How can intrinsic and extrinsic noise be more accurately estimated in single-cell gene expression data with limited sample sizes?
• RQ2 What is the statistical basis for the widely used ELSS formulas for noise decomposition?
• RQ3 How do the assumptions of the hierarchical model affect the bias and variance of noise estimators?
• RQ4 Why is quantile normalization necessary in two-reporters assays, and how does it improve noise estimation?
• RQ5 What is the justification for using the sample correlation between reporter expression levels to estimate extrinsic noise contribution?

Key findings

• The authors derive corrected estimators for intrinsic and extrinsic noise that minimize mean squared error, particularly beneficial for small sample sizes.
• The hierarchical model justifies the use of the law of total variance and provides a formal statistical basis for the ELSS formulas.
• Quantile normalization is shown to be essential for accurate noise estimation, as it corrects for systematic differences in reporter expression levels.
• The sample correlation between reporter pairs is mathematically justified as a valid estimator for the proportion of extrinsic noise in total variance.
• Geometric interpretations clarify the relationship between intrinsic noise, extrinsic noise, and total variance in the context of expression vector projections.
• Under normality and zero-mean assumptions, the variance of the intrinsic noise estimator is shown to be proportional to σ⁴, with explicit expressions derived for its mean squared error.