Toward a neutral evolutionary model of gene expression

Abstract

We introduce a stochastic model that describes neutral changes of gene expression over evolutionary time as a compound Poisson process where evolutionary events cause changes of expression level according to a given probability distribution. The model produces simple estimators for model parameters and allows discrimination between symmetric and asymmetric distributions of evolutionary expression changes along an evolutionary lineage. Furthermore, we introduce two measures, the skewness of expression difference distributions and relative difference of evolutionary branch lengths, which are used to quantify deviation from clock-like behavior of gene expression distances. Model-based analyses of gene expression profiles in primate liver and brain samples yield the following results: (1) The majority of gene expression changes are consistent with a neutral model of evolution; (2) along evolutionary lineages, upward changes in expression are less frequent but of greater average magnitude than downward changes; and (3) the skewness measure and the relative branch length difference confirm that an acceleration of gene expression evolution occurred on the human lineage in brain but not in liver. We discuss the latter result with respect to a neutral model of transcriptome evolution and show that a small number of genes expressed in brain can account for the observed data.

Figure 1.—

Predicted distributions of expression differences between samples 1 and 2 for two different mutational effect distributions X, given t₁ = t₂. The top row indicates predictions for a normally distributed effect model: Distributions of expression differences of all genes (Z_1,2), of sample 1-intermediate genes only Z_1,2¹, and of sample 2-intermediate genes only Z_1,2² are all symmetric (γ₁ = 0). The bottom row shows predictions for a positively skewed extreme value distribution effect model: Distribution of differences over all genes is symmetric; distributions of sample 1-intermediate and sample 2-intermediate genes are negatively (γ₁ < 0) and positively (γ₁ > 0) skewed, respectively, and are therefore indicative for an asymmetric effect model distribution.

Figure 2.—

Evolutionary tree of three taxa.

Figure 3.—

Performance of estimators for indicated parameters. Data were generated under the proposed model with an extreme value mutation effect distribution along a three-species tree (see Figure 2), using six parameter sets (cases a–e). The number of genes (in thousands) simulated per data set is given by the index of the cases. Dashed horizontal lines correspond to parameters used for simulation. Means (circles) and 95% probability intervals (solid vertical lines) generated for the estimators from 10,000 simulated data sets per case are shown.

Figure 4.—

Distribution of human-chimpanzee expression differences (on log scale) from liver95 (A), liver133 (B), brain95 (C), and brain133 (D).

Figure 5.—

Transcriptome distance measured as averaged pairwise variance of expression differences (y-axis) as a function of time since divergence in millions of years (x-axis) (Glazko and Nei 2003), for (A) liver95, (B) liver133, (C) brain95, and (D) brain133. Numeral code: 0, comparison within orangutan; 1, within chimpanzees; 2, within humans; 3, between human and chimpanzee; 4, between chimpanzee and orangutan; 5, between human and orangutan.

Figure 6.—

Illustration of a mixture of two models to explain the significant skewness in brain133 analysis. The two models differ only in the assumed evolution. Model A has branch lengths t_Human and t_C; in model B both branches are of equal length t_C. (A) Solutions of the mixture of two models (see appendix, Equation A1). The branch length t_Human of model A is shown as a function of the fraction q of genes that evolve according to model A. The dashed lines indicate parameters for the example depicted in B and C. (B) Contribution of 10% of genes evolving according to model A (solid line) and 90% of genes evolving according to model B (dashed line) to the human-chimpanzee difference distribution. (C) Histogram of human-chimpanzee expression differences in brain133 and fitted mixture distribution with the same variance and skewness as brain133 data.