Together, we show selection on the protein level is sufficient for producing negative transcription-translation correlation and weak mRNA-protein correlation among species, helping explain patterns formerly seen in empirical studies. In addition, we show the same evolutionary processes generates among-species correlation and among-gene correlation that are drastically different, reconciling difference between correlations observed at these different scales. 8/n

When multiple genes with different optimal protein levels are considered, there's negative transcription-translation correlation and weak mRNA-protein correlation within-gene among species, but positive correlations among genes. 7/n

Similar correlations arise when the protein level is under directional selection: evolutionary changes in transcription and translation are generally concordant in direction but complementing each other in magnitude. 6/n

Using simulations, we demonstrate a negative correlation between the mRNA level and the per-transcript rate of translation when the protein level is under stabilizing selection, which also leads to weak mRNA-protein correlation. 5/n

The view of compensatory evolution is supported by some empirical studies but has remained mostly a post hoc explanation for observed patterns. Here we develop a theoretical framework to explicitly connect evolutionary mechanisms to patterns of mRNA-protein level coevolution. 4/n

One hypothesis is compensatory evolution of transcription and translation: evolutionary change in one of them get buffered by compensatory change in the other, such that the protein level is conserved despite variation in the mRNA level. 3/n