The inherent stochasticity generates substantial gene expression variation among isogenic cells under identical conditions, which is frequently referred to as gene expression noise or cell-to-cell expression variability. Similar to (average) expression level, expression noise is also subject to natural selection. Yet it has been observed that noise is negatively correlated with expression level, which manifests as a potential constraint for simultaneous optimization of both. Here, we studied expression noise in human embryonic cells with computational analysis on single-cell RNA-seq data and in yeast with flow cytometry experiments. We showed that this coupling is overcome, to a certain degree, by a histone modification strategy in multiple embryonic developmental stages in human, as well as in yeast. Importantly, this epigenetic strategy could fit into a burst-like gene expression model: promoter-localized histone modifications (such as H3K4 methylation) are associated with both burst size and burst frequency, which together influence expression level, while gene-body-localized ones (such as H3K79 methylation) are more associated with burst frequency, which influences both expression level and noise. We further knocked out the only "writer" of H3K79 methylation in yeast, and observed that expression noise is indeed increased. Consistently, dosage sensitive genes, such as genes in the Wnt signaling pathway, tend to be marked with gene-body-localized histone modifications, while stress responding genes, such as genes regulating autophagy, tend to be marked with promoter-localized ones. Our findings elucidate that the "division of labor" among histone modifications facilitates the independent regulation of expression level and noise, extend the "histone code" hypothesis to include expression noise, and shed light on the optimization of transcriptome in evolution.

Nonconvex nonsmooth regularization method has been shown to be effective for restoring images with neat edges. Fast alternating minimization schemes have also been proposed and developed to solve the nonconvex nonsmooth minimization problem. The main contribution of this paper is to show the convergence of these alternating minimization schemes, based on the Kurdyka-Łojasiewicz property. In particular, we show that the iterates generated by the alternating minimization scheme, converges to a critical point of this nonconvex nonsmooth objective function. We also extend the analysis to nonconvex nonsmooth regularization model with box constraints, and obtain similar convergence results of the related minimization algorithm. Numerical examples are given to illustrate our convergence analysis.