Proteins rely on associations to improve packing quality and thus maintain structural integrity. This makes packing deficiency a likely determinant of dosage sensitivity, that is, of the fitness impact of concentration imbalances relative to the stoichiometry of the protein complexes. This hypothesis was validated by examining evolution-related dosage imbalances: Duplicates of genes encoding for deficiently packed proteins are less likely to be retained than genes coding for well-packed proteins. This selection pressure is apparent in unicellular organisms, but is mitigated in higher eukaryotes. In human, this effect reveals a capacitance toward dosage imbalance. This capacitance is not expected in organisms with larger population size, where evolutionary forces are more efficient at promoting adaptive functional innovation and purifying selection, thus curbing the concentration imbalance arising from gene duplication. By examining miRNA target dissimilarities within human gene families, we show that the capacitance is operative at a post-transcriptional regulatory level: The higher the packing deficiency of a protein, the more likely that its paralogs will be dissimilarly targeted by miRNA to mitigate dosage imbalance. For families with low capacitance, paralog sequence divergence and family size correlate tightly with packing deficiency, just like in unicellular eukaryotes. Thus, a major component of human tolerance toward dosage imbalances is rooted in the paralog-discriminating capacity of miRNA regulation. The results may clarify the evolutionary etiology of aggregation-related diseases, since aggregation is often promoted by overexpression (a dosage imbalance) and aggregation propensity is associated with extreme packing deficiency.