Ohno's "peril of hemizygosity" revisited: gene loss, dosage compensation, and mutation

Abstract

We explore the evolutionary origins of dosage compensation (DC) in sex chromosomes in the context of metabolic control theory. We consider first the cost of gene loss (hemizygosity) per se in reducing flux, and examine two relationships between flux and fitness (linear and Gaussian) to calculate a fitness cost of hemizygosity. Recognizing that new sex chromosomes are derived from autosomes, we also calculate the cost of unmasking deleterious mutations segregating on the nascent sex chromosomes as loci become hemizygous. The importance of deleterious mutations to the fitness cost of hemizygosity depends on their frequency, and on the relative costs of halving gene dose for wild-type alleles. We then consider the evolution of DC in response to gene loss, and include a cost of overexpression (i.e., DC such that expression exceeds the wild-type homozygote). Even with costs to excess flux, hypomorphic mutations can cause the optimal level of DC to be higher than 2-fold when the absolute cost of hemizygosity is small. Finally, we propose a three-step model of DC evolution: 1) once recombination ceases and the Y begins to deteriorate, genes from longer metabolic pathways should be lost first, as halving these genes does not drastically reduce flux or, thereby, fitness; 2) both the cost of hemizygosity and the presence of hypomorphic mutations will drive an increase in expression, that is, DC; 3) existing DC will now permit loss of genes in short pathways.

Fig. 1.—

The relationship between E_i for a focal enzyme in a pathway of length 1, 5, or 10 enzymatic steps, and the relative flux through the pathway. All E_i in the pathway, except the one associated with the focal enzyme, and C_x are set to 1. Relative flux is calculated by dividing absolute flux with one altered enzyme by the absolute flux with all enzymes identical, where flux is obtained from equation (2). To achieve a 50% reduction in flux in a pathway of 1, 5, or 20 enzymatic steps would require a reduction in E_i of 50%, 83%, or 95% in E_i, respectively. Modified from Kacser and Burns (1981).

Fig. 2.—

The relationship between flux and fitness under two models of fitness. Black curve is for linear fitness versus flux relationship (eq. 4), and bluish curves are for a Gaussian relationship (eq. 5), with σ = 0.3 (dark blue), 0.6 (intermediate blue), and 1.2 (light blue). (A) There is no cost (m = 0) to flux above the optimal level, which is set to 1. (B) The cost is maximal (m = 1), such that the fitness functions are symmetric around the optimal flux. (C) The cost is intermediate (m = 0.3).

Fig. 3.—

Fitness decrease due to hemizygosity as a function of the length of the pathway. Dotted lines, μ = 0.0001; solid lines, μ = 0 (no hypomorphic mutations segregating in the population). Note for shortest pathways, the effects of mutation are negligible and thus dotted and solid lines are indistinguishable. Line color indicates fitness–flux relationship: black for linear and blue for Gaussian (with σ = 0.6). (A and B) Mutant allele produces 2% (E_m = 0.02) as much enzyme as wild-type (per allele). (C and D) Mutant allele produces 10% (E_m = 0.1) as much enzyme as wild-type (per allele). (E and F) Mutant allele produces 50% (E_m = 0.5) as much enzyme as wild-type (per allele). (A, C, and E) Fitness reduction when the focal enzyme is in a short pathway (10 steps or less); (B, D, and E) for long pathways (15–25 steps). Note change in scale of y axis in two sets of panels. In long pathways, where the overall fitness decline is small, just a few percent in the examples shown, the effect of hypomorphic mutations is relatively large. For example in (B), well over half the reduction in fitness caused by hemizygosity can be attributed to hypomorphic mutations when the fitness–flux relationship is Gaussian (i.e., the dotted blue line, μ = 0.0001, is more than twice the value of the solid blue line, μ = 0). Allele frequency of deleterious mutation in males is equal to that expected under autosomal inheritance, and thus representing young sex chromosomes (see text for details).

Fig. 4.—

The effect of the rate of mutation on the fitness reduction due to hemizygosity with Gaussian fitness (σ = 0.6) in pathways of 15–25 steps. Hypomorphic mutation produces 2% as much gene product as wild-type, that is, E_m = 0.02. Solid line is no mutation (μ = 0); short dashes is μ = 10⁻⁶, medium dashes is μ = 10⁻⁵, and long dashes is μ = 10⁻⁴ mutations per allele per generation. As the mutation rate declines to zero, the effect of hypomorphic mutations becomes negligible.

Fig. 5.—

Fitness increase as a function of DC in a hemizygous male. Dotted lines are for short pathways (length 5) and solid are for long pathways (length 25). Line color indicates fitness–flux relationship: black for linear, and blue for Gaussian (with σ = 0.6). Mutant allele produces 2% as much enzyme as wild-type allele, that is, E_m = 0.02. (A–D) No cost to flux in excess of optimum (m = 0). (E–H) Cost to flux with m = 0.3 (fig. 2). (A and E) The gain in fitness for levels of DC (c) up to 110-fold. Other panels restrict the range of the x and/or y axes so that the shape of the curves can be seen. Points representing the level of DC that results in the highest possible fitness in males are shown by solid dots of matching color to the curves. In (A–D), these points occur at 100-fold compensation, which represents the level necessary to compensate for the hypomorphic mutation, if present. DC above this level gives the same fitness because there is no cost to overexpression. In curves (E–H), points are strict maxima, such that increased levels of DC reduce fitness.