Heterosis describes the phenomenon whereby a hybrid population has higher fitness than an inbred population, and has previously been explained by either Mendelian dominance or overdominance, where it is generally assumed that one gene controls one trait. How-ever, recent studies have demonstrated that genes interact through a complex gene regulatory network (GRN). Furthermore, phenotypic variance due to noise is reportedly lower for heterozygotes, whereas the origin of such variance-related heterosis remains elusive. There-fore, a theoretical analysis linking heterosis to GRN evolution and stochastic gene expression dynamics is required. Here, we investigate heterosis related to fitness and phenotypic variance in a system with interacting genes, by numerically evolving diploid GRNs. According to the results, the heterozygote population exhibited higher fitness than the homozygote population, that is, fitness-related heterosis resulting from evolution. In addition, the heterozygote population expressed lower noise-related phenotypic variance in expression levels than the homozygous population, implying that the heterozygote population is more robust to noise. Furthermore, the distribution of the ratio of heterozygote phenotypic variance to homozygote phenotypic variance exhibited quantitative agreement with previous experimental results. By applying dominance and over-dominance to the gene expression pattern rather than only a single gene expression, we confirmed the correlation between heterosis and overdominance. We explain our results by proposing that the convex high-fitness region is evolutionarily shaped in the genetic space to gain noise robustness under genetic mixing through sexual reproduction.Significance StatementHeterosis, that is higher fitness in hybrid populations than inbred populations, is a long-standing problem in genetics, evolution, and breeding studies. Heterosis is not necessarily the result of a trait uniquely determined by a single gene, but likely involves stochasticity and interactions among genes. Through numerical evolution of the gene regulatory network, we demonstrate that heterosis is shaped by evolution to achieve robustness against noise and genetic mixing by sexual recombination in gene expression dynamics. That is, a mixed population is more robust to noise than inbred populations, as revealed experimentally by reduced phenotypic variance. The observed link between heterosis and phenotypic robustness, Mendelian dominance, and the convex single-humped fitness landscape represents a novel avenue in evolution and genetics research.

The fitness landscape is a critical concept in evolutionary biology and genetics that depicts fitness in the genotype space and visualizes the relationship between genotype and fitness. However, the fitness landscape is challenging to characterize because the quantitative relationships between genotype and phenotype and their association to fitness has not been comprehensively well described. To address this challenge, we adopted gene regulatory networks to determine gene expression dynamics. We analyzed how phenotype and fitness are shaped by the genotype in two-gene networks. A two-by-two matrix provided the two-gene regulatory network in which a vector with two angle values (Θ) was introduced to characterize the genotype. Mapping from this angle vector to phenotypes allowed for the classification of steady-state expression patterns of genes into seven types. We then studied all possible fitness functions given by the Boolean output from the on/off expression of the two genes. The possible fitness landscapes were obtained as a function of the genetic parameters Θ. Finally, the evolution of the population distribution under sexual reproduction was investigated in the obtained landscape. We found that the distribution was restricted to a convex region within the landscape, resulting in the branching of population distribution, including the speciation process.

Mendelian inheritance is a fundamental law of genetics. Considering two alleles in a diploid, a phenotype of a heterotype is dominated by a particular homotype according to the law of dominance. This picture is usually based on simple genotype-phenotype mapping in which one gene regulates one phenotype. However, in reality, some interactions between genes can result in deviation from Mendelian dominance.Here, by using the numerical evolution of diploid gene regulatory networks (GRNs), we discuss whether Mendelian dominance evolves beyond the classical case of one-to-one genotype-phenotype mapping. We examine whether complex genotype-phenotype mapping can achieve Mendelian dominance through the evolution of the GRN with interacting genes. Specifically, we extend the GRN model to a diploid case, in which two GRN matrices are added to give gene expression dynamics, and simulate evolution with meiosis and recombination. Our results reveal that Mendelian dominance evolves even under complex genotype-phenotype mapping. This dominance is achieved via a group of genotypes that differ from each other but have a common phenotype given by the expression of target genes. Calculating the degree of dominance shows that it increases through the evolution, correlating closely with the decrease in phenotypic fluctuations and the increase in robustness to initial noise. This evolution of Mendelian dominance is associated with phenotypic robustness against meiosis-induced genome mixing, whereas sexual recombination arising from the mixing of chromosomes from the parents further enhances dominance and robustness. Owing to this dominance, the robustness to genetic differences increases, while the optimal fitness is sustained up to a large difference between the two genomes. In summary, Mendelian dominance is achieved by groups of genotypes that are associated with the increase in phenotypic robustness to noise.Author summary Mendelian dominance is one of the most fundamental laws in genetics. When two conflicting characters occur in a single diploid, the dominant character is always chosen. Assuming that one gene makes one character, this law is simple to grasp. However, in reality, phenotypes are generated via interactions between several genes, which may alter Mendel’s dominance law. The evolution of robustness to noise and mutations has been investigated extensively using complex expression dynamics with gene regulatory networks. Here, we applied gene-expression dynamics with complex interactions to the case of a diploid and simulated the evolution of the gene regulatory network to generate the optimal phenotype given by a certain gene expression pattern. Interestingly, after evolution, Mendelian dominance is achieved via a group of genes. This group-based Mendelian dominance is shaped by phenotype insensitivity to genome mixing by meiosis and evolves concurrently with the robustness to noise. By focusing on the influence of phenotypic robustness, which has received considerable attention recently, our result provides a novel perspective as to why Mendel’s law of dominance is commonly observed.

Background Mendelian inheritance is a fundamental law of genetics. When we consider two genomes in a diploid cell, a heterozygote’s phenotype is dominated by a particular homozygote according to the law of dominance. Classical Mendelian dominance is concerned with which proteins are dominant, and is usually based on simple genotype–phenotype relationship in which one gene regulates one phenotype. However, in reality, some interactions between genes can exist, resulting in deviations from Mendelian dominance. Whether and how Mendelian dominance is generalized to the phenotypes of gene expression determined by gene regulatory networks (GRNs) remains elusive. Results Here, by using the numerical evolution of diploid GRNs, we discuss whether the dominance of phenotype evolves beyond the classical Mendelian case of one-to-one genotype–phenotype relationship. We examine whether complex genotype–phenotype relationship can achieve Mendelian dominance at the expression level by a pair of haplotypes through the evolution of the GRN with interacting genes. This dominance is defined via a pair of haplotypes that differ from each other but have a common phenotype given by the expression of target genes. We numerically evolve the GRN model for a diploid case, in which two GRN matrices are added to give gene expression dynamics and simulate evolution with meiosis and recombination. Our results reveal that group Mendelian dominance evolves even under complex genotype–phenotype relationship. Calculating the degree of dominance shows that it increases through the evolution, correlating closely with the decrease in phenotypic fluctuations and the increase in robustness to initial noise. We also demonstrate that the dominance of gene expression patterns evolves concurrently. This evolution of group Mendelian dominance and pattern dominance is associated with phenotypic robustness against meiosis-induced genome mixing, whereas sexual recombination arising from the mixing of genomes from the parents further enhances dominance and robustness. Due to this dominance, the robustness to genetic differences increases, while optimal fitness is sustained to a significant difference between the two genomes. Conclusion Group Mendelian dominance and gene-expression pattern dominance are achieved associated with the increase in phenotypic robustness to noise.