Stochasticity or randomness is ubiquitous in biological systems and contributes to a variety of phenotypic diversity [1] . Intrinsic molecular stochastic phenomenon (“gene expression noise” and its random fluctuation) can be beneficial to an isogenic bacterial colony which has genetic factors for the wide-range phenotypic noise covering an ability to survive [1] - [4] . In diploid eukaryotic organisms, epigenetically-driven “monoallelic expression” including autosomal random allelic exclusion is closely associated with organ functions and dysfunctions [5] - [7] . The stochasticity originates in circuits of interacting genes and proteins or feedback loops [8] [9] and the need for an enormous assortment of physiological responses in complexity and specificity depends on stochasticity- based phenotypic diversity [9] . Therefore, many complex biological functions in higher eukaryotes may derive from the stochastic interactions [9] . There may be no human trait being outside of complex diversity. Even the degree of phenotypic penetrance of a major variant gene effect, the liability to dichotomous diagnosis, the latent period duration of later-onset diseases, clinical response to treatment, and both resistance and vulnerability to the pathogens, toxins, or mental stress can be illustrated by quantitative complexity [10] . The stochastic epistasis perspective was introduced to explain such human complex conditions [10] [11] , and it is characterized by the phenotypic concordance in monozygotic twins in contrast to the discordance of “gene expression noise” and “random monoallelic expression” (Table 1). Although involvement of environmental factors and the epigenetic factors is sometimes emphasized to explain the presence of discordant identical twin cases in some complex conditions [12] , the stochastic epistasis can have a great role in complex conditions which have a disparity between monozygotic and dizygotic concordances.

The frontiers of research in human complex traits and conditions still survive in the black box between genotype and phenotype. Because co-segregation between phenotypic characteristics and the related genetic variants is a prerequisite for speciation and evolution, dissection of the processes that maintain genetic variation persists as a major challenge in evolutionary biology [13] [14] . The source of replenishment for genetic individual variation is undoubtedly an ongoing load of rare genetic variants [14] , and importantly the fitness significance of a variant can be varied among species from beneficial to deleterious [15] . The difficulty in explaining the persistence of genetic variation also manifests itself in a mating arena (lek). Even though the female choice should result in genetic benefits to offspring, the powerful directional selection cannot immediately drive the beneficial alleles to fixation in the population [16] - [18] . In human behavioral and cognitive liability to complex conditions which have sizable heritable component, the prevalence rate of the extreme cases with obvious reproductive demerits strangely seldom decline [19] - [21] . Such conditions including autism and schizophrenia are characterized by a high concordance in monozygotic twins and a low concordance in dizygotic twins [22] [23] . Hundreds of candidate variants associated with complex conditions have recently been found, and most of these variants explain only a modest amount of the observed heritability (“missing heritability”) [24] - [26] . The reported gene variants including de novo mutations are detected in a minor part of cases and sometimes show incomplete penetrance in family members and low result reproduction rate in different samples [11] . In addition, the modest genetic overlap among sub-domains in a condition [27] [28] may be indistinguishable from that among multiple other complex conditions [29] [30] . Although the genetic contribution to a quantitative complex trait can theoretically be attributable to the cumulative effect of a set of quantitative trait loci (QTLs) [31] , the delay and

^aThe discordance in genetically identical pairs had been exemplified by human fingerprints for the gene expression noise [4] and demonstrated in the mosaic-like coat-coloration pattern of a kitten clone for the random monoallelic expression [5] .

difficulty in detecting the causal variant alleles at QTLs is a common problem for complex traits [32] -[34] . The involvement of stochastic epistasis in the genetic architecture of complex traits is an attractive candidate of solution for these outstanding paradoxes in contrast to the insufficient interpretability by traditional perspectives in which the presence of extreme-specific genetic factors is hypothetically underlined [35] [36] . In some of these paradoxes, the stochastic epistasis model provides the sole explanation.

Epistasis can be explained as the relationship between a modifier variant and the genetic background in which the variant occurs [37] . The genetic background itself consists of other modifier variants and each modifier is a constituent of a gene network module [38] [39] . In experimental investigations to show the interactions between epistasis-related loci, a remarkable function (an epistatic switch) had been demonstrated at some loci in plant strains [40] - [42] . Although the locus sometimes has little or no effect on trait variation, the phenotypic diversity determined by epistatic modifiers is conditional upon the specific switch allele at the locus (Figure 1). Because the presence of epistatic switch is a prerequisite for the manifestation of epistatic interactions, the switch locus may be closely associated with the outcome functions and at least one switch should be involved in each epistatic module [10] . The presence of switch alleles which are necessary for the emergence of environmental fitness effects are also known in association with local adaptation [13] .

Genetic individuality is mainly sustained by an ongoing load of rare genetic variants [14] . Except for pairs of identical twin, the genetic individuality is furthermore warranted by meiotic replacement of the variant assortment in a gamete. These meiotic processes, chromosome recombination and shuffling (independent assortment), provide genetically individual gametes for the following genetically unique zygosis. Because the phenotypic effect size and direction of a modifier allele depend on the individual genetic background, the high individuality of the genetic backgrounds in the modifier carriers allows stochastic epistatic outcomes [39] [42] [43] . The stochastic epistasis components (modules) include stochastic modifiers (C, D, and E in Figure 2) and an invisible epistatic switch, and the mean value of the carrier outcomes can be approximately the same as that in the former generation (zero). In traditional (non-stochastic) epistasis model, scattering of the carrier outcomes is limited and the involvement of an epistatic switch may be not necessary.

The only ground for evolutionary selection of an allele is the fitness value of phenotypic outcomes in the allele

carrier. If the epistatic switch locus is originally polymorphic, in carriers of the switch-on allele the module-related phenotypic outcome can have any fitness value from the unfavorable extreme to the advantageous extreme according to the individual genetic background. This extensive distribution or the full phenotypic diversity of the switch-on carriers can confer strong evolutionary advantages in socio-ecological changes which affect the fecundity of a part of the population. It can be simply exemplified and simulated in a series of severe environmental changes, by which partial extinction of the population is introduced and the extinct counterpart is different in each case. The presence of an identical-by-descent switch allele causes an intergenerational fluctuation of the fitness outcomes in members of the descent [10] [44] , and the fixation of the switch can be facilitated by the intermittent succession of different types of ecological changes and the following rapid restoration of the extinct part through the stochastic epistasis. This restoration process to keep the bell-shape, which is immediately launched after the distributional damage, can be achieved by the survived population and may be one of the switch-specific characteristics. If the epistatic switch locus is originally monomorphic, it ought to have another quite essential function. Although traditional genetic models deal only with segregating genetic variation and thus effects of monomorphic loci are invisible in the research field, both monomorphic and polymorphic loci are involved in the actual gene networks for complex traits [45] [46] . Assuming that the switches are fixed (monomorphic) and the genetic individuality is sufficiently stochastic in the population, the individual stochasticity of the phenotypic outcomes in carriers of a modifier variant also makes it difficult to detect the modifier. The mean genetic effect of a modifier variant in the carriers can be approximately the same as the population mean through generations because of the stochastic individuality (Figure 2). This cryptic collaboration between the switch and modifiers can accomplish the maintenance of phenotypic diversity under selection. Non-switch monomorphic involvement is also plausible, but interactions between the non-switch monomorphic loci and polymorphic modifiers are again invisible.

This perspective can explain the situation that mapping epistatic interactions is still challenging, experimentally, statistically, and computationally [47] . However, there are some clues that suggest where the epistatic switch locus is and how the modifier members are. As mentioned above, because the presence of epistatic switch is a prerequisite for the manifestation of epistatic interactions, the function of monomorphic switch may be closely associated with the module phenotypic functions. In stochastic epistasis model, genetic investigations with partial sample sizes may accidentally have valuable information on the candidates of growing modifier variants (Figure 2). Therefore, functions and structures derived from the variants which were demonstrated in association with human complex conditions may be informative in spite of the low reproducibility of the association. RNA processing, transcription, cell to cell interaction (including immune reactions), signal transduction, synaptic infrastructures, and neuronal development may be involved in the modifier functions of human complex conditions [48] -[56] . If there are simple functional links between a switch and modifier variants in a module, the functions of modifier candidates may suggest the function of the switch locus. Because the known stochastic mechanisms originate in circuits of interacting genes and proteins or feedback loops [8] [9] , the switch- related network may be associated with multi-subunit molecules, complex feedback loops, cascade structure, hierarchical structure, conformational structure, or the presence of alternative pathways. The reported mutations may not be the causal variants of diseases, but they are mere nonspecific modifiers at the predetermined loci. The evolutionary significance of the epistatic switch implies that some of the modifier variants may hitchhike alongside the monomorphic switch locus [57] . Hot spots for de novo germinal mutation in relation to complex conditions [58] may be derived from such hitchhiking. Because genes that act as genetic hubs are also important for buffering stochastic genetic variation [59] , genes that interact with many different loci with different functions might have the epistatic switch. Epistasis that involves essential reactions often occurs between reactions without overlapping functions [60] . A long string of complex links or biological endo-phenotypes may intermediate between a switch and modifiers. If the functional link between the switch and modifiers is unintelligible, it may be difficult to detect the monomorphic switch locus.

The bell-shaped distribution of quantitative traits can be well simulated by the involvement of multiple stochastic components [10] , and stochastic genetic individuality is reflected in the stochastic phenotypic outcomes. The stochastic constituent is a modifier in a module, and phenotypic outcomes sometimes need multiple gene modules as stochastic components. If there was no interaction with the environment, the stochastic outcome through stochastic epistasis was never accidental. As described above, both germinal genetic alterations and stochastic epistasis are concordant in identical twins (Table 1). In the real natural world, however, the environment influences every level of genotype-phenotype relationships (Figure 3). Especially, the behavioral and cognitive world provides complex gene-environment interactions. Even if a pair of identical twins in the same ecological niche has extrinsically the same behavioral experiences, they each intrinsically experience the individually original non-shared environment, and the non-shared environment can be accidentally variable. Individual phenotypes themselves can influence the accidentally variable non-shared environments. The developmental trajectory is one of the complex phenotypes interacting with the environment and can also be highly individualized by the non-shared environments. The network module structure plays a key role at each level of genotype-phenotype relationships [38] [61] [62] (Figure 3), and stochastic epistatic gene modules affect individual responses to environmental cues through the hierarchical cascades of the overlapped modules. In multi-dimensions of trait vectors, a change in modifier members or an ecological cue may nonlinearly move the position of phenotypic distribution [63] .

The stochastic epistasis model never dismiss the comprehensive view of the known genetic contribution, including major variant effects, additive networks, and non-stochastic epistasis [10] [11] [44] . This model was originally described as fitness uncertainty of the phenotypic consequences whose underpinnings involve additive genetic factors, gene-environment interactions, and epistasis [44] . To simulate the significance of stochastic epistatic gene components, the stochasticity was substituted by the random contribution from a module with a switch locus [10] . Random effects of associated modifiers additively constitute the module stochastic contribution and the sum total of each module contribution in an individual can be regarded as the overall phenotypic outcome. For evolutional considerations, the dimensional phenotypic distribution is viewed in a fitness vector. From a comprehensive view, the effect value of a variant can be evaluated as the mean of the individual phenotypic outcomes in variant carriers in the population, and the difference between the variant mean and the population mean value is assigned to show the genetic effect (Figure 2 and Figure 4). In this article, the involvement of non-stochastic epistasis is taken for granted and the non-stochastic participants are included in variants with certain level of mean effects. Variants with major genetic effects have big mean effects, but are exposed to big selection pressures, resulting in low contribution to the maintenance of phenotypic diversity under selection (Figure 4). The mean genetic effect of a modifier variant that is a component of the stochastic epistatic module can be approximately the same as the population mean through generations, and the modifier can contribute to

the maintenance of phenotypic diversity under selection. Major variants can take one variant-one condition manner and stochastic epistatic modules underpin complex conditions.

Physiological and cognitive functions need an enormous assortment of responses that are available as necessary to deal with a variety of targets and stimulations [9] . Both complexity and specificity is required for the stock or supplies, and stochasticity-based phenotypic diversity is an efficient strategy for the complexity. Furthermore, the stochasticity eventually plays a role in building up the repertoire of specificity [9] and inter-module complexness. This diversification strategy is, no doubt, the only way to win the chance at survival [64] [65] . Interactions between polymorphic loci can make the organisms insensitive to the impact of novel mutations or environmental perturbation [59] [66] [67] , and stochastic epistatic modules may have critical roles in this buffer system. A variant which was buffered and harbored in the stochastic genetic network is one of the modifiers of the module and can participate in an active evolutionary phenotypic change through a nonlinear alteration of the mean value or the position of the distribution in multi-dimensions of trait vectors [63] [68] . Like uncovering cryptic genetic variation [69] [70] , a genetic or ecological change may unveil the phenotypic or evolutionary

significance of an old member in the module. The accumulation of transiently concealed genetic alterations may be associated with the evolution of complexity. Moreover, the phenotypic complexation increases the complexity of the social-ecological niche (social structure) and individual non-shared environments. The evolution of social complexity needs a more enormous assortment of individual responses to the local ecological conditions [13] and changing human activities [71] , resulting in an evolutionary cycle or spiral. “Genes do not exist to cause diseases” [72] , but perhaps one of the purposes of the genetic architecture is to cope with complexity.

The stochastic epistasis mechanism warrants evolutionary robustness of bell-shaped distribution of a complex trait. In other words, it reproduces the bilateral extreme tails of the population distribution in each generation as well as the major part around the mean value. On the other hand, the stochastic epistasis may also explain the phenotype changes. In the fitness-related dimension including fecundity, selection pressures can rapidly purge the unfavorable tail. Therefore, one extreme tail may include the non-fittest who cannot leave offspring and the other tail or the border flanking to the non-fittest tail may provide the founder or pioneer who is the key person for human terrestrial migration and local adaptation. In spite of the stochastic manner of the stochastic epistasis, the repeated purge of the same non-fittest extremes may be another process of evolutionary trends in which the phenotypic mean value can be passively changed [67] . In a liability-probability model, the non-fittest extreme tail possesses extremely low fecundity and the fecundity is continuously and sharply recovered at the majority of the bell-shaped population [73] . Both the active directional trend by de novo epistatic modifiers and the passive directional trend by repeated purge of the non-fittest tail may affect the position of the distribution in the nonlinear relation to fecundity probability, especially through a population bottleneck or reproductive isolation of a smaller population. To predict the ongoing trends or future directional changes of the complex trait distributions, the presence of fitness benefits of border cases who is located in the transitional zone flanking to the non-fittest tail [11] , a trade-off between the non-fittest outcomes and fitness benefits by expanding the range of complexity [74] , and the role of female carriers of modifiers to maintain the non-fittest tail in the population distribution [75] should be involved in the consideration.