The fitness of an individual can be defined in terms of the number of its fertile offspring, and is a function of its survival and reproductive performance. Fitness can be regarded as a quantitative trait, but in natural populations it is difficult to determine how other traits influence fitness. Although it is clear that in *Drosophila*, for example, there is an overt relationship between female egg production and fitness, the relationship between bristle number and fitness is less obvious. For some traits there is likely to be a monotonic relationship with fitness, for others it may be quadratic with an intermediate optimum (Falconer and Mackay 1996; Walsh and Lynch 2009). How phenotype determines fitness is one of the fundamental challenges to developing the evolution theory of adaptation (Stearns and Hoekstra 2000; Orr 2005; Wagner and Zhang 2011).

Fisher's (1958) fundamental theorem shows that the rate of change in fitness is equal to the additive genetic variance in fitness. Thus natural selection would be expected to use up the useful variance (Crow 2008). However, variation in fitness remains at high levels in populations (Mousseau and Roff 1987; Merila and Sheldon 1999; Hill and Zhang 2008; Long et al. 2009). For example, Fowler et al. (1997) allowed replicated competition of a sample of wild-type third chromosome of *D. melanogaster* against balancer chromosomes in a population cage, and estimated fitness by recording changes in chromosome frequency. They obtained an estimate of the genetic variance in log_{e} fitness for the whole genome as high as 0.45. Genetic variation in fitness can be maintained by mutation and other factors such as heterozygote superiority and heterogeneous environments; genetic variation in other traits can remain through these factors and also because the covariance with fitness may be small (for discussion see, e.g., Falconer and Mackay 1996; Lynch and Walsh 1998; Bürger 2000; Zhang and Hill 2005a; Crow 2008). Although there are some theoretical models to provide qualitative understanding of why genetic variance in fitness remains (Charlesworth and Hughes 2000), satisfactorily quantitative models are not available. To understand the maintenance of genetic variance in fitness, we need a model to describe how fitness is controlled by mutant genes and genotypes and how natural selection acts on them.

Fisher (1930) was one of the first to consider this relationship. He introduced a geometrical model of natural selection acting on multiple quantitative traits that characterize an individual: each trait has an environment-dependent optimal value and the fitness of a complete phenotype is jointly determined by the distances of the traits from their optimal values. For phenotypes near this optimum, selection is of stabilizing type while phenotypes far from it are consequently subject to directional selection (Barton 1998). There is some evidence for stabilizing selection and consequent directional selection on various traits (Kingsolver et al. 2001; Elena and Lenski 2003; Hereford et al. 2004). Although this selection model appears simplistic, it has been widely applied to different aspects of evolutionary biology (e.g., Poon and Otto 2000; Orr 2000; Barton 2001; Welch and Waxman 2003; Waxman and Welch 2005; Martin and Gandon 2010). Importantly, an analysis of the fitness effects of mutations across environments shows that predictions from Fisher's geometrical model are consistent with empirical estimates, implying that multivariate stabilizing selection is a reasonable fitness landscape model (Martin and Lenormand 2006b).

A phenotype can be characterized by a very large number of traits (Zhang and Hill 2003; Wagner and Zhang 2011). Because the total number of genes is limited, most genes must affect more than one trait (i.e., pleiotropy is common) (Barton and Keightley 2002; Mackay 2004; Ostrowski et al. 2005; Weedon and Frayling 2008; Wagner et al. 2008; Wang et al. 2010). Further, environmental constraints on traits are not independent, so selection on these traits is also expected to be correlated (Welch and Waxman 2003; Zhang and Hill 2003; Blows 2007). The complicated situation can be reduced, however, by simultaneous diagonalization of the mutation matrix and the selection matrix (Zhang and Hill 2003; Hine and Blows 2006) such that the number of traits that are under independent selection and mutationally independent is much smaller (Waxman and Welch 2005; Hine and Blows 2006; McGuigan et al. 2011). By comparing theoretical predictions with empirical data on the distribution of mutational effects on fitness, Martin and Lenormand (2006a) suggest there could be less than three independent traits for some model species.

In the discussion of adaptation, it is usual to assume that the population has suffered a one time environmental change such that it departs from the current optimum for many generations (Orr 2005). The question of interest is how often the optimum changes and what kind of mutation emerges to help the population catch up with the changing optimum (Bello and Waxman 2006; Kopp and Hermisson 2007). A large change is rare but small changes must be frequent. The idea that an environmental change determines a new phenotypic optimum is supported by the long-term adaptation of experimental populations to new environments (for review see Elena and Lenski 2003 for microbes and Gilligan and Frankham 2003 for *Drosophila*). By comparing predictions for a fitness landscape model (Martin and Lenormand 2006a) and observations from mutation accumulation experiments, Martin and Lenormand (2006b) concluded that a Gaussian fitness function with a constant width across environments and with an environment-dependent optimum is consistent with the observed patterns of mutation accumulation experiments. As environment may change randomly or directionally, so does the movement in the optimum. With changes such as these in the position of the optimum, the effect of a mutant on fitness varies over generations even though the mutational effect on quantitative traits may not (Martin and Lenormand 2006b; cf. Hermisson and Wagner 2004).

Theoretical studies show that stabilizing selection toward a directionally moving optimum under recurrent mutation is an important mechanism for maintaining quantitative genetic variation (Bürger 1999; Waxman and Peck 1999; Bürger and Gimelfarb 2002). With a directionally moving optimum, mutant alleles suffer both stabilizing and directional selection; mutants become beneficial if they draw phenotypes toward the moving optimum and rise in frequency, which then increases the genetic variance. Previous investigations based on a quadratic fitness function, however, show that a randomly fluctuating optimum can hardly increase the quantitative genetic variance (Turelli 1988). Based on a Gaussian fitness function, Bürger and Gimelfarb (2002) showed that stochastic perturbation of a periodic optimum, in combination with recurrent mutation, can increase genetic variance substantially; but Bürger (1999) showed that purely fluctuating selection cannot increase genetic variance.

In this study we employ Fisher's (1958) model for the map between mutational effects on quantitative traits and the fitness effect with a changing optimum to investigate the variance in fitness under mutation-selection balance (MSB). Any realistic scenario may be complicated, but here we consider an idealized situation of additive genes, free recombination, and a large population with random mating. Results from these simplified models should provide essential and useful information for more complicated and realistic situations. Moreover, taking into account empirical data for mutation and selection parameters, we test whether this model can provide a quantitative interpretation for the high level of genetic variance in fitness. Based on these results, we further discuss how pleiotropy affects the genetic variance in fitness and the so-called “cost of complexity” (Orr 2000), and review the validity of the Fisher geometric model of fitness.