The Problem. Complex traits do not follow simple Mendelian rules of inheritance. Mutations that affect complex traits have variable expressivity and incomplete penetrance. Typically, a given mutation will affect only a certain percentage of carriers and that percentage is different in different populations.

Simple Mendelian traits are rare; the vast majority of traits are complex.

The complicated inheritance of complex traits is often ascribed to the fact that such traits are affected by the simultaneous variation of many genes that interact in complex ways. These genes are called the genetic background, and variation in the genetic background often has a major effect on the expression of the gene of interest. In addition, there may be various non-genetic so-called "predisposing factors" that affect the expression of a complex trait.

In reality, the complicated genetics of complex traits comes about through the fact that the relationship between genetic variation and trait variation is nonlinear. This non-linearity has many non-intuitive consequences for the relationship between genes and traits.

 

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A (very) brief primer on nonlinear genetics

First a look at a linear and additive system. In such a system the relationship between genetic variation and trait variation is a straight line. The relationship between simultaneous variation in more than one gene is a flat plane ( in [n+1] space, where n = number of genes). In such a system the amount of variation attributable to one gene is independent of the amount attributable to other genes. The genetics of linear systems is well understood.

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In a nonlinear and non-additive system, the relationship between genetic variation and trait variation is not a straight line (hence, nonlinear). The effect of simultaneous variation in more than one gene is described by a curved surface. In such a system, the effect of variation of one gene on a trait is highly dependent on the value of the other genes (as illustrated below by the two sections through the surface).

The trait below is a very simple complex system, namely, a 3 enzyme pathway, where the trait of interest is the flux through the pathway (after Kacser and Burns, 1981). The flux is a nonlinear function of the activity of any one of the enzymes. Here we look at the effect of variation of only two of the genes.

One interesting consequence of non-linearity is illustrated by the individuals represented by X and Y on the surface. These individuals have exactly the same trait value but different genotypes. In individual X variation in gene A has a big effect on the value of the trait whereas variation in gene B has little or no effect. The reverse is true for individual Y. This means that for individual X, A is a major gene and B a gene of small effect, and the reverse is true for individual Y. For more complex systems (e.g. see below) this means that a different set of QTLs (quantitative trait loci) will be detected for individuals in the neighborhood of X than for individuals in the neighborhood of Y.

So, what makes systems nonlinear?

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Here is an example of the non-intuitive consequence of non-linearity in a genetic system. It concerns the response to selection of a trait produced by a diffusion-gradient-threshold mechanism (for details see Nijhout and Paulsen, 1997). The value of the trait is controlled by 6 genes that affect the processes shown in the top panel.

Selection on the trait produces an orderly phenotypic response (top panel), but a complex genetic response (middle panel). Initially only a few genes respond to selection. Other genes respond only after the frequency of the first ones has changed substantially. The reason for this response is that not all genes are equally correlated with the phenotype. Those that are most highly correlated respond first to selection (bottom panel). As their frequency changes, they become less correlated with the phenotype, and another gene now becomes most highly correlated. During selection, each gene takes a turn, as it were, in being the most strongly correlated with the trait. The correlation of a gene with a trait is determined by the frequencies of all the other genes that affect the trait. Thus an observer encountering this population, say in generation 7, would perceive a different gene as being the most responsible for variation in the trait than an observer encountering the population in generation 11. At any one time, most genes are uncorrelated with the trait, even though their contribution to the ontogeny of the trait remains the same. This is called "contingent neutrality." These pseudo-neutral genes would be detected as QTLs, and this is why different combinations of QTLs are detected for the same trait in different populations.

Can this pattern of variable correlations between genes and traits be deduced form first principles? Yes it can. Look here.

Biological traits come about through developmental processes and physiological regulatory mechanisms. Most of these processes are nonlinear. Examples of nonlinear processes are:

• The sensitivity of reaction rate to substrate concentration
• inhibition
• negative feedback
• positive feedback
• cooperativity
• most non-steady state processes
• any process that depends on diffusion

Any mechanism that contains one or more of these processes (and most regulatory mechanisms in biology do) will have a nonlinear relationship between variation in its determinants and variation in the trait affected by the process.

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