PKU: Heritability, changeability, and gene-environment interaction

PKU (Phenylketonuria) is a condition that is often invoked to demonstrate that genetic does not mean unchangeable. It is also often cited as a trait with 100% heritability in normal environments that can, nevertheless, be changed and in the right environment have zero heritability. This post explains why these two sentences are not synonymous and how the second is flawed and reinforces an incorrect idea of heritability. I also examine PKU in terms of measured genetic and environmental factors, arriving at the counter-intuitive idea that no gene x environment interaction is involved.

PKU in humans is associated with having two copies of a malfunctioning allele for the enzyme phenylalanine hydroxylase (PAH). The cognitive development of such individuals is extremely impaired by the level of phenylalanine present in normal diets. With either one or two copies of the functioning allele, cognitive development is not affected (with caveats that I put aside for simplicity). However, given a special low-phenylalanine diet and adherence to that diet, individuals who are homozygous for the malfunctioning allele have more or less normal cognitive development (again with caveats that I put aside for simplicity; see Paul and Bosco 2013). In short, PKU is clearly associated with genes and is clearly changeable. Given that we know this, the following examination of the heritability of PKU and then of the trait in relation to measured levels of genetic and environmental factors is warranted only to clarify the concepts and their relationship.

For a given trait heritability is the technical term for the fraction of the total variation due to variation among the mean values of the trait for individuals having a given genotype. The three genotypes are (PAH+, PAH+), (PAH+, PAH-),(PAH-, PAH-), where + denotes functioning and – non-functioning. In circumstances where the special diet is not available, people with the first two genotypes will have normal cognitive development and people with the third genotype will not. If we put aside all other sources that influence cognitive development, we can classify a person as either a 1 or a 0 and group together as the first two genotypes as the non-PKU genotype. The heritability is then 1, i.e., 100%. There is, after all, no other source of variability in the trait other than the variability associated with the different genotypes. Yet, in circumstances where the special diet is available, people with non-PKU and PKU genotypes will have normal cognitive development.

Should the 100% heritability make us surprised by the change that is possible in the new circumstances? No, because the 100% heritability is within-location heritability, so it says nothing about any other location (i.e., circumstances). Now, it is possible to calculate across-locations heritability, which provides a prediction of how closely the mean trait value for the genotypes will be from one data set to the next when we do not insist that the genotypes are raised in the same circumstances. (This figure is always less than the average of within-location heritability values.) This is possible given that there is more than one circumstance. The technicalities are given in the next paragraph; readers may choose to fast forward to the implications that follow that.

Assume that access to the special diet occurs half the time and the frequency of the non-PKU genotype in the population is P. The overall variance for the trait, cognitive impairment, is .25(1-P^2). The mean trait value across both locations (circumstances) for the normal (non-PKU) genotype is 1; for PKU is .5. The variance of these means is .25P(1-P). Divide this by the overall variance and this is the (across-locations) heritability of the trait, i.e., P/(1+P). The mean across both genotypes, weighted by the frequency of the genotypes, is P when there is no access to the special diet and 1 when there is. The variance of these means is.25(1-P)^2. The remaining variance is of the residuals, that is, the values for each genotype-location combination after subtracting the genotype and location means and adding back the overall mean (because it has been subtracted twice). The variance of those values is .25P(1-P). This last quantity is the genotype-location interaction variance. When added to the variance of the genotype means and divided by the 1- variance of the means for the locations, the result is the average within-locations heritability. In this case .5P(1-P)/.25(1-P^2)/(1-.25(1-P)^2) = 8P/(1+P)/(3+2P-P^2). When P approaches 1 (it is about .9999 in U.S. populations), (across-locations) heritability of the trait approaches .5, as does the genotype-location interaction variance. That leaves the variance among the means for the locations approaching zero. The average within-locations heritability approaches 1. It might also be noted that the within-location heritability for the circumstances where the special diet is available, and thus there is no variation in that trait at all, is not zero, but 1. (Technically it is 0/0, which is undefined, but the limit of that heritability as the trait of PKU genotype moves from 0 to 1 is defined, and it is, as is the case in the circumstances when there is no special diet, 1.)

As suggested earlier, there is no reason to calculate heritability for the PKU case given that we know exactly what is behind the trait values observed. Indeed, heritability arose in agricultural sciences in order to make predictions when researchers did not know the genetics underlying the varieties and the environmental factors underlying the locations, moreover the different ways that different varieties elicited environmental factors from the locations. (On this last point, the analogy would be that, in the circumstances where a special diet is available, non-PKU individuals elicit the normal diet, whereas PKU individuals elicit the special diet.) Nevertheless, given that PKU or, similarly, Wilson’s disease, are often invoked in critical discussions of heritability, let us see what can be seen from partitioning variance into fractions:
1. Across-locations heritability of the trait, here cognitive impairment, is less than or almost equal to 1/2 and is the same as genotype-location interaction variance.
2. The difference between locations (circumstances) is mostly reflected in the genotype-location interaction variance, not the variance among the location means.
3. In each location, the within-location heritability is 1 (given that the only variation is among the genotypes), but the average within-location heritability for the full data set is slightly less.
4. The difference between the across-locations heritability and average within-location heritability comes from a) the latter subsuming genotype-location interaction into its estimation and b) the divisor being a smaller quantity (given that the variation of the location means is taken out of consideration).

Lewontin (2000, 38) states that “heritability of a characteristic has no predictive… value.”  This is not correct.  If we want to make predictions outside a given circumstance, the relevant quantity for thinking about how differences among genotypes combine with differences among locations to generate variation in a trait is across-locations heritability, not average within-location heritability or heritability within any specific location.  (Lewontin’s comment is justified about predictions for circumstances outside the range of locations that went into the estimation of across-location heritability.)

Suppose we put aside heritability for the PKU case given that we know exactly what is behind the trait values observed and examine the relationship of the trait to the measured genes or genetic factors and measured environmental factors. We might look to fit the observations to an equation of the form:
trait value for a given level of the genetic factor and the environmental factor =


+ coefficient1 * value for genetic factor

+ coefficient2 * value for environmental factor

+ coefficient3 * value for genetic factor * value for environmental factor

+ residual

If the non-PKU genotype is a 1 and the PKU genotype is 0 and the normal diet is 0 with the special diet 1 then the PKU-related cognitive impairment observations can be fit to the simple equation:
trait value for a given level of the genetic factor and the environmental factor =

value for genetic factor + value for environmental factor
(In other words: constant = 1; coefficient1 = coefficient2 = 1; coefficient3 = 0; residuals are 0)

1. In symbols, this might be simply expressed as cognitive trait = g + e.
2. This equation holds independently of the frequency of non-PKU and PKU genotypes (which is not the case for the earlier partitioning of variance).
3. There are no observations of individuals having the non-PKU genotype being raised on the special diet, so we do not have to worry whether the equation correctly predicts the trait in that case.
4. A non-zero coefficient3 would mean the equation includes a significant gene-environment interaction term, which is a conceptually and empirically distinct quantity from genotype-location interaction variance. However, in the PKU case, there is no gene-environment interaction, while there is a significant genotype-location interaction variance. Similarly, the equation has a significant term for the measured environmental factor, but a small variance among location means.

The contrast in the PKU case between partitioning variation and deriving an equation that relates the trait to the measured genetic factors and measured environmental factors should remind us that, when researchers do not know the genetics underlying the varieties and the environmental factors underlying the locations, it is incorrect and misleading to interpret heritability as the “contribution of genetic differences to observed differences among individuals.” (The quote is from Plomin et al. 1997, 83, but the interpretation is widespread; see Taylor 2014a, 24ff.)

In closing, it is true that PKU can be pinned down to having two PAH- alleles passed down to the individual from their parents. However, that is an issue of the trait being heritable. When someone uses PKU to show how a trait can have high or 100% heritability but the trait is nonetheless changeable, they are overlooking or incorrect about relevant aspects of calculating heritability for PKU and they are perpetuating the confusion between the heritability of a trait and the trait being influenced by heritable genetic factors (Taylor 2014).


Lewontin, R. C. (2000). It Ain’t Necessarily So: The Dream of the Human Genome and Other Illusions. New York, New York Review of Books.

Paul, D. B. and J. P. Brosco (2013). The PKU Paradox: A Short History of a Genetic Disease. Baltimore, Johns Hopkins University.

Plomin, R., J. C. Defries, et al. (1997). Behavioral Genetics. New York, Freeman.

Taylor, P. J. (2014). Nature-Nurture? No: Moving the Sciences of Variation and Heredity Beyond the Gaps. Arlington, MA, The Pumping Station.

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