32 Continuous traits from discrete genes

32.1 Background

This activity uses a fast classroom simulation to show how many small genetic effects, together with some environmental variation, can generate a continuous, approximately normal trait distribution. It connects to the heritability practical in Chapter 20.

Reading: Neal, chapter Polygenic Inheritance and Quantitative Genetics (Nilsson-Ehle’s wheat kernel colour example makes the same point).

Learning outcomes:

  • Explain how many genes with small additive effects create many trait classes, and how environmental variation blurs these classes into a continuous distribution.
  • Compare a classroom simulation with a simple R simulation.
  • Relate the simulation to the partition \(V_P = V_G + V_E\) and to heritability.

32.2 Worked example

32.2.1 Inputs

  • Baseline trait value: 35
  • Number of genes: 25
  • Each gene contributes 1 unit when ON, and each gene is ON with probability 0.5

32.2.2 Steps

  1. Compute expected number of ON genes under 50:50 ON/OFF: 25 * 0.5 = 12.5.
  2. Add expected genetic contribution to baseline.

32.2.3 Output and interpretation

Expected trait value is 35 + 12.5 = 47.5. Individual values vary around this because each gene is ON or OFF at random. The genetic variance is \(25 \times 0.5 \times 0.5 = 6.25\) (the variance of a binomial distribution). Notice that, with genes alone, every trait value is a whole number: there are many classes (35 to 60), but they are still discrete.

32.3 Your task

You will need the instructions and the Excel spreadsheet (download them):

  1. Classroom simulation. Each student is a gene. In each round, every student flips a coin: heads = ON (1), tails = OFF (0). Count the number of ONs in the class and add it to a baseline of 35. That is the trait value of one individual. Repeat for at least 30 rounds (30 individuals), and record each value. (Alternatively, use the spreadsheet to generate at least 30 individuals.)
  2. Plot a quick histogram of the observed trait values. Describe the shape.
  3. Now simulate the same process in R for 100 individuals and 25 genes (first chunk below). Compare your histogram with the classroom version.
  4. Real traits are also affected by the environment (food, temperature, chance events during development). Add a small random environmental effect to each individual (second chunk below) and compare the two histograms.
  5. Relate the outcome to the concepts of genetic and environmental variance in Chapter 20.
baseline <- 35
n_genes <- 25
n_individuals <- 100

gene_on <- rbinom(n_individuals, size = n_genes, prob = 0.5)
trait_value <- baseline + gene_on

hist(trait_value, breaks = seq(min(trait_value) - 0.5, max(trait_value) + 0.5, 1),
     col = "gray80", main = "Genes only", xlab = "Trait")

Now add an environmental effect, drawn from a normal distribution with mean 0 and standard deviation 1.5 (so the environmental variance, \(V_E\), is \(1.5^2 = 2.25\)):

environment_effect <- rnorm(n_individuals, mean = 0, sd = 1.5)
trait_value_env <- trait_value + environment_effect

hist(trait_value_env, breaks = 15, col = "gray80",
     main = "Genes + environment", xlab = "Trait")

var(trait_value)       # genetic variance (expected 6.25)
## [1] 6.290404
var(trait_value_env)   # phenotypic variance, V_P = V_G + V_E (expected 8.5)
## [1] 8.240729
  • Are the trait values in the second histogram still whole numbers? Why does the environment make the distribution continuous?
  • In this model all the genetic variance is additive. What is the expected heritability, \(h^2 = V_A/V_P\)? (Use the expected variances: 6.25 and 8.5.)
  • What happens to the heritability if you increase the standard deviation of the environmental effect to 4?

32.4 Takeaways

  • Many genes with small additive effects produce many trait classes and a roughly bell-shaped distribution; environmental variation blurs the classes into a continuous distribution. (Normality is an approximation, not a law.)
  • Phenotypic variance is the sum of genetic and environmental variance (\(V_P = V_G + V_E\)), so the same genes give a lower heritability in a more variable environment.
  • Random genetic variation produces a spread of trait values even with identical environments.