Linkage, Epistasis, Chi-squared and Sex Linkage

When the ratio is not what you expected

Two genes give $9:3:3:1$ only if they are on different chromosomes, or far apart on the same one. If they are linked — close together on the same chromosome — they tend to be inherited together, so the parental combinations are far more common than expected and the recombinant combinations far rarer. Recombinants appear only when crossing over separates the linked alleles, and the closer the loci, the rarer this is. Epistasis is different again: one gene masks or modifies the expression of another, so the four phenotypic classes collapse into fewer. Recessive epistasis gives $9:3:4$, dominant epistasis $12:3:1$, and complementary gene action $9:7$. When a question gives you an odd ratio, first check whether the numbers add to $16$ — if they do, suspect epistasis rather than linkage.

Testing a ratio with chi-squared

The chi-squared test asks whether the difference between observed and expected numbers is small enough to be due to chance. Calculate $\chi^2 = \sum \dfrac{(O-E)^2}{E}$, using actual numbers, never percentages. The degrees of freedom are the number of classes minus one. Compare the value with the critical value at $p = 0.05$: if $\chi^2$ is less than the critical value, the difference is not significant and the null hypothesis — that the observed results fit the expected ratio — is accepted. If it is greater, the difference is significant and something other than chance is at work, such as linkage.

Sex linkage and control of gene expression

A gene on the $X$ chromosome with no equivalent on $Y$ is sex-linked. A male has only one copy, so a single recessive allele is expressed; this is why haemophilia and red-green colour blindness are far commoner in males, and why an affected male passes the allele to all his daughters but none of his sons. In prokaryotes, gene expression is controlled by operons: in the lac operon, a repressor protein normally binds the operator and blocks transcription, but when lactose is present it binds the repressor and changes its shape, so transcription of the enzymes for lactose metabolism begins. This is an example of an enzyme being inducible — made only when its substrate is present.