Breaking proteins with mutations: threads and thresholds in evolution

A common high school science experiment involves anchoring one end of a rubber band to a desk and then attaching a small weight to the other end. The weight stretches the rubber band, and adding another weight causes the rubber band to dangle even lower. More weights can be added, and each one pulls the rubber band a little further towards the floor. Now, instead imagine attaching the weights to a thread. The thread stretches only slightly; so the first couple of weights have just a small effect. But if you add enough weights, the thread suddenly breaks and the weights fall to the floor. In the first case, each additional weight stretches the rubber band by the same amount, whereas in the second, it is the combination of several weights that breaks the thread. Mutating proteins is like adding weights, as mutations eventually ‘break’ the individual proteins, dragging down the fraction of proteins that still function (this fraction is the average fitness). The question becomes, does each mutation decrease the average fitness by the same amount (like weights on a rubber band), or are several mutations worse than their combined individual effects (like weights on a thread)? The latter situation is called ‘negative epistasis,’ and measuring the extent of epistasis is important for understanding evolution. If mutating a protein is like stretching a rubber band, then there is no epistasis and each random mutation has the same probability n of leaving a protein’s function intact. So the average fitness W(1) after each protein is mutated once is n, as this is the fraction of proteins that are still functional. Because there is no epistasis, the second mutation is just as likely as the first to inactivate the protein; so the average fitness W(2) after each protein is mutated twice is n. After n mutations, the average fitness is just