Branching processes and Apert syndrome

Apert syndrome is a birth defect caused by mutation of either of two specific base pairs. The syndrome occurs at a rate of about 1 in 200,000 live births, much higher than the estimated 1 in 100,000,000 expected from the average estimated human mutation rate. Mutation rates do vary significantly across the genome (e.g. “hot spots”); but such a large variation is still very unexpected. It has also been observed that the father’s age has an effect on the likelihood of occurrence – older fathers are much more likely to have offspring with the syndrome. There is an alternate explanation, but first, some background. Here is a cartoon of how sperm is produced in the testes. There is a large population of stem cells which usually divide asymmetrically, producing one sperm and one stem cell, but may occasionally divide symmetrically, producing two stem cells. They may also, eventually, die. Suppose, then, that the mutation causing Apert syndrome also causes the stem cells to divide symmetrically more often. In an individual with the mutation, then, this would eventually lead to a disproportionately large number of mutated sperm, and thus a much larger chance of offspring with the syndrome than we’d expect from the mutation rate. We will make assumptions that allow us to treat the number of stem cells present as a branching process (ignoring the sperm), and will denote the probabilities of symmetric division and death, respectively, by b and d. Since the number of stem cells is more-or-less constant, we assume that b = d for nonmutated cells. This then suggests two models: