We measured the frequency of the 755C>G and 758C>G mutations in the sperm of 314 donors. Either mutation will cause Apert syndrome. On average, the frequency of both of these mutations increases with the age of the donor. We compared the actual distribution of the ages of fathers of affected children to that expected from the sperm data, namely, the normalized product of the distribution of the ages of all fathers in the population (the vast majority of whose children do not have the disease) and the observed sperm mutation frequency as a function of the donors' ages. We then used these distributions to test the hypothesis that the sperm data is consistent with the data on sporadic births of affected children in men of different ages, and we were not able to reject this hypothesis. Our results provide strong statistical support for the idea that the Apert syndrome PAE results from an increase in sperm mutation frequency. These results disagree with one paper 
, presumably due to the lack of sensitivity in that study's assay, but agree with the conclusions of another paper 
, even though these authors did not have enough data to test this hypothesis statistically. Despite this agreement, the increase from the youngest age category to the oldest age category is greater for the birth data than for the sperm data: 26-fold for the birth data and 9-fold for the average sperm mutation frequency or 18-fold for the median sperm mutation frequency. One possible explanation for this discrepancy is uncertainty in the estimates of the fold-increase for both the birth data and the sperm data. Although a selective advantage in fertilization for sperm carrying either Apert mutation over wild-type sperm is not required to explain this data, such a possibility could be another contributing factor 
It is perhaps most clear from , that the PAE for Apert syndrome is most likely not due only to more germ-line divisions in older males than younger males. A possible additional contributing factor is that the mutation rate per cell division may increase as men age, perhaps due to a decline in the efficiency of DNA repair mechanisms or an increase in mutagen exposure. While this mutation rate per cell division increase is possible, it would result in a uniform elevation of mutation frequencies throughout the testes. Therefore, it would not explain the clustering of cells harboring the Apert mutations in the testes of older donors that we have observed in previous publications 
. Moreover, it would not explain the apparent growth of these mutation clusters as adult men age, since when we previously studied the testes from two younger individuals, ages 19 and 23 years, the mutation clusters were either non-existent or had far fewer mutant cells than in the older donors' testes 
. We have previously proposed a model for mutation in the male germ-line 
. This model includes positive selection, in the form of occasional symmetric divisions, on the mutant premeiotic testis cells. These occasional symmetric divisions allow the number of mutant cells to grow locally in the testis, similar to a tumor. Crow 
independently proposed a similar selective mechanism. In this paper and our previous publications 
, we estimate the mutant cells divide symmetrically approximately 1% of the time; this value is sufficient to explain both the mutation clusters observed in the older donors' testes and the growth of these clusters as adult men age. We are now arguing that this same selection scheme can also help to explain the increase in the PAE that is greater than would be expected just from the larger number of germ-line divisions in older males than younger males.
Both the sperm data and the birth data, as demonstrated by the O/E ratio, show a non-monotonic increase with age. For the birth data, researchers had previously declared this observation “difficult to explain” with any model that accumulates mutations with age during spermatogenesis 
. These researchers suggested two possibilities: (1) heterogeneous division rates for the stem cells where those cell lines undergoing more divisions would both be more likely to harbor mutations and to die earlier, and/or (2) an introduction of fresh, undivided reserve stem cells at middle age. We have now incorporated the reserve stem cell possibility into our model for mutation in the germ-line, and simulations from the modified model reproduce the middle age decrease. In addition to this decrease, in both the sperm data and the birth data, there is a dramatic increase in frequency by the very next age category. While possibilities (1) and (2) cannot explain this dramatic increase, simulations of our model do reproduce it, due to the selection on premeiotic testis cells carrying the Apert mutations.
Not all PAE diseases feature a non-monotonic increase 
. The type of causal mutation most likely has an effect on this property (base substitution, deletion, duplication, etc., reviewed in 
). Interestingly, even for the Apert syndrome sperm data, the two causal base substitution mutations display different non-monotonic patterns. These two mutations are just three base pairs apart within the same gene, but subtle phenotypic differences (severity of syndactyly and the presence of cleft palate) 
exist between them. In addition, functional differences between the two mutants in in vitro
assays are known 
The dip for the sum of the two Apert mutation frequencies in sperm is five years later than for the birth data. An implicit assumption of the generalized likelihood ratio test is that the increase in the sperm mutation frequencies for contemporary donors is similar to what would have been found for the population of fathers in the earlier birth studies. There is some evidence 
that contemporary male youths begin puberty at younger ages (resulting in more mutation events at an earlier ages), but this evidence is not conclusive 
. Moreover, 
report that there has been a decrease in semen quality over the last fifty years (also see 
and the contained references for more analysis). The effects of these or other possible generational changes (including environmental exposures) are not clear, but it is conceivable that they could influence the observed difference in the timing of the non-monotonic increase between the sperm data and the birth data.
In a previous publication we showed that the two Apert mutations arise independently within any one testis 
. It was therefore unexpected that we observed a significant correlation, after normalizing for age, between the two mutation frequencies within individual sperm donors. Perhaps this correlation observation is due to some property which is heterogeneous across the population but similar for cells within a single individual. Variation in properties such as the mutation rate per cell division, the cell division rate, or the selection coefficient for cells harboring the mutations could result from genetic or environmental causes. It would be interesting to collect mutation frequencies at additional nucleotide sites from these same individuals, to examine whether some individuals have higher overall mutation frequencies than others.
Finally, achondroplasia, the most common form of dwarfism (OMIN # 100800) also shows a PAE. Almost all cases are due to a spontaneous mutation of paternal origin, at a single nucleotide site in the fibroblast growth factor receptor 3 gene (FGFR3) 
. This disease however features a monotonic increase in both birth data 
and sperm data 
. Also, unlike Apert syndrome, the age-dependent increase in the sperm mutation frequency for achondroplasia is not sufficient to explain the birth data (
, for another interpretation see 
). However, additional studies are needed (including complete testis dissection analysis and the development of more sensitive assay methods) before the mechanisms behind this disease's PAE can conclusively be thought as being the same or different from Apert syndrome.