MEN2B testes data
We follow our recently developed approach 
by measuring the spatial distribution of the MEN2B c.2943T>C mutation in fourteen testes from normal men. We then quantitatively test whether or not these distributions are consistent with the hot spot model that predicts a uniform distribution of SrAp with new mutations or the selection model that predicts these mutant cells will be clustered. Both models assume that the germ cells that undergo mutation are uniformly distributed in the testis (for details supporting this assumption see Text S2
). Each testis was cut into 6 slices and each slice into 32 pieces of approximately equal size. The amount of DNA in each piece was quantitated and the frequency of mutant MEN2B molecules was established for each piece using a highly sensitive modification of allele-specific PCR called PAP 
that gave a false positive rate of 4.7×10−7
(based on analysis of ~2.7×108
control genomes). For each testis piece we estimate the mutation frequency p
enomes (pmg). In , this frequency is represented by a heat map with colors ranging from light gray to dark brown. Dataset S1
contains mutation frequency estimates for each testis piece. In , we use several statistics to summarize this data. For each testis, we consider the average mutation frequency of all the pieces (Av). Many testes have individual pieces with frequencies that are very different from the average. For each testis, we also identify the piece with the maximum mutation frequency (Mx). In order to normalize for the varying average frequencies among different testes, we consider the ratio of Mx to Av in each testis (Mx/Av). In addition, we consider the fraction of testis pieces with mutation frequencies less than 50 pmg (F<50); in these pieces are colored light or dark gray.
Distribution of the MEN2B mutations in 14 human testes.
MEN2B mutation frequency summaries from 14 testes.
The youngest age group is made up of three individuals 19, 21, and 23 years of age. For this age group, the Av ranges from 1 to 15 pmg. The Mx ranges from 13 to 65 pmg. shows that all the pieces' mutation frequencies are colored light gray (<25 pmg), dark gray (25 to 50 pmg), or pink (50 to 500 pmg). The few pieces colored pink in this age group are in the low end of the pink range, since the one with the greatest frequency is only 65 pmg. For each testis the F<50 ranges from 95% to 100%.
Six individuals, aged 36 to 68 years, comprise the middle-aged group. For these testes, the Av ranges from 19 to 1,188 pmg. In contrast to the youngest age group, each testis has a small number of pieces with mutation frequencies that are several orders of magnitude greater than the remaining pieces. The Mx ranges from 643 to 48,884 pmg. These high frequency pieces are more darkly colored in , and are often clustered together in the same slice or in adjacent slices. The sample with the lowest Av (#59089) also has the lowest Mx, and the sample with the highest Av (#854-2) also has the highest Mx. The Mx/Av ratio ranges from 34 to 139. The F<50 fraction is still high, ranging from 76% to 99%.
Both the Av and the Mx are greater for the middle-aged group than the youngest age group. However, within the middle-aged group there is no obvious correlation between frequency and age. Indeed the testis with both the lowest Av and Mx is from a 45 year old (#59089), and the testis with both the highest Av and Mx is from a 54 year old (#854-2). So the extreme frequencies come from individuals with ages in the middle of the group, and with ages that are close to each other.
The oldest age group containing five individuals aged 75 to 80 years is heterogeneous. Two of the individuals (#64302 and #60954) have frequency values typical of the middle-aged group: the Av are 75 and 203 pmg, the Mx are 4,372 and 6,673 pmg, the Mx/Av are 58 and 33, and the F<50 are 98% and 84%. The remaining three individuals (#60955, #57650, and #60507) have much lower frequency values typical of the youngest age group: the Av are 10 pmg or less, the Mx are 56 pmg or less, and the F<50 are 99% and 100%. The three low frequency old samples will be further discussed later in the Results section. For discussion purposes, we define a testis as having “substantial” mutation clusters if Mx is greater than 500 pmg: this group includes all of the middle-aged samples and the first two from the oldest age group, while excluding all of the youngest samples and the last three from the oldest age group.
Hot spot model
Previously, we developed a model based on what is known about human germ-line development and maturation to quantitatively test whether the mutation distribution in a testis is consistent with a hot spot model 
. Here we briefly review the model, apply it to the c.2943T nucleotide site in the RET
gene, and discuss a new variant to the model. The computer programs to simulate all the models discussed here and elsewhere in the paper can be found in Protocol S1
The hot spot model has two phases that we call the growth-phase and the adult-phase. The growth-phase models the testis from zygote formation to puberty. During this phase, divisions of the male germ-line cells are symmetric and self-renewing, and the number of such cells increases exponentially. Similar to a Luria and Delbruck “mutation jackpot” in bacteria 
, a mutation arising early in this phase will be shared by more descendent germ-line cells than will later mutations. The primordial germ cells migrate to the site of gonad formation and form the seminiferous cords early in fetal development 
and since germ cells are expected to remain physically close to their ancestors once the chords are formed, further cell divisions of early mutations can result in mutation clusters. There are approximately 30 growth-phase generations 
The germ-cells originating during the growth-phase eventually form the adult SrAp. These cells cycle throughout a man's life providing many opportunities for new mutations. The adult-phase portion of the model considers the testis after puberty. During this phase, the SrAp divide asymmetrically to produce a daughter SrAp (self-renewal) and another daughter cell whose descendants, after a few additional divisions, will produce sperm. In an adult male, the SrAp divide every 16 days 
, and therefore from an individual's age we can estimate the number of adult phase generations that his SrAp cells have experienced. In our model any new mutation in the adult phase can produce only one mutant SrAp self-renewing cell lineage. The model has only one free parameter: the mutation rate per cell division.
For each testis, the data is the mutation frequencies of the 192 testis pieces. In order to test the hot spot model using the maximum likelihood approach one would need to calculate the probability, as a function of the model parameters, of the mutation frequencies for all 192 testes pieces. Unfortunately, none of the models we consider are amenable to such calculations. One could estimate this probability function by counting the number of computer simulations of the model that exactly match all 192 frequencies. However, the probability of exactly matching all 192 frequencies is so low that this approach is not feasible. The goodness-of-fit strategy that we pursue instead is we test whether there are values of the model parameters such that computer simulations of the model can approximately match the three summary statistics Av, Mx/Av, and F<50 simultaneously. These statistics attempt to summarize both the mutation frequency and the clustering observed in the testes. Say, for example, a model predicts a more uniform distribution of frequencies than was observed so that simulations which approximate the observed Av statistic also feature much lower than observed Mx/Av ratios. Since this model fails to capture both the mutation frequency and the clustering observed in the testis, we would reject such a model. Alternatively, suppose another model approximately matches the three summary statistics simultaneously. Since this model reproduces both the mutation frequency and the clustering observed in the testis, we would declare such a model consistent with the data.
Let us consider the example of testis #374-1 from a 62 year old. The observed Av is 68 pmg (). In simulations, we vary the mutation rate per cell division until we find the value of this model parameter such that the simulated Av best matches the observed Av. We simulate the model using this parameter value until we have one million simulations where the simulated Av is within 5% of the observed Av, and then we compare the other statistics for these simulations () to the actual data. For the observed data the Mx/Av ratio is 85, while in 95% of simulations the ratio is between 2.1 and 4.3. Indeed, in one million simulations this ratio is always less than was observed in the data. Similarly the observed Mx is 5,784 pmg, while in 95% of simulations the Mx is between 144 and 288 pmg. Since we only consider those simulations such that the simulated Av is within 5% of the observed Av, the results for the two statistics Mx and Mx/Av closely correspond. Since we find the ratio Mx/Av more intuitive, we will only consider it subsequently. Likewise, for the data the F<50 statistic is 90%, while in 95% of simulations this fraction is between 25% and 35%. In one million simulations this fraction is always less than was observed in the data. Thus we are able to strongly reject the hot spot model with p-value less than 10−6. In , we see the same conclusion holds for the remaining seven testes with substantial mutation clusters. Note that for testis #59089 in 95% of simulations the F<50 statistic is between 99% and 100% because the Av (19 pmg, ) is less than 50 pmg.
Hot spot model parameter and simulation results for those testes with substantial MEN2B mutation clusters.
In the hot spot model, a mutation early in the growth phase could produce a mutation cluster. In order to match the observed Av, however, the inferred value of the mutation rate per cell division model parameter is low enough such that mutations early in the growth phase are rare. Since the SrAp divide every 16 days, in a 62 year old male there have been approximately 500 times more adult phase cell divisions than growth phase divisions 
and mutations in the adult phase do not produce mutation clusters. Consequently, in simulations of the hot spot model that match the observed Av the distribution of mutations is more uniform than observed. Furthermore, even if one does not agree with the modeling details, the youngest age group has markedly lower Av and Mx statistics than the middle-aged group (). Therefore, the increase in the mutation frequencies and the growth of the mutation clusters occurs in the adult, not during development.
Finally, we previously examined the distribution of a likely neutral mutation in testis samples using the same approach 
. We assayed a C to G mutation in the intron of the CAV1 gene on chromosome 7. This presumably neutral mutation was studied in testes 374-1 and 374-2 (62 years of age) and involved the same DNA samples we used for the MEN2B analysis. The summary statistics are identical for both testes (Av
6.67 and F<50
100%) and similar to the MEN2B data from much younger donors. Simulations showed that the relatively uniform distribution of mutations was consistent with the hot spot model.
Symmetric hot spot model variant
Based on work in the mouse 
and human 
, we also consider a variant to the hot spot model where the SrAp in the adult phase, independent of whether or not they have acquired the disease mutation, may divide symmetrically. As in the original model, each SrAp in the adult phase divides every 16 days, but now there are three possible types of divisions (the probabilities of these types sum to one). This variant introduces a second model parameter q. With probability 1-2q, the SrAp cells divide asymmetrically as in the original hot spot model. However, now with probability q, the SrAp cells divide symmetrically producing two SrAp cells: both daughter SrAp cells share any accumulated mutations and since these cells remain physically near each other, multiple symmetric divisions would produce a mutation cluster. Also with probability q, in order to keep the number of SrAp cells approximately constant 
, an SrAp cell can produce two differentiated daughter cells (B spermatogonia) that both go on to make sperm thus eliminating one SrAp cell lineage.
For a given mutation rate per cell division the Mx/Av and F<50 statistics increase with the value of the symmetric parameter q, therefore to make the test as conservative as possible we only consider the case where q equals the maximum possible value 0.5 (so one-half of the divisions produce two SrAp cells and one-half produce two B spermatogonia). As in the test of the original hot spot model, we simulate the model with the mutation rate per cell division that best matches the observed Av until we have one million simulations with Av within 5% of the observed Av. We again consider sample #374-1. For the data the Mx/Av ratio is 87, while in 95% of simulations this ratio is between 5.6 and 12.9. For the data F<50 is 90%, while in 95% of simulations this fraction is between 55% and 70%. The symmetric variant to the hot spot model increases these statistics greater than the level achieved by the original hot spot model, but not as high as is observed for the data. Since in one million simulations both the Mx/Av ratio and the F<50 fraction were always less than was observed in the data, this variant is also strongly rejected with p-value less than 10−6. As shown in , the same conclusion holds for the other testes with substantial mutation clusters.
Previously, in order to explain the mutation clustering for the Apert syndrome mutations, we had proposed a role for selection 
. The selection model is based on the original hot spot model, and adds a selection parameter p: at each adult phase generation, a mutated SrAp divides symmetrically with probability p and asymmetrically with probability 1-p (after a symmetric division, each daughter SrAp reverts to asymmetric divisions until the next rare symmetric division). A similar model was proposed by Crow 
. Unlike the symmetric hot spot model considered above, non-mutated SrAp cells in the adult phase can only divide asymmetrically. Since the SrAp daughter of an SrAp cell is expected to remain near its progenitor, these rare symmetric divisions can cause mutation clusters to form and grow locally over time. The motivation for the selection model is that in model organisms it has been shown that stem cells can switch from asymmetric to symmetric divisions and back again, and that such behavior can depend on factors intrinsic and extrinsic to the stem cells 
Consider again sample #374-1 for the MEN2B mutation. The selection model has two free parameters: the mutation rate per cell division and the selection parameter p. With these two free parameters, we can now try to match both the Av and the Mx. As before, we only consider those simulations such that the simulated Av is within 5% of the observed Av. For the data the Mx/Av ratio is 85, and in 95% of simulations this ratio is between 26 and 92. For the data the F<50 fraction is 90%, and in 95% of simulations this fraction is between 86% and 93%. Therefore the selection model is consistent with the data for this testis. The inferred selection parameter p is 0.0084, so if the mutated SrAp cells divide symmetrically approximately 1% of the time, this is sufficient to form mutation clusters similar to what is observed in the testes. Moreover, if we now take the inferred mutation rate per cell division (4.4×10−11
) and set the selection parameter p equal to zero, then simulations of the model produce mutation frequencies similar to the already established genome averages 
, implying that the mutation rate per cell division is not elevated at this nucleotide 
The selection model can explain the paternal age effect since the mutation clusters will grow as the man ages and the male mutation bias since this growth is only in the male germline. The selection model can also explain the elevated mutation frequencies and the clustering observed for all the other testes with substantial mutation clusters (results not shown). However, this model predicts that the samples in the oldest age group will have the highest mutation frequencies and the most intense mutation clusters, and thus cannot explain the low mutation frequencies and lack of mutation clusters observed in three of the testes from this age group (see next heading).
We have also considered a “combined” model which merges the symmetric variant to the hot spot model with the original selection model: all SrAp randomly divide asymmetrically, divide symmetrically or divide to produce two differentiated daughter cells, but the mutant SrAp are more likely than the wild type SrAp to divide symmetrically. However, we did not pursue this combined model further since it introduces an additional model parameter without improving the fit of the selection model.
Heterogeneous oldest age group
Our results for the oldest individuals were surprising in that three (#60955, #57650, and #60507) of the five samples had unexpectedly low levels of the c.2943T>C MEN2B mutation similar to young testes (). One trivial explanation for such low levels of mutation was germ cell degradation in these three older samples and that this data should be discarded. To examine this question we looked at the distribution of a different mutation. We used the version of our assay originally designed for the Apert syndrome c.755C>G mutation 
on the same 14 testes we studied for the MEN2B mutation (plus one 21 year old sample, #63205, which we had not studied for MEN2B). Figure S1
shows the Apert syndrome mutation distribution for all the testes and Dataset S2
contains the mutation frequency estimates for every piece. summarizes the mutation frequency statistics for each testis and shows the mutation distribution for the five testes in the oldest age group. The results showed substantial Apert mutation clusters in all five older testes including those with the fewest MEN2B mutations. Therefore general germ cell degradation in the three testes cannot explain the heterogeneity in the MEN2B data. Another observation, which will play a part in the subsequent modeling, is that for the middle-aged group of testes the median Av is ~4-fold higher and the median Mx is ~3-fold higher for the Apert mutation compared to the MEN2B mutation (see and ).
Apert syndrome c.755C>G mutation frequency summaries from 15 testes.
Distribution of the Apert syndrome c.755C>G mutation for the oldest age group (75–80 years).
Selection model incorporating cell death
To try and explain the MEN2B data on the oldest age group, we concluded that the only acceptable model modification was to incorporate age-dependent cell death. Researchers have shown that the number of SrAp cells decreases as men grow old 
: from the ages of 31–40 to 61–70 there is a slight decrease from 162 cells per mm2
of seminiferous tubule cross section to 120 per mm2
, but from the ages of 61–70 to ages 81–90 there is a much more rapid decrease from 120 mm2
to 57 per mm2
. There is a similar pattern of decrease for the A-dark spermatogonia (Ad). Believed to act as “reserve” stem cells, the Ad remain quiescent until the number of SrAp cells is sufficiently diminished to activate the Ad to replace the SrAp 
. Since the Ad have not been cycling as frequently as the SrAp until this point, they are less likely to have acquired any mutations, and thus the pool of SrAp cells is replenished with a fresh supply of primarily non-mutated cells. We have incorporated cell death into the selection model by assuming that all SrAp, whether or not they are mutated, die at the same rate. The details of this new model can be found in Text S3
The selection model incorporating cell death can explain all of the testes data for both MEN2B and Apert syndrome. For those testes with substantial MEN2B mutation clusters, as before, we varied the mutation rate per cell division and the selection parameter to try to match both the Av and the Mx, and we only considered those simulations such that the simulated Av was within 5% of the observed Av. shows that this model is consistent with these testes. For those testes without substantial mutation clusters, we did not fit each testis separately (many low values of the model parameters would suffice) but rather for a given age and set of parameter values we simulated the model many times to see how often the simulations were typical of a young donor and how often they were typical of a middle-aged donor (see Text S3
for details). For MEN2B, we found that when we set the selection parameter at the low end of the range in then most simulations of an older individual were typical of a young donor. However, when we increased the selection parameter to the median value in then most simulations of an older individual were typical of a middle-aged donor. Thus a relatively slight variation in the selection parameter between individuals can explain the heterogeneity in the older donors for MEN2B. As for the Apert syndrome mutation, the Av and Mx values in and are greater for the Apert mutation than the MEN2B mutation, leading to slightly higher inferred values for the selection parameter for the Apert mutation (see Table S1
). When we increased the selection parameter to the median value for the Apert mutation in Table S1
then almost all of the simulations of an older individual were typical of a middle-aged donor in agreement with . The slight increase in the value of the selection parameter for Apert syndrome compared to MEN2B can explain the difference in the oldest age groups for these two mutations. Finally for MEN2B and the youngest donors, even using the greatest values of both the mutation rate per cell division and the selection parameter from , almost all simulations of a 23 year old are typical of a young donor in agreement with . For these parameter values, the probability of a substantial mutation cluster developing in a 23 year old is very small due to the relatively low number of adult phase generations.
Selection model incorporating cell death model parameters and simulation results for those testes with substantial MEN2B mutation clusters.