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## Combining data with different mutation rates

LAMARC offers three ways to handle variation in mutation rate among markers. This file describes all three and gives some guidelines on their correct use. Note that we will use the term 'segment' for a contiguous stretch of sites with markers of the same data type, and 'region' to indicate one or more linked segments of the same or different data types.

Variation within a contiguous segment. If the mutation rate (or fixation rate) may vary from site to site within a single contiguous genetic segment, such as a DNA sequence or group of linked microsatellites, the best approach is to use the "Multiple rate categories" option of the appropriate data model. This option is described in the data models section of the documentation.

Known variation among segments, regions, and/or data types. If you know in advance that one region has, say, a tenfold higher mutation rate than another, the best approach is to set the "Relative mutation rate" option of the appropriate data model. Choose one region as the standard, and set its relative mutation rate to 1.0; set the others proportionally. This is a good approach if you have, for example, a DNA sequence and some microsatellites, and are fairly sure that microsatellite shift mutations are about 1000 times more common than single base pair substitutions. The unit of comparison is the single marker: one microsatellite, one base pair of DNA, one SNP.

This approach can also be used for areas where the mutation rate variation is known for a contiguous stretch of markers, for example a DNA sequence containing both introns and exons. If each intron and exon is assigned to a unique segment, the known relative mutation rates can be set explicitly.

Even if you are not perfectly sure of the ratio between your data, using a reasonable guess will still be better than allowing the default of identical rates everywhere. If you are not sure whether microsatellites mutate 1000x or only 100x faster than DNA, pick an intermediate value. Assuming that they mutate at the same rate will definitely give bad results.

Unknown variation among regions. If you suspect that your regions vary in mutation rate, but you don't have any information on their specific rates, you can assume that these rates are drawn from a gamma distribution. The gamma distribution is a somewhat arbitrarily-chosen, flexible statistical distribution which varies from looking exponential when its scaled shape parameter α is low, to looking like an increasingly narrow bell curve as α increases. Low values of α correspond to cases in which most regions are nearly invariant, and a few evolve rapidly. High values of α correspond to cases in which the single-region mutation rates are approximately normally distributed about the mean single-region mutation rate. (The gamma distribution actually has two parameters, a "shape parameter" α and a "scale parameter" β, but LAMARC sets β = 1/α to avoid overparameterization, and to allow it to work with a distribution whose mean, the product αβ, is 1.)

LAMARC can estimate α if you have no prior conception of what a good value here would be (though a reasonable starting guess will speed up maximization). In practice, it needs more than two or three regions to make a reasonable estimate of α. If you have only 2-3 regions, it is best to guess at their ratio, or fix α to a value you find reasonable; estimation of α is likely to fail because not enough information is available. (With only one region, α cannot be estimated and will not be used.)

If your data consist of several microsatellites and several DNA or SNP regions, the real distribution of mutation rates probably resembles a two-humped camel and not a gamma distribution at all. You can try fitting a gamma anyway, but be aware that you are fitting an inappropriate model. A better alternative is to guess the relative mutation rates: the large difference between microsatellites and DNA data probably trumps any differences among each group. You can even do both, giving a mutation rate constant for each region and then adding a gamma on top. We believe that this has the effect of drawing the different regions from versions of the same gamma but with its mean shifted by the given mutation rate ratio. However, this combination has not been extensively tested: use it at your own risk. It assumes that α is the same for DNA and microsatellites, which is probably not the case, but sometimes a shaky assumption is better than nothing.

Please note that Lamarc can only apply a gamma distribution to single-region relative mutation rates if all populations are assumed to remain constant in their respective sizes. This is due to a mathematical complication in the way Lamarc implements the gamma distribution (in this case, Lamarc does not approximate the gamma distribution by a histogram of relative rates). This means that Lamarc cannot simultaneously model the gamma "force" and the force of exponential population growth, even for fixed values of α or g. If you believe one or more of your populations is rapidly growing or shrinking, and you think the single-region relative mutation rates are approximately gamma-distributed for your data, then your best bet is to estimate the relative rates by some other method and supply these to Lamarc as constants, and then to proceed to estimate growth rates.

Also, because of the way Lamarc implements this feature, it can only be used for maximum-likelihood analyses. If you want to perform a Bayesian analysis, and you think the single-region relative mutation rates are approximately gamma-distributed for your data, then your best bet is to estimate the relative rates by some other method and supply these to Lamarc as constants, and then proceed with your Bayesian analysis.

Bottom line. If your data has mutation rate variation within a segment, use the "Multiple rate categories" option of the mutation model. If it has variation between segments and you know the relative rate of each, use the "Variable mutation rate" option of the mutation model. If it has variation among regions and you don't know the rates of individual regions, you can assume that they are drawn from a gamma, but this is likely to work well only if you have more than 3 regions, and is not ideal if the regions fall into large classes with distinctly different rates. It is best suited for large collections of one data type, such as multiple regions with DNA.

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