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## Data Models (models of the mutational process)

This article describes available mutational models and how to use them. The first section covers basic capabilities shared by all models. Nucleotide models are covered next, followed by models suitable for microsatellites and general allelic data.

The models supported at this time are:

- F84
- GTR
- Stepwise
- Brownian-Motion
- K-Allele
- Mixed Stepwise/K-Allele

### General features of all mutational models

All mutational models offer multiple mutation-rate categories using the Hidden Markov Model of Felsenstein and Churchill. In this model, we do not know what rate each site has, but we assume that we know how many rates there are, the relative value of each rate, and the proportion of sites that are expected to have each rate. For example, we might assume that our data has 10% fast-evolving sites and 90% slow-evolving sites, and that the fast-evolving sites are 5 times faster than the others.

This would be indicated using the "Categories" option of the mutational model, giving 2 categories, the first category with a probability of 0.1 and relative rate of 5, and the second with a probability of 0.9 and relative rate of 1.

Additionally, you can indicate that the rates at adjacent sites are correlated by setting the auto-correlation coefficient. If you expect that, on average, runs of 10 sites will have the same rate, set this to 10. The default is 1--every site chooses its rate independently. Coding sequence has a different kind of correlation--every third site is likely to be much faster than the rest--and unfortunately LAMARC does not yet model this type of correlation. Coding sequence is currently best handled by two or three rate categories with no correlation.

Programs such as PAUP* can be useful in fine-tuning the rate model. In general, if you suspect your data has multiple rates, a multiple-rate model, even if not fully correct, is better than a single-rate model which is sure to be wrong.

The program will slow down in proportion to the number of rate categories; it is seldom useful to have more than 3-4 categories.

If you use multiple categories, the "mu" in your estimate of Theta=4N(mu) will be the weighted average of the rates. In our given example, that would be 1.4 times the slow rate.

Sometimes you will know that an entire region or segment has a significantly higher or lower mutation rate than the rest of your data. This is best dealt with by setting that segment's relative mutation rate, not by using categories. (This option is found in the data model menu as "R Relative mutation rate".) Be aware that the parameter estimates produced by LAMARC will be scaled proportional to a segment of relative mutation rate 1, even if no such segment is included in the data. That is, if you tell LAMARC that you have two segments, one with a relative mutation rate of 5 and the other with a relative mutation rate of 50, your final estimate of Theta will describe a fictional segment with a relative mutation rate of 1, and you will need to multiply by 5 or 50 to find the Theta of your actual segments.

### Nucleotide models

LAMARC provides two models suitable for nucleotide (DNA, RNA or SNP) data. The F84 model allows transitions and transversions to differ in rate, and also allows for different nucleotide frequencies. It can be used to emulate simpler models such as the Kimura 2-parameter or Jukes-Cantor models. The GTR model allows every combination of bases to have its own rate, and can emulate any simpler model, but is much slower and should be used only when it's really necessary. The Modeltest utility of Posada, used in conjunction with PAUP*, can be used to select the most appropriate model. Both models allow the user to specify data uncertainty.

#### F84 Model

The F84 or Felsenstein 84 nucleotide model distinguishes transition mutations (purine to purine, pyrimidine to pyrimidine) from transversion mutations and allows them to have different rates. This is controlled by the "ttratio" parameter, which is the expected ratio of transitions to transversions. Due to the way that these rates are handled internally, the ttratio must be strictly greater than 0.5. If you want the Jukes-Cantor model, in which transitions and transversions are equally probable (because there are twice as many possible transversions, this corresponds to a ratio of 0.5), set the ttratio to a value such as 0.50001.

PAUP* can be useful in estimating the ttratio. The value does not have to be very precise, but you risk misleading results if you analyze, for example, human mtDNA (whose correct ttratio may be around 30) using the default ttratio of 2.

The F84 model also allows the frequencies of the four bases to vary. LAMARC can estimate these frequencies from the data, or you can specify values (for example, obtained from PAUP* or from a larger data set). If your data set is very small, it is best to specify the values as the ones calculated from the data may be misleading.

#### GTR (General Time-Reversible) Model

The GTR model is the most complex tractable model of nucleotide evolution; it allows for six different rates (each base to each other base) and unequal base frequencies. The combination of rates and base frequencies must describe an equilibrium. We recommend use of PAUP* to estimate the rates and base frequencies. GTR is a slow model and should only be used if it's really necessary, but it frequently is necessary for virus data and other high-mutation situations.

Some releases of PAUP* provide only five rates for GTR. The sixth rate is 1 by convention.

### Microsatellite models

The microsatellite models are appropriate for data which is expected to vary up and down a ladder. They can also be used for elecrophoretic data. The K-Allele model is, in addition, appropriate for allelic data where nothing is known about the expected direction of mutations: an example would be presence/absence of some chromosomal rearrangement.

#### Stepwise Model

The stepwise model assumes that each mutation increases or decreases the number of repeats by one; larger changes always result from multiple mutations. It is suitable for microsatellite or electrophoretic data. Even if the microsatellite does not evolve in a perfectly stepwise fashion, this model may be adequate unless the violations are fairly common or large.

We allow for the possibility of alleles in the ancestry of the sample which were up to 5 repeats larger or smaller than the most extreme alleles observed in the data. Currently this cannot be adjusted by the user.

This model offers no user-settable parameters except for the rate categories options common to all data models.

#### Brownian-Motion Model

The Brownian model is a mathematical approximation of the stepwise model. It considers the mutational process as a continuous random walk around the starting point. This is much faster than the full stepwise calculation, but can break down if the number of mutations is very low. Breakdown will be signalled by data likelihoods of zero. We normally try the Brownian model first and resort to the Stepwise only if bad data likelihoods are observed. Results should be extremely similar.

#### K-Allele Model

This model treats all alleles as equivalent, with mutation from any allele to any other allele equally probable. It is the only model LAMARC offers for allelic data which are definitely not stepwise. The program assumes that the alleles observed in the data are the only possible alleles, so that if you present data with three different alleles, K=3. This can be a problem if you know that there are actually 4 alleles but one failed to occur in your data. At present there is no workaround for this.

#### Mixed Stepwise/K-Allele Model

This is a hybrid of the Stepwise and K-Allele models and may be appropriate for data in which most mutations are stepwise but a few much larger mutations occur. It has a parameter 'percent_stepwise' which controls the proportion of Stepwise changes: percent_stepwise of zero is a K-Allele model, percent_stepwise of one is a Stepwise model, and percent_stepwise of 1/2 asserts that K-Allele and Stepwise mutations are equally probable. LAMARC's implementation can attempt to optimize percent_stepwise to maximize the likelihood of the observed data.

Mixed K/S is a new and experimental model with which we have little experience. One difficulty is that the stepwise model generally considers the possibility of alleles a bit larger or smaller than any observed, but allowing those unobserved alleles in the K-Allele model makes K very large and may cause the model to rule out K-Allele type mutations. We have allowed the Mixed model to consider only alleles one step larger or smaller than the largest or smallest observed alleles. Even so, if the microsatellite being studied has only two, adjacent alleles, the Mixed model will consider 4 states whereas a normal K-Allele model would consider only 2. As a result, we expect LAMARC's estimates of percent_stepwise to be somewhat higher than the truth.

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