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## What do the population parameters mean?

This page explains what each of the population parameters estimated by LAMARC means, and what its units are. The same information is available elsewhere in the documentation but it is collected here for convenience.

- Theta
- M (migration rate)
- Epoch boundary time
- r (recombination rate)
- g (exponential growth rate)
- alpha (shape parameter of gamma distribution)

### Theta

Theta is two times the mutation rate (per site per generation) times the number of heritable units in the population. Thus, if you are considering diploid individuals (with two heritable units each) Theta is 4Nmu. If you have haploids, it is 2Nmu. For mitochondrial DNA, which is heritable only in females, it is Nmu (or 2N(f)mu).

If you wish to combine genes with different copy numbers, be sure to set the relative N appropriately or you will get a confused result. For example, if you are combining mitochondrial DNA and nuclear DNA, either set the relative N as 1 for mtDNA and 4 for nuclear (you will then estimate Theta on the mtDNA scale) or as 0.25 for mtDNA and 1 for nuclear (you will then estimate Theta on the nuclear scale).

Similarly, if you are combining data with different mutation rates, be sure to set the relative mu appropriately. Theta will be scaled by whichever mutation rate you select to be 1. So if you believe your microsatellites mutate 100 times faster than your DNA, either set the msat rate to 100 and the DNA rate to 1 (you will then estimate Theta in the DNA scale) or set the msat rate to 1 and the DNA rate to 0.01 (you will then estimate Theta in the msat scale).

If you wish to convert Theta to a headcount of individuals, you will need both an external estimate of the mutation rate, and an idea of the ratio between headcount population size and effective (breeding) population size. LAMARC cannot help you with either of these issues.

### M (migration rate)

LAMARC measures migration rate as M=m/mu, where m is the chance for a lineage to immigrate per generation, and mu is the mutation rate per site per generation. An M of 1 means that it is as likely for the sequence to migrate as it is for a site on the sequence to mutate. If there are different mutation rates for different genes, M will be given relative to the gene (if any) given a mutation rate of 1.0.

Biologists often want to measure migration rate as 4Nm. This is useful because when 4Nm is higher than one, the force of migration becomes strong enough to compete with genetic drift. To get 4Nm, multiply LAMARC's estimate of M by its estimate of Theta for the recipient population.

LAMARC is the wrong tool to estimate migration rates if 4Nm is larger than about 5, as it will go crazy trying to keep track of so many migration events. If the program STRUCTURE sees no structure in your populations, there is probably too much migration for LAMARC to succeed, and you may need to pool your populations together.

### Epoch boundary time

Epoch boundary times are estimated as number of generations back from the present (the time at which the data were collected) to the population splitting time, scaled by the mutation rate mu per site per generation. Thus, an epoch boundary time of 1 means that on average each site will have accumulated one mutation since the populations split, and would represent a very ancient split. If you want to know how many generations or years ago the populations split, you will need external information about mu.

Note that not all scenarios will allow inference of epoch boundary times, migration rates, and population sizes. For example, if the populations split yesterday there will be no information about their post-split sizes, and if they split very long ago there will be no information about their pre-split sizes. If migration between them is very high there will be no information about the split time. It is important to compare the Bayesian posteriors with their priors; a posterior that closely resembles its prior indicates a parameter about which the data can say nothing.

### r (recombination rate)

LAMARC measures recombination rate as r=C/mu, where C is the rate of recombination per inter-site link per generation, and mu is the mutation rate per site per generation. An r of 1 means that it is as likely for a recombination to occur next to a site as it is for that site to mutate, and is a rather high rate of recombination. If there are different mutation rates for different genes, r will be given relative to the gene (if any) given a mutation rate of 1.0.

LAMARC does not allow r to vary among populations.

For many comparative purposes you will want the recombination rate per locus rather than per site; you can obtain this by multiplying LAMARC's r by the number of sites minus 1 (as no meaningful recombination can occur rightward of the final site).

### g (exponential growth rate)

This is the hard one!

The parameter g is the exponent of an exponential growth rate formula which gives the Theta at a time t in terms of the modern-day Theta and the growth rate:

Theta(t) = Theta(modern) exp(-gt)

Time is measured in mutational units; that is, one unit of time is the time needed for each site to experience, on average, one mutation. The units of our mutation rate are mutations per generations, so the units of g are 1/generations. Almost no one finds this intuitive.

We define time as increasing into the past (the present is time 0) and as a result, a negative value of g indicates that the population has been shrinking (it was bigger in the past than it is now) and a positive value indicates that it has been growing (it was smaller in the past than it is now). A g of zero indicates constant size. This much we know even without knowing the mutation rate, but to say anything more we need to either know the mutation rate, or be comparing two populations with the same mutation rate (in which case, the one with higher g is growing faster). If we know the mutation rate, we can plug it into the equation above to get a feeling for what this implies in terms of Theta or effective population size.

Be aware that LAMARC's estimates of g are biased upwards (due to the shape of the likelihood surface) especially with only one or a few genes. If the estimate of g is positive and big, but the confidence intervals include zero, it's quite likely that there is in fact little or no growth. If the intervals exclude zero, that finding is generally reliable.

### alpha (shape parameter of gamma distribution)

LAMARC can use the gamma distribution to represent the unknown variation of mutation rate among genes. Gamma was chosen because it is a simple distribution that ranges from looking rather like an exponential (most genes mutate very little, a few mutate much more rapidly) to looking rather like a bell curve (there is an average mutation rate and genes are spread nearly symmetrically around it).

The gamma distribution has two parameters, but we fix the mean of the gamma to 1, which determines one parameter. This leaves only the shape parameter alpha (α). Low values of alpha, below 1, describe a gamma distribution which looks somewhat exponential. High values describe a gamma which looks somewhat like a normal. So if you estimate alpha as 0.3, this means that most of your genes have a low mutation rate but a few are much higher; if you estimate gamma as 20, this means that your genes are tightly clustered around their mean mutation rate. If all your regions mutate at exactly the same rate, the proper estimate of alpha would be infinite, but LAMARC will estimate an arbitrary high value that is practically equivalent to infinity.

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