By right tail being "heavy" do you mean a gamma distribution?

http://en.wikipedia.org/wiki/Gamma_distribution

Gamma distribution do not have numeric solution. I might choose it if I cannot find any other distribution. Thanks Rodrigo!

You can use the Gumbel distribution

http://en.wikipedia.org/wiki/Gumbel_distribution

or the Bastien-Ortet distribution which can fit better for bioinformatics score count. The R code is given in the paper:

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3023300/pdf/ebo-2010-159.pdf

Common heavy-tailed distributions[edit]All commonly used heavy-tailed distributions are subexponential.

Those that are one-tailed include:

the Pareto distribution;

the Log-normal distribution;

the Lévy distribution;

the Weibull distribution with shape parameter less than 1;

the Burr distribution;

the log-gamma distribution;

the log-Cauchy distribution, sometimes described as having a "super-heavy tail" because it exhibits logarithmic decay producing a heavier tail than the Pareto distribution.

Those that are two-tailed include:

The Cauchy distribution, itself a special case of both the stable distribution and the t-distribution;

The family of stable distributions, excepting the special case of the normal distribution within that family. Some stable distributions are one-sided (or supported by a half-line), see e.g. Lévy distribution. See also financial models with long-tailed distributions and volatility clustering.

The t-distribution.

The skew lognormal cascade distribution.

http://en.wikipedia.org/wiki/Heavy-tailed_distribution

I think it should be a log-normal. So just log it and throw a normality test at it. I know there's still a lot of debate about how legitimate a log transformation really is, but i think it is fine.

Also, this publication may help:

http://www.nature.com/msb/journal/v7/n1/full/msb201128.html