Improved inference of mutation rates: I. An integral representation for the Luria-Delbruck distribution

The estimation of mutation rates is ordinarily performed using results base d on the Luria-Delbruck distribution. There are certain difficulties associ ated with the use of this distribution in practice, some of which we addres s in this paper (others in the companion paper, Oprea and Kepler, Theor. Po pul. Biol., 2001). The distribution is difficult to compute exactly, especi ally for large values of the random variable. To overcome this problem, we derive an integral representation of the Luria-Delbruck distribution that c an be computed easily for large culture sizes. In addition, we introduce th e usual assumption of very small probability of having a large proportion o f mutants only after the generating function has been computed. Thus, we ob tain information on the moments for the more general case. We examine the a symptotic behavior of this system. We find a scaling or "standardization" t echnique that reduces the family of distributions parameterized by three pa rameters (mutation rate, initial cell number, and final cell number) to a s ingle distribution with no parameters, valid so long as the product of the mutation rate and the final culture is sufficiently large. We provide a pai r of techniques for computing confidence intervals for the mutation rate. I n the second paper of this series, we use the distribution derived here to find approximate distributions for the case where the cell cycle time is no t well-described as an exponential random variable as is implicitly assumed by Luria-Delbruck distribution. (C) 2001 Academic Press.