New algorithms for Luria–Delbrück fluctuation analysis

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Abstract

Fluctuation analysis is the most widely used approach in estimating microbial mutation rates. Development of methods for point and interval estimation of mutation rates has long been hampered by lack of closed form expressions for the probability mass function of the number of mutants in a parallel culture. This paper uses sequence convolution to derive exact algorithms for computing the score function and observed Fisher information, leading to efficient computation of maximum likelihood estimates and profile likelihood based confidence intervals for the expected number of mutations occurring in a test tube. These algorithms and their implementation in SALVADOR 2.0 facilitate routine use of modern statistical techniques in fluctuation analysis by biologists engaged in mutation research.

Introduction

Estimation of mutation rates plays an indispensable role in genetic and evolutionary studies. The most widely used methods for estimating mutation rates have been those based on the fluctuation test protocol devised by Luria and Delbrück [1] about 60 years ago. Although the major concern of Luria and Delbrück at the time was to address the issue of the origin of certain type of bacterial mutation, the Luria–Delbrück fluctuation test protocol has left indelible imprint on almost every present-day method for estimating mutation rates. Successful use of the fluctuation test protocol in estimating mutation rates depends not only on efficient methods for computing the probability distribution of mutants, but also on statistically sound techniques for computing point and interval estimates of mutation rates. The past half a century has seen vigorous mathematical developments in tackling the first issue, but much less effort has been expended on the second issue. (Most of the theoretical contributions in the field are summarized in Ref. [2].) Lea and Coulson [3] were the first to outline a method of finding the maximum likelihood estimate (m.l.e.) of the fundamental parameter m – the expected number of mutations occurring in a parallel culture. They gave no specific algorithms but offered numeric tables to assist the user in finding m.l.e.s of m. Jones et al. [4] appear to be the first to give an explicit and practical algorithm for computing the m.l.e. of m. The Jones algorithm for computing m.l.e. is inexact in the sense that the log likelihood function and its derivatives are computed by truncating an infinite series representing the mutant distribution. This algorithm has not been extended to the case where growth rate of non-mutants can differ from that of mutants. (We shall refer to this case as the differential growth case.) Stewart [5] is the first to seek a systematic method for constructing confidence intervals (CIs) for m. The Stewart approach relies on computer simulation; the user consults computer generated charts or tables to construct a CI. This approach is not convenient to apply and has not extended to the differential growth case either. The first computationally feasible method for constructing CIs for m appears to be that proposed by Zheng [6]. The Zheng method uses sequence convolution to compute expected Fisher information, whence Wald type asymptotic CIs can be obtained. Theoretical considerations and simulations have shown that this method is efficient in computing CIs when growth rates of non-mutants and mutants are the same (the equal growth case) or when the two growth rates differ but the ratio is known in advance. However, there is an uncertainty in computing the expected Fisher information for the differential growth case when the ratio of the two growth rates is unknown; this uncertainty is due to lack of knowledge about the rate of convergence of certain infinite series employed by the algorithm to compute expected Fisher information.

A major goal of the present report is to propose new algorithms that effectively obviate the above difficulty. The algorithms to be presented use a sequence convolution technique to compute likelihood ratio based CIs for the equal growth case and profile likelihood based CIs for the differential growth case. The new methods for computing CIs also provide a statistical procedure that can help choose between the equal growth model and the differential growth model. This paper also gives an explicit and more efficient method for computing m.l.e.s based on the Newton–Raphson algorithm. An important feature of this method is its use of the exact mutant distribution, the exact log likelihood function and its exact derivatives – a desirable feature lacking in the Jones algorithm. Finally, unlike the Jones algorithm and the Stewart approach, the methods presented here embrace the differential growth case. All these new algorithms have been incorporated into version 2.0 of SALVADOR (available at http://library.wolfram.com/infocenter).

This paper does not address the issue of plating efficiency [7], [8], for which two sources can cause concern. The first is deliberate partial plating. For example, in some of the first fluctuation tests [1], only a 0.05 ml sample from a 10 ml broth culture was plated. The practice of partial plating was common until the late 1980s when various assumptions and laboratory procedures were scrutinized to enhance the validity of conclusions drawn from fluctuation experiments. As a result, complete plating now appears to be the norm. The second cause is cell death. This is a relevant issue in experiments involving eukaryotic cells, but effects of cell death seem negligible in experiments involving bacterial cells. It is straightforward in principle to extend most of the results in this paper to the case of imperfect plating, but satisfactory solution of the problem requires a separate investigation.

Section snippets

Nomenclature and notation

In a slightly idealized fluctuation test experiment a small number, N0, of non-mutant cells are seeded into each of n test tubes containing broth or a similar liquid culture. After an incubation period the population of non-mutant cells in each tube expands exponentially to a size of NT, and the whole contents of each tube are then plated onto a selective and solid culture in a dish. Each dish is then incubated until every mutant cell transferred from the tube grows into a visible mutant

The equal growth case

When non-mutants and mutants grow at the same rate, the p.g.f. of the mutant distribution is given by (2) or (3). Because ϕ is either assumed to be known beforehand or set to unity for simplicity, m is the only unknown parameter. The difficulty that is to be overcome by the present paper arises from the fact that the p.m.f. p(k; m, ϕ) is not amenable to the usual algebraic manipulation. As first noticed by Ma et al. [12], the p.m.f. is defined only by a recursive relationp(0;m,ϕ)=e-mp(k;m,ϕ)=mkj=

The mutant distribution

When growth rate of non-mutants β1 differs from that of mutants β2, the mutant distribution is often identified by the ratio ρ = β2/β1 along with m and ϕ. Denoted by M(m, ρ, ϕ), this distribution has its p.g.f. of the form (e.g. Eq. (23) in Ref. [6])G(z;m,ρ,ϕ)=exp-m+mρϕk=1Bωk,1+1ρzk,where ω = 1  (1  ϕ)ρ and Bz(a,b)=0zta-1(l-t)b-1dt is the incomplete beta function. We shall confine our attention to the commonly used approximate case where ϕ = 1 (and hence ω = 1). Because the reciprocal of ρ, denoted

A case study

To determine whether theoretical mutant distributions were in agreement with experimental observations, Boe et al. [18], [19] conducted perhaps the largest and most meticulously designed fluctuation experiment in history. The mutants studied were Escherichia coli cells resistant to nalidixic acid. Twenty three tests (48 cultures each) were performed under slightly different conditions, and the data were then pooled for goodness of fit test. The data are quoted below, with the notation x(k)

Concluding notes

Interval estimation of mutation rates has received scanty attention. The recent approach to computing Wald type asymptotic CIs for m is practical for the equal growth case, but, due to difficulties in computing infinite series, the approach is not convenient to use in the differential growth case. The present approach to computing likelihood ratio based CIs is practical for both types of cases. In practice these two methods complement each other. The method proposed here to compute m.l.e.s

Acknowledgments

I extend grateful appreciation to two anonymous reviewers and the handling editor. I in particular acknowledge cordially that one reviewer’s detailed comments enabled me to improve the presentation significantly.

References (19)

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