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A Bayesian two-level model for fluctuation assay


The fluctuation experiment is an essential tool for measuring microbial mutation rates in the laboratory. When inferring the mutation rate from an experiment, one assumes that the number of mutants in each test tube follows a common distribution. This assumption conceptually restricts the scope of applicability of fluctuation assay. We relax this assumption by proposing a Bayesian two-level model, under which an experiment-wide average mutation rate can be defined. The new model suggests a gamma mixture of the Luria-Delbrück distribution, which coincides with a recently discovered discrete distribution. While the mixture model is of considerable independent interest in fluctuation assay, it also offers a practical Markov chain Monte Carlo method for estimating mutation rates. We illustrate the Bayesian approach with a detailed analysis of an actual fluctuation experiment.

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I am most grateful to Professor P. Foster who drew my attention to the problem addressed here and who kindly provided unpublished experimental data. I am also indebted to two conscientious reviewers for many insightful comments that led to a substantially improved manuscript. The bulk of the simulations were generated on an IBM iDataPlex machine managed by Texas A&M Supercomputing Facility.

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Correspondence to Qi Zheng.



We begin with the p.g.f. of the Luria-Delbrück distribution (e.g., Ma et al. 1992)

$$ G(z;m)= \exp\left\{ m \left(\frac{1}{z}-1\right)\log(1- z)\right\}. $$

Because m is a positive parameter, it is customary to assume that m obeys a gamma distribution having the probability density function

$$ g(m;\alpha,\beta)=\frac{\beta^{\alpha}}{\Upgamma(\alpha)} m^{\alpha-1} e^{-\beta m}. $$

The p.g.f. of the LD-mixture distribution can then be derived as follows:

$$ \begin{aligned} G_1(z;\alpha,\beta) =\int\limits_{0}^{\infty} G(z;m) g(m;\alpha,\beta) dm\\=\frac{\beta^{\alpha}}{\Upgamma(\alpha)}\int\limits_{0}^{\infty}m^{\alpha-1}\exp[-\{\beta-(z^{-1}-1)\log(1-z)\} m ]dm\\=\left(\frac{1}{1-\beta^{-1}(z^{-1}-1)\log(1-z)}\right)^{\alpha}. \end{aligned} $$

If \(z\in(0,1)\), then (z −1 − 1)log(1 − z) < 0. Therefore, the above integral converges for \(z\in(0,1)\). As a result, the last expression in (15) is the p.g.f. of the mixture distribution. On the other hand, the p.g.f. of the \({B}^0\) distribution having parameters A and k is of the form (Zheng 2010)

$$ G_2(z;A,k) = \left( \frac{1}{1-A(z^{-1}-1)\log(1-z)} \right)^k. $$

Therefore, the p.g.f. \(G_1(z;\alpha,\beta)\) in (15) coincides with the p.g.f. of a \({B}^0(\beta^{-1},\alpha)\) distribution, as is claimed in the proposition. In addition, the probability mass function b 0(n;A,k) can be calculated recursively with the help of two auxiliary sequences. The first auxiliary sequence, denoted {η n }, is defined by

$$ \left. \begin{array}{ll} \eta_0=1+A\\ \eta_n=\frac{-A}{n(n+1)} & \hbox{ for } n\geqslant1 \end{array} \right\}. $$

Using this sequence, one can calculate the second auxiliary sequence {ξ n } as follows.

$$ \left. \begin{array}{ll} \xi_0=\log(1+A)\\ \xi_n=\frac{1}{n(1+A)}\left( n\eta_n-\mathop{\sum}\limits_{j=1}^{n-1}j\xi_j\eta_{n-j}\right) &\hbox{ for } n\geqslant 1 \end{array} \right\}. $$

The probability mass function can be computed by the following recursive algorithm (Zheng 2010):

$$ \left. \begin{array}{ll} b^0(0;A,k)=\exp(-k\xi_0)=1/(1+A)^k\\ b^0(n;A,k)=\frac{-k}{n}\mathop{\sum}\limits_{j=1}^n j\xi_j b^0(n-j; A,k) & \hbox{ for } n\geqslant 1 \end{array} \right\}. $$

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Zheng, Q. A Bayesian two-level model for fluctuation assay. Genetica 139, 1409–1416 (2011).

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  • Mutation rate
  • Fluctuation experiment
  • Bayesian model