A Bayesian two-level model for fluctuation assay

Abstract

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.

Appendix

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

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

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

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

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

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

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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

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

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

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

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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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