Continuum approximation of the Luria-Delbrück distribution

The Luria-Delbrück distribution came out of a study designed to test whether mutations in a bacterial population subject to strong selection arise in response to the selective agent, or independently of it. The distribution of the number of mutants, M, in a culture of size N, that has been grown under conditions in which these mutations did not confer a selective advantage to the cells bearing them, came to be known as the Luria-Delbrück distribution. It constitutes the basis for mutation rate estimation using the so-called "fluctuation analysis". Such an experiment involves growing a number of bacterial cultures from one cell to a final culture size N, and estimating the number of mutants in each of the parallel cultures. The mean or median number of mutants, or the proportion of cultures with no mutants are the statistics generally employed for mutation rate estimation. Fluctuation analysis has been applied to the study of mutational processes not only in prokaryotic, but also eukaryotic genomes (1994). The mathematical study of the L-D distribution, initiated by Luria and Delbrück themselves, was elaborated by Bartlett (1978); Kendall (1948); Lea and Coulson (1949). More recently, a revisiting of the mutational processes in bacteria initiated by Cairns et al. (1988) caused another wave of mathematical exploration of the L-D distribution (). These efforts made the numerical computation of the L-D distribution reasonably efficient, though no closed form solution for it has been found.

In section , I described the basic setup for fluctuation analysis, which I used to construct my computational model. This setup is assumed in the derivation of L-D as well, with the restriction that at any moment, all replicators have equal probability of dividing. This can be shown to be equivalent of assuming an exponential distribution for the cell cycle time (recall that in my simulations I allowed for more general forms of the cell cycle time distribution). If we work in the regime where the product of mutation rate and culture size, $\mu N$, is large, but the product $\mu \log(N)$, giving the probability that any given cell is a mutant, is small, we can use the following approximation. Instead of taking the number of mutants, M as the random variable of interest, we take the proportion of mutants in the culture, $X \equiv M/N$, and approximate it as a continuous random variable. The validity of this approximation follows from the prior assumption $\mu N \gg 0$. We then determine the density function for X, and attempt to generalize this distribution for non-exponential cell-cycle time distributions.

We start from the generating function of the Luria-Delbrück distribution, defined as

$$g_N(s) \equiv \mbox{E}[s^M\vert N].$$ (6.29)

Where E$[\cdot\vert N]$ is the conditional expectation. For L-D, this generating function was found by Bartlett:

$$\log g_N(s) = \frac{\mu N (1-s)}{s} \log (1 - s(1-\frac{N_0}{N})),$$ (6.30)

with $\mu$ being the mutation rate per cell per division, N₀ the initial number of cells and N the final number of cells in the culture. Retrieving the probability distribution from the generating function is non-trivial, and much of recent work has been focussed on ways of producing efficient means for doing so in the absence of closed-form solutions. It turns out that, if we work in the continuum limit, we can derive an integral form of the distribution. Stated formally, the conditions that need to be fulfilled for the continuum approximation to hold are:

$$\mu (N- N_0) \gg 0$$ (6.31)

and

$$\mu \log(N/N_0) \ll 1.$$ (6.32)

The continuum version of L-D will be designated cLD. The characteristic function of cLD is obtained from the generating function by substituting s = e^{-i z}. This being the Fourier transform of the density function, one could use the Fourier theorem to recover the probability distribution, p(x):

$$p(x) = \frac{1}{2 \pi} \int_{-\infty}^{+\infty} g(e^{-iz/N}\vert N)e^{ixz}\: dz$$ (6.33)

Through complex integration, and a change of variable, one arrives at an integral form in terms of a scaled variable

$$\xi = \frac{x}{\mu} - \frac{1}{\beta},$$ (6.34)

where x is the proportion of mutants and $\beta = \mu N$. The distribution of $\xi$ is given by:

$$p(\xi) = \frac{1}{\pi} \int_{0}^{\beta - \epsilon} {\left... ...right)}^{\beta \xi} \sin\left(\pi(w + \epsilon)\right) \:dw,$$ (6.35)

where $\epsilon = \mu N_0$. This integral form can be used directly to retrieve the cLD distribution.