A model for mutation in bacterial populations

We describe the evolution of E.coli populations through a Bak-Sneppen type model which incorporates random mutations. We show that, for a value of the mutation level which coincides with the one estimated from experiments, this model reproduces the measures of mean fitness relative to that of a common ancestor, performed for over 10,000 bacterial generations. The last decade has seen a renewed interest in the study of biological evolution. Besides the painstaking work of analyzing fossil records, spanning 10 8−9 years, there are now experiments performed by Lenski and co-workers with E.coli, which already comprise tens of thousands of generations [1]. This has opened an entire " experimental evolution " area, and their data are extremely useful to study the long-term evolutionary dynamics of populations. We first briefly review the essentials of Lenski's experiment. It considered 12 initially identical populations, each of them founded by a single cell from an asexual clone, propagating in similar environments during 1500 days, in the following manner. At the beginning of each 24 hour period, an initial batch of around 5×10 6 bacteria is placed in a growth medium, and, 24 hours later, when the population has increased by a factor of about 100, which implies log 2 100 ≈ 6.6 generations, the process is repeated by starting a new batch with 1% of the bacteria present. The mean cell volume and mean fitness of bacterial populations relative to the ancestor RF were measured every 100 generations. The RF of these populations, in its g−th generation is measured by placing a sample of each of them in contact with its original ancestor (un-freezing a sample taken at time 0, the first generation), and measuring the ratio of their rates of increase. In all the experiments the RF shows a rapid increase for ≈ 2000 generations after its introduction into the experimental environment, and then becomes practically constant. The average asymptotic value of the relative fitness is ¯ RF ∞ ≃ 1.48 [1] (the bar denotes the average over the 12 populations). This behavior may be parameterized by an hyperbolic fit, f = (A + Bg)/(C + Dg), where f represents the relative fitness of the g-th generation and A, B, C, D are constants. On the theoretical side, several approaches to the evolutions of species in interactive systems have been proposed: the well known models of Kauffman and collaborators for co-evolving species operating at …