Note: this post is a guest post by Rohan Maddamsetti, posted by the regular blog author, Titus Brown. Typos are Titus's fault. Flaws in logic are Rohan's ;). See the paper on arXiv  and the discussion on Haldane's Sieve, also.
I recently wrote a short paper explaining some interesting results reported by Inigo Martincorena and coauthors . Unfortunately for me, Nature rejected it last Friday. I still stand by my conclusions, but you should take these thoughts with a grain of salt -- in this blog post, I'm going to talk about why I think these results are important and interesting, but it's clear that I'm going to have to do more science to support or reject these ideas.
In my paper, I replicated the main result of the original paper, which is that neutral (nearly-neutral if you want to be a stickler) variation across E. coli genes varies over several orders of magnitude. Neutral variation is measured by $\theta_s = 2 N_e \mu$, where $N_e$ is the coalescent effective population size, and $\mu$ is the mutation rate per nucleotide. Inigo and I disagree about the source of this variation - I believe it is in $N_e$, while Inigo believes it is due to $\mu$.
A very important semantic point needs to be made first. Inigo and I examine the rate of synonymous substitutions, and use this as a proxy for the rate of neutral mutations. This is why we talk about $\theta_s = 2N_e\mu$, rather than $\theta = 2N_e\mu$ -- because neither of us are actually measuring the de novo mutation rate. Drawing conclusions about the mutation rate from the rate of substitutions is justified by a classic argument which goes as follows: in an idealized population with $N_e$ individuals, with a neutral mutation rate $\mu$ per generation, new mutations enter the population at a rate of $N_e\mu$ per generation. Since each individual is as likely as any other to eventually be the last common ancestor of the population (probability $1/N_e$), new mutations substitute (end up in the last common ancestor of the population) at a rate $N_e\mu*(1/N_e) = \mu$.
Why talk about $\theta_s = 2N_e\mu$ in the first place? This is because $\theta_s$ measures the number of neutral mutations that separate two lineages on an evolutionary tree since the last common ancestor (Figure 1). So, $\theta_s$ is a natural measure of genetic variation, and can be estimated on a gene-by-gene basis. Inigo and I did this for several thousand E. coli genes using a program called OmegaMap .
Time is scaled by $N_e$ generations so that the formidable mathematical machinery of coalescent theory (see the discussion of the coalescent in Rice's Evolutionary Theory , or Wakeley's Introduction to Coalescent Theory ) can be used.
Inigo and I both show that some genes in E. coli have far more synonymous variation than other genes. Inigo thinks that this is caused by the molecular clock of neutral mutation ticking faster in some genes than others. I disagree with Inigo; instead, I believe that some gene genealogies in E. coli have much longer branches than others. In other words, more time has passed since the last common ancestor of Individual 1 and Individual 2 in some genes than in others.
This does not make sense assuming that all genes traveled together in the lineages of Individuals 1 and 2 since they diverged; however, this is not the case for bacteria, due to recombination and horizontal gene transfer. Genes are fairly free to "jump ship" as it were; any two different E. coli strains can differ by up to thousands of genes! This means that more diverged, older versions of genes can jump into an E. coli genome, bringing much more synonymous genetic variation than would be expected if all genes diverged at exactly the same time.
Horizontal gene transfer can occur through a number of mechanisms, some of which are "selfish" (hitchhiking through viruses, plasmids, transposons, or other selfish genetic elements); others of which are "cooperative" (homologous recombination among related bacteria, conjugation, or transformation from environmental DNA sequences). I like to imagine the following scenario: imagine a "wind" of diverse alleles blowing into a population of E. coli (this "wind" is non- conservative migration of alleles into the population). Genes under purifying selection can resist this wind more strongly, and will have a shorter coalescence time than genes that cannot effectively resist the input of diverse alleles from horizontal gene transfer. Under this model, I predict that there should be an association between $\theta_s$ and HGT in all clades of bacteria, and not simply in Escherichia. Further, the strength of this association should vary with the complexity of the bacterial community from which the genome was isolated. Perhaps obligate symbiont bacteria with small population sizes will show little variation in $\theta_s$ if they don't have many potential partners for horizontal gene transfer.
However, I think the coolest prediction of this model is that some genes might have much larger population sizes than other genes. I think this has deep implications for how speciation works in bacteria. Perhaps the "core genome" is made up of cooperative genetic elements that need each other to survive, while the "pangenome" contains a lot of genetic cruft; commensal and parasitic gene sequences that have recently entered the genome, and that selection will soon wash out. One prediction of this hypothesis is that genes with high $\theta_s$ should tend to be "commensal genetic elements" that spread easily among genomes without contributing much to adaptation. In contrast, horizontally-transferred genes that are important in adaptation, say for antibiotic resistance, should have a short coalescence time in the population (because they have been selected for recently), and so should have low $\theta_s$. Are multi-level selection approaches necessary to understand bacterial speciation and genome evolution? I don't know. I think that collecting more genome data from finer and finer timescales (going from 'Comparative genomics' to 'Population genomics' to 'Meta-population genomics' to 'Community genomics') will help shed light on how complex mutational processes beyond point mutation affect genomes over short time-scales, and how much of these mutational effects end up being purged by selection over longer time-scales.
As an example of complex mutational processes, the point mutation rate across the 12 lineages of Rich Lenski's long-term E. coli evolution experiment (LTEE - see ) is not constant! This is due to the fixation of mutations in some lineages (but not others) that knock out different DNA repair pathways, increasing the mutation rate in some lineages by 100-fold. In fact, the vast majority of the synonymous substitutions that I observe come from these "mutator" lineages. Mutator phenotypes are often observed in natural bacterial populations as well. I think it's generally believed that mutator lineages tend to burn out over long periods of evolutionary time, so that mutator subpopulations don't contribute much to long- term adaptation. But given that over 50,000 generations of evolution, 98% of synonymous substitutions have occurred in mutator lineages, I wonder how much synonymous diversity in wild bacterial populations originated in mutators. Perhaps some genes mutated heavily in a mutator, then were able to jump into a non- mutator background. Or, perhaps some mutator lineages were able to fix their defective DNA repair machinery through horizontal gene transfer or homologous recombination with a non-mutator. I think this is an interesting open question.
There are also mutational processes in the LTEE that do not act like a Poisson process (any random process with a low rate of events occurring, where the events are independent from each other, converges to a Poisson process in the limit; point mutation is a great example of this). Even though the 12 lineages of the LTEE started from a common ancestor in 1988, there is an incredible amount of variation in the number and type of transposition events involving IS elements (IS elements are a kind of selfish genetic element in bacterial genomes) that occurred after 40,000 generations of evolution. These events are certainly important in the evolution observed in the LTEE, but they behave very differently from point mutations (Figure 2). Why this is, I really don't know.
Finally, I want to mention that a paper published this week in the Proceedings of the National Academy of Sciences by Heewook Lee and co-authors  also argues against Inigo's hypothesis that higher expressed genes have lower mutation rates. Unfortunately, the state of the art in sequencing technology doesn't allow for direct measurements of the de novo mutation rate (much lower error rates are needed). Instead, Lee et al. perform mutation accumulation experiments to measure substitution rates in very small populations of E. coli, thus minimizing the effects of selection. This paper in particular does a very nice job of reviewing the literature on bacterial mutation rates and their determinants. I recommend this paper for further reading if you're interested.
Thanks for reading!
p.s. Note from Titus -- the math in this blog post was formatted using the super-awesome MathJax system; see relevant posts on Circles and Tangents and Amic Frouvelle re configuring it for Pelican, the blogging system I use. The source for this post is on github if you want to see exactly how to put in formulas.