This post follows up on my post from yesterday, which was about choosing a dilution factor in a microbial evolution experiment that avoids the loss of too many beneficial mutations during the transfer bottleneck.

If we only want to maximize the cumulative supply of beneficial mutations that survive dilution, then following the reasoning in yesterday’s post, we would choose the dilution factor (D) to maximize g N_{e} = (g^{2}) N_{min} = (g^{2}) N_{max} / (2^{g}), where N_{max} is a constant (the final population size) and D = 1 / (2^{g}). Thus, we want to maximize (g^{2}) / (2^{g}) for g > 0, which gives g = ~2.885 and D = ~0.1354, which is in agreement with the result of Wahl et al. (2002, Genetics), as noted in a tweet by Danna Gifford.

The populations would therefore be diluted and regrow by ~7.4-fold each transfer cycle. But as discussed in my previous post, this approach does not account for the effects of clonal interference, diminishing-returns epistasis, and perhaps other important factors. And if I had maximized this quantity, the LTEE would only now be approaching a measly 29,000 generations!

So let’s not be purists about maximizing the supply of beneficial mutations that survive bottlenecks. There’s clearly also a “wow” factor associated with having lots and lots of generations. This wow factor should naturally and powerfully reflect the increasing pleasure associated with more and more generations. So let’s define wow = g^{e}, which is both natural and powerful. Therefore, we should maximize wow (g^{2}) / (2^{g}), which provides the perfect balance between the pleasure of having lots of generations and the pain of losing beneficial mutations during the transfer bottlenecks.

It turns out that the 100-fold dilution regime for the LTEE is almost perfect! It gives a value for wow (g^{2}) / (2^{g}) of 75.93. You can do a tiny bit better, though, with the optimal ~112-fold dilution regime, which gives a value of 76.03.

Every day, we propagate the E. coli populations in the long-term evolution experiment (LTEE) by transferring 0.1 ml of the previous day’s culture into 9.9 ml of fresh medium. This 100-fold dilution and regrowth back to stationary phase—when the bacteria have exhausted the resources—allow log_{2} 100 = 6.64 generations (doublings) per day. We round that to six and two-thirds generations, so every 15 days equals 100 generations and every 75 days is 500 generations.

A few weeks ago, I did the 10,000^{th} daily transfer, which corresponds to 66,667 generations. Not bad! But as I was walking home today, I thought about one of the decisions I had to make when I was designing the LTEE. What dilution factor should I use?

If … if I had chosen to use a 1,000-fold dilution instead of a 100-fold dilution, the LTEE would be past 100,000 generations. That’s because log_{2} 1,000 = ~10 generations per day. In that case, we’d have reached a new power of 10, which would be pretty neat. As it is, it will take us (or rather the next team to take over the LTEE) another 14 years or so to get there.

I’ll discuss my thinking as to why I chose a 100-fold dilution factor in a bit. But first, here’s a question for you, which you can vote on in the poll below.

Let’s say that we had done a 1,000-fold daily dilution all along. And let’s say we measured fitness (relative to the ancestral strain, as we usually do) after 10,000 days. Do you think that the mean fitness of the evolved populations subjected to 1,000-fold dilutions after 100,000 generations (on day 10,000) would be higher or lower than that of the evolved populations subjected to 100-fold dilutions after 66,667 generations (also day 10,000)?

I’ll begin by mentioning a couple of practical issues, but then set them aside, as they aren’t so interesting. First, a 100-fold dilution is extremely simple to perform given the volumes involved (i.e., 0.1 and 9.9 ml). And the LTEE was designed to be simple, in order to increase its reliability. A 1,000-fold dilution isn’t quite as easy, as it involves either an intermediate dilution or the transfer of a smaller volume (0.01 ml), which in my experience tends to be a bit less accurate. Second, the relative importance of the various phases of growth—lag, exponential, transition, and stationary—for fitness would change a bit (Vasi et al., 1994).

Setting those issues aside, here was my thinking about the dilution factor when I planned the LTEE. In asexual populations that start without any standing genetic variation, the extent of adaptive evolution depends on both the number of generations and the supply rate of beneficial mutations. The supply rate of beneficial mutations, in turn, depends on the mutation rate (m) times the fraction of mutations that are beneficial (f) times the effective population size (N_{e}).

There are many different uses and meanings of effective population size in population genetics, depending on the problem at hand: the question is “effective” with respect to what process? Without going into the details, we would like to express N_{e} such that it takes into account the expected loss of beneficial mutations during the daily dilutions. To a first approximation, theory shows that the relevant N_{e} is equal to the product of the “bottleneck” population size right after the dilution (N_{min}) and the number of generations (g) between N_{min} and the final population size during each transfer cycle (Lenski et al., 1991).

The final population size in the LTEE is ~5 x 10^{8} cells (10 ml x 5 x 10^{7} cells per ml), and it is the same regardless of the dilution factor, provided that the bacteria have enough time to reach that density between transfers. The 1,000-fold dilution regime would reduce N_{min} by 10-fold relative to the 100-fold regime, although the 50% increase in the number of generations per cycle would offset that reduction with respect to the effective population size. Nonetheless, N_{e} would be ~6.7-fold higher in the 100-fold regime than in the 1,000-fold regime.

The greater number of generations in 10,000 days under the 1,000-fold regime would also increase the cumulative supply of beneficial mutations by 50%. Nonetheless, the extent of adaptive evolution, which is (under this simple model) proportional to the product of the elapsed generations and N_{e}, would be ~44% greater under the 100-fold dilution regime than the 1,000-fold dilution regime. So that’s why I chose the 100-fold dilution regime … I was more interested in making sure we would see substantial adaptation than in getting to a large number of generations.

Now you know why the LTEE has only reached 67,000 or so generations.

Of course, I could also have chosen a 10-fold regime, and by this logic the populations might have achieved even higher fitness levels. I could also have chosen a much higher dilution factor; even with a 1,000,000-fold dilution the ancestral strain could double 20 times in 24 h, allowing them to persist. Or at least they could persist for a while. With severe bottlenecks, natural selection becomes unable to prevent the accumulation of deleterious mutations by random drift, so that fitness declines. And if fitness declines to the degree that the populations can no longer double 20 times in 24 h, then the bacteria would go extinct as the result of a mutational meltdown.

Returning to the cases where the bottlenecks are not so severe, the theory that led me to choose the 100-fold dilution regime ignores a number of complicating factors, such as clonal interference (Gerrish and Lenski, 1998; Lang et al., 2013; Maddamsetti et al., 2015) and diminishing-returns epistasis (Khan et al., 2011; Wiser et al., 2013; Kryazhimskiy et al., 2014). It’s predicated, I think, on the assumption that the supply rate of beneficial mutations limits the speed of adaptation.

When the LTEE started, I had no idea what fraction of mutations would be beneficial. I think it was generally understood that beneficial mutations were very rare. But the LTEE and other microbial evolution experiments have shown that beneficial mutations, while rare, are not so rare as we once thought, especially once an experiment has run long enough (Wiser et al., 2013) or otherwise been designed (Perfeito et al., 2007; Levy et al., 2015) to allow beneficial mutations with small effects to be observed and counted.

So I think it remains an open question whether my choice of the 100-fold dilution regime was the right one, in terms of maximizing fitness gains.

And that makes me think about redoing the LTEE. OK, maybe not starting all over, as we do have a fair bit invested in the last 29 years of work. But maybe expanding the LTEE on the fly, as it were. We could, for example, expand from 12 populations to 24 populations without too much trouble. We’d keep the 12 original populations going, of course, but we’d spin off 12 new ones in a paired design (i.e., one from each of the 12 originals) where we changed the dilution regime. What do you think? Is this a good idea for a grant proposal? And if so, what dilution factor would you suggest we add?

Feel free to expand on your thoughts in the comments section below!

Note: See my next post for a bit more of the mathematics, along with a tongue-in-cheek suggestion for combining the effects of the beneficial mutation supply rate and a “wow” factor associated with having lots of generations.