Optimizing the product of the wow factor and the beneficial mutation supply rate

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 = (g²) N_min = (g²) N_max / (2^g), where N_max is a constant (the final population size) and D = 1 / (2^g). Thus, we want to maximize (g²) / (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 (g²) / (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 (g²) / (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.

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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 log₂ 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 log₂ 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 10⁸ cells (10 ml x 5 x 10⁷ 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.

References

Gerrish, P. J., and R. E. Lenski. 1998. The fate of competing beneficial mutations in an asexual population. Genetica 102/103:127-144.

Khan, A. I., D. M. Dinh, D. Schneider, R. E. Lenski, and T. F. Cooper. 2011. Negative epistasis between beneficial mutations in an evolving bacterial population. Science 332: 1193-1196.

Kryazhimskiy, S., D. P. Rice, E. R. Jerison, and M. M. Desai. 2014. Global epistasis makes adaptation predictable despite sequence-level stochasticity. Science 344: 1519-1522.

Lang, G. I., D. P. Rice, M. J. Hickman, E. Sodergren, G.M. Weinstock, D. Botstein, and M. M. Desai. 2013. Pervasive genetic hitchhiking and clonal interference in forty evolving yeast populations. Nature 500: 571-574.

Lenski, R. E., M. R. Rose, S. C. Simpson, and S. C. Tadler. 1991. Long-term experimental evolution in Escherichia coli. I. Adaptation and divergence during 2,000 generations. American Naturalist 138: 1315-1341.

Levy, S. F., J. R. Blundell, S. Venkataram, D. A. Petrov, D. S. Fisher, and G. Sherlock. 2015. Quantitative evolutionary dynamics using high-resolution lineage tracking. Nature 519: 181-186.

Maddamsetti, R., R.E. Lenski, and J. E. Barrick. 2015. Adaptation, clonal interference, and frequency-dependent interactions in a long-term evolution experiment with Escherichia coli. Genetics 200: 619-631.

Perfeito, L., L. Fernandes, C. Mota, and I. Gordo. 2007. Adaptive mutations in bacteria: high rate and small effects. Science 317: 813-815.

Vasi, F., M. Travisano, and R. E. Lenski. 1994. Long-term experimental evolution in Escherichia coli. II. Changes in life-history traits during adaptation to a seasonal environment. American Naturalist 144: 432-456.

Wiser, M. J., N. Ribeck, and R. E. Lenski. 2013. Long-term dynamics of adaptation in asexual populations. Science 342: 1364-1367.

Some Wrinkles in Time

Today is another milestone for the E. coli long-term evolution experiment—the LTEE, for short. I did the 10,000^th daily transfer today at about noon.

[Yours truly, doing the 10,000^th LTEE transfers. Technician Neerja Hajela is keeping a close eye on me, and with good reason. Photo by Thomas LaBar.]

Some of you will remember we just celebrated the LTEE’s 29th birthday a few weeks ago, on February 24^th. And if you’re quick with math, you might be thinking: “Wait a second: 29 years times 365 days per year is a lot more than 10,000 days. Have Lenski and his team screwed up?”

The answer is both yes and no. Let me explain.

The LTEE began on February 24, 1988 [1, 2].

From February 24, 1988, to March 13, 2017, equals 10,609 days on which we could have done transfers. But we’ve only had 10,000 transfers. What happened to those other days?

In short, the bacteria spent the 609 “lost” days in a freezer at –80°C or in a refrigerator at 4°C.

One chunk of days was lost when the LTEE was moved from my lab at UC-Irvine, where I started the experiment, to MSU, where it is today. Moving a lab is difficult: it requires moving people, moving equipment and materials, often renovating space, obtaining new supplies and equipment, hiring new people, and trouble-shooting and otherwise getting everything organized to resume work [3].

We lost 191 days from April 8, 1992, when the 10,000-generation samples went into the freezer at UCI, to October 16, 1992, when the LTEE restarted from the frozen samples at MSU.

Most of the other days have been lost as a result of various accidents. I’m often asked, when I give talks on the LTEE, how we’ve kept the experiment going so long without contamination, broken flasks, equipment failure, etc.

The short answer is that we haven’t. Many accidents have happened along the way.

There are 3 main types of accidents, each of which involves a different sort of interruption and recovery.

Little mistakes: Sometimes a flask has a hairline crack; when you take it out of the incubator the next day, there’s just a puddle of salt on the bottom. Or maybe someone knocked over a flask while doing the daily transfers. In cases like these where a mistake occurs that is immediately recognized, we go back in time (and lose) one day.

How do we do that? Each day, after the transfers have been made, we don’t immediately discard the previous day’s cultures. Instead, we put them in a refrigerator, where we can use them to restart the experiment after these little mistakes. The bacteria have finished growing long before each day’s transfer, so they are in stationary phase, and their metabolic activity is even lower sitting there at 4°C. Restarting the populations from the refrigerated cultures is a perturbation, of course, but a tiny one in the scheme of things.

When these little mistakes happen to one population, we go back a day for all the populations. We do that so that the rhythm of the experiment, which involves quality-control checks and freezing samples at regular intervals, is the same for all of the populations.

Bigger slipups: Another sort of problem can occur if the entire experiment is compromised in a way that is not immediately recognized. For example, the autoclave might not be working properly, and we realize that bottles of media that we’ve been using for a few days are contaminated. In that case, the cultures stored in the refrigerator won’t help us.

But we don’t have to start the LTEE all over at t = 0. (If we did, then the experiment wouldn’t be here today!) Instead, we go back to the last time that we froze samples, just like we did when we restarted the experiment after the move from UCI to MSU. Importantly, we restart the LTEE from whole-population samples, not individual clones, so that we do not lose the diversity that is present in an evolving population.

Of course, moving the bacteria into and out of the freezer is a perturbation, involving the addition of a cryoprotectant, freezing the cells, thawing them, and re-acclimating them to the conditions of the LTEE. Still, it happens only occasionally. Moreover, all of the samples used in competitions or other assays go into the freezer, come out, and are re-acclimated to the relevant conditions before measurements are made.

Dreaded cross-contamination: The third kind of accident is when bacteria from one LTEE population “migrate” into another population. That’s not supposed to happen, because it compromises the statistical independence of the populations, which are units of replication on which many analyses rest. I worried about this issue before I started the LTEE, because one of the central questions that motivated the experiment is the reproducibility of evolution. And I’m glad I worried about it. Fortunately, there was a pretty easy way of dealing with this concern from the outset.

Six of the 12 populations started from cells of an ancestral strain, REL606, that cannot grow on the sugar arabinose; they are phenotypically Ara^–. The others started from cells of a mutant, REL607, that can grow on arabinose; these populations are Ara⁺. There is no arabinose in the LTEE environment, and the mutation that allows growth on arabinose has no measurable affect on fitness in that environment. However, when Ara^– and Ara⁺ cells grow on Tetrazolium Arabinose (TA) agar in a petri dish, they make red and white (or pink) colonies, respectively.

[Mix of Ara^– and Ara⁺ colonies on TA agar.]

The arabinose phenotype serves two important purposes in the LTEE. First, we use it to estimate the abundance of competitors in the assays we perform to measure relative fitness. To that end, we typically compete an evolved Ara^– population sample against the Ara⁺ ancestor, and vice versa. Second, with respect to the possibility of cross-contamination, we alternate Ara^– and Ara⁺ populations during the daily transfers. The idea is that, if an accidental cross-contamination does occur, it will likely involve adjacent populations and lead to cells that have the wrong phenotype (i.e., produce the wrong-colored cells on TA agar) in a population. So we check each population for that phenotype whenever we freeze samples.

When we find one or more cells that produce the wrong-colored colony, we have to figure out what to do. There are various additional checks that we can perform, especially nowadays when DNA sequencing has allowed us to discover many mutations—additional markers—that uniquely identify each population. In particular, these extra markers have, in recent years, let us distinguish between “false alarms” (new mutations that affect colony color on the TA agar) and actual cross-contamination events. In any case, when we’ve had suspected or confirmed cross-contamination events, we restart the invaded population from the previous sample [4]. We then typically monitor that population by plating samples periodically on TA agar, to make sure it didn’t have a low frequency of cross-contaminating invaders even before that earlier sample was frozen. As a consequence of restarting invaded populations, some of the LTEE populations are 500 generations (or multiples thereof) behind the leading edge.

So today’s 10,000^th daily transfer applies to some, but not all, of the LTEE populations.

Despite these precautions and procedures, I worried that somehow we had slipped up and there were undetected cross-contamination events. Maybe there had been an especially fun party one Friday night … and on Saturday someone forgot the protocol and transferred all six red Ara^– populations in a row before moving on to the six white Ara⁺ populations. In that case, a cross-contamination might occur but not be detected. So I was thrilled when we sequenced hundreds of genomes from different generations of the LTEE populations and there was no evidence of any cross-contamination. Have I mentioned all the terrific people who have worked with me?

One of the unsung heroes of the LTEE is my technician and lab manager, Neerja Hajela. She has worked with me for over 20 years now, and she’s probably done more daily transfers than everyone else combined.

[Neerja Hajela, technician and lab manager extraordinaire.]

By the way, there were not 12, but 15, flasks in the trays while I was doing the transfers. What’s going on with that?

[The 15 LTEE flasks in the incubator.]

One of the extras is a blank—a culture without bacteria. If the medium in that flask is turbid the next day, then “Houston, we have a problem.” Another of the extras is a population we’re calling Ara–7. It was spun off population Ara–3 after we discovered—many thousands of generations later—that one lineage in that population had gone extinct for some reason that we do not understand. You can read more about that here. Ara–7 doesn’t count as one of the “real” LTEE populations, but it might prove useful in comparison with Ara–3 at some point in the future.

And the third extra? Remember what I said about cross-contamination? Well, we recently discovered a cross-contamination event in which cells that made red colonies on TA agar were found among the white-colony-forming cells of the Ara+1 population. Postdoc Zachary Blount confirmed they weren’t new mutants that made the wrong-colored colonies in Ara+1; instead, those cells had specific mutations that showed they came from population Ara–1, meaning they were cross-contaminating invaders.

[Zachary Blount, aka Dr. Citrate.]

So we restarted Ara+1 from its previous frozen sample, monitored it by plating cells on TA agar, and … alas, up came some more of those red invaders. It’s interesting, in a way, because Ara–1 is one of the most fit LTEE populations, while Ara+1 is the very least fit, which means Ara+1 is especially susceptible to invasion from its Ara–1 neighbor in the daily transfers. Anyhow, we then restarted Ara+1 going back in time 1000 and 1500 generations—hence, the extra flask—and we will monitor those for a while by plating samples on TA agar. If neither of them shows any sign of invaders for several weeks, then we will continue only the one with the fewer “lost” generations and drop the other.

There’s one other little issue related to keeping time in the LTEE. Every day, we remove 0.1 mL from each flask culture and transfer it to 9.9 mL of fresh medium. That 100-fold dilution allows the bacterial population to grow 100-fold before it depletes the available resources. And that 100-fold growth corresponds to log₂ 100 ≈ 6.64 generations. But we round it up a tad to 6.67 generations, so that every 15 transfers equals 100 generations [5].

In any case, our fielding percentage (baseball jargon for the ratio of plays without errors to total chances on defense) is 10,000 / 10,609 ≈ 0.943. If we exclude the lost days associated with the move from UCI to MSU, then the percentage rises to 0.960. Not bad, not bad at all. Did I mention the terrific people who have worked, and are working, on the LTEE?

This post’s title is a play on the novel A Wrinkle in Time by Madeleine L’Engle.

[1] I first started the LTEE on February 15, 1988, but I then restarted it on February 24, because I got worried that the first arabinose-utilization mutation I had selected, which serves as a neutral marker, wasn’t quite neutral.

[2] So the LTEE experienced a leap day in its very first week!

[3] I was fortunate that three experienced graduate students—Mike Travisano, Paul Turner, and Farida Vasi—moved to MSU even before I did to help set up the lab, and that our research was allowed to continue in my UCI lab—led by technician Sue Simpson and John Mittler, who was finishing his PhD—after I moved in late December, 1991.

[4] To keep all the populations in sync with respect to the freezing cycle, we restart the others at the same time, too. Of course, for the others, we don’t go back in time—we use the latest sample, where the cross-contaminated population was discovered during the quality-control checks associated with the freezing cycle.

[5] In fact, 6.67 generations per day might be a slight underestimate given the possibility of turnover during stationary phase. Moreover, every lineage with a beneficial mutation that sweeps to fixation goes through more than the average number of generations, since each mutant lineage starts as one cell among millions.