Our May Insights puzzle was inspired by recent discoveries of some rare, intriguing patterns of inheritance that hark back to Jean-Baptiste Lamarcks theory of evolution and its emphasis on the inheritance of acquired characteristics. Elementary textbooks often present Lamarcks theory as a failed 19th-century rival to Charles Darwins theory of evolution by natural selection. But reality, as usual, is far more complicated. There is indeed a great deal of evidence that most acquired characteristics are not inherited, but as the new findings have shown, this proscription is not absolute. The famous verkalix study, for example, showed that men who were exposed to a poor food supply between the ages of 9 and 12 were found, two generations later, to have conferred a measurably lower risk of diabetes and cardiovascular death to their grandchildren. Adaptive Lamarckian inheritance does seem to be possible, and epigenetic mechanisms for it have been found. These mechanisms modify DNA in ways that differ from those of heredity.

But at a deeper level this kind of inheritance can be naturally selected for in the traditional Darwinian way, provided certain environmental conditions are satisfied. So Darwinian natural selection remains the fundamental basis of evolution and can produce Lamarckian inheritance: The theories are not rivals after all! Using simple models, our puzzles show how natural selection can sustain Lamarckian inheritance. The requirement is that environmental conditions, such as famines, follow patterns that persist across several generations and are repeated over long stretches of evolutionary time.

Imagine there exists an animal that has a new generation every year. Every normal individual has an average of 1.6 surviving offspring in a normal year, which can be defined as the animals fitness (lets call itf), after which the animal dies. During a famine year,ffalls to 1.3.Now suppose there are a bunch of smaller individuals whosefvalues are 1.5 in normal years but 1.35 in famine years: Their smaller food requirement helps them survive famines better. How long would a famine have to last for the small individuals to do better than normal ones? How many famine years before small individuals make up 90 percent of the population?

The basic mathematics of natural selection is simple. For every group, you just multiply the fitness numbers across multiple generations. You then find the ratio between the numbers you obtain for the different groups. This gives you their relative populations, assuming that the initial numbers were the same. (Note that these numbers dont signify the actual populations of each group, but they indicate their relative success. If f is larger than 1, then the product may grow extremely large after many generations. In the real world, there are many checks on the population of a species, so at equilibrium, the population is actually stable. What does change are the relative ratios between the populations of the different groups, which are accurately reflected in the above calculation.)

For Question 1, assuming we start from a normal year, we have to find a positive integer n such that 1.5 x 1.35n > 1.6 x 1.3n. You can do this analytically using logarithms or by setting up a spreadsheet and reading off the values. After two years of famine, the smaller individuals already have a population over 50 percent. (If you want to bookend the famine with normal years on either side, then it requires four years of famine for the small individuals to be ahead of the normal ones a year after the famine is over.)

As Ty Rex noted, for smalls to make up more than 90 percent of the population, the number of famine years needs to be greater than [log(9) + log(1.6/1.5)]/log(1.35/1.3) ~ 59.9. So, 60 famine years are needed for smalls to make up 90 percent of the population.

Suppose there exists an initially normal mutant group of individuals called Epi2s, whose germ cells are affected by a year of famine in such a way that their progeny changes to the small type for two generations before they revert back to normal in the third generation, through epigenetic mechanisms. Consider a 13-year period that starts and ends with normal years but has a one-year famine, two two-year famines and a three-year famine in between. Which of the three groups (normals, smalls, Epi2s) will be most successful? Are there famine patterns in which Epi2s overwhelm the other two groups over the very long term?

As a couple of commenters noted, there is an ambiguity here: What happens when Epi2s that have changed to the small type encounter a year of famine? Is their status reset and do they continue to be smalls for another two years, or do they continue on their original timetable and revert to normals two years after the original famine year? Most commenters assumed the former. I had the latter in mind, because otherwise the Epi2s behave very much like smalls in extended famines. In any case, the choice of the assumption does not change the answer to this question. As Ty Rex noted, if we start with equal populations, the ratios between the normals, smalls and Epi2s become 85.5 to 83.8 to 86.1, assuming Epi2s reset, so the Epi2s do best by a small margin. If there are no resets, then Epi2s do even better, their relative ratio going up to 87.4. With no resets, Epi2s are adapted to famines that are three years long, so the pattern NFFFNFFF gives them an even larger advantage over the other two groups. With this pattern, Epi2s will make up 90 percent of an initially evenly divided population in 329 years.

Lets add another type of animal to the above: the Epi1s, which like the Epi2s switch to small progeny after a famine, but in this case the progeny revert back to normal after just one generation. Over a period of 20 years, can you come up with a famine-year schedule such that all four types of animals (normals, smalls, Epi1s and Epi2s) exist in virtual equilibrium over this time period?

For this question, note that the numbers of the normal and the small groups are only affected by the number of famine and nonfamine years and not their temporal arrangement. So we have to find a positive number of nonfamine years n such that 1.6n1.320-n is as close as possible to 1.5n1.3520-n. This happens for seven nonfamine and 13 famine years, which gives a relative ratio of 813 to 845 for normals to smalls. How do the years need to be arranged to equalize the numbers of Epi1s and Epi2s? As noted above, without resets, Epi2s are best adapted to famines that last three years, and similarly, Epi1s are best adapted to famines that last two years. So our 20-year pattern needs to have famines of both these durations. The pattern NFF NFFF NFF NFFF NFF NFN meets all the conditions mentioned and gives relative scores of 809 for Epi1s and 817 for Epi2s on the above scale, which are both within 0.5 percent of the number for normals. This seems to be the best approach to virtual equilibrium.

So what these simple models teach us is that it is possible to come up with environmental conditions that will lead natural selection to favor epigenetic inheritance across generations if the selecting factor (here, famine) occurred frequently enough in an animals evolutionary history in the right pattern. Furthermore, there can be patterns that maintain different groups of the species at relatively constant numbers, ready to take advantage of a change in climate, rendering the species as a whole more stable and prepared for several different eventualities.

As I discussed in the puzzle column, scientists have found molecular mechanisms that can implement these transgenerational changes by suppressing the activity of certain genes through the attachment of methyl groups (DNA methylation) or through changes in the configuration of the protein that packages the DNA (histone modification). Transgenerational inheritance is even easier in small organisms that do not go through a germ cell stage, such as bacteria. These organisms can use even more efficient mechanisms that have allowed the evolution of the spectacular DNA-cutting system called CRISPR, which is currently revolutionizing genetic engineering. This system uses bits of DNA called transposons or jumping genes that can jump around from one location to another in genomes. Its amazing what natural selection can achieve in evolutionary time!

Thanks to all who participated in this Insights puzzle. I enjoyed reading your comments and especially my dialogue with Josh Mitteldorf. The Quanta T-shirt goes to Ty Rex. Congratulations!

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Solution: 'Darwinian Evolution Explains Lamarckism' - Quanta Magazine