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https://www.quantamagazine.org/mathematics-shows-how-to-ensure-evolution-20180626/ June 26, 2018

Mathematics Shows How to Ensure EvolutionNew results emerging from graph theory prove that the way a population is organized can guaranteethe eventual triumph of natural selection — or permanently thwart it.

By John Rennie

Timothy Reynolds for Quanta Magazine

Migration patterns and other factors that shape the organization, or structure, of a population can determine howwell advantageous mutations spread within it.

Natural selection has been a cornerstone of evolutionary theory ever since Darwin. Yet mathematicalmodels of natural selection have often been dogged by an awkward problem that seemed to makeevolution harder than biologists understood it to be. In a new paper appearing in CommunicationsBiology, a multidisciplinary team of scientists in Austria and the United States identify a possibleway out of the conundrum. Their answer still needs to be checked against what happens in nature,but in any case, it could be useful for biotechnology researchers and others who need to promote

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https://www.quantamagazine.org/mathematics-shows-how-to-ensure-evolution-20180626/ June 26, 2018

natural selection under artificial circumstances.

A central premise of the theory of evolution through natural selection is that when beneficialmutations appear, they should spread throughout a population. But this outcome isn’t guaranteed.Random accidents, illnesses and other misfortunes can easily erase mutations when they are newand rare — and it’s statistically likely that they often will.

Mutations should theoretically face better odds of survival in some situations than others, however.Picture a huge population of organisms all living together on one island, for example. A mutationmight get permanently lost in the crowd unless its advantage is great. Yet if a few individualsregularly migrate to their own islands to breed, then a modestly helpful mutation might have abetter chance of establishing a foothold and spreading back to the main population. (Then again, itmight not — the outcome would depend entirely on the precise details of the scenario.) Biologistsstudy these population structures to understand how genes flow.

Sharona Jacobs Photography

Martin Nowak, the director of the Program for Evolutionary Dynamics at Harvard University, became interested inthe effects of population structures on natural selection while studying cancer.

Martin Nowak, who is today the director of Harvard University’s Program for EvolutionaryDynamics, began thinking about how population structures could potentially affect evolutionaryoutcomes in 2003 while studying the behavior of cancer. “It was clear to me then that cancer is anevolutionary process that the organism does not want,” he said: After malignant cells arise throughmutation, competition among those cells selects for the ones best able to run rampant through thebody. “I asked myself, how would you get rid of evolution?” Attacking mutations was one solution,

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Nowak realized, but attacking selection was another.

The problem was that biologists had only loose ideas about how specific population structures mightaffect natural selection. To find more generalizable strategies, Nowak turned to graph theory.

Mathematical graphs are structures that represent the dynamic relations among sets of items:Individual items sit at the vertices of the structure; the lines, or edges, between every pair of itemsdescribe their connection. In evolutionary graph theory, individual organisms occupy every vertex.Over time, an individual has some probability of spawning an identical offspring, which can replacean individual on a neighboring vertex, but it also faces its own risks of being replaced by someindividual from the next generation. Those probabilities are wired into the structure as “weights”and directions in the lines between the vertices. The right patterns of weighted connections canstand in for behaviors in living populations: For example, connections that make it more likely thatlineages will become isolated from the rest of a population can represent migrations.

With graphs, Nowak could depict diverse population structures as mathematical abstractions. Hecould then rigorously explore how mutants with extra fitness would fare in each scenario.

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https://www.quantamagazine.org/mathematics-shows-how-to-ensure-evolution-20180626/ June 26, 2018

Those efforts led to a 2005 Nature paper in which Nowak and two colleagues showed how stronglycertain population structures can suppress or enhance the effects of natural selection. In populationsthat have “burst” and “path” structures, for example, individuals can never occupy positions in thegraph that their ancestors held. Those structures stymie evolution by denying advantageousmutations any chance to take over a population.

The opposite is true, however, for a structure dubbed the Star, in which fitter mutations spreadmore effectively. Because the Star magnifies the effects of natural selection, the scientists labeled itan amplifier. Even better is the Superstar, which they called a strong amplifier because it ensuresthat mutants who are even slightly more fit will eventually replace all other individuals.

“A strong amplifier is an amazing structure because it guarantees the success of the advantageousmutation, no matter how small the advantage is,” Nowak said. “Everything about evolution isprobabilistic, and here we somehow turn probability into near certainty.”

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Yet that certainty came with a catch. Most potential population structures didn’t seem theoreticallycapable of being strong amplifiers. A few others looked like possibilities, but they seemed contrivedrather than realistic, and they were so complex that their status as amplifiers couldn’t be proved. (Aformal proof that the Superstar works came out just two years ago from a group at the University ofOxford, and Nowak described it as an intricate paper “with about a hundred pages of densemathematics.”) It was hard to see how population structure could boost natural selection among realliving creatures except under highly unusual circumstances.

Not quite a decade ago, however, one of Nowak’s collaborators, Krishnendu Chatterjee, a computerscience researcher at the Institute of Science and Technology Austria, also became interested in thisproblem. He and his group had already spent years developing an understanding of similar problemsinvolving graph theory and probabilities, and they thought the intuitions and insights they haddeveloped might prove useful on this evolution problem.

The key to constructing amplifiers, Chatterjee and his students Andreas Pavlogiannis (now at theÉcole Polytechnique Fédérale de Lausanne, EPFL) and Josef Tkadlec learned, was in the weights ofthe connections within the graphs. They realized that all potential strong amplifiers would havecertain features in common, such as hubs and self-loops. They then showed that by assigning theright weights to the connections, they could create strong amplifiers inside even simple populationstructures. “It came as a very big surprise to show that almost any population structure can becomea strong amplifier by adjusting weights,” Nowak said.

IST Austria

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https://www.quantamagazine.org/mathematics-shows-how-to-ensure-evolution-20180626/ June 26, 2018

Krishnendu Chatterjee (center), a professor and computer science researcher at the Institute of Science andTechnology Austria, and his students Andreas Pavlogiannis (right) and Josef Tkadlec developed a way to efficientlyconstruct population structures that amplify natural selection.

All told, the recent and previous papers make a case for population structure as a meaningful forcein evolution. Any populations that act like the “burst” will be evolutionary dead ends —advantageous mutations that appear within them will never take off, no matter what the details ofthe interrelationships might be. Other population structures may not automatically enhance naturalselection, but most of them at least have the potential to amplify advantageous mutations and giveevolution a helping hand.

The scientists’ findings come with some important caveats. One is that the population models inthese studies apply only to asexual organisms like bacteria and other microbes. Taking into accountthe wholesale reshuffling of genes that occurs in sexual reproduction would massively complicatethe models, Nowak and Chatterjee said, and to their knowledge, no one has yet seriously taken onthat challenge. The consequences of allowing the modeled populations to grow or shrink also needto be determined.

Another issue is that although strong amplifiers guarantee that useful mutations will spreadinexorably through a population, they don’t ensure it will happen quickly, Nowak said. It’s entirelypossible that some populations might benefit from structures in which natural selection is lesscertain but more swift.

That’s an important consideration, agreed Marcus Frean, an associate professor at VictoriaUniversity of Wellington in New Zealand. Work that he and his colleagues presented in 2013 showsthat the rate of evolution can slow down substantially even in population structures that amplifynatural selection. The certainty that a mutation will take over a population and the speed with whichit does might often oppose each other. “The thing we really care about — the rate of evolution —involves both,” Frean explained by email.

Nevertheless, Nowak, Chatterjee and their colleagues suggest in their paper that their algorithm forconstructing strong amplifiers might still be useful to researchers working with cell cultures whowant to foster the emergence of desirable mutants or to screen for faster-growing strains of cells.Microfluidic growth systems could be adjusted to produce any desired population structure bycontrolling how cells mix and migrate.

Perhaps a more intriguing application of their work, however, might be to identify where thesestrong amplifiers can already be found in nature. Nowak and his colleagues suggest that, forinstance, immunologists could check whether populations of immune cells in the spleen and lymphnodes show these structural features, which might help speed up how quickly the body fights backagainst infections. If they do, it could prove that natural selection sometimes favors itself as a goodsolution to life’s challenges.

This article was reprinted on Wired.com.