[WP] Because of the intrinsic randomness of the evolutionary process, a mutant with a fitness advantage has some chance to be selected but no certainty. Any experiment that searches for advantageous mutants will lose many of them due to random drift.
It is therefore of great interest to find population structures that improve the odds of advantageous mutants. Such structures are called amplifiers of natural selection: they increase the probability that advantageous mutants are selected. Arbitrarily strong amplifiers guarantee the selection of advantageous mutants, even for very small fitness advantage. Despite intensive research over the past decade, arbitrarily strong amplifiers have remained rare.
Here we show how to construct a large variety of them.
Our amplifiers are so simple that they could be useful in biotechnology, when optimizing biological molecules, or as a diagnostic tool, when searching for faster dividing cells or viruses. They could also occur in natural population structures.
In the evolutionary process, mutation generates new variants, while selection chooses between mutants that have different reproductive rates. Any new mutant is initially present at very low frequency and can easily be eliminated by random drift. The probability that the lineage of a new mutant eventually takes over the entire population is called the fixation probability. It is a key quantity of evolutionary dynamics and characterizes the rate of evolution.
…In this work we resolve several open questions regarding strong amplification under uniform and temperature initialization. First, we show that there exists a vast variety of graphs with self-loops and weighted edges that are arbitrarily strong amplifiers for both uniform and temperature initialization. Moreover, many of those strong amplifiers are structurally simple, therefore they might be realizable in natural or laboratory setting. Second, we show that both self-loops and weighted edges are key features of strong amplification. Namely, we show that without either self-loops or weighted edges, no graph is a strong amplifier under temperature initialization, and no simple graph is a strong amplifier under uniform initialization.
…In general, the fixation probability depends not only on the graph, but also on the initial placement of the invading mutants…For a wide class of population structures17, which include symmetric ones28, the fixation probability is the same as for the well-mixed population.
… A population structure is an arbitrarily strong amplifier (for brevity hereafter also called “strong amplifier”) if it ensures a fixation probability arbitrarily close to one for any advantageous mutant, r > 1. Strong amplifiers can only exist in the limit of large population size.
Numerical studies30 suggest that for spontaneously arising mutants and small population size, many unweighted graphs amplify for some values of r. But for a large population size, randomly constructed, unweighted graphs do not amplify31. Moreover, proven amplifiers for all values of r are rare. For spontaneously arising mutants (uniform initialization): (1) the Star has fixation probability of ~1 − 1⁄r2 in the limit of large N, and is thus an amplifier17, 32, 33; (2) the Superstar (introduced17, see also34) and the Incubator (introduced in35, 36), which are graphs with unbounded degree, are strong amplifiers.
…In this work we resolve several open questions regarding strong amplification under uniform and temperature initialization. First, we show that there exists a vast variety of graphs with self-loops and weighted edges that are arbitrarily strong amplifiers for both uniform and temperature initialization. Moreover, many of those strong amplifiers are structurally simple, therefore they might be realizable in natural or laboratory setting. · Second, we show that both self-loops and weighted edges are key features of strong amplification. Namely, we show that without either self-loops or weighted edges, no graph is a strong amplifier under temperature initialization, and no simple graph is a strong amplifier under uniform initialization.
…Intuitively, the weight assignment creates a sense of global flow in the branches, directed toward the hub. This guarantees that the first 2 steps happen with high probability. For the third step, we show that once the mutants fixate in the hub, they are extremely likely to resist all resident invasion attempts and instead they will invade and take over the branches one by one thereby fixating on the whole graph. For more detailed description, see Methods § Construction of strong amplifiers.
Necessary conditions for amplification: Our main result shows that a large variety of population structures can provide strong amplification. A natural follow-up question concerns the features of population structures under which amplification can emerge. We complement our main result by proving that both weights and self-loops are essential for strong amplification.
Thus, we establish a strong dichotomy. Without either weights or self-loops, no graph can be a strong amplifier under temperature initialization, and no simple graph can be a strong amplifier under uniform initialization. On the other hand, if we allow both weights and self-loops, strong amplification is ubiquitous.
…Some naturally occurring population structures could be amplifiers of natural selection. For example, the germinal centers of the immune system might constitute amplifiers for the affinity maturation process of adaptive immunity46. Habitats of animals that are divided into multiple islands with a central breeding location could potentially also act as amplifiers of selection. Our theory helps to identify those structures in natural settings.