A PDE Model for Protocell Evolution and the Origin of Chromosomes via Multilevel Selection

Evolutionary dynamics within and among competing groups

“…The long-time behavior of solutions f ( t , x ) depends on the initial distribution of the strategic compositions across groups, f 0 ( x ), which we assume follows a uniformly continuous probability density. In particular, the dynamics depend on the density of the initial distribution near the full-cooperation equilibrium, x = 1, as quantified by the Hölder exponent θ at x = 1 ( 16 , 62 ). A probability distribution with density function u ( t , x ) has Hölder exponent θ near x = 1 provided …”

“…We prove this result in SI Appendix , Proposition 2.1 , for point-wise convergence of density-valued solutions. Conversely, when among-group selection is too weak, λ ≤ λ * , all groups become pure defectors in the long-time limit ( 16 )—that is, f ( t , x ) converges to the delta-function δ ( x ) concentrated at x = 0. The threshold selection strength that governs the long-term survival of cooperation is given by the following formula * …”

“…Competition at multiple scales of organization is especially acute in biological systems on the cusp of major evolutionary transitions ( 8 , 9 ), when the interests of individuals are at odds with the group incentives to form cooperative collectives ( 10 , 11 ). This tension arises, for example, in the trade-off between an individual gene’s incentive for rapid replication versus the collective need to establish a balanced ensemble of genes for healthy cellular function—a conflict that can be mediated by genetic linkage, that is, by the origin of chromosomes ( 12 – 16 ), hypercycles ( 17 ), or package replication ( 18 , 19 ) and stochastic correctors ( 20 , 21 ). Multilevel competition continues to play a role in transitions to yet higher levels of biological organization, including the origins of multicellularity ( 22 – 24 ), such as the formation of multicellular colonies in Volvox ( 25 ) and slime molds ( 26 , 27 ).…”

“…Evolutionary game theory can be generalized to describe natural selection operating simultaneously at multiple levels of organization in a group-structured population, characterizing both competition among individuals within a group and competition among groups of individuals ( 16 , 53 – 68 ). This generalization provides a framework for exploring the countervailing effects of the individual incentive to defect and the collective incentive to cooperate, highlighting the evolutionary tug-of-war between levels of selection.…”

“…This generalization provides a framework for exploring the countervailing effects of the individual incentive to defect and the collective incentive to cooperate, highlighting the evolutionary tug-of-war between levels of selection. Multilevel evolutionary game theory is relevant to questions as microscopic as the origins of chromosomes and cellular life, and as macroscopic as the dynamics of cultural evolution, cooperation, and conflict in human societies ( 16 , 49 , 61 – 66 , 69 – 71 ).…”