A scaling law of multilevel evolution: how the balance between within- and among-collective evolution is determined

Numerous living systems are hierarchically organised, whereby replicating components are grouped into reproducing collectives -- e.g., organelles are grouped into cells, and cells are grouped into multicellular organisms. In such systems, evolution can operate at two levels: evolution among collectives, which tends to promote selfless cooperation among components within collectives (called altruism), and evolution within collectives, which tends to promote cheating among components within collectives. The balance between within- and among-collective evolution thus exerts profound impacts on the fitness of these systems. Here, we investigate how this balance depends on the size of a collective (denoted by $N$) and the mutation rate of components ($m$) through mathematical analyses and computer simulations of multiple population genetics models. We first confirm a previous result that increasing $N$ or $m$ accelerates within-collective evolution relative to among-collective evolution, thus promoting the evolution of cheating. Moreover, we show that when within- and among-collective evolution exactly balance each other out, the following scaling relation generally holds: $Nm^{\alpha}$ is a constant, where scaling exponent $\alpha$ depends on multiple parameters, such as the strength of selection and whether altruism is a binary or quantitative trait. This relation indicates that although $N$ and $m$ have quantitatively distinct impacts on the balance between within- and among-collective evolution, their impacts become identical if $m$ is scaled with a proper exponent. Our results thus provide a novel insight into conditions under which cheating or altruism evolves in hierarchically-organised replicating systems.