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

The origin of chromosomes was a major transition in the evolution of complex cellular life. To model this phenomenon, we consider a simple protocell composed of two types of genes: a ``fast gene'' with an advantage for gene-level self-replication and a ``slow gene'' that confers an advantage for protocell-level reproduction. Using a PDE describing the dynamics genetic composition of protocells under within-cell and between-cell competition, we find that the gene-level advantage of fast replicators casts a long shadow on the dynamics of protocell evolution: no level of between-protocell competition can produce coexistence of the fast and slow replicators when the two genes are equally needed for protocell-level reproduction. By introducing a ``dimer replicator'', a linked pair of the slow and fast genes, we see that the possibility of genetic linkage can help to overcome this shadow of lower-level selection and promote coexistence of the fast and slow genes via multilevel selection.