  The evolution of complex cellular life involved two major transitions: the
encapsulation of self-replicating genetic entities into cellular units and the
aggregation of individual genes into a collectively replicating genome. In this
paper, we formulate a minimal model of the evolution of proto-chromosomes
within protocells. We model 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 replicates more slowly at the gene level, but which confers an
advantage for protocell-level reproduction. Protocell-level replication
capacity depends on cellular composition of fast and slow genes. We use a
partial differential equation to describe how the composition of genes within
protocells evolves over time under within-cell and between-cell competition. We
find that the gene-level advantage of fast replicators casts a long shadow on
the multilevel 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" consisting of a linked pair of the slow and fast genes, we
show analytically that coexistence between the two genes can be promoted in
pairwise multilevel competition between fast and dimer replicators, and provide
numerical evidence for coexistence in trimorphic competition between fast,
slow, and dimer replicators. Our results suggest that dimerization, or the
formation of a simple chromosome-like dimer replicator, can help to overcome
the shadow of lower-level selection and work in concert with deterministic
multilevel selection to allow for the coexistence of two genes that are
complementary at the protocell-level but compete at the level of individual
gene-level replication.
