The deep differences between the three domains of life are puzzling

and unexplained. Bacteria and archaea are almost indistinguishable

in their size and morphology, but are strikingly distinct in their

biochemical and metabolic properties, differing, for example, in the

genes responsible for glycolysis, cell membrane, cell wall, and even

DNA replication. In contrast, eukaryotes are chimeric, with most

of their informational genes deriving from archaea, and genes for

intermediary metabolism, as well as membranes, deriving from

bacteria. Eukaryotes apparently arose just once in 4 billion years of

evolution, around 2 billion years ago. Genetic divergence alone cannot

explain this evolutionary trajectory, as bacteria and archaea evolved

nothing resembling the morphological complexity of eukaryotes,

despite virtually unlimited exploration of genetic space. I will argue

that the differences between bacteria, archaea and eukaryotes are

not attributable to selection (or neutral evolution) operating on

populations of cells, but rather to the structural constraints imposed

by membrane bioenergetics. The divergence of bacteria and archaea

could have arisen from the dependence of LUCA, the last universal

common ancestor, on geologically sustained proton gradients in

alkaline hydrothermal vents at the origin of life. The constraints

imposed by membrane bioenergetics then prevented both prokaryotic

groups from becoming larger and more complex over 4 billion years of

evolution. These constraints were relaxed, at the origin of eukaryotes,

by a singular endosymbiosis between an archaeal host cell (probably

related to the Lokiarchaeota), and a proteobacterium, the ancestor

of mitochondria. This rare endosymbiosis restructured the topology

of eukaryotic genomes in relation bioenergetic membranes, with the loss of mitochondrial genes enabling the accumulation of genes and

gene families in the nucleus. In effect, eukaryotes have multi-bacterial

power without the huge genomic overheads of copying thousands of

full bacterial genomes. This extreme genomic asymmetry ultimately

gave eukaryotes 3-5 orders of magnitude more energy availability

per nuclear gene. The acquisition of mitochondria enabled a 15,000-

fold increase in maximal genome size, encompassing around a 5-10-

fold expansion in the number of protein-coding genes, coupled with

a 1000-10,000-fold expansion in gene expression. Many basal

eukaryotic traits, such as sex, probably evolved in the context of this

singular endosymbiosis between two prokaryotes.