The evolution of multicellularity in the volvocine algae appears to have happened primarily through co-option of existing genes for new functions. For example, the initial transition from a unicellular life cycle to a simple multicellular one involved the retinoblastoma gene, as Hanschen and colleagues elegantly demonstrated (see “The evolution of undifferentiated multicellularity: the Gonium genome“). A Volvox gene involved in cellular differentiation, regA, was likely co-opted from an ancestral role in environmental sensing, and a similar origin appears to explain the use of cyclic AMP for the signaling that causes multicellular aggregation in cellular slime molds (see “Volvox 2015: evolution“).
Some of the changes leading to complex multicellularity, though, clearly did involve new genes. Two gene families involved in building the extracellular matrix that makes up most of a Volvox colony, the pherophorins and metalloproteinases, have undergone multiple duplication events leading to greatly expanded gene families (see “Heads I win; tails you lose: Evolution News & Views on Gonium, part 2“). One mechanism by which genes are duplicated is retroposition, in which a messenger RNA is reverse transcribed into DNA and inserted into the genome:

Fig S1A from Jakalski et al. 2016. Basic mechanism of retroposition. DNA is transcribed into a pre-mRNA by RNA polymerase, introns are spliced out, and a poly(A) tail is added to the 3′ end, resulting in a mature messenger RNA. The mRNA is then reverse-transcribed to DNA and inserted into a new genomic location.