Understanding the emergence and evolution of multicellularity and cellular differentiation is a core problem in biology. We develop a quantitative model that shows that a multicellular form emerges from genetically identical unicellular ancestors when the compartmentalization of poorly compatible physiological processes into component cells of an aggregate produces a fitness advantage. This division of labour between the cells in the aggregate occurs spontaneously at the regulatory level owing to mechanisms present in unicellular ancestors and does not require any genetic predisposition for a particular role in the aggregate or any orchestrated cooperative behaviour of aggregate cells. Mathematically, aggregation implies an increase in the dimensionality of phenotype space that generates a fitness landscape with new fitness maxima, in which the unicellular states of optimized metabolism become fitness saddle points. Evolution of multicellularity is modelled as evolution of a hereditary parameter: the propensity of cells to stick together, which determines the fraction of time a cell spends in the aggregate form. Stickiness can increase evolutionarily owing to the fitness advantage generated by the division of labour between cells in an aggregate.