Research teams are the fundamental social unit of science, and yet there is currently no model that describes their basic property: size. In most fields teams have grown significantly in recent decades. We show that this is partly due to the change in the character of team-size distribution. We explain these changes with a comprehensive yet straightforward model of how teams of different sizes emerge and grow. This model accurately reproduces the evolution of empirical team-size distribution over the period of 50 years. The modeling reveals that there are two modes of knowledge production. The first and more fundamental mode employs relatively small, core teams. Core teams form by a Poisson process and produce a Poisson distribution of team sizes in which larger teams are exceedingly rare. The second mode employs extended teams, which started as core teams, but subsequently accumulated new members proportional to the past productivity of their members. Given time, this mode gives rise to a power-law tail of large teams (10-1000 members), which features in many fields today. Based on this model we construct an analytical functional form that allows the contribution of different modes of authorship to be determined directly from the data and is applicable to any field. The model also offers a solid foundation for studying other social aspects of science, such as productivity and collaboration.