  Nuclei import and export proteins, including cell cycle regulators. These
import-export processes are modulated periodically by the cell cycle, for
example due to the periodic assembly and breakdown of the nuclear envelope. As
such, replicated DNA can be segregated between the two daughter cells and the
proteins that were localized in the nucleus are free to diffuse throughout the
cytoplasm. Here, we study a mathematical import-diffusion model to show how
proteins, i.e. cell cycle regulators, could be redistributed in the cytoplasm
by nuclei that periodically toggle between interphase and mitosis. We show that
when the cell cycle period depends on the local concentration of regulators,
the model exhibits mitotic waves. We discuss how the velocity and spatial
origin of these mitotic waves depend on the different model parameters. This
work is motivated by recent in vitro experiments reporting on mitotic waves in
cycling cell-free extracts made with Xenopus laevis frog eggs, where multiple
nuclei share the same cytoplasm. Such experiments have shown that nuclei act as
pacemakers for the cell cycle and thus play an important role in collectively
defining the spatial origin of mitotic waves.
