Nuclear shapes are geometrically determined by the excess surface area of the nuclear lamina.

Nuclei have characteristic shapes dependent on cell type, which are critical for proper cell function, and nuclei lose their distinct shapes in multiple diseases including cancer, laminopathies, and progeria. Nuclear shapes result from deformations of the sub-nuclear components-nuclear lamina and chromatin. How these structures respond to cytoskeletal forces to form the nuclear shape remains unresolved. Although the mechanisms regulating nuclear shape in human tissues are not fully understood, it is known that different nuclear shapes arise from cumulative nuclear deformations post-mitosis, ranging from the rounded morphologies that develop immediately after mitosis to the various nuclear shapes that roughly correspond to cell shape ( elongated nuclei in elongated cells, flat nuclei in flat cells). We formulated a mathematical model to predict nuclear shapes of cells in various contexts under the geometric constraints of fixed cell volume, nuclear volume and lamina surface area. Nuclear shapes were predicted and compared to experiments for cells in various geometries, including isolated on a flat surface, on patterned rectangles and lines, within a monolayer, isolated in a well, or when the nucleus is impinging against a slender obstacle. The close agreement between predicted and experimental shapes demonstrates a simple geometric principle of nuclear shaping: the excess surface area of the nuclear lamina (relative to that of a sphere of the same volume) permits a wide range of highly deformed nuclear shapes under the constraints of constant surface area and constant volume. When the lamina is smooth (tensed), the nuclear shape can be predicted entirely from these geometric constraints alone for a given cell shape. This principle explains why flattened nuclear shapes in fully spread cells are insensitive to the magnitude of the cytoskeletal forces. Also, the surface tension in the nuclear lamina and nuclear pressure can be estimated from the predicted cell and nuclear shapes when the cell cortical tension is known, and the predictions are consistent with measured forces. These results show that excess surface area of the nuclear lamina is the key determinant of nuclear shapes. When the lamina is smooth (tensed), the nuclear shape can be determined purely by the geometric constraints of constant (but excess) nuclear surface area, nuclear volume, and cell volume, for a given cell adhesion footprint, independent of the magnitude of the cytoskeletal forces involved.