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'Invagination' means to fold inward or to sheath. In biology, this can refer to a number of processes.
(1) Invagination is the morphogenetic processes by which an embryo takes form, and is the initial step of gastrulation, the massive reorganization of the embryo from a simple spherical ball of cells, the blastula, into a multi-layered organism, with differentiated germ layers: endoderm, mesoderm, and ectoderm. More localized invaginations also occur later in embryonic development, to form coelom, etc.
(2) Invagination is the formation of a cleavage furrow during cytokinesis in animal cells.
(3) Invagination in some bacteria (also called a mesosome) is a tightly-folded region of the cell membrane containing all the membrane-bound proteins required for respiration and photosynthesis. It can also be associated with the nucleoid.
(4) The inner membrane of a mitochondrion 'invaginates' to form cristae, thus providing a much greater surface area to accommodate the protein complexes and other participants that produce ATP.