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Branching in trichomes (no sheath) or filaments (sheath)

Trichomes in cyanobacteria elongate by cell proliferation (simple fission), that involves division of a cell along an axis perpendicular to the trichome, and the adhesion of the descendant cells to each other in the plane of the trichome.

Pseudobranches can develop in two ways.  In one way, the external sheath of a filament is sufficiently adherent to the trichome cells, continued elongation of the trichome by intercalary growth can cause a bend to develop, and eventually the “bend” splits through the sheath.  Continued cell division causes the bend to expand, and eventually the trichome breaks in two, leaving a double strand that resembles a branch.

The second way to form pseudobranches is in the case of filaments with heterocysts.  When the vegetative cell adjacent a heterocyst divides, it often breaks contact with the heterocyst so that descendant cells continue to proliferate, usually at an acute angle away from the original plane of the filament.  So the pseudobranch originates next to the heterocyst.

True branching is the result of a 90 degree change in the axis of cell division, a major step in evolution of morphology.  I am unaware of any bacteria other than a few cyanobacteria that have advanced to true branching.

Without doubt the most advanced bacterial branching occurs in Stigonema mamillosa that has a combination of true branches and multiserate filaments.

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