Is it possible that nuclear division in mushrooms represents an intermediate mechanism in the evolution of mitosis

Possibly, because karyokinesis, the division of a nucleus, is similar with that of the processes of mitosis and cytokinesis. It might be that nuclear division was an intermediate to mitosis which later developed to other cells. This means that because this replication is essential for the proliferation of cells, essentially, the mechanism that is mitosis should be vitally involved. 

step 1: replication fork formation.

step 2: primer binding.

step 3: elongation.

step 4: termination.

explanation:

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