What normally regulates the rate of cell division in typical cells

Cyclins

Explanation:

Cyclins and cyclin-dependent protein kinases (CDKs, cell proteins) also function to control the cell cycle. A group of cyclins: the G1 cyclins, are synthesized during G1 phase and function to activate CDKs which initiate DNA synthesis at the G1/S checkpoint. The cell fails to progress to S phase if there is no sufficient synthesis of G1 cyclins. After a cell passes through this point, the G1 cyclins are degraded, allowing for another group of cyclins: the M cyclins (mitotic cyclins) to be synthesized. M cyclins activate a second group of CDKs which allow the cell to pass the G2/M control point and into mitosis.

from the original order (top to bottom) the new order will go

1 (founder)

4 (hawk)

2 (mutate)

3 (next generation)