In eukaryotic cells (your cells) the cytoskeleton is needed to shape the cell, position DNA, and most importantly for this study, separate daughter cells allowing replication. Think of the complexity here, you need to make compartments to separate the copies of the genetic material, physically separated during division. Microtubules assemble the "mitotic spindle" and then pulls the sister chromatids apart from each other. After the chromosomes separate, other cytoskeletal filaments (actin and myosin) form a contractile ring, which tightens to create a cleavage furrow. The membrane pinches inward until the cell splits in two.
Bacteria work slightly differently, since they don't have a eukaryotic cytoskeleton, but they do have cytoskeletal-like proteins (FtsZ), since they divide by building the cell wall inward (I am not an expert on bacteria lol).
SpudCell doesn't have a cytoskeleton, so instead it relies on a physical membrane-rupture strategy. It makes membrane proteins from its own DNA (a-hemolysin), which inserts into the membrane. They help fuse with feeder liposomes for growth. For division, these proteins crowd on the membrane surface, creating mechanical stress which leads to membrane instability, which then splits on its own.