The flexible nature of a cell membrane results from its PHOSPHOLIPIDS.
Cell membrane is made up of four different types of molecules; namely phospholipids, cholesterol, proteins, and carbohydrates. Its flexibility is from phospholipids because phospholipids are made up of two layers; one layer is hydrophilic and the other is hydrophobic. Just like a vegetable oil, the hydrophobic layer of the phospholipid bilayer does not like water and is able to move sideways with in the cell membrane to avoid contact with water.
