The movements of amoeba are based on the actin cytoskeleton (background here).

Regions of the cell that contain short filaments and actin monomers form a sol.

Generally, an actin gel underlies (and is attached to) the plasma membrane, this is the cell cortex.

In free living amoeba, movement of the cell can occur without the cell being attached to a solid surface.

One way that this can be done is to contract a specific region of the cell.

This contraction involves the actin-associated protein myosin -- two-headed myosin molecules use energy to walk along oppositely oriented actin filaments, pulling the filaments (and whatever they are attached to closer together).

As this region of the cell contracts, the contents of the cell are squeezed.

If the actin cortex were uniform, little would happen -- the cell would just get rigid. However, what does happen is that in local regions, the cortical actin gel dissolves, creating a 'weak spot'.

Another set of actin binding proteins is involved in dissolving actin filament networks. These proteins bind to and sever actin filaments.

The cytoplasmic contents of the cell are extruded through this cortical weak spot and a cellular process moves forward.

If a rigid cortex reforms, movement will stop until another cortical breech appears in another part of the cell.

By regulating when and where the cortical actin network is partially dissolved, the cell can produce complex and directed movements, such as phagocytosis.

Regulating of actin-binding proteins (and so actin filament organization) depends on various factors - including local changes in intracellular Ca2+ concentration.