Does depolymerisation take place at the minus end of microtubule?
Often microtubule (MT) dynamics and actin dynamics are taught side by side, which can be quite confusing as MTs display unique characteristics relative to actin filaments. Indeed, treadmilling is an in vivo behavior primarily associated with actin, while dynamic instability characterizes MT growth. But why?

Both actin filaments and MTs are "sided" (polar), with actin having barbed- and pointed-ends and MTs having plus- and minus-ends. This polarity emerges from two properties: 1) the energetic state of the individual monomers that make up the filament and 2) the rate of growth. 

For actin, the barbed-end is compromised mostly of ATP-actin (high energy), while the pointed-end is made up of ADP-actin (low energy). This composition preferentially adds new ATP-monomers to the ATP-rich barbed-end, yielding a faster rate of growth than at the pointed-end. Disassembly at the pointed-end occurs more readily as ATP-actin is converted to lower energy ADP-actin. Treadmilling occurs when the rate of assembly at the barbed-end is equal to the rate of disassembly at the pointed-end (Pollard and Borisy, 2003). 

Similarly to actin filaments, MTs are polar, with their plus-ends made up of GTP-tubulin monomers and GDP-tubulin concentrated at their minus-ends (Desai and Mitchison, 1997). Unlike actin filaments, which can form sporadically almost anywhere in the cell (Pollard and Borisy, 2003), MTs are initiated at what we call a microtubule organizing center, or MTOC (Kollman, 2011). 

The bulk of MTs within the cell are attached to the MTOC through their minus-ends, though there are unattached sub-populations. Both in vivo and in vitro data indicate that growth and shrinkage at the minus-ends does occur, but that the rates are slow relative to those occurring at the plus-ends (Hendershott, 2014). In these instances, MTs can undergo treadmilling in a manner similar to that as actin and this behavior is readily observed in vitro (Waterman-Storer, 1997). Recent studies have suggested that MT minus-ends are under tight regulation in vivo, likely limiting their dynamics (Akhmanova, 2015). Therefore, MT growth and shrinkage primarily occurs at thee plus-ends. Enter dynamic instability.

MT plus-ends undergo a cyclical process of growth and shrinkage known as dynamic instability (Mitchison and Kirschner, 1984). The key to this process is the difference in shape between GDP-tubulin and GTP-tubulin (Buey, 2006). One can imagine GTP-tubulin as a rectangle, with perfectly straight lines connecting its angles. On the other hand, GDP-tubulin has a small bend in the center of one of its long ends.   

As a MT grows, GTP-tubulin is added to the plus-end forming a "GTP cap." The role of this cap is to counteract the inherent desire of GDP-tubulin to bend outward, thereby forming a straight filament and ensuring continued growth of the MT (Desai and Mitchison, 1997). However, the MT will lose its cap when the rate of GTP-tubulin addition drops below the rate of GTP to GDP conversion, thereby exposing GDP-tubulin. This event characterizes the "catastrophe" stage of dynamic instability (Mitchison and Kirschner, 1984). The shape of GDP-tubulin is no longer constrained in this stage, and instead, bends outward causing rapid disassembly of the MT. Because the cell is chock full of GTP-tubulin, recovery of the GTP cap rapidly occurs and the MT re-enters productive growth. This cycle of growth-catastrophe-recovery is the hallmark of MT dynamic instability.