One possibility is the loss of cofactors/ prostetic groups as the concentration is lowered below their binding constant. A multisubunit protein may dissociate as mentioned by A.Lane. Another factor is the loss or denaturation of the protein as it is adsorbed to the plastic or glass surface of the vial. For proteins of very limited stability, the crowding effects by proteins in the physiological environment may be essential for stability.
It may lose the cofactors that may be needed for full activity. second the protein may not be able to maintain the conformation in dilute solutions without the proper osmoticum
At the begining it should be clear wheather we are discussing about activity of an enzyme molecule or specific activity of the enzyme? If we are concerned with the activity of the enzyme, than it will not be altered under any dilution, provided all additional factors are constant during determination of the activity.
But if we are considering the change of activity of an enzyme preparation or sample, under conditions of dilution of the source keeping all other parameters fixed, or rather identical, than the explaination is very simple.
With dilution, there is less number of available enzyme molecules to encounter and accociate with a fixed number of substrate molecules to do the catalysis even if all other factors are present sufficiently above their individual half saturation constant under the setup.
The turn over number of each enzyme protein will not change but the enzyme-substrate association will be reduced drastically leading to reduced enzyme- substrate complex formation essential for catalysis.
In determining enzyme activity, we always count either the numbe of substrate molecules lost or the number of product molecules obtained during the catalysis. We do not count the number of enzyme molecule involved. That means we do not consider the molecular weight of the enzyme protein in determination of enzyme activity. This situation generally generate the confusion over relation ship between dilution and activity of the enzyme source and enzyme molecule.
It may be a practical problem: high dilutions of proteins may cause adsorption on recipient walls.This can be avoided/diminiushed by diluting in speciallity eppendorf tubes (eg Protein LoBind tubes) or adding BSA (make sure it does not contain enzyme activity!) or tween20 (check that it does not inhibit).
Adsoption onto the walls of the tube can happen even at high concentrations. It is rather difficult to maintain the protein conformation in diute solutions not having the proper ionic environment. That is we add BSA to increase the general protein concentration which gives stability to the enzyme but not Tween 20