Re: How do ligases 'tape' the sticky ends of DNA molecules?

That is an excellent question! Ligases form phosphodiester bonds between 
neighboring nucleotides in DNA. This reaction is more efficient if the two 
ends to be ligated are complementary (“sticky”), but some enzymes will 
also join blunt ends (without one of the strands being longer by a few 
nucleotides than the other). Ligases can, of course, also join the ends of 
a nick in DNA (one phosphodiester bond broken somewhere within a double 
stranded piece of DNA). At any rate, one of the two nucleotides to be 
joined needs to have a hydroxyl on carbon 3 of its deoxyribose sugar. The 
other ligation partner needs to provide a phosphate group linked to carbon 
5 of its deoxyribose sugar.  Formation of a phosphodiester bond requires 
the input of energy, so the ligation reaction is coupled with an exergonic 
(energy releasing) reaction, such as the hydrolysis of a phosphoanhydride 
bond. Some ligases use the general “energy currency” adenosine 
triphosphate (ATP) for that purpose, others another nucleotide (NAD+ = 
nicotinamide adenine dinucleotide). Such molecules are generally referred 
to as coenzymes, indicating that they are essential participants in the 
chemical reaction (ligation in this case) that is catalyzed by the enzyme. 
That is the short version of the answer. If you have a good background in 
chemistry, you will understand the following summary of the chemical 
reaction sequence going on as the enzyme links the two nucleotides.
The adenosine monophosphate portion of the ATP or NAD+ is first covalently 
linked to the enzyme itself, then transferred to the phosphate end of the 
two nucleotides to be joined, to create a phosphoanhydride bond. This bond 
is then attacked by the hydroxyl group on the other of the two nucleotides 
to be linked and is replaced by a phosphodiester bond between the hydroxyl 
and the phosphate of the neighboring nucleotides, thereby ligating the 
two. Adenosine monophosphate leaves the enzyme, which can now catalyze 
another round of ligating two nucleotides.