When DNA replicates, one strand is continuous, while the other is a lagging strand?

Why is it necessary for this to occur, and why do they travel in opposite directions?

This process occurs because of the polarity of DNA.
Note that you have anti-parallel strands (5'-->3' and 3'-->5') DNA replication is done within the same polarity, 5'-->3' because that is where the 3' OH is located for nucleophilic attack.

Look up the Trombone model (see youtube video below). You will see that the lagging strand is looped. It is synthesized for a few base pairs and then released. The loop can only be of a certain length. So when the DNA polymerase is finished synthesizing the DNA strand within the loop, it lets go and grabs another region of the DNA. Hence lagging strand.

http://www.youtube.com/watch?v=nIwu5MevZyg&feature=related

good luck.

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