You can look at this question strategically.
That's important, because now you know they are looking at the constant region of immunoglobulin aka Fc of the heavy chain.
So why is this cDNA that encodes Fc, lighting up only once in other tissues, but the same cDNA lights up multiple times in lane 2 (bone marrow)? This is because, at the bone marrow random recombination of light or heavy chain occurs, and our cDNA is present in different random combinations with light chains. That specific cDNA sequence they used didn't change between the tissues or even within the bone marrow, and it will only bind to the one gene sequence. The fact it's lighting up multiple times tells you that the same sequence is present, your question is why so many times.
Also a side note, southern blots are time consuming, so labs use other methods to do the same thing (like PCR), but southern blots are still the best when it comes to checking immunoglobulin and T cell receptor gene rearrangement.
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I believe (pls correct me if I'm wrong) you would have similar Southern blot results seen in B cells undergoing somatic hypermutation, but that takes place in the secondary lymphoid tissue, not in the bone marrow.
I read this question as analyzing immunoglobulins in the bone marrow vs out circulating peripherally. I now realize this is wrong, and they are just analyzing the DNA of different cell population, like a hepatocyte for example, not immunoglobulins present in the liver.
With this cleared up, I can see how V(D)J recombination could make the original DNA section shorter. But how could V(D)J recombination possibly make the DNA section longer? There are bands on the gel that are higher molecular weight in the bone marrow. Recombination only shortens the DNA, someone please correct me.
Also why is there no band in the bone marrow that is the same size as peripheral bands? Surely the HSCs have not undergone recombination, where is their DNA???
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Echoing @waterloo's comment.
This Q is essentially asking "What is the function of Southern blots?"
Answer: The function of Souther blots is to check for BCR or TCR gene rearrangements.
See @waterloo's explanation for what is specifically going on as far as why there are different bands at different places. Again, the answer to this is "gene rearrangement" but @waterloo explains in with more detail.