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I was reading Watson and Crick's article on DNA structure, and the diagram on the lower left of the first page had something called the fiber axis going through the DNA. This axis isn't in modern models of DNA, so what is it?

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Short answer: The term fiber axis is not in reference to the DNA model, it comes from the experiments that Watson and Crick used to guess their model. The fiber axis is basically the dimension along the length of the DNA strand.

Longer:

Watson and Crick created their model of DNA based on the somewhat esoteric experiment called X-ray fiber diffraction. For the experimental data (Rosalind Franklin was actually gathering the fiber diffraction data) DNA is pulled out of a cell lysate and washed with buffer. It looks like a clear, liquid string of snot (a technical term that is). Its stringy though because the long DNA molecules pull out and create viscous fluid where the DNA is pulled along the length of the fiber. Because the DNA is ordered, if you shoot a beam of X-rays through it, it creates a x-pattern on film that looks like this:

fiber diffraction data by Rosalind Franklin

As you can see its got a nice X-shaped pattern. This is because the DNA in the fiber is mostly aligned along the fiber, in the up and down direction. Because this is so, the spacings between the different layers is due to the spacings between the DNA bases and the x-pattern comes from the fact that DNA forms a double helix.

Some misc details - you have to use X-rays because their wavelength is about the size of an chemical bond and this is the sort of model Watson and Crick and Franklin and Wilkinson were trying to find.

The fiber axis is along the y axis (top to bottom) because the trail of DNA snot hangs down - it will sag if you hold it an angle.
Rosalind Franklin's contribution was that she was a great experimentalist and she realized that the DNA snot trail dries out over time and the experiment would sometimes go on for days. She had a moist stream of hydrogen gas blowing over the DNA strand during the X-ray experiment and so was the only experimentalist who obtained what we now know to be the true result - double helical DNA. As it dries out, DNA interconverts to the Z-DNA form I believe, which does not have the x-pattern and also confuses things because the result is a smear of the DNA helix converting from one form to another.

Francis Crick had actually predicted the x-pattern for helices. The structure actually required these two to come together to get the answer while the American genius Linus Pauling was working hard in California on the structure as well.

The reference for nearly all of this is Watson's somewhat self-centered but scientifically accurate book "The Double Helix". After some surprisingly difficult Googling, I can't found a picture of DNA snot fiber... picture below. You can make some yourself from the protocol listed here and some saliva, soap and other common household items.

DNA snot

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Your statement "For their experiment (and for Rosalind Franklin who was also doing Fiber diffraction) " implies that W & C were actually doing experiments, but of course they weren't - they relied entirely upon Rosalind Franklin's data, supplied to them by Maurice Wilkins who shared the Nobel prize with W & C. –  Alan Boyd Jan 22 '13 at 14:25
    
I adjusted the wording, but didn't feel the need to get into every detail of the story - link to the book is provided. –  shigeta Jan 22 '13 at 15:13
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The obituary of the person who took the key photograph has just appeared in The Guardian "...Freda Collier [was]...part of the core team of Maurice Wilkins, John Randall and Rosalind Franklin (plus Franklin's PhD student, Raymond Gosling) working at King's College London on the structure of DNA. Freda...headed the photographic laboratory at King's that produced the famous "photo 51" seen by James Watson from Cambridge University. Watson immediately realised that the molecule revealed was a double helix." –  Alan Boyd Jan 22 '13 at 18:27
    
@AlanBoyd what was your opinion of Maurice Wilkins? It seems to that Franklin had little or no discussion with him - whether right or wrong - about the work. I know he's on the paper, but what was his contribution really? Did he train her in fiber work? I've not found an answer to that question and certainly sometimes it does not really seem he helped. I know it was his lab and all, but I mean intellectually or in terms of direct work... –  shigeta Jan 22 '13 at 20:34
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