Thursday, December 10, 2009

Fractal DNA

Packing meters of DNA into a nucleus with a diameter a million times smaller is quite a challenge. Wrapping the DNA around nucleosomes, and arranging these nucleosomes into 30nm fibers, both help, but these structures must themselves be packed densely. Beautiful new research, reported in Science in October, supports a 20-year old idea that some DNA is arranged in an exotic knot-free fractal structure that is particularly easy to unpack.

Alexander Grosberg, now at New York University, predicted (1M pdf) in 1988 that a polymer would initially collapse into a "crumpled globule," in which nearby segments of the chain would be closer to each other than they would be in the final, equilibrium globule. Creating the equilibrium structure requires "reptation," in which the polymer chain threads its way through its own loops, forming knots. This gets very slow for a long chain like DNA. Grosberg also applied (1M pdf) these ideas to DNA, and explored whether fractal patterns in the sequence could stabilize it. But experimental evidence was limited.

Now Erez Lieberman-Aiden and his coworkers at MIT and Harvard have devised a clever way to probe the large-scale folding structure of DNA, and found strong support for this picture.

The experiment is similar to chromatin immunoprecipitation techniques that look for DNA regions that are paired to target proteins by crosslinking and precipitating the pairs and then sequencing the DNA. In this case, however, the researchers crosslink nearby sections of the collapsed DNA to each other. To sequence both sections of DNA, they first splice the ends of the pairs to each other to form a loop, and then break them apart at a different position in the loop. The result is a set of sequence pairs that were physically adjacent in the cell; their positions along the DNA are found by matching them to the known genome.

The researchers found that the number of neighboring sequences decreases as a power law of their sequence separation, with an exponent very close to -1, for sequence distances in the range of 0.5 - 7 million bases. This is precisely the expected exponent for the crumpled--or fractal--globule. This structure is reminiscent of the space-filling Peano curve with its folds, folds of folds, and folds of folds of folds forming a hierarchy. In contrast, the equilibrium globule has an exponent of -3/2.

As a rule, I don't put a lot of stock in claims that a structure is fractal simply by seeing a power law, or a straight line on a double-logarithmic plot, unless the data cover at least a couple of orders of magnitude. After all, a true fractal is self-similar, meaning that the picture looks exactly the same at low resolution at high resolution, and in many cases there's no reason to think that fine structure resembles the coarse structure at all.

But when there's a good theoretical argument for similar behavior at different scales, I relax my standards of evidence a bit. For example, there's a good argument that rate the random walk of a diffusing molecule looks into neighboring volumes looks similar, whatever the size of the volume you consider--this is a known fractal. The standard polymer model is just a self-avoiding random walk, which adds the constraint that two parts of the chain can't occupy the same space. The DNA data are different in detail, but the mathematical motivation is similar.

At the conference I covered last week in Cambridge, MA, Lieberman-Aiden noted that the fractal structure has precisely the features you would want for a DNA library: it is compact, organized, and accessible. The densely packed structure keeps nearby sequence regions close in space, and parts of it can easily be unfolded to allow the transcription machinery to get access to it. Co-author Maxim Imakaev has verified all of these features with simulations of the collapsing DNA.

These experiments and simulations are fantastic, and the fractal globule structure makes a lot of sense. But this dense structure makes it all the more amazing what must happen when cells divide, making a complete copy of each segment of DNA (except the telomeres), and ensuring that the epigenetic markers on the DNA and histones of one copy are replicated on the other. It's still an awesome process.

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