Eric Lander (Science Adviser to the President and Director of Broad Institute) et al. delivered the message
on Science Magazine cover (Oct. 9, 2009) to the effect:
"Mr. President; The Genome is Fractal !"
On this joyful occasion, when first with the Personal Genomes conference at Cold Spring Harbor (Sept. 14-17, 2009) and now on Science Cover Article (Oct. 9, 2009) the establishment embraced fractal recursive algorithm, the International HoloGenomics Society may wish to make some technical comments, as well as a projection.
First, it may seem to be trivial to rectify the statement in "About cover" of Science Magazine by AAAS. The statement "the Hilbert curve is a one-dimensional fractal trajectory" needs mathematical clarification. While the paper itself does not make this statement, the new Editorship of the AAAS Magazine might be even more advanced if the previous Editorship did not reject (without review) a Manuscript by 20+ Founders of (formerly) International PostGenetics Society in December, 2006.
Second, it may not be sufficiently clear for the reader that the reasonable requirement for the DNA polymerase to crawl along a "knot-free" (or "low knot") structure does not need fractals. A "knot-free" structure could be spooled by an ordinary "knitting globule" (such that the DNA polymerase does not bump into a "knot" when duplicating the strand; just like someone knitting can go through the entire thread without encountering an annoying knot):
Just to be "knot-free" you don't need fractals
Note, however, that the above "strand" can be accessed only at its beginning - it is impossible to e.g. to pluck a segment from deep inside the "globulus".
This is where certain fractals provide a major advantage - that could be the "Eureka" moment for many readers. For instance, the mentioned Hilbert-curve is not only "knot free" - but provides an easy access to "linearly remote" segments of the strand. This fractal property is sketched below:
If the Hilbert curve starts from the lower right corner (see turquoise circle) and ends at the lower left corner, for instance the red path shows the very easy access of what would be the mid-point if the Hilbert-curve was measured by the Euclidean distance along the zig-zagged path. Likewise, even the path (shown in blue) from the beginning of the Hilbert-curve is about equallly easy to access - in fact it is easier than to reach from the origin a point that is about 2/3 down the path. The path shown in purple illustrates that the Hilbert-curve provides an easy access between two points within the "spooled thread"; from a point that is about 1/5 of the overall length to about 3/5 is also in a "close neighborhood". This may be the "Eureka-moment" for some readers, to realize that the strand of "the Double Helix" requires quite a finess to fold into the densest possible globuli (the chromosomes) in a clever way that various segments can be easily accessed, moreover - as we'll see - in a way that distances between various segments are minimized. To better illustrate this marvellous fractal structure, look at the 3D rendering of the Hilbert-curve below (source). If your browser can show the animated .gif (left, below), you can observe the "slits of the kimono" feature of the complex 3D structure. As it rotates, in some position you can "see through", for easy access of otherwise remote points. In case your browser will not show the rotating "ultra-dense, but ultra see-through" 3D structure, a still picture on the right will demonstrate how easily one can cut through the entire structure to reach even segments at the "most distant" positions. Once you'll observe such fractal structure, you'll never again think of a chromosome as a "brillo mess", would you? It will dawn on you that the genome is orders of magnitudes more finessed than we ever thought so.
Most readers should probably stop here, with the basic lesson learned that Fractal DNA, while it may appear fanciful, is most importantly an ultimately practical structure - and non-Euclidean geometries require much attention if the instruction of ENCODE is to be headed ("the scientific community will need to rethink some long-held views" - Francis Collins, June 14, 2007).
Those who are ready to engage in the "why" and "how" might wish to contact the Authors - and even this researcher has his answers for future advancements; pellionisz_at_junkdna.com.
Those embarking at a somewhat complex review of some historical aspects of the power of fractals may wish to consult the ouvre of Mandelbrot (also, to celebrate his 85th birthday this November 20th). For the more sophisticated readers, even the fairly simple Hilbert-curve (a representative of the Peano-class) becomes even more stunningly brilliant than just some "see through density". Those who are familiar with the classic "Traveling Salesman Problem" know that "the shortest path along which every given n locations can be visited once, and only once" (a very practical task for Salesmen and Presidents) requires fairly sophisticated algorithms (and tremendous amount of computation if n>10 (or much more). Some readers will be amazed, therefore, that for n=9 the underlying Hilbert-curve helps to provide an empirical solution:
On the source page of this diagram (Takahashi), one can click any number of "destinations" on the Hilbert-curve (here, if your browser is Java-enabled) - and see the resulting path as a solution to the Traveling Salesman Problem emerge.
Unification of Recursive Algorithms of Fractal DNA and Neural Networks that the Genome governs to physiologically (or pathologically) grow
Briefly, the significance of the above realization, that the (recursive) Fractal Hilbert Curve is intimately connected to the (recursive) solution of Traveling Salesman Problem, a core-concept of Artificial Neural Networks can be summarized as below.
Accomplished physicist John Hopfield (already a member of the National Academy of Science) aroused great excitement in 1982 with his (recursive) design of artificial neural networks and learning algorithms which were able to find reasonable solutions to combinatorial problems such as the Traveling Salesman Problem. (Book review Clark Jeffries, 1991, see also 2. J. Anderson, R. Rosenfeld, and A. Pellionisz (eds.), Neurocomputing 2: Directions for research, MIT Press, Cambridge, MA, 1990):
"Perceptrons were modeled chiefly with neural connections in a "forward" direction A -> B -* C -- D. The analysis of networks with strong backward coupling proved intractable. All our interesting results arise as consequences of the strong back-coupling" (Hopfield, 1982).
The Principle of Recursive Genome Function surpassed obsolete axioms that blocked, for half a Century, entry of recursive algorithms to interpretation of the structure- and function of (Holo)Genome.
This breakthrough, by uniting the two largely separate fields of Neural Networks and Genome Informatics, is particularly important for those who focused on Biological (actually occurring) Neural Networks (rather than abstract algorithms that may not, or because of their core-axioms, simply could not represent neural networks under the governance of DNA information).
Tensor Network Theory of the Central Nervous System (see Encyclopedia of Neuroscience, Figs. 1. and 2., 1987) took the structural geometry of existing cerebellar neural networks (with principal components of so-called Purkinje brain cells), and explained the well-known function of the cerebellum (space-time coordination of sensori-motor activities) in terms of the extrinsic Euclidean (Minkowski) space-(time) embedded into a curved multidimensional functional geometry, where the cerebellar neural networks act as the metric tensor of the intrinsic space; thus capable of transforming non-executable sensory-vectors (that are, in the slang of tensor-geometry "covariant expressions") into precisely executable motor-vectors (that are "contravariant"). The author resigned to the general observation that appreciation of geometrization of biology (following the required understanding) may call for a period of 20-25 years in experimental neuroscience. Thus, meanwhile focused on utilization of an artificial electronic cerebellum to automatically regain the balance of severely misconfigured fighters, such as F15 for NASA. Since such compute-intensive operation literally "on the fly- with supersonic speeds" the approach used parallel computers "Transputers" for the project based on at that time classified defense info [see a near-disaster, initiating the project, on YouTube]).
Unification in functional geometry of Genome Informatics and Neural Networks, in significance goes way beyond the elegance of unified algorithms and thus better computation-performance. There is no denial that (biological) Neural Networks develop under the governance of Genome Information. Thus, the embedding of fractal curves in metrical (albeit often curved) spaces is not just an opening of an avalanche of algorithms of novel genome informatics, but ultimately unifies Fractal DNA (structure and function) with Fractal development of organelles, organs and organisms (FractoGene, 2002, 2008, 2009).
With the "affordable full DNA sequences" already here, deploying tools of advanced mathematics for interpretation (see also here) to the very same parallel computing means already developed for earlier defense applications) is even more vital than earlier. This time, not only those in the frontline, but all "in the homelands" (our health and prevention of diseases) are affected.
Computing architecture for The Genome Based Economy will hinge on algorithmic understanding and building the most effective implementation.