Arcadian Functor

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Marni D. Sheppeard

Monday, June 11, 2007

M Theory Lesson 66

Recall that Joan Birman et al studied knots in the Lorenz template with two generating holes X and Y. So knots are expressed as words in X and Y. In Robert Ghrist's paper Branched two-manifolds supporting all links he shows that the template $\mathcal{V}_0$ on more letters contains an isotopic copy of every (tame) knot and link. More specifically, for a parameter range $\beta \in [6.5,10.5]$ every link appears as a periodic solution to the equation which is used to model an electric circuit. This is cool stuff. In M Theory we like ribbon diagrams which are twisted into loops like in the Lorenz template diagram. The universal template $\mathcal{V}_0$ can be embedded in an infinite sequence of more complicated templates, which in turn are embeddable in $\mathcal{V}_0$. Ghrist also considers flows arising from fibrations, such as the 1-punctured torus fibration for the figure 8 knot complement. This fibration flow is also an example of a universal flow.

I was quite intrigued when a mathematical biologist at a conference told me recently that no one really knew why DNA had four bases rather than two. Apparently it isn't clear why self-replicating molecules fail to adopt a binary code in X and Y. Somebody else muttered something about hydrogen bonds and then, inspired and ignorant, I started rambling on about knot generation in templates. After all, DNA molecules need to know how to knot themselves.

8 Comments:

Blogger CarlBrannen said...

You should be getting ready to defend yourself! The equations you gave remind me of what you get when you toss together a diode and some resistors, and combine it with an opamp.

It turns out that the simple circuit of a D flip-flop with its output connected back to its input with an inverter (not gate) will produce a transition to chaos as the frequency is increased. One could do the same thing with an opamp, provided you have a time delay built in.

Back when I was interested in the transition to chaos, I wrote down a very simple set of differential equations and got chaos to simulate on Mathematica. I don't know if this is a different set of differential equations than what is well known, since I never really studied the subject.

June 11, 2007 1:42 PM  
Blogger Doug said...

Hi Kea and CarlB,

I looked more closely at the Ghrist web page. In his CV, under referenced publication, he lists two papers with LaValle.
http://www.math.uiuc.edu/~ghrist/cv.pdf

One paper, ’Nonpositive curvature and pareto-optimal coordination motion planning', SIAM Journal of Control and Optimization, 45(5), 1697-1713, 2006
[I could not find this on the web]
http://www.math.uiuc.edu/~ghrist/preprints/pareto.pdf

But a similar [perhaps identical] paper ’Nonpositive curvature and pareto-optimal coordination of robots’, SIAM Journal of Control and Optimization, 2007
[Is on the web]
http://msl.cs.uiuc.edu/~lavalle/mulrob.html

“Pareto-optimal” is game theory terminology.
http://en.wikipedia.org/wiki/Pareto_efficiency

I am more familiar with LaValle, “Planning Algorithms”.
This lead to my reading the Basar and Olsder book that I have often referenced.
[LaValle book available on-line]
http://planning.cs.uiuc.edu/

I am not famililiar with Joan Birman but I am intrigued.

Could the Lorentz knots be 'stringing loops' or 'looping strings' playing games?

Did Yau derive the concept of flop transitions from EM flip-flops?

Check out the Rossler Attractor, especially the section 'Links to other topics',
"The banding evident in the Rössler attractor is similar to a Cantor set rotated about its midpoint. Additionally, the half-twist in the Rössler attractor makes it similar to a Möbius strip."
http://en.wikipedia.org/wiki/R%C3%B6ssler_map

June 11, 2007 1:59 PM  
Blogger Doug said...

RE: "... why DNA had four bases rather than two ..."

U_Utah has a great web page on ‘Purine and Pyrimidine Metabolism’.
http://library.med.utah.edu/NetBiochem/pupyr/pp.htm

Note that precursors of Purines could be”
Hypoxanthine = 6-oxy purine
or
Xanthine = 2,6-dioxy purine

Precursor(s) of Pyrimidines could be Orotic acid = 2,4-dioxy-6-carboxy pyrimidine.

This is not my final answer, because who knows for certain?

The binary code may be nested:
RNA or DNA
then
Purine or Pyrimidine
If Purine
then generally Adenine or Guanine
If Pyrimidine
then generally Cytosine or
Uracil if RNA or
Thymine if DNA.

Note:
Uracil is more specific to RNA than Adenine and
Thymine is more specific to RNA than Adenine.
Why - Oxy or deoxy presence?

June 11, 2007 2:15 PM  
Anonymous Anonymous said...

Kea, I am a bit surprised that such a simple thing like pairing of AT and GC is unknown on this blog. Surely genetic code is binary because of such a pairing and surely information is read off in binary code. In addition, please, take a look at gr-qc/040029 and take a look how REAL alloy structures are related to gravity and Yamabe functionals

June 11, 2007 2:32 PM  
Anonymous Anonymous said...

Addendum. Sorry, it is late here, in US. The correct reference is gr-qc/0410029. In addition to this, you may take a look at hep-th/0701084 where these ideas are developed further in the style of Grisha's Perelman work(s)

June 11, 2007 3:08 PM  
Blogger Mitchell said...

Anonymous - DNA is quaternary, not binary. AT is different from TA.

June 12, 2007 2:44 PM  
Blogger Matti Pitkanen said...

I have considered some number theory inspired models for genetic code. Quaternary code is one of them studied also by Khrennikov.

Also 5-adicity has been suggested and I constructed for half year ago a model in which codons correspond to 5-adic numbers with non-vanishing digits: this means that codons correspond to numbers in interval [31,126].

The basic observation is that there are 20 primes in this interval. They would correspond naturally to primes labelling aminoacids. The variational principle states that the p-adic negentropy (log(x) is replaced with log(|x|_p) in Shannon entropy) associated with thermodynamical state in 5-adic thermodynamics and assigned to the partitions of integer n labelling a given codon is maximized as a function of p. Hence correspondence n-->p(n) characterizing code results.

There also other constraints and only few solutions are satisfying the constraints are obtained. See this.

June 12, 2007 6:28 PM  
Blogger CarlBrannen said...

Kea, I've posted a blog on why DNA uses 4 nucleotides . Great topic.

June 14, 2007 6:28 PM  

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