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 Responses so far »

  1. 1

    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.

  2. 2

    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

  3. 3

    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?

  4. 4

    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

  5. 5

    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)

  6. 6

    Mitchell said,

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

  7. 7

    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.

  8. 8

    CarlBrannen said,

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


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