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.

    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]

    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]

    “Pareto-optimal” is game theory terminology.

    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]

    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.”

  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’.

    Note that precursors of Purines could be”
    Hypoxanthine = 6-oxy purine
    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
    Purine or Pyrimidine
    If Purine
    then generally Adenine or Guanine
    If Pyrimidine
    then generally Cytosine or
    Uracil if RNA or
    Thymine if DNA.

    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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