

Unknotting DNA
6^{th} April 2018

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"You should let the world know that the true source of mathematical methods as applicable to physics is to be found in the Proceedings of the Royal Society of Edinburgh. The volume surface and line integrals of vectors and quaternions and their properties as in the course of being worked out by Tait is worth all that is going on in other seats of learning."  James Clerk Maxwell, 1871. 
In this blog we discuss the application of the knot theory in DNA structure. We will continue our discussions in Quantum Computing series afterwards. We had earlier described the importance of the measurement capacity of a macroscopic observer in jspace and how it results in an Unknot (Shallow Well) and Trefoil knot (Box). We next developed the description of a Trefoil knot in jspace. The description was in form of Laurent polynomials and it satisfied the following properties:
The knot diagram for the figure8 knot, is shown as following: The knot polynomial for the figure8 knot can be written, in the manner similar to the one introduced for Trefoil knot earlier as following: As discussed earlier while drawing the Figure8 knot, we noticed that the direction of sectional loops is Clockwise and Anticlockwise both with in the same knot. In Trefoil knot the direction is exclusively either Clockwise and Anticlockwise. We therefore need to introduce a representation which along with the knotpolynomial, provides a complete picture to the observer by including the directions of sectional loops within a knot. An example based on Figure8 knot, is shown below:
The
knot description consists of the knot polynomial and representation
columns. For example (+, 1, 3) in above table, describes a knot
whose first crossing is overcrossing (+). One of its crossings,
is the result of a loop with ACW orientation, whereas three of its
crossings are result of loops with ACW orientation.
If we compare the Trefoil
and Figure8 knots, we notice the absence of constant in the knot
polynomial for Figure8 knot. The Trefoil knot in contrast has
for example, a value "3" representing the contributions of 0_{j}. This is an important difference as the absence of the contribution of 0_{j}
in the knot polynomial for figure8 knot, represents
a corresponding physical structure, which has no memory of the
initial state.
The
structures with no memory of the initial state, are statistical in
nature described by the random variables. Thus if we write a knot
polynomial which shows no contribution of 0_{j},
then it is likely to be of statistical nature. Such structures
can be further reduced by a process similar to symmetry breaking, until
contribution of 0_{j}
is accounted for, in the knot polynomials. The most fundamental
structure in jspace is represented by Trefoil knot, which has
three alternating crossings, the required minimum, to form a stable
knot.
A Comparison Criterion for DNA and Enzymes: A fantastic application will be to apply jspace knot polynomials, to the mechanism used by enzymes to unpack DNAs for replication. In jspace, enzymes and DNA both are observers, who are making measurements and being measured at the same time. Each molecule in a DNA represents a measurement. Subsequently by stringing together these measurements, a knot polynomial can be written for a specific DNA. The DNA represented by knot polynomials with no contribution from 0_{j} term, are more likely to be affected by enzymes and hence easy to replicate. If the enzyme is related to a potentially catastrophic disease then such DNA are at risk. Similarly the DNA represented by very high contributions of 0_{j} term in their knot polynomials, are likely to be much more robust. DNA and Enzymes Taking
the argument further we can possibly also write similar knot
polynomials for enzymes in jspace. If we compare the knot
polynomials written for enzymes and DNA, the enzymes corresponding to
the polynomials of lower order (t^{n}, t^{+n}) and with high contributions of 0_{j} terms, are more likely to succeed in breaking down DNA with higher order polynomials which have small contributions of 0_{j} terms.
In similar fashion, the DNA with lower order polynomials and with high contributions from 0_{j} terms, should be immune to the effects of enzymes corresponding to higher order polynomials and with little contributions from 0_{j} terms. ***

Previous Blogs: Chiral Symmetry
Sigmaz and I Spin Matrices Rationale behind Irrational Numbers The Ubiquitous zAxis Majorana ZFC Axioms Set Theory Nutshell2014 Knots in jSpace Supercolliders Force Riemann Hypothesis Andromeda Nebula Infinite Fulcrum Cauchy and Gaussian Distributions Discrete Space, bField & Lower Mass Bound Incompleteness II The Supersymmetry The Cat in Box The Initial State and Symmetries Incompleteness I Discrete Measurement Space The Frog in Well Visual Complex Analysis The Einstein Theory of Relativity *** 

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