Significance of Knotted Structures for Function of Proteins and Nucleic Acids
Poster Session II
46 – POS
Board 18
How Molecular Knots can pass through Each Other
Benjamin Trefz
1,2
, Jonathan Siebert
1
, Peter Virnau
1
.
1
Johannes Gutenberg University Mainz, Mainz, Germany,
2
Graduate School Materials Science in
Mainz, Mainz, Germany.
We suggest and explain a mechanism in which two molecular knots on a single DNA strand can
pass through each other and swap positions along the strand: One of the two knots expands in
size and the other diffuses along the contour of the former. This peculiar mechanism, which only
requires a few k
B
T, is not only interesting from an aesthetic point of view, but may also play a
role in future technological applications such as nanopore sequencing once strand sizes exceed
100,000 base pairs and knots in DNA become likely.
47 – POS
Board 19
Progressive Factorization of Composite Knots into Isolated Prime Components: A
Systematic Computational Investigation
Luca Tubiana
.
Institut Jožef Stefan, Ljubljana, Slovenia.
Composite knots are known to dominate the knot spectrum of long polymer rings, but in spite of
their ubiquity their behavior remains largely unexplored. One of the few standing points is that,
in the limit of long polymers rings, their knotting probability tends to the product of the knotting
probabilities of the single factor knots composing them. This factorization of the knotting
probability has been justified with the assumption that in long polymers knots become localized
and therefore behave like point-like decorations on the rings.
Here, using Monte Carlo simulations and advanced knot localization methods, we analyze the
length and distribution of prime components in composite knots tied on Freely Jointed Polymer
Rings. For increasing contour length, we observe the progressive factorization of composite
knots into separated prime components. However, we observe that a complete factorization,
equivalent to the "decorated ring'' picture, is not obtained even for rings of contour lengths up to
tens of times the most probable length of the prime knots tied on the rings.
Following our results, we suggest that the "decorated ring'' hypothesis may not be necessary to
explain the factorization of the knotting probabilities, at least when polymers excluded volume is
not relevant. We rationalize the behavior of the system through a simple one dimensional model
in which prime knots are replaced by sliplinks randomly placed on a circle, with the only
constraint that the length of the loops has the same distribution of the length of the corresponding
prime knots.
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