Untangling Knots in Lattices and Proteins

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Untangling Knotsin Lattices and Proteins
A Computational Study

• By Rhonald Lua
• Adviser Alexander Yu. Grosberg

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Human Hemoglobin(oxygen transport protein)
(Structure by G. FERMI and M.F. PERUTZ)
Globular proteins have dense, crystal-like
packing density. Proteins are small biomolecular
machines responsible for carrying out many life
processes.
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Hemoglobin Protein Backbone (string of a-carbon
units)
One chain
ball of yarn
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4x4x4 Compact Lattice Loop
Possible cube dimensions 2x2x2,4x4x4,6x6x6,,LxLx
L,
No. of distinct conformations
(Flory)
NL3, z 6 in 3D
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14x14x14 Compact Lattice Loop
What kind of knot is (in) this?
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Subchains
contour length l lt N2/3
What are the statistics of the end-to-end
distance?
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In this talk

• Knots and their relevance to physics
• Virtual tools to study knots
• Results for lattices
• Results for proteins

• Knotting probabilities
• Subchain statistics

• Knot closure methods
• Knotting probabilities
• Subchain statistics

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Knot a closed curve in space that does not
intersect itself.
The first few knots
Trivial knot (Unknot) 0-1
3-1 (Trefoil)
4-1 (Figure-8)
5-1 (Cinquefoil, Pentafoil Solomons seal)
5-2
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Knots in Physics

• Lord Kelvin (1867) Atoms are knots (vortices) of
  some medium (ether).
• Knots appear in Quantum Field Theory and
  Statistical Physics.
• Knots in biomolecules. Example The more
  complicated the knot in circular DNA the faster
  it moves in gel-electrophoresis experiments.

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Protein Folding
denatured state
collapsed native state (functioning protein)

• Knots present a barrier to folding.
• If knots are rare, is protein folding ergodic?
  (Mansfield 1994)
• Are proteins selected by evolution to preclude
  knots?

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A Little Knot Math
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Reidemeister Moves
Reidemeisters Theorem Two knots are equivalent
if and only if any diagram of one may be
transformed into the other via a sequence
of Reidemeister moves.
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Compounded Reidemeister Moves
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Knot Invariants -Mathematical signatures of a
knot.
D(-1)1 v20 v30
Examples
Trivial knot 0-1
D(-1)3 v21 v31
Trefoil knot 3-1
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Prime and Composite Knots
Composite knot, K
K1
K2
Alexander
Vassiliev
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Method to Determine Type of Knot
Project 3D object into 2D diagram.
Inflation/tightening for large knots.
Preprocess and simplify diagram using
Reidemeister moves.
Compute knot invariants.
Give object a knot-type based on its signatures.
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A. Projection
2D knot projection
3D conformation
projection process
Projected nodes and links
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B. Preprocessing
Using Reidemeister moves
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C. Knot Signature Computation
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Caveat!

• Knot invariants cannot unambiguously classify a
  knot.
• However
• For prime knots with 10 crossings or fewer (249
  knots in all), knots invariants for knots
  0-1,3-1,4-1,5-1,5-2 are unique with the exception
  (5-1 and 10-132)
• Reidemeister moves and knot inflation can
  considerably reduce the number of possibilities.

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Knot Inflation
Monte Carlo. Points remain on a lattice.
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Knot Tightening
Shrink-On-No-Overlaps (SONO) method of Piotr
Pieranski. Scale all coordinates slt1, keep bead
radius fixed.
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Results
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Knotting Probabilities for Compact Lattice Loops
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Chance of getting an unknot
for several cube sizes
Mansfield slope -1/270
slope -1/196
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Chance of getting the first few simple knots for
different cube sizes
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Subchain statistics
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14x14x14 Compact Lattice LoopMean square
end-to-end distance versus length of subchain
Subchain (fragment)
Fragments of trivial knots are more crumpled
compared to fragments of all knots.
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Contrast with Non-compact, Unrestricted
LoopAverage gyration radius (squared) versus
length
Closed random walk with fixed step length
(N. Moore)
Trivial knots swell compared to all knots for
non-compact chains. This topologically-driven
swelling is the same as that driven
by self-avoidance (Flory exponent 3/5 versus
gaussian exponent 1/2).
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Compact Lattice Loops
General scaling of subchains (mean-square
end-to-end) versus length
Over all knots
i.e. Gaussian Florys result for chains in a
polymer melt.
Trivial knots
?
(A. Borovinskiy)
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Knots in Proteins
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Previous work
1. M.L. Mansfield (1994) Approx. 400 proteins,
with random bridging of terminals, using
Alexander polynomial. Found at most 3 knots. 2.
W.R. Taylor (2000) 3440 proteins, fixing the
terminals and smoothing (shrinking) the segments
in between. Found 6 trefoils and 2
figure-eights. 3. J.R. Banavar, T.X. Huang, A.
Maritan (2005) (Not about knots) Flory theorem
holds for proteins. 4. K. Millet, A. Dobay, A.
Stasiak (2005) (Not about proteins) A study of
linear random knots and their scaling behaviour.
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Finding knots in proteins
Protein Data Bank (PDB/PAPIA)
4716 protein chains
Extract coordinates of backbone protein chain
Close the knot with DIRECT,CENTER,RANDOM
Project, simplify and compute knot
invariants/signatures
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Protein gyration radius versus length
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Distribution of the distance of the protein
terminals from the center-of-mass
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DIRECT closure method
T1, T2 protein terminals
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CENTER closure method
C center of mass S1, S2 - located on surface of
sphere surrounding the protein F- point at some
large distance away from C
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RANDOM closure method
(random)
(random)
Study statistics of knot closures after
generating 1000 pairs of points (S1 and S2).
Determine the dominant knot-type.
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Knot probabilities in RANDOM closures for protein
1ejg chain A
N46
next dominant
dominant
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Knot probabilities in RANDOM closures for protein
1xd3 chain A
N229
next dominant
dominant
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Distribution of the of RANDOM closures giving
the dominant knot-type
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Knot counts of the 4716 protein chains in the
three closure methods

• RANDOM and CENTER methods gave the same
  predictions for all but 5 chains.
• DIRECT method gave significantly more knots.

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Unknotting probabilities versus length for
proteins and for compact lattice loops
Total of 19 non-trivial knots in the RANDOM
method. Knots in proteins occur much less often
than in compact lattice loops.
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Scaling of subchains in proteins and in compact
lattice loops
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Degree of interpenetration of subchains
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Secondary and Tertiary structures
b strand
a helix
Basic units
aa
bab
bb
Supersecondary
bbbb (Greek key)
aaaa (4-helix Bundle)
babab (Rossman fold)
Domain folds (examples)
Based on http//www.med.unibs.it/marchesi/pps97/c
ourse/section9/9_term.html and http//swissmodel.e
xpasy.org/course/text/chapter4.htm
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Summary of scales in proteins
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Summary of Results

• Unknotting probability drops exponentially with
  chain length.
• Compact lattice subchains satisfy Florys
  theorem. Subchains of trivial knots are
  consistently smaller than subchains of
  non-trivial knots. For noncompact conformations,
  the opposite is observed. The fragments seem to
  be aware of the knottedness of the whole thing
  (AYG).
• Knots in proteins are rare, compatible with the
  compactness of protein subchains.

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Acknowledgments
A. Yu. Grosberg. A. Borovinskiy, N. Moore. P.
Pieranski and associates for SONO
animation. Minnesota Supercomputer Institute,
DTC. DDF support, UMN Graduate School. Knot
Mathematicians. Biologists, Chemists and other
researchers for making protein structures
available.