untangling knots in lattices and proteins knots in lattices and proteins by rhonald lua adviser:...
TRANSCRIPT
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Untangling Knotsin Lattices and Proteins
By Rhonald Lua
Adviser: Alexander Yu. Grosberg
University of Minnesota
A Computational Study
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Human Hemoglobin(oxygen transport protein)
Globular proteins have dense, crystal-like packing density.Proteins are small biomolecular “machines” responsible
for carrying out many life processes.
(Structure byG. FERMI andM.F. PERUTZ)
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Hemoglobin Protein Backbone (string of α−carbon units)
“ball of yarn”
One chain
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4x4x4 Compact Lattice Loop
Possible cube dimensions: 2x2x2,4x4x4,6x6x6,…,LxLxL,…
Ne
z )( 1−No. of distinct conformations: (Flory)
z = 6 in 3D
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Hamiltonian Path Generation(A. Borovinskiy, based on work by R. Ramakrishnan, J.F. Pekny, J.M.
Caruthers)
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14x14x14 Compact Lattice Loop
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In this talk…
• knots and their relevance to physics
• “virtual” tools to study knots
• knotting probability of compact lattice loops
• statistics of subchains in compact lattice loops
• knots in proteins
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Knot – a closed curve in space that does not intersect itself.
The first few knots:
3-1 (Trefoil)
4-1 (Figure-8)
5-1 (Cinquefoil, Pentafoil
Solomon’s seal)5-2
Trivial knot (Unknot)0-1
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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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A Little Knot Math
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Reidemeister’s Theorem:Two knots are equivalent if and only ifany diagram of one may be transformed
into the other via a sequence ofReidemeister moves.
Reidemeister Moves
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Compounded Reidemeister Moves
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Knot Invariants -Mathematical signatures of a knot.
Trefoil knot
3-1
Trivial knot0-1
Examples: ∆(-1)=1v2=0v3=0
∆(-1)=3v2=1v3=1
v3Vassiliev degree 3
v2Vassiliev degree 2
∆(-1)AlexanderSymbolName
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Alexander Polynomial, ∆(t)(first knot invariant/signature)
u1
u2 u3
g2
g3g1
start
Alexander matrix for this trefoil:
1t-10
-t1t-1
t-1-t1
∆(-1) = det1-2
11 = 3Alexander invariant:
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In the followingindex k corresponds to kth underpass and
index i corresponds to the generator numberof the arc overpassing thekth underpass
For row k:1) when i=k or i=k+1 then���=-1, �����=1
2) when i equals neither k nor k+1:If the crossing has sign -1:
���=1, �����=-t, ���=t-1If the crossing has sign +1:���=-t, �����=1, ���=t-13) All other elements are zero.
Recipe for Constructing Alexander Matrix, ���n x n matrix where n is
the number of underpasses
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Gauss Code and Gauss Diagram
Gauss Diagram for trefoil:
Gauss code for left-handed trefoil:b - 1, a - 2, b - 3, a - 1, b - 2, a – 3
(Alternatively…)
sign:
1, (-)
2, (-)3, (-)
a – ‘above’b – ‘below’
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Vassiliev Invariants(Diagram methods by M. Polyak and O. Viro)
Degree two (v2): Look for this pattern:
Degree three (v3): Look for these patterns:
+2
1
e.g. trefoil
v2 =1v3 =1
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Prime and Composite Knots
)()()(
)()()(
)()()(
23133
2212
21
2
KvKvKv
KvKvKv
tttKKK
+=+=
∆∆=∆
Composite knot, K
K1 K2
Alexander:
Vassiliev:
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Method to Determine Type of Knot
Project 3D objectinto 2D diagram.
Preprocess and simplify diagramusing Reidemeister moves.
Compute knot invariants.
Inflation/tighteningfor large knots.
Give object a knot-type
based on its signatures.
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A. Projection
Projected nodes and links
3D conformation 2D knot projection
projection process
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B. PreprocessingUsing Reidemeister moves
537205712
1219389614
18796910
403838
41146
0204
…and after reduction
Average crossings before…
L
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C. Knot Signature Computation
YES3275-2
YES5355-1
NO0-154-1
YES1133-1
NO001Trivial
Chiral?|v3|v2|∆(-1)|Knot
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Caveat!Knot invariants cannot unambiguously classify a
knot.However• knot invariants of the trivial knot and the first four
knots are distinct from those of other prime knots with 10 crossings or fewer (249 knots in all), with one 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
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Knot Tightening
Shrink-On-No-Overlaps (SONO) method of Piotr Pieranski.Scale all coordinates s<1, keep bead radius fixed.
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Results
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Knotting Probabilities for Compact Lattice Loops
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12x12x12
14x14x14
10x10x10
8x8x86x6x6
4x4x4
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
0 500 1000 1500 2000 2500 3000
length
frac
tion
of u
nkno
ts
Chance of getting an unknot for several cube sizes:
slope -1/196
Mansfieldslope -1/270
0/0 )( NNeNP −≈
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0
0.05
0.1
0.15
0.2
0.25
0 500 1000 1500 2000
length
frac
tio
n o
f kn
ot
trefoil (3-1)
figure-eight(4-1)
star (5-1)
(5-2)
Chance of getting the first few simple knots for different cube sizes:
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Subchain statistics
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Fragments of trivial knots are more crumpled compared to fragments of all knots.
Sub-chain:(fragment)
14x14x14 Compact Lattice LoopAverage size of subchain (mean-square end-to-end) versus
length of subchain
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Noncompact, Unrestricted LoopAverage gyration radius (squared) versus length
Trivial knots swell compared to all knots for noncompact chains.This topologically-driven swelling is the same as that driven byself-avoidance (Flory exponent 3/5 versus gaussian exponent 1/2).
(N. Moore)
Closed random walkwith fixed step length
53
NR≈
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0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
0 50 100 150 200 250
segment length
rati
o m
ean
-sq
uare
end
-to-
end
of u
nkn
ot v
s al
l kn
ots 4x4x4
6x6x6
8x8x8
10x10x10
12x12x12
14x14x14
Fragments of trivial knots are consistently more compactcompared to fragments of all knots.
Compact Lattice LoopsRatios of average sub-chain sizes, trivial/all knots
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Compact Lattice Loops
(A. Borovinskiy)
Over all knots: 21
tR ≈i.e. Gaussian;Flory’s result for chains ina polymer melt.
Trivial knots: 31
tR ≈ ?
General scaling of subchains (mean-square end-to-end) versus length
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Knot (De)Localization
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Localized or delocalized?
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What have been shown computationally…
*Katritch,Olson, Vologodskii, Dubochet, Stasiak (2000). Preferred size of ‘ core’ of trefoil knot is 7 segments.**Orlandini, Stella, Vanderzande (2003). Localization to delocalization transition below a θ−point temperature.
Delocalized**?Compact (collapsed)
Localized**Localized*Noncompact(swollen)
Flat knots
(Polymer on sticky surface)
Random circular chains
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Knot Renormalization
g=1 g=2
Localized trefoil
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Renormalization trajectory space
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Renormalization trajectoryInitial state: Noncompact loop, N=384
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Renormalization trajectoryInitial state: 8x8x8 compact lattice loop
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Renormalization trajectoryInitial state: 12x12x12 compact lattice loop
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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. K. Millet, A. Dobay, A. Stasiak (2005)(Not about proteins) A study of linear random knots
and their scaling behaviour.
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1. Obtain protein structural information (.pdb files) from the Protein Data Bank. 4716 id’s of representative protein chains obtained from the Parallel Protein Information Analysis (PAPIA) system’s Representative Protein Chains from PDB (PDB-REPRDB).
2. Extract coordinates of protein backbone
3. Close the knot (3 ways)
4. Calculate knot invariants/signatures
Steps
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Protein gyration radius versus length
31
aNR =
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CM-to-terminals distance versus gyration radius
RTC 5.1≈−
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DIRECT closure method
T1, T2 – protein terminals
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CM-AYG closure method
C – center of massS1, S2 - located on surface of sphere surrounding the protein
F- point at some large distance away from C
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RANDOM2 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 RANDOM2 closures for protein 1ejg chain A
N=46
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Knot probabilities in RANDOM2 closures for protein 1xd3 chain A
N=229
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Knot counts in the three closure methods
•RANDOM2 and CM-AYG methods gave the same predictions for 4711 chains (out of 4716).•RANDOM2 and DIRECT methods gave the same predictions for 4528 chains (out of 4716).
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Distribution of the % of RANDOM2 closures giving the dominant knot-type
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Unknotting probabilities versus length for proteins and for compact lattice loops
Total of 19 non-trivial knots in the RANDOM2 method.Knots in proteins occur much less often than in compact lattice loops.
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Summary of Results
• Unknotting probability drops exponentially with chain length.
• For compact conformations, 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.
196N
ePtrivial−≈
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Unresolved issues…
•Are knots in compact loops delocalized? To what degree?•Theoretical treatment of the scaling of subchainsin compact loops with trivial knots.•Theoretical prediction for the characteristic length of knotting N0. 0/
0 )( NNeNP −≈
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Acknowledgments
A. Yu. Grosberg.A. Borovinskiy, N. Moore.
P. Pieranski and associates forSONO animation.
DDF support, UMN Graduate SchoolKnot Mathematicians.
Biologists, Chemists and other researchers formaking protein structures available.