protein folding. a theoretical view alexei finkelstein institute of protein research, russian...
TRANSCRIPT
![Page 1: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/1.jpg)
Protein folding. A theoretical view
Alexei Finkelstein
Institute of Protein Research, Russian Academy of Sciences,
Pushchino, Moscow Region, Russia
University of Orange Free StateBloemfontain, South Africa
September 5, 2007
![Page 2: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/2.jpg)
TWO protein folding problemsTWO protein folding problems::
1)1) How does protein structure fold?How does protein structure fold? √
2)2) How to predict protein structure How to predict protein structure from the chain’s a. a. sequence?from the chain’s a. a. sequence?
U NU N
![Page 3: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/3.jpg)
BASIC FACTSBASIC FACTS::Protein chains has unique sequenceProtein chains has unique sequence & unique 3D structure& unique 3D structure
Protein chain can fold Protein chain can fold spontaneouslyspontaneously (RNase, (RNase, Anfinsen, 1961Anfinsen, 1961; ; RNase, RNase, Merrifield, 1969Merrifield, 1969))
Folding time: Folding time: in vivo: in vivo: BiosynthesisBiosynthesis ++ Folding < 10–20 min Folding < 10–20 min in vitroin vitro:: from microseconds to hoursfrom microseconds to hours
![Page 4: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/4.jpg)
BASIC FACTSBASIC FACTS::Protein chains has unique sequenceProtein chains has unique sequence & unique 3D structure& unique 3D structure
Protein chain can fold Protein chain can fold spontaneouslyspontaneously (RNase, Anfinsen, 1961; (RNase, Anfinsen, 1961; RNase, Merrifield, 1969)RNase, Merrifield, 1969)
Folding time: Folding time: in vivo: in vivo: BiosynthesisBiosynthesis ++ Folding < 10–20 min Folding < 10–20 min in vitroin vitro:: from microseconds to hoursfrom microseconds to hours
For:For:Water-solubleWater-solublesingle-domain proteins;single-domain proteins;or separate domainsor separate domains
![Page 5: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/5.jpg)
How CAN protein fold in a “bio-reasonable” time?How CAN protein fold in a “bio-reasonable” time?
Levinthal paradox (1968):Levinthal paradox (1968):
Random exhaustive enumerationRandom exhaustive enumerationSpecial pathway?Special pathway?Folding intermediates?Folding intermediates?
Native protein structure Native protein structure refolds from various starts, refolds from various starts, i.e., it behaves as i.e., it behaves as thermodynamically thermodynamically stablestable..
HOW CAN it be found - HOW CAN it be found - within seconds - among within seconds - among zillions of the others?zillions of the others?
U NRANDOMRANDOM
![Page 6: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/6.jpg)
Is “Levinthal paradox” a paradox at all?
![Page 7: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/7.jpg)
Is “Levinthal paradox” a paradox at all?
LL-dimensional-dimensional““Golf course”Golf course”
![Page 8: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/8.jpg)
Zwanzig, 1992; Bicout & Szabo, 2000
Is “Levinthal paradox” a paradox at all?
…any tilt of energy surface solves this “paradox”… (?)
““Funnel”Funnel”LL-dimensional-dimensional
LL-dimensional-dimensional““Golf course”Golf course”
![Page 9: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/9.jpg)
Cunning simplicity of a “funnel” Cunning simplicity of a “funnel” (without phase separation) folding(without phase separation) folding
- NO- NO simultaneous simultaneous explanation to explanation to (I) “all-or-none” transition(I) “all-or-none” transition(II) folding within non-astron. time(II) folding within non-astron. time at mid-transitionat mid-transition
UU NN
EE
EE
LL-dimensional “folding funnel”?-dimensional “folding funnel”?
~L~L
L-L-
STST
Resistance of entropy at T>0Resistance of entropy at T>0
All-or-none transition All-or-none transition for 1-domain proteins for 1-domain proteins (in thermodynamics: Privalov,1974;(in thermodynamics: Privalov,1974;in kinetics: Segava, Sugihara,1984)in kinetics: Segava, Sugihara,1984)
Funnel helps, but ONLY when Funnel helps, but ONLY when NN is much more stable than is much more stable than U U !!!!
![Page 10: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/10.jpg)
Phillips (1965) hypothesis:Phillips (1965) hypothesis: folding nucleus is formed by the N-endfolding nucleus is formed by the N-end of the nascent protein of the nascent protein
chain, and the remaining chain wraps around it.chain, and the remaining chain wraps around it.
for single-domain proteins:for single-domain proteins: NO: NO:Goldenberg & Creighton, 1983: Goldenberg & Creighton, 1983: circular permutants: circular permutants: N-end has no special role in the N-end has no special role in the in vitro in vitro folding.folding.
A special pathway?A special pathway?
![Page 11: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/11.jpg)
Phillips (1965) hypothesis:Phillips (1965) hypothesis: folding nucleus is formed by the N-endfolding nucleus is formed by the N-end of the nascent protein of the nascent protein
chain, and the remaining chain wraps around it.chain, and the remaining chain wraps around it.
for single-domain proteins:for single-domain proteins: NO: NO:Goldenberg & Creighton, 1983: Goldenberg & Creighton, 1983: circular permutants: circular permutants: N-end has no special role in the N-end has no special role in the in vitro in vitro folding.folding.
A special pathway?A special pathway?
HoweverHowever, , for for manymany-domain-domain proteins: proteins: Folding from N-end Folding from N-end domaindomain, , domain after domain domain after domain
DO NOT CONFUSEDO NOT CONFUSE N-ENDN-END DRIVEN FOLDING DRIVEN FOLDING WITHIN DOMAINWITHIN DOMAIN(which seems to be absent)(which seems to be absent)and and N-DOMAIN DRIVENN-DOMAIN DRIVEN FOLDING IN FOLDING IN MANYMANY-DOMAIN PROTEIN-DOMAIN PROTEIN(which is observed indeed)(which is observed indeed)
![Page 12: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/12.jpg)
NOW: NOW:NOW: NOW:pre-molten MOLTENpre-molten MOLTEN globuleglobule GLOBULE
HYPOTHESIS:HYPOTHESIS:Stages in the mechanism of self-organization of protein molecules O.B.Ptitsyn, Dokl. Akad. Nauk SSSR. 1973; 210:1213-1215.
Folding intermediates?Folding intermediates?
![Page 13: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/13.jpg)
PROTEINPROTEINFOLDING:FOLDING:
current picturecurrent picture(Dobson, 2003)(Dobson, 2003)
ee
![Page 14: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/14.jpg)
UU
NN
MGMG
pre-MGpre-MG
TRUE: FOLDING with observable (accumulating in experiment) intermediates
UU
NN
= MG= MG
INDEED, INDEED, NO exhaustive enumerationNO exhaustive enumerationwhenwhen NN is much more stable thenis much more stable then UU
Hierarchic (stepwise) foldingHierarchic (stepwise) foldingavoids many “bad” pathways avoids many “bad” pathways
![Page 15: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/15.jpg)
UU
NN
MGMG
pre-MGpre-MG
TRUE: FOLDING with observable (accumulating in experiment) intermediates
UU
NN
= MG= MG
Special pathway -Special pathway -Folding intermediates -Folding intermediates -they help, but ONLY when they help, but ONLY when NN is much more stable than is much more stable than U U !! !!
INDEED, INDEED, NO exhaustive enumerationNO exhaustive enumerationwhenwhen NN is much more stable thenis much more stable then UU
Hierarchic (stepwise) foldingHierarchic (stepwise) foldingavoids many “bad” pathways avoids many “bad” pathways
![Page 16: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/16.jpg)
UU
NN
BUT ALSO: FOLDING WITHOUT ANY observable intermediates
UU NN
NO hierarchic foldingNO hierarchic folding – –NO “special pathways”, NO “special pathways”, NONO explanation ofexplanation ofnon-astron. folding time at non-astron. folding time at ““all-or-none” transition,all-or-none” transition,especially close to mid-transitionespecially close to mid-transition
Cunning simplicity of Cunning simplicity of hierarchic folding hierarchic folding as applied to resolve as applied to resolve the Levinthal paradoxthe Levinthal paradox
All-or-none transition All-or-none transition for 1-domain proteins for 1-domain proteins (in thermodynamics: Privalov,1974;(in thermodynamics: Privalov,1974;in kinetics: Segava, Sugihara,1984)in kinetics: Segava, Sugihara,1984)
![Page 17: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/17.jpg)
How CAN protein fold in a “bio-reasonable” time?How CAN protein fold in a “bio-reasonable” time?
Levinthal paradox (1968):Levinthal paradox (1968):
Special pathway?Special pathway?Folding intermediates?Folding intermediates?““Funnel”?Funnel”?Can Can helphelp…, but ONLY when …, but ONLY when NN is much more stable then is much more stable then UU … …
Native protein structure Native protein structure refolds from various starts, refolds from various starts, i.e., it behaves as if i.e., it behaves as if thermodynamically thermodynamically stablestable..
HOW can it be found - HOW can it be found - within seconds - among within seconds - among zillions of the others?zillions of the others?
SEARCH TIME AT SEARCH TIME AT MID-TRANSITION= ???MID-TRANSITION= ???
U NRANDOMRANDOM
![Page 18: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/18.jpg)
Kinetics vs. stability:Kinetics vs. stability: Native protein structure:Native protein structure: That, which folds most rapidly?That, which folds most rapidly? That, which is the most stable?That, which is the most stable?
Practical questions:Practical questions:What to predict? What to design?What to predict? What to design?
![Page 19: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/19.jpg)
Kinetics vs. stability:Kinetics vs. stability: Native protein structure:Native protein structure: That, which folds most rapidly?That, which folds most rapidly? That, which is the most stable?That, which is the most stable?
Practical questions:Practical questions:What to predict? What to design?What to predict? What to design? ((railway? railway? airport?airport?))
![Page 20: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/20.jpg)
However: However: Is there a contradiction between the “foldable” Is there a contradiction between the “foldable” structure and the “most stable” structure?! structure and the “most stable” structure?!
NO!NO!
Computer experiments (Shakhnovich et al, 1993-96); Computer experiments (Shakhnovich et al, 1993-96); general theory (Finkelstein et al., 1995-97) general theory (Finkelstein et al., 1995-97) √
Kinetics vs. stability:Kinetics vs. stability: Native protein structure:Native protein structure: That, which folds most rapidly?That, which folds most rapidly? That, which is the most stable?That, which is the most stable? √
Practical questions:Practical questions:What to predict? What to design?What to predict? What to design?
![Page 21: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/21.jpg)
NucleationNucleation:: Folding with phase separation Folding with phase separation
folding interm.
L
1
![Page 22: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/22.jpg)
NucleationNucleation occurs at theoccurs at the““all-or-none” transitionall-or-none” transition((NN and and UU states are observed only): states are observed only):
NucleationNucleation results from the “ results from the “energy gapenergy gap””
Energy landscapeEnergy landscape
The “The “energy gapenergy gap” is” is: - necessary for unique protein structure: - necessary for unique protein structure - necessary for fool-proof protein action- necessary for fool-proof protein action - necessary for direct - necessary for direct UUNN transition transition - - necessary for fast foldingnecessary for fast folding
UU NN
gapgap
![Page 23: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/23.jpg)
NucleationNucleation:: Folding with phase separation Folding with phase separation
folding interm.
L
1
![Page 24: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/24.jpg)
NucleationNucleation:: Folding with phase separation Folding with phase separation“Detailed Balance”: at given conditions, folding pathway = unfolding pathway
folding interm. = unfolding interm.
L
1
![Page 25: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/25.jpg)
NucleationNucleation:: Folding with phase separation Folding with phase separation“Detailed Balance”: at given conditions, folding pathway = unfolding pathway
folding interm. = unfolding interm.
L
1
folding pathway = unfolding pathway at mid-transition TtrS = Hfolding pathway unfolding pathway close to mid-transition TS 90%H “close to” T 90%Ttr
indeed: T 300oK, Ttr 330oK
![Page 26: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/26.jpg)
NucleationNucleation:: Folding with phase separation Folding with phase separation“Detailed Balance”: at given conditions, folding pathway = unfolding pathway
F # ~ L2/3 surface tension
a) micro-; b) loops [from melting] [from Flory]
F #/RT ~ (1/2 3/2) L2/3 Ln(kf ) ~
folding interm. = unfolding interm.
L
1
![Page 27: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/27.jpg)
↓ ↓
Corr. = 0.7
loops
At mid-transition
intermediatesdo not matter…
![Page 28: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/28.jpg)
↓ ↓ ↓ ΔFN ↓ ↓
ΔFN ↓
Any stable fold is automatically a focus of rapid folding pathways. No “special pathway” is needed.
U N
![Page 29: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/29.jpg)
When globules (N & M) become more stable than U:
a
b
a
b
GAP
1) Acceleration: lnkf -1/2FN/RT
2) Large gap large acceleration before “rollover” caused by intermediates M at “bio-conditions”
↓ ↓ ↓ ΔFN ↓ ↓
ΔFN ↓
GAP
![Page 30: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/30.jpg)
α-helices decreaseeffective chain length. THIS HELPS TO FOLD!
Corr. = 0.84
α-HELICESAREPREDICTEDFROM THEAMINO ACID SEQUENCE
In water
Ivankov D.N., Finkelstein A.V. (2004) Prediction of protein folding rates from the amino-acid sequence-predicted secondary structure. - Proc. Natl. Acad. Sci. USA, 101:8942-8944.
![Page 31: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/31.jpg)
choice of choice of oneone structure out of structure out of zillionszillions REQUIRESREQUIRES very precise estimate of very precise estimate of interactionsinteractions
choice of choice of oneone structure out of structure out of twotwoDOES NOTDOES NOT require too precise estimate of interactionsrequire too precise estimate of interactions
2) One still cannot predict protein structure from the a. a. 2) One still cannot predict protein structure from the a. a. sequence without homologues…sequence without homologues… WHY??WHY??
GAP
GAP
![Page 32: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/32.jpg)
Protein folding. A theoretical view
Alexei Finkelstein
Institute of Protein Research, Russian Academy of Sciences,
Pushchino, Moscow Region, Russia
Gratitude to: D.A. Dolgikh, R.I. Gilmanshin, A.E. Dyuysekina, V.N. Uversky, E.N. Baryshnikova, B.S. Melnik, V.A. Balobanov, N.S. Katina, N.A. Rodionova, R.F. Latypov, O.I Razgulyaev, E.I. Shakhnovich, A.M. Gutin, A.Ya. Badretdinov, O.V. Galzitskaya, S.O. Garbuzynskiy, D.N.Ivankov, N.S. Bogatyreva, V.E. Bychkova, G.V. Semisotnov
The Russian Acad. Sci. Program “Mol. & Cell Biology”, The Russian Foundation for Basic Research, ISSEP, HFSPO, CRDF, INTAS, The Howard Hughes Medical Institute
University of Orange Free StateBloemfontain, South Africa
September 4, 2007
![Page 33: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/33.jpg)
U: stable N: stable
unstablesemi-folded
Consider sequential folding (with phase separation)
M: all unstable
? HOW FAST the most stable state is achieved ?
ESTIMATE free energy barrier F #
Experiment: F # ~ L2/3
Rearrangement of 1 residue takes 1-10 ns
#
L
1ns
Detailed Balance: at given conditions,folding pathway = unfolding pathway
Consider thermodynamic mid-transition U ↔ N.
![Page 34: Protein folding. A theoretical view Alexei Finkelstein Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russia University](https://reader035.vdocument.in/reader035/viewer/2022062305/5697bf831a28abf838c867ab/html5/thumbnails/34.jpg)
L
1ns
F # ~ (1/2 3/2) L2/3
micro loops
Any stable fold is automatically a focus of rapid folding pathways. No “special pathway” is needed.
HOW FAST the most stable state is achieved? free energy barrier
F # ~ L2/3 surface tension
F (U) a) micro-; b) loops
= compact folded nucleus: ~1/2 of the chain
F (N)
micro: F # L2/3 [/4]; 2RT0
[experiment]loops: F
# ≤ L2/31/2[3/2RTln(L1/3)]e-N/(100)
[Flory] [knots]