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Lee Nam-Kyung Department of Physics Sejong University. Optimal confinement for internal polymer binding. Outline. Introduction Loop formation Kinetics of an ideal chain Kinetics of a Excluded volume chain Diffusion/Reaction under confinement Confinement - PowerPoint PPT Presentation
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Lee Nam-KyungDepartment of Physics
Sejong University
Optimal confinement for internal polymer binding
Outline Introduction
Loop formation Kinetics of an ideal chain Kinetics of a Excluded volume chain
Diffusion/Reaction under confinement Confinement Optimal confinement for Cyclization
Formation of higher order vertices Excluded volume barrier for higher order vertex formation Effects of confinement
Conclusions
Binding Phenomena in softmatter
Biology specific bindings: Proliferation, differentiation,
migration of cells Technology
Polymerization, cyclization, Nanostructure-fabrication via self assembly
cyclization reactions of DNA, biopolymers and other synthetic polymers
synthesis of AB diblock copolymers by block reaction at the AB interface
connection by polymers via biotin/streptavidin complexes
adhesion of vesicles on substrates
Binding •Hydrophobic moiety•Hydrogen bonding•Chemical reaction
•Molecular recognition
Diffusion of
Moiety+fluctuating object
Irreversible
Binding
Small length scales Intermediate length scales
☞The kinetics is governed by the diffusion of some groups connected to a strongly fluctuating object toward its target.
The whole fluctuation spectrum of the polymer is potentially involved in the diffusion.
Binding of Biotin and Streptavidin ~the strongest non-covalent biological interaction known.
DNA, RNA probes – detecting complementary target system:-A short sequence of labelled DNA the detection of a complementary nucleotide sequence.
Labeling DNA or RNA probes with biotin: heat or light - Biotin labeling of nucleic acids
Streptavidin coated colloidal particle, nanoparticle: detecting probes
streptavidin
biotin
Cyclization : mRNA loop formation
(A) An electron micrograph of polysomes on mRNA. (B) An AFM micrograph of circularization of mRNA mediated by loop forming proteins. From Wells et al. (1998) .
T. Chou (UCLA)
DNA loop formation
Telomeric DNA +TRFsYoshimura et al. Gene to Cells 9 (2004)
Protein folding
Trp cage protein folding NCSA, UIUC Caponi et al.
62 residue intrachain formation rate ~( Hagen et al. JMB 2000)
What ultimately limits the speed of protein folding? Upper limit : Intrachain contact formation?
15106 s
Dominated by Diffusion ? or Equilibrium Statistics ?
Spacer ~sstickers
Internal Cyclization
From the view of Partition Functions:
Zc~ZasdZa~N2 Zb~Zasd
Loop: sd
(Duplantier,1988)
R ~ NZ~ N
s s
s
Connectivity and Anomalous Diffusion de Gennes J.Chem .Phys 76 (1992)
• Free reactants
– Fickian diffusion x2(t) ~ D t
Connected reactants Anomalous diffusionShort time exploration (t < tR) : dense, marginally dense x2(t) ~ t
Ideal chain + Rouse dynamics
Friction grows with s’(t) t~ x2(t)/(1/s’) ~ x4(t) (t < ts~s2)
x2(t) ~s’ ( x(t) < Rs~s The volume explored x3(t) ~ t3/4 (xd(t) <t ) compact exploration
in 3-d : DC Accumulated time at contact
→ P(t) ~ t (b/x(t))3
txV
ttxD
tx
4/3
2
4/1
)(~
/1~/)(~
~)(
short times (t< r~N2t0) r :Rouse time
correlated linear length s’(t) which diffuse together increases with time
s
Binding Kinetics (RouseDynamics)
Q : binding rate at contact
Reaction Controlled typical time for binding (contact probability) -1 tc~1/Q~ s3/2/Q (small Q)
Diffusion Controlled Longest Relaxation time of “s” first passage time ts ~ s20
(Large Q)
The cross over from RC to DC at Qs1/20 ~ 1
t < s
Zimm Dynamics
Friction grows with x(t) x(t) ~s’
t~ x2(t)/(1/x(t)) ~ x3(t) (t < ts)
The volume explored x3(t) ~ t Accumulated contact probability
→ P(t) ~ t (b/x(t))3
(xd(t) ~t ) exploration is marginally compact longest relaxation time for spacer s (ts~s3)
Excluded Volume
Random Walks <R2> ~ N
Self avoiding Walks <R2>~N2
Contact Exponent (Fixmann)
• Probability to be at contact distance a: P ~1/Rd f(a/R) ~ R-(+d)
• Universal contact exponents f(x) ~x
• statistical weight of conformations at contact is reduced by s
Flog(s)) •Large barrier for internal stickers.
2 2 21 4 21
21 2
2 21 3 1
•self avoiding chain: L
Zimm+excluded volume
Excluded volume : contact conformations has reduced statistical weight by ~s
•Reaction controlled• tr ~ P-1(t) ~ s(R/b)3/Q
~sdt0/Q
)()(
d
d
seq
MF QsR
aQQk
dt
tdr tcyc~ s(d+Q
kinetic rate of a internal cyclization Internal Cyclization time
Confinement
Klimov, Newfield, Thirumalai PNAS (2002)
Takagi et al.
Kinetics of blobs
1
2
0
2~
~
bb
bs
db
QgQ
g
s
g
Osmotic Pressure sets a correlation length for density fluctuation
)1/(1
)1/(
~ d
ddB
cg
Tck
s an ideal chain of s/g swollen blobs of size g
c g
Blob diffusion time
Spacer relaxation time
Blob Binding rate
• Blob dynamics : Rouse• The volume spanned by a spacer• The probability for contact • The accumulated time at contact• The reaction probability
• After the longest relaxation time of the spacer (t=tR)
• P>1 BDL
• P<1 BRC )()/(~ )21/(132/3 22 sgggstr
)()/(~ )21/(132 2 sgggstr
Dynamics under Confinement
4/33 )/( btg
)()/(~ 4/3Rb ttt
)()/(~)( 4/1Rbb ttttA
)()/(~)(~)( 4/1Rbb ttgttAQtP
Large Qb (small cavity, small blob size g) : Blob Diffusion Limited :BDL
Small Qb
(large cavity, large blob size g) : Blob Reaction controlled :BRC
BDL negative exponent on g! BRC
Optimal Confinement at
screened hydrodynamics (Rouse dynamics) SLOW dynamics
screened excluded volume FAST Dynamics
)()/(~ )21/(132/3 22 sgggstr
)()/(~ )21/(132 2 sgggstr
)21/(1 2~ sg
c g tR
c g tR
Geometrical Confinement
• Blobs are space filling: Blob size
• Critical cavity size reflecting spacers
• Optimal cavity size
• Smaller Cavity size (R < Rc) : spacer reflection – The longest Relaxation time is set by BOX size– Confinement always accelerate kinetics
)21(3/)13(3/1
21132/10
313
2~
~)/(~
~/
sNR
NsggsR
gRN
c
BRCBDLNo optimum
Optimum at osmotic regime
Spacer never feel boundary
Screening of Hydrodynamics & Excluded Volume barrier
Formation of higher order vertices
Excluded volume barrier W: ',~/ ppB se TkW ''' pppppp
11 1,~ pNZ ppp
p
Extracted from results in Grassberger et al. Macromolecules (2004)
Kinetic rate: k(t)~exp(Eb)
Higher Order Vertex formation under confinement
•Binding rate •Optimum at g~s1/(1+2)p,p’ )
1'.~ bb
ppgQ
Energy Barrier (Daoud-Cotton limit)
pr
rpr
Tkrf
r
rTpkF
B
mB
/~
4)(
/~)(
)1ln(
22
3
0
2/3
Star of p-arms
Local concentration blob
Free energy
Energy Barrier)'log()')'((2.0~
,2/32/32/3
0
ppgpppp
pgrrr mm
2/31 2.01 pppp
5+5
4+6
Summary The confinement can accelerate the intrachain binding • By cutting long internal relaxation modes• By suppressing late stage energy barrier• By increasing the initial concentration of reacting sites
Optimum confinement: Interplay between excluded volume and the screening
of hydrodynamic interactions
References N.-K. Lee, C.F. Abrams and A. Johner Europhys. Lett. (2005) Macromolecules (2006)
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