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)