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  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+fluctu ating object Irreversible Binding Small length scalesIntermediate 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.

My tail Seems to have its own will …. ☞ The kinetics is governed by the diffusion of some groups connected to a strongly fluctuating object (the polymer or the membrane) 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 +TRFs Yoshimura 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?

Dominated by Diffusion ? or Equilibrium Statistics ? Spacer ~s stickers

Internal Cyclization From the view of Partition Functions: Z c ~Z a s    d Z a ~N 2   Z b ~Z a s    d Loop : s  d (Duplantier,1988) R ~ N  Z~   N  s s s

Connectivity and Anomalous Diffusion de Gennes J.Chem.Phys 76 (1992) Free reactants – Fickian diffusion x 2 (t) ~ D t Connected reactants Anomalous diffusion Short time exploration (t < t R ) : dense, marginally dense x 2 (t) ~ t 

Ideal chain + Rouse dynamics  Friction grows with s’(t)  t~ x 2 (t)/(1/s’) ~ x 4 (t) (t < t s ~s 2 )  x 2 (t) ~s’ ( x(t) < R s ~s     The volume explored x 3 (t) ~ t 3/4  (x d (t) <t ) compact exploration in 3-d : DC  Accumulated time at contact → P(t) ~ t (b/x(t)) 3  short times ( t<  r ~N 2 t 0 )  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 t c ~1/Q  ~ s 3/2 /Q ( small Q)  Diffusion Controlled Longest Relaxation time of “s” first passage time t s ~ s 2  0 (Large Q)  The cross over from RC to DC at Qs 1/2  0 ~ 1

 t <  s Zimm Dynamics  Friction grows with x(t) x(t) ~s’  t~ x 2 (t)/(1/x(t)) ~ x 3 (t) (t < t s )  The volume explored x 3 (t) ~ t  Accumulated contact probability → P(t) ~ t (b/x(t)) 3  (x d (t) ~t ) exploration is marginally compact  longest relaxation time for spacer s (t s ~s 3 )

Excluded Volume Random Walks ~ N Self avoiding Walks ~N 2 

Contact Exponent (Fixmann) Probability to be at contact distance a : P ~1/R d f(a/R) ~ R -(  +d) Universal contact exponents  f(x) ~x  statistical weight of conformations at contact is reduced by s   F  log(s)) Large barrier for internal stickers. self avoiding chain:  L                                  

Zimm+excluded volume  Excluded volume : contact conformations has reduced statistical weight by ~s  Reaction controlled t r ~ P -1 (t) ~ s  (R/b) 3 /Q ~s  d  t 0 /Q t cyc ~ s (d+   Q kinetic rate of a internal cyclization Internal Cyclization time

Confinement

Klimov, Newfield, Thirumalai PNAS (2002)

Takagi et al.

Kinetics of blobs  Osmotic Pressure sets a correlation length for density fluctuation  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=t R ) P>1  BDL P<1  BRC Dynamics under Confinement

 Large Q b (small cavity, small blob size g) : Blob Diffusion Limited :BDL  Small Q b (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 c  g  t R  c  g  t R 

Geometrical Confinement Blobs are space filling: Blob size Critical cavity size reflecting spacers Optimal cavity size Smaller Cavity size (R < R c ) : spacer reflection – The longest Relaxation time is set by BOX size –Confinement always accelerate kinetics

BRC BDL No optimum

Optimum at osmotic regime

Spacer never feel boundary

Screening of Hydrodynamics & Excluded Volume barrier

Formation of higher order vertices Excluded volume barrier W: Extracted from results in Grassberger et al. Macromolecules (2004)

Kinetic rate: k(t)~exp(E b ) Higher Order Vertex formation under confinement Binding rate Optimum at g~s 1/(1+2  )    p,p’ )

Energy Barrier (Daoud-Cotton limit) Star of p-arms Local concentration blob Free energy Energy Barrier

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)