Optimal confinement for internal polymer binding

1. Lee Nam-Kyung Department of Physics Sejong University Optimal confinement for internal polymer binding

2. 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

3. 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

4. Hydrophobic moiety • Hydrogen bonding • Chemical reaction • Molecular recognition Diffusion of Moiety+fluctuating object Irreversible Binding Small length scales Intermediate length scales Binding • 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.

5. biotin streptavidin • Binding of Biotin and Streptavidin ~the strongest non-covalent biological interaction known. • DNA, RNA probes – detecting complementary target system: -A short sequence of labelledDNA 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

6. Cyclization : mRNA loop formation T. Chou (UCLA) (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) .

7. DNA loop formation Telomeric DNA +TRFs Yoshimura et al. Gene to Cells 9 (2004)

8. Protein folding 62 residue intrachain formation rate ~ ( Hagen et al. JMB 2000) What ultimately limits the speed of protein folding? Upper limit : Intrachain contact formation? Trpcage protein folding NCSA, UIUC Caponi et al.

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

10. Internal Cyclization From the view of Partition Functions: R ~ Nn Z~ mNNg-1 s s s Zc~Zass6-2nd Za~N2s1 Zb~Zass4-nd Loop: s-nd (Duplantier,1988)

11. Connectivity and Anomalous Diffusionde Gennes J.Chem .Phys 76 (1992) • Free reactants • Fickian diffusion x2(t) ~ D t • Connected reactants • Anomalous diffusion • Short time exploration (t < tR) • : dense, marginally densex2(t) ~ t d

12. Ideal chain + Rouse dynamics • short times (t< tr~N2t0) tr :Rouse time correlated linear length s’(t) which diffuse together increases with time • Friction grows with s’(t) • t~ x2(t)/(1/s’) ~ x4(t) (t < ts~s2) • x2(t) ~s’ ( x(t) < Rs~sn ) • 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 s

13. Binding Kinetics (RouseDynamics) Q : binding rate at contact • Reaction Controlled typical time for binding (contactprobability) -1 tc~1/Qr ~ s3/2/Q (small Q) • Diffusion Controlled Longest Relaxation time of “s” first passage timets ~ s2t0(Large Q) • The cross over fromRC to DC at Qs1/2t0 ~1

14. Zimm Dynamics • Friction grows with x(t)x(t) ~s’n • 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~s3n) a= 3, t < ts

15. Excluded Volume Random Walks <R2> ~ N Self avoiding Walks <R2>~N2n n=0.588

16. Contact Exponent (Fixmann) • self avoiding chain: nq=SsL-sS q0=(2s1-s2)/n , q1=(2s1-s1 -s3)/n , q2=(2s1-s4 )/n • Probability to be at contact distance a: P ~1/Rd f(a/R) ~ R-(q+d) • Universal contact exponentsq : f(x) ~xq • statistical weight of conformations at contact is reduced bysnq (F~nqlog(s)) • Large barrier for internal stickers. q0 = 0.27 q1= 0.46 q2= 0.71

17. Zimm+excluded volume • Excluded volume : contact conformations has reduced statistical weight by ~snq • Reaction controlled • tr ~ P-1(t) ~ snq(R/b)3/Q • ~sn(d+q)t0/Q kinetic rate of a internal cyclization Internal Cyclization time tcyc~ sn(d+q) /Q

18. Confinement

19. Klimov, Newfield, Thirumalai PNAS (2002)

20. Takagi et al.

21. Kinetics of blobs • Osmotic Pressure sets a correlation length for density fluctuation c h g i • s a an ideal chain of s/g swollen blobs of size g • Blob diffusion time • Spacer relaxation time • Blob Binding rate

22. Dynamics under Confinement • 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 aBDL • P<1 aBRC

23. 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 c h g 6 tRh c h g 6 tR6

24. 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

25. BRCnBDL No optimum

26. Optimum at osmotic regime

27. Spacer never feel boundary

28. Screening of Hydrodynamics & Excluded Volume barrier

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

30. Kinetic rate: k(t)~exp(Eb) • Higher Order Vertex formation under confinement • Binding rate • Optimum at g~s1/(1+2nq)(q2 -> qp,p’ )

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

33. 5+5 4+6

35. 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)