Structures induced from polymer-membrane interactions: Monte Carlo simulations How much can we torture the axisymme

1. Structures induced from polymer-membrane interactions: Monte Carlo simulations • How much can we torture the axisymmetric bending-energy model? Jeff Z. Y. Chen (陈征宇) Department of Physics & Astronomy University of Waterloo, CANADA

2. Multi-var min • Simulated MC Annealing • Hamiltonian “dynamics” method • Monte Carlo

3. Membrane physics • Polymer physics + Entropy-dominate Energy-dominate ?

4. Polymer adsorption to a flat surface • Second-order transition; relationship with quantum physics formalism, etc. • or • Strong adsorption: any DRASTIC conformational change? • Polymer confined in a cylinder • de Gennes’ scaling theory… • Membrane confinement: any DRASTIC conformational change?

5. Membrane confinement: any DRASTIC conformational change?

6. Swollen-to-globular transition of a self-avoiding polymer confined in a soft tube Experiments: membrane tubes • Borghi, Rossier and Brochard-Wyart, Europhys Lett 64, 837 (2003) • Tokarz, etc, PNAS 102, 9127 (2005) • Borghi, Kremer, Askvic and Brochard-Wyart, Europhys Lett 75, 666 (2006) Soft tube lipid bilayer

7. DNA in a soft tube Michal Tokarz, Bjorn Akerman, Jessica Olofsson, Jean-Francois Joanny, Paul Dommersnes, and Owe Orwar, PNAS 102, 9127 (2005). • case 1: fluorescence light intensity = constant • case 2: fluorescence light intensity ~ DNA length

8. L F(poly) ~ N /R2/3 • Swollen (elongated) “Snake eating a swollen sausage (Brochard-Wyart et al, 2005)” • Globular R2 = k/2s F(poly) ~ Ns4/3 F(mem)~0 KITPC 2009 Swollen to globular transition • Swollen (elongated) “Snake eating a swollen sausage (de Gennes, 2005)” • Globular F(poly) ~ weaker N dependence F(mem) ~ s R2

9. L • Swollen state • Globular state F(poly) ~ Ns4/3 F(poly) ~ weaker N dependence F(mem) ~ s R2 KITPC 2009

10. Swollen to globular transition by Brochard-Wyart, Tanaka, Borghi and de Gennes, Langmuir (2005) N

11. Energy model: Helfrich model for a fluid membrane • Geometrical: area: DA principal curvatures: 1/r1 and 1/r2 • Physical parameters surface tension: s bending energy: k • In the following, the sontaneous curvature and Gaussian curvature are ignored Helfrich energy = DA [s + (k/2) (1/r1+1/r2)2 ]

12. Helfrich energy = DA [s + (k/2) (1/r1+1/r2)2 ] bk =20 • sa2/k

13. 2R Energy model: Helfrich model for a perfect cylinder Helfrich energy = DA [s + (k/2) (1/r1+1/r2)2 ] • 1/r1= 1/R • 1/r2 =0 E = 2pLR [s + k/(2R2)] Equilibrium: R02 = k/2s Minimization: dE/dR = 0

14. Derenyi, Julicher and Prost, PRL 88, 238101 (2002) Smith, Sackmann, and Seifert, PRL 92, 208101 (2004) Chen, PRE, in press (2012)

15. Step k Monte Carlo simulation of a self-avoiding chain Basic parameters Bond length a Total no. of monomers: N Excluded-volume diameter: D(=a) Step k+1 …… entropic effects can be easily modeled

16. Coarse grained • All-atom model • Elastic sheet Lipid bilayer MC models

17. Monte Carlo simulation by Avramova and Milchev Avramova and Milchev,J. Chem. Phys. 124, 024909 (2006) • Tube = 3D mesh system • “Expensive” in modeling the tube (a M*M problem; M=number of nodes) • Relatively small N (<400) • No observations of the structural transition

18. Monte Carlo simulation of an axi-symmetric tube… J. Z. Y. Chen, PRL 98, 088302 (2007) • Write down the Helfrich energy directly • Exploit the built-in symmetry of the problem • The globular state does exist

19. Ei = DA [s + (k/2) (1/r1+1/r2)2 ] = function of Ri-1,Ri,Ri+1 with two physical parameters k and s M • E = S Ei i=1 Helfrich energy for a soft tube • Geometrical: area: DA(Ri-1,Ri,Ri+1) inverse curvatures: r1 (Ri-1,Ri,Ri+1) r2 (Ri-1,Ri,Ri+1) Radius: Ri-1 Ri Ri+1

20. Parameters in the model • N, a (polymer) • k, s (membrane) • b (inverse temperature introduced in Monte Carlo ) Reduced Parameters in the model • N (polymer) • bsa2 • bk (we FIX bk =10)

21. Equilibrium: R02 = k/2s Increases KITPC 2009

22. Power law for the extension in the swollen state • Swollen-to-globular transition point L = N R-2/3 N=1200 N=400 KITPC 2009

23. Phase diagram Globular Swollen

24. Other related problems

25. Electrophoretic transport --- charge effects? • velocity ~ V curve? • More direct observation of theoretical predictions? • More-than-one polymers confined? S. Jun & B. Mulder, PNAS 103, 12388(06)

26. Polymer on a hard/soft surface (MC) • N • bu • bsa2 • bk=20

27. u • Gaussian chain: the theory is a Schrodinger eigenvalue problem of potential well! Density is yy*!; Free energy is the eigenvalue [De Gennes, 1993] • Self-avioding chain: Second-order phase transition, 1/N finite-size effects in MC… [Lai, PRE 49, 5420 (1994); Metzger, Macromol. Theo. Simul. 11, 985 (2002)…]

28. Tethered polymer • Lipowsky, et al. Physica A 249, 536 (1998) • Auth and Gompper, PRE 72, 031904 (2005) • Breidenich, Netz, and Lipowsky, EPL 49, 431 (2000) • M. Laradji, JCP 121, 1591 (2004).

29. Polymer adsorption • Rodgornik, EPL 21, 245 (1993) • Lipowsky, et al. Physica A 249, 536 (1998) • Kim and Sung, PRE 63, 041910 (2001) • Chen, PRE, 82, 06080 (2010)

30. MC simulations Chen, PRE, 82, 06080 (2010) • Modeling the shape of potential well. • Keep the polymer’s center of mass on the axis. • Properly account for the weight of different concentric “rings” and move them correctly.

31. Transition 1: adsorption transition • Membrane always bend towards the polymer; • First-order characteristics: stronger as N and membrane becomes soft.

32. Central membrane height Adsorption fraction at the transition

33. Beyond adsorption: bu R|| Entropy suffers! R|| Transition 2: ads-tube transition • First-order characteristics: transition takes place as the adhesion energy increases; • The transition is entropy-driven; • A tube is extracted from the surface. • Larger u, longer the tube.

34. Transition 3: budding transition • First-order characteristics: transition takes place as the adhesion energy further increases; • The transition is a result of a balance between entropy and energy; • The bud is almost spherical

35. Phase diagram

36. Simple adsorption model: 3 transitions. • Budding and tube-formation can occur in a simple model • Analytic theory?

37. Polymer confined by a membrane • Experiment: Hisette et al, Soft Matter 4, 828 (2008) … “pushed” a GUV on sparsely grafted DNA molecules. • Theoretical model: Thalman, Billot, Marques, PRE 83, 061922 (2011) • Monte Carlo: Su and Chen, (2012) GUV

38. Polymer confined by a membrane

39. Polymer confined by a membrane bk=16, bs a2=1 Adsorption energy/unit area

40. Other related problems

41. Balloon bulge… • Thalman, Billot, Marques, PRE 83, 061922 (2011) • Su and Chen (2012)

42. Short summary • Trade-off between the membrane’s energy and the polymer’s entropy (excluded volume). • Induced structural transitions (usually first order). • The need of more serious scaling theory/self-consistent field theory.

43. Structure induced by the interaction between a hard particle and a membrane via a contact attraction energy per unit area: w • R • wR2/k • sR2/k (or the reduced volume) E(total) = Emembrane – wAcontact

44. Seifert and Lipowsky, PRA 42, 4769 (1990) • Deserno, PRE 69, 031903 (2004)

45. Typical numerical approach • Hamiltonian “dynamics” • Shooting may be needed for boundary conditions

46. Adsorption of two colloid particles on a soft membrane

47. Chen et al, PRL 2009

48. Energy model: Helfrich energy • Analytic approach: fluctuations of the membrane are ignored • The energy is minimized with respect to a andy(s) • Constraint governing the variables r(s) and y(s):

49. Phase diagram

50. + Two spheres =