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									 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
                                      Multi-var min
                                      Simulated MC
                  Hamiltonian       Annealing
  Monte Carlo
                 “dynamics” method
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    Polymer physics            Membrane physics




                       +
    Entropy-dominate            Energy-dominate




                           ?
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 Polymer confined in a cylinder    Polymer adsorption to a flat surface

 de Gennes’ scaling theory…        Second-order transition; relationship
                                   with quantum physics formalism, etc.
 Membrane confinement: any
DRASTIC conformational
change?
                                                     or


                                    Strong adsorption: any DRASTIC
                                   conformational change?
 Membrane confinement: any DRASTIC
conformational change?
            lipid bilayer
                                Experiments:
                                membrane tubes
                    Soft tube
                                 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)




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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




  9/14/2012
 F(poly) ~ N /R2/3



                              Swollen
      R2 = k/2s              (elongated)

                             “Snake eating a swollen
                             sausage
                             (Brochard-Wyart et al, 2005)”
F(poly) ~ Ns4/3   F(mem)~0
                                  Swollen state
                                 F(poly) ~ Ns4/3



             L


                                  Globular state



F(poly) ~
weaker N         F(mem) ~ s R2
dependence
Swollen to globular transition
by Brochard-Wyart, Tanaka, Borghi and de Gennes, Langmuir (2005)




                                                            N


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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 ]

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   Helfrich energy = DA [s + (k/2) (1/r1+1/r2)2 ]




             bk = 20

             sa2/k



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 Energy model: Helfrich model for a perfect cylinder

 Helfrich energy = DA [s + (k/2) (1/r1+1/r2)2 ]


                                             1/r1= 1/R
                          2R
                                             1/r2 =0



E = 2pLR [s + k/(2R2)]
                               Equilibrium: R02 = k/2s
Minimization: dE/dR = 0



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Derenyi, Julicher and Prost, PRL 88,
238101 (2002)




                                       Smith, Sackmann, and
                                       Seifert, PRL 92,
                                       208101 (2004)

 Chen, PRE, in press (2012)
Monte Carlo simulation of a self-avoiding chain

Step k

                            Basic parameters

                            Bond length a

Step k+1                    Total no. of monomers: N

                            Excluded-volume diameter: D(=a)



               …… entropic effects can be easily modeled

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                         Coarse grained
   Lipid bilayer
   MC models

      All-atom model




                             Elastic sheet




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 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

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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



9/14/2012
       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)


                                          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
Radius:       Ri-1   Ri     Ri+1
  9/14/2012
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)



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                         Equilibrium:
                         R02 = k/2s




                         Increase s



9/14/2012   KITPC 2009
     Power law for the extension in the swollen state

                  L = N R -2/3
     Swollen-to-globular transition point




9/14/2012                      KITPC 2009                22/47
      Phase diagram



                                Globular




                      Swollen




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   Other related problems




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      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)




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Polymer on a hard/soft surface (MC)




                                      N

                                       bu

                                       bsa2

                                       bk=20
                                                                                 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)…]
 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).
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)
  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.
Transition 1: adsorption transition




 Membrane always bend towards the polymer;

 First-order characteristics: stronger as N
and membrane becomes soft.
                  Central
                  membrane
                  height


Adsorption
fraction at the
transition
                                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.
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
Phase diagram
 Simple adsorption model: 3 transitions.

 Budding and tube-formation can occur in a
simple model

 Analytic theory?
Polymer confined by a
membrane

Experiment: Hisette et al, Soft
Matter 4, 828 (2008)
         … “pushed” a GUV on
sparsely grafted DNA molecules.
                                       GUV
Theoretical model: Thalman, Billot,
Marques, PRE 83, 061922 (2011)




 Monte Carlo: Su and Chen, (2012)
Polymer confined by a membrane




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




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   Other related problems




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   Balloon bulge…




  Thalman, Billot,    Su and Chen (2012)
  Marques, PRE 83,
  061922 (2011)
9/14/2012
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.
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 – w Acontact
 Seifert and Lipowsky, PRA    Deserno, PRE 69, 031903 (2004)
42, 4769 (1990)
 Typical numerical approach




 Hamiltonian “dynamics”

 Shooting may be needed for
boundary conditions
Adsorption of two colloid particles on a soft membrane




9/14/2012                                                46
9/14/2012    Chen et al, PRL 2009
Energy model: Helfrich energy




   Analytic approach: fluctuations of the membrane are ignored

   The energy is minimized with respect to a and y(s)

   Constraint governing the variables r(s) and y(s):




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  Phase diagram




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Two spheres




              =

                  +


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Internal part




                 Handling the BC carefully…




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  Phase diagram: one sphere   Phase diagram: two spheres




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            D   s adjustable




B




A




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  Adsorption of colloid spheres on a soft membrane



                                        3D: M. Muller, M. Deserno and
                                        J. Guven, PRE (2007);

                                        Reynwar and Deserno, Soft
                                        Satter (2011)




 After the adsorption, is there an
membrane induced capillary force
between the particles? Attraction?
Repulsion?
                                         Reynwar et al. Nature 447,
                                         461 (2007)


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      Rigid cylinder adhered to a soft tube




     Mkrtchyan and Chen, PRE 81, 041906 (2010)
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Rigid cylinder adhered to a soft tube




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9/14/2012
 A sphere adsorbed on a vesicle



                                  Deserno and Gelbart
                                  JPC B106, 5543 (2002)

                                  J. Bernoit and Saxena,
                                  PRE 76, 041912 (2007)

                                  W. T. Gozdz, Langmuir
                                  23, 5665 (2007)




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A sphere adsorbed on a vesicle


                                  Parameters




  Constraints   V and A

  Introduce     P and s

  Convert back to V and A
A sphere adsorbed on a vesicle (v=0.6)



                    prolate




                  stomatocyte




                                Oblate




               Cao, Wei, and Chen, PRE 84, 050901 (2011)
A sphere adsorbed on a vesicle

 Other v?




 Non-axisymmetric solution is needed…
Short summary

 Continuous adsorption transition.

 First-order transitions between shallow
and deep membrane wrapping.

 Non-axisymmetric solution: open
Pulling a vesicle at the north/south poles with a force

                                                               Smith, Sackmann,
                     from flat                                and Seifert, PRL 92,
                                                               208101 (2004)
                    surface
                                                                very strong
                                                               pulling limit




Derenyi, Julicher and Prost, PRL 88,
238101 (2002)




                                        Parameters:



                                        Done in the [v, Z]-ensemble (a
                                       constraint), not [v, F]-ensemble.
Numerical method          Chen, PRE, inpress




  Independent variables r(s), y(s), S.

  Specified parameters: v, Z, V, A

  Target energy




  Multi-variable minimization problem (BFGS)
 Moving from [v, Z] ensemble to [v, F] ensemble




 Stabilization of a structure is determined from
the Gibbs energy

         G(v, F) = Eb – F Z
 v=0.95

                    Bozic, Svetina, Zeks,
                    PRE 55, 5834 (1997);
                    Fygenson, Marko,
                    Libchaber, PRL 79, 4497
                    (1997)


                            X

           Heinrich et al, Biophys J
           76, 2056 (1999)
           Chen, PRE, in press
           (2012).
Experimental…

                Shitamichi,
                Ichikawa, and
                Kimura, CPL
                479, 274 (2009)
Force-extension curve




Koster et al, PRL, 94,      Shitamichi, Ichikawa, and
068101 (2005)               Kimura, CPL 479, 274
                            (2009)




                         distance
Phase diagram   Chen, PRE, in press (2012)
Strong-F region
                                               Smith, Sackmann,
                                               and Seifert, PRL 92,
                                               208101 (2004)




                  Chen, PRE, in press (2012)
Pipette aspiration

                                    Parameters




  Analysis   can be done in the                  ensemble,

 which is then converted to the               ensemble
 Theoretical work
                     Das, PRE 82, 021908 (2010)
 Direct bending energy minimization



 Use a target energy directly




 Move the shape by the MC algorithm.

 Simulated annealing: T goes to ~ 0
a/r0=3/8
a/r0=3/8

            Comparison with
           the van der Waals
           equation of state for
           gas-liquid transition.

           First-order
           transition: binodal

            Stability limits:
           spinodal

            The stability limits
           were called “critical
           aspiration pressure”
           and releasing
           aspiration pressure”
           in experiments [e.g.,
           Tian, PRL 98, 208102
           (2007)]
Short summary
 Free-weak aspiration:   continuous
transition.

 Strong first-order transitions between
weak and strong aspired states.

 Non-axisymmetric solution: open

 Combined aspiration/point-force
stretching: open

 Add the effects of inner/outer surface
difference (Bozic 1997 , Heinrich 1999):
open

 Going for the higher-order bending
energy terms?
            ACKNOWLEDGMENT

             Yu-cheng Su (Waterloo, Canada)
             Sergey Mkrtchyan (Waterloo,Canada)
             Christopher Ing (Waterloo, Canada)

             Yuan Liu (USTC,中科大)
             H.J. Liang (USTC,中科大)

             Siqin Cao (Fudan,复旦)
             Guanghong Wei (Fudan, 复旦)

             NSERC
             SHARCNET


9/14/2012
9/14/2012
Thank you!        www.science.uwaterloo.ca/~jeffchen


            谢谢

9/14/2012                                  79/47

								
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