structures induced from polymer- membrane interactions: monte carlo simulations how much can we...

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

04/21/23

Monte Carlo Hamiltonian “dynamics” method

Multi-var min Simulated MC Annealing

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+

?

Polymer physics Membrane physics

Entropy-dominate Energy-dominate

Polymer confined in a cylinder

de Gennes’ scaling theory…

Membrane confinement: any DRASTIC conformational change?

Polymer adsorption to a flat surface

Second-order transition; relationship with quantum physics formalism, etc.

or

Strong adsorption: any DRASTIC conformational change?

Membrane confinement: any DRASTIC conformational change?

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Swollen-to-globular transition of a self-avoiding polymer confined in a soft tube

lipid bilayer 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

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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

F(poly) ~ N4/3

F(mem)~0

F(poly) ~ N /R2/3

Swollen(elongated)

“Snake eating a swollen sausage (Brochard-Wyart et al, 2005)”

Globular

R2 = /2

F(poly) ~ weaker N dependence

F(mem) ~ R2

F(poly) ~ N4/3

Swollen state

L

Globular state

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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: A

principal curvatures: 1/r1 and 1/r2

Physical parameters

surface tension:

bending energy:

Helfrich energy = A [ + (/2) (1/r1+1/r2)2 ]

In the following, the sontaneous curvature and Gaussian curvature are ignored

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a

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Energy model: Helfrich model for a perfect cylinder


1/r1= 1/R

1/r2 =0

E = 2LR [ + /(2R2)]

Minimization: dE/dR = 0Equilibrium: R0

2 = /2

2R

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

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

Chen, PRE, in press (2012)

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Monte Carlo simulation of a self-avoiding chain

Step k

Step k+1

Basic parameters

Bond length a

Total no. of monomers: N

Excluded-volume diameter: D(=a)

…… entropic effects can be easily modeled

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Lipid bilayer MC models

All-atom model

Coarse grained

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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Write down the Helfrich energy directly

Exploit the built-in symmetry of the problem

The globular state does exist

Monte Carlo simulation of an axi-symmetric tube…

J. Z. Y. Chen, PRL 98, 088302 (2007)

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Helfrich energy for a soft tube

Radius: Ri-1 Ri Ri+1

Geometrical:

area: A(Ri-1,Ri,Ri+1)

inverse curvatures: r1 (Ri-1,Ri,Ri+1) r2 (Ri-1,Ri,Ri+1)

Ei

= A [ + (/2) (1/r1+1/r2)2 ]= function of Ri-1,Ri,Ri+1 with two

physical parameters and

E = Eii=1

M

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Parameters in the model

N, a (polymer) (membrane) (inverse temperature introduced in Monte Carlo )

Reduced Parameters in the model

N (polymer) a2

(we FIX )

04/21/23 KITPC 2009

Equilibrium: R0

2 = /2

Increase

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N=400

N=1200

L = N R -2/3

Power law for the extension in the swollen state

Swollen-to-globular transition point

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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)

Polymer on a hard/soft surface (MC)

N

u

a2

=20

Gaussian chain: the theory is a Schrodinger eigenvalue problem of potential well! Density is *!; 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)…]

u

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

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.

Chen, PRE, 82, 06080 (2010)

Transition 1: adsorption transition

Membrane always bend towards the polymer;

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

Adsorption fraction at the transition

Central membrane height

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.

R||

Beyond adsorption:

u R||

Entropy suffers!

Transition 2: ads-tube 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

Transition 3: budding transition

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.

Theoretical model: Thalman, Billot, Marques, PRE 83, 061922 (2011)

Monte Carlo: Su and Chen, (2012)

GUV

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Ads

orpt

ion

ener

gy/u

nit

area

a2=1

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Other related problems

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Thalman, Billot, Marques, PRE 83, 061922 (2011)

Su and Chen (2012)

Balloon bulge…

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.

Short summary

Structure induced by the interaction between a hard particle and a membrane via a contact attraction energy per unit area: w

R

wR2

R2(or the reduced volume)

E(total) = Emembrane – w Acontact

Seifert and Lipowsky, PRA 42, 4769 (1990)

Deserno, PRE 69, 031903 (2004)

Hamiltonian “dynamics”

Shooting may be needed for boundary conditions

Typical numerical approach

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Adsorption of two colloid particles on a soft membrane

04/21/23 Chen et al, PRL 2009

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Energy model: Helfrich energy

Analytic approach: fluctuations of the membrane are ignored

The energy is minimized with respect to and (s)

Constraint governing the variables r(s) and (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 adjustable

A

B

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Adsorption of colloid spheres on a soft membrane

After the adsorption, is there an membrane induced capillary force between the particles? Attraction? Repulsion?

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

Reynwar and Deserno, Soft Satter (2011)

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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Deserno and GelbartJPC B106, 5543 (2002)

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

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

A sphere adsorbed on a vesicle


Parameters

Constraints V and A

Introduce P and

Convert back to V and A

A sphere adsorbed on a vesicle (v=0.6)

Cao, Wei, and Chen, PRE 84, 050901 (2011)

Oblate

stomatocyte

prolate


Non-axisymmetric solution is needed…

Other v?

Continuous adsorption transition.

First-order transitions between shallow and deep membrane wrapping.

Non-axisymmetric solution: open

Short summary

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

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

Pulling a vesicle at the north/south poles with a force

Parameters:

Done in the [v, Z]-ensemble (a constraint), not [v, F]-ensemble.

from flat surface

very strong pulling limit

Numerical method

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

Specified parameters: v, Z, V, A

Target energy

Multi-variable minimization problem (BFGS)

Chen, PRE, inpress

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)

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

XFirst-order

transition

Experimental…

Shitamichi, Ichikawa, and Kimura, CPL 479, 274 (2009)

Force-extension curve

Shitamichi, Ichikawa, and Kimura, CPL 479, 274 (2009)

Koster et al, PRL, 94, 068101 (2005)

distance

Phase diagram Chen, PRE, in press (2012)

Strong-F regionSmith, 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 workDas, 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)]

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?

Short summary

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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

04/21/23

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Thank you!

谢谢　

79/47

www.science.uwaterloo.ca/~jeffchen

structures induced from polymer- membrane interactions: monte carlo simulations how much can we...

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