structures induced from polymer- membrane interactions: monte carlo simulations how much can we...
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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
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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Helfrich energy = A [ + (/2) (1/r1+1/r2)2 ]
a
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Energy model: Helfrich model for a perfect cylinder
Helfrich energy = A [ + (/2) (1/r1+1/r2)2 ]
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 )
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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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Polymer confined by a membrane
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Polymer confined by a membrane
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
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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
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
A sphere adsorbed on a vesicle
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
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Thank you!
谢谢
79/47
www.science.uwaterloo.ca/~jeffchen