1. ORBITAL DYNAMICS 1.1. Orbits in an axisymmetric potential
Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we...
-
Upload
candice-crystal-chambers -
Category
Documents
-
view
212 -
download
0
Transcript of Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we...
![Page 1: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/1.jpg)
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
![Page 2: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/2.jpg)
04/21/23
Monte Carlo Hamiltonian “dynamics” method
Multi-var min Simulated MC Annealing
![Page 3: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/3.jpg)
04/21/23
+
?
Polymer physics Membrane physics
Entropy-dominate Energy-dominate
![Page 4: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/4.jpg)
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?
![Page 5: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/5.jpg)
Membrane confinement: any DRASTIC conformational change?
![Page 6: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/6.jpg)
04/21/23
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
![Page 7: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/7.jpg)
04/21/23
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
![Page 8: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/8.jpg)
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
![Page 9: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/9.jpg)
F(poly) ~ weaker N dependence
F(mem) ~ R2
F(poly) ~ N4/3
Swollen state
L
Globular state
![Page 10: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/10.jpg)
04/21/23
Swollen to globular transition by Brochard-Wyart, Tanaka, Borghi and de Gennes, Langmuir (2005)
N
![Page 11: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/11.jpg)
04/21/23
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
![Page 12: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/12.jpg)
04/21/23
Helfrich energy = A [ + (/2) (1/r1+1/r2)2 ]
a
![Page 13: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/13.jpg)
04/21/23
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
![Page 14: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/14.jpg)
Derenyi, Julicher and Prost, PRL 88, 238101 (2002)
Smith, Sackmann, and Seifert, PRL 92, 208101 (2004)
Chen, PRE, in press (2012)
![Page 15: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/15.jpg)
04/21/23
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
![Page 16: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/16.jpg)
04/21/23
Lipid bilayer MC models
All-atom model
Coarse grained
Elastic sheet
![Page 17: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/17.jpg)
04/21/23
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
![Page 18: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/18.jpg)
04/21/23
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)
![Page 19: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/19.jpg)
04/21/23
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
![Page 20: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/20.jpg)
04/21/23
Parameters in the model
N, a (polymer) (membrane) (inverse temperature introduced in Monte Carlo )
Reduced Parameters in the model
N (polymer) a2
(we FIX )
![Page 21: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/21.jpg)
04/21/23 KITPC 2009
Equilibrium: R0
2 = /2
Increase
![Page 22: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/22.jpg)
04/21/23 KITPC 2009 22/47
N=400
N=1200
L = N R -2/3
Power law for the extension in the swollen state
Swollen-to-globular transition point
![Page 23: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/23.jpg)
04/21/23
Phase diagram
Globular
Swollen
![Page 24: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/24.jpg)
04/21/23
Other related problems
![Page 25: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/25.jpg)
04/21/23
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)
![Page 26: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/26.jpg)
Polymer on a hard/soft surface (MC)
N
u
a2
=20
![Page 27: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/27.jpg)
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
![Page 28: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/28.jpg)
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).
![Page 29: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/29.jpg)
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)
![Page 30: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/30.jpg)
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)
![Page 31: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/31.jpg)
Transition 1: adsorption transition
Membrane always bend towards the polymer;
First-order characteristics: stronger as N and membrane becomes soft.
![Page 32: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/32.jpg)
Adsorption fraction at the transition
Central membrane height
![Page 33: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/33.jpg)
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
![Page 34: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/34.jpg)
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
![Page 35: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/35.jpg)
Phase diagram
![Page 36: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/36.jpg)
Simple adsorption model: 3 transitions.
Budding and tube-formation can occur in a simple model
Analytic theory?
![Page 37: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/37.jpg)
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
![Page 38: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/38.jpg)
04/21/23
Polymer confined by a membrane
![Page 39: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/39.jpg)
04/21/23 39/47
Polymer confined by a membrane
Ads
orpt
ion
ener
gy/u
nit
area
a2=1
![Page 40: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/40.jpg)
04/21/23 40/47
Other related problems
![Page 41: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/41.jpg)
04/21/23
Thalman, Billot, Marques, PRE 83, 061922 (2011)
Su and Chen (2012)
Balloon bulge…
![Page 42: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/42.jpg)
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
![Page 43: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/43.jpg)
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
![Page 44: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/44.jpg)
Seifert and Lipowsky, PRA 42, 4769 (1990)
Deserno, PRE 69, 031903 (2004)
![Page 45: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/45.jpg)
Hamiltonian “dynamics”
Shooting may be needed for boundary conditions
Typical numerical approach
![Page 46: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/46.jpg)
04/21/23 46
Adsorption of two colloid particles on a soft membrane
![Page 47: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/47.jpg)
04/21/23 Chen et al, PRL 2009
![Page 48: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/48.jpg)
04/21/23
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):
![Page 49: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/49.jpg)
04/21/23 49/47
Phase diagram
![Page 50: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/50.jpg)
04/21/23
Two spheres
=
+
![Page 51: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/51.jpg)
04/21/23
Internal part
Handling the BC carefully…
![Page 52: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/52.jpg)
04/21/23 52/47
Phase diagram: one sphere Phase diagram: two spheres
![Page 53: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/53.jpg)
04/21/23 53
D adjustable
A
B
![Page 54: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/54.jpg)
04/21/23 54/79
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)
![Page 55: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/55.jpg)
04/21/23
Rigid cylinder adhered to a soft tube
Mkrtchyan and Chen, PRE 81, 041906 (2010)
![Page 56: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/56.jpg)
04/21/23
Rigid cylinder adhered to a soft tube
![Page 57: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/57.jpg)
04/21/23
![Page 58: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/58.jpg)
04/21/23
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
![Page 59: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/59.jpg)
A sphere adsorbed on a vesicle
Parameters
Constraints V and A
Introduce P and
Convert back to V and A
![Page 60: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/60.jpg)
A sphere adsorbed on a vesicle (v=0.6)
Cao, Wei, and Chen, PRE 84, 050901 (2011)
Oblate
stomatocyte
prolate
![Page 61: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/61.jpg)
A sphere adsorbed on a vesicle
Non-axisymmetric solution is needed…
Other v?
![Page 62: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/62.jpg)
Continuous adsorption transition.
First-order transitions between shallow and deep membrane wrapping.
Non-axisymmetric solution: open
Short summary
![Page 63: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/63.jpg)
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
![Page 64: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/64.jpg)
Numerical method
Independent variables r(s), (s), S.
Specified parameters: v, Z, V, A
Target energy
Multi-variable minimization problem (BFGS)
Chen, PRE, inpress
![Page 65: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/65.jpg)
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
![Page 66: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/66.jpg)
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
![Page 67: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/67.jpg)
Experimental…
Shitamichi, Ichikawa, and Kimura, CPL 479, 274 (2009)
![Page 68: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/68.jpg)
Force-extension curve
Shitamichi, Ichikawa, and Kimura, CPL 479, 274 (2009)
Koster et al, PRL, 94, 068101 (2005)
distance
![Page 69: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/69.jpg)
Phase diagram Chen, PRE, in press (2012)
![Page 70: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/70.jpg)
Strong-F regionSmith, Sackmann, and Seifert, PRL 92, 208101 (2004)
Chen, PRE, in press (2012)
![Page 71: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/71.jpg)
Pipette aspiration
Parameters
Analysis can be done in the ensemble,
which is then converted to the ensemble
![Page 72: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/72.jpg)
Theoretical workDas, PRE 82, 021908 (2010)
![Page 73: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/73.jpg)
Direct bending energy minimization
Use a target energy directly
Move the shape by the MC algorithm.
Simulated annealing: T goes to ~ 0
![Page 74: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/74.jpg)
a/r0=3/8
![Page 75: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/75.jpg)
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)]
![Page 76: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/76.jpg)
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
![Page 77: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/77.jpg)
04/21/23
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
![Page 78: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/78.jpg)
04/21/23
![Page 79: Structures induced from polymer- membrane interactions: Monte Carlo simulations How much can we torture the axisymmetric bending-energy model? Jeff.](https://reader036.fdocument.pub/reader036/viewer/2022070412/5697bf881a28abf838c8937a/html5/thumbnails/79.jpg)
04/21/23 79/47
Thank you!
谢谢
79/47
www.science.uwaterloo.ca/~jeffchen