Large-scale simulations of fluctuating biological membranes

G.

 

Brannigan

,

L. C.-L.

 

Lin

, and

F. L. H.

 

Brown

,  , ().

O.

 

Edholm

, in , edited by

S. E.

 

Feller

(, , ).

J. C.

 

Shelley

,

M. Y.

 

Shelley

,

R. C.

 

Reeder

,

S.

 

Bandyopadhyay

, and

M. L.

 

Klein

,  , ().

S. J.

 

Marrink

,

A. H.

 

de Vries

, and

A. E.

 

Mark

,  , ().

G. S.

 

Ayton

and

G. A.

 

Voth

,  , ().

G. S.

 

Ayton

,

E.

 

Lyman

, and

G. A.

 

Voth

,  , ().

I. R.

 

Cooke

,

K.

 

Kremer

, and

M.

 

Deserno

,  , ().

G.

 

Brannigan

,

P. F.

 

Philips

, and

F. L. H.

 

Brown

,  , ().

M.

 

Venturoli

,

M. M.

 

Sperotto

,

M.

 

Kranenburg

, and

B.

 

Smit

,  , ().

J. -M.

 

Drouffe

,

A. C.

 

Maggs

, and

S.

 

Leibler

,  , ().

G.

 

Gompper

and

D. M.

 

Kroll

,  , ().

G.

 

Ayton

and

G. A.

 

Voth

,  , ().

H.

 

Noguchi

and

G.

 

Gompper

,  , ().

G. S.

 

Ayton

,

S.

 

Izvekov

,

W. G.

 

Noid

, and

G. A.

 

Voth

,  , ().

T.

 

Kohyama

,  , ().

W.

 

Helfrich

,  , ().

S. A.

 

Safran

, (, , ).

G. S.

 

Ayton

,

P. D.

 

Blood

, and

G. A.

 

Voth

,  , ().

G. S.

 

Ayton

,

E.

 

Lyman

,

V.

 

Krishna

,

R. D.

 

Swenson

,

C.

 

Mim

,

V. M.

 

Unger

, and

G. A.

 

Voth

,  , ().

G.

 

Gompper

and

D. M.

 

Kroll

,  , ().

B.

 

Alberts

,

A.

 

Johnson

,

J.

 

Lewis

,

M.

 

Raff

,

K.

 

Roberts

, and

P.

 

Walter

, , 4th ed. (, , ).

D.

 

Chandler

,  , ().

The elastic behavior of the model membrane is dominated by entropic effects, yielding a nearly temperature-independent reduced bending rigidity $κ/kBT$⁠. Translation and rotation of finite membrane patches, on the other hand, proceed by displacements of individual particles, which are suppressed at low temperatures. To accelerate the self-assembly process the trajectory shown in Fig. 1 was generated at a higher temperature $ϵ=25kBT$⁠.

W.

 

Rawicz

,

K. C.

 

Olbrich

,

T.

 

McIntosh

,

D.

 

Needham

, and

E.

 

Evans

,  , ().

J. -B.

 

Fournier

and

C.

 

Barbetta

,  , ().

To compute the fluctuation spectrum of a membrane that spans the simulation box in the $xy$-plane we define the height function $h(x,y)=∑i=1NziΘ(xi−x)Θ(yi−y)/∑i=1NΘ(xi−x)Θ(yi−y)$ where $Θ(x)=1$ if $|x|≤Δ/2$ and zero otherwise. Evaluating this function for points $(x,y)$ that form a square grid of resolution $Δ$ and taking the two-dimensional discrete Fourier transform, we obtain the Fourier amplitude $ĥq$⁠. This procedure generates a systematic bias at large wavevectors (Ref. 33). Taking this bias into account does not change the value of $κ$ for our model.

A. P.

 

Liu

,

D. L.

 

Richmond

,

L.

 

Maibaum

,

S.

 

Pronk

,

P. L.

 

Geissler

, and

D. A.

 

Fletcher

,  , ().

X.

 

Chen

,

A.

 

Kis

,

A.

 

Zettl

, and

C. R.

 

Bertozzi

,  , ().

I.

 

Derényi

,

F.

 

Jülicher

, and

J.

 

Prost

,  , ().

T. R.

 

Powers

,

G.

 

Huber

, and

R. E.

 

Goldstein

,  , ().

E.

 

Atilgan

,

D.

 

Wirtz

, and

S. X.

 

Sun

,  , ().

G.

 

Koster

,

A.

 

Cacciuto

,

I.

 

Derenyi

,

D.

 

Frenkel

, and

M.

 

Dogterom

,  , ().

I. R.

 

Cooke

and

M.

 

Deserno

,  , ().

K. A.

 

Brakke

,  , ().

K. A.

 

Brakke

, Surface Evolver 2.30c. http://www.susqu.edu/brakke/evolver.

Continuum calculations were performed by finding the ground state energy of a membrane under the Helfrich energy functional, given by the surface integral $∫SdS(σ+2κH2)$⁠, where $H$ is the local mean curvature. Calculation were performed using the Surface Evolver software package (Refs. 34 and 35), which employs discretized approximations for both the membrane shape and the operators in the energy functional. To mimic the geometry of our particle simulation as closely as possible, we constrained the membrane height to zero on a square frame of side length $60d$⁠, and set the Cartesian coordinates of a single vertex to $(0,0,l)$⁠. The force $f(l)$ was computed as the derivative of the ground state energy with respect to $l$⁠.