-
3
-
-
0041103603
-
-
edited by, E. Clementi, S. Chin, Plenum, New York
-
(1986)
Structure and Dynamics of Nucleic Acids, Proteins and Membranes
, pp. 251
-
-
Sackmann, E.1
Duwe, H.P.2
Zeman, K.3
Zilker, A.4
-
8
-
-
84931552507
-
-
H.-G. Döbereiner, W. Rawicz, M. Wortis, and E. Evans (unpublished).
-
-
-
-
11
-
-
84931552516
-
-
Canham introduced the bending elasticity as a contribution to the vesicle energy. His form, ( κ /2 ) ( C12+ C22), for the local elastic energy differs from Eq. (1) by (a) assuming zero spontaneous curvature and (b) lacking the cross term κ C1C2, which is proportional to the local Gaussian curvature. The integral of the Gaussian curvature over a closed surface is a topological invariant (Gauss-Bonnet theorem), so this latter term cannot influence either the Euler shapes or their relative energies at given (spherical) topology.
-
-
-
-
13
-
-
84931552515
-
-
A Gaussian-curvature term is omitted here and henceforth, since we shall deal throughout with vesicles of spherical topology.
-
-
-
-
23
-
-
84931552514
-
-
Enforcing the constraint (2) introduces a Lagrange multiplier which plays the role of a spontaneous curvature, thus any physical asymmetry of the bilayer is irrelevant in this model.
-
-
-
-
28
-
-
84931552513
-
-
Even more exotic shapes have been observed experimentally, such as a spherical vesicle with tetherlike protrusions or spherical vesicles with small blebs distributed nonaxisymmetrically. These shapes are not yet understood theoretically in the context of bending energy models. Indeed, it is possible that further contributions to the energy, such as higher-order-curvature terms and van der Waals attraction of the membrane, become relevant for these shapes.
-
-
-
-
29
-
-
0042006785
-
-
edited by, D. Beysens, N. Boccara, G. Forgacs, Nova, New York
-
(1993)
Proceedings of the Workshop on Dynamical Phenomena at Interfaces, Surfaces, and Membranes
, pp. 221-236
-
-
Wortis, M.1
Seifert, U.2
Berndl, K.3
Fourcade, B.4
Miao, L.5
Rao, M.6
Zia, R.K.P.7
-
31
-
-
84931552512
-
-
Note that these authors use a definition of alpha which is pi times our alpha.
-
-
-
-
33
-
-
84931552493
-
-
U. Seifert, L. Miao, H.-G. Döbereiner, and M. Wortis, in Structure and Conformation of Amphiphilic Membranes, edited by R. Lipowsky, D. Richter, and K. Kremer (Springer, Berlin, 1992), pp. 93–96.
-
-
-
-
38
-
-
84931552494
-
-
Berndl and co-workers [6,23] have noted previously in the context of the Δ A model that a very small asymmetry (one part in 103) in the thermal expansion coefficients of the two monolayers can have a significant effect on the temperature trajectories and, hence, on the sequence of shape transformations. This remark applies equally to the ADE model. There are as yet no measurements demonstrating such asymmetry.
-
-
-
-
42
-
-
84931552495
-
-
Strictly speaking, a limit shape for v < sqrt 2 /2 could also be a vesiculated shape involving several segments connected by microscopic necks. In this paper, we use the term vesiculation only for the limit shape of two spheres of different radius.
-
-
-
-
44
-
-
84931552496
-
-
In recent work [53] Käs et al. suggest the inclusion of a third-order term of the form γ ( Δ a - Δ a0)3. By choosing a coefficient γ apeq 200, they can generate a discontinuous transition from a weak pear to a strong pear. It is hard to regard this model as a credible explanation of the observed behavior, since the authors do not provide any rationale for the large value of gamma required. In fact, the estimates sketched in Appendix A would yield a value γ apeq D/R apeq 10-3.
-
-
-
-
45
-
-
84931552492
-
-
Once the neck closes, there are strong hysteresis effects, probably connected with short-range van der Waals forces in the region of the (microscopic) neck [52,53]. Thus reproducible cycling through the transition has not yet been achieved.
-
-
-
-
46
-
-
84931552502
-
-
Hans-Günther Döbereiner (unpublished).
-
-
-
-
47
-
-
84931552501
-
-
J. Käs and H.-G. Döbereiner (private communication).
-
-
-
-
48
-
-
84931552498
-
-
Interestingly, although these fluctuations go in both ``up'' and ``down'' directions, preliminary analysis of the fluctuations indicates that they usually do not quite average to zero, so some source of a small up-down asymmetry apparently remains [39].
-
-
-
-
50
-
-
84931552497
-
-
The dominant interactions include the (attractive) hydrophobic interaction, the (repulsive) hydration force between head groups, the (repulsive) chain-chain steric interaction, and (less importantly) the van der Waals interaction.
-
-
-
-
52
-
-
84931552500
-
-
This picture neglects the energy associated with the forces (e.g., van der Waals attraction) which hold the monolayers at separation D. If included explicitly in Eq. (A13), such forces induce local correlations between the separate monolayer densities. Correlated lateral density modes are presumably present in the dynamical spectrum of the bilayer; however, at the mean-field level it is easy to show that there is no effect on the shape mechanics, provided that the range of the forces is small on the scale of the local curvatures C1,2.
-
-
-
-
54
-
-
84931552499
-
-
This is exactly the kissing condition we found in Ref. [22], although the form is slightly different from that in Ref. [22], which is given in its dimensionless form with the normalization chosen so that C bar0=1.
-
-
-
-
55
-
-
84931552475
-
-
In Appendix E in Ref. [23], it was proved that, for a smooth and deformed sphere, the geometric area difference m has the leading terms given here.
-
-
-
-
60
-
-
84931552476
-
-
This fact can be traced back to the perturbative development of the l=2 symmetric ellipse, which can be expressed in powers of the fractional excess area Δ1/2 (see Ref. [22]). Note that DELTA is linear in (1-v) to lowest order as v -> 1.
-
-
-
|
