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0004097995
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33645090257
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note
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The product of the two principal curvatures is called the "Gaussian curvature." Its integral over a surface 5 can be rewritten as a line integral of the geodesic curvature over the boundary ∂S of S (Gauss-Bonnet theorem [30]). It is thus invariant under deformations of S which change neither its boundary nor its topology, Under these conditions the Gaussian part in Eq. (1) only results in an additive constant in the energy which can be ignored in what follows.
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33
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0031982879
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BT - see, for instance, R. Simson, E. Wallraff, J. Faix, J. Niewöhner, G. Gerisch, and E. Sackmann, Biophys. J. 74, 514 (1998). However, on length scales of only a few tens of nanometers, which do not exceed the mesh size of the cortex, the lipid bilayer can deform independently of the actin network and will be characterized by its own, much softer modulus. Note that, for instance, the spectrin network of human erythrocytes has a mesh size of about 75 nm [see D. Boal, Mechanics of the Cell (Cambridge University Press, Cambridge, 2002)].
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Simson, R.1
Wallraff, E.2
Faix, J.3
Niewöhner, J.4
Gerisch, G.5
Sackmann, E.6
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34
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0004150063
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Cambridge University Press, Cambridge
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BT - see, for instance, R. Simson, E. Wallraff, J. Faix, J. Niewöhner, G. Gerisch, and E. Sackmann, Biophys. J. 74, 514 (1998). However, on length scales of only a few tens of nanometers, which do not exceed the mesh size of the cortex, the lipid bilayer can deform independently of the actin network and will be characterized by its own, much softer modulus. Note that, for instance, the spectrin network of human erythrocytes has a mesh size of about 75 nm [see D. Boal, Mechanics of the Cell (Cambridge University Press, Cambridge, 2002)].
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Mechanics of the Cell
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Boal, D.1
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35
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33645061588
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note
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[n Ref. [14] it is suggested that for partially wrapped spheres the deformation of the membrane along the contact line will contribute an additional bending term to the contact line energy which acts as an energy barrier for the encapsulation process and which should scale ∼√z(2-z). However, if the membrane is tensionless, such a term is rigorously absent (due to its catenoid shape). Under nonvanishing tension there is indeed a barrier, but it does not have the proposed form, since the bending energy is generally not localized at the rim.
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39
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33645061080
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note
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Observe that the stronger requirement h(s) → const is not used. In fact, for zero tension the solution of the shape equations is the catenoid, which diverges logarithmically in the limit s →∞. However, this is a special case, and for any nonzero tension the profile height h(s) remains bounded and converges toward a constant.
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41
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33645082302
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note
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0 >1/a are no longer inaccessible and may become physically relevant.
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42
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33645067457
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note
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BT.
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43
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33645078970
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note
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Recall that the adhesion energy contributes a linear term - w̃z to the total energy of the complex, see Eq. (4). Increasing w̃ hence implies "pulling the energy curve Ẽ(z) down at the right-hand side," thus the enveloped state becomes increasingly favorable.
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44
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0000219094
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B. Fourcade, L. Miao, M. Rao, M. Wortis, and R.K.P. Zia, Phys. Rev. E 49, 5276 (1994).
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Fourcade, B.1
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Zia, R.K.P.5
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45
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33645070028
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note
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0 converges toward - 1/a, i.e., the negative of the azimuthal contact curvature. Of course, once the deformation of the membrane occurs on length scales comparable to its thickness (typically a few nanometers), neither linear elasticity theory underlying Eq. (1) nor the description of the membrane as an idealized surface are appropriate, and the bilayer structure needs to be modeled in more detail.
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46
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33645055742
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note
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The negative slope of Ẽ(z) at z = 2 [see inset of Fig. 3] implies that the regions of the bilayer at the upper neck are pushed against each other. This may potentially facilitate a subsequent fusion event.
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48
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0003603424
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Structure and Dynamics of Membranes, edited by R. Lipowsky and E. Sackmann, Elsevier, New York/ North-Holland, Amsterdam
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U. Seifert and R. Lipowsky, in Structure and Dynamics of Membranes, edited by R. Lipowsky and E. Sackmann, Handbook of Biological Physics, Vol. 1A (Elsevier, New York/ North-Holland, Amsterdam, 1995).
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Seifert, U.1
Lipowsky, R.2
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49
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33645076098
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note
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Equation (17) can also be derived by a direct small gradient approximation of the shape equations (9). This approach is less elegant, but it illustrates quite vividly how many "unpleasant" terms drop out in lowest order - and only in lowest order. Hence, extending the small gradient expansion to the next order appears about as hard as treating the full nonlinear problem.
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51
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21444436092
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R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, and D.E. Knuth, Adv. Comput. Math. 5, 329 (1996).
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Corless, R.M.1
Gonnet, G.H.2
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Knuth, D.E.5
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52
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33645047500
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note
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-1(x)≈ln|x|-ln|lnx||.
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54
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33645078215
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note
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r(0) would be infinite.
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55
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33645053596
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note
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2σ̃).
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57
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0000693828
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T. Kawakatsu, D. Andelman, K. Kawasaki, and T. Taniguchi, J. Phys. II 3, 971 (1993); T. Taniguchi, K. Kawasaki, D. Andelman, and T. Kawakatsu, ibid. 4, 1333 (1994).
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Kawakatsu, T.1
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58
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0000174468
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T. Kawakatsu, D. Andelman, K. Kawasaki, and T. Taniguchi, J. Phys. II 3, 971 (1993); T. Taniguchi, K. Kawasaki, D. Andelman, and T. Kawakatsu, ibid. 4, 1333 (1994).
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Taniguchi, T.1
Kawasaki, K.2
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63
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0003437218
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Addison-Wesley, Reading, MA, Chap. 8-6
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The only subtle point is that the arclength s is varied as well, which, for instance, implies that the Hamiltonian H enters into Eq. (A2), and not the Lagrangian L. See, for instance, the discussion in H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, MA, 1980), Chap. 8-6.
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Classical Mechanics, 2nd Ed.
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Goldstein, H.1
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