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Unformatted text preview: hich we will take
as the bending energy since we are assuming inextensibility. Nevertheless, in the ﬁxed γ ensemble, the total material
area ﬂuctuates as lipids are expelled and readsorbed. Thus the partition function is
e−H/kB T , Z (γ, Ap ) = (2) C where the sum is performed over all possible conﬁgurations with ﬁxed Ap , but any A. The free energy in this case is
then
Go (γ, Ap ) = −kB T ln Z (γ, Ap ),
and the membrane mechanical tension is (3) 3 τ = lim Ap →∞ Go (γ, Ap )
.
Ap (4) In the case of an isolated, unframed membrane, (A, τ )ensemble, we can ﬁnd the relevant thermodynamic potential
by applying Legendre transforms stepwise, and transform the (γ, Ap )ensemble ﬁrst to the (A, Ap )ensemble, then
from the (A, Ap )ensemble to the (A, τ )ensemble. The ﬁrst transform is eﬀected by
F (A, Ap ) = Go (γ, Ap ) − γA. (5) The value of γ that ﬁxes the speciﬁc value of A is found implicitly by solving ∂F/∂γ = 0, or A = (∂G/∂γ )Ap . The
next Legendre transformation is
Gi (A, τ ) = F (A, Ap ) − τ Ap , (6) ∂F
∂Ap (7) where τ is found from ∂Gi /∂Ap , or τ= .
A What is the meaning of tension τ for an isolated, unframed membrane patch? The mean projected area is can be
found from ∂F/∂Ap = 0. If F (A, Ap ) has a minimum for 0 < Ap /A < 1, then τ = 0. Although the membrane is
shrunk due to thermal ﬂuctuations, it is nonetheless a ﬂat, extended object and has τ = 0 since
∂F
∂Ap = 0. (8) Ap = Ap Only when Ap = 0 can τ = 0.
B Energetics Now consider the bending part of the energy Hb . The bilayer material as a wide aspect ratio, that is, its thickness
is about 45nm and much smaller than the lateral size of the surfaces of interest. Therefore, we can treat the bending
deformations of the membrane with ideas from plate or thin shell theory. Figure 3: Typical constituents of lipid bilayers
A standard model for this is one ﬁrst written by Canham called the Helfrich free energy density: Hb [S] = κ(S)
(C (S) − C0 (S))2 + κG (S)K (S),
2 (9) 4
which is a local energy density due to local bilayer bending at surface position S. In (9), C (S) − C0 (S) is the mean
curvature (measured from the midplane of the lipid) at S, K (S) is the local...
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This note was uploaded on 11/14/2013 for the course MATH 201 taught by Professor Tomchou during the Fall '12 term at UCLA.
 Fall '12
 TomChou
 Math

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