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area Ap . Various free energies, or thermodynamic potentials, can be derived and used depending on the experimental
Consider the patch of membrane shown in Fig. 2(b). It has a fairly ﬁxed material area A since membranes are nearly
inextensible. Therefore, to a very good approximation, A will be proportional to the total number of lipids in the patch 2 A A’>A (a) A Ap
(b) Figure 2: (a) Molecular surface tension. (b) Fluctuations of a ﬁxed material surface.
of membrane. This patch is ﬁxed on a frame of area Ap , the projected area of the membrane patch. Both areas may be
relevant. When one is considering mechanical pulling experiments on a thermally ﬂuctuating membranes, the pulling
force is the conjugate variable to Ap , while if one is concerned with say adsorption of protein on a membrane surface,
the relevant area is A. An inextensible material area A can ﬂuctuate only if the lipid constituents are exchanging
with lipid molecules in the bulk solution. Let us call the thermodynamic variables conjugate to A and Ap , γ and τ ,
respectively. The “mechanical” tension is τ , while the “material” surface tension is γ , much like an air-water surface
tension. Therefore, we can distinguish four thermodynamic ensembles:
I. (A, Ap )-ensemble, isolated and framed (both A and Ap ﬁxed)
II. (A, τ )-ensemble, isolated, unframed membrane
III. (γ, Ap )-ensemble, open, framed membrane
IV. (γ, τ )-ensemble, open, unframed membrane.
Isolated membranes cannot exchange lipid constituents with lipid molecules in the surroundings. Whether or not the
bilayer eﬀectively exchanges lipids depends on the time scale for expulsion and reincorporation of lipid molecules. If
this is slow compared to the relevant experimental time scale, then one can assume A is ﬁxed. Experimentally, one
typically has open, framed systems such as a long experiment in which a lipid is stretched across an oriﬁce. The
ensemble relevant for a biological vesicle in which lipids exchange slowly is the isolated, unframed system.
In the (γ, Ap ) ensemble, the energy is
H (γ, A, Ap ) = Hb + γA, (1) and depends on both the precise values of γ and A. Here Hb is the elastic energy of the membrane w...
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- Fall '12