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Keeping the same physical picture and ensemble, where the number of lipids and proteins is fixed, we now consider a more crowded vesicle where the area available to each protein to wander around is actually reduced by the presence of the other proteins. The area of the lipids in such a vesicle is

A`=Ao(1 +φ), (2.55)

whereAois the unstressed area of thelipids. The total area of the proteins is then simplyAp =N α,

whereα is the area per protein, such that the total area of the vesicle isAtot=A`+Ap. Then the

available area for any one protein to wander around is Aent =Atot−N α pmax =A`−N α p−max1 −1 =A` 1₋N α A` p−_max1 ₋1 , (2.56)

the constant pmax=π √

3/6_'0.907 is the maximum 2D packing fraction for discs. Effectively, this is subtracting from the total area of the vesicle N hexagonal unit cells of area α/pmax, to give a

more accurate measure of the area in which any one protein can wander. From this equation, we can see that the ideal gas assumption is indeed the dilute limitN α/A` 1. Then the partition

function of this system is

Z = A`−N α p−max1 −1

N

. (2.57)

This partition function is the popular approximation [135] that leads to the equation of state for a 2D hard disc gas

τc=kBT

c 1_−cc

max

, (2.58)

when the derivative is taken with respect to Atot, but we are asserting that proteins are incom-

pressible, and it is onlyA`that can change. The parametercmax =pmax/αis the maximum packing

concentration of discs in 2D. Then the entropic component of the free energy is Gnon-ideal=−N kBTln A`−N α p−max1 −1

Like before, the elastic energy stored in the bilayer is Gstretch=Ao

KA

2 φ

2_{, (2.60)}

so that the total free energy is

G=Gideal+Gstretch =−N kBTln Ao(1 +φ)−N α p−max1 −1

+Ao

KA

2 φ

2_{. (2.61)}

Again we solve∂G/∂φ= 0 to find the entropic areal strain

φ= 1 2 "r (₋1)2₊4 δ +−1 # , (2.62)

where the new constant,

= N α Ao

p−_max1 ₋1, (2.63)

is the ratio of the total close-packed interstitial area to the total area of unstressed lipids, and δ is defined immediately after eqn.2.53. The value= 1 signifies that all available lipid is packed into the interstitial sits of the hexagonal protein crystal. The value of is generally between 0 and 1, however, values greater than 1 are possible, simply corresponding to the case where all protein is in a close-packed disc crystal and there is stressed lipid in the interstitial area. For a situation where the proteins are nearly close-packed, and hence _'1, φ_'δ−1/2. Generally, this equation of state does a good job of accounting for the first few terms of the real Virial expansion of the non-ideal hard disc gas, and contains a singularity in the right place (i.e.when the packing fraction reaches pmax), however, the order of the singularity is not correct [135].

Examiningδwe find it is a constant,KAao/kBT ∼35, connected to the stretch stiffness per lipid

(ao is the unstressed area per lipid), multiplied by the molar ratio of lipid to protein,N`/N. If the

area per protein is 10 nm2 and each protein is occupying twice as much area as the minimum unit cell (α/pmax), then the lipid associated with each protein uses∼12 nm2, or∼20 lipids. Using these

values, the molar ratio of lipid to protein is twenty, and then the δ _' 700, _' 0.09, φ = 0.0016, and τc = 0.09kBT /nm2.

Let us estimate what this means for the crowding tension felt by lipids in the bilayer. If the proteins are fairly densely packed (only 25% more area than the minimum unit cell) then for a 10 nm2 protein, the area of lipid is_∼3.8 nm2 or about 6 lipids per protein. Thenδ _'220,_'0.27, φ_'0.0062, and τc '0.36kBT /nm2. Already this suggests some interesting possibilities for lipids

in the bilayer, if there are organizational principles in membranes that are dependent on the lateral tension felt by lipids, this protein induced crowding tension may be important. Additionally, this means that in a protein dense membrane, there will be a chemical potential benefit for lipids to join the bilayer, distinct from the aforementioned hydrophobic effects. This crowding tension is felt by the lipids, but the overall effect on embedded proteins remains unclear, although we will explore

0 0.5 1 200 400 600 800 1000 0.04 0.08 0.12 0.16 0

Figure 2.10: Plot showing the effects of material constants and protein concentration on the osmotic areal strain in the bilayer. This plots show how the areal strain (φ) in the slightly non-ideal crowding tension model depends on the dimensionless material constantδ, and the dimensionless protein area occupancy . Multiplyφby _∼60kBT /nm2 to get the corresponding osmotic lateral tension.

two physical scenarios below.