2.6 Further Discussion
2.6.4 Implications of Bilayer Volume Conservation
When examining the mechanics of bilayer deformations around a channel, we considered the bilayer to be a volume preserving material. In the first part of this section, we quantitatively show that this is the case, using values of the isothermal compressibility of bilayers [60,61], the area stretch modulus, and bilayer thickness [26]. Imagine we compress the bilayer thickness by an amount u and ask the question — given the elastic parameters, does the bilayer preserve volume and hence change area, or does the area remain relatively constant at the cost of changing volume? If volume is preserved, the area change of the membrane will be proportional to the compressionu. Indeed, this was the model espoused throughout this work, but we can be more explicit as to why it is a good approximation.
Consider a patch of membrane with undeformed areaAoand undeformed volumeVo. We write the change in patch area as
∆A=Ao−A, (2.37)
and the change in patch volume as
∆V =A(l+u)−Aol. (2.38)
Treating the bilayer as a linear elastic material, the energy in both area changing and volume changing deformation modes is a quadratic function of the strain in both modes, written as
Gvol= B 2
∆V Vo
2
Vo+KA 2
∆A Ao
2
Ao, (2.39)
whereKA'60kBT /nm2 is the area stretch modulus [26] and B'2 GPa is the bulk modulus of a typical bilayer known from previous measurements [60,61]. Equation 2.39 can be rearranged into a dimensionless form, written as
Gvol= KAAo 2
"
f ∆V
Vo 2
+ ∆A
Ao 2#
, (2.40)
where the dimensionless material constantf = KBl
A is a relative measure of the area compressibility and volume compressibility – if this value is greater than one, area change is the preferred mode of deformation, if the values is less than one, volume change is the preferred mode of deformation.
With the aforementioned values of the elastic constants f '40. Using eqn. 2.37 and 2.38, we can rewrite the patch deformation free energy as a function of patch area
Gvol = KAAo 2
"
f A
Ao(1 +µ)−1 2
+ A
Ao −1 2#
(2.41) where the dimensionless constant µ=u/l is simply a normalized value of the compression. This free energy is then minimized with respect to the only variable, the patch areaA, to find
A Ao
min
= f(1 +µ) + 1
f(1 +µ)2+ 1 (2.42)
In this same notation, the areal strain is written as
∆A Ao =
A Ao
min
−1 (2.43)
and the volumetric strain is written as
∆V Vo =
A Ao
min
(1 +µ)−1. (2.44)
-0.2 -0.1 0 0.1 0.2 0.3 0.4
-0.2 0.2 0.3
-0.3 -0.1 0 0.1
Figure 2.8: Comparison of areal and volumetric strain. Upon compressing a bilayer patch by an amount µ, the bilayer both changes volume and area, however the bilayer thickness, stretch modulus, and bulk modulus conspire to conserve volume and allow the membrane to change area, as shown by the relatively small volumetric strain (green) and the relatively large areal strain (red).
Finally, we are in a position to quantitatively comment on the way area and volume of a bilayer patch change with compression µ. It is likely that MscL represents a protein whose structural change is accompanied by a severe change in membrane thickness. As we showed earlier, the anticipated compression in the open state is µ'0.3, and hence we plot the areal and volumetric strain over a range of −0.3 < µ < 0.3, as shown in Fig. 2.8, demonstrating that under bilayer compression, volume is conserved and area is not.
The fact that bilayer is generally volume conserving has implications for our understanding of mechanosensation in general. Up until now, we modeled single channel gating statistics by the equation
Popen= 1
1 +e(∆Gmem+∆Gprot−τ∆A) (2.45)
where here ∆A refers to the change in channel area upon gating, ∆Gmem is the component of the free energy change from membrane deformation, and ∆Gprotis the component from internal changes in the protein itself. However, operating under the assumption of volume conservation, we derived the bilayer deformation free energy
Gsingle =πκb
uo λ + τ
KA l λ
2 1 +√
2ro λ
(2.46) in eqn.2.8. Expanding the term in parentheses we find both a linear and quadratic term in bilayer tension. Considering that l/λ = O(1) and τ /KA 1, only the linear term is important for our energetic analysis. The linear term can be interpreted as the component of the free energy that is
due to the area change of the lipidssurrounding the protein, and not the protein itself. Consider the free energy change for a cylindrical protein whose hydrophobic mismatch has an initial value u(i)o and final value u(fo ) and, for simplicity, whose radius is constant. In addition to the expected term that is quadratic inuo/λ, there is a term linear in the tension, given by
∆Glinear =τ ·2π κbl KAλ2
1 +√ 2ro
λ u(fo )−u(i)o
| {z }
lipid area change
, (2.47)
that corresponds to the area change of the lipids surrounding the protein. Thus the perceived area change during an experiment is partly composed of a change in the area of the protein, and partly composed of a change in the area of the lipids surrounding the protein. This means that changing bilayer thickness changes both the tension independent contribution to the gating free energy, and adds a tension dependent contribution that depends on the bilayer thickness relative to the change in hydrophobic thickness of the protein. Hence we would expect that changing bilayer thickness changes not only the nominal gating tension of the channel, but also adjusts the sensitivity to tension upon gating, that is, the slope of the Popen curve. Indeed, the experiments by Perozo et al. [22] showed both of these effects, a decrease in gating tension and corresponding decrease in channel sensitivity. The change in sensitivity, and hence change in total gating area, was somewhat unexpected because the open state conductance properties and hence open state area were identical regardless of bilayer thickness — alternatively stated — if open state structure is essentially the same, why would the perceived area change with lipid thickness? The simple argument presented here qualitatively explains these results, however, experiments that measure the membrane tension itself (as opposed to the trans-patch pressure), as well as more precise information about channel structure, would be needed to make a quantitative comparison with this theoretical insight.