it depends linearly on the membrane tension. (See Fig.2.3.) This is exactly the tension dependence one would na¨ıvely expect for a two state system if each state had a fixed area under external tension.
The free energy difference between states would then be
∆GOC = ∆G0−α∆AOC, (2.7)
where ∆G0 is constant with respect to the tension and ∆AOC is the area difference between states the open and closed states. Clearly high tension stabilizes the open state with its larger radius. The sensitivity of the channel, the slope of the free energy with respect to tension, is the area difference between states ∆AOC = 6.5 nm2. This measurement of the area change is large for a channel protein.3 This large area change is a consequence of opening a 3 nm pore in channel core and must correspond to a very significant conformational change in the protein.
Figure 2.9: The MscL channel crystal structure. The channel is a homopentamer with two trans- membrane helices per subunit. In the figure above, each subunit, although identical, is colored differently to distinguish the individual protein subunits. The transmembrane helices are labeled TM1 and TM2. There is a third cytoplasmic alpha helix which protrudes into the cell interior. This five helix complex is thought to function as a filter [13]. Figure from Ref. [17].
membrane, we were struck by how large these interaction energies were for the MscL channel; they were on order the gating free energy. The importance of these interactions was soon demonstrated experimentally by Perozo and coworkers [21] who showed that the gating tension depends sensitively on the membrane characteristics. Fig. 4.4.2 shows the channel opening probability as a function of the applied pressure for the MscL protein reconstituted into several different lipid membranes.
The thickness of the lipid membrane is roughly proportional to the number of carbons in the tail group (acyl chain length) [22]. Perozo and coworkers demonstrated that the gating tension rises dramatically with the thickness of the bilayer.
2.5.1 An analogy to nucleation
Motivated by the sensitivity of the channel function to the membrane characteristics demonstrated by the experiments of Perozo and coworkers [21] and by our own estimates of the membrane-protein interaction energy, we proposed a very simple model for the gating of mechanosensitive channels.
This model harnessed the membrane-protein interaction energy as the spring that opposes tension and keeps the channel from opening spontaneously.
High sensitivity to membrane tension implies that the area change between the open and closed states (∆A) is large. This large expansion during channel gating implies a significant restructuring of the membrane, since the interface with the protein has also significantly grown with the areal expansion of the channel. As we mentioned above, since the membrane must typically be deformed to accommodate the channel, these membrane-protein interactions act to stabilize the closed state of the channel. This interface energy is proportional to the length of the membrane-protein interface.
Figure 2.10: The MscL gating tension depends on the thickness of the lipid bilayer. In the plot above, the open-state probability of the channel is plotted as a function of the applied pressure difference over the bilayer for three different bilayers. The bilayer lipids are phosphatidylcholine with acyl chain lengths of 16, 18, and 20. It is assumed that the radius of curvature of the membrane patch, though unobserved, is constant and identical in each of these experiments. Under these assumptions the tension is proportional to the pressure difference over the membrane. This data therefore shows a dramatic rise in the gating tension (the tension at whichPO = 0.5) as a function of the bilayer thickness (acyl chain length). Figure from Ref. [21].
Energy contributions with this scale are generically called line tensions. To summarize, the channel experiences a competition between the tensile forces from membrane tension and compressive forces due to the line-tension induced by membrane-protein interaction.
An analogous competition occurs in the canonical nucleation problem. (When a small region of the nucleating phase forms, for example a precipitate in solution, there is a competition between a surface tension, which in turn scales as the area of the interface with solution (R2) that favors the decay of the precipitate, and a bulk term which scales like the volume (R3) that favors the growth of the precipitate.) Due to the difference in the radial dependence of these two competing energetic contributions, there is an energetic barrier to nucleation. Below a critical radius, the interface energy dominates and the radius of precipitate is unstable to decay. But once the precipitate reaches this critical radius, the bulk term dominates and the precipitate is unstable to further growth.
A similar competition exists for channel gating. The energetic contribution from the line tension scales as the interface size (R) and the contribution due to the tension scales as the area (R2) of the channel. In the next chapter we will show that this nucleation picture can both describe many qualitative features of the channel function, for instance the short lifetime of the sub-states, as well as making quantitative predictions about the size of the opening tension and its dependence on the properties of the lipid bilayer, in agreement with recent experiments [21]. These calculations high- light the importance of membrane-protein interaction in describing the function of transmembrane proteins and in particular mechanosensitive proteins.