La cooperación internacional en el marco de la gobernanza global para el

Deformation of the midplane of the bilayer involves a cost to bend the midplane from its flat, equilibrium position [46, 44,32]. We use the function h(r) to denote this change in height of the bilayer midplane as a function of the position ras shown in Fig. 1.7. The energy cost associated with bending the membrane away from its flat configuration can be written as

G(bendmid)=

κb

2 Z

∇2h(r)₋cb_o2d2r, (1.9) where the bilayer bending modulusκb '20kBT [51,26] andcbois the bilayer spontaneous curvature.

Throughout this chapter the gradient operator is defined by_∇= (∂/∂x, ∂/∂y), and the Laplacian operator by _∇2=∂2/∂x2+∂2/∂y2, in Cartesian coordinates. In general, bilayers with symmetric leaflet compositions have zero midplane spontaneous curvature, although we will discuss the origins of spontaneous curvature and quote the relevant energetic contribution later in this chapter. Tension also plays a role in the energetics of midplane deformation because any bend in the midplane results in a reduction in the projected area of the membrane, which couples directly to an increase in the energy of the loading device, resulting in a contribution to the free energy of the form

G(mid) ten = τ 2 Z (_∇h(r))2d2r, (1.10) where the tension,τ, ranges from zero up to the nominal membrane lytic tension of_∼3.5kBT /nm2

[26]1. In general, the elastic parameters we use are representative of a typical phosphatidylcholine (PC) lipid. Thus the total energy expended to deform the midplane over an areaA is

G(mid)₌Z A _τ 2(∇h(r)) 2₊κb 2 (∇ 2_h(r))2_d2_{r. (1.11)}

The logic behind this kind of analysis is to find the free energy minimizing function h(r). One way to carry out this minimization is by solving a partial differential equation that is generated by formally minimizing the free energy, the details of which are discussed in the second half of chap.4. An alternative (and approximate) scheme to be explored later in this section is to make a guess for the functional form ofh(r) and to minimize with respect to some small set of parameters. This approach is called a variational method and can be quite useful for developing intuition.

In the midplane-deforming model, the protein can dictate the slope of the membrane at the protein-lipid interface which, in addition to the protein radius, will determine the deformation energy. The length scale over which the membrane returns to its unperturbed state is given by

1_{The lytic tension of a bilayer is technically a stochastic quantity [53], however, we quote the lytic tension as the} tension at which bilayer lysis is a rapid, spontaneous process.

λτ =

p

κb/τ and the energy for this type of deformation is

G(mid)_{(R, τ) =πκ} bθ2 R λτ · K0(R/λτ) K1(R/λτ) , (1.12)

where R is the radius of the protein, θ is the slope of the membrane at the protein-lipid interface as shown in Fig. 1.7, and Ki are modified Bessel functions of the second kind of orderi [54,32].

Given a protein with a particular radius and fixed boundary slope, an increase in tension will make any deformationmore costly. Hence, for midplane deformation, increased tension prefers a flatter membrane and/or smaller protein radius. To get a feel for the energy scale of this deformation several examples for different parameter values are summarized in Table1.2.

With the contribution to the free energy difference arising from midplane deformation in hand, we can now explore the competition between applied tension and the energetics of membrane deformation in dictating channel gating. The key to understanding the interplay between tension and deformation energetics lies in the scaling of these two effects with protein radius. The midplane deformation energy scales roughly linearly with the radius of the protein and is unfavorable. On the other hand, the term proportional to the applied tension scales as the square of the protein radius and favors the open state. If we fix the membrane slope, then the energy of a midplane deforming protein as a function of protein radius and tension is

G(R, τ)_'G(mid)_{(R, τ)−τ πR}2_{. (1.13)}

As tension increases, the potential energy of the loading device eventually overcomes the deforma- tion energy and a larger protein radius and/or more cylindrical protein shape is the preferred state. Indeed, midplane deformations have been hypothesized to be an important functional mechanism of MscL [54]. One of the uncertainties that accompanies a model of this type is the fact that there is some function that connects the slope of the membrane at the protein-lipid interface (θ) with the current radius of the channel, that is, there is some unknown function θ(R) [55]. Future experiments will be necessary to further clarify this point. If we make the simplest approximation thatθ(R) = constant and look at two reasonable values ofθ= 0.6 andθ= 0.8 [54], using eqn.1.13 and the parameters in Table 1.1, we find the rather small critical tensions _∼0.004kBT /nm2 and

∼0.06kBT /nm2, respectively, compared to the known critical tension of MscL at ∼2.5kBT /nm2

[23, 34]. Though we have shown that midplane deformations are capable of endowing a channel protein with bistability, the scale of the critical tension and the free energy difference between con- formations indicates that, at least for MscL, an additional kind of deformation might be important as well.

a) b)

d) c)

Thickness Deformation (MscL) Midplane Bending (MscS)

Loc al Ene rg y Den sity Hydrophobic Mismatch ( ) Loc al Ene rg y Den sity

Bilayer Midplane Slope ( )

Figure 1.7: Elastic idealizations of deformations and protein-lipid boundary conditions. a) Atomic- level structure [11] and an elastic idealization of the mechanosensitive channel of large conductance (MscL) as a rigid cylinder of radiusR with hydrophobic mismatch at the proteinlipid interface,uo,

and the bilayer thickness deformation described by the functionu(r). For r λk the membrane

thickness is at its equilibrium value. b) Atomic-level structure [18] and an elastic idealization of the mechanosensitive channel of small conductance (MscS) as a protein that deforms the membrane midplane by an angle θ at the protein lipid interface, causing a midplane height deformation described by the function h(r). For r λτ the bilayer midplane is flat. c) Membrane distortion

and corresponding free energy of deformation per unit area of membrane surrounding MscL. d) Membrane distortion and corresponding free energy of deformation per unit area of membrane surrounding MscS. In (c) and (d) the elastic response of the lipids is captured with springs, and the color coding indicates the local strain energy density at different distances from the proteins.

Fixed Parameters: Dynamic Parameter: Free Energy Difference: R= 3 nm, θ= 0.5 τ = 0_→2kBT /nm2 10kBT R= 3 nm, τ = 2kBT /nm2 θ= 0→0.5 10kBT R= 3 nm, τ = 2kBT /nm2 θ= 0→0.8 26kBT θ= 0.5,τ = 2kBT /nm2 R= 3→6 nm 14kBT θ= 0.8,τ = 2.5kBT /nm2 R= 2.5→3.5 nm 13kBT

Table 1.2: Typical free energies for midplane deformation. The first row indicates how tension leads to an increase in deformation energy. The second and third rows show the sensitivity to the boundary slope. The fourth row indicates how protein radius changes deformation energy. The last row is a comparison with the known radius change and critical tension of MscL.