Análisis de los diarios de campo y/o lista de cotejo u otros

The basic structure of biomembranes comprises of a large number of discrete lipid molecules, which move laterally in the plane of the membrane, with a diffusion constant,D_lipids ∼10−8cm2/s, that is larger by nearly two orders of magnitude than

the diffusion of transmembrane proteins, namelyDproteins ∼10−10cm2/s [141–143].

As a consequence, during the time that a membrane protein diffuses an average distance of one lipid diameter, many lipid molecules will interchange places near the protein, coarse-graining out the lipidic discreteness of the membrane. Moreover, the typical transition time for conformational changes of the protein (about 5 µs) is much slower than the characteristic diffusion time of the lipids. This gives us a strong indication that a lipid bilayer can be effectively approximated as a continuous medium in the vicinity of a membrane protein (see Section 2.2.2). In addition, the transmembrane proteins can be regarded as rigid inclusions and embedded into the bilayer, such that their hydrophobic and hydrophilic parts match with those of the adjacent lipids. This fitting usually disturbs the equilibrium configuration of the protein-free membrane and changes the free energy of the lipid bilayer [139].

As a result, the mechanical deformations of the lipid environment in the vicinity of a transmembrane protein can be quantitatively described by local field variables, such as the height and/or the thickness of the bilayer [31–35, 144–154]. These examples of structural variables correspond to the two main classes of defor- mations induced by transmembrane proteins, namely the mid-plane bending and the hydrophobic mismatch, respectively [139, 148]. The free-energy cost associated with these deformation modes are completely decoupled on symmetry grounds and they can be independently analysed provided that the perturbations are small [146–148]. Furthermore, the deformation fields of neighbouring proteins can overlap and in- duce membrane-mediated interactions between the proteins, which may be either attractive or repulsive depending on their shape and orientation [31–35, 144–154]. The characteristic length scale of the mid-plane bending interactions is generally longer than the length scale of the deformations in the membrane thickness, but the interaction strength of the former deformation mode is typically much weaker than the latter [147,148]. Consequently, many theoretical studies have been devoted to understand how membrane-mediated interactions may affect the spatial organi- zation of proteins, and their ability to respond and communicate conformational changes to each other [34, 151–157].

Here, we consider an additional deformation mode that results from the enrichment of curvature sensitive inclusions in the vicinity of a membrane protein, such as non-lamellar-forming lipids or any kind of molecules smaller than the typical size of a protein. The addition of these molecular particles is usually characterized by a spontaneous curvature. This is a thermodynamic property of lipid monolayers that is operationally defined as their preferred mean curvature in the absence of external mechanical stresses [36, 42]. As discussed in Section 2.2.3, two opposed

monolayers that have the same lipid curvature will always form a planar bilayer, even though the monolayers may have a non-zero intrinsic curvature [138, 140]. Hence, in the context of bilayers, it is more appropriate to consider a composite spontaneous curvature given by the difference between the monolayer spontaneous curvature of each leaflet [51]. This suggests that asymmetrically doped bilayers generate a non-zero mean curvature by bending the membrane away from one of the aqueous surroundings. As a result, the spontaneous curvature can be used to quantitatively describe the asymmetry in the distribution of molecular inclusions between the two lipid layers of the membrane [138, 140].

Irrespective of its microscopic origin, the spontaneous curvature is normally treated as a well-defined global property, where a uniform distribution is assumed across the different leaflets of the bilayer. However, this assumption is not generally valid and consequently a local description is needed to account for non-homogeneous regions of membranes. This is particularly the case for the environment around transmembrane proteins, where the membrane-induced deformation fields provide the possibility of selection and enrichment of certain lipids, or surfactants, near the protein faces. As an illustrative example, cone-shaped molecules (such as lysophos- pholipids) can laterally and transversely diffuse within the bilayer, and localise to energetically favourable regions near proteins, where the membrane has complemen- tary curvature [36, 158]. Because of their curvature preference, this leads to a local compositional asymmetry in the vicinity of the membrane protein, which subse- quently generates a local spontaneous curvature, as shown in Figure 3.1. The main purpose of this study is to investigate such situations within a continuum theory of membranes, where the transmembrane proteins are treated as rigid inclusions.

In the next section, an analytic methodology is described which can be used for estimating the membrane energy, their shape, and the local phase behaviour, near a transmembrane protein. Subsequently, in Section 3.3, we apply this model to a number of biologically relevant problems. In particular, in Section 3.3.1, we examine the regime in which the membrane can become unstable and how this may be used to estimate the unknown parameters in our framework. In Section 3.3.2, the methodology is applied to a simple model of transmembrane proteins which display an asymmetrical shape. Lastly, in the final sections, we show how this model can be used to extend the gating-by-tilt mechanism for mechanosensitive channels of large conductance [159], and, furthermore, we investigate the effect due to the membrane compositional asymmetry on the early stages of protein coat assembly [160].

Figure 3.1: Schematic diagrams of a single transmembrane protein embedded into a two-component fluid membrane composed by lysophospholipids (red) and bilayer- forming lipids (blue). By assuming no hydrophobic mismatch, the deformation is characterised by the functions u(r) and ϕ(r), which are the deviation from flatness of the mid-plane of the bilayer, and its local leaflet asymmetry, respectively. The radial distance r is measured from the centre of the protein. Here, (a) depicts one extreme possibility, where the cone-shaped protein induces a mid-plane bending of the bilayer without any changes in the compositional asymmetry between the leaflets. (b) shows another extreme possibility, where the membrane only locally demixes to accommodate the membrane protein, leading to a non-zero ϕ near the inclusion (namely, a local spontaneous curvature). In practice, the membrane is expected to partially bend and partially demix, as illustrated in (c).