Figure 1.
Membrane embedding of a hydrophobic insertion.
A cylindrical insertion of a circular cross-section of radius is embedded to a depth
into a lipid monolayer, the neutral plane of the latter lying at two thirds of its thickness (left cartoon). Membrane embedding is presented in two steps. The first step is generation of a void necessary for the insertion embedding without altering the membrane shape. The second step represents the membrane shape relaxation. The color code represents the intra-membrane stresses.
Figure 2.
Five ways of generating membrane stress preceding the insertion embedding.
A qualitative trans-monolayer stress profile, incorporating the main characteristics, is shown for illustrative purposes. (A) Addition of lipids with inverse conical molecular shape (lysolipids, depicted in blue) to the outer monolayer. The induced positive monolayer spontaneous curvature of the outer monolayer, , results in a positive bilayer curvature
and the corresponding trans-membrane stress profile. (B) Insertion of molecules with conical molecular shape (e.g. DAG, depicted in red) to the inner monolayer. The corresponding negative monolayer spontaneous curvature
induces a positive bilayer curvature
and the corresponding trans-membrane stress profile. (C) Symmetric enrichment of the two membrane monolayers in inverted conical (left cartoon) or conical (right cartoon) lipids. The bilayer remains flat,
, but the trans-membrane stress profile develops. (D) Membrane bending by the action of an externally applied torque that induces a positive bilayer curvature J>0 and the corresponding trans-membrane stress profile. (E) Membrane stretching (right cartoon) or compression (left cartoon) by external force and the corresponding trans-membrane stress profiles. In all panels, red crosses and green ticks illustrate, respectively, low and high binding affinities of protein insertions for differently stressed membranes.
Figure 3.
Stress sensing by hydrophobic insertions for laterally uncoupled monolayers.
The elastic binding energy of insertion was computed using the elastic model of a lipid bilayer for the insertion length of 2 nm, the insertion cross-sectional radius of 0.5 nm and the insertion embedding depth of 0.8 nm, as it has been estimated based on structural data for typical amphipathic helices. The monolayer thickness is taken to be 2 nm. The membrane stress was generated in five ways, as presented in Fig. 4 below. (A) The elastic binding energy
as a function of the void energy
, all points laying approximately on a straight line of slope one (black line). (B) The relative binding constant
as a function of the void energy
. The black line shows the expected exponential profile. (C) The elastic binding energy
(left) and the relative binding constant
(right) as functions of the void energy
for two different physiologically relevant cross-sectional radii (
in blue, and
in black) and the embedding depth
. (D) The elastic binding energy
(left) and the relative binding constant
(right) as functions of the void energy
for the insertion cross-sectional radius
and different biologically feasible values of the embedding depth (
in blue,
in black, and
in red).
Figure 4.
Computed membrane shapes for different ways of stress generation before (left column) and after (right column) the insertion embedding.
(A) Changing the spontaneous curvature of the outer monolayer; (B) Changing the spontaneous curvature of the inner monolayer; (C) Application of an external torque; (D) Symmetric generation of the spontaneous curvature of both monolayers; (E) Action of an externally applied stretching force. The color code represents the value of the lateral stress profile at each point of the membrane in all panels (different scales on left and right panels).
Figure 5.
Insertion binding as a function of the initial membrane curvature for the three scenarios of stress generation accompanied by creation of membrane curvature.
The monolayers are laterally uncoupled. The elastic binding energy (A) and the relative binding constant
(B) are presented as functions of the membrane curvature before insertion J. (C) The elastic binding energy
as a function of the membrane curvature J for insertions of different cross-sectional radii (
in blue, and
in black) embedded to the depth
. (D) The elastic binding energy
as a function of the membrane curvature J for insertions of the cross-sectional radius
embedded to three different depths (
in blue,
in black, and
in red). (E–F) The curvature sensitivity
as a function of the insertion radius r (E) and the insertion embedding depth d (F).
Table 1.
Relative binding constants of ALPS1 and ALPS2 to liposomes of different sizes.
Figure 6.
Treatment of experimental data on binding of ArfGAP1 ALPS motifs to liposomes of different sizes.
(A) The relative binding constant was numerically computed as a function of the liposome radius for laterally coupled monolayers bent by the action of an externally applied torque, and plotted as a function of the liposome radius R for the insertion cross-sectional radius
, the insertion depth
, and three different insertion lengths (
, solid line;
, dashed line; and
, dotted line). (B) The same quantities as in (A) for a
long insertion embedded to three different depths (
, solid line;
, dashed line; and
, dotted line). (C) The optimal insertion length, L, as a function of the insertion embedding depth, d, that best fits the experimental results presented in Table 1 for both ArfGAP1 ALPS1 (red line) and ALPS2 (blue line). The shaded regions represent the range of ALPS lengths estimated for each motif based on structural data.
Figure 7.
Treatment of experimental data on the binding of ArfGAP1 to liposomes of different lipid compositions.
(A) The fraction of bound ArfGAP1 to liposomes of different lipid composition as a function of the estimated monolayer spontaneous curvature , as taken from the experimental study [29] (see Table 2). (B) The relative binding constant as a function of the monolayer spontaneous curvature
, as taken from the experimental study [29] (circles), and a comparison with a fit using our model, for an insertion length of
and a depth of insertion of
(solid line). (C) The relative binding constant numerically computed for laterally coupled symmetric monolayers as a function of the monolayer spontaneous curvature,
, for the insertion cross-sectional radius
, the insertion depth
, and different insertion lengths (
, solid line;
, dashed line; and
, dotted line). (D) The same quantities as in (C) plotted for a
long insertion embedded to three different depths (
, solid line;
, dashed line; and
, dotted line). (E) The optimal insertion length, L, as a function of the insertion embedding, d, that best fits the experimental results presented in Table 2 for ArfGAP1 (solid line). The shaded region represents the estimated range of ArfGAP1 insertion lengths.
Table 2.
Liposome monolayer spontaneous curvatures and relative binding constants of ArfGAP1 to liposomes of different lipid compositions.