many other factors such as local lipid packing contribute to the effective potential of
mean force, which cannot easily be separated from the pure hydrophobic mismatch
contribution [30]. Except for inclusions with very large radii [251], the hydrophobic
mismatch contribution to the effective interactions was generally found to be
attractive.
3.2 Curvature-Mediated Interactions Between Proteins
3.2.1 The Mystery of the Sign
A very striking experimental demonstration of membrane curvature-mediated
interactions was given by Koltover et al. in 1999 [252]. These authors mixed
micron-sized colloidal particles with giant unilamellar vesicles to which they
could adhere. In the absence of vesicles, the colloidal particles showed no tendency
to aggregate in solution, whereas they quickly did once they adsorbed onto the
vesicles. Since it was also evident from many micrographs that the colloids induced
local bending of the vesicle’s membrane, the experiment strongly pointed towards
membrane curvature-mediated attractions between the adhering colloids. This,
however, was very surprising. Although interactions were indeed expected, the
force should have been repulsive, as predicted 6 years earlier by Goulian
et al. [253]. Interestingly, the prefactor of this interaction had to be corrected
twice [254, 255], but this did not change the outcome: the colloids should have
repelled. It was soon understood that objects that cause anisotropic deformations
could in fact orient and then attract [256–258], but the colloids of Koltover
et al. were isotropic (as far as one could tell experimentally).
We will try to provide a glimpse into this mystery. A big part of it has to do with
careless use of the statement “theory has predicted.” Theory always deals with
model systems and makes simplifying assumptions, and this particular problem is
fraught with seemingly inconsequential details that could and sometimes do matter.
3.2.2 The Nonlinear Ground State: Take I
The relevant field Hamiltonian pertaining to the curvature-mediated interaction
problem is Eq. (1), minus several terms that will not matter. For a start, the last term
involving the edge tension γ does not arise in the absence of any membrane edge.
The spontaneous bilayer curvature K 0 usually vanishes for symmetry reasons. If
lipids can flip between the two leaflets, their chemical potential must be the same in
both, and if no other symmetry-breaking field is present, this means that K 0 ¼ 0.
Unfortunately, membrane curvature itself breaks the bilayer symmetry, and
any existing lipid composition degree of freedom must couple to the geometry
[75, 259–263]. So let us for now assume that this is not the case and take a note of
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M. Deserno et al.
mean force, which cannot easily be separated from the pure hydrophobic mismatch
contribution [30]. Except for inclusions with very large radii [251], the hydrophobic
mismatch contribution to the effective interactions was generally found to be
attractive.
3.2 Curvature-Mediated Interactions Between Proteins
3.2.1 The Mystery of the Sign
A very striking experimental demonstration of membrane curvature-mediated
interactions was given by Koltover et al. in 1999 [252]. These authors mixed
micron-sized colloidal particles with giant unilamellar vesicles to which they
could adhere. In the absence of vesicles, the colloidal particles showed no tendency
to aggregate in solution, whereas they quickly did once they adsorbed onto the
vesicles. Since it was also evident from many micrographs that the colloids induced
local bending of the vesicle’s membrane, the experiment strongly pointed towards
membrane curvature-mediated attractions between the adhering colloids. This,
however, was very surprising. Although interactions were indeed expected, the
force should have been repulsive, as predicted 6 years earlier by Goulian
et al. [253]. Interestingly, the prefactor of this interaction had to be corrected
twice [254, 255], but this did not change the outcome: the colloids should have
repelled. It was soon understood that objects that cause anisotropic deformations
could in fact orient and then attract [256–258], but the colloids of Koltover
et al. were isotropic (as far as one could tell experimentally).
We will try to provide a glimpse into this mystery. A big part of it has to do with
careless use of the statement “theory has predicted.” Theory always deals with
model systems and makes simplifying assumptions, and this particular problem is
fraught with seemingly inconsequential details that could and sometimes do matter.
3.2.2 The Nonlinear Ground State: Take I
The relevant field Hamiltonian pertaining to the curvature-mediated interaction
problem is Eq. (1), minus several terms that will not matter. For a start, the last term
involving the edge tension γ does not arise in the absence of any membrane edge.
The spontaneous bilayer curvature K 0 usually vanishes for symmetry reasons. If
lipids can flip between the two leaflets, their chemical potential must be the same in
both, and if no other symmetry-breaking field is present, this means that K 0 ¼ 0.
Unfortunately, membrane curvature itself breaks the bilayer symmetry, and
any existing lipid composition degree of freedom must couple to the geometry
[75, 259–263]. So let us for now assume that this is not the case and take a note of
258
M. Deserno et al.