Interface mediated interactions
What forces do particles exert on each other when adhering to a surface (e.g. a lipid membrane or a water surface)?
Apart from gravity or electromagnetic forces, deformations of the surface which are caused by the adhering
particles can also lead to an attraction or a repulsion between them.
One example of such a surface mediated interaction can be found in this Video
(3.4 MB): two sewing needles are adhering to a water surface. By deforming the surface due to their weight they attract each other. You can try this at home, too. All you need is a bowl of water and two needles.
Relevant publications
Computer simulations
 Aggregation and vesiculation of membrane proteins by curvaturemediated
interactions
Benedict J. Reynwar, Gregoria Illya, Vagelis A. Harmandaris, Martin Michael Müller, Kurt Kremer, Markus DesernoMembrane remodelling plays an important role in cellular tasks such as endocytosis, vesiculation and protein sorting, and in the biogenesis of organelles such as the endoplasmic reticulum or the Golgi apparatus. It is well established that the remodelling process is aided by specialized proteins that can sense as well as create membrane curvature, and trigger tubulation when added to synthetic liposomes. Because the energy needed for such largescale changes in membrane geometry significantly exceeds the binding energy between individual proteins and between protein and membrane, cooperative action is essential. It has recently been suggested that curvaturemediated attractive interactions could aid cooperation and complement the effects of specific binding events on membrane remodelling. But it is difficult to experimentally isolate curvaturemediated interactions from direct attractions between proteins. Moreover, approximate theories predict repulsion between isotropically curving proteins. Here we use coarsegrained membrane simulations to show that curvatureinducing model proteins adsorbed on lipid bilayer membranes can experience attractive interactions that arise purely as a result of membrane curvature. We find that once a minimal local bending is realized, the effect robustly drives protein cluster formation and subsequent transformation into vesicles with radii that correlate with the local curvature imprint. Owing to its universal nature, curvaturemediated attraction can operate even between proteins lacking any specific interactions, such as newly synthesized and still immature membrane proteins in the endoplasmic reticulum.
Nature 447(7143): pp. 461464, 2007.
Theory
 Interfacemediated interactions: Entropic forces of curved membranes
Pierre Gosselin, Hervé Mohrbach, Martin Michael MüllerParticles embedded in a fluctuating interface experience forces and torques
mediated by the deformations and by the thermal fluctuations of the medium.
Considering a system of two cylinders bound to a fluid membrane we show that
the entropic contribution enhances the curvaturemediated repulsion between
the two cylinders. This is contrary to the usual attractive Casimir force in
the absence of curvaturemediated interactions. For a large distance between
the cylinders, we retrieve the renormalization of the surface tension of a
flat membrane due to thermal fluctuations.
Phys. Rev. E, 83(5): 051921, 2011. See also arXiv:1011.1221.
 Cell Model Approach to Membrane Mediated Protein Interactions
Martin Michael Müller, Markus DesernoMembranedeforming proteins can interact through the curvature
fields they create. In the case of many such proteins a cell model
approach can be used to calculate the energy per protein and
predict, whether it would lead to phase segregation or
budformation. Using covariant differential geometry exact results
are derived for the lateral pressure in terms of geometric
properties at the cell boundary. Numerical solutions of the exact
shape equations in the highly nonlinear regime are found and it is
seen that both phase segregation and bud formation can occur.
Prog. Theor. Phys. Suppl., 184: pp. 351363, 2010.
 Balancing torques in membranemediated interactions: Exact results and
numerical illustrations
Martin Michael Müller, Markus Deserno, Jemal GuvenTorques on interfaces can be described by a divergencefree tensor
which is fully encoded in the geometry. This tensor consists of two
terms, one originating in the couple of the stress, the other capturing
an intrinsic contribution due to curvature. In analogy to the description
of forces in terms of a stress tensor, the torque on a particle can be
expressed as a line integral along any contour surrounding the particle.
Interactions between particles mediated by a fluid membrane are studied
within this framework. In particular, torque balance places a strong
constraint on the shape of the membrane. Symmetric twoparticle
configurations admit simple analytical expressions which are valid
in the fully nonlinear regime; in particular, the problem may be
solved exactly in the case of two membranebound parallel cylinders.
This apparently simple system provides some flavor of the remarkably
subtle nonlinear behavior associated with membranemediated interactions.
Phys. Rev. E, 76(1): 011921, 2007. See also condmat/0702340. Also featured in the Virtual Journal of Biological Physics Research.  Interface mediated interactions between particles  a geometrical approach
Martin Michael Müller, Markus Deserno, Jemal GuvenParticles bound to an interface interact because they deform its shape.
The stresses that result are fully encoded in the geometry and described
by a divergencefree surface stress tensor. This stress tensor can be
used to express the force on a particle as a line integral along any
conveniently chosen closed contour that surrounds the particle. The
resulting expression is exact (i.e., free of any 'smallness' assumptions)
and independent of the chosen surface parametrization. Additional surface
degrees of freedom, such as vector fields describing lipid tilt, are readily
included in this formalism. As an illustration, we derive the exact force
for several important surface Hamiltonians in various symmetric twoparticle
configurations in terms of the midplane geometry; its sign is evident in
certain interesting limits. Specializing to the linear regime, where the
shape can be analytically determined, these general expressions yield
forcedistance relations, several of which have originally been derived
by using an energy based approach.
Phys. Rev. E, 72(6): 061407, 2005. See also condmat/0506019. Also featured in the Virtual Journal of Biological Physics Research.  Geometry of surfacemediated interactions
Martin Michael Müller, Markus Deserno, Jemal GuvenSoft interfaces can mediate interactions between particles bound to
them. The force transmitted through the surface geometry on a
particle may be expressed as a closed line integral of the surface
stress tensor around that particle. This contour may be deformed to
exploit the symmetries present; for two identical particles, one
obtains an exact expression for the force between them in terms of
the local surface geometry of their midplane; in the case of a
fluid membrane the sign of the interaction is often evident. The
approach, by construction, is adapted directly to the surface and is
independent of its parameterization. Furthermore, it is applicable
for arbitrarily large deformations; in particular, it remains valid
beyond the linear smallgradient regime.
Europhys. Lett., 69(3): pp. 482488, 2005. See also condmat/0409043.
