Geometry in Nature


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 curvature-mediated interactions
    Benedict J. Reynwar, Gregoria Illya, Vagelis A. Harmandaris, Martin Michael Müller, Kurt Kremer, Markus Deserno

    Membrane 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 large-scale 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 curvature-mediated attractive interactions could aid cooperation and complement the effects of specific binding events on membrane remodelling. But it is difficult to experimentally isolate curvature-mediated interactions from direct attractions between proteins. Moreover, approximate theories predict repulsion between isotropically curving proteins. Here we use coarse-grained membrane simulations to show that curvature-inducing 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, curvature-mediated 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. 461-464, 2007.



  • Interface-mediated interactions: Entropic forces of curved membranes
    Pierre Gosselin, Hervé Mohrbach, Martin Michael Müller

    Particles 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 curvature-mediated repulsion between the two cylinders. This is contrary to the usual attractive Casimir force in the absence of curvature-mediated 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 Deserno

    Membrane-deforming 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 bud-formation. 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. 351-363, 2010.


  • Balancing torques in membrane-mediated interactions: Exact results and numerical illustrations
    Martin Michael Müller, Markus Deserno, Jemal Guven

    Torques on interfaces can be described by a divergence-free 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 two-particle 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 membrane-bound parallel cylinders. This apparently simple system provides some flavor of the remarkably subtle nonlinear behavior associated with membrane-mediated interactions.

    Phys. Rev. E, 76(1): 011921, 2007. See also cond-mat/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 Guven

    Particles bound to an interface interact because they deform its shape. The stresses that result are fully encoded in the geometry and described by a divergence-free 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 two-particle 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 force-distance relations, several of which have originally been derived by using an energy based approach.

    Phys. Rev. E, 72(6): 061407, 2005. See also cond-mat/0506019.
    Also featured in the Virtual Journal of Biological Physics Research.


  • Geometry of surface-mediated interactions
    Martin Michael Müller, Markus Deserno, Jemal Guven

    Soft 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 mid-plane; 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 small-gradient regime.

    Europhys. Lett., 69(3): pp. 482-488, 2005. See also cond-mat/0409043.







     © Martin Michael Müller