Lipid membranes can spontaneously form in liquids

Crooked membranes make protein

Biological membranes, flexible bilayers made of lipid molecules, form the outer shell of every living cell. Many of their important properties are based on principles that can be understood using methods from soft matter physics. For example, adhering proteins can interact with one another after a local membrane curvature has been imprinted, as was recently shown in computer simulations.

Life in a soap bubble

All living cells on earth are surrounded by a lipid membrane. This separates it from its external environment, but at the same time allows a controlled transport of matter and energy so that - in accordance with the second law of thermodynamics - order can be generated and maintained inside. The construction principle of these membranes is impressively simple: They consist of amphiphilic lipid molecules that combine a water-soluble head group with two water-insoluble hydrocarbon chains. Lipids aggregating in water therefore automatically turn into two-dimensional layers in which the insoluble chains are shielded from the water by the head groups, as shown in FIG illustration 1 outlined. These layers can form closed "vesicles", so to speak soap bubbles under water, the rudimentary beginnings of a cell wall. Because this happens spontaneously, not only are there no major control efforts, the resulting membranes are also stable against localized "injuries". What forms spontaneously can also repair itself spontaneously!

Cell membranes may once have started so simply - today they are highly complex structures, consisting of hundreds of types of lipids and thousands of embedded proteins. They take on countless tasks, from controlled substance transport to protein sorting and signal transmission. Their basic structure, however, has not changed. And just like at the beginning of life, certain physical aspects of the lipid membrane - such as the ability to self-assemble or its elastic properties - are of the greatest biological importance.

How membranes transport forces

In the theory department of the MPI for Polymer Research, in the subgroup of Markus Deserno, the biophysics of cell membranes is researched using theoretical and computer-aided approaches. The main question is what consequences the interaction of the membrane with proteins has on a length scale that is larger than the proteins themselves. When a protein binds to a membrane or is embedded in it, it locally changes certain physical properties of the membrane. These include, for example, the order of the lipids, their molecular composition, the thickness of the membrane, their fluctuations or their local curvature. These changes spread around the protein in the membrane and can also be felt at some distance from the protein. In particular, they can be "felt" by other proteins and in this way lead to the proteins, for example, attracting or repelling each other. Such membrane-mediated physical interactions therefore specifically complement biological protein-protein bonds. The distribution and aggregation of proteins on a membrane, which is so important for its function, results from an interplay between these biophysical and biochemical processes.

Simple physics, difficult equations

Deserno's group looked in particular at the type of interaction mediated by the curvature of the lipid membrane. The effect that curved surfaces can transport forces is well known from everyday life: two cornflakes floating on milk attract each other via the capillary forces of the distorted milk surface. Two people lying on a mattress “get dressed” because each person rolls into the hollow formed by the other. In fact, even since Einstein, gravity itself has been understood as an interaction based on a curvature of space-time caused by masses. In all these cases it is the curved geometry of the milk surface, the mattress or the space-time that is responsible for the interaction, and differential geometry is therefore always the mathematical tool of choice to describe these phenomena. This therefore also applies to the forces induced by curved membranes. Unfortunately, however, the "field equations" resulting from such a geometric theory are non-linear. This is very uncomfortable for the theorist because then a whole series of familiar techniques no longer work - starting with the seemingly obvious “superposition principle” that the deformation caused by two objects is simply the sum of the individual deformations. In fact, this non-linearity is precisely why Einstein's theory of gravity is so much more arduous than Newton's. Since, for the same reason, the equations for membranes cannot generally be solved analytically (except for the case of weak deformations, in which they can be linearized with a good approximation), it is therefore not easy to fathom their fundamental character.

Attraction or repulsion?

There is, however, a complementary approach to the problem that avoids solving complicated field equations. The trick is to use the membrane stress tensor. It creates a direct connection between the forces and the membrane geometry, and its coordinate-free description was presented a few years ago by Capovilla and Guven [1]. In collaboration with Guven, the Deserno group succeeded in analytically correlating membrane-mediated interactions with the membrane geometry [2, 3] (see Fig. 2). It turned out that even such elementary questions as “Will the proteins attract or repel each other?” Cannot be answered in general. Here, too, one ultimately cannot avoid solving the equations if one is aiming for quantitative results; but the access allows a clearer view of physics due to its geometric immediacy, which is not obscured from the start by whatever parameterization of the membrane surface is chosen in potentially complex coordinates.

This realization that the direction of the force between two proteins is not an obvious matter, was important because there was confusion about the point of the "sign" of this force: On the one hand, there were experiments in both cell biological systems [4] and in colloidal model systems [5] suggested an attraction between the objects attached to the membrane. On the other hand, approximate linearized calculations showed that in the completely symmetrical case there should be repulsion [6]. Since both experiments and calculations could only be carried out and interpreted with certain simplifications, this question could not be clarified at first.

Impressionistic lipids on the computer

In the Deserno group, this question was therefore examined from a completely different angle, namely by means of computer simulations - the tool that has been an integral part of physical research for several decades. In the present case, however, it should be noted that even with the high-performance computers available today, it is still impossible to map a sufficiently large section of a membrane together with the proteins of interest in the computer in detail and to track its behavior over time. Instead, Deserno and his colleagues have developed a highly coarse model in which lipids are represented by just three spheres stuck together, while the surrounding water is completely dispensed with [7], see Figure 3.

The membrane-bending proteins were accordingly also only represented by curved disks consisting of about 100 similar spheres. What looks daring at first glance is actually quite quantitative: It can be verified in advance that this model correctly reproduces the essential properties of a membrane on the relevant large length scales [7 - 9]. One might think of an impressionistic painting that, viewed from a distance, conveys the correct visual impression, despite the striking lack of detail on closer inspection.
In such crude simulations, Deserno and co-workers found that membrane-bending proteins can actually attract each other. Even more, if a sufficiently large number of such proteins is available, they can all come together to form a large aggregate [10]. However, since this consists of only pre-curved subunits, it is not simply embedded flat in the membrane, but induces a protuberance; a small bubble that hangs on the otherwise still flat membrane (see Fig. 4).

This not only clarifies that the membrane can actually induce attractive interactions between proteins through its curvature; the simulations also suggest a mechanism by which vesicles can form on cell membranes with the help of proteins. These vesicles are of vital importance for the functioning of every cell, as they are required, for example, for the controlled transport of substances enclosed in them. It is fascinating to see that very elementary and unspecific physical processes can lead to their formation. But this raises the question of which specific biological control mechanisms the cell uses to ensure that vesicle formation occurs exactly where and where genetically coded necessities dictate - a problem that can only be solved in cooperation between biologists and biophysicists .

Original publications

Stresses in lipid membranes.
Journal of Physics A: Mathematical and General 35, 6233–6247 (2002).
M. M. Müller, M. Deserno, J. Guven:
Geometry of surface mediated interactions.
Europhysics Letters 69, 482–488 (2005).
M. M. Müller, M. Deserno, J. Guven:
Interface mediated interactions between particles - a geometrical approach.
Physical Review E. 72, 061407 (2005).
K. Takei, V. I. Slepnev, V. Haucke, P. De Camilli:
Functional partnership between amphiphysin and dynamin in clathrin-mediated endocytosis.
Nature Cell Biology 1, 33–39 (1999).
I. Koltover, J. O. Rädler, C. R. Safinya:
Membrane mediated attraction and ordered aggregation of colloidal particles bound to giant phospholipid vesicles.
Physical Review Letters 82, 1991–1994 (1999).
T. R. Weikl, M. M. Kozlov, W. Helfrich:
Interaction of conical membrane inclusions: Effect of lateral tension.
Physical Review E. 57, 6988–6995 (1998).
I. R. Cooke, K. Kremer, M. Deserno:
Tunable generic model for fluid bilayer membranes.
Physical Review E. 72, 011506 (2005).
Solvent free model for self-assembling fluid bilayer membranes: Stabilization of the fluid phase based on broad attractive tail potentials.
Journal of Chemical Physics 123, 224710 (2005).
V. A. Harmandaris, M. Deserno:
A novel method for measuring the bending rigidity of model lipid membranes by simulating tethers.
Journal of Chemical Physics 125, 204905 (2006).
B. J. Reynwar, G. Illya, V. A. Harmandaris, M. M. Müller, K. Kremer, M. Deserno:
Aggregation and vesiculation of membrane proteins by curvature-mediated interactions.
Nature 447, 461–464 (2007).