Have you ever tried to bend a piece of paper in two different directions at once? If not, try it with a piece of paper about 5 cm by 5 cm: hold the sheet of paper in front of you, bend the top left and bottom right corners up, and the other two corners down (Figure 1, left and centre). You may need someone else to help you, but either way, you will quickly find that it is impossible to bend the piece of paper in this way without it either tearing or creasing. This is in contrast to the effect of bending the paper across its middle, i.e., bending it when holding the two pairs of adjacent corners, where it happens readily and without tearing (Figure 1, right).

Figure 1. Diagram of the effect of trying to bend a square sheet in two different directions simultaneously (left and centre). Does it work with a sheet of paper?

It is easy to imagine that the first kind of bending, the one that tore or creased the paper, will work better with a flexible material such as rubber.  It is also obvious why: the rubber can stretch, it is elastic.  Paper lacks this quality, and as the applied force overcomes the forces holding the sheet together, the sheet breaks.  We know that bilayers are sheets, and so this begs the question ‘What happens when we try to bend a sheet of lipids?’

It would obviously be useless if the lipid bilayer were like the sheet of paper: moving your hands to type, or moving your eyes to read this sentence, would result in the rupture of many millions of your cells.  That does not have an evolutionary advantage and so, by this stage in cellular evolution, it no longer occurs.  Nature has developed a way of ensuring that cells have a measure of flexibility in their membranes to cope with applied forces.  This includes forces from within the body, such the pressure between forearm and upper arm, of folding an elbow, or peristaltic activity in the small intestine.  It can also be external ones such as air, or water, pressure.

Figure 2. Left, cell division at the stage shortly before membrane fission. Image courtesy of Harold C. Smith, University of Rochester, Rochester, NY, USA. Right, unduloid surface analogous to that observed in vivo, with curvature in opposite directions simultaneously, shown here in a repeating manner. Image courtesy of David Dumas, University of Illinois at Chicago (2010).

Figure 3. Left, saddle shape shown on a leather dressage saddle, with the curve that is perpendicular to the plane of the paper shown in white. Right, saddle formation of the lipid molecules, where the curvatures are at perpendicular planes (J. M. Seddon, R. H. Templer, 1995, Polymorphism of Lipid-Water SystemsfromThe Handbook of Biological Physics, ed. R. Lipowsky, E. Sackmann, Elsevier Science). Points: (a) Saddle point or seat, i.e., where the two curves meet and there is curvature of both negative and positive sign at once (In practice this is restricted to no more than one lipid molecule), (b) Left (arbitrarily assigned name) apex of the upper curve, (c) Right apex of the upper curve, (d) and (e) are the apices of the lower curve.

Clearly there are limits to what we can endure. Other species have different adaptations, however.  Anybody who likes penguins will know that they are quite happy with the (air) pressure on land, but also can dive and swim at considerable depths in the sea.  This is a huge range of pressures, and it is not an accident that penguins are adapted for this breadth of conditions.  The physical requirements are not just for strength, e.g., to breathe despite the external pressure, but also for physical resistance to pressure at a cellular level.

Although this may seem unexpected, there is more (and in my opinion this is the most interesting bit).  Whilst we understand how the cell membrane is constructed (well beyond the point of knowing that biological membranes are not flat), and what a membrane does, we also know that in order for you to be reading this now, millions upon millions of cells have had to reproduce, fight infection and communicate with one another.  And that is just in your body, carries on all of the time and hopefully without your being aware of it.

What all these processes have in common is that they involve membrane-dividing, or membrane-fusing, events as a crucial stage.  And all of these things must occur despite applied pressure, and changes in this applied pressure.  There is no evolutionary advantage to forcing a penguin to stand still for a while, such that its cells can divide without changes in pressure.  So how does the body do it?  Well, there are several things, not all of which are lipid-based.  Mammals and birds have a skeleton, which helps give support to the whole body. Although not the full answer at a cellular level, it is undoubtedly a helping hand.  At a lipid level, the rest mainly consists of enzyme (protein) action causing changes in the desired shape of lipid molecules.  So, if the action of an enzyme means that a lipid is no longer cylindrical, but wedge-shaped, the thermodynamic driving force for bending the membrane will start to emerge.  This can be particularly intricate, and can even be responsible for driving the membrane to bend in two directions at once (Figure 2).

A surface bending in opposite directions simultaneously is described as having negative Gaussian curvature. Despite its obscure-sounding name, the concept of something flat bending in two directions at once is not unique to cells.  Figure 3 shows an example of this, the seat of a saddle. A bilayer can be draped over the same shaped surface in order to give this shape.  Cells exploit this topology as an intermediate in division (Figure 2).  Not only does the cell exercise control it for this process to occur, but it does it despite external pressures, both internal, and external.