by Samuel Foley

Lipid membranes are fundamentally important structures in all branches of life. Not only do they form the boundary of living cells, they also perform a multitude of roles within them. To deeply understand these fluid membranes and their properties is then essential for explaining the mechanics of life at these scales. An important property of biological membranes which has recently moved to the forefront of research is the asymmetry of their constituent layers, as discussed in a previous post on this blog.

While experiment will always be the gold standard of scientific knowledge, it often becomes advantageous to turn to computer simulation for tasks which are experimentally infeasible for whatever reason. Although they cannot reproduce real-world systems with exact precision, molecular simulations allow direct inspection of their current configuration at any given moment—something which would be impossible in an experimental setting.

However, atomic-level detail needn’t be the goal of computer simulation, and can actually be detrimental due to the time and computer resources required to carry out such detailed calculations. In fact, if what you’re interested in is large-scale, long-time behavior, it’s useful—or even necessary—to simplify the model you simulate so that you still capture the essential features you care about, while not having to waste CPU cycles calculating details that are irrelevant to what you’re investigating. What you get is a “coarse-grained” model of the system of interest, where many degrees of freedom have been “integrated out”.

This brings us back to the important point at the start: biomembranes tend to be asymmetric, with asymmetric lipid leaflet compositions persisting for hours even without the help of active biological processes keeping them that way. But in a very useful coarse-grained simulation model for lipids called the Cooke model, the process of lipid “flip-flop”, where lipids flip from one side of the membrane to the other, is so fast that any initially asymmetric membrane decays to a mixed symmetric one long before any statistically significant measurements can be recorded. This can be useful for relaxing membrane leaflets that were initially set up in strained states, but in a model with such rapid flip-flop, there’s no way to study asymmetric membranes. This is the problem that Markus Deserno and I sought to tackle in our latest paper [J. Chem. Theory Comput. 2020, 16, 11, 7195–7206].

The Cooke model takes the complicated chemical structure of a lipid molecule and simplifies it down to just three effective beads, with interactions between them tuned just right to produce the formation of stable fluid membranes, as real lipids do in water. Since there are no water molecules in a Cooke model simulation, this cohesion is accomplished through attractive interactions between the tail beads of lipids, pulling them together. 

In a previous paper [Commun. Comput. Phys. 2013, 13, 1093−1106], Han Wang and coworkers came up with a scheme for suppressing flip-flop in these membranes by artificially labeling the lipids in the two sides of membrane as different “types” (call them A and B, for simplicity). The attractive interaction between the middle beads of these opposite-type lipids was then disabled, so that it was no longer favorable for an A and B lipid to be next to each other. Interactions of A and B lipids amongst themselves are left alone, so that as long as they stay where they’re supposed to, everything should be (nearly) the same as before. While this worked well enough for the particular use case that interested Wang and coworkers, we found that this “fix” didn’t cut it when it came to a crucial scenario we want to study: differentially stressed membranes.

A differentially stressed membrane is one in which the two leaflets of the bilayer have different tensions. A state like this can be arrived at through a variety of ways, such as over- or under-filling of leaflets, or mismatch of lipid curvatures. When the fix by Wang and coworkers is used to try to simulate such systems, it fails, and asymmetry is destroyed. In our paper we quantified exactly when and how this failure occurs, and developed a simple theory which explains the break-down process. This allowed us to better understand how to remedy the problem, and to even understand the limitations of our own subsequent modifications—which I’ll now explain.

With only three beads per lipid, there’s not much room to make dramatic changes to the model without detrimentally impacting usability. Our solution involves augmenting the Cooke lipids with an additional tail bead to open up some extra playing field. With the addition of this bead, we now have two cross-leaflet middle beads whose attraction we can disable, which allows us to more than double the penalty for an A lipid to have a B lipid neighbor. With this change, we found that our simulated membranes can support differential stresses well beyond what is expected to be possible in biological settings, and well into the territory of new and interesting biophysics. With our newly modified model in hand, we went on to explore two new applications now accessible, and made an interesting discovery in the process.

First, we analyzed what happens to the lateral stress profile of an asymmetric membrane. The lateral stress profile basically tells you what the local tension is at a particular depth in the membrane. That is, even if you have a membrane which overall has zero tension, at varying depths through the membrane you generally have pulling and pushing in a way that overall balances out to zero in the end. In terms of the lateral stress profile, this statement is that the integral of the stress profile over some region tells you the net tension in that region. This means that we can integrate the two halves of the stress profile separately to get the tension in each of the individual leaflets of the membrane, giving us direct access to the current state of differential stress. By analyzing the resulting stress vs. asymmetry data in the context of simple elastic theory, we were able to measure the stretching moduli of the individual monolayers directly. This allowed us to confirm that the value of the monolayer modulus is compatible with simply dividing the bilayer modulus in half, which had been questioned in some recent work by others.

Our second application revolved around the question of what area a differentially stressed membrane will occupy when at rest (i.e., under no net tension). When one leaflet of the membrane is overfilled with lipids, while the other is under-filled, and the system is allowed to rest at overall zero tension, you might expect that the area of the resulting membrane would simply be the same as the original symmetric system. However, the most basic elastic theory you could write down, which essentially treats the two leaflets as two springs in parallel, actually predicts that the area of the bilayer will shrink quadratically with increasing asymmetry.

Surprisingly, our simulations indicate that this is not the case, and in fact the complete opposite is true: as you increase the asymmetry of the membrane, the rest area increases quadratically! So it appears that the simple lowest-order theory for the elastic behavior of membrane area is not sufficient to explain what happens when the two leaflets compete, causing a cancellation of the lowest-order effects. To remedy this, we permit the area modulus of a monolayer to actually have some dependence on the current monolayer strain. Re-analyzing our data in the context of this modified theory results in very good agreement, and produces the first ever measurement of a higher order strain-dependent correction to the monolayer area modulus. The ability to access this non-Hookean behavior depends critically on the ability to simulate asymmetric membranes, as this is what allowed for the cancellation of the lowest order effects which ordinarily dominate and render this correction invisible.

I’m personally very excited about what might lie in the future for this new modified model. We have already demonstrated with our first two examples that there’s a whole world of interesting physics previously hidden behind the veil of asymmetry. In this work, we haven’t even begun to explore what happens if you introduce more than one chemically distinct lipid species into an asymmetric system, but this too is now accessible. The flip-fixed Cooke model gives us access to this realm of asymmetry on extraordinary length- and time-scales not feasible for more highly resolved models, putting it in a unique position for answering future questions regarding asymmetric membrane phenomena.

Samuel Foley is a graduate student in the Department of Physics at Carnegie Mellon University, pursuing his degree in biophysics with Markus Deserno. His field of study is computational membrane biophysics, with a focus on coarse-grained lipid membrane modeling.