How can I model cellular membranes in terms of topology?

Question

I'm reading from Principles of Physical Biochemistry by Kensal E van Holde (1998-01-06) in order to review some biochemistry and work my way toward a more mathematical approach to understanding chemistry.

On page 18, chapter 1 (Biological Macromolecules), I read

"The structure of an integral membrane protein can be thought of as being inverted relative to the structure of a water soluble-protein, with the hydrophobic groups now exposed to the solvent, while the hydrophilic atoms form the core. An example of this inverted topology is an ionic channel (Figure 1.12). The polar groups that line the internal surface of the channel mimic the polar water solvent, thus alloweding charged ions to pass readily through an otherwise impenetrable bilayer."

I don't have a picture handy to show you Figure 1.12, but essentially it is a channel protein.

What's interesting to me about this passage is their use of the word topology. It reminded me of algrabraic topology's notion of a topology on a set, but I can't seem to state this membrane chemistry in terms of the mathematical formalism.

In algebraic topology, a topology $\tau$ on a set $X$ is a subset of the powerset $\mathcal{P}(X)$ where the elements of $\tau$ are open, meaning they satisfy

1. Containing the original set and the empty set:$$X \in \tau, \emptyset \in \tau$$
2. Unions of sets in the topology are closed. $$\{ U_i \}_{i \in I} \subseteq \tau \implies \bigcup_{i \in I} U_i \in \tau$$
3. Closure under finite union. $$\{ U_i \}_{i=1}^n \subseteq \tau \implies \bigcap_{i=1}^n U_i \in \tau$$

My initial thinking on this is that the membrane could be one set, the extracellular matrix a second set, and the intracellular matrix as a third set. Each of these sets might be thought of as their own topologies, and that their mutual intersections should be empty so long as you cleanly partition the spaces. But then again, membranes have a lot of intermolecular spaces within them; seems structurally complicated.

While knowingly extrapolating beyond the original intended meaning of the authors, how can I model cellular membranes in terms of topology?

Answer 1

As Buck Thorn and MaxW already said in the comments, there is less to it than the textbook excerpt might suggest. The use of the term topology in this context is non-mathematical, similar to when someone says there has been a quantum leap in product development.

The actual topology of most proteins is the same as that of a sphere, and the topology of the typical cartoon channel is that of a donut. Having hydrophilic residues in the channel or outside of the channel is not an inversion because those parts of the surface are topologically equivalent.

[OP] I can't seem to state this membrane chemistry in terms of the mathematical formalism.

That's because it is non-mathematical. There is some real topology associated with membranes, and that is the concept of inside and outside. Membrane proteins insert in one direction, and they don't "flip" directions over time, even though they are perfectly mobile to move around in the membrane (or have lipids move around them) while pointing in one direction or the other one. During endocytosis and exocytosis, the relative orientation of membrane proteins is conserved but inverted, so that parts that are inside organelles (such as glycosylation) end up on the outside of the plasma membrane.

Answer 2

The original context of the textbook is almost surely being used in a non-mathematical sense. But since the question is looking for some mathematical way of looking at it, I will suggestion one.

Given a point cloud of the locations of each membrane's atoms, or alternatively of meshes approximately something like a space-filling model, you could induce a topological space using a metric. Considering the unions of $\epsilon$-close atoms will only be as interpretable as your choice of $\epsilon$ is meaningful.

Just considering such unions will inform you more about the induced connected components at some distance more than anything else. If you are interested identifying holes (like the hierarchy of compartments found in cells), you will need to go further. Given a similar input as above, you can compute can compute a sequence of simplicial complexes of a chosen type (e.g. Čech or Vietoris-Rips) induced by an interval of $\epsilon \in [a,b]$. From these simplicial complexes the persistent Betti numbers can be estimated. For more information see An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists.

A potential application can be found in (rather computationally expensive) simulations. If you simulate the positions of atoms in a membrane being disrupted by a force, you could track their persistent homology. This approach could also be applied to simulations of protein folding to track whether transient pockets form. Sufficiently small systems have to be chosen in order for such calculations to be feasible.