How can I model cellular membranes in terms of topology?

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? The questions and comments on this question seem to ignore this extrapolation.

• While an analysis in terms of formal topology is certainly possible, the word is here used more to mean "shape" and specifically where hydrophilic and hydrophobic residues are located relative to the surrounding medium. You have in essence two sets of residues (hydrophilic and hydrophobic) and one topological property (inside or outside). You might be overthinking this, depending on your objective. Apr 27 '20 at 6:27
• I would agree that you're over thinking this. As Buck indicated the only topological property is inside or outside, as for a circle.
– MaxW
Apr 27 '20 at 7:28