Instant Notes - Genus

Here are the notes on the topological properties of biomolecular structure.

Backgrounds

The biopolymers such as proteins, RNA, and DNA are composed of a finite set of basic building blocks linked with covalent bonds forming a sequential order. Their native states are often involved with intra-molecular non-covalent interactions except that some residues would form covalent disulfide bonds toward low-energy conformations. Not to mention those high energy states and interactions with partners and solvent molecules for now. People have come up with many ways to describe those interactions. The concept of “interaction” (i.e. graph) itself has sound mathematical formulations, and the genus is such a quantitative measurement. And we can apply it to analyze the topological properties of biomolecular structures.

Euler Characteristic

From graph duality, we know that for any convex polyhedron’s surface, we have:

\[\begin{aligned} E+2 &= V+F \\ 2 &= V-E+F \end{aligned}\]

And the Euler characteristic is formulated as:

\[\chi= V-E+F\]

for the surfaces of polyhedra, where

• \(V\): the numbers of vertices
• \(E\): the numbers of edges
• \(F\): the numbers of faces

For general surfaces, we can also calculate their Euler characteristic by deriving a polygonization of the surfaces. Besides, we can consider a closed orientable surface as a “convex” polyhedron surface but with holeswith one hole, such a surface is a torus.. The number of holes is called genus and denoted as \(g\). Thus:

\[\begin{aligned} \chi &= V-\underbrace{(E-n+n)}_{E'}+\underbrace{(F - 2g)}_{F'} \\ &=2-2g \end{aligned}\]

If such surfaces with \(r\) boundary components:

\[\chi = 2-2g-r\]

…

\[b − n = 2 − 2g − r\]

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Cited as:

@online{zhu2022instant-notes-on-genus,
        title={Instant Notes - Genus},
        author={Zefeng Zhu},
        year={2022},
        month={July},
        url={https://naturegeorge.github.io/blog/2022/07/genus/},
}