Commenting on the work of Terasaki et al. revealing ER’s helicoidal structure, Wallace Marshall writes in the same issue of Cell a Leading Edge report suggesting this sort of study has been neglected in the ‘omics era. Highlighting the work of pioneering biomathematician D’Arcy Wentworth Thompson, he suggests the author’s field has been neglected with the discovery of the genetic code,
...and with it the desire to explain away questions of cellular structure by telling ourselves that geometry is encoded in the genome. Although the genome is not a blueprint that explicitly encodes shape, the genome does encode proteins that sculpt cellular structures, for example by dictating membrane curvature. The existence of such proteins goes against the concepts of D’Arcy Thompson and appeared to be a ﬁnal nail in the cofﬁn of his Pythagorean approach to cell biology.
On the differential equations that helped resolve the ER’s structure, Marshall notes:
The fact that the helicoid shape is predicted without having to know the value of any physical constants is one of the most beautiful and surprising results of this paper. It is not very shocking that physical properties of cellular components contribute to their shape... but usually when one talks about modeling the relation between physical forces and biological structures, one ends up having to know, or at least estimate, the value of various physical constants like elastic moduli, rate constants, and so on.
In this sense, it may be said that the shape of the ER connectors comes from mathematics rather than physics.
This simple mathematics emerges due to the helicoid’s minimal surface: i.e. average curvature of the sheets is zero (as they are saddle-shaped).
Whereas helicoids are not seen that often, another minimal surface, the gyroid, arises in a huge number of contexts. The gyroid contains helical twists similar to the helicoid but, whereas the helicoid is periodic along one axis, the gyroid is periodic in three perpendicular axes.
This gives it the abstract (cubic) forms pictured above in diagrams of gyroid formations observed in butterfly scales, which “divide the cubic structure into two regions forming two continuous interpenetrating networks in the system" (maximising the band gap of the scales, related to their interaction with light).
The scales are formed of chitin (a ubiquitous structural biomolecule found across life’s kingdoms), but this is deposited into gaps formed by folds in the plasma membrane separated by smooth ER tubules,
suggesting that the elastic properties of biological membranes may drive the formation of complex-looking minimal surfaces. Under conditions of stress or viral infection, ER can form periodic structures, some of which represent triply periodic minimal surfaces.
The fact that Terasaki et al. needed advanced microscopy methods to visualize their structure suggests that we might not see recognizable mathematical forms because we don’t yet know how to look for them.
Read Thompson’s 1917 masterpiece of biophysics, On Growth and Form (extract pictured above) in full at archive.org.