Read the term differential geometry, and usually what comes to mind are black holes and Einstein’s field equations for the curvature of space-time.
It turns out, though, that differential geometry may also dictate the forms of organelles inside our cells.
The latest issue of Cell highlights a new study that has mapped the geometry of one of the cell’s key components: the endoplasmic reticulum (ER), which is, in a sense, the grand central station for all of the cell’s biochemical compounds.
According to Wallace F. Marshall, Associate Professor, Department of Biochemistry & Biophysics, University of California, San Francisco, the team of scientists headed by Mark Terasaki of University of Connecticut Health Center, explored the structure of the endoplasmic reticulum using a new imaging method, and were surprised by the results:
In professional secretory cells, the ER forms stacks of membrane cisternae (fluid sacs), apparently because this is an efficient way to pack a lot of ER membrane into a small volume inside the cell. Although these stacks have been seen for decades, it was never quite clear how the edges of the cisternae were connected together into one continuous lumen. Using a new method of serial section scanning electron microscopy, the authors discovered a novel arrangement of membranes forming the connectors, which turned out to resemble a mathematical object known as a Helicoid, discovered by Jean Baptiste Meusner in 1776.
A rough way to visualize this, is to see the endoplasmic reticulum as a parking garage, with the helicoid acting as the connection ramps between each floor.
The helicoid, Marshall adds, is well known in the classical literature of differential geometry. And its properties predicted something that turned out to be true for the ER.
Generally, in modeling the relation between physical forces and biological structures, “one ends up having to know, or at least estimate, the value of the various physical constants, like elastic moduli, rate constants, and so on.”
But Terasaki’s team showed that the form of the helicoidal shape in the ER does not depend on knowing any constants; the shape of the ‘ramps’ between membranes in the ER appears to be dictated purely by mathematics.
To be sure, this work examines just one structure of the cell’s many organelles, but it will be interesting to see if the serial section scanning method of electron microscopy uncovers more hidden mathematical structures elsewhere in the cell.
The ghost of Plato lives on.