6.5310: Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Spring 2025)

Prof. Erik Demaine TAs Josh Brunner, Jenny Diomidova

[+] Protein folding: Fixed-angle linkages, tree, and chains; span; flattening; flat-state connectivity, disconnectivity of orthogonal partially rigid trees, connectivity of orthogonal open chains; locked fixed-angle chains; producible protein (fixed-angle) chains, ribosome, β-producible chains, helix-like canonical configuration, flat states are producible, producible states are connected.
This lecture is about fixed-angle linkages in 3D, which have the constraint that the angles (in addition to the lengths) must remain fixed at all times. Fixed-angle linkages model the mechanics of chemical bonds between atoms in a molecule. In particular, the backbone of a biological protein is a fixed-angle tree in this model, and can be approximated by a fixed-angle chain. We'll cover several results about fixed-angle linkages, all motivated by questions about protein folding: Span: What's the farthest or nearest you can fold the endpoints of a fixed-angle chain? Flattening: When does a fixed-angle chain have a non-self-intersecting flat state? Flat-state connectivity: When can a fixed-angle chain be folded without self-intersection between every pair of flat states? Locked: When is a fixed-angle linkage locked? In particular, we'll show that all states of a fixed-angle chain producible by a simple model of the ribosome (which includes all flat states) can reach each other. A lot is known about each question, though many problems remain open.

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Enzyme, classified as “actin fold protein”, in every organism checked. Image released in the public domain. http://en.wikipedia.org/wiki/File:Hexokinase_ball_and_stick_model,_with_substrates_to_scale.png

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