6.849: Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Spring 2017)

Prof. Erik Demaine; Martin Demaine; Dr. Jason Ku; TAs Adam Hesterberg & Jayson Lynch

[Home] [Problem Sets] [Project] [Lectures] [Coauthor] [Accessibility]

Lecture 19 Video     [previous] [next] [completion form]

[+] 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.

This class is the last one in which we solve problems. Specifically, we have a couple of problems related to protein folding in the HP model, and a few about interlocked linkages. In addition, I'll summarize the past and future of the class:

PAST: I'll review what we accomplished in all of our problem solving during the semester.

FUTURE: I'll present the 1-slide summaries of the presentations to come in the final two class sessions (C12 and C13).

Download Video: 360p, 720p

Handwritten notes, page 1/12 • [previous page] • [next page] • [PDF]

Handwritten notes, page 1/12 • [previous page] • [next page] • [PDF]

Slides, page 1/21 • [previous page] • [next page] • [PDF]

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

Slides, page 1/21 • [previous page] • [next page] • [PDF]

The video above should play if your web browser supports either modern Flash or HTML5 video with H.264 or WebM codec. The handwritten notes and slides should advance automatically. If you have any trouble with playback, email Erik.