Part I: Paper
d: Protein Folding
Erik Demaine, MIT
Stefan Langerman, U. Bruxelles
Joseph O’Rourke, Smith College
Outline
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Interlocked Chains
Fixed-angle chains
Producible chains
Flattenable
Proof Outline
Consequence?
Definitions
 Open vs. closed chains. (Closed chains are
more constrained.)
 Flexible chains: no constraints on joint
motion (each joint universal).
 Rigid chains: each joint is frozen, and the
entire chain is rigid.
 Fixed-angle chains: maintain angle between
links incident to each joint.
Crosstable of results
Rigid 2-chains cannot interlock
Flexible 2-chain can interlock with
rigid 5-chain
Open Problem
What is the smallest value of k that permits
a flexible 2-chain to interlock with a
flexible k-chain? Theorem 10.1.2 shows
that a rigid 5-chain suffices; presumably k
> 5 is needed for a flexible chain.
Demaine, Langermann, JOR:
Main Theorem
Theorem 1:
A fixed angle polygonal (≤)-chain is
-producible (  ≤ 90º ),
if and only
if it is flattenable.
Consequence
Theorem 2:
The -producible configurations of chains
are rare:
The probability that a random configuration
of a random chain is -producible
approaches 0 as n∞.
Protein
Folding
Main Theorem
Theorem 1:
A fixed angle polygonal (≤)-chain is
-producible (  ≤ 90º ),
if and only
if it is flattenable.
Fixed-angle chain
(≤)-chain
Locked 3D Chains
[Cantarella & Johnston
1998; Biedl, Demaine, Demaine, Lazard, Lubiw, O’Rourke,
Overmars, Robbins, Streinu, Toussaint, Whitesides 1999]
Cannot straighten some chains,
even with universal joints.
Ribosome
http://www.biochimie.univ-montp2.fr/maitrise/ribosome/50s_letunnel.htm
“The majority of the
surface of the tunnel is
trained by field I (yellow)
and V (red) of 23S and by
the nonglobular areas of the
proteins L4, L22 and L39e.
Incipient polypeptide first
meets field V then field II
and IV with the proteins L4
and L22. Half of the tunnel
is constituted by field I and
III and the L39e protein.”
Ribosome (closeup)
“The 2 proteins, L22 and
L4 (in dark blue) form
what appears to be an
open door. This crossing
point could be the place
where the nature of
incipient polypeptide is
detected and from which
information would be
transmitted to the
surface of ribosome,
perhaps through proteins
L22 and L4.”
Constraint: Cone
Main Theorem
Theorem 1:
A fixed angle polygonal (≤)-chain is
-producible (  ≤ 90º ),
if and only
if it is flattenable.
-production
Lemma 1
An (≤)-chain can be produced only in a cone
with (whole) apex angle of ≥ .
B: Emergence cone
-chain
Canonical Configuration
Lemma 2.
If a configuration of a chain is producible, then it can be moved inside the
cone to a canonical coiled configuration,
the -CCC.
-CCC
Proof figure
Proof Idea
 Replay production movements in time
reversal, coiling the chain inside the cone.
Main Theorem
Theorem 1:
A fixed angle polygonal (≤)-chain is
-producible (  ≤ 90º ),
if and only
if it is flattenable.
Flattenable
A configuration of a chain if flattenable if it
can be reconfigured, without selfintersection, so that it lies flat in a plane.
Otherwise the configuration is
unflattenable, or locked.
Every 90º-angle chain has a
flattenable configuration.
Unflattenable chain
Main Theorem (revisited)
Theorem 1:
All -producible (≤)-chains are flattenable,
provided  ≤ 90º.
All flat configurations of (≤)-chains are producible, for  ≤ 90º.
Logical Flow of Ideas
 -producible 
-CCC canonical configuration
 flattened → -CCC
 -producible  flattenable
 flattenable → not locked
 locked → abundant
 not locked → rare
 rare → search easier?
Consequence (revisited)
Theorem 2:
The -producible configurations of chains
are rare:
The probability that a random configuration
of a random chain is -producible
approaches 0 as n∞.
Configuration Space
All configurations
Flattenable
configurations
Why restriction to  ≤ 90º ?
Protein Sidechains
Tunnel Exit
“Localization of
proteins at the exit
of the tunnel.”
Open Problems:
Locked Equilateral Chains?
(1) Is there a configuration of a chain with
universal joints, all of whose links have
the same length, that is locked?
Perhaps: No?
(2) Is there a configuration of a 90o fixedangle chain, all of whose links have the
same length, that is locked?
Perhaps: Yes for 1+e?
Ribosome structure
“The figure at bottom
represents the interactions
allowing pairing codonanticodon. The elements of
contact are marked (A) with
(c). The anticodon of ARNt
is in dark blue and the codon
of ARNm in the site P is in
red.”
http://www.biochimie.univ-montp2.fr/maitrise/ribosome/sommaire.htm