Part I: Paper d: Protein Folding

1
Part I Paperd Protein Folding
Erik Demaine, MIT Stefan Langerman, U.
Bruxelles Joseph ORourke, Smith College
2
Outline

• Interlocked Chains
• Fixed-angle chains
• Producible chains
• Flattenable
• Proof Outline
• Consequence?

3
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.

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Crosstable of results
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Rigid 2-chains cannot interlock
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Flexible 2-chain can interlock with rigid 5-chain
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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 gt 5 is needed for
  a flexible chain.

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Demaine, Langermann, JORMain Theorem

• Theorem 1
• A fixed angle polygonal (?)-chain is
• ?-producible ( ? 90º ),
• if and only
• if it is flattenable.

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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?8.

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Chymotrypsin unfolded folded
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ProteinFolding
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Main Theorem

• Theorem 1
• A fixed angle polygonal (?)-chain is
• ?-producible ( ? 90º ),
• if and only
• if it is flattenable.

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Fixed-angle chain
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(?)-chain
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Locked 3D Chains Cantarella Johnston 1998
Biedl, Demaine, Demaine, Lazard, Lubiw, ORourke,
Overmars, Robbins, Streinu, Toussaint, Whitesides
1999

• Cannot straighten some chains,
• even with universal joints.

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Ribosome
http//www.biochimie.univ-montp2.fr/maitrise/ribos
ome/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.
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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.
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Constraint Cone
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Main Theorem

• Theorem 1
• A fixed angle polygonal (?)-chain is
• ?-producible ( ? 90º ),
• if and only
• if it is flattenable.

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?-production
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Lemma 1

• An (?)-chain can be produced only in a cone with
  (whole) apex angle of ?.

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B Emergence cone
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?-chain
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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.

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?-CCC
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Proof figure
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Proof Idea

• Replay production movements in time reversal,
  coiling the chain inside the cone.

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Main Theorem

• Theorem 1
• A fixed angle polygonal (?)-chain is
• ?-producible ( ? 90º ),
• if and only
• if it is flattenable.

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Flattenable

• A configuration of a chain if flattenable if it
  can be reconfigured, without self-intersection,
  so that it lies flat in a plane.
• Otherwise the configuration is unflattenable, or
  locked.

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Every 90º-angle chain has a flattenable
configuration.
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Unflattenable chain
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Main Theorem (revisited)

• Theorem 1
• All ?-producible (?)-chains are flattenable,
  provided ? 90º.
• All flat configurations of (?)-chains are
  ?-producible, for ? 90º.

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Logical Flow of Ideas

• ?-producible ?
• ?-CCC canonical configuration
• flattened ? ?-CCC
• ?-producible ? flattenable
• flattenable ? not locked
• locked ? abundant
• not locked ? rare
• rare ? search easier?

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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?8.

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Configuration Space
All configurations
Flattenable configurations
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Why restriction to ? 90º ?
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Protein Sidechains
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Tunnel Exit
Localization of proteins at the exit of the
tunnel.
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Open Problems Locked Equilateral Chains?

• Is there a configuration of a chain with
  universal joints, all of whose links have the
  same length, that is locked?
• Is there a configuration of a 90o fixed-angle
  chain, all of whose links have the same length,
  that is locked?

Perhaps No?
Perhaps Yes for 1e?
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Ribosome structure
The figure at bottom represents the interactions
allowing pairing codon-anticodon. 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/ribo
some/sommaire.htm