# Part I: Paper d: Protein Folding

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Part I: Paper d: Protein Folding
Erik Demaine, MIT Stefan Langerman, U. Bruxelles Joseph O’Rourke, Smith College

Outline 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 shows that a rigid 5-chain suffices; presumably k > 5 is needed for a flexible chain.

Demaine, Langermann, JOR: Main Theorem
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∞.

Chymotrypsin: unfolded & folded

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 “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 self-intersection, 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)
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?
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 1+e?

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.”

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