Part I: PaperPart I: Paperd: Protein Foldingd: Protein Folding

Erik Demaine, MITErik Demaine, MIT

Stefan Langerman, U. Stefan Langerman, U. BruxellesBruxelles

Joseph O’Rourke, Smith Joseph O’Rourke, Smith CollegeCollege

OutlineOutline

Interlocked ChainsFixed-angle chainsProducible chainsFlattenableProof OutlineConsequence?

DefinitionsDefinitions

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 resultsCrosstable of results

Rigid 2-chains cannot Rigid 2-chains cannot interlockinterlock

Flexible 2-chain can interlock Flexible 2-chain can interlock with rigid 5-chainwith rigid 5-chain

Open ProblemOpen 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:Demaine, Langermann, JOR:Main TheoremMain Theorem

Theorem 1: A fixed angle polygonal (≤)-chain is

-producible ( ≤ 90º ),

if and only if it is flattenable.

ConsequenceConsequence

Theorem 2: The -producible configurations of chainsare rare:

The probability that a random configuration of a random chain is -producible approaches 0 as n∞.

ProteinProteinFoldingFolding

Main TheoremMain Theorem

Theorem 1: A fixed angle polygonal (≤)-chain is

-producible ( ≤ 90º ),

if and only if it is flattenable.

Fixed-angle chainFixed-angle chain

(≤(≤)-chain)-chain

Locked 3D Chains Locked 3D Chains [Cantarella & [Cantarella & Johnston 1998; Johnston 1998; Biedl, Demaine, Demaine, Lazard, Lubiw, Biedl, Demaine, Demaine, Lazard, Lubiw, O’Rourke, Overmars, Robbins, Streinu, Toussaint, O’Rourke, Overmars, Robbins, Streinu, Toussaint, Whitesides 1999]Whitesides 1999]

Cannot straighten some chains, even with universal joints.

RibosomeRibosomehttp://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)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: ConeConstraint: Cone

Main TheoremMain Theorem

Theorem 1: A fixed angle polygonal (≤)-chain is

-producible ( ≤ 90º ),

if and only if it is flattenable.

-production-production

Lemma 1Lemma 1

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

B: Emergence coneB: Emergence cone

-chain-chain

Canonical ConfigurationCanonical 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-CCC

Proof figureProof figure

Proof IdeaProof Idea

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

Main TheoremMain Theorem

Theorem 1: A fixed angle polygonal (≤)-chain is

-producible ( ≤ 90º ),

if and only if it is flattenable.

FlattenableFlattenable

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 90Every 90ºº-angle chain has a -angle chain has a flattenable configuration.flattenable configuration.

Unflattenable chainUnflattenable chain

Main Theorem (revisited)Main Theorem (revisited)

Theorem 1:

All -producible (≤)-chains are flattenable, provided ≤ 90º.

All flat configurations of (≤)-chains are -producible, for ≤ 90º.

Logical Flow of IdeasLogical Flow of Ideas

-producible -CCC canonical configurationflattened → -CCC-producible flattenableflattenable → not lockedlocked → abundant

not locked → rarerare → search easier?

Consequence (revisited)Consequence (revisited)

Theorem 2: The -producible configurations of chainsare rare:

The probability that a random configuration of a random chain is -producible approaches 0 as n∞.

Configuration SpaceConfiguration Space

All configurations

Flattenable configurations

Why restriction to Why restriction to ≤ 90 ≤ 90º ?º ?

Protein SidechainsProtein Sidechains

Tunnel ExitTunnel Exit

“Localization of proteins at the exit of the tunnel.”

Open Problems: Open Problems: Locked Equilateral Locked Equilateral Chains?Chains?

(1)Is there a configuration of a chain with universal joints, all of whose links have the same length, that is locked?

(2)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+?

Ribosome structureRibosome 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/ribosome/sommaire.htm