Introduction

• What are proteins ?

• can be conceptualised as
  strings of amino-acids.

• there are 23 amino-acids
  that are proteinogenic

Introduction

• H

• P

Introduction

• Protein Folding

Introduction

• Protein Folding

• So in this model, folding of a protein
  sequence is defined as a self-avoiding walk
   in a 3D lattice.

• The HP model abstracts the hydrophobic
  interaction in protein folding by labeling
  the amino acids as:

• hydrophobic (H)

• hydrophilic (P)

Proof

The main result is that HP STRING-FOLD is NP-complete when G is Z3. The proof follows by showing that the following folding problem is NP-hard.

• HP STRING-FOLD

Instance: A finite sequence S over the alphabet {H, P},
an integer m, and a graph G.

Question: Is there a fold of S in G where the number of H-H bonds is at least m?

Proof

The proof that PERFECT HP STRING-FOLD is NP-hard involves a transformation from the (strongly) NP-complete problem of BIN PACKING ...

• PERFECT HP STRING-FOLD

Instance: An integer n and a finite sequence S over the alphabet {H, P} which contains n^3 H's.

Question: Is there a fold of S in Z3 for which the H's are perfectly packed into an n x n x n cube?

Proof

To simplify matters, the paper uses a variation of BIN PACKING which is more easily shown to be strongly NP-complete.

• BIN PACKING

Instance: A finite set U of items, a size s(u) ∈ Z+ for each u ∈ U, a positive integer bin capacity B, and a positive integer K. 

Question: Is there a partition of U into disjoint sets U1, U2, ... ,Uk such that the sum of the sizes of the items in each U¡ is B or less? 

Proof

• MODIFIED BIN PACKING

Instance: A finite set U of items, a size s(u) that is a positive even integer for each u ∈ U, a positive integer bin capacity B, and a positive integer K, where ∑ u ∈ U s(u) = BK. 

Question: Is there a partition of U into disjoint sets U1, U2, ... ,Uk such that the sum of the sizes of the items in each U¡ is precisely B ?

Proof

• INTUITION

• if the H's in a String  S are to be perfectly packing into a cub, then any H in S which is adjacent to a P in S must be packed on to the surface of the cube. 

• This is because any H which is mapped to an internal node of the cube must have only H's for neighbours, because adjacent items in S must be mapped to adjacent locations in the cube. 

Proof

• REDUCTION

• Consider any MODIFIED BIN PACKING problem β. Let :

q = 2 max K, [ β^(1/3) ] )

T = (2q - 1)(n - 2)

n = q (2q + 1) + 2 

Proof

• REDUCTION

• Arrows indicate the order in which the first nine edges are filled:

Proof

• REDUCTION

• ​The back-face is filled with
  ( n - 2 )^(2) / 2  P H H P's

Proof

• REDUCTION

• ​The n x n x n cube with paths to indicate the order in which the left, bottom, right, and top faces are filled:

Proof

• REDUCTION

• The front face of the cube partitioned into q "bins." The top and bottom edges of the face have previously been filled in.

Proof

• Any single H surrounded by P's in S mapped to β  (eg. P H P) must be in a node that is on the edge of the cube.

• Any H that is adjacent to a P in S mapped to β  (eg. H P) must be embedded on a face (including its edges) of the cube.

• If a pair of H's are separated by two P's S mapped to β  (eg. H P P H), then the H's are embedded in adjacent positions on a face of the cube.