1. HP model for protein folding
K. Dill, Theory for the folding and stability of globular proteins.
Biochemistry, 24: , 1985
red - Hydrophobic ; blue - Polar (hydrophylic) molecules

2. Reasons why HP model can explain
some key features of protein folding

3. HP model “game”

7. red - Hydrophobic ; blue - Polar (hydrophylic) molecules
N=14 beads, 7 H-H contacts
R.Hayes, Prototeins, American Scientist, Volume 86 , 216

8. 21 beads - “folded” proteins (11 H-H contacts – maximum number) vs
“unfolded” proteins (0 or 1 H-H contacts – minimum number)

9. The aminoacid sequence for the protein
P H H H H P
a=[ ] n
2 configurations at length n
The folding sequence for the protein
1 (east), i (north), -1 (west), and -i (south).
f = [ ] (n-1 bonds)
f = [1 1 i 1 1] or
f(x+iy)= [ i 3+1i 4+1i]
n /32-1
(2.3683) * n configurations
at length n for large n
Lattice walks can be represented as a list of x,y coordinates, as compass directions or as left, right and forward commands. The last representation can also be encoded in a ternary (base 3) number

10. Self-avoiding walks
Brian Hayes, “How to Avoid Yourself,” American Scientist, July-August 1998,
Volume 86, p. 314

11. Combinatorial explosion
Berger B and Leighton T (1998) Protein folding in the hydrophobic-hydrophilic (HP) model is NP-complete. J Comput Biol 5:27-40
Combinatorial explosion
Number of HP model
configurations at length n+1
n n /32
2 * (2.3683) * n

12. Millennium Prize Problems
P versus NP problem
Hodge conjecture
Poincaré conjecture (solved by Grigori Pelerman in 2003)
Riemann hypothesis
Yang–Mills existence and mass gap
Navier–Stokes existence and smoothness
Birch and Swinnerton-Dyer conjecture
P versus NP problem is a major unsolved problem in computer science
P - decision problems solved on a deterministic sequential machine in an amount of time that is polynomial in the size of the input
NP -"- exponential time

13. NP-complete problems are a set of problems to each of which any other NP-problem can be reduced in polynomial time, and whose solution may still be verified in polynomial time. That is, any NP problem can be transformed into any of the NP-complete problems. Informally, an NP-complete problem is an NP problem that is at least as "tough" as any other problem in NP.

17. Pitfalls of HP model
Unrealistic feature – H ends are usually buried unlike real proteins
Reason: ends can form 3 HH contacts, instead middle H beads can form maximum 2 HH contacts

18. Unrealistic feature –
unnatural extended polar loops
Original HP model does not take
in account compactness

19. Folding process

20. During the folding process the scaning of many configurations can slow folding.

21. HPPPPH protein folding How to do it
c1 = [ ]
Apply folding at third bead
f1 = [1 1 i i i]
c2 =c1*f1=[ 1 1 i i i] or
C2(x+iy)= [ i 2+2i 2+3i]
Apply folding at fourth bead
f2 = [1 1 1 i i]
c3 = c2*f2 = [1 1 i ] or
C3(x+iy) =[ i 1+1i 1i]

22. Basic operations:
Bending
Flipping
c1 = [ ]
Apply bending at third bead
f1 = [1 1 i i i]
c2 =c1*f1=[ 1 1 i i i] or
C2(x+iy)= [ i 2+2i 2+3i]
c2 = [1 1 i i i]
Apply flipping at second and third bond
f2 = [1 i 1 i i]
c3 =c1*f2=[ 1 i 1 i i] or
C3(x+iy)= [0 1 1+i 2+i 2+2i 2+3i]

23.  Suboptimal folded configuration with E = - 3
Native configuration with E = - 4