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Protein Folding in the 2D HP Model
Alexandros Skaliotis – King’s College London
Joint work with:
•Andreas Albrecht (University of Hertfordshire)
•Kathleen Steinhöfel (King’s College London)
Overview
1.
2.
3.
4.
5.
6.
7.
8.
9.
Proteins
Protein Folding
2D HP Model
Simple Example
Local Search for Protein Folding
Set of Moves
Logarithmic Cooling Schedule
Selected Benchmarks
Experiment
1. Proteins
• A protein is a sequence of amino acids encoded by a
gene in a genome.
• There are 20 different amino acids.
• The length of the sequence can range from about 20 to
3500.
• The function of a protein is determined by its threedimensional structure.
• Predicting this structure is quite daunting and very
expensive.
2. Protein Folding
• Protein Folding is the process by which a sequence of
amino acids conforms to a three-dimensional shape.
• Anfinsen’s hypothesis suggests that proteins fold to a
minimum energy state.
• So, our goal is to find a conformation with minimum
energy.
• We want to investigate algorithmic aspects of
simulating the folding process.
• We need to simplify it.
3.1 2D HP Model [Dill et al. 1985]
1. Classify each amino acid as hydrophobic (H) or
hydrophilic (P).
2. Confine consecutive amino acids to adjacent nodes in
a lattice (Treat search space as a grid).
3. Flatten the search on a 2D lattice.
•
Function HHc: Number of new HH contacts
•
Parameter ξ < 0: Influence ratio of the new HH
contacts (usually ξ = -1)
•
Objective Function = HHc * ξ = -HHc
3.2 2D HP Model [Dill et al. 1985]
• Protein Folding in the 2D HP Model is NP-Hard for a
variety of lattice structures [Paterson/Przytycka 1996;
Hart/Istrail 1997; Berger/Leighton 1998; Atkins/Hart
1999].
• Constant factor approximations in linear time but not
helpful for predictions of real protein sequences
[Hart/Istrail 1997].
• Exact methods work only for sequences up to double
digits length.
4. Simple Example
•
•
Normally the energy is a positive number
But we have a minimisation problem, so we talk about
negative energies
H = RED
P = PINK
Energy = 0
Energy = -3
5. Local Search for Protein Folding
• A wide range of heuristics have been applied to find
optimal HP structures, especially evolutionary
algorithms.
• Lesh et. Al (2003) and Blazewicz et al. (2005) applied
tabu search to the problem.
• We apply Logarithmic Simulated Annealing.
• To move in the search space we employ a complete
and reversible set of moves proposed by Lesh et al. in
2003 and Blazewicz et al. in 2005.
6. Set of Moves
1
L
2
C
3
L
4
1
2
L
L
5
6
3
L
4
5
6
7. Logarithmic Cooling Schedule
• Cooling Function:
g
c(k ) 
, k  0, 1,...
ln( k  2)
• Following Hajek’s theorem (1988), we are guaranteed
to find the optimal solution after an infinite number of
steps if and only if g  .
•  is the maximum value of the minimal escape heights
from local minima.
O( g )
• Albrecht et al. show that after (n /  )
transitions,
the probability to be in a minimum energy conformation
is at least 1   , where n is the maximum size of the
neighbourhood of sequences.
8. Selected Benchmarks
•
•
S36:
S60:
•
S64:
3P 2H 2P 2H 5P 7H 2P 2H 4P 2H 2P 1H 2P
2P 3H 1P 8H 3P 10H 1P 1H 3P 12H 4P 6H
1P 2H 1P 1H 1P
12H 1P 1H 1P 1H 2P 2H 2P 2H 2P 1H 2P
2H 2P 2H 2P 1H 2P 2H 2P 2H 2P 1H 1P
1H 1P 12H
9.1 Experiment
•
Estimate  experimentally.
20 runs
20 runs
20 runs
Sequence
g n
g  n /2
g  n /4
Time
Frame
S36
Optimal
Optimal
Optimal
10 min
S60
Optimal
Optimal
No
30 min
S64
Optimal
Optimal
No
90 min
Processor:
2.2 GHz AMD Athlon
9.2 Experiment
• We found that n / 2 is a good estimated upper
bound for .
• We checked this against S85 and got the best known
results in 10 / 10 runs.
• Of course we need more benchmarks.
• But this can be a good starting point in trying to
develop a formal proof for the value of .
