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An Optimization Approach to Protein Structure Prediction

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### An Optimization Approach to Protein Structure Prediction

Richard Byrd

Betty Eskow

Robert Schnabel

Brett Bader

Lianjun Jiang

University of Colorado

Teresa Head-Gordon

Univ. of California, Berkeley

Silvia Crivelli

Lawrence Berkeley Laboratory

Problem Definition

Predict the 3-dimensional shape, or

native state, of a protein given its

sequence of constituent amino acids.

Approach

Assuming the native state of a protein corresponds to its minimumfree energy state, use a global optimization method to find the minimum energy configuration of the target protein.

Importance of Protein Folding

- 3-Dimensional structure useful in molecular drug design.

- Laboratory experiments are expensive:
- X-ray crystallography
- NMR
- Genome projects are providing sequences for many proteins whose structure will need to be determined.

Protein Structures

Proteins consist of a long chain of

amino acids called the primary structure.

Gly

Leu

Ser

Pro

The constituent amino acids may encourage hydrogen bonding and form regular structures, called secondary structures.

a-helix

b-sheet

The secondary structures fold

together to form a compact

3-dimensional or tertiary structure.

H

H

R

H

O

H

O

N

N

N

N

O

H

O

H

H

R

H

R

H

R

H

R

H

O

H

O

N

N

N

N

O

O

H

H

H

R

H

R

Chemistry of ProteinsSide chain

Amino acid

Backbone

H-bond

Hydrogen bonds strongly influence a protein’s shape. They largely occur in secondary structures and help hold the protein together.

Computational Approaches to Protein Structure Prediction

- Comparative Modeling
- Compares and aligns to a known protein sequence of amino acids
- Fold Recognition
- Searches for the best fitting fold template from a library of known protein folds
- New Fold Methods
- Not based on knowledge of complete protein sequences or folds
- e.g. energy minimization

- Very large parameter space

e.g., modestly sized protein

- 100-300 amino acids
- ~ 1,600 atoms
- ~ 4,800 variables
- Model of the energy surface may not match nature

The 3-dimensional structure of the protein found in nature is

believed to minimize potential energy:

Min V(x)

where x = atom coordinates

Challenges:

Amber Energy Function

V(x) =

S

cl(b-b0)2

(b = bond length)

bonds

(q = bond angle)

S

ca(q-q0)2

+

bond angles

+

S

cd[1 + cos(n +)]

(w = dihedral angle)

dihedral angles

+

S

(rij = distance)

charged pairs

S

+

cwj(rij)

(j = Lennard-Jones potential)

nonbonded pairs

Internalcoordinates are determined using bonds, bond angles

and dihedral angles.

Internal coordinates are determined using bonds, bond angles and dihedralangles

Additional energy terms to model protein behavior in an aqueous environment

- Formulated from simulations of pairs of hydrophobic molecules in water
- ESOLVATION =
- Advantages of this model:
- Provides stabilizing force for forming hydrophobic cores.
- Well defined model of the hydrophobic effect of small hydrophobic groups in water.
- Computationally tractable and differentiable

i,j are aliphatic carbons, M Gaussians with position(ck ), depth(hk) and width(wk) describe 2 minima: (1) molecules in contact and (2)mol-ecules separated by a distance of 1 water molecule.

Global Optimization Approaches

- Deterministic methods
- Branch and bound, interval methods
- Very reliable, deterministic guarantees
- Too expensive for more than 20-50 variables
- Stochastic methods
- Random steps or sampling
- Probabilistic guarantees
- Practical for < 300 variables
- Heuristic search
- e.g. Simulated annealing, Tabu search, Genetic algorithms
- Effective on some very large problems
- No practical guarantees

A Stochastic-Perturbation Global Optimization Approach

- Generate and maintain a pool of candidates (configurations), as in genetic algorithms.
- Solve the full-dimensional problem as a series of small-dimensional ones.
- Use protein database information to bias toward likely substructures.

Algorithm Phases

Given the amino acid sequence of a

protein, find the 3-dimensional

structure likely to be found in nature.

Simplify problem by utilizing domain-specific knowledge

Generate

Initial

Population

Global

Optimization

Phase 1

Phase 2

Phase 1: Create Initial Population

- Submit amino acid sequence to server:
- EFIAIYDYKAETEEDLTIKKGEKLEIIEKEGDWWKAKAIGSGEIGY
- IPANYIAAAE
- Use server predictions to determine the location of α-helices, β-strands, and coils :
- CCCCHHHHHHEEEEEEEEEEEECCEEEEEEEEEEEHHHHHHHHCCC
- HHHHHHCCCC
- Use ProteinShop visualization tool to form configurations with secondary structure:
- Assign ideal values to the dihedral angles in the sequence according to the predictions. Manipulate β-strands to form β-sheets.
- Perform Energy Minimizations

Phase 2:Improve Local Minima

Select a protein

and a subset of

dihedral angles

- Uses a combination of breadth-first and depth-first searches from initial pool
- Dihedral angles act as “internal coordinates” and reduce the number of variables, speeding an optimization run

Small-scale global

optimization

Full-dimensional

local optimization

iterate

Cluster minima and test stopping criteria

Small Scale Global Optimization in Phase 2

- Minimize energy over 5-20 torsion angles’
- Use a stochastic global optimization algorithm base on sampling, sample pruning and local minimization (Rinooy-Kan et al).
- From best start points, do local minimizations using quasi-Newton

Full-scale local minimizations

- Using best points from small-scale global, do local minimizations.
- Because of problem size we use limited-memory quasi-Newton.
- Best local minimizers are added to pool.

Biasing functions

- Used to form secondary structure during in first phase and sometimes infull-dimensional local minimizations.
- Dihedral angle biasing:

E= dihedrals kf[1 – cos(f - f0)] + k[1 – cos( - 0)]

- Hydrogen Bond biasing
- For -helices:

EHB=wiwi+4 / Dri,i+4 (w’s are weights from the server for residues i and i+4 in the helix)

- To form -sheets from -strands:

EHB= wiwj / Dri,j

Neural Network Predictions

Sequence:

SKIGIDGFGRIGRLVLRAALSCGAQ

Neural nets trained on a large database of proteins can predict secondary structure likely to be in a target protein.

Sequence:

Type:

Weight:

SKIGIDGFGRIGRLVLRAALSCGAQ

BBBB B AAAAAAA BBBBB

13552 6789992 56673

Forming β-sheets from the predicted -strands is a combinatorial problem.

Which strands are paired?

?

?

?

Which orientation?

anti-parallel

parallel

Which residues are paired?

even

odd

Distribution of Beta Sheets in Proteins with Applications to Structure Prediction

Ruckzinski, Kooperberg, Bonneau, and Baker

Proteins 48, 2002

ParallelOrganization

- Select k subsets of dihedral angles
- Maintain a queue of (configuration,subspace) for k optimization crews to work on
- Each optimization crew performs a small-scaleglobal optimization of its assigned configuration and subspace.
- Gather intermediate results and re-insert them into the work queue. Idle optimization crews do full-dimensional local minimizations oradditionalsmall-scale global optimization.

Massively parallel exploration of optimization space

- Automatic load balancing

2UTG_A: 7.5Å R.M.S.D. from Crystal

1POU: 6.3Å R.M.S.D. from NMR structure

CASP competition

- Community-wide experiment on the Critical Assessment of Techniques for Protein Structure Prediction
- Protein crystallographers and NMR spectroscopists provide structures prior to their publication for blind prediction by participants.
- Biannual competition open to all computationalmethods – including servers.
- Difficulty of targets assessed by which type of methods work to predict the structure – CM, FR, NF.
- We participated in CASP4 (Dec. 2000) and CASP5 (Dec. 2002).

Our submitted CASP4 models ranked by target difficulty and relative accuracy

Results on Phospholipase C beta C-terminus, turkey (containing 242 amino acids). Ribbon structure comparison between experiment (center), submitted M1 prediction (right), our lowest energy submission, had an RMSD with experiment of 8.46Å, and next generation run of the global optimization algorithm (left). This new run lowered the energy of our previous best minimizer, resulting in a new structure with an RMSD of 7.7Å.

CASP4 Results Summary

Best structure predicted on one of the hardest targets

Our method is more effective than some knowledge-based methods on targets for which less information from known proteins is available.

Global optimization algorithm is very effective at improving structures from a small initial population.

Our submitted CASP5 models ranked by target difficulty and relative accuracy

Our submitted CASP5 models of targets (domains) that were assessed in the CASP5 NEW FOLD category.

CASP5 Results Summary

- Ranked ~15/165 groups in assessments of New Fold (and NF/FR) Results.
- Our method uses less knowledge from known protein structures than most other (New Fold) methods participating in CASP5
- More diverse starting populations (especially for -sheet proteins) using the visualization tool led to better performance in some cases.

Future Research Directions

- Simpler energy models for early stages of the algorithm, and alternative models of solvation.
- New techniques for choosing -strand pairings.
- Improve our techniques for maintaining existing secondary structure in our models.

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