Dukka

Application of Monte Carlo Simulation: Removing averaging artifacts in protein

structure prediction

• Background

• Protein Structure Prediction and CASP

• TASSER algorithm

• MCORE algorithm

Outline

• Experimental or computational method often output results as an ensemble of protein structures.

– NMR, Protein Structure Prediction, Protein Docking, RNA Structure Prediction

• A single representative structure is required to compare or do further analysis.

• Representative structure (consensus structure) = a centroid structure by averaging the Cartesian coordinates of the ensemble of superimposed structures.

• RMSD between the ‘averaged structure’ and any reference structure is always less than or equal to the average RMSD of the individual members. (Zagrovic et al.)

• However, the centroid structure has averaging artifacts rendering bond angles and bond lengths to be unphysical.

Background

• Critical Assessment of Structure prediction of Proteins (CASP) is a biannual contest where different groups try to predict structure of a protein whose structure is not released to the outside world.

• One of the most popular and objective contest in the bioinformatics field.

• CASP8 just over.

• Major observations from CASP7:– Methods are more or less ripe enough

– Consensus servers usually outperform individual servers

– A lot of work needed to be done in the refinement step

Protein Structure Prediction and CASP

• Given a set of conformations obtain a conformation that is closest to the native structure.

• Molecular force fields like AMBER, CHARMM can be utilized but as we know they are not perfect.

• Furthermore, still lack of perfect definition of “closest”. Hence, CASP coming up with new ideas of other measures to measure the closeness to the native like HB score and so on.

• Often, the ‘most closest prediction’ is not ranked top 1. Hence, ‘Refinement’ is getting a lot of attention.

Refinement

TASSER algorithm(Threading/ASSembly/Refinement)

Zhang & Skolnick, 2004

Centroid Structure

• TASSER is one of the best prediction server in both CASP7 and CASP8.

• A large number of conformations is generate after the assembly step. However, we can submit only a couple of models.

• Clustering is utilized and the centroid of the largest cluster (Combo model) is predicted as the output and has proven to be successful.

• Artifacts in ‘Tasser (combo) output’– Unrealistic bond lengths and bond angles due to averaging artifactsScope– To fix these unrealistic bond lengths and bond angles

Problem Identification

C-alpha Space

Energy Minimization!

Combo and Closc Models

COMBO model : The centroid structure of the most dense cluster.CLOSC model : The structure that is closest to the centroid of the most dense cluster.

Fra

ctio

n of

cla

shes

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• PULCHRA - based on steepest descent minimization and a simple force field.

• Sometimes, can not come out of the kinetic trap.

• Heavily distorted chain, the minimization procedure does not converge or the optimized model still exhibits irregularities.

PULCHRA

Rotkiewicz and Skolnick, 2008

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Generate an extended structure based on Combo model

Monte-Carlo Minimization

MCORE

Output the best structure

Start from a ‘close-by model’

Using the distance distribution from the PDB, mainly three types: x-Pro = 3.77, x-{ALA|ARG|ASN|LEU|LYS|MET} = 3.81, and x-{ASP|CYS|GLU|GLY|HIS|ILE|PHE|SER|THR|TRP|TYR|VAL} = 3.80

Generation of Extended Structure

• Two major components of any Monte-Carlo Approach

– Energy Function

• Can be generic force field or any combination of terms

– Move Sets

• Critical to the performance of the algorithm, more of an art(?)

– Convergency Criteria

• Naïve way (Run for certain number of steps)

• Introduce some criteria based on the generated conformations

Monte-Carlo

• Starting from a state A, make a change in the configuration to obtain a new (nearby) configuration B.

• Compute EB

• If (EB < EA), assume the new configuration, since it is a desirable thing.

• If (EB > EA), calculate the probability p

• Draw r from uniform distribution [0,1], if r < p then accept the new configuration B else reject the new configuration B.€

p = e−(EB −EA ) /T

Monte-Carlo: Metropolis Criteria

• Move Sets

– Global move-set

• Rest-all bead move

– Local move-set

• 1-bead move

• 2-bead move

• 3-bead move

• 4-bead move

• 5-bead move

– End-bond move

• 1,2,3-bead C-terminal end bond move

• 1,2,3-bead N-terminal end bond move

Move Sets

• Calculate the unit vector along axis defined by i-1 and i+1

• Calculate the rotation matrix around this vector

• Calculate the new position of i

• Important thing is to preserve the bond length i.e. to preserve the distance between consecutive C-alphas.

i-1

i

i+1

Move Sets

One bead move

Two bead move

Four-bead move

Three-bead move

Five-bead move

Rest-bead move

Axis of rotation

End-bond Move Sets

Axis of rotation

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kexcl (rkl − ro_ excl )2 + kang (ϕ i,i+1,i+2 −ϕ o _ ang )2 + kclosk=1

N −2∑

l=k+2

N∑

k=1

N −2∑ (dkkt

− do _ clo )2

k=1

N∑

otherwise same and150 if 150 70 if 70 2i1,ii,_oo ><= ++ andango ϕϕ

075.0=angk 4.5=closk001.0_ =clood0.4_ =exclor

9.2=exclk

Energy Function

Excluded volume Bond angle Closeness to target

Penalize if the distance is less than 4.0A

Penalize if the angle is not between 70 and 150

Penalize if the difference in C-alpha position between the target and starting structure is not with-in certain cutoff

N: Number of C-alpha atoms

• Before doing the actual computation, have to test whether the move sets and energy function is properly working or not.

• So, have to design some test cases. Positive test cases would be to drive extended structure to native structure.– Desired results:

should be able to drive ‘very close’ to extended structure to native structure in relatively short number of steps

Assessment of Move Sets and Energy Function

• 1363 proteins less than 200 residues and the combo RMSD to the native is lesser than 6.5 Å.

• 1363 Centroid structures (COMBO models)

• 1363 CLOSC models

• 1363 Close-by structures (CLOSC models + Pulchra Refinement)

• 1363 Native structures.

Data Set

Driving Extended to NativeA

vera

ge E

nerg

y

Ave

rage

RM

SD

to N

AT

IVE

(Å

)

Steps Steps

0.060.045

0.041

10000 steps RMSD = 0.039

Ext-refined Vs CA

Driving Extended to Native

0.033Å

i l

| rmsd_diff((i –l))| < Tolerance value, where l = i+j , j=1,…,L

Tried with different value of L and L=49 and Tolerance value = 0.005 seems reasonable.

Convergency criteria

• MCORE: Start from a ‘close-by model and drive it towards the COMBO model.

• CLOSC models as the close-by models.

– When close-by model is readily available

• MCORE-EXT: Start from an extended structure and drive it towards the COMBO model.

– When close-by model is not readily available

Propose two algorithms

MCORE: Driving Close-by models to COMBO A

vera

ge E

nerg

y

Steps

Why cannot go much closer to COMBO?F

ract

ion

of A

tom

s C

lash

ing

in C

OM

BO

Fra

ctio

n of

Ato

ms

Cla

shin

g in

M

CO

RE

RMSD of MCORE to COMBO (Å) RMSD of MCORE to COMBO (Å)

38 proteins had even lesser RMSD than the respective combo model

MCORE Vs Combo

RMSD of COMBO to NATIVE

RM

SD

of

MC

OR

E to

NA

TIV

E (Å

)

Comparison of Different Models

3.35 3.36 3.54 3.28

Fra

ctio

n of

Ato

ms

in C

lash

es

0.010 0.065 0.000 0.63

RM

SD

to N

ativ

e (Å

)

0.770 0.746 0.747 0.754

TM-score of four models

Avg. RMSD to Native (Å)

Avg clash < 1.9 Avg clash < 3.6

Combo 3.28 0.03 0.630

Closc 3.54 0 0.614

MCORE 3.35 0 0.010

Pulchra (Closc) 3.54 0 0

MCORE(EXT, 2000 steps)

3.35 0 0.011

Pulchra (Combo) 3.36 0.005 0.065

Results

Some Examples

1akhA refined Vs native 1akhA com Vs refined

1akhA com Vs native

12 clashes

0.354Å

0.68Å

0.78Å

1akhA pulchra Vs native

0.674Å

3bbn_ refined Vs Native3bbn_ pulchra Vs Native

3bbn_ comboVs Native3bbn_ refined Vs combo

2.918Å

2.948Å

3.099Å

0.852Å

• Built the main chain atoms of the refined Cα trace.

• Rebuilt side-chains using two methods

– Pulchra (-c)

– Scwrl

3.92(all) 2.868(cα) 3.95(all)

All-atom model reconstruction

• Designed an algorithm to remove averaging artifacts and applied it to refine combo model.

• Acknowledgments

– Dr. Jeff Skolnick and all the members of the Skolnick Lab, especially Lila, Shashi, Hongyi, Seung Yup,…….

– Dr. Dennis Livesay

• Future Works

– Refinement in All-atom space

Conclusion