Modeling Macromolecular Machines Using Rigid-Cluster Networks Moon - - PowerPoint PPT Presentation

Modeling Macromolecular Machines Using Rigid-Cluster Networks

Moon K. Kim, Robert Jernigan and Gregory S. Chirikjian Department of Mechanical Engineering The Johns Hopkins University

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Coarse-grained elastic network model

Cα N H H C OH O R H

animo group carboxyl group side chain

Cα N H H C H2O O R H Cα N H C OH O R H

peptide bond

• Only Cα atoms in a protein are treated as point masses and

spatially proximal points are assumed to be linked with linear springs. DOF : 90% ↓

3

Normal mode analysis (NMA)

• NMA is a conventional harmonic analysis to calculate

vibrational and thermal behaviors of macromolecules around a low energy equilibrium conformation.

• However, it is not able to predict large anharmonic

motions and pathways.

• A simplified harmonic potential is derived from a coarse-

grained elastic network model.

• The generalized eigenvalue problem results in

– Eigenvalues: the vibrational frequencies – Eigenvectors: the details of corresponding motions

4

NMA for GroEL-GroES complex

Twisting (mode 1) Sliding (mode 2) Squeezing (mode 6)

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Elastic network interpolation (ENI)

• Large anharmonic motions and pathways can be generated

by using the simplest cost function based on coarse- grained models.

• The key idea is to interpolate two values of the distances

between residues in both conformations.

• Unrealistic conformations and steric clashes do not happen.
• ENI pathway monotonically minimizes the root-mean-

square-deviation (RMSD) between the two end conformations.

• Realistic change of internal variables is observed during

the transition along the ENI pathway.

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ENI derivation

• Cost function is
• Desired distance is
• The union liking matrix is used to force the initial conformation to

move toward the final conformation.

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• The simplified cost function is
• The solution is obtained from
• should be reduced to an invertible matrix.

– Linear momentum conservation

• Intermediate conformations are iteratively calculated.

and

• RMS superposition is needed for smooth animation of conformational

transition.

Lactoferrin Transition from 1lfg.pdb to 1lfh.pdb

Lactate Dehydrogenase Transition from 1ldm.pdb to 6ldh.pdb

Citrate Synthase Transition from 4cts.pdb to 1cts.pdb

Conformational Transitions in Virus Capsid HK97 Primary (1FH6 to 1IF0)

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ENI for GroEL-GroES complex

• ENI represents an improvement over simplified linear interpolation in

terms of the realism of intermediate conformations, and over all atom based MD and NMA, in terms of computational efficiency.

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Rigid-cluster systems

• Most conformational changes in macromolecules can be

resolved into hinge and shear motions which are associated with the collective behavior of atoms.

• Some macromolecules could be represented by a set of

rigid-clusters.

xi(t) di,a(t) ith cluster

mi,a

• The center of mass of the ith cluster is

where

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Rigid-cluster ENI

• The position of residue a at time t is
• Assuming small rigid-body displacement,

where and

translational

• rientational
• The new ENI cost function is defined as

xi(t1) di,a(t1)

ith cluster

t0 t1 di,a(0) xi(0) vi(t1) ωi(t1) ∆t

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Lactoferrin

Ratio of DOF = =

20 40 60 80 100 1 2 3 4 5 6 7 RMS[Angstrom] Index of conformations Target Initial

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Virtual torsion angle

20 40 60 80 100

• 150
• 100
• 50

50 100 Index of conformations Torsion Angle[Degree] Thr90 Val250

• This monotonic angle changes indicate that the computed conformational

transition seems to be naturally favorable and energetically feasible.

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Conclusions (Part 2)

• Coarse-grained elastic network models are addressed as a new tool for

the study of biomolecular structure and dynamics.

• Only Cα atoms are treated as representatives of each residue of a

protein and the interaction between proximal residues is modeled with a linear spring.

• The ENI method is developed for the realistic simulation of

conformational transition between the two different conformations in macromolecules.

• This method is extended to rigid-cluster systems. The system modeling

is much simplified when using rigid clusters instead of Cα coarse graining.

• The rigid-cluster ENI method is computationally faster than the

conventional Cα ENI, still observing steric constraints.