Normal mode analysis (NMA) tutorial and lecture notes

by K. Hinsen

Serkan Apaydın

Protein flexibility

Frequency spectrum of a proteinOver half of the 3800 known protein movements can be

modelled by displacing the studied structure using at most two low-frequency normal modes. Gerstein et al. 2002

Outline

• NMA– What it is– Vibrational dynamics– Brownian modes– Coarse grained models– Essential dynamics

Harmonic approximation

Conformation (r)

Energy (U)

0

Rmin

Harmonic approximation

U(r) =0.5 (r − Rmin)’ · K(Rmin) · (r − Rmin)

0

U

rRmin

NMA

U(r) =0.5 (r − Rmin)’ · K(Rmin) · (r − Rmin)

NMA

Normal mode direction 1

U(r) =0.5 (r − Rmin)’ · K(Rmin) · (r − Rmin)

NMA

-e2

Normal mode direction 2

U(r) =0.5 (r − Rmin)’ · K(Rmin) · (r − Rmin)

NMA (2)

O(n3)

U(r) =0.5 (r − Rmin)’ · K(Rmin) · (r − Rmin)

min

min

Properties of NMA

• The eigenvalues describe the energetic cost of displacing the system by one length unit along the eigenvectors.

• For a given amount of energy, the molecule can move more along the low frequency normal modes

• The first six eigenvalues are 0, corresponding to rigid body movements of the protein

4 ways of doing NMA

A. Using minimization to obtain starting conformation, and computing the Hessian K:

1. Vibrational NMA2. Brownian NMA

B. Given starting structure:1. Coarse grained models

C. Given set of conformations corresponding to the motion of the molecule:

1. Essential Dynamics

1. Vibrational NMA

•derived from standard all-atom potentials by energy minimization•time scale: < residence time in a minimum•appropriate for studying fast motions•Useful when comparing to spectroscopic measurements•Requires minimization and Hessian computation

1. Vibrational NMA

Vibrational frequency spectrum

2. Brownian NMA

•derived from standard all-atom potentials by energy minimization•time scale: > residence time in a minimum•appropriate for studying slow motions•Requires minimization and Hessian computation

2. Brownian NMA

The friction coefficients

• describe energy barriers between conformational substates

• Can be obtained from MD trajectories (<xi

2>)

• Depend on local atomic density (not a solvent effect)

http://dirac.cnrs-orleans.fr/plone/Members/hinsen/

3. Coarse grained models

•Around a given structure•time scale: >> residence time in a minimum•appropriate for studying slow, diffusive motions (jump between local minima)•Does not require expensive minimization and Hessian computation

3. Coarse grained models (2)

• Capture collective motions

– Specific to a protein

– Usually related to its function

– Largest amplitudes• Atoms are point

masses• Springs between

nearby points

Coarse grained models (3)

f can be a step function or may have an exponential dependence.

Elastic network model NMA (aka ANM)

Find Hessian of V, then eigendecomposition

Gaussian network model

•Or a step function…

Coarse grained models (4)All atom or C-alpha based models…

Equilibrium fluctuations

Ribonuclease T1

Gaussian network model: Theory and applications. Rader et al. (2006)

Disulphide bond facilitator A (DsbA)

Difference between ENM NMA and GNM

• GNM more accurate in prediction of mean-square displacements

• GNM does not provide the normal mode directions

Lower resolution models

• Groups of residues clustered into :

• unified sites

• Rigid blocks (rotation and translation of blocks (RTB) model)

• To examine larger biomolecular assemblies

G Li, Q Cui - Biophysical Journal, 2002

4. Essential dynamics

•Given a set of structures that reflect the flexibility of the molecule•Find the coordinates that contribute significantly to the fluctuations•time scale: >> residence time in a minimum

Essential Dynamics(2)

Angel E. García, Kevin Y. Sanbonmatsu Proteins. 2001 Feb 15;42(3):345-54.

<r> = R<(r − R) (r − R)'> =kBT inv(K)

Essential dynamics(3)

• Cannot capture the fine level intricacies of the motion

• Freezing the small dofs make small energy barriers insurmountable

• Need to run MD for a long time in order to obtain sufficient samples

38, 150, 199 dofs

Applications of normal modes

• Use all modes or a large subset– Analytical representation of a potential well

– Limitations: • approximate nature of the harmonic approximation

• Choice of a subset

• Properties of individual modes– Must avoid overinterpretation of the data

• E.g., discussing differences of modes equal in energy

• No more meaningful than discussing differences between motion in an arbitrarily chosen Cartesian coord. system

Applications of normal modes (2)

• Explaining which modes/frequencies are involved in a particular domain’s motion

• Answered using projection methods:– Normal modes form a basis of the config. space of the

protein

– Given displacement d, pi = d · ei

• Contribution of mode i to the motion under consideration

– Cumulative contribution of modes to displacement

NkpCk

iik 3..1,

6

2

Cumulative projections of transmembrane helices in Ca-

ATPase

Comparison chart

NGivenLongLargeEssential

YGivenLongLargeCoarse grained

Y/NBy Minimization

LongLargeBrownian

NBy Minimization

ShortSmallVibrational

PracticalStarting structure

Time scale

Amplitude

Summary

NMA:• no sampling problem• computational efficiency, especially for

coarse-grained models

• simplicity in application

• Predicts experimental quantities related to flexibility, such as B-factors, well.

•http://igs-server.cnrs-mrs.fr/elnemo/ (all atom)

WebNM: (C-alpha based) http://www.bioinfo.no/tools/normalmodes

http://promode.socs.waseda.ac.jp/pages/jsp/index.jsp

(all-atom)

Ignm (C-alpha based): http://ignm.ccbb.pitt.edu/

http://molmovdb.org/nma/ (C-alpha based)

http://lorentz.immstr.pasteur.fr/nomad-ref.php (all atomic or just C-alpha)

Protein Flexibility Predictions Using Graph Theory

Jacobs, Rader, Kuhn and ThorpeProteins: Structure, function and genetics 44:150-165 (2001)

Serkan Apaydın

Characterizing intrinsic flexibility and rigidity within a protein

1. Compares different conformational states

Limited by the diversity of the conformational states

Characterizing intrinsic flexibility and rigidity within a protein

2. Simulates molecular motion using MD

Limited by the computational time

Characterizing intrinsic flexibility and rigidity within a protein

3. Identifies rigid protein domains or flexible hinge joints based on a single conformation

Can provide a starting point for more efficient MD or MCS

Outline

• The main idea: constraint counting• Brute force algorithm• Rigidity theory • Pebble game analysis• Rigid cluster decomposition• Flexibility Index• Examples

Overview of FIRST

• Floppy Inclusion and Rigid Substructure Topography

• Given constraints:– Covalent bonds– hydrogen bonds– Salt bridges

• Evaluate mechanical properties of the protein:

Find regions that are:– rigid– move collectively– move independently of other

regionsCompute a relative degree of

flexibility for each region

Rigidity in Networks – a history

• 1788: Lagrange introduces constraints on the motions of mechanical systems

• 1864: Maxwell determined whether structures are stable or deformableapplications in engineering,

such as the stability of truss configurations in bridges

• 1970: Laman’s theorem: determines the degrees of freedom within 2D networks and allow rigid and flexible regions to be foundextended to bond-bending

networks in 3D

http://unabridged.m-w.com

Brute force algorithm to test rigidity

ORACLE

INDEPENDENT

REDUNDANT

Brute force algorithm to test rigidity

ORACLE

INDEPENDENT

REDUNDANT

•Compute normal modes w/ and w/o the constraint

•If the number of zero eigenvalues remains constant, then the constraint is redundant.

Complexity? O(n2 .n3)

O(n5)

Laman’s theorem accelerates constraint counting

• Constraint counting to all the subgraphs– Applying directly, complexity is O(exp(n))– Applying recursively, pebble game algorithm.

Complexity is O(n2), O(n) in practice.

Pebble Game

•3 pebbles per node

•Each edge must be covered by a pebble if it is independent

•Pebbles remaining with nodes are free and represent DOFs of the system

•An edge once covered should stay covered but pebbles can be rearranged.

The Pebble Game: A Demonstration

Mykyta Chubynsky and M. F. Thorpe

Arizona State University

Pebble game

Flexible hinges

Hyperstatic

Pebble game

Final arrangement of pebbles

Blue: Free pebble, one DOF

Red: Associated with an edge, a “used” DOF by the constraint

This arrangement determines the flexible regions and rigid clusters

In 2D, 2 pebbles / node.

Finding rigid clusters

•A rigid cluster can have a maximum of 3 pebbles in 2D

•Rearrange the pebbles to obtain > 3 pebbles in a connected region

Finding rigid clustersThis is not a rigid cluster since there are 4 pebbles here

Hydrogen bonds•Selection of a cut-off energy for hydrogen bonds

•Selected based on agreement of hydrogen bonds within a family of protein structures

Hydrogen bond energy computation

d<= 3.6 Å

r <=2.6 Å

90 <= <= 180

sp3 donor-sp3 acceptor F=cos2 cos2(-109.5)

V0 = 8 kcal/mol d0 = 2.8 Å

Flexibility Index

– #(independent DOFs)/#(rotatable bonds)

– #(redundant constraints)/#(distance constraints)

•4-3 =1 DOF

•3 rotatable bonds

•F = 1/3•1 redundant

constraint

•6 distance constraints

•F = -1/6

Application to HIV protease (unbound)

Agreement with experiment

Comparison of the open (L) and and closed (R) structures of HIV

protease

Dihydrofolate reductase

Rigid cluster decomposition

barnase

Maltodextrin binding protein

Gohlke and Thorpe. Biophysical Journal 91:2115-2120 (2006)

FIRST/FRODA predictions

barnase

Maltodextrin binding protein

Gohlke and Thorpe. Biophysical Journal 91:2115-2120 (2006)

Rigid cluster NMA (RCNMA)

• Protein decomposed into rigid

clusters• Better than ad-hoc definition of blocks

• Rotation-Translation Block Analysis

for the resulting network

• 9-27 times less memory

• 25-125 times faster

Comparison of RCNMA w/ ENM

Barnase

r2 0.56 vs. 0.50

Maltodextrin binding protein

r2 0.62 vs. 0.55

Gohlke and Thorpe. Biophysical Journal 91:2115-2120 (2006)

Comparison of FIRST and NMA

All frequencies

YY (N for VNMA, Brownian, ED)

Y / N (N for coarse grained)

NMA

Low frequency motion

YYYFIRST

Freq. spectrum?

Given starting pt.

Speed?All-atom?

Comparison of FIRST and NMA (2)

YYNMA

Y*Y (with ROCK or FRODA)

FIRST

Flexibility/mobility index

Way of generating new conformations?

*: incorrect for rigid regions flanked by flexible hinges

http://flexweb.asu.edu

Conclusion

• Rigidity theory • Constraint counting• Based on a single structure• Fast• Available on the web: http://flexweb.asu.

edu• Tools using FIRST to generate new

conformations: ROCK, FRODA