Transcript Normal mode analysis (NMA) tutorial and lecture notes by K. Hinsen Serkan Apaydın

Normal mode analysis (NMA)
tutorial and lecture notes
by K. Hinsen
Serkan Apaydın
Protein flexibility
Frequency spectrum of a protein
Over 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
Energy (U)
0
Rmin
Conformation (r)
Harmonic approximation
U
0
Rmin
r
U(r) =0.5 (r − Rmin)’ · K(Rmin) · (r − Rmin)
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
Normal mode direction 2
-e2
U(r) =0.5 (r − Rmin)’ · K(Rmin) · (r − Rmin)
NMA (2)
U(r) =0.5 (r − Rmin)’ · K(Rmin) · (r − Rmin)
min
min
O(n3)
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.
2.
Vibrational NMA
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
(<xi2>)
• 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 
Coarse grained models (4)
All atom or C-alpha based models…
•Or a step function…
Equilibrium fluctuations
Ribonuclease T1
Disulphide
bond
facilitator A
(DsbA)
Gaussian network model: Theory and applications. Rader et al. (2006)
Difference between ENM NMA
and GNM
• GNM more accurate in prediction of meansquare displacements
• GNM does not provide the normal mode
directions
Lower resolution models
• Groups of residues clustered
into :
• unified sites
G Li, Q Cui - Biophysical Journal, 2002
• Rigid blocks (rotation and translation of blocks
(RTB) model)
• To examine larger biomolecular
assemblies
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)
<r> = R
<(r − R) (r − R)'> =kBT inv(K)
Angel E. García, Kevin Y. Sanbonmatsu Proteins. 2001 Feb 15;42(3):345-54.
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
38, 150, 199 dofs
obtain sufficient
samples
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
k
Ck   pi2 , k  1..3N
i 6
Cumulative projections of
transmembrane helices in CaATPase
Comparison chart
Amplitude Time Starting
scale structure
Vibrationa Small
Short By
l
Minimization
Brownian Large
Long By
Minimization
Coarse
Large
Long Given
grained
Essential Large
Long Given
Practical
N
Y/N
Y
N
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 Thorpe
Proteins: 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
regions
Compute a relative degree of
flexibility for each region
Rigidity in Networks – a history
•
•
introduces
constraints on the motions
of mechanical systems
1864: Maxwell determined
whether structures are stable
or deformable
1788: Lagrange
applications 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 found
extended to bond-bending
networks in 3D
http://unabridged.m-w.com
Brute force algorithm to test rigidity
INDEPENDENT
ORACLE
REDUNDANT
Brute force algorithm to test rigidity
INDEPENDENT
ORACLE
REDUNDANT
•Compute normal modes w/
and w/o the constraint
Complexity? O(n2 .n3)
O(n5)
•If the number of zero
eigenvalues remains
constant, then the
constraint is redundant.
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
In 2D, 2 pebbles / node.
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
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 clusters
This 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
r <=2.6 Å
90 <=  <= 180
d<= 3.6 Å
V0 = 8 kcal/mol d0 = 2.8 Å
sp3 donor-sp3 acceptor
F=cos2  cos2(-109.5)
Flexibility Index
•4-3 =1 DOF
•3 rotatable
bonds
•F = 1/3
•1 redundant
constraint
•6 distance
constraints
•F = -1/6
–
#(independent DOFs)/#(rotatable bonds)
–
#(redundant constraints)/#(distance constraints)
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
Allatom?
Speed?
Given
starting
pt.
Freq.
spectrum?
FIRST
Y
Y
Y
Low frequency
motion
NMA
Y / N (N Y (N for
for coarse VNMA,
grained) Brownian,
Y
All
frequencies
ED)
Comparison of FIRST and NMA
(2)
Way of generating
new conformations?
Flexibility/mobili
ty index
FIRST
Y (with ROCK
or FRODA)
Y*
NMA
Y
Y
*: 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