Motion prediction:
Molecular Dynamics (MD)
and Normal Mode Analysis
(NMA)
Lecture 10
Structural Bioinformatics
Dr. Avraham Samson
81-871
1
Normal mode analysis
Basic theory of normal modes
• Normal modes analysis (NMA) is a technique for
calculating long time dynamics of biomolecules.
• In contrast to molecular dynamics which solves
the exact equation of motion approximately, NMA
solves the approximate equation of motion
exactly.
• Even with its limitations, such as the neglect of
the solvent effect and the use of harmonic
approximation of the potential energy function,
NMA have provided much useful insight into
protein dynamics.
3
Basic theory of normal modes (I)
F = ma (Newton’s second law)
F = -Cx (Hooke’s law)
-Cx = m d2x/dt2
The solution is a periodic
function:
x(t) = A cos (ωt + δ)
Where ω = √C/m
C
-x
m
0
x
V=½Cx2
For an harmonic oscillator with a mass, m, and a spring constant, C,
there is one normal mode with frequency ω= √C/m
4
Basic theory of normal modes (II)
m dx12/dt2 = -C x1 + C(x2-x1)
m dx22/dt2 = -C x2 + C(x1-X2)
x1(t) = A1 cos (ωt +δ)
x2(t) = A2 cos (ωt +δ)
-ω2 m A1 cos
-ω2 m A2 cos
(ωt + δ) = -2C A1 cos (ωt +δ) + C A2 cos (ωt +δ)
(ωt + δ) =
C A1 cos (ωt +δ) – 2C A2 cos (ωt +δ)
( ω2 m -2 C) A1 + C A2 =0
C A1 + ( ω2 m -2 C) A2 =0
[ω2m C-
2C
C
ω2m - 2C
ω1 = sqrt(C/m)
ω2 = sqrt(3C/m)
][ AA12]=0
C m C m C
1
[ XX12(t)
]
=
A
[
(t)
1]cos (ωt +δ)
1
1
[ XX12(t)
]
=
A
[
(t)
-1]cos (ωt +δ)
2
For two masses, m, and a spring constant, C, there are two normal
modes. One when they move in the same direction [1,1], and another
when they move in opposite directions [1,-1].
5
Basic theory of normal modes (III)
ma = F
TAω2=VA
For proteins, there are
many normal modes.
On average:
30 per residue for (x,y,z)
4 per residue for (φ,ψ,χ)
6
Normal mode analysis tools
• ElNemo
http://igs-server.cnrs-mrs.fr/elnemo
• Nomad Ref
http://lorentz.immstr.pasteur.fr/nma/submissi
on.php
http://lorentz.immstr.pasteur.fr/gromacs/nma
_submission.php
• STAND (by appointment only)
7
Example 1. Normal Mode Dynamics
of the Acetylcholine Receptor
Samson A. & Levitt M. (2008) Biochemistry 47:4065-70
Normal mode of acetylcholine receptor
•The receptor displays an twist like motion, responsible for
the axially symmetric opening and closing of the ion channel
9
10
Opening of ion channel
•The ion channel diameter increases to allow influx of Na+ and K+ ions.
•The ion channel of unbound free receptor opens to a larger extent than that of the
toxin bound receptor.
11
Dynamics of AChR are
inhibited by α-Bungarotoxin
•Twist like motion is not observed in the modes of the toxin bound receptor.
•The toxins seem to lock together adjacent α and δ, and α and γ subunits
•One toxin is sufficient to prevent the opening motion.
12
Example 2. Normal Mode Dynamics
of Prion Proteins
Samson A. & Levitt M. (2011) Biochemistry 50:2243-8
Prion proteins
• Prion proteins (PrP) are the proteinaceous infectious particle in
neurodegenerative TSE (i.e. mad cow disease)
• In its native state the PrP is a cell surface receptor
• Conversion of cellular PrP (PrPC) into scrapie PrP (PrPSc) and
aggregation into cytotoxic oligomers leads to a loss of neuronal
cells and brain spongiosis with debilitating, dementing and fatal
consequences.
• CD spectroscopy showed a substantial difference in the
secondary structure content of PrPC and PrPSc (Pan et al.1993).
β-sheet
α-helix
Turn
Coil
PrPC
3%
42%
32%
23%
PrPSc
43%
30%
11%
16%
14
Normal mode of M129 prion (WT)
(PDB ID 3HES)
_________________________________________________________________
.
.
.
.
.
SEQ 125
LGGYMLGSAMSRPIIHFGSDYEDRYYRENMHRYPNQVYYRPMDEYSNQNN 174
Native
EE
HHHHHHHHHHGGG
EE GGGGG HHH
Distorted
EE
HHHHHHHHHHGGG
EE GGGGG HHH
.
.
.
.
.
SEQ 175
FVHDCVNITIKQHTVTTTTKGENFTETDVKMMERVVEQMCITQYERESQA
Native
HHHHHHHHHHHHHHHHHHH
HHHHHHHHHHHHHHHHHHHHHHHHH
Distorted HHHHHHHHHHHHHHHHHHH
HHHHHHHHHHHHHHHHHHHHHHHH
224
15
Normal mode of V129 prion (variant)
(PDB ID 3HAK)
_________________________________________________________________
.
.
.
.
.
SEQ 125
LGGYVLGSAMSRPIIHFGSDYEDRYYRENMHRYPNQVYYRPMDEYSNQNN 174
Native
EE
HHHHHHHHHHGGG
EE GGGGG HHH
Distorted
EEEEE
HHHHHHHGGG EEEEE GGGGG HHH
.
.
.
.
.
SEQ 175
FVHDCVNITIKQHTVTTTTKGENFTETDVKMMERVVEQMCITQYERESQA
Native
HHHHHHHHHHHHHHHHHHH
HHHHHHHHHHHHHHHHHHHHHHHHH
Distorted HHHHHHHHHHHHHHHHHHH
HHHHHHHHHHHHHHHHHHHHHHHH
224
16
Trajectory path using NMA
http://lorentz.immstr.pasteur.fr/joel/index.php
Coiled coils of hemagglutinin!
17
Molecular dynamics
Molecular dynamics
• Why simulate motion of molecules?
• Energy model of conformation
• Two main approaches:
– Monte Carlo - stochastic
– Molecular dynamics - deterministic
• Applications of MD
• Speeding up MD
19
Why simulate motion?
•
•
•
•
•
•
Predict structure
Understand interactions
Understand properties
Learn about normal modes of vibration
Design of bio-nano materials
Experiment on what cannot be studied
experimentally
• Obtain a movie of the interacting molecules
20
Aquaporin simulation
2004 Winner of Visualization Challenge in Science and Engineering,
Organized by the National Science Foundation and Science Magazine.
21
Aquaporin simulation
Structure, Dynamics, and Function of
Aquaporins
http://www.ks.uiuc.edu/Research/aquaporins/
22
Method of Molecular Dynamics
1. Use physics to find the potential energy
between all pairs of atoms.
2. Move atoms to the next state.
3. Repeat.
HOW???
23
Kinetics
Potential Energy
•


dvi
Newton:  Fij  mi
dt
j

ij
Fij  
rˆij
rij
j
Fij
Force
Mass
Acceleration
Unit vector from i
to j
i
(t)
vi
24
Kinetics
•
•


dvi
Newton:  Fij  mi
dt
j

ij
Fij  
rˆij
rij
Updated Position:




1
2 F (t )
ri (t  t )  ri (t )  t vi (t )  t  i
2
mi
j
Fij
(t + t)
i
(t)
new
position
i
vi
old force
old
position
old
velocity
25
Kinetics
•
•
•


dvi
Newton:  Fij  mi
dt
j

ij
Fij  
rˆij
rij
vi
Updated Position:




1
2 F (t )
ri (t  t )  ri (t )  t vi (t )  t  i
2
mi
Updated Velocity:

  t  
1
F (t )
vi  t    vi (t )  t  i
2
2
mi

first half of time step
Fij
j
Fij
i
(t + t)
i
(t)
vi

  t  1
F (t  t )
vi (t  t )  vi  t    t  i
2 2
mi

second half of time step
26
Kinetics
•
•
•
•


dvi
Newton:  Fij  mi
dt
j

ij
Fij  
rˆij
rij
vi
Updated Position:




1
2 F (t )
ri (t  t )  ri (t )  t vi (t )  t  i
2
mi
Fij

  t  
1
F (t )
vi  t    vi (t )  t  i
2
2
mi

i
(t + t)
i
(t)
Updated Velocity:
Fij
j
vi

  t  1
F (t  t )
vi (t  t )  vi  t    t  i
2 2
mi

System Temperature:
N
1 2 3
mvi  NkBT

2
i 1 2
system kinetic
energy
1
T
3NkB
equipartition
theorem
N
m v
i 1
2
i i
27
Intermolecular Potentials
•
•
Gas
Dense
Gases,
Electric dipole ~ r
Interact
2 dipoles ~ r and
esvia elastic collisions• Liquids
“Hard Sphere” Potential
• Repulsive nuclear forces ~ r
• Solids
Total Potential = (Attractive) +
No long range interactions •
-3
-6
-12
(Repulsive)
“Lennard-Jones 6-12 Potential”
Carey, V.P., Statistical Thermodynamics and Microscale Thermophysics, New York: Cambridge
University Press, 1999.
28
Energy Function
• Target function that MD tries to optimize
• Describes the interaction energies of all
atoms and molecules in the system
• Always an approximation
– Closer to real physics --> more realistic, more
computation time (I.e. smaller time steps and
more interactions increase accuracy)
29
The energy model
• Proposed by Linus
Pauling in the 1930s
• Bond angles and lengths
are almost always the
same
• Energy model broken up
into two parts:
– Covalent terms
• Bond distances (1-2
interactions)
• Bond angles (1-3)
• Dihedral angles (1-4)
– Non-covalent terms
• Forces at a distance
between all non-bonded
atoms
http://cmm.cit.nih.gov/modeling/guide_documents/molecular_me
chanics_document.html
The NIH Guide to Molecular Modeling
30
The energy equation
Energy =
Stretching Energy +
Bending Energy +
Torsion Energy +
Non-Bonded Interaction Energy
These equations together with the data (parameters) required to
describe the behavior of different kinds of atoms and bonds, is
called a force-field.
31
Bond Stretching Energy
kb is the spring constant of the
bond.
r0 is the bond length at
equilibrium.
Unique kb and r0 assigned for each bond
pair, i.e. C-C, O-H
32
Bending Energy
k is the spring constant of the
bend.
0 is the bond length at
equilibrium.
Unique parameters for angle bending are
assigned to each bonded triplet of atoms
based on their types (e.g. C-C-C, C-O-C,
C-C-H, etc.)
33
The “Hookeian” potential
kb and k broaden or steepen the slope of the parabola.
The larger the value of k, the more energy is required to deform an
angle (or bond) from its equilibrium value.
34
Torsion Energy
A controls the amplitude of the
curve
The parameters are determined from
curve fitting.
n controls its periodicity
Unique parameters for torsional
rotation are assigned to each bonded
quartet of atoms based on their types
(e.g. C-C-C-C, C-O-C-N, H-C-C-H, etc.)
 shifts the entire curve along the
rotation angle axis ().
35
Torsion Energy parameters
A is the amplitude.
n reflects the type symmetry in the dihedral angle.
 used to synchronize the torsional potential to the initial rotameric
state of the molecule
36
Non-bonded Energy
A determines the degree the
attractiveness
A determines the degree the
attractiveness
B determines the degree of repulsion
B determines the degree of repulsion
q is the charge
q is the charge
37
Non-bonded Energy parameters
38
Simulating In A Solvent
• The smaller the system, the more particles on
the surface
–
–
1000 atom cubic crystal, 49% on surface
10^6 atom cubic crystal, 6% on surface
• Would like to simulate infinite bulk
surrounding N-particle system
• Two approaches:
– Implicitly
– Explicitly
• Ex. Periodic boundary
conditions
Schematic representation of periodic
boundary conditions.
http://www.ccl.net/cca/documents/molecul
ar-modeling/node9.html
40
Monte Carlo
Explore the energy surface by randomly probing
the configuration space
Metropolis method (avoids local minima):
1. Specify the initial atom coordinates.
2. Select atom i randomly and move it by random
displacement.
3. Calculate the change of potential energy, E corresponding
to this displacement.
4. If E < 0, accept the new coordinates and go to step 2.
5. Otherwise, if E  0, select a random R in the range [0,1]
and:
1. If e-E/kT < R accept and go to step 2
2. If e-E/kT  R reject and go to step 2
41
Properties of MC simulation
• System behaves as a Markov process
– Current state depends only on previous.
– Converges quickly.
• System is ergodic (any state can be reached
from any other state).
• Accepted more by physicists than by chemists.
– Not deterministic and does not offer time evolution of
the system.
42
Deterministic Approach
• Provides us with a trajectory of the
system.
• Typical simulations of small proteins
including surrounding solvent in the picoseconds.
E
Fi 
x i
F  ma
43
Deterministic / MD
methodology
• From atom positions, velocities, and
accelerations, calculate atom positions and
velocities at the next time step.
• Integrating these infinitesimal steps yields the
trajectory of the system for any desired time
range.
• There are efficient methods for integrating
these elementary steps with Verlet and
leapfrog algorithms being the most
commonly used.
44
MD algorithm
• Initialize system
{r(t+t), v(t+t)}
{r(t), v(t)}
– Ensure particles do not overlap in initial
positions (can use lattice)
– Randomly assign velocities.
• Move and integrate.
Leapfrog
algorithm
45