Download Introduction to Molecular Dynamics Simulations

Introduction to Molecular
Dynamics Simulations
Roland H. Stote
Institut de Chimie LC3-UMR 7177
Université Louis Pasteur
Strasbourg France
1EA5
Title
Native Acetylcholinesterase (E.C. 3.1.1.7) From Torpedo Californica At 1.8A Resolution
Classification
Cholinesterase
Compound Mol_Id: 1; Molecule: Acetylcholinesterase; Chain: A; Ec: 3.1.1.7
Exp. Method
X-ray Diffraction
1
Macromolecules in motion
• Local motions
–
–
–
–
(0.01 à 5 Å, 10-15 à 10-1 s)
Atomic Fluctuations
Sidechain motions
Loop motions
• Rigid body motions
–
–
–
–
(1 à 10 Å, 10-9 à 1 s)
Helix motions
Domain motions
Subunit motions
• Large scale motions
–
–
–
–
(> 5 Å, 10-7 à 104 s)
helix-coil Transitions
Dissociation/Association
Folding and unfolding
• Biological function requires flexibility (dynamics)
Energy Minimization
"E a # b # "E b # c # "E c ### "E MIN $ 0 # fin
!
c
b
2
Central idea of Molecular
Dynamics simulations
• Biological activity is the result of time dependent interactions
between molecules and these interactions occur at the
interfaces such as protein-protein, protein-NA, protein-ligand.
• Macroscopic observables (laboratory) are related to microscopic
behavior (atomic level).
• Time dependent (and independent) microscopic behavior of a
molecule can be calculated by molecular dynamics simulations.
Molecular Dynamics Simulations
• One of the principal tools for modeling proteins, nucleic acids and
their complexes.
• Stability of proteins
• Folding of proteins
• Molecular recognition by:proteins, DNA, RNA, lipids, hormones
STP, etc.
• Enzyme reactions
• Rational design of biologically active molecules (drug design)
• Small and large-scale conformational changes.
• determination and construction of 3D structures (homology, Xray diffraction, NMR)
• Dynamic processes such as ion transport in biological systems.
3
Molecular dynamics simulations
• Approximate the interactions in the system using simplified
models (fast calculations). Include in the model only those
features that are necessary to describe the system.
• In the case of molecular dynamics simulations, this means a
potential energy function that models the basic interactions.
• Allows one to gain insight into situations that are impossible to
study experimentally
• Run computer experiments. Ask the question
« What if…? »
• The method allows the prediction of the static and dynamic
properties of molecules directly from the underling interactions
between the molecules.
Classical Dynamics
• Newton’s Equations of motion
dvi
d 2ri
Fi = mi ! ai = mi !
= m! 2
dt
dt
Fi = !"i E
• Position, speed and acceleration are functions of time
ri(t); vi(t); ai(t)
• The force is related to the acceleration and, in turn, to the
potential energy
• Integration of the equations of motion =>
initial
structure : ri(t=0); initial distribution of velocities: vi(t=0)
4
Dynamics: calculating trajectories
• Trajectory: positions as function of time: ri (t)
• How does one determine ri (t) from Fi = mi ai ?
dvi
d 2ri
Fi = mi ! ai = mi !
= m! 2
dt
dt
• Simple case where acceleration is constant
a=
dv
dt
v = at + v0
Simple case:
motion of a particle in one dimension
• Acceleration:
• If a is constant a≠f(t)
a=
dv
dt
v(t) = at + v0
• Speed:
v(t) =
• Position:
• The trajectory x(t) obtained
by integration taking into
account the initial positions
and velocities (x0 et v0)
dx(t)
dt
t2
x(t) = v ! t + x0 = a ! + v0t + x0
2
5
Balistic trajectory
Initial conditions are
x(0) = z(0) = 0
vx (0) = vo cos
vz (0) = vo sin
Z
V0
In the x direction
ax = 0
vx (t) = vo cos
x(t) = vo cos t
X
In the z direction, one has to take into account gravity az = g
vz (t) = vo sin - gt
z(t)= vo sin t – g t2 /2
z = ax -b x2 : the trajectory in the (x,z) plane is parabolic
Potential Energy
E(R) =
1
2
1
2
Kb ( b " b0 ) + ! K# (# " # 0 ) + ! K$ (1 + cos(n$ " % ))
1, 2 pairs 2
angles 2
dihedrals
!
4
8
.( + 1 2 ( + 6 1
6
'
'
q
q
6
0
ij
ij
3
i
j
9
+ ! 54& ij 0** -- " ** -- 3 +
r
r
&
Dr
6:
i, j 6
0
)
ij
,
)
ij
,
3
ij
/
2
7
• The energy is a function of the positions ri
• Therefore the acceleration is a function of the positions
• Since the positions vary as a function of time ri(t), so
does the acceleration, ai(t)
6
Numerical Integration
• Taylor series development
t2
t3
'
x(t) = x0 + v0 t + a0 + a 0 + O(t 4 )
2
3!
• If we know x at time t, after passage of a certain time, Δt, we
can find x(t+Δt)
F(t) !t 2 F ' (t) !t 3
x(t + !t) = x(t) + v(t)!t +
+
+ O(!t 4 )
m 2
m 3!
• We restart from the coordinates x(t+Δt) to get x(t+2Δt)
• To pass from x(t) to x(t+Δt) is to carry out 1 step of dynamics
• The change in velocity v(t) to v(t+Δt) can be calculated in the
same manner
• The acceleration is recalculate from E(r) at each step
Acceleration as a function of time
• Acceleration: calculated from the force, that is, from the
derivative of the potential energy, including at t=0
1 dE(RN )
ai (t) = !
m dri (t)
• Potential Energy
E(RN ) =
!
1
1,2 pairs 2
2
Kb (b " b0 ) +
!
1
2
K# (# " # 0 ) +
angles 2
! K$ (1 + cos(n$ " % ))
dihedrals
4
.( ' + 12 ( ' + 6 1 q q 8
6
ij
ij
i j 6
9
+ ! 54& ij 0** -- " ** -- 3 +
0) rij ,
) rij , 32 &Drij 6:
i, j 6
/
7
7
Principle of the trajectory
t0+4 Δt
t0+2Δt
t0+Δt
t0+7Δt
t0
Integration algorithms
Verlet, Velocity Verlet
LeapFrog, Beeman
•Choice of the algorithm:
–Energy conservation
–Calculation time (least expensive)
–Integration time step as large as possible
8
Trajectory of a macromolecule
• Initial positions x0
PDB file
• Xray
• NMR
• Model
• Initial velocities v0
Coupled to the temperature
3
m v2
NkT = ! i i
2
2
i
• Acceleration
Calculated from the force, that is,
from the derivative of the potential
energy.
a=!
1 dE
m dr
Relationship between velocities
and temperature
• Temperature specifies the thermodynamic state of the system
• Important concept in dynamics simulations.
• Temperature is related to the microscopic description of
simulations through the kinetic energy
• Kinetic energy is calculated from the atomic velocities.
3
m v2
NkT = ! i i
2
2
i
9
Molecular Dynamics Simulation programs
AMBER
CHARMM
NAMD
POLY-MD
etc
Potential energy function
parameter files contain the numerical constants needed to
evaluate forces and energies
http://www.pharmacy.umaryland.edu/faculty/amackere/research.html
10
Molecular Dynamics
Calculation of forces
Displacement
t=Δt
New set of coordinates
Practical Aspects
• Choice of integration timestep Δt
> As long as possible compatible with a correct numerical integration
> 1 to 2 fs (10-15 s)
• Calculating nonbonded Interactions: consumes the most
CPU time
> The cost (CPU) is proportional to N2 (N number of atoms)
> Truncation
0
4
*# & 1 2 # & 6 2
"
"
q
q
2
,
ij
ij
/
i
j
14! ij ,%% (( ) %% (( / +
5
7
r
r
!
r
i, j 2
$
'
$
'
ij
ij 2
+ ij
.
3
6
11
Nonbonded Energy Terms
Electrostatic Forces
r
+
-
van der Waals
1 Forces
E(R) =
!
r
1, 2 pairs
r
+
2
2
Kb ( b " b0 ) +
+
1
! 2 K (# " # )
2
+
0
#
angles
!
K$ (1 + cos(n$ " % ))
dihedrals
4
8
.( + 1 2 ( + 6 1
6
'
'
q
q
6
0
ij
ij
3
i
j
9
+ ! 54& ij 0** -- " ** -- 3 +
r
r
&
Dr
i, j 6
0/) ij ,
) ij , 32
ij 6
7
:
Truncation
•
Switch
Bring the potential to zero between
ron and roff. The potential is not
modified for r < ron and equals zero
for r > roff
•
Shift
Modify the potential over the entire
range of distances in order to bring
the potential to zero for r > rcut
•
Long-range electrostatic
interactions
Ewald summation
Multipole methods (Extended
electrostatics model)
12
Treatment of solvent
•
Implicit: The macromolecule
interacts only with itself, but the
electrostatic interactions are
modified to account for the
solvent
E elec (r ) = A
•
qi q j
!r
All solvent effects are contained in
the dielectric constant ε
Vacuum ε =1
Proteins ε = 2-20
Water ε = 80
Treatment of solvent
• Explicit representation
The macromolecule is surrounded by
solvent molecules (water, ions) with
which the macromolecule interacts.
Specific nonbond interactions are
calculated
0
4
*# & 1 2 # & 6 2
" ij (
" ij ( / qi q j 2
,
%
%
14! ij ,% ( ) % ( / +
5
7
r
r
r
26
i, j 2
$
'
$
'
ij
ij
ij
+
.
3
• In this case, one must use ε =1.
• More correct (fewer approximations)
but more expensive
13
Periodic boundary conditions
• For explicit representation of
solvent
• The boundaries of the
system must be represented
• For periodic system
Permits the modeling of very
large systems, but introduces
a level of periodicity not
present in nature.
Boundary Conditions
Solvation sphere: finite system
Around the entire macromolecule
Around the active site
14
Some properties that can be calculated from a
trajectory
•
Average Energie moyenne
•
RMS between 2 structures
(ex : initial structure)
•
Fluctuations of atomic des positions
•
Temperature Fators
•
Radius of gyration
Copyright " www.ch.embnet.org/MD_tutorial"
Reproduction ULP Strasbourg. Autorisation CFC - Paris
Protocol for an MD simulation
• Initial Coordinates
– X-ray diffraction or NMR coordinates from the Protein Data Bank
– Coordinates constructed by modeling (homology)
• Treatment of non-bonded interactions
– Choice of truncation
• Treatment of solvent
– implicit: choice of dielectric constant
– Implicit: advanced treatment of solvent: Generalized Born, ACE, EEF1
– explicit: solvation protocol
• If using explicit treatment of solvent ->boundary condition
–
–
–
–
Periodic boundary conditions (PBC)
Solvation sphere
Active site dynamics
Time step for integration of equations of motion
15
Steps of a molecular dynamics
simulation
An application of Molecular
Dynamics Simulations
The acetylcholinesterase story
16
Acetylcholinesterase
• Acetylcholinesterase (AChE) is an enzyme that hydrolyzes
ACh to acetate and choline to inactivate the
neurotransmitter
• A very fast enzyme, approaching diffusion controlled.
• Inhibitors are utilized in the treatment of various
neurological diseases, including Alzheimer’s disease.
• Organophosphorus compounds serve as potent insecticides
by selectively inhibiting insect AChE.
Neuromuscular junction: motor neurons : muscle cells
17
1EA5
Title
Native Acetylcholinesterase (E.C. 3.1.1.7) From Torpedo Californica At 1.8A Resolution
Classification
Cholinesterase
Compound Mol_Id: 1; Molecule: Acetylcholinesterase; Chain: A; Ec: 3.1.1.7
Exp. Method
X-ray Diffraction
18
Access of
ligands to the
active site is
blocked -->
requires
fluctuations
Secondary channels open transiently: Identified by MD simulations
Molecular Dynamics Simulation of
Acetylcholinesterase
•
•
•
•
•
•
•
•
10 ns simulations
Protein obtained from the Protein Data Bank (PDB)
Structure solved by x-ray crystallography
Solvated in a cubic box of water
Ions added to neutralize the system
Periodic Boundary Conditions
Treatment of Long-Range electrostatic interactions
Total of 8289 solute atoms and 75615 solvent atoms
• Biophysical Journal Volume 81 715-724 (2001)
• Acc. Chem. Research 35 332-340 (2002)
19
Molecular Dynamics Simulation of
Acetylcholinesterase
20
Effect of the His44Ala mutation on the Nucleocapsid
protein from the HIV virus - NC(35-50)
Working at the interface of theory and experiment
21
Primary function of NC is to bind nucleic acids
The life cycle of the HIV-1 retrovirus and the multiple roles of the nucleocapsid protein
NC
NC
NC
NC
22
Structural determinants for the specificity of NC for DNA
The structure of the mutant His44Ala:NC(35-50):an NMR, MM and FL study
NMR and Fluorescence studies demonstrate
•
•
•
Mutant protein binds zinc.
Mutant protein maintains some structure
Binding to nucleic acids is less strong.
Biochemistry (2004) Stote RH et al, 43,7687-7697
•Two-dimensional 1H NMR
•pH 6.5 at 274K
E. Kellenberger and B. Kieffer, ESBS
Answer the questions left unanswered by experiment
•How does mutant protein bind zinc ion?
•If folded, why is the activity diminished?
Can simulations can predict the structural effects of point mutations?
23
From NMR
From MD
angular S
rmsd (Å)
1.2
0.8
0.7
1
0.6
0.8
0.5
0.6
Ensemble of structures from MD
0.4
0.3
0.4
0.2
0.2
0.1
0
35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
residue number
Biochemistry (2004) Stote RH et al, 43,7687-7697
Structural Chemical Shifts : Δδ
Shifts Ösapay & Case, J. Am. Chem. Soc. 113
Free
H
Complex
1991
H
• Structural Chemical Shift (Δδ)
–
Δδ(Η) = δ(Η)complex - δ(Η)Random Coil
• Semi-empirical model for the calculation of Δδ
Δδ
–
–
–
divided into different contributions
Magnetic anisotropy
Ring Current
Electrostatics
24
Difference between calculated and experimental Δδ
Δ !" (ppm)
1
G35
C36
W37 K38
C39
G40
K41
E42
G43
A44
Q45
M46 K47
D48
C49
T50
0.5
0
-0.5
-1
-1.5
-2
Zinc binding by the mutant protein
Reorientation of mainchain carbonyl oxygens stabilizes the ion zinc.
In more unfolded protein, water molecules move in to form hydrogen bonds
25
Study of the DNA/NC complex. Free energy decomposition.
LYS 47
MET 46
TRP 37
Decomposition of the binding free energy by amino acid for the native protein
Amino acids that contribute significantly to DNA binding are those most affected by the
mutation
26
Conclusions
Since molecules are dynamic, experimental structures alone can not give the
entire picture.
An interdisciplinary approach is required.
Molecular simulations are a necessary complement to the experimental
studies.
Computer Simulation of Liquids
Edition New ed
Allen, M. P., Tildesley, D. J.
Computational Chemistry
Grant, Guy H., Richards, W. Graham
Molecular Modelling: Principles and Applications
(2nd Edition) (Paperback)
by Andrew Leach
http://www.ch.embnet.org/MD_tutorial/
27
Acknowledgements
•
•
Hervé Muller
Elyette Martin
•
•
•
•
•
•
Prof. Bruno Kieffer (ESBS/IGBMC, Illkirch)
Dr. Esther Kellenberger (ULP, Illkirch)
Marc-Olivier Sercki (ESBS, Illkirch)
Prof. Yves Mély (ULP, Illkirch)
Dr. Elisa Bombarda (ULP, Illkirch)
Prof. Bernard Roques (INSERM/CNRS, Paris)
28