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

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 10}4 _s)

– helix-coil Transitions – Dissociation/Association – Folding and unfolding

• Biological function requires flexibility (dynamics)

Energy Minimization

c b

!

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, X-ray diffraction, NMR)

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

• Position, speed and acceleration are functions of time

r

_i

(t); v

_i

(t); a

_i

(t)

• The force is related to the acceleration and, in turn, to the

potential energy

• Integration of the equations of motion =>

initial

F

i

=

m

i

!

a

i

=

m

i

!

dv

i

dt

=

m

!

d

2

r

i

dt

2

Dynamics: calculating trajectories

• Trajectory: positions as function of time:

r

_i

(t)

• How does one determine

r

_i

(t)

from

F

_i

= m

_i

a

_i

?

• Simple case where acceleration is constant

a

=

dv

dt

v

=

at

+

v

0

F

i

=

m

i

!

a

i

=

m

i

!

dv

i

dt

=

m

!

d

2

r

i

dt

2

v

(

t

)

=

dx

(

t

)

dt

x(t)

=

v

!

t

+

x

₀

=

a

!

t

2

2

+

v

0

t

+

x

0

Simple case:

motion of a particle in one dimension

• Acceleration:

• If a is constant a≠f(t)

• Speed:

• Position:

• The trajectory x(t) obtained by integration taking into account the initial positions and velocities (x0et v0)

a

=

dv

dt

Z

X V0

Initial conditions are

x(0) = z(0) = 0

vx(0) = vo cos

vz (0) = vo sin

In the x direction

ax = 0

vx(t) = vo cos

x(t) = vo cos t

In the z direction, one has to take into account gravity az = g

vz (t) = vo sin - gt

z(t)= vo sin t – g t

2

/2

z = ax -b x2 : the trajectory in the (x,z) plane is parabolic

Balistic trajectory

E

(

R

)

=

1

2

1, 2pairs

!

K

_b

(

b

"

b

₀

)

2

+

1

2

angles

!

K

_#

(

# " #

0

)

2

+

dihedrals

!

K

_$

(

1

+

cos

(

n

$ " %

)

)

+

4

&

ij

'

_ij

r

_ij

(

)

*

*

+

,

-1 2

"

'

ij

r

_ij

(

)

*

*

+

,

-6

.

/

0

0

0

1

2

3

3

3

+

q

_i

q

_j

&

Dr

_ij

4

5

6

7

6

8

9

6

:

6

i,j

!

Potential Energy

• The energy is a function of the positions

r

_i

Numerical Integration

• Taylor series development

• If we know x at time t, after passage of a certain time, Δt, we can find x(t+Δt)

• 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

x

(

t

)

=

x

₀

+

v

₀

t

+

a

₀

t

2

2

+

0

'

a

t

3

3!

+

O

(

t

4

)

x

(

t

+

!

t

)

=

x

(

t

)

+

v

(

t

)

!

t

+

F

(

t

)

m

!

t

2

2

+

F

'

(

t

)

m

!

t

3

3!

+

O

(

!

t

4

)

Acceleration as a function of time

• Acceleration: calculated from the force, that is, from the

derivative of the potential energy, including at t=0

• Potential Energy

a

i

(

t

)

= !

1

m

dE

(

R

N

)

dr

i

(

t

)

E(R_N)= 1 2 1,2!pairs K_b(b"b₀)2+ 1 2 angles! K_#(# "#0)2+ dihedrals! K_$(1+cos(n$ " %)) + 4&ij 'ij r_ij ( ) * * + , -12 " 'ij r_ij ( ) * * + , -6 . / 0 0 1 2 3 3+ q_iq_j &Dr_ij 4 5 6 7 6 8 9 6 : 6 i,j !

Principle of the trajectory

t

₀

t

₀

+

Δ

t

t

₀

+2

Δ

t

t

₀

+4

Δ

t

t

0

+7

Δ

t

Integration algorithms

Verlet, Velocity Verlet

LeapFrog, Beeman

•Choice of the algorithm:

–Energy conservation

–Calculation time (least expensive) –Integration time step as large as possible

Trajectory of a macromolecule

• Initial positions x0 PDB file • Xray • NMR • Model • Initial velocities v0

Coupled to the temperature

• Acceleration

Calculated from the force, that is, from the derivative of the potential energy.

3

2

NkT

=

m

i

v

i 2

2

i

!

a

= !

1

m

dE

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

2

NkT

=

m

_i

v

_i2

2

i

!

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

Molecular Dynamics

Calculation of forces

Displacement

t=

Δ

t

New set of coordinates

Practical Aspects

• Choice of integration timestep

Δ

t

> 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

4

!

_ij

"

ij

r

_ij

#

$

%

%

&

'

(

(

1 2

)

"

ij

r

_ij

#

$

%

%

&

'

(

(

6

*

+

,

,

-.

/

/

+

q

_i

q

_j

!

r

_ij

0

1

2

3

2

4

5

2

6

2

i,j

7

Electrostatic Forces

+

-

+

+

van der Waals Forces

r

r

r

E

(

R

)

=

1

2

1, 2

!

pairs

K

_b

(

b

"

b

₀

)

2

+

1

2

angles

!

K

_#

(

# " #

₀

)

2

+

dihedrals

!

K

_$

(

1

+

cos

(

n

$ " %

)

)

+

4

&

ij

'

_ij

r

_ij

(

)

*

*

+

,

-1 2

"

'

ij

r

_ij

(

)

*

*

+

,

-6

.

/

0

0

0

1

2

3

3

3

+

q

_i

q

_j

&

Dr

_ij

4

5

6

7

6

8

9

6

:

6

i,j

!

Nonbonded Energy Terms

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)

Treatment of solvent

• Implicit: The macromolecule interacts only with itself, but the electrostatic interactions are modified to account for the solvent

• All solvent effects are contained in the dielectric constant ε

Vacuum ε =1 Proteins ε = 2-20 Water ε = 80

E

elec

( )

r

=

A

q

i

q

j

!

r

Treatment of solvent

• Explicit representation

The macromolecule is surrounded by solvent molecules (water, ions) with which the macromolecule interacts. Specific nonbond interactions are calculated

• In this case, one must use ε =1.

• More correct (fewer approximations) but more expensive

4

!

_ij

"

ij

r

_ij

#

$

%

%

&

'

(

(

1 2

)

"

ij

r

_ij

#

$

%

%

&

'

(

(

6

*

+

,

,

-.

/

/

+

q

_i

q

_j

r

_ij

0

1

2

3

2

4

5

2

6

2

i,j

7

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

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

Steps of a molecular dynamics

simulation

An application of Molecular

Dynamics Simulations

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.

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

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)

Molecular Dynamics Simulation of

Acetylcholinesterase

Effect of the His44Ala mutation on the Nucleocapsid

protein from the HIV virus - NC(35-50)

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

NMR and Fluorescence studies demonstrate

• Mutant protein binds zinc.

• Mutant protein maintains some structure • Binding to nucleic acids is less strong.

Structural determinants for the specificity of NC for DNA

The structure of the mutant His44Ala:NC(35-50):an NMR, MM and FL study

Biochemistry (2004) Stote RH

et al

, 43,7687-7697

E. Kellenberger and B. Kieffer, ESBS

•Two-dimensional

1

H NMR

•pH 6.5 at 274K

Answer the questions left unanswered by experiment •How does mutant protein bind zinc ion? •If folded, why is the activity diminished?

Biochemistry (2004) Stote RH

et al

, 43,7687-7697

0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 3 5 3 6 3 7 3 8 3 9 4 0 4 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 5 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 residue number angular S rmsd (Å) From NMR From MD

Ensemble of structures from MD

H

Free

Complex

Structural Chemical Shifts :

Δδ

Shifts Ösapay & Case, J. Am. Chem. Soc. 113

1991

• 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

Difference between calculated and experimental

Δδ

-2 -1.5 -1 -0.5 0 0.5 1 C36 W37 K38 C39 G40 K41 E42 G43 A44 Q45 M46 K47 D48 C49 T50 G35 !" (ppm)

Δ

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

TRP 37

LYS 47

MET 46

Study of the DNA/NC complex. Free energy decomposition.

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.

Conclusions

Molecular Modelling: Principles and Applications (2nd Edition) (Paperback)

by Andrew Leach

Computer Simulation of Liquids

Edition New ed

Allen, M. P., Tildesley, D. J.

Computational Chemistry

Grant, Guy H., Richards, W. Graham

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)