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Introduction to MolecularDynamics Simulations

Roland H. StoteInstitut de Chimie LC3-UMR 7177

Université Louis PasteurStrasbourg France

1EA5

Title Native Acetylcholinesterase (E.C. 3.1.1.7) From Torpedo Californica At 1.8A ResolutionClassification CholinesteraseCompound Mol_Id: 1; Molecule: Acetylcholinesterase; Chain: A; Ec: 3.1.1.7Exp. Method X-ray Diffraction

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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

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Central idea of MolecularDynamics simulations

• Biological activity is the result of time dependent interactionsbetween molecules and these interactions occur at theinterfaces such as protein-protein, protein-NA, protein-ligand.

• Macroscopic observables (laboratory) are related to microscopicbehavior (atomic level).

• Time dependent (and independent) microscopic behavior of amolecule can be calculated by molecular dynamics simulations.

Molecular Dynamics Simulations

• One of the principal tools for modeling proteins, nucleic acids andtheir 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)• Dynamic processes such as ion transport in biological systems.

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Molecular dynamics simulations

• Approximate the interactions in the system using simplifiedmodels (fast calculations). Include in the model only thosefeatures that are necessary to describe the system.

• In the case of molecular dynamics simulations, this means apotential energy function that models the basic interactions.

• Allows one to gain insight into situations that are impossible tostudy experimentally

• Run computer experiments. Ask the question « What if…? »

• The method allows the prediction of the static and dynamicproperties of molecules directly from the underling interactionsbetween the molecules.

Classical Dynamics

• Newton’s Equations of motion

• Position, speed and acceleration are functions of timeri(t); vi(t); ai(t)

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

• Integration of the equations of motion => initialstructure : ri(t=0); initial distribution of velocities: vi(t=0)

Fi = mi !ai = mi !dvi

dt= m !

d2ri

dt2

Fi= !"

iE

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Dynamics: calculating trajectories

• Trajectory: positions as function of time: ri (t)

• How does one determine ri (t) from Fi = mi ai ?

• Simple case where acceleration is constant

a =dv

dtv = at + v

0

Fi = mi !ai = mi !dvi

dt= m !

d2ri

dt2

v(t) =dx(t)

dt

x(t) = v ! t + x0 = a !t2

2+ v0t + x0

Simple case:motion of a particle in one dimension

• Acceleration:

• If a is constant a≠f(t)

• Speed:

• Position:

• The trajectory x(t) obtainedby integration taking intoaccount the initial positionsand velocities (x0 et v0)

a =dv

dt

v(t) = at + v0

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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 t2/2 z = ax -b x2 : the trajectory in the (x,z) plane is parabolic

Balistic trajectory

E(R) =1

21,2pairs

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Potential Energy

• 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)

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Numerical Integration• Taylor series development

• If we know x at time t, after passage of a certain time, Δt, wecan 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) = x0 + v0t + a0t2

2+ 0

'at3

3!+O(t

4)

x(t + !t) = x(t) + v(t)!t +F(t)

m

!t2

2+F'(t)

m

!t3

3!+O(!t

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Acceleration as a function of time

• Acceleration: calculated from the force, that is, from thederivative of the potential energy, including at t=0

• Potential Energy

ai (t) = !1

m

dE(RN )

dri(t)

E(RN ) =1

21,2 pairs

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Principle of the trajectory

t0

t0+Δt

t0+2Δt

t0+4 Δt

t0+7Δt

Integration algorithms

Verlet, Velocity VerletLeapFrog, Beeman

•Choice of the algorithm:–Energy conservation–Calculation time (least expensive)–Integration time step as large as possible

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Trajectory of a macromolecule

• Initial positions x0PDB file

• Xray• NMR• Model

• Initial velocities v0Coupled to the temperature

• AccelerationCalculated from the force, that is,from the derivative of the potentialenergy.

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2NkT =

mivi

2

2i

!

a = !1

m

dE

dr

Relationship between velocitiesand 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.

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2NkT =

mivi

2

2i

!

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Molecular Dynamics Simulation programsAMBERCHARMMNAMDPOLY-MDetc

Potential energy functionparameter files contain the numerical constants needed toevaluate forces and energies

http://www.pharmacy.umaryland.edu/faculty/amackere/research.html

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Molecular Dynamics

Calculation of forces Displacementt=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 mostCPU time> The cost (CPU) is proportional to N2 (N number of atoms)> Truncation

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Electrostatic Forces

+ - + +

van der Waals Forces

r

r r

E(R) =1

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Nonbonded Energy Terms

Truncation• Switch

Bring the potential to zero betweenron and roff. The potential is notmodified for r < ron and equals zerofor r > roff

• ShiftModify the potential over the entirerange of distances in order to bringthe potential to zero for r > rcut

• Long-range electrostaticinteractions

Ewald summationMultipole methods (Extendedelectrostatics model)

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Treatment of solvent

• Implicit: The macromoleculeinteracts only with itself, but theelectrostatic interactions aremodified to account for thesolvent

• All solvent effects are contained inthe dielectric constant ε

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

Eelec

r( ) = Aqiqj

!r

Treatment of solvent• Explicit representation

The macromolecule is surrounded bysolvent molecules (water, ions) withwhich the macromolecule interacts.Specific nonbond interactions arecalculated

• In this case, one must use ε =1.• More correct (fewer approximations)

but more expensive

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Periodic boundary conditions

• For explicit representation ofsolvent

• The boundaries of thesystem must be represented

• For periodic system

Permits the modeling of verylarge systems, but introducesa level of periodicity notpresent in nature.

Boundary Conditions

Solvation sphere: finite system

Around the entire macromolecule Around the active site

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Some properties that can be calculated from atrajectory

• 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

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Steps of a molecular dynamicssimulation

An application of MolecularDynamics Simulations

The acetylcholinesterase story

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Acetylcholinesterase

• Acetylcholinesterase (AChE) is an enzyme that hydrolyzesACh to acetate and choline to inactivate theneurotransmitter

• A very fast enzyme, approaching diffusion controlled.

• Inhibitors are utilized in the treatment of variousneurological diseases, including Alzheimer’s disease.

• Organophosphorus compounds serve as potent insecticidesby selectively inhibiting insect AChE.

Neuromuscular junction: motor neurons : muscle cells

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1EA5

Title Native Acetylcholinesterase (E.C. 3.1.1.7) From Torpedo Californica At 1.8A ResolutionClassification CholinesteraseCompound Mol_Id: 1; Molecule: Acetylcholinesterase; Chain: A; Ec: 3.1.1.7Exp. Method X-ray Diffraction

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Access ofligands to theactive site isblocked -->requiresfluctuations

Secondary channels open transiently: Identified by MD simulations

Molecular Dynamics Simulation ofAcetylcholinesterase

• 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)

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Molecular Dynamics Simulation ofAcetylcholinesterase

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Effect of the His44Ala mutation on the Nucleocapsidprotein from the HIV virus - NC(35-50)Working at the interface of theory and experiment

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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

NCNC

NC

NC

NC

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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 DNAThe 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 1H 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?

Can simulations can predict the structural effects of point mutations?

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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 00

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

HFree Complex

Structural Chemical Shifts : ΔδShifts 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 ΔδΔδ divided into different contributions– Magnetic anisotropy– Ring Current– Electrostatics

H

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Difference between calculated and experimental Δδ

-2

-1.5

-1

-0.5

0

0.5

1

C36 A44G43E42K41G40C39K38W37 C49D48K47M46Q45 T50G35

!" (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

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TRP 37

LYS 47

MET 46

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

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 themutation

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Since molecules are dynamic, experimental structures alone can not give theentire picture.

An interdisciplinary approach is required.

Molecular simulations are a necessary complement to the experimentalstudies.

Conclusions

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

Computer Simulation of LiquidsEdition New edAllen, M. P., Tildesley, D. J.

Computational Chemistry Grant, Guy H., Richards, W. Graham

http://www.ch.embnet.org/MD_tutorial/

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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)