Introduction to Molecular Dynamics Simulations

Transcription

1 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 ) From Torpedo Californica At 1.8A Resolution Classification Cholinesterase Compound Mol_Id: 1; Molecule: Acetylcholinesterase; Chain: A; Ec: Exp. Method X-ray Diffraction 1

2 Macromolecules in motion Local motions (0.01 à 5 Å, 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 "E a # b # "E b # c # "E c ### "E MIN $ 0 # fin! c b 2

3 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) Dynamic processes such as ion transport in biological systems. 3

4 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 F i = m i! a i = m i! dv i dt = m! d 2 r i dt 2 F i =!" i E 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 structure : r i (t=0); initial distribution of velocities: v i (t=0) 4

5 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 F i = m i! a i = m i! dv i dt = m! d 2 r i dt 2 v = at + v 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 (x 0 et v 0 ) a = dv dt v(t) = at + v 0 v(t) = dx(t) dt x(t) = v! t + x 0 = a! t2 2 + v 0 t + x 0 5

6 Initial conditions are Balistic trajectory x(0) = z(0) = 0 vx(0) = vo cos vz (0) = vo sin Z In the x direction ax = 0 vx(t) = vo cos x(t) = vo cos t V 0 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 t 2 /2 z = ax -b x2 : the trajectory in the (x,z) plane is parabolic E(R) = 1! K b b " b 0 1,2pairs2 Potential Energy ( ) 2 + 1! K # # "# 0 angles2 ( ) 2 +! K $ 1 + cos n$ "% dihedrals ( ( )) 4. 6 ( ' + + 4& 0! 5 ij ij * ) r - i, j 6 7 / 0 ij, 12 " ' 6 ( + 1 ij 3 * ) r - + q 8 iq j 6 9 ij, 23 &Dr ij 6 : The energy is a function of the positions r i Therefore the acceleration is a function of the positions Since the positions vary as a function of time r i (t), so does the acceleration, a i (t) 6

7 Numerical Integration Taylor series development x(t) = x 0 + v 0 t + a 0 t a ' t3 0 3! + O(t4 ) If we know x at time t, after passage of a certain time, Δt, we can find x(t+δt) x(t +!t) = x(t) + v(t)!t + F(t) m!t F' (t) m!t 3 3! + O(!t 4 ) 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 Potential Energy a i (t) =! 1 m de(r N ) dr i (t) 1 E(R N ) =! 1,2 pairs 2 K b ( b " b 0 ) 2 1 +! angles 2 K # (# "# 0 ) 2 +! K $ ( 1+ cos( n$ " %)) dihedrals 4.(' + 12 ( ij + 4& ij * ) r - " ' ij * 3 0 ij, ) r - / ij, 3 + q 8 6 iq j 6! 5 9 i, j 2 &Dr 76 ij : 6 7

8 Principle of the trajectory t 0 +4 Δt t 0 +2Δt t 0 +Δt t 0 +7Δt t 0 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

9 Trajectory of a macromolecule Initial positions x 0 PDB file Xray NMR Model Initial velocities v 0 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 a =! 1 m i de dr 2 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! i 2 i 2 9

10 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 10

11 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 N 2 (N number of atoms) > Truncation 7 i, j 0 * 2 #" & 1, ij 4! ij,% $ r ( ij ' 12 ) " 6 # & - ij / % $ r ( / ij ' + q 4 iq j 2 5.!r ij

12 Nonbonded Energy Terms Electrostatic Forces r r van der Waals 1 Forces E(R) =! K b b " b 0 r 1,2pairs2 ( ) 2 + 1! K # # "# 0 angles2 ( ) 2 +! K $ 1 + cos n$ "% dihedrals ( ( )) 4. 6 ( ' + + 4& 0! 5 ij ij * ) r - i, j 7 6 / 0 ij, 12 " ' 6 ( + 1 ij 3 * ) r - ij, 23 + q q 8 i j 6 9 &Dr ij : 6 Truncation Switch Bring the potential to zero between r on and r off. The potential is not modified for r < r on and equals zero for r > r off Shift Modify the potential over the entire range of distances in order to bring the potential to zero for r > r cut Long-range electrostatic interactions Ewald summation Multipole methods (Extended electrostatics model) 12

13 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 q q i 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 7 i, j 0 * 2 #" & 1, ij 4! ij,% $ r ( ij ' 12 ) " 6 # & - ij / % $ r ( / ij ' + q 4 iq j 2 5. r ij 6 2 In this case, one must use ε =1. More correct (fewer approximations) but more expensive 13

14 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

15 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 " 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

16 Steps of a molecular dynamics simulation An application of Molecular Dynamics Simulations The acetylcholinesterase story 16

17 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

18 1EA5 Title Native Acetylcholinesterase (E.C ) From Torpedo Californica At 1.8A Resolution Classification Cholinesterase Compound Mol_Id: 1; Molecule: Acetylcholinesterase; Chain: A; Ec: Exp. Method X-ray Diffraction 18

19 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 solvent atoms Biophysical Journal Volume (2001) Acc. Chem. Research (2002) 19

20 Molecular Dynamics Simulation of Acetylcholinesterase 20

21 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

22 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

23 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, Two-dimensional 1 H 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

24 From NMR angular S 1.2 From MD rmsd (Å) Ensemble of structures from MD residue number 0 Biochemistry (2004) Stote RH et al, 43, Structural Chemical Shifts : Δδ Shifts Ösapay & Case, J. Am. Chem. Soc Free H Complex 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

25 Difference between calculated and experimental Δδ Δ!" (ppm) 1 G35 C36 W37 K38 C39 G40 K41 E42 G43 A44 Q45 M46 K47 D48 C49 T 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

26 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

27 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 27

28 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