The Molecular Dynamics Method

Transcription

1 The Molecular Dynamics Method Thermal motion of a lipid bilayer Water permeation through channels Selective sugar transport

2 Potential Energy (hyper)surface What is Force? Energy U(x) F = d dx U(x) Conformation (x)

3 Classical Molecular Dynamics r ( t + δ t) = r( t) + v( t) δt v ( t + δ t) = v( t) + a( t) δt a ( t ) = F(t) / m d F = U (r) dr

4 Classical Molecular Dynamics U ( r) = 1 4πε 0 q q Coulomb interaction i r ij j U ( r) = ε ij &, R $ * $ % + r min, ij ij ) ' ( 12, R 2* + r van der Waals interaction min, ij ij ) ' ( 6 #!! "

5 Classical Molecular Dynamics U(r) = 1 + q i q $ j + ε ij - R min,ij 4πε 0 r ij - & % r, ij ' ) ( 12 $ 2 R min,ij & % r ij ' ) ( / % F(r) = ' 1 q i q j 2 ' 4πε 0 r & ij 12 ε ij r ij + % - - ',& R min,ij r ij ( * ) 12 % R min,ij ' & r ij ( * ) 6.( 0 * 0 * ˆ /) r ij

6 Classical Molecular Dynamics

7 Classical Molecular Dynamics Bond definitions, atom types, atom names, parameters,.

8 What is a Force Field? In molecular dynamics a molecule is described as a series of charged points (atoms) linked by springs (bonds). To describe the time evolution of bond lengths, bond angles and torsions, also the non-bonding van der Waals and elecrostatic interactions between atoms, one uses a force field. The force field is a collection of equations and associated constants designed to reproduce molecular geometry and selected properties of tested structures.

9 Energy Functions U bond = oscillations about the equilibrium bond length U angle = oscillations of 3 atoms about an equilibrium bond angle U dihedral = torsional rotation of 4 atoms about a central bond U nonbond = non-bonded energy terms (electrostatics and Lenard-Jones)

10 Energy Terms Described in the CHARMm Force Field Bond Angle Dihedral Improper

11 Interactions between bonded atoms V angle = K θ ( ) 2 θ θ o V bond = K b ( b b o ) 2 V dihedral = K ( 1+ cos( nφ δ )) φ

12 Classical Dynamics F=ma at 300K Energy function: used to determine the force on each atom: yields a set of 3N coupled 2 nd -order differential equations that can be propagated forward (or backward) in time. Initial coordinates obtained from crystal structure, velocities taken at random from Boltzmann distribution. Maintain appropriate temperature by adjusting velocities.

13 Langevin Dynamics Langevin dynamics deals with each atom separately, balancing a small friction term with Gaussian noise to control temperature:

14 The most serious bottleneck steps Rotation of buried sidechains Local denaturations Allosteric transitions s ms µs 10 9 (year) Hinge bending Rotation of surface sidechains Elastic vibrations ns ps 10 6 (day) 10 3 SPEED LIMIT Bond stretching Molecular dynamics timestep fs 10 0 δt = 1 fs

15 Potential Energy (hyper)surface r ( t + δ t) = r( t) + v( t) δt Energy U(x) Conformation (x)

16 V bond = K b ( b b ) 2 o Chemical type K bond b o C-C 100 kcal/mole/å Å C=C 200 kcal/mole/å Å C=C 400 kcal/mole/å Å Bond Energy versus Bond length 400 Potential Energy, kcal/mol Single Bond Double Bond Triple Bond Bond length, Å Bond angles and improper terms have similar quadratic forms, but with softer spring constants. The force constants can be obtained from vibrational analysis of the molecule (experimentally or theoretically).

17 Dihedral Potential V dihedral = K ( 1+ cos( nφ δ )) φ Dihedral energy versus dihedral angle 20 Potential Energy, kcal/mol K=10, n=1 K=5, n=2 K=2.5, N= Dihedral Angle, degrees δ = 0

18 van der Waals interaction Lennard-Jones Energy versus Distance 0.9 Interaction Energy, kcal/mol e=0.2,rmin= eps,i,j Rmin,i,j Distance, Å ε ij *#,, % + $ R min,ij r ij & ( ' 12 # 2 R min,ij % $ r ij & ( ' 6 - / /. Short range

19 Electrostatic Energy versus Distance Interaction energy, kcal/mol q1=1, q2=1 q1=-1, q2= Distance, Å From MacKerell Note that the effect is long range.

20 Steps in a Typical MD Simulation 1. Prepare molecule Read in pdb and psf file 2. Minimization Reconcile the structure with force field used (T = 0) 3. Heating Raise temperature of the system 4. Equilibration Ensure system is stable 5. Dynamics Simulate under desired conditions (NVE, NpT, etc) Collect your data 6. Analysis Evaluate observables (macroscopic level properties) Or relate to single molecule experiments Solvation

21 Preparing Your System for MD Solvation Biological activity is the result of interactions between molecules and occurs at the interfaces between molecules (protein-protein, protein- DNA, protein-solvent, DNA-solvent, etc). Why model solvation? many biological processes occur in aqueous solution solvation effects play a crucial role in determining molecular conformation, electronic properties, binding energies, etc How to model solvation? explicit treatment: solvent molecules are added to the molecular system implicit treatment: solvent is modeled as a continuum dielectric

22 Maxwell Distribution of Atomic Velocities

23 CHARMM Potential Function PDB file geometry Topology PSF file parameters Parameter file

24 File Format/Structure The structure of a pdb file The structure of a psf file The topology file The parameter file Connection to potential energy terms

25 Structure of a PDB file resname index name chain resid X Y Z segname ATOM 22 N ALA B BH ATOM 23 HN ALA B BH ATOM 24 CA ALA B BH ATOM 25 HA ALA B BH ATOM 26 CB ALA B BH ATOM 27 HB1 ALA B BH ATOM 28 HB2 ALA B BH ATOM 29 HB3 ALA B BH ATOM 30 C ALA B BH ATOM 31 O ALA B BH ATOM 32 N ALA B BH ATOM 33 HN ALA B BH ATOM 34 CA ALA B BH ATOM 35 HA ALA B BH ATOM 36 CB ALA B BH ATOM 37 HB1 ALA B BH ATOM 38 HB2 ALA B BH ATOM 39 HB3 ALA B BH ATOM 40 C ALA B BH ATOM 41 O ALA B BH >>> It is an ascii, fixed-format file <<< No connectivity information

26 Checking file structures PDB file Topology file PSF file Parameter file

27 Parameter Optimization Strategies Check if it has been parameterized by somebody else Literature Google Minimal optimization By analogy (i.e. direct transfer of known parameters) Quick, starting point - dihedrals?? Maximal optimization Time-consuming Requires appropriate experimental and target data Choice based on goal of the calculations Minimal database screening NMR/X-ray structure determination Maximal free energy calculations, mechanistic studies, subtle environmental effects

28 Major Difficulties in Simulating Size problem Biological Systems Biological Systems are Complex Natural environment: Membrane, solvent, ions, Many biosystems function in assemblies Photosynthetic apparatus Nuclear receptors, GPCRs, Time scale problem Many biological events happen at µs ms time scales Signaling and other regulatory mechanisms Protein folding

29 Large Systems F 1 unit: 330,000 atoms β α β γ δ ε F 0 unit: 110,000 atoms a c 10 ~ 10 millisecond timescale ATP synthase

30 Size scale can be addressed effectively by larger clusters of computers HP 735 cluster 12 processors (1993)? Blue Waters (UIUC) 200,000+ processors (2012) SGI Origin processors (1997) Anton/DESHAW/PSC 512 processors (2010) PSC LeMieux AlphaServer SC 3000 processors (2002) Ranger/Kraken ~60,000 processors (2007)

31 Lipid Diffusion in a Membrane D lip = 10-8 cm 2.s -1 (50 Å in ~ 5 x 10-6 s) D wat = 2.5 x 10-5 cm 2.s -1 Modeling mixed lipid bilayers! Once in several hours! (~ 50 Å in ~ 10 4 s) ~9 orders of magnitude slower ensuring bilayer asymmetry

32 Technical difficulties in Simulations of Biological Membranes Time scale Heterogeneity of biological membranes L 60 x 60 Å Pure POPE 5 ns ~100,000 atoms

33 Coarse-grained modeling of lipids 150 particles 9 particles! Also, increasing the time step by orders of magnitude.

34 by: J. Siewert-Jan Marrink and Alan E. Mark, University of Groningen, The Netherlands

35 Steered Molecular Dynamics Single Molecule Experiments: AFM Accelerating Events Ligand Binding/Unbinding Unfolding Experiments Applying Surface Tension Torque Application Pressure induction Inducing Large Domain Conformational Changes Constant-force: f(t) = C Constant-velocity: f(t) = k [vt (x t - x 0 )]

36 displacement applied force A wide variety of events that are inaccessible to conventional molecular dynamics simulations can be probed. The system will be driven, however, away from equilibrium, resulting in problems in describing the energy landscape associated with the event of interest. Second law of thermodynamics W ΔG

37 Jarzynski s Equality Transition between two equilibrium states λ = λ i T λ = λ(t) T work W heat Q λ = λ f e -βw p(w) ΔG W W ΔG Δ G = G G f i p(w) C. Jarzynski, Phys. Rev. Lett., 78, 2690 (1997) C. Jarzynski, Phys. Rev. E, 56, 5018 (1997) e βw = e βδg In principle, it is possible to obtain free energy surfaces from repeated nonequilibrium experiments. β = 1 k T B

38 Aquaporins Membrane water channels

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40 Water Bipolar Configuration in Aquaporins

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42 R E M E M B E R: One of the most useful advantages of simulations over experiments is that you can modify the system as you wish: You can do modifications that are not even possible at all in reality! This is a powerful technique to test hypotheses developed during your simulations. Use it!

43 Electrostatic Stabilization of Water Bipolar Arrangement