Molecular Dynamics. A very brief introduction

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1 Molecular Dynamics A very brief introduction Sander Pronk Dept. of Theoretical Physics KTH Royal Institute of Technology & Science For Life Laboratory Stockholm, Sweden

2 Why computer simulations? Two primary roles: Allen&Tildesley Numerical experiments needs accuracy Model testing needs reductionism Computers are fast enough for numerical experiments Most models are too complicated for purely theoretical reasoning

Molecular Dynamics Initial input data: Interaction function V(r) - "force field" coordinates r, velocities v Compute potential V(r) and forces F i = i V(r) on atoms Update coordinates &

3 Molecular Dynamics Initial input data: Interaction function V(r) - "force field" coordinates r, velocities v Compute potential V(r) and forces F i = i V(r) on atoms Update coordinates & velocities according to equations of motion Collect statistics and write energy/coordinates to trajectory files Repeat for millions of steps More steps? Yes No Done! Newtonian mechanics, integrated.

4 Why Molecular Dynamics? Many of the world s supercomputing cycles are spent on this. Why is this so important? Because of its domain: Anything dominated by statistical mechanics, which is too big for quantum weirdness

5 Time and energy scales The energy scales are that of thermal motion: on the order of 1 k B T per particle, or ~ 2 kj/mol and time scales are longer than ~1ps (individual steps 1-5 fs)

6 Why Molecular Dynamics? This means: most biological materials And many other soft materials

7 Time scales Biological Experiments s s 10-9 s 10-6 s 10-3 s 10 0 s 10 3 s Coarse-grained models (Whole proteins) Molecular dynamics (Atomic detail) QM simulations (Electrons)

8 Why no QM? So what is too big for quantum weirdness? Take the energy, time uncertainty principle: DEDt h 2 and put in numbers for thermal energy at room temperature (k B T), and we get h 2DE Dt s = 12 fs And we re simulating systems with many particles and therefore energies of many times k B T

9 Why no QM? But that is for the system as a whole! What about the individual particles and their forces? Surely, those must be given by QM! We can coarse-grain those.

10 How does coarse graining work? If we re simulating, we re pretending to obey physics, and take measurements Everything obeys thermodynamics and statistical mechanics So that s where we ll start: in stat mech, we measure averages hai

11 Averages And in general Phase space (all possible configurations) R dr A (r) p(r) hai = Q Normalization constant Boltzmann distribution p(r) =e E(r)/k BT This integrates over all possible configurations and follows the Boltzmann distribution

12 Averages, coarse grained But we can coarse grain this hai = R dr A (r) p(r) Q p(r) =e E(r)/k BT By introducing a variable q(r) where many values r have the same q and still have averages R dq A (q) p(q) hai = Q for example, by collapsing a few degrees of freedom in r onto the same q

13 Averages, coarse grained This is only allowed: R dq A (q) p(q) hai = Q if All values A(r) for the same q have the same A and p(q) =e F (q)/k BT F (q) = where k B T ln Z q dr e E(q)/k BT is the free energy of q

14 New length scales So we went from this hai = R dr A (r) p(r) Q To this p(r) =e E(r)/k BT hai = R dq A (q) p(q) Q p(q) =e F (q)/k BT and we know classical dynamics of many particles obeys statistical mechanics, so it s fine to use it (though caveats apply).

15 Atomic-scale simulations Model atoms as classical point particles Coarse-grained, empirical approach: models not generally valid Adjust model to fit reality, not theory Goal: predict real structure & motion, not to explain it from first principles Example: Adjust the size of atoms to reproduce experimental density (instead of calculating the size from quantum chemistry)

16 Interactions For efficiency, only a few types of interactions are allowed non-bonded

17 Atom types Different atom types for a given element (e.g. carbon) depending on the environment They have different partial charges

18 Lennard-Jones Models the Pauli exclusion principle (repulsive) and van der Waals forces (attractive) 2.0 V LJ (kj mole -1 ) U µ 1 r 12 1 r r (nm) Usually cut off at ~1nm

19 Electrostatics Electrostatic interactions very long range U µ q 1q 2 r So long range that cutting off anywhere is incorrect. What to do with infinite number of periodic images? Use Ewald methods, specifically Particle Mesh Ewald (PME)

20 Ewald summation It is actually possible to solve the full electrostatic problem Add a screening function around charges to make their interactions decay fast Solve the rest in reciprocal space by using Fourier transforms

21 Force fields Popular force fields are: AMBER (AMBER03): Proteins, DNA, RNA Charmm (Charmm27): Proteins, DNA, RNA OPLS: Small organic molecules Don t mix force fields

22 Force Fields A set of interactions constitutes a force field

23 Force Fields A set of interactions constitutes a force field

24 Time steps The time step should be as long as possible, but not cause instabilities. For most atomic-scale simulations 2-4 fs

26 Environments unrealistic (biologically) realistic (biologically) Each force field has their own water that works best (AMBER&Charmm: TIP3P). Choose that.

27 Boundary conditions Simulation boundary: This is a tiny system. Number of waters: 10x10x10 = 1000 Number of edge waters: x8x8 = 488 (48%) Large system: Number of waters: 10 8 x10 8 x10 8 = Number of edge waters: (10 8-2) 3 ~ (0%)

28 Periodic Boundary Conditions Small systems: edge dominated So we pretend boundaries don t exist; like in pacman

29 Simulation box shapes Triclinic box shapes Maximizes periodic separation distance Can be viewed compact/triclinic/rectangular

30 Ensembles Still missing: Newtonian mechanics conserves energy, but we need to control the ensemble: temperature and pressure

31 Temperature Temperature kinetic energy we can impose the speed of molecules Thermostats: Berendsen ( incorrect but stable) v-rescale (thermodynamically correct & pretty stable) Couple separate large groups (proteins, water) to separate thermostats

32 Pressure Pressure can be coupled by changing the box size (and shape) Barostats: Berendsen (again, incorrect but stable) Parrinello-Rahman (correct, but relatively finicky) Volume changes slowly!

33 Imposed coupling time scales Both have time scales; need to be slow enough to control oscillations, but not too slow typically, temperature: 0.1 ps, pressure: 5 ps beware of oscillations:

34 The scales of thermal motion Statistical mechanics applies directly This energy scale applies: k B T Free energy determines what happens Statistics is everything! If you see something once, it didn t happen! All results are measurements with errors

35 Sampling a landscape 3N-dimensional space Native structure is the free energy minimum Ideally, we would sample all of phase space exhaustively In practice we have to make do with the most populated parts

36 What can we expect? Timescales: ps to μs Length scales: nm to ~100 nm Free energies ~1 kj/mol at best Force field limitations Insufficient sampling (it takes time to visit each state) But much easier to do (and to repeat) than experiments!

Thinking about the scale Atomic scale is not always best: more coarse grained simulations are cheaper to run Coarse graining works well if large-scale processes don t depend

37 Thinking about the scale Atomic scale is not always best: more coarse grained simulations are cheaper to run Coarse graining works well if large-scale processes don t depend on small scale details (see previous slides). Also, think about what you want to get from the simulation: predictions or explanations Examples: lipids, fluids, nano-particles

38 Doing simulations Think first, then simulate Ask specific questions Which of these molecules binds best? Simulate 10 models: do they act the same? Try to answer A/B type questions: Does a His-Arg mutation affect stability?

Gromacs Workshop Spring CSC

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