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Computational Methods

Computational Methods. Matt Jacobson matt.jacobson@ucsf.edu. Some slides borrowed from Jed Pitera (IBM, Adjunct Faculty UCSF). r. Bonds. Angles. q. Torsions. f. Nonbonded:. N. Lennard-Jones. H. O. Electrostatic. C. Molecular Mechanics Models of Macromolecules.

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Computational Methods

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  1. Computational Methods Matt Jacobson matt.jacobson@ucsf.edu Some slides borrowed from Jed Pitera (IBM, Adjunct Faculty UCSF)

  2. r Bonds Angles q Torsions f Nonbonded: N Lennard-Jones H O Electrostatic C Molecular Mechanics Models of Macromolecules • Sources of parameters: • Gas-phase QM • Macroscopic properties via liquid state simulation, e.g., density, heat capacity, compressibility (esp. OPLS) • Spectroscopic and crystallographic data (small molecules) All-Atom Force Fields: e.g., CHARMM, AMBER, OPLS, GROMOS

  3. r Bonds Angles q Torsions f Nonbonded: N Lennard-Jones H O Electrostatic C Putting It All Together: Molecular Mechanics Models of Macromolecules • Sources of parameters: • Gas-phase QM • Macroscopic properties via liquid state simulation, e.g., density, heat capacity, compressibility (esp. OPLS) • Spectroscopic and crystallographic data (small molecules) All-Atom Force Fields: e.g., CHARMM, AMBER, OPLS, GROMOS

  4. Covalent forces are very strong 1 s, 1 p bonds 1 s, 2 p bonds

  5. Torsion potentials

  6. r Bonds Angles q Torsions f Nonbonded: N Lennard-Jones H O Electrostatic C Putting It All Together: Molecular Mechanics Models of Macromolecules • Sources of parameters: • Gas-phase QM • Macroscopic properties via liquid state simulation, e.g., density, heat capacity, compressibility (esp. OPLS) • Spectroscopic and crystallographic data (small molecules) All-Atom Force Fields: e.g., CHARMM, AMBER, OPLS, GROMOS

  7. VDW Part 1: Dispersion Forces • Consider 2 He atoms – the least chemically reactive, most “ideal” gas. They still interact with each other! • Quantum mechanical effect • Long-range, weak attraction • Can be described classically as a spontaneously induced dipole-induced dipole interaction • As r∞, the interaction scales as 1/r6 • Magnitude of force: obviously depends strongly on distance; generally small relative to kT. But it adds up (N2 interactions in protein).

  8. VDW Part 2: Close-Range Repulsion • Direct consequence of Pauli exclusion principle: 2 electrons (which necessarily have same spin) cannot simultaneously occupy same space • Formally increases exponentially with decreasing internuclear separation • However, frequently modeled as 1/r12. • Magnitude of force: gets extremely large very quickly (“steric clash”)

  9. VDW Part 3: Complete Potential • Dispersion and short-range repulsion are then combined in the Lennard-Jones formula: A/r12 – C/r6 • Narrow, rather shallow minimum at the sum of the “VDW” radii (when the atoms are just touching).

  10. r Bonds Angles q Torsions f Nonbonded: N Lennard-Jones H O Electrostatic C Putting It All Together: Molecular Mechanics Models of Macromolecules • Sources of parameters: • Gas-phase QM • Macroscopic properties via liquid state simulation, e.g., density, heat capacity, compressibility (esp. OPLS) • Spectroscopic and crystallographic data (small molecules) All-Atom Force Fields: e.g., CHARMM, AMBER, OPLS, GROMOS

  11. Where do the partial charges come from? Electrostatic potential (ESP) from QM calculation: Basic idea: Fit this quantity with point charges. This idea has been elaborated by a number of workers, including RESP (Kollman): “Restrained electrostatic potential fit”. Gas phase vs. condensed phase ...

  12. r Bonds Angles q Torsions f Nonbonded: N Lennard-Jones H O Electrostatic C Putting It All Together: Molecular Mechanics Models of Macromolecules • Sources of parameters: • Gas-phase QM • Macroscopic properties via liquid state simulation, e.g., density, heat capacity, compressibility • Spectroscopic and crystallographic data (small molecules) All-Atom Force Fields: e.g., CHARMM, AMBER, OPLS, GROMOS This is sufficient to describe a macromolecule by itself; but what about solvent?

  13. Implicit/Continuum Explicit O - + - + + - + - - + O H - + + H - H H H O O O H H Pro: solvation free energy estimates cheap and generally accurate Con: dynamics, first shell effects ??? Poisson-Boltzmann Generalized Born Adjustable parameters: radii Pro: water models fairly mature Con: ensemble averaging extremely expensive for large system SPC, TIP4P, etc. Adjustable parameters: Partial charges, bond lengths, etc. H H H H H O O e=80 e=1 – – + + Semi-analytical approximation O O O O H H H H H H H H H H Models of Solvation Heuristic Distance-dependent dielectric Surface-area based methods

  14. Electrostatics in Solution • Simplest way to introduce effect of water is to use screening that depends on dielectric constant: q1q2/er12 • But what is the dielectric, especially for a partially solvated group in a protein?? • Effective dielectric should depend on both a) what’s in-between the charges, and b) the location of the charges. • More complex theories attempt to reproduce the free energy of solvation. Simplest example: Born equation for monoatomic ions (charge in sphere) … DGsolv  q2/R (where R is atomic radius) this is a useful formula to remember!

  15. Ionic Contributions to Charge Density Debye-Huckel theory gives the density of ions as Ionic density in bulk solution This just gives a model for the enrichment of, e.g., negative ions in places where the potential is positive. So, for a 1:1 salt solution, we have

  16. Many other types of “energy models” and “scoring functions” exist and are used for many applications • small molecule docking • protein-protein docking • homology modeling • membrane permeability • protein dynamics/flexibility Very frequently these models do attempt to capture certain aspects of the physics, but not generally as directly, or with as much generality, as force fields. Empirical parameterization in most cases. Main motivation: computational speed Physics vs. empiricism

  17. Questions I am frequently asked • How good can I expect results to be from an MD simulation? • Surely you should be able to compute a factor of 2 difference in binding affinity ... • What’s the best force field?

  18. Outline • Force fields • Molecular dynamics • integrators • explicit solvent, periodic boundary conditions • a few applications • Free energy methods • theory • alchemical perturbations • applications

  19. Molecular Dynamics • Very simple idea: Just use the simple molecular mechanics models of forces, then feed them into Newton’s equations of motion, basically F=ma. Then watch the molecules move! • In the realm of biology, Martin Karplus deserves a lot of credit for early work that convinced people to think about macromolecules as dynamic, not static, structures. His program, Charmm, is still widely used. • Now there are many thousands of papers using molecular dynamics, and lots of widely used programs. • Some of the areas of current interest where MD continues to play an important role: • Mechanisms of action of membrane proteins • Mechanisms of allostery • Protein folding • Quantitative prediction of binding affinities • A nice review article: Karplus and McCammon, “Molecular dynamics simulations of biomolecules”, Nat Struct Biol. 2002 Sep;9(9):646-52.

  20. Molecular dynamics integrators Variable definitions x: position v: velocity a: acceleration (All of these are obviously vectors of size 3N) Basic idea: If we know (x,v,a) at time t, estimate their values at time t+Dt There are many integrators, and they basically all start from Taylor expansions of position and/or velocity:

  21. Velocity Verlet integrator Position is updated first, based on current x, v, a Velocity updates are more accurate if you use both the current/future acceleration. Q: How to get a(t + Dt)?? A: Update positions, calculate new forces, use F=ma This is not quite how it works in practice, but this is good enough for the problem set.

  22. Possible to play some tricks to get beyond 1 fs, e.g., freezing bonds, multiple timescale methods. Currently limited to ~microsecond simulations now, soon getting up to milliseconds, probably, although these will be huge calculations, not something everyone can do. Typical simulations: nanoseconds.

  23. Different representations of water

  24. Some properties are reproduced very well ...

  25. Others are less so ...

  26. What’s missing from these water models? • Polarizability • Bond flexibility • Dissociation • Multibody effects • Purely quantum effects Dipole Moments An ongoing challenge: Water in first solvation shell of protein (or “trapped” in an active site or the interior) is rather different than bulk water.

  27. This leads to some challenges in computing electrostatic interactions

  28. Can in principle extract two types of information from MD simulations • Thermodynamic properties: e.g., • DH, DS, DG, etc. • experimental observables, e.g., NOEs An important point about these is that they are properties of the ensemble, not a single snapshot. • Kinetic properties: How long does a process take? This can be very tricky to predict accurately, for several reasons. Kinetic properties are also ensemble properties, in general.

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