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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

Computational Methods

Matt Jacobson

matt.jacobson@ucsf.edu

Some slides borrowed from Jed Pitera (IBM, Adjunct Faculty UCSF)

molecular mechanics models of macromolecules

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

putting it all together molecular mechanics models of macromolecules

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

covalent forces are very strong
Covalent forces are very strong

1 s, 1 p bonds

1 s, 2 p bonds

putting it all together molecular mechanics models of macromolecules1

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

vdw part 1 dispersion forces
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).
vdw part 2 close range repulsion
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”)
vdw part 3 complete potential
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).
putting it all together molecular mechanics models of macromolecules2

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

where do the partial charges come from
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 ...

putting it all together molecular mechanics models of macromolecules3

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?

slide13

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

electrostatics in solution
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!

ionic contributions to charge density
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

many other types of energy models and scoring functions exist and are used for many applications
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

questions i am frequently asked
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?
outline
Outline
  • Force fields
  • Molecular dynamics
    • integrators
    • explicit solvent, periodic boundary conditions
    • a few applications
  • Free energy methods
    • theory
    • alchemical perturbations
    • applications
molecular dynamics
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.
molecular dynamics integrators
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:

velocity verlet integrator
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.

slide23

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.

what s missing from these water models
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.

can in principle extract two types of information from md simulations
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.