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
kr (r  r0 )2
Angles
k (  0 )2
Torsions
 kn (cos n )
n
Nonbonded:
Lennard-Jones
Electrostatic
Sources of parameters:

f
  12   6 
 ij  ij    ij  
 rij  
 rij 
  

N
H
qi q j
O
rij
C
• 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
kr (r  r0 )2
Angles
k (  0 )2
Torsions
 kn (cos n )
n
Nonbonded:
Lennard-Jones
Electrostatic
Sources of parameters:

f
  12   6 
 ij  ij    ij  
 rij  
 rij 
  

N
H
qi q j
O
rij
C
• 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
Bond
Bond energy, kJ/mol
1 , 1 p bonds
C–C
347
C=C
615
1 , 2 p bonds
C≡C
812
C–O
360
C=O
728
F–F
158
Cl–Cl
244
C–H
414
H–H
436
H–O
464
O=O
498
Torsion potentials
Putting It All Together: Molecular
Mechanics Models of Macromolecules
r
Bonds
kr (r  r0 )2
Angles
k (  0 )2
Torsions
 kn (cos n )
n
Nonbonded:
Lennard-Jones
Electrostatic
Sources of parameters:

f
  12   6 
 ij  ij    ij  
 rij  
 rij 
  

N
H
qi q j
O
rij
C
• 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
• 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
• 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
• Dispersion and shortrange 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 Macromolecules
r
Bonds
kr (r  r0 )2
Angles
k (  0 )2
Torsions
 kn (cos n )
n
Nonbonded:
Lennard-Jones
Electrostatic
Sources of parameters:

f
  12   6 
 ij  ij    ij  
 rij  
 rij 
  

N
H
qi q j
O
rij
C
• 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?
Electrostatic potential (ESP) from QM calculation:
 



Z
dr  r 
f r   fN r   fe r         
r  r
 r  R
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 Macromolecules
r
Bonds
kr (r  r0 )2
Angles
k (  0 )2
Torsions
 kn (cos n )
n
Nonbonded:
Lennard-Jones
Electrostatic
Sources of parameters:

f
  12   6 
 ij  ij    ij  
 rij  
 rij 
  

N
H
qi q j
O
rij
C
• 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?
Models of Solvation
Explicit
Implicit/Continuum
O
H
+
+ + =80
+
=1
+ – +
+
- + +
Heuristic
- -
H
–
H
O
H
Pro: water models fairly
mature
Con: ensemble averaging
extremely expensive
for large system
Pro: solvation free energy
estimates cheap and
generally accurate
Con: dynamics, first shell
effects ???
SPC, TIP4P, etc.
Poisson-Boltzmann
Adjustable parameters:
Partial charges, bond
lengths, etc.
Generalized Born
Semi-analytical
approximation
Adjustable parameters: radii
Distance-dependent
dielectric
 r   r
Surface-area
based methods
Gsolv    i Ai
i
Electrostatics in Solution
• Simplest way to introduce effect
of water is to use screening that
depends on dielectric constant:
q1q2/r12
• But what is the dielectric,
especially for a partially
solvated group in a protein??
• Effective dielectric should
depend on both a) what’s inbetween 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) …
Gsolv  q2/R
(where R is atomic radius)
this is a useful formula to remember!
Ionic Contributions to Charge Density
Debye-Huckel theory gives the density of ions as

i r   i0e

 qi  r 
kT
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



ionic r     r     r    0e

  r 
kT
  0e

  r 
kT

  r  
0
 2  sinh

 kT 
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
• 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
• Force fields
• Molecular dynamics
– integrators
– explicit solvent, periodic boundary conditions
– a few applications
• Free energy methods
– theory
– alchemical perturbations
– applications
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
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+t
There are many integrators, and they basically all start from
Taylor expansions of position and/or velocity:
xt  t   xt   t  vt   t  at   
1
2
2
Velocity Verlet integrator
xt  t   xt   t  vt   12 t 2  at   
Position is updated first, based on current x, v, a
vt  t   vt   t 12 at   at  t 
Velocity updates are more accurate if you use both the
current/future acceleration.
Q: How to get a(t + t)??
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.
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.
Different representations of water
Some properties are
reproduced very well ...
Others are less so ...
What’s missing from these water
models?
•
•
•
•
•
Polarizability
Bond flexibility
Dissociation
Multibody effects
Purely quantum effects
Dipole Moments
Gas phase
1.85 D
TIP4P
2.18 D
SPC
2.27 D
Liquid water
~2.5 D
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.
This leads to some challenges in computing electrostatic interactions
Can in principle extract two types of
information from MD simulations
• Thermodynamic properties: e.g.,
– H, S, G, 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.