lecture10_handouts.pdf

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Theory
Energy functions
Algorithms
Contents of the lecture
1
Theory
Physical chemistry
Overview of methods
Energy functions
Atomic forcefields
Simplified energy functions
Other important energy terms
Algorithms
Optimization methods
Molecular dynamics
Monte Carlo
2
3
Theory
Energy functions
Algorithms
Introduction
Applications of molecular modeling in bioinformatics:
Structure recognition and optimization
Evaluation of structures by energetics
Function recognition (interfaces, binding sites, mechanics)
Ligand design and docking (interaction energies)
Protein & NA engineering
Theory
Energy functions
Algorithms
Physical chemistry
Overview of methods
Physical chemistry
In spontaneous processes (protein folding, NA base pairing,
intermolecular recognition and binding, enzymatic reactions):
∆
G
<
0.
G
=
H
−
TS
=
U
+
pV
−
TS
F
=
U
−
TS
S
=
k
B
ln
Ω
(
F
≡
A
)
The
pV
component represents
work done on the system
and this
one makes sense for gases; however,
any type of work
should be
included in
H.
For processes in solution, a justified approximation is
F
≈
G.
The most troublesome quantity is
S;
in practice, it is rarely
evaluated directly.
Theory
Energy functions
Algorithms
Physical chemistry
Overview of methods
The energy of the system
U
=
E
+
K
+
...
(
E
≡
V
)
E
is potential energy:
∆
E
= ∆
U
(
T
=
const)
K
is kinetic energy:
m
i
v
i
2
— at isotermic conditions
K
=
const
,
∑
i
m
i
|
v
i
|
2
3Nk
B i
The potential energy of a molecular system can be evaluated:
from physics: Schrödinger or Kohn-Sham equations (QM,
ab
initio,
non-empirical methods)
using parametrized QM (semi-empirical methods)
using
some parametrized functions
(MM, FF, empirical methods)
by look-up in a table (scoring functions)
because it is proportional to temperature
T
=
1
∑
Theory
Energy functions
Algorithms
Physical chemistry
Overview of methods
Accuracy and computational cost
QM
Cost
Precision
Reliability
Function
Scaling
highest
highest
highest
analytic
(
AO
)
≥
3
semiempirical
high
moderate
limited
analytic
(
AO
)
1
...
3
MM/FF
moderate to low
moderate to low
limited
analytic
N
2
scoring
functions
low
low
limited
discrete
N
1
...
2
AO
– number of atomic orbitals;
N
– number of atoms.
discrete models are not differentiable (no gradients)
all parametrized models are only reliable for specific cases.
Theory
Energy functions
Algorithms
Atomic forcefields
Simplified energy functions
Other important energy terms
MM: Mechanics on atomic level
A forcefield (FF) is an arbitral, parametrized function representing
energy of atomic interactions as a sum of contributions of various
nature. The FF term is used to name both the parameter set and the
functional form itself.
There is a variety of interaction types, which must be included in
a proper description of molecular energy.
”van der Waals” interactions;
electrostatic interactions;
bond (distortion) energies;
valence angle (distortion) energies;
dihedral angle (rotational, torsional) correction;
out of plane distortion energies (”improper” dihedrals)
Theory
Energy functions
Algorithms
Atomic forcefields
Simplified energy functions
Other important energy terms
Types of interatomic interactions
Dispersive
Pauli repulsion
Coulomb
Bonding
Valence angles
Dihedral distortions
Theory
Energy functions
Algorithms
Atomic forcefields
Simplified energy functions
Other important energy terms
Lennard-Jones and 6-exp functions
E
LJ
=
A
r
12
−
B
r
6
Arbitral form
Easy for computer
Historical but still used
E
6
−exp
=
Ae
−
Cr
−
B
r
6
Better functional form
Slower to calculate
More parameters
Theory
Energy functions
Algorithms
Atomic forcefields
Simplified energy functions
Other important energy terms
Lennard-Jones parameters
E
LJ
=
ε
LJ
R
0
r
12
−
2
R
0
r
6
ε
LJ
– binding energy
R
0
– equilibrium
distance
Theory
Energy functions
Algorithms
Atomic forcefields
Simplified energy functions
Other important energy terms
Electrostatics
Coulomb energy:
E
qq
=
q
i
q
j
4
πε
0
r
ij
Physical law
Parameters:
q
i
,
q
j
–
non-physical!
ε
(”dielectric constant”)
– not constant at all!
Theory
Energy functions
Algorithms
Atomic forcefields
Simplified energy functions
Other important energy terms
Bonding energies
Bond distortions:
E
R
=
k
R
2
(
R
0
−
R
)
2
Angle distortions:
E
α
=
k
α
2
(α
0
− α)
2
physical law (Hooke’s),
but not adequate
idealized elasticity
simple
works well enough
Theory
Energy functions
Algorithms
Atomic forcefields
Simplified energy functions
Other important energy terms
Rotation barriers
Correction to the torsional rotation
is a function of a dihedral angle:
E
Θ
=
k
Θ
2
(
1
+
cos
(
n
Θ − ω))
Θ
is the independent variable;
k
Θ
is the (correction to)
barrier height;
n
is responsible for the correct
number of extrema;
ω
(phase shift) ensure 0
correction at equilibrium
Θ
.
Theory
Energy functions
Algorithms
Atomic forcefields
Simplified energy functions
Other important energy terms
Out of plane (OOP) distortions
Function of a dihedral angle
composed by planar
(
sp
2
)
group:
E
Φ
=
k
Φ
2
(π
−
Φ)
2
this is the Hooke law
Φ
is the dihedral angle
Theory
Energy functions
Algorithms
Atomic forcefields
Simplified energy functions
Other important energy terms
The forcefield in its full beauty
E
=
+
+
+
+
∑
i
,
j
bonds
ε
LJ
(
i
,
j
)
k
R
2
k
α
R
0
r
ij
12
−
2
R
0
r
ij
6
+
q
i
q
j
4
πε
0
r
ij
i
angles
∑
(
R
0
−
R
i
)
2
(α
0
−
α
i
)
2
(
1
+
cos
(
n
Θ
i
−
ω))
2
i
torsions
i
OOP
∑
∑
k
Θ
2
∑
i
k
Φ
2
(π
−
Φ
i
)
2
Theory
Energy functions
Algorithms
Atomic forcefields
Simplified energy functions
Other important energy terms
United and coarse-grained forcefields
To reduce the computations, there are forcefields which represents
groups of atoms as single interacting points
An
united atom FF
incorporates hydrogen atoms into the atoms
which they bond to.
A
coarse-grained FF
represents larger groups of atoms –
residues or sidechains are common – as single beads or
polyhedra with aggregate mass, charge and other properties.
Theory
Energy functions
Algorithms
Atomic forcefields
Simplified energy functions
Other important energy terms
Scoring functions
Fixed number of "energy score" if a pair of entities (atoms,
pseudo-atoms, residues) satisfies a condition (e.g. distance).
electrostatic
H-bonds
hydrophobic
solvent
Theory
Energy functions
Algorithms
Atomic forcefields
Simplified energy functions
Other important energy terms
Statistical potentials
Also referred to as „knowledge based potentials”, calculated from
distribution of (sub)structures in the PDB database.
Used extensively in structure prediction methods and modeling of
folding
Usually calculated from distribution of residue-residue contacts
or conformers/rotamers
Frequently correlate with
∆
F, but
Their physical interpretation is unclear and disputed.
Statistical potentials are introduced by analogies to the potential of
mean force (PMF) or with Bayesian statistics, but due to unfounded
assumptions their interpretation as approximation to Boltzmann PMF
or to free energy is wrong.
See:
http://en.wikipedia.org/wiki/Statistical_potential
Theory
Energy functions
Algorithms
Atomic forcefields
Simplified energy functions
Other important energy terms
Solvent effects
explicit solvent
very expensive, but the only way to model specific interactions
implicit solvent
polarizable continuum (Poisson-Boltzmann, GB)
proportional to the surface
discrete, e.g. burial index
Theory
Energy functions
Algorithms
Optimization methods
Molecular dynamics
Monte Carlo
Energy minimization
In general: search for the minimum of an energy function.
In our case: search for conformation of the lowest energy (geometry
optimization)
A general problem: local minima
There are examples of algorithms designed to overcome this
limitation, generally referred to as "global optimization" methods.
Theory
Energy functions
Algorithms
Optimization methods
Molecular dynamics
Monte Carlo
Optimization algorithms
Zeroth order (e.g. simplex)
First order (based on first derivatives)
steepest descents
conjugate gradients
Second order or newtonian (Hessian required)
Newton-Raphson
pseudo-newton, e.g. BFGS
Other
Often, molecular dynamics with cooling is used for optimization
Monte Carlo techniques

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