Molecular Mechanics

1
Molecular Mechanics

• a. Force fields
• b. Energy minimization /
• Geometry optimization
• c. Molecular mechanics examples

2
(No Transcript)
3
(No Transcript)
4
(No Transcript)
5
Nonbonded energy Van der Waals
force Lenard-Jones type potential Short-distance
Electrostatic forces Long-distance
6
Bonded energy
7
(No Transcript)
8
Some popular force fields are AMOEBA (Atomic
Multipole Optimized Energetics for Biomolecular
Applications) force field developed by Pengyu Ren
(University of Texas at Austin) and Jay W. Ponder
(Washington University). AMBER (Assisted Model
Building and Energy Refinement) - widely used for
proteins and DNA CHARMM (Chemistry at HARvard
Molecular Mechanics) - originally developed at
Harvard, widely used for both small molecules and
macromolecules OPLS (Optimized Potential for
Liquid Simulations) (variations include OPLS-AA,
OPLS-UA, OPLS-2001, OPLS-2005) - developed
by William L. Jorgensen at the Yale University
Department of Chemistry MMFF (Merck Molecular
Force Field)- developed at Merck, for a broad
range of molecules (MMFF94 for molecular
dynamics, MM94s static for molecular
mechanics). MM2, MM3, MM4 - developed by Norman
Allinger, parametrized for a broad range of
molecules.MM2 was developed by Norman
Allinger primarily for conformational analysis
of hydrocarbons and other small organic
molecules. UFF - A general force field with
parameters for the full periodic table up to and
including the actinoids - developed at Colorado
State University
9
(No Transcript)
10
(No Transcript)
11
(No Transcript)
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
Simplex method
16
(No Transcript)
17
Simplex Methods
Minimization Bounds ? Polygon of N1 vertices
Solution is a vertex of N1-d
polygon Procedure (Downhill Simplex Method) Begin
with simplex for input coordinate values Find
lowest point on simplex (best) Find highest point
on simplex (worst) (Intuition Move away from
high point, towards low point) Reflect
(x1-xo) If E(x1)ltE(xo) then expand
(xxl) Else Try
intermediate point If
E(xnew)ltE(xo) expand If
E(xnew)gtE(xo) contract
18
(No Transcript)
19
Simplex downhill (amoeba) method Nelder, Meade
1965
20
(No Transcript)
21
(No Transcript)
22
Newton-Raphson method
23
(No Transcript)
24
(No Transcript)
25
(No Transcript)
26
(No Transcript)
27
(No Transcript)
28
An energy contour surface.
The minimizer must find the direction to the
minimum and its distance from the initial guess
Improve efficiency by finding how the derivatives
change
29
Most minimizers use line searches
line search - one-dimensional minimization along
a direction vector determined at each iteration.
(x, y) are coordinates along the line away
from the current point (x0, y0) in the direction
of the derivative at (x0, y0). For the path
shown, it would be along derivative vector
(2x,10y)
30
shows the minimization path followed by a
steepest-descents approach for the simple
quadratic function. As expected, each line search
produces a new direction that is perpendicular to
the previous gradient however, the directions
oscillate along the way to the minimum. This
inefficient behavior is characteristic of
steepest descent method, especially on energy
surfaces having narrow valleys.
31
Conjugate Gradient method
It would be preferable to prevent the next
direction vector from undoing earlier progress.
This means using an algorithm that produces a
complete basis set of mutually conjugate
directions such that each successive step
continually refines the direction toward the
minimum. If these conjugate directions truly span
the space of the energy surface, then
minimization along each direction in turn must by
definition end in arriving at a minimum. The
conjugate gradient algorithm constructs and
follows such a set of directions.
32
Model size and distance from the minimum The
choice of which algorithm to use depends on two
factors--the size of the model and its current
state of optimization. The conjugate gradient and
steepest descents methods can be used with models
of any size. Most Newton-Raphson methods cannot
be used with very large models, because they need
sufficient disk space to store a
second-derivative matrix. (the storage
requirements scale as approximately 3N
x3N for N atoms) Until the derivatives are well
below 100 kcal/mol/1Å, it is likely that the
point is sufficiently distant from a minimum that
the energy surface is far from quadratic.
Algorithms that assume the energy surface to be
quadratic (Newton-Raphson, conjugate gradient)
can be unstable when the model is far from the
quadratic limit. The Newton-Raphson method is
particularly sensitive because it must invert the
Hessian matrix. Therefore, as a general rule,
steepest descents is often the best minimizer to
use for the first 10-100 steps, after which the
conjugate gradient and/or a Newton-Raphson
minimizer can be used to complete the
minimization to convergence.
33
Aspirin
Molecular docking simulation of aspirin with
phospholipase A
34
Phospholipase cleavage sites Phospholipase A2 is
an enzyme present in the venom of bees and viper
snakes, mammalian digestive juices and
inflammatory exudates.
35
Optimization problems
 knapsack problem or rucksack problem which boxes
should be chosen to maximize the amount of money
while still keeping the overall weight under or
equal to 15 kg? A multiple constrained
problem could consider both the weight and volume
of the boxes.(Solution if any number of each
box is available, then three yellow boxes and
three grey boxes if only the shown boxes are
available, then all but the green box.)
ant colony optimization algorithm Dorigo
1992 search for an optimal path in a graph, based
on the behavior of ants seeking a path between
their colony and a source of food The ants
prefer the smaller drop of honey over the more
abundant, but less nutritious, sugar
36
In the natural world, ants (initially)
wander randomly, and upon finding food return to
their colony while laying down pheromone trails.
If other ants find such a path, they are likely
not to keep travelling at random, but to instead
follow the trail, returning and reinforcing it if
they eventually find food. Over time, however,
the pheromone trail starts to evaporate, thus
reducing its attractive strength. The more time
it takes for an ant to travel down the path and
back again, the more time the pheromones have to
evaporate. A short path, by comparison, gets
marched over more frequently, and thus the
pheromone density becomes higher on shorter paths
than longer ones. Pheromone evaporation also has
the advantage of avoiding the convergence to a
locally optimal solution. If there were no
evaporation at all, the paths chosen by the first
ants would tend to be excessively attractive to
the following ones. In that case, the exploration
of the solution space would be constrained. Thus,
when one ant finds a good (i.e., short) path from
the colony to a food source, other ants are more
likely to follow that path, and positive
feedback eventually leads to all the ants'
following a single path. The idea of the ant
colony algorithm is to mimic this behavior with
"simulated ants" walking around the graph
representing the problem to solve.