Word version - Colby College

Colby College Molecular Mechanics Tutorial
QUANTA Version
November 2003
Thomas W. Shattuck
Department of Chemistry
Colby College
Waterville, Maine 04901
Please, feel free to use this tutorial in any way you wish ,
provided that you acknowledge the source
and you notify us of your usage.
Please notify us by e-mail at twshattu@colby.edu
or at the above address.
This material is supplied as is, with no guarantee of correctness.
If you find any errors, please send us a note. Table of Contents
Introduction to Molecular Mechanics Section 1: Steric Energy
Section 2: Enthalpy of Formation
Section 3: Comparing Steric Energies
Section 4: Energy Minimization
Section 5: Molecular Dynamics
Section 6: Distance Geometry and 2D to 3D Model Conversion
Section 7: Free Energy Perturbation Theory, FEP
Section 8: Normal Mode Analysis
1: MOE: Molecular Operating Environment
2: Conformational Preference of Methylcyclohexane
3: Geometry (or How Does Molecular Mechanics Measure Up?)
4: Building More Complex Structures: 1-Methyl-trans-Decalin
5: Conformational Preference for Butane
6: MM2
7: Comparing Structures
8: Plotting Structures
9: Conformational Preference of Small Peptides
10: Dynamics in Small Peptides
11. Solvation and ?-Cyclodextrin
12. Docking: ?-Cyclodextrin and ?-Naphthol
13. FEP and Henry's Law Constants and Gibb's Free Energy of Solvation
14. Distance Geometry
15. Protein Structure and Gramicidin-S Introduction to Molecular Mechanics
Section 1 Summary The goal of molecular mechanics is to predict the detailed
structure and physical properties of molecules. Examples of physical
properties that can be calculated include enthalpies of formation,
entropies, dipole moments, and strain energies. Molecular mechanics
calculates the energy of a molecule and then adjusts the energy through
changes in bond lengths and angles to obtain the minimum energy structure. Steric Energy
A molecule can possess different kinds of energy such as bond and
thermal energy. Molecular mechanics calculates the steric energy of a
molecule--the energy due to the geometry or conformation of a molecule.
Energy is minimized in nature, and the conformation of a molecule that is
favored is the lowest energy conformation. Knowledge of the conformation of
a molecule is important because the structure of a molecule often has a
great effect on its reactivity. The effect of structure on reactivity is
important for large molecules like proteins. Studies of the conformation of
proteins are difficult and therefore interesting, because their size makes
many different conformations possible.
Molecular mechanics assumes the steric energy of a molecule to arise
from a few, specific interactions within a molecule. These interactions
include the stretching or compressing of bonds beyond their equilibrium
lengths and angles, torsional effects of twisting about single bonds, the
Van der Waals attractions or repulsions of atoms that come close together,
and the electrostatic interactions between partial charges in a molecule
due to polar bonds. To quantify the contribution of each, these
interactions can be modeled by a potential function that gives the energy
of the interaction as a function of distance, angle, or charge1,2. The
total steric energy of a molecule can be written as a sum of the energies
of the interactions: Esteric energy = Estr + Ebend + Estr-bend + Eoop + Etor + EVdW + Eqq
(1) The bond stretching, bending, stretch-bend, out of plane, and torsion
interactions are called bonded interactions because the atoms involved must
be directly bonded or bonded to a common atom. The Van der Waals and
electrostatic (qq) interactions are between non-bonded atoms. Bonded Interactions
Estr represents the energy required to stretch or compress a bond between
two atoms, Figure 1. A bond can be thought of as a spring having its own equilibrium length, ro,
and the energy required to stretch or compress it can be approximated by
the Hookian potential for an ideal spring:
Estr = 1/2 ks,ij ( rij - ro )2
(2) where ks,ij is the stretching force constant for the bond and rij is the
distance between the two atoms, Figure 1. Ebend is the energy required to bend a bond from its equilibrium angle, ?o.
Again this system can be modeled by a spring, and the energy is given by
the Hookian potential with respect to angle:
Ebend = 1/2 kb,ijk ( ?ijk - ?? )2
(3) where kb,ijk is the bending force constant and ?ijk is the instantaneous
bond angle (Figure 2).
[pic] Figure 2. Bond Bending Estr-bend is the stretch-bend interaction energy that takes into account
the observation that when a bond is bent, the two associated bond lengths
increase (Figure 3). The potential function that can model this interaction
is:
Estr-bend = 1/2 ksb,ijk ( rij - ro ) (?ijk - ?o )
(4) where ksb,ijk is the stretch-bend force constant for the bond between atoms
i and j with the bend between atoms i, j, and k.
[pic]
Figure 3. Stretch-Bend Interaction Eoop is the energy required to deform a planar group of atoms from its
equilibrium angle, (o, usually equal to zero.3 This force field term is
useful for sp2 hybridized atoms such as doubly bonded carbon atoms, and
some small ring systems. Again this system can be modeled by a spring, and
the energy is given by the Hookian potential with respect to planar angle: Eoop = 1/2 ko,ijkl ( (ijkl - (? )2
(5) where ko,ijkl is the bending force constant and (ijkl is the instantaneous
bond angle (Figure 4).
Figure 4. Out of Plane Bending
The out of plane term is also called the improper torsion in some force
fields. The oop term is called the improper torsion, because like a
dihedral torsion (see below) the term depends on four atoms, but the atoms
are numbered in a different order. Force fields differ greatly in their use
of oop terms. Most force fields use oop terms for the carbonyl carbon and
the amide nitrogen in peptide bonds, which are planar (Figure 5). Figure 5. Peptide Bond is Planar. Torsional Interactions: Etor is the energy of torsion needed to rotate
about bonds. Torsional energies are usually important only for single bonds
because double and triple bonds are too rigid to permit rotation. Torsional
interactions are modeled by the potential: Etor = 1/2 ktor,1 (1 - cos ? ) +1/2 ktor,2 (1 - cos 2 ? ) + 1/2
ktor,3 ( 1 - cos 3 ? ) (6) The angle ? is the dihedral angle about the bond. The three-fold term, that
is the term in 3?, is important for sp3 hybridized systems ( Figure 6a and
b ). The two-fold term, in 2?, is needed for halogens, for example F-C-C-F,
and sp2 hybridized systems, such as C-C-C=O and vinyl alcohols1. The one-
fold term in just ? is useful for alcohols with the C-C-O-H torsion,
carbonyl torsions like C-C-C(carbonyl)-C, and to a lesser extent even the
central bond in molecules such as butane that have C-C-C-C
frameworks(Figure 6c). The constants ktor,1, ktor,2 and ktor,3 are the
torsional constants for one-fold, two-fold and three-fold rotational
barriers, respectively. [pic] [pic] [pic]
a. b. c. Figure 6. Torsional Interactions, (a) dihedral angle in sp3systems. (b)
three-fold, 3?, rotational energy barrier in ethane. (c) butane, which also
has a contribution of a one fold, ?, barrier. The origin of the torsional interaction is not well understood. Torsion
energies are rationalized by some authors as a repulsion between the bonds
of groups attached to a central, rotating bond ( i.e., C-C-C-C frameworks).
Torsion terms were originally used as a fudge factor to correct for the
other energy terms when they did not accurately predict steric energies for
bond twisting. For example, the interactions of the methyl groups and
hydrogens on the "front" and "back" carbons in butane were thought to be
Van der Waals in nature (Figure 7). However, the Van der Waals function
alone gives an inaccurate value for the steric energy.
Bonded Interactions Summary: Therefore, when intramolecular interactions
stretch, compress, or bend a bond from its equilibrium length and angle,
the bonds resist these changes with an energy given by the above equations
summed over all bonds. When the bonds cannot relax back to their
equilibrium positions, this energy raises the steric energy of the entire
molecule.
Non-bonded Interactions
Van der Waals interactions, which are responsible for the liquefaction of
non-polar gases like O2 and N2, also govern the energy of interaction of
non-bonded atoms within a molecule. These interactions contribute to the
steric interactions in molecules and are often the most important factors
in determining the overall molecular conformation (shape). Such
interactions are extremely important in determining the three-dimensional
structure of many biomolecules, especially proteins. A plot of the Van der Waals energy as a function of distance between two
hydrogen atoms is shown in Figure 7. When two atoms are far apart, an
attraction is felt. When two atoms are very close together, a strong
repulsion is present. Although both attractive and repulsive forces exist,
the repulsions are often the most important for determining the shapes of
molecules. A measure of the size of an atom is its Van der Waals radius.
The distance that gives the lowest, most favorable energy of interaction
between two atoms is the sum of their Van der Waals radii. The lowest point
on the curve in Figure 7 is this point. Interactions of two nuclei
se