Molecular Modeling
The compendium of methods for mimicking the
behavior of molecules or molecular systems
Points for Consideration
Remember
• Molecular modeling forms a model of the real world
• Thus we are studying the model, not the world
• A model is valid as long as it reproduces the real world
Why Use Molecular Modeling?
(and not deal directly with the real world?)
Fast, accurate and relatively cheap way to:
•
•
•
•
•
Study molecular properties
Rationalize and interpret experimental results
Make predictions for yet unstudied systems
Study hypothetical systems
Design new molecules
And
• Understand
reactants
probe
?
products
Some Molecular Properties
Energy
•
•
•
•
Molecular Spectroscopy
• NMR (Nuclear Magnetic
Resonance)
• IR (Infra Red)
• UV (Ultra Violate)
• MW (Microwave)
Free energy (DG)
Enthalpy (DH)
Entropy (DS)
Steric energy (DG)
3D Structure
• Distances
• Angles
• Torsions
Kinetics
• Reaction Mechanisms
• Rate constants
Electronic Properties
• Molecular orbitals
• Charge Distributions
• Dipole Moments
Molecular Structure: Saccharin
A single 1D structure
C7H5NO3S
O
A single 2D structure
N
S
O
Many 3D structures
O
H
Molecular Structure and
Molecular Properties
Structure

Property
Activity
Cell Permeability
Toxicity
Descriptors
1D: e.g., Molecular weight
2D: e.g., # of rotatable bonds
3D: e.g., Molecular volume
Molecular mechanics
Introduction to Force Fields
Molecular mechanics
• Ball and spring description of molecules
• Better representation of equilibrium geometries than
plastic models
• Able to compute relative strain energies
• Cheap to compute
• Lots of empirical parameters that have to be carefully
tested and calibrated
• Limited to equilibrium geometries
• Does not take electronic interactions into account
• No information on properties or reactivity
• Cannot readily handle reactions involving the making
and breaking of bonds
Energy as a function of geometry
• Polyatomic molecule:
– N-degrees of freedom
– N-dimensional potential energy surface
http://www.chem.wayne.edu/~hbs/chm6440/PES.html
A Molecule is a Collection of
Atoms Held Together by Forces
Intuitively forces act between bonded atoms
Forces act to return structural parameters to their equilibrium
values
stretch
bend
torsion
And More Forces…
Forces also act between non-bonded atoms
Cross terms couple the different types of interactions
Stretch-bend
Non-bonded
Forces are Described by Potential
Energy Functions
k
v(l )  (l  l0 ) 2
2
k
v( )  (   0 ) 2
2
N
Vn
(1  cos( n   )]
n 0 2
v( )  
And More Functions…

  12    6  
 4  ij    ij   
ij 
 r  
N
N 
 rij 
 ij   
v   
i 1 j i 1

qi q j
 

 4 0 rij

v(l1 , l2 , ) 
kl1l2
2
[(l1  l1, 0 )  (l2  l2, 0 )](   0 )
A Force Field is a Collection of
Potential Energy Functions
A force field is defined by the functional forms of the energy
functions and by the values of their parameters.
ki
ki
Vn
2
2
(
l

l
)

(



)

(1  cos( n   ))



i
i ,0
i
i ,0
bonds 2
angles 2
torsions 2
N
N 
  12    6  q q 
    4 ij  ij    ij    i j   cross terms

rij 
rij   4 0 rij 

i 1 j i 1






V (r N ) 
Molecular Mechanics Energy (Steric
Energy) is Calculated from the Force
Field
A molecular mechanics program will return an energy value for
every conformation of the system.
Steric energy is the energy of the system relative to a reference
point. This reference point depends on the bonded interactions and
is both force field dependent and molecule dependent.
Thus, steric energy can only be used to compare the relative
stabilities of different conformations of the same molecule and can
not be used to compare the relative stabilities of different
molecules.
Further, all conformational energies must be calculated with the
same force field.
Steric Energy
Steric Energy = Estretch + Ebend + Etorsion + EVdW + Eelectrostatic +
Estretch-bend + Etorsion-stretch + …
Estretch
Ebend
Etorsion
EVdW
Eelectrostatic
Estretch-bend
Etorsion-stretch
Stretch energy (over all bonds)
Bending energy (over all angles)
Torsional (dihedral) energy (over all dihedral angles)
Van Der Waals energy (over all atom pairs > 1,3)
Electrostatic energy (over all charged atom pairs >1,3)
Stretch-bend energy
Torsion-stretch energy
EVdW + Eelectrostatic are often referred to as non-bonded energies
Force Field and Potential Energy
Surface
A force field defines for each molecule a unique PES.
Each point on the PES represents a molecular conformation
characterized by its structure and energy.
Energy is a function of the coordinates.
Coordinates are function of the energy.
CH3
energy
CH3
CH3
coordinates
Moving on (Sampling) the PES
Each point on the PES is represents a molecular conformation
characterized by its structure and energy.
By sampling the PES we can derive molecular properties.
Sampling energy minima only (energy minimization) will lead to
molecular properties reflecting the enthalpy only.
Sampling the entire PES (molecular simulations) will lead to
molecular properties reflecting the free energy.
In both cases, molecular properties will be derived from the PES.
Force Fields: General Features
Force field definition
• Functional form (usually a compromise between accuracy and
ease of calculation.
• Parameters (transferability assumed).
Force fields are empirical
• There is no “correct” form of a force field.
• Force fields are evaluated based solely on their performance.
Force field are parameterized for specific properties
• Structural properties
• Energy
• Spectra
Force Fields: Atom Types
In molecular mechanics atoms are given types - there are
often several types for each element.
Atom types depend on:
• Atomic number (e.g., C, N, O, H).
• Hybridization (e.g., SP3, SP2, SP).
• Environment (e.g., cyclopropane, cyclobutane).
MM2 type 2
C
O
C
C
MM2 type 3
“Transferability” is assumed - for example that a C=O bond
will behave more or less the same in all molecules.
A General Force Field Calculation
Input
• Atom Types
• Starting geometry
• Connectivity
Energy minimization / geometry optimization
Calculate molecular properties at final geometry
Output
•
•
•
•
Molecular structure
Molecular energy
Dipole moments
etc. etc. etc.
A Simple Molecular Mechanics
Force Field Calculation
V (r N ) 
ki
ki
Vn
2
2
(
l

l
)

(



)

(1  cos( n   ))



i
i ,0
i
i ,0
bonds 2
angles 2
torsions 2
  12    6  q q 
   4 ij  ij    ij    i j 
i 1 j i 1
 rij 
 rij   4 0 rij 
N
N
Bonds
• C-C x 2
• C-H x 8
Angles
• C-C-C x 1
• C-C-H x 10
• H-C-H x 7
Torsions
• H-C-C-H x 12
• H-C-C-C x 6
Non-bonded
• H-H x 21
• H-C x 6
Bond Stretching
Morse potential
v(l )  De{1  exp[ a(l  l0 )]}2
a    2 De
E
 k 
De = Depth of the potential energy minimum
l0 = Equilibrium bond length (v(l)=0)
 = Frequency of the bond vibration
 = Reduced mass
k = Stretch constant
• Accurate.
• Computationally inefficient (form, parameters).
• Catastrophic bond elongation.
r
Bond Stretching
Harmonic potential (AMBER)
k
v(l )  (l  l0 ) 2
2
E
r
• Inaccurate.
• Coincides with the Morse potential at the bottom of the well.
• Computationally efficient.
Bond Stretching
Cubic (MM2) and quadratic (MM3) potentials
k1
k2
v(l )  (l  l0 ) 2  (l  l0 )3
2
2
k1
k 2 amber 3 k 3
2
v(l )  (l  l0 )  (mm2
l  l0 )  (l  l0 ) 4
2
2 mm3 C-C bond2
600.0
harmonic
cubic
quadratic
500.0
energy
400.0
300.0
200.0
100.0
0.0
-100.0
1
1.2
1.4
1.6
bond leng th
1.8
2
2.2
Bond Stretching Parameters
(MM2)
-1
-2
l0 (A)
k (kcal mol A )
Csp -Csp
Csp3-Csp2
1.523
1.497
317
317
Csp2=Csp2
1.337
690
1.208
777
1.438
1.345
367
719
Bond
3
3
2
Csp =O
3
3
Csp -Nsp
C-N (amide)
Hard mode.
Bond types correlate with l0 and k values.
A 0.2Å deviation from l0 when k=300 leads to an energy
increase of 12 kcal/mol.
Angle Bending
k
v( )  (   0 ) 2
2
k1
k2
v( )  (   0 ) 2  (   0 ) 6
MM2:
2
2
k1
k2
k3
k4
k5
3
4
5
6
(



)

(



)

(



)

(



)
MM3: v( )  (   0 ) 2  amber
0
0
0
0
mm2
2
2
2
2
2
mm3
AMBER:
C-C-C angle
25.0
MM3
MM2
AMBER
20.0
energy
energy
15.0
10.0
5.0
0.0
-5.0
80
90
100
110
angle
bond
ang le
120
130
Angle Bending Parameters (MM2)
q0
Angle
-1
Csp3-Csp3-Csp3 109.47
0.0099
Csp3-Csp3-H
109.47
0.0079
H-Csp -H
109.47
Csp3-Csp2-Csp3 117.2
0.007
0.0099
3
3
2
3
2
Csp -Csp =Csp
Csp -Csp =O
2
-1
k (kcal mol deg )
121.4
0.0121
122.5
0.0101
Hard mode (softer than bond stretching)
Torsional (Dihedral) Terms
Reflect the existence of barriers to rotation around chemical bonds.
Used to set the relative energies of the rotational minima and
maxima.
Together with the non-bonded terms are responsible for most of the
structural and energetic changes (soft mode).
Usually parameterized last.
Soft mode.
General Functional Form
N
Vn
(1  cos( n   )]
n 0 2
v( )  
: Torsional angle.
n (multiplicity): Number of minima in a 360º cycle.
Vn: Correlates with the barrier height.
(phase factor): Determines where the torsion passes through its
minimum value.
.
Vn = 4, n = 2,  = 180
Torsion energy
.
.
.
Vn = 2, n = 3,  = 60
.
Torsion angle (  )
Functional Form: AMBER
N
Vn
v( )   (1  cos( n   )]
n 0 2
Preference to single terms.
Usage of general torsional parameters (i.e., torsional
potential solely depends on the two central atoms of the
torsion).
• C-C-C-C = O-C-C-C = O-C-C-N = …
Example: Butane
The barrier to rotation around the C-C bond in butane is ~20kJ/mol.
All 9 torsional interactions around the central C-C bond should be
considered for an appropriate reproduction of the torsional barrier.
rotational profile of butane
#
1
4
4
21.0
energy
15.7
10.5
5.2
0.0
0
90
180
dihedral ang le
270
360
Type
C-C-C–C
C-C-C–H
H-C-C–H
V1
0.200
0.000
0.000
V2
0.270
0.000
0.000
V3
0.093
0.267
0.237
Out-of-Plane Bending
Allows for non-planarity when required (e.g., cyclobutanone).
Prevents inversion about chiral centers (e.g., for united atoms).
Induces planarity when required.
O
O
correct
United Atoms
Absorb non-polar hydrogen atoms into respective carbons.
Greatly reduces computational cost.
Out-of-Plane Bending
j
Wilson Angle
l
i
Between ijk plane and i-l bond
v(  )  k (    0 ) 2
k
i
Pyramidal Distance
j
Between atom i and jkl plane
v(d )  k (d  d 0 ) 2
Improper torsion
1-2-3-4 torsion
v(  )  k[1  cos( n   0 )]
l
k
Cross Terms: Stretch-Bend
Coupling between internal coordinates.
Important for reproducing structures of unusual (e.g., highly
strained ) systems and of vibrational spectra.
Stretch-Bend: As a bond angle decreases, the adjacent bonds
stretch to reduce the interaction between the 1,3 atoms.
v(l1 , l2 , ) 
kl1l2
2
[(l1  l1, 0 )  (l2  l2, 0 )](   0 )
Cross Terms: Stretch-Torsion
Stretch-Torsion: For an A-B-C-D torsion, the central B-C
bond elongates in response to eclipsing of the A-B and C-D
terminal bonds.
v(l , )  k (l  l0 ) cos n
Non-Bonded Interactions
Operate within molecules and between molecules.
Through space interactions.
Modeled as a function of an inverse power of the distance.
Soft mode.
Divided into:
• Electrostatic interactions.
• VdW interactions.
Electrostatic Interactions
A
B
Vintermoleclar  
i 1 j 1
qi q j
4 0 rij
A
qi q j
A
Vintramoleclar   
4 0 rij
i 1 j i 1
qi, qj are point charges.
Charge-charge interactions are long ranged (decay as r-1).
When qi, qj are centered on the nuclei they are referred to as partial
atomic charges.
• Fit to known electric moments (e.g., dipole, quadrupole etc.)
• Fit to thermodynamic properties.
• Ab initio Calculations
– Electrostatic potential
Rapid Methods for Calculating
Atomic Charges
Partial Equalization of Orbital Electronegativity
(Gasteiger and Marsili)
http://www2.chemie.uni-erlangen.de/software/petra/manual/manual.pdf
Electronegativity (Pauling):
“The power of an atom to attract an electron to itself”
Orbital electronegativity depends upon:
• Valence state (e.g., sp > sp3).
• Occupancy (e.g., empty > single > double).
• Charges in other orbitals.
Electrons flow from the less electronegative atoms to the more
electronegative atoms thereby equalizing the electronegativities.
Partial Equalization of Orbital
Electronegativity
For the dependency of orbital electronegativity on charge assume:
 A  aA  bAQA  cAQA2
Theoretically correct for orbitals but formally applied to atoms.
Values of a, b and c were obtained for common elements in their
usual valence states (e.g., for atom types).
Apply an interactive process:
• Assign each atom its formal charge.
• Calculate atomic electronegativities based on above assumption.
• Calculate the electron charge transferred from atom A to the
more electronegative atom B bonded to it by:
Q
(k )
 B( k )   A( k ) k
Dumping factor



A
Cation Electronegativity of the less electronegative atom
Solvent Dielectric Models
A
qi q j
A
V 
4 0 rij
i 1 j i 1
0 = Dielectric constant of vacuum (1).
For a given set of charges and distance, 0 determines the strength
of the electrostatic interactions.
Solvent effect dampen the electrostatic interactions and so can be
modeled by varying 0:
 eff = 0r
 r(protein interior) = 2-4
 r(water) = 80
Solvent Dielectric Models:
Distance Dependence
Dielectric constant increases as a function of distance
A
A
 eff   eff (r )  r  V   
i 1
qi q j
2
4

r
j i 1
0 ij
Smooth increase in eff form 1 to r as the distance increases
eff
 eff   r 
distance
 r 1
2
[(rS ) 2  2rS  2]e rS
eff varies from 1 at zero separation to the
bulk permitivity of the solvent at large
separation. As the separation between the
interacting atoms increases, more solvent
can “penetrate” between them.
Van Der Waals (VdW) Interactions
Electrostatic interactions can’t account for all non-bonded
interactions within a system (e.g., rare gases).
VdW interactions:
• Attractive (dispersive) contribution (London forces)
– Instantaneous dipoles due to electron cloud fluctuations.
– Decays as r6.
• Repulsive contribution
– Nuclei repulsion.
– At short distances (r < 1) rises as 1/r.
– At large distances decays as exp(-2r/a0); a0 the Bohr radius.
Van Der Waals (VdW) Interactions
The observed VdW potential results from a balance
between the attractive and repulsive forces.
Lennard-Jones Potential
  12    6 
v(r )  4      
r 
 r 
• Rapid to calculate
• Attractive part theoretically sound
• Repulsive part easy to calculate but
too steep
Alternative forms
rm  21 6   sum of VdW radii
energy
Modeling VdW Interactions


rm
separation
v(r )   {( rm r )12  2(rm r ) 6 }
v(r )  A r12  C r 6 A  rm12  4 12 C  2rm6  4 6
Hydrogen Bonding
H-bond geometry independent:
A C
v(r )  12  10
r
r
• r is the distance between the H-bond donor and H-bond acceptor.
H-bond geometry dependent (co-linearity with lp preferred):
vHB
 A
A  2
  12
 10  cos  Don...H ... Acc cos 4  H ... Acc ...LP
 rH ... Acc rH ... Acc 
Acc


H
N
Don
Force Field Parameterization
Choosing values for the parameters in the potential
function equations to best reproduce experimental data.
Parameterization techniques
• Trial and error
• Least square methods
Types of parameters
•
•
•
•
•
•
Stretch: natural bond length (l0) and force constants (k).
Bend: natural bond angles (0) and force constants (k).
Torsions: Vi’s.
VdW: (, VdW radii).
Electrostatic: Partial atomic charges.
Cross-terms: Cross term parameters.
Parameters - Quantity
For N atom types require:
•
•
•
•
N non-bonded parameters
N*N stretch parameters
N*N*N bend parameters
N*N*N*N torsion parameters
Example
MacroModel MM2*: 39 atom types
Stretch
Bend
Torsion
Required
1521
58,319
2,313,441
Actual
164
357
508
Parameters - Source
A force field parameterized according to data from one source (e.g.,
experimental gas phase, experimental solid phase, ab initio) will fit
data from other sources only qualitatively.
Experiment: geometries and non-bonded parameters
•
•
•
•
X-Ray crystallography
Electron diffraction
Microwave spectroscopy
Lattice energies
Advantages
• Real
Disadvantages
• Hard to obtain
• Non-uniform
• Limited availability
Parameterization - Example
O
H
O
O
H
ab initio
-0.69
ABMER
-1.25
parameterized AMBER -0.68
5.00
Energy (kcal/mol)
5.00
Energy (kcal/mol)
O
relative energy (kcal/mol)
4.00
3.00
2.00
1.00
0.00
4.00
3.00
2.00
1.00
0.00
0
90
180
torsion
270
360
0
90
180
torsion
270
360
Transferability of Parameters
Assumption: The same set of parameters can be used to
model a related series of compounds:
• Prediction
• Missing parameters
– Educated guess
– Parameters derived from atomic properties (e.g., UFF)
• Reducing number of parameters
– Generalized torsions (e.g., depend only on two central atoms)
– Generalized VdW parameters (same for SP, SP2, SP3)
Above assumption breaks down for close interaction
function groups:
O
X
O
O
Existing Force Fields
AMBER (Assisted Model Building with Energy Refinement)
• Parameterized specifically for proteins and nucleic acids.
• Uses only 5 bonding and non-bonding terms along with a
sophisticated electrostatic treatment.
• No cross terms are included.
• Results can be very good for proteins and nucleic acids, less so
for other systems.
CHARMM (Chemistry at Harvard Macromolecular Mechanics)
• Originally devised for proteins and nucleic acids.
• Now used for a range of macromolecules, molecular dynamics,
solvation, crystal packing, vibrational analysis and QM/MM
studies.
• Uses 5 valence terms, one of which is electrostatic term.
• Basis for other force fields (e.g., MOIL).
Existing Force Fields
GROMOS (Gronigen molecular simulation)
• Popular for predicting the dynamical motion of molecules and
bulk liquids.
• Also used for modeling biomolecules.
• Uses 5 valence terms, one of which is an electrostatic term.
MM1, 2, 3, 4
• General purpose force fields for (mono-functional) organic
molecules.
• MM2 was parameterized for a lot of functional groups.
• MM3 is probably one of the most accurate ways of modeling
hydrocarbons.
• MM4 is very new and little is known about its performance.
• The force fields use 5 to 6 valence terms, one of which is an
electrostatic term and one to nine cross terms.
Existing Force Fields
MMFF (Merck Molecular Force Field)
• General purpose force fields mainly for organic molecules.
• MMFF94 was originally designed for molecular dynamics
simulations but is also widely used for geometry optimization.
• Uses 5 valence terms, one of which is an electrostatic term and
one cross term.
• MMFF was parameterized based on high level ab initio
calculations.
OPLS (Optimized Potential for Liquid Simulations)
• Designed for modeling bulk liquids.
• Has been extensively used for modeling the molecular dynamics
of biomolecules.
• Uses 5 valence terms, one of which is an electrostatic term but no
cross terms.
Existing Force Fields
Tripos (SYBYL force field)
• Designed for modeling organic and biomolecules.
• Often used for CoMFA analysis (QSAR method).
• Uses 5 valence terms, one of which is an electrostatic term.
MOMEC
• A force field for describing transition metals coordination
compounds.
• Originally parameterized to use 4 valence terms but not an
electrostatic term.
• Metal-ligand interactions consist of bond stretch only.
• Coordination sphere is maintained by non-bonding interactions
between ligands.
• Work reasonable well for octahedrally coordinated compounds.
Existing Force Fields
UFF (Universal Force Field)
• Designed for coverage of the entire periodic table.
• All force field parameters are atomic based.
• Parameters for specific interactions are derived from atomic
parameters based on a series of mixing rules, thereby
addressing the problem of parameter explosion.
• Uses 4 valence terms but not an electrostatic term.
YATI
• Designed for the accurate representation of non-bonded
interactions.
• Most often used for modeling interactions between biomolecules
and small substrate molecules.
Existing Force Fields
CVFF (Consistent Valence Force Field)
• Parameterized for small organic (amides, carboxylic acids, etc.)
crystals and gas phase structures.
• Handles peptides, proteins, and a wide range of organic systems.
• Primarily intended for studies of structures and binding
energies, although it predicts vibrational frequencies and
conformational energies reasonably well.
Existing Force Fields
CFF, CFF91, CFF95 (Consistent Force Field)
• Parameterized (based on quantum mechanics calculations and
molecular simulations) for acetals, acids, alcohols, alkanes,
alkenes, amides, amines, aromatics, esters, and ethers.
• Useful for predicting:
– Small molecules: gas-phase geometries, vibrational
frequencies, conformational energies, torsion barriers,
crystal structures.
– Liquids: cohesive energy densities; for crystals: lattice
parameters, RMS atomic coordinates, sublimation energies.
– Macromolecules: protein crystal structures, polycarbonates
and polysaccharides (CFF91,95).
Existing Force Fields
Dreiding
• Parameters derived by a rule based approach.
• Good coverage of the periodic table.
• Good coverage for organic, biological and main-group inorganic
molecules.
• Only moderately accurate for geometries, conformational energies,
intermolecular binding energies, and crystal packing.
Energy Minimization
Introduction
A system of N atoms is defined by 3N Cartesian coordinates or
3N-6 internal coordinates. These define a multi-dimensional
potential energy surface (PES).
• Minima (stable conformations)
• Maxima
• Saddle points (transition states)
energy
A PES is characterized by
stationary points:
coordinates
Classification of Stationary Points
1st Derivative
0
0
0
Type
Minimum
Maximum
Saddle point
2nd Derivative*
positive
negative
1 negative
20.0
16.0
energy
transition state
12.0
8.0
local minimum
4.0
global minimum
0.0
0
90
180
270
360
coordinate
*
Refers to the eigenvalues of the second derivatives (Hessian) matrix
Population of Minima
Active Structure
Most populated minimum
Global minimum
Most minimization method can only go downhill and so locate
the closest (downhill sense) minimum.
No minimization method can guarantee the location of the
global energy minimum.
No method has proven the best for all problems.
A General Minimization Scheme
Starting point x0
Minimum?
No
Calculate
xk+1 = f(xk)
yes
Stop
Sequential Univariate Method
For each coordinate:
• Generate two new points (xi+dxi, xi+2dxi) and calculate their
energies.
• Fit a parabola to xi (1), xi+dxi (2), xi+2dxi (3) and locate its
minimum (4).
• Set the coordinate of xi to the parabola minimum.
• When the change in all coordinates is small enough, stop.
This method requires
fewer function evaluations
than the Simplex method
but is still slow to
converge.
Summary
Simplex, Sequential Univariate (memory requirements scale as N)
• Robust (i.e., can be initiated from poor starting geometries).
• Slow to converge even on quadratic surfaces.
• Require many function evaluations (especially Simplex).
Steepest Descent (memory requirements scale as 3N)
• Robust.
• Convergence guaranteed on quadratic surfaces.
• Slow to converge especially near the minimum.
Conjugate gradient, Quasi NR (memory requirements scale as (3N)2)
• Converges in approximately N steps for N degrees of freedom.
• Convergence guaranteed on quadratic surfaces.
• Unstable away from the minimum (i.e., when surface not quadratic).
• CG is the method of choice for large systems.
Newton-Raphson (memory requirements scale as (3N)2)
• Converges in one step on quadratic surfaces.
• Unstable away from the minimum.
• Computationally expensive.