Molecular Modeling

1
Molecular Modeling

• Part I.
• A Brief Introduction to
• Molecular Mechanics

2
Molecular Modeling (Mechanics)

• Calculation of preferred (lowest energy)
  molecular structure and energy based on
  principles of classical (Newtonian) physics
• Steric energy based on energy increments due to
  deviation from some ideal geometry
• Components include bond stretching, bond angle
  bending, torsional angle deformation,
  dipole-dipole interactions, van der Waals forces,
  H-bonding and other terms.

3
Components of Steric Energy

• E steric E stretch E bend E torsion
  E vdW E stretch-bend E H- bonding
  E electrostatic E
  dipole-dipole E other

4
Bond Stretching Energy

• A Morse potential best describes energy of bond
  stretching ( compression), but it is too complex
  for efficient calculation and it requires three
  parameters for each bond.
• n(l) De1- exp -a (l - l0)2
• if De depth of potential energy minimum,
• a w?(m/2De) where m is the reduced mass
  and w is related to the bond stretching
  frequency by w ?(k/m)

5
Morse potential Hookes Law

• Most bonds deviate in length very little from
  their equilibrium values, so simpler mathematical
  expressions, such as the harmonic oscillator
  (Hookes Law) have been used to model the bond
  stretching energy
• n(l) k/2(l - l0)2

6
Bond Stretching Energy

• Estretch ks/2 (l - l0)2
• (Hookes law force...
• harmonic oscillator)
• graph C-C CO

7
Higher order terms give better fit

• With cubic and higher terms
• n(l) k/2(l - l0)2 1- k(l - l0)
• - k(l - l0)2
• - k(l - l0)3 -

(cubic terms give better fit in region near
minimum inclusion of a fourth power term
eliminates the maximum in the cubic fcn.)
8
Bond Angle Bending Energy

• Ebend kb/2 (q - q0)2
• graph sp3 C-C-C

(Likewise, cubic and higher terms are added for
better fit with experimental observations)
9
Torsional Energy

• Related to the rotation barrier (which also
  includes some other contributions, such as van
  der Waals interactions).
• The potential energy increases periodically as
  eclipsing interactions occur during bond rotation.

gauche
Eclipsed
eclipsed
Anti
10
Torsional Energy

• Etorsion 0.5 V1 (1 cos f) 0.5 V2 (1 cos
  2f)
• 0.5 V3 (1 cos 3f)

11
Torsional Barrier in n-Butane
12
Butane Barrier is Sum of Two Terms V1(1
cos f) V3(1 cos 3f)
13
van der Waals Energy

• EvdW A/r12 - B/r6
• Lennard-Jones or
• 6-12 potential

combination of a repulsive term A and an
attractive term B
14
van der Waals Energy...

• EvdW A (B/r ) - C/r6
• Buckingham potential
• (essentially repulsion only, especially at
  short distances)

15
Hydrogen Bonding Energy

• EH-Bond A/r12 - B/r10
• (Lennard-Jones type,
• with a 10, 12 potential)

16
Electrostatic Energy

• E electrostatic q1q2 / ce r
• (attractive or repulsive, depending on
  relative signs of charge value depends
  inversely on permitivity of free space, or the
  dielectric constant of the hypothetical medium)

17
Dipole-Dipole Energy

• Calculated as the three dimensional vector
  sum of the bond dipole moments, also considering
  the permitivity (related to dielectric constant)
  of the medium (typical default value is 1.5)
• (this is too complicated to demonstrate!!!)

18
Use of Cut-offs

• Van der Waals forces, hydrogen bonding,
  electrostatic forces, and dipole-dipole forces
  have dramatic distance dependencies beyond a
  certain distance, the force is negligible, yet it
  still costs the computer to calculate it.
• To economize, cut-offs are often employed for
  these forces, typically somewhere between 10 and
  15Å.

19
Properties Calculated

• Optimized geometry (minimum energy conformation)
• Equilibrium bond lengths, bond angles, and
  dihedral (torsional) angles
• Dipole moment (vector sum of bond dipoles)
• Enthalpy of Formation (in some programs).

20
Enthalpy of Formation

• Equal to steric energy plus sum of group
  enthalpy values (CH2, CH3, CO, etc.), with a few
  correction terms
• Not calculated by all molecular mechanics
  programs (e.g., HyperChem and Titan)
• Calculated values are generally quite close to
  experimental values for common classes of
  organic compounds.

21
Enthalpy of Formation...
22
Enthalpy of Formation...
23
Bond Lengths

• Sybyl MM MM3 Expt
• CH3CH3
• C-C 1.554 1.532 1.531 1.526
• C-H 1.095 1.115 1.113 1.109
• CH3COCH3
• C-C 1.518 1.517 1.516 1.522
• C-H 1.107 1.114 1.111 1.110
• CO 1.223 1.210 1.211 1.222

24
Bond Angles

• Sybyl MM MM3
• CH3CH3
• H-C-C 109.5 111.0 111.4
• H-C-H 109.4 107.9 107.5
• CH3COCH3
• C-C-C 116.9 116.6 116.1
• H-C-H 109.1 108.3 107.9
• C-C-O 121.5 121.7 122.0

25
Common Force Fields

• MM2 / MM3 (Allinger) best general purpose
• MMX (Gilbert) added TSs, other elements good
• MM (Ostlund) in HyperChem general good
• OPLS (Jorgenson) proteins and nucleic acids
• AMBER (Kollman) proteins and nucleic acids
• BIO (Karplus) CHARMm nucleic acids
• MacroModel (Still) biopolymers, general good
• MMFF (Merck Pharm.) general newer good
• Sybyl in Alchemy2000, general (poor).

26
Molecular Modeling Programs

• HyperChem (MM, OPLS, AMBER, BIO)
• Spartan (MM3, MMFF, Sybyl on SGI or via
  x-windows from pc)
• Titan (like Spartan, but faster MMFF)
• Alchemy2000 (Sybyl)
• Gaussian 03 (on our SGIs linux cluster and on
  unix computers at NCSU and ECU no graphical
  interface not for molecular mechanics MO
  calculations only)

27
Steps in Performing Molecular Mechanics
Calculations

• Construct graphical representation of molecule to
  be modeled (front end)
• Select forcefield method and termination
  condition (gradient, cycles, or time)
• Perform geometry optimization
• Examine output geometry... is it reasonable?
• Search for global minimum.

28
Energy Minimization

• Local minimum vs global minimum
• Many local minima only ONE global minimum
• Methods Newton-Raphson (block diagonal),
  steepest descent, conjugate gradient, others.

local minima
global minimum