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Molecular Modeling

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Molecular Modeling

Part I.

A Brief Introduction to

Molecular 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.

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

- 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) = De{1- 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)

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

Estretch = ks/2 (l - l0)2

(Hooke’s law force...

harmonic oscillator)

graph: C-C; C=O

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.)

Ebend = kb/2 (q - q0)2

graph: sp3 C-C-C

(Likewise, cubic and higher

terms are added for better fit

with experimental observations)

- 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

Etorsion = 0.5 V1 (1 + cos f) + 0.5 V2 (1 + cos 2f) +

0.5 V3 (1 + cos 3f)

EvdW = A/r12 - B/r6

Lennard-Jones or

6-12 potential

combination of a repulsive

term [A] and an attractive term [B]

EvdW = A (B/r ) - C/r6

Buckingham potential

(essentially repulsion only, especially at short distances)

EH-Bond = A/r12 - B/r10

(Lennard-Jones type,

with a 10, 12 potential)

E electrostatic = q1q2 / cer

(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)

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!!!)

- 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.

- 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).

- Equal to “steric energy” plus sum of group enthalpy values (CH2, CH3, C=O, 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.

SybylMM+MM3Expt

CH3CH3

C-C1.554 1.5321.531 1.526

C-H1.095 1.1151.113 1.109

CH3COCH3

C-C1.518 1.5171.516 1.522

C-H1.107 1.114 1.111 1.110

C=O1.223 1.210 1.211 1.222

SybylMM+MM3

CH3CH3

H-C-C109.5111.0111.4

H-C-H109.4107.9107.5

CH3COCH3

C-C-C116.9116.6116.1

H-C-H109.1108.3107.9

C-C-O121.5121.7122.0

- MM2 / MM3 (Allinger) best; general purpose
- MMX (Gilbert) added TS’s, 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).

- 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)

- 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.

- 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