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Molecular Modeling

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

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.

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

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)

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

Bond Stretching Energy

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

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.)

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)

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

Torsional Energy

- Etorsion 0.5 V1 (1 cos f) 0.5 V2 (1 cos

2f) - 0.5 V3 (1 cos 3f)

Torsional Barrier in n-Butane

Butane Barrier is Sum of Two Terms V1(1

cos f) V3(1 cos 3f)

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

van der Waals Energy...

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

short distances)

Hydrogen Bonding Energy

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

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)

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!!!)

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.

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).

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.

Enthalpy of Formation...

Enthalpy of Formation...

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

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

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).

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)

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.

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

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