Molecular Modeling

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) = 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)

Morse potential & Hooke’s Law • 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

Bond Stretching Energy Estretch = ks/2 (l - l0)2 (Hooke’s law force... harmonic oscillator) graph: C-C; C=O

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 / 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)

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, 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.

Enthalpy of Formation...

Bond Lengths Sybyl MM+ MM3Expt 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 C=O 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 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).

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

Molecular Modeling