Molecular ModelingMolecular Modeling

Part I.Part I.

A Brief Introduction to A Brief Introduction to

Molecular MechanicsMolecular Mechanics

Molecular Modeling (Mechanics)Molecular Modeling (Mechanics)

Calculation of preferred (lowest energy) molecular Calculation of preferred (lowest energy) molecular structure and energy based on principles of structure and energy based on principles of classical (Newtonian) physics classical (Newtonian) physics

““Steric energy” based on energy increments due to Steric energy” based on energy increments due to deviation from some “ideal” geometrydeviation from some “ideal” geometry

Components include bond stretching, bond angle Components include bond stretching, bond angle bending, torsional angle deformation, dipole-dipole bending, torsional angle deformation, dipole-dipole interactions, van der Waals forces, H-bonding and interactions, van der Waals forces, H-bonding and other terms.other terms.

Components of “Steric Energy”Components of “Steric Energy”

E E stericsteric = E = E stretch stretch + E + E bendbend + E + E torsiontorsion + E + E vdWvdW

+ E + E stretch-bendstretch-bend + E + E H- bondingH- bonding

+ E + E electrostatic electrostatic + + E E dipole-dipole dipole-dipole + E + E other other

Bond Stretching EnergyBond Stretching Energy

A Morse potential best describes energy of bond A Morse potential best describes energy of bond stretching (& compression), but it is too complex stretching (& compression), but it is too complex for efficient calculation and it requires three for efficient calculation and it requires three parameters for each bond. parameters for each bond.

(l) = D(l) = Dee{1- exp [-a (l - l{1- exp [-a (l - l00)]})]}2 2

if: Dif: Dee = depth of potential energy minimum, = depth of potential energy minimum,

a = a = ((/2D/2Dee) where ) where is the reduced mass and is the reduced mass and

is related to the bond stretching frequency by is related to the bond stretching frequency by (k/(k/))

Morse potential & Hooke’s LawMorse potential & Hooke’s Law

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

(l) = k/2(l - l(l) = k/2(l - l00))22

Bond Stretching EnergyBond Stretching Energy

EEstretchstretch = k = kss/2 (l - l/2 (l - l00))22

(Hooke’s law force...(Hooke’s law force...

harmonic oscillator)harmonic oscillator)

graph: C-C; graph: C-C; C=OC=O

Bond Stretching Energy

0

50

100

150

200

250

300

350

0 1 2 3

Internuclear Distance

En

erg

y, k

cal/

mo

l

Higher order terms give better fitHigher order terms give better fit

With cubic and higher terms:With cubic and higher terms:

(l) = k/2(l - l(l) = k/2(l - l00))2 2 [1- k’(l - l[1- k’(l - l00))

- k’’(l - l- k’’(l - l00))22

- k’’’(l - l- k’’’(l - l00))33 - …] - …]

(cubic terms give better fit(cubic terms give better fitin region near minimum; inclusionin region near minimum; inclusionof a fourth power term eliminates the maximum in the cubic fcn.)of a fourth power term eliminates the maximum in the cubic fcn.)

Bond Angle Bending EnergyBond Angle Bending Energy

EEbend bend = k= kbb/2 (/2 ( - - 00))22

graph: graph: spsp33 C-C-C C-C-C

Bond Angle Deformation, C-C-C

0

5

10

15

20

25

30

35

106 108 110 112

Bond Angle

En

erg

y, k

cal/

mo

l

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

Torsional EnergyTorsional Energy

Related to the rotation Related to the rotation “barrier” (which also “barrier” (which also includes some other includes some other contributions, such as van contributions, such as van der Waals interactions).der Waals interactions).

The potential energy The potential energy increases periodically as increases periodically as eclipsing interactions eclipsing interactions occur during bond occur during bond rotation.rotation.

CH3

H H

H

H

CH3

CH3

H H

H

HCH3

CH3

H H

H

HCH3

CH3

H H

CH3

HH

CH3

H H

H

CH3

H

gauche Eclipsed

eclipsed Anti

Torsional EnergyTorsional Energy

EEtorsiontorsion = = 0.5 V0.5 V11 (1 + cos (1 + cos )) + + 0.5 V0.5 V22 (1 + cos 2 (1 + cos 2)) + +

0.5 V0.5 V33 (1 + cos 3 (1 + cos 3))

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 30 60 90 120 150 180 210 240 270 300 330 360

Torsion Angle

En

erg

y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 30 60 90 120 150 180 210 240 270 300 330 360

Torsion Angle

En

erg

y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 30 60 90 120 150 180 210 240 270 300 330 360

Torsion Angle

En

erg

y

Torsional Barrier in n-ButaneTorsional Barrier in n-Butane

Rotational Barrier in n-Butane

0

1

2

3

4

5

6

7

0 30 60 90 120

150

180

210

240

270

300

330

360

Torsion Angle

Kcal

/ m

ol

Butane Barrier is Sum of Two Terms: V1(1+ cos + V3(1 + cos 3Butane Barrier is Sum of Two Terms: V1(1+ cos + V3(1 + cos 3

0

0.5

1

1.5

2

2.5

0 30 60 90 120 150 180 210 240 270 300 330 360

Torsion Angle

En

erg

y

van der Waals Energyvan der Waals Energy

EEvdWvdW = A/r = A/r12 12 - B/r- B/r66

Lennard-Jones or Lennard-Jones or

6-12 potential6-12 potential

van der Waals Energy

-0.5

0

0.5

1

1.5

2

2.5

0 1 2 3 4

Nonbonded Internuclear Distance

En

erg

y, k

cal/

mo

l

combination of a repulsive combination of a repulsive term [A] and an attractive term [B]term [A] and an attractive term [B]

van der Waals Energy...van der Waals Energy...

EEvdWvdW = A = A (B/r ) (B/r ) - C/r- C/r66

Buckingham potentialBuckingham potential

(essentially repulsion (essentially repulsion only, especially at only, especially at short distances)short distances)

Buckingham Potential

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4

Nonbonded Internuclear Diatance

En

erg

y, k

cal/

mo

l

Hydrogen Bonding EnergyHydrogen Bonding Energy

EEH-BondH-Bond = A/r = A/r12 12 - B/r- B/r1010

(Lennard-Jones type,(Lennard-Jones type,

with a 10, 12 potential)with a 10, 12 potential)

Hydrogen Bonding

-10

0

10

20

30

40

0 2 4 6 8 10 12 14 16

Internuclear Distance

En

erg

y, k

cal/m

ol

Electrostatic EnergyElectrostatic Energy

E E electrostaticelectrostatic = q = q11qq2 2 / c/ crr

((attractiveattractive or or repulsiverepulsive, , depending on relative signs of depending on relative signs of charge; value depends charge; value depends inversely on inversely on permitivity of free permitivity of free spacespace, or the , or the dielectric dielectric constant constant of the hypothetical of the hypothetical medium)medium)

Electrostatic Energy

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

Internuclear Distance

En

erg

y, k

ca

l/m

ol

Dipole-Dipole EnergyDipole-Dipole Energy

Calculated as the three dimensional vector Calculated as the three dimensional vector

sum of the bond dipole moments, also sum of the bond dipole moments, also considering the considering the permitivitypermitivity (related to (related to dielectric constant)dielectric constant) of the medium (typical of the medium (typical default value is 1.5)default value is 1.5)

(this is too complicated to demonstrate!!!)(this is too complicated to demonstrate!!!)

Use of Cut-offsUse of Cut-offs

Van der Waals forces, hydrogen bonding, Van der Waals forces, hydrogen bonding, electrostatic forces, and dipole-dipole forces electrostatic forces, and dipole-dipole forces have dramatic distance dependencies; beyond have dramatic distance dependencies; beyond a certain distance, the force is negligible, yet a certain distance, the force is negligible, yet it still “costs” the computer to calculate it.it still “costs” the computer to calculate it.

To economize, “cut-offs” are often employed To economize, “cut-offs” are often employed for these forces, typically somewhere between for these forces, typically somewhere between 10 and 15Å.10 and 15Å.

Properties Calculated Properties Calculated

Optimized geometry (minimum energy Optimized geometry (minimum energy conformation)conformation)

Equilibrium bond lengths, bond angles, and Equilibrium bond lengths, bond angles, and dihedral (torsional) anglesdihedral (torsional) angles

Dipole moment (vector sum of bond dipoles)Dipole moment (vector sum of bond dipoles) Enthalpy of Formation (in some programs).Enthalpy of Formation (in some programs).

Enthalpy of FormationEnthalpy of Formation

Equal to “steric energy” plus sum of group Equal to “steric energy” plus sum of group enthalpy values (CHenthalpy values (CH22, CH, CH33, C=O, etc.), with a , C=O, etc.), with a

few correction termsfew correction terms Not calculated by all molecular mechanics Not calculated by all molecular mechanics

programs (e.g., programs (e.g., HyperChemHyperChem and and TitanTitan)) Calculated values are generally quite close Calculated values are generally quite close

to experimental values for common classes to experimental values for common classes of organic compounds.of organic compounds.

Enthalpy of Formation...Enthalpy of Formation...

MMX calc. Exp. Hf (kcal/mol)

-29.53 -29.5

-18.26 -18.4

+5.96 +6.8

+13.37 +12.7

Enthalpy of Formation...Enthalpy of Formation...

O

CH3CH2CH3

MMX calc. Exp.

Hf (kcal/mol)

CH3

-44.09 -44.02

-24.77 -24.82

-37.02 -36.99

Bond LengthsBond Lengths

SybylSybyl MM+MM+ MM3MM3 Expt Expt

CHCH33CHCH33

C-CC-C 1.5541.554 1.532 1.532 1.531 1.5261.531 1.526

C-HC-H 1.0951.095 1.115 1.115 1.113 1.1091.113 1.109

CHCH33COCHCOCH33

C-CC-C 1.5181.518 1.517 1.517 1.5161.516 1.522 1.522

C-HC-H 1.1071.107 1.114 1.114 1.111 1.1101.111 1.110

C=OC=O 1.2231.223 1.210 1.210 1.211 1.2221.211 1.222

Bond AnglesBond Angles

SybylSybyl MM+MM+ MM3MM3

CHCH33CHCH33

H-C-CH-C-C 109.5109.5 111.0111.0 111.4111.4

H-C-HH-C-H 109.4109.4 107.9107.9 107.5107.5

CHCH33COCHCOCH33

C-C-CC-C-C 116.9116.9 116.6116.6 116.1116.1

H-C-HH-C-H 109.1109.1 108.3108.3 107.9107.9

C-C-OC-C-O 121.5121.5 121.7121.7 122.0122.0

Common Force FieldsCommon Force Fields

MM2 / MM3 MM2 / MM3 (Allinger) (Allinger) bestbest; general purpose; general purpose MMXMMX (Gilbert) added TS’s, other elements; good (Gilbert) added TS’s, other elements; good MM+ MM+ (Ostlund) in HyperChem; general; good (Ostlund) in HyperChem; general; good OPLS OPLS (Jorgenson) proteins and nucleic acids(Jorgenson) proteins and nucleic acids AMBERAMBER (Kollman) proteins and nucleic acids + (Kollman) proteins and nucleic acids + BIO+ BIO+ (Karplus) CHARMm; nucleic acids(Karplus) CHARMm; nucleic acids MacroModelMacroModel (Still) biopolymers, general; good (Still) biopolymers, general; good MMFFMMFF (Merck Pharm.) general; newer; good (Merck Pharm.) general; newer; good SybylSybyl in Alchemy2000, general (poor). in Alchemy2000, general (poor).

Molecular Modeling ProgramsMolecular Modeling Programs

HyperChem HyperChem (MM+, OPLS, AMBER, BIO+)(MM+, OPLS, AMBER, BIO+) SpartanSpartan (MM3, MMFF, Sybyl; on SGI or (MM3, MMFF, Sybyl; on SGI or viavia

x-windows from pc) x-windows from pc) Titan Titan (like (like Spartan,Spartan, but faster; MMFF)but faster; MMFF) Alchemy2000 Alchemy2000 (Sybyl)(Sybyl) Gaussian 03 Gaussian 03 (on our SGIs linux cluster and (on our SGIs linux cluster and

on unix computers at NCSU and ECU; no on unix computers at NCSU and ECU; no graphical interface; not for molecular graphical interface; not for molecular mechanics; MO calculations only)mechanics; MO calculations only)

Steps in Performing Molecular Mechanics CalculationsSteps in Performing Molecular Mechanics Calculations

Construct graphical representation of Construct graphical representation of molecule to be modeled (“front end”)molecule to be modeled (“front end”)

Select forcefield method and termination Select forcefield method and termination condition (gradient, # cycles, or time)condition (gradient, # cycles, or time)

Perform geometry optimizationPerform geometry optimization Examine output geometry... is it reasonable?Examine output geometry... is it reasonable? Search for Search for globalglobal minimum. minimum.

Energy MinimizationEnergy Minimization

Local minimum vs Local minimum vs globalglobal minimum minimum Many local minima; only ONE Many local minima; only ONE globalglobal minimum minimum Methods: Newton-Raphson (block diagonal), Methods: Newton-Raphson (block diagonal),

steepest descent, conjugate gradient, others.steepest descent, conjugate gradient, others.

global minimumglobal minimumlocal minimalocal minima