Molecular ModelingMolecular ModelingFundamentals: Fundamentals: Modus in Modus in

SilicoSilico

C372C372

Introduction to Introduction to Cheminformatics IICheminformatics II

Kelsey ForsytheKelsey Forsythe

"Every attempt to employ mathematical methods in the study of "Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and chemical questions must be considered profoundly irrational and

contrary to the spirit in chemistry. If mathematical analysis contrary to the spirit in chemistry. If mathematical analysis should ever hold a prominent place in chemistry - an aberration should ever hold a prominent place in chemistry - an aberration which is happily almost impossible - it would occasion a rapid which is happily almost impossible - it would occasion a rapid

and widespread degeneration of that science." A. Comte and widespread degeneration of that science." A. Comte (1830)(1830)

1992 Nobel Prize in Chemistry1992 Nobel Prize in Chemistry

Rudolph Marcus (Theory of Electron Rudolph Marcus (Theory of Electron Transfer)Transfer)

1998 Nobel Prize in Chemistry1998 Nobel Prize in Chemistry

John Pople (John Pople (ab initioab initio))

Walter Kohn (DFT-density functional Walter Kohn (DFT-density functional theory)theory)

Characteristics of Molecular Characteristics of Molecular ModelingModeling

Representing behavior of molecular Representing behavior of molecular systemssystems Visual rendering of moleculesVisual rendering of molecules

Tinker toysTinker toys Tinker Program (Washington Univ. St. Louis)Tinker Program (Washington Univ. St. Louis)

Mathematical rendering of molecular Mathematical rendering of molecular interactionsinteractions Newton’s Laws - Kinetic Theory of GasesNewton’s Laws - Kinetic Theory of Gases Matrix Algebra - Quantum TheoryMatrix Algebra - Quantum Theory

Graph Theory? Informatics!!Graph Theory? Informatics!!

Molecular Modeling Molecular Modeling

++ ==

Underlying equations:Underlying equations:empirical (approximate, soluble)empirical (approximate, soluble)

--Morse Potential Morse Potential

ab initioab initio (exact, insoluble (exact, insoluble (less hydrogen atom)(less hydrogen atom)))--Schrodinger Wave EquationSchrodinger Wave Equation

VHH D0(1 e a(R R0 ))2

ˆ H E

Valence Valence Bond Bond TheoryTheory

EnergyEnergyEnergy = ?Energy = ?E=KE + PEE=KE + PE

Depends on underlying equations/assumptions:Depends on underlying equations/assumptions:

Energy of all/some of particles?Energy of all/some of particles?Energy = 0?Energy = 0?

EEMMFFMMFF NOTNOT E EHFHF

E H G

ElectrostaticsElectrostatics Coulombs Law Coulombs Law

Permittivity used for vacuumPermittivity used for vacuum Point particles?Point particles? Solvent effectsSolvent effects

Poisson EquationPoisson Equation Used to calculate electronic propertiesUsed to calculate electronic properties

PE qiq j

40rij

F

(PE)

2

Atomic UnitsAtomic Units

PE qiq j

40rij

PE qiq j

rij

ThermodynamicsThermodynamics

How might we compute relevant How might we compute relevant thermodynamic quantities?thermodynamic quantities? Equipartition TheoremEquipartition Theorem Harmonic Oscillator ApproximationHarmonic Oscillator Approximation

Quantum MechanicsQuantum Mechanics All chemical properties for a system are given by the Schrodinger equationAll chemical properties for a system are given by the Schrodinger equation

No closed form solutions for systems of more than two-bodies (H-atom)No closed form solutions for systems of more than two-bodies (H-atom) Number of equations too numerous for computation/storage (informatics problem?)Number of equations too numerous for computation/storage (informatics problem?)

ˆ H E

Schrodinger’s EquationSchrodinger’s Equation

- Hamiltonian operator- Hamiltonian operator

Gravity? Gravity?

ˆ H E

ˆ H

ˆ H ˆ T ˆ V

2

2mi

2

i

N

Ceie j

ri rji j

N

Hydrogen Molecule Hydrogen Molecule HamiltonianHamiltonian

Born-Oppenheimer ApproximationBorn-Oppenheimer Approximation

Now Solve Electronic ProblemNow Solve Electronic Problem

221221112121

22

21

22

21

2

111111

2ˆ

ˆˆˆ

epepepepppee

e

e

e

e

p

p

p

p

rrrrrrC

mmmmH

VTH

ˆ H el ˆ T el ˆ V el nuclei Vnuclei

ˆ H el 2

2

e12

me

e 2

2

me

C

1

re1e 2

1

rp1e1

1

rp1e 2

1

rp2e1

1

rp 2e 2

C

1

rp1p 2

cons tan t

Electronic Schrodinger Electronic Schrodinger EquationEquation

Solutions:Solutions:

, the basis set, are of a known form , the basis set, are of a known form Need to determine coefficients (cNeed to determine coefficients (cm)

Wavefunctions gives probability ( ) of Wavefunctions gives probability ( ) of finding electrons in space (e. g. s,p,d and f finding electrons in space (e. g. s,p,d and f orbitals)orbitals)

Molecular orbitals are formed by linear Molecular orbitals are formed by linear combinations of electronic orbitals (LCAO)combinations of electronic orbitals (LCAO)

(r ) cm m (

r )

m

F

m (r )

ˆ O * ˆ O d

*

Statistical MechanicsStatistical Mechanics Molecular description of thermodynamicsMolecular description of thermodynamics

Temperature represents average state for system of moleculesTemperature represents average state for system of molecules

Energy of system is not energy of each molecule - distributionEnergy of system is not energy of each molecule - distribution

Condensed Phase - Ideal Gas Law not applicable. Condensed Phase - Ideal Gas Law not applicable. Boltzmann averaging Boltzmann averaging Use Monte Carlo for spatial/configurational averaging or molecular dynamics to average a property Use Monte Carlo for spatial/configurational averaging or molecular dynamics to average a property

(ergodic hypothesis)(ergodic hypothesis)

1

N

1

2m v 2

3

2kT

w(i)e E i / kT

Geometry OptimizationGeometry Optimization

First Derivative is Zero - At First Derivative is Zero - At minimum/minimum/maximummaximum

As N increases so does As N increases so does dimensionality/complexity/beauty/difficuldimensionality/complexity/beauty/difficultyty Multi-dimensional (macromolecules, Multi-dimensional (macromolecules,

proteins)proteins) Conjugate gradient methodsConjugate gradient methods Monte Carlo methodsMonte Carlo methods

dV (r )

dr

0

Empirical ModelsEmpirical Models Simple/Elegant?Simple/Elegant? Intuitive?-Vibrations ( ) Intuitive?-Vibrations ( ) Major Drawbacks:Major Drawbacks:

Does not include quantum mechanical effectsDoes not include quantum mechanical effects No information about bonding (No information about bonding (e) Not generic (organic inorganic)Not generic (organic inorganic)

InformaticsInformatics Interface between parameter data sets and Interface between parameter data sets and

systems of interest systems of interest Teaching computers to develop new potentials Teaching computers to develop new potentials

from existing math templatesfrom existing math templates

rkF

MMFF PotentialMMFF Potential

E = E = EEbondbond + + EEangleangle + + EEangleangle

-bond -bond + + EEtorsiontorsion + + EEVDW VDW + + EEelectrostaticelectrostatic

Merck Molecular Force FieldMerck Molecular Force Field-Common organics/biopolymers-Common organics/biopolymers

MMFF EnergyMMFF Energy

StretchingStretching

202020 )(

12

7)(1*)( ijijijijijijbondbond rrcsrrcsrrKE

MMFF EnergyMMFF Energy

BendingBending

)(1*)( 020ijkijkijkijkangle cbKE

MMFF EnergyMMFF Energy

Stretch-Bend InteractionsStretch-Bend Interactions

000 )()( ijkijkkjkjkjiijijijkanglebond rrKrrKE

MMFF EnergyMMFF Energy

Torsion (4-atom bending)Torsion (4-atom bending)

3cos12cos1cos15.0 321 VVVEtorsion

MMFF EnergyMMFF Energy

Analogous to Lennard-Jones 6-12 Analogous to Lennard-Jones 6-12 potentialpotential London Dispersion ForcesLondon Dispersion Forces Van der Waals RepulsionsVan der Waals Repulsions

2

07.0

07.1

07.0

07.17*7

7*7

*

*

ijij

ij

ijij

ijijVDW

RR

R

RR

RE

Intermolecular/atomic Intermolecular/atomic modelsmodels

General form:General form:

Lennard-Jones Lennard-Jones

V V (r) V (ri,rj ) V (ri,rj ,rk ) .....i jjk

N

i j

N

V (rij )4 r

12

r

6

Van derWaals repulsionVan derWaals repulsion London AttractionLondon Attraction

MMFF EnergyMMFF Energy

Electrostatics (ionic compounds) Electrostatics (ionic compounds) D – Dielectric ConstantD – Dielectric Constant - electrostatic buffering constant- electrostatic buffering constant

nij

jiticelectrosta

RD

qqE

8.35E-28 8.77567E+14 20568787140 2.03098E-18 1.05374E-188.35E-28 8.77567E+14 20568787140 1.77569E-18 9.66155E-198.35E-28 8.77567E+14 20568787140 1.54682E-18 8.82365E-198.35E-28 8.77567E+14 20568787140 1.34201E-18 8.02375E-198.35E-28 8.77567E+14 20568787140 1.15913E-18 7.26185E-198.35E-28 8.77567E+14 20568787140 9.96207E-19 6.53795E-198.35E-28 8.77567E+14 20568787140 8.51451E-19 5.85205E-198.35E-28 8.77567E+14 20568787140 7.23209E-19 5.20415E-198.35E-28 8.77567E+14 20568787140 6.09973E-19 4.59425E-198.35E-28 8.77567E+14 20568787140 5.10362E-19 4.02235E-198.35E-28 8.77567E+14 20568787140 4.2311E-19 3.48845E-198.35E-28 8.77567E+14 20568787140 3.47061E-19 2.99255E-198.35E-28 8.77567E+14 20568787140 2.81155E-19 2.53465E-198.35E-28 8.77567E+14 20568787140 2.24426E-19 2.11475E-198.35E-28 8.77567E+14 20568787140 1.75987E-19 1.73285E-198.35E-28 8.77567E+14 20568787140 1.35031E-19 1.38895E-198.35E-28 8.77567E+14 20568787140 1.0082E-19 1.08305E-198.35E-28 8.77567E+14 20568787140 7.26787E-20 8.15147E-208.35E-28 8.77567E+14 20568787140 4.99924E-20 5.85247E-208.35E-28 8.77567E+14 20568787140 3.22001E-20 3.93347E-208.35E-28 8.77567E+14 20568787140 1.87901E-20 2.39447E-208.35E-28 8.77567E+14 20568787140 9.29638E-21 1.23547E-208.35E-28 8.77567E+14 20568787140 3.29443E-21 4.56475E-21

Empirical Potential for Hydrogen Molecule

0

2E-19

4E-19

6E-19

8E-19

1E-18

1.2E-18

1.4E-18

0 0.5 1 1.5 2 2.5 3 3.5 4

8.35E-28 8.77567E+14 20568787140 2.03098E-18 1.05374E-188.35E-28 8.77567E+14 20568787140 1.77569E-18 9.66155E-198.35E-28 8.77567E+14 20568787140 1.54682E-18 8.82365E-198.35E-28 8.77567E+14 20568787140 1.34201E-18 8.02375E-198.35E-28 8.77567E+14 20568787140 1.15913E-18 7.26185E-198.35E-28 8.77567E+14 20568787140 9.96207E-19 6.53795E-198.35E-28 8.77567E+14 20568787140 8.51451E-19 5.85205E-198.35E-28 8.77567E+14 20568787140 7.23209E-19 5.20415E-198.35E-28 8.77567E+14 20568787140 6.09973E-19 4.59425E-198.35E-28 8.77567E+14 20568787140 5.10362E-19 4.02235E-198.35E-28 8.77567E+14 20568787140 4.2311E-19 3.48845E-198.35E-28 8.77567E+14 20568787140 3.47061E-19 2.99255E-198.35E-28 8.77567E+14 20568787140 2.81155E-19 2.53465E-198.35E-28 8.77567E+14 20568787140 2.24426E-19 2.11475E-198.35E-28 8.77567E+14 20568787140 1.75987E-19 1.73285E-198.35E-28 8.77567E+14 20568787140 1.35031E-19 1.38895E-198.35E-28 8.77567E+14 20568787140 1.0082E-19 1.08305E-198.35E-28 8.77567E+14 20568787140 7.26787E-20 8.15147E-208.35E-28 8.77567E+14 20568787140 4.99924E-20 5.85247E-208.35E-28 8.77567E+14 20568787140 3.22001E-20 3.93347E-208.35E-28 8.77567E+14 20568787140 1.87901E-20 2.39447E-208.35E-28 8.77567E+14 20568787140 9.29638E-21 1.23547E-208.35E-28 8.77567E+14 20568787140 3.29443E-21 4.56475E-21

Empirical Potential for Hydrogen Molecule

0

2E-19

4E-19

6E-19

8E-19

1E-18

1.2E-18

1.4E-18

0 0.5 1 1.5 2 2.5 3 3.5 4

Hydrogen MoleculeHydrogen Molecule

Bond DensityBond Density