Molecular Modeling and Molecular Modeling and InformaticsInformatics

C371C371

Introduction to Introduction to CheminformaticsCheminformatics

Kelsey ForsytheKelsey Forsythe

Characteristics of Molecular Characteristics of Molecular ModelingModeling

Representing behavior of molecular Representing behavior of molecular systemssystems Visual (tinker toys – LCDs) rendering of Visual (tinker toys – LCDs) rendering of

moleculesmolecules Mathematical rendering (differential Mathematical rendering (differential

equations, matrix algebra) of molecular equations, matrix algebra) of molecular interactionsinteractions

Time dependent and time independent realmsTime dependent and time independent realms

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

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VHH = D0(1− e−a(R−R0 ))2

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ˆ H Ψ = EΨ

Valence Valence Bond Bond TheoryTheory

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

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

rkFvv

−=

≠

MMFF PotentialMMFF Potential

E = E = EEbonbondd + + EEangleangle + + EEangleangle

-bond-bond + + EEtorsiontorsion + + EEVDWVDW + + EEelectrostaticelectrostatic

Atomistic Model HistoryAtomistic Model History

Atomic SpectraAtomic Spectra Balmer (1885)Balmer (1885)

Plum-Pudding ModelPlum-Pudding Model J. J. Thomson (circa 1900)J. J. Thomson (circa 1900)

QuantizationQuantization Planck (circa 1905)Planck (circa 1905)

Planetary ModelPlanetary Model Neils Bohr (circa 1913)Neils Bohr (circa 1913)

Wave-Particle DualityWave-Particle Duality DeBroglie (circa 1924)DeBroglie (circa 1924)

Schrodinger Wave EquationSchrodinger Wave Equation Erwin Schrodinger and Werner Heisenberg Erwin Schrodinger and Werner Heisenberg

Classical vs. QuantumClassical vs. Quantum TrajectoryTrajectory Real numbersReal numbers

Deterministic (“The Deterministic (“The value is ___”)value is ___”)

VariablesVariables Continuous energy Continuous energy

spectrumspectrum

WavefunctionWavefunction Complex (Real and Complex (Real and

Imaginary Imaginary components)components)

Probabilistic (“The Probabilistic (“The average value is __ ”average value is __ ”

OperatorsOperators Discrete/Quantized Discrete/Quantized

energyenergy TunnelingTunneling Zero-point energyZero-point energy

Schrodinger’s EquationSchrodinger’s Equation

- Hamiltonian operator- Hamiltonian operator

Gravity? Gravity? €

ˆ H Ψ = EΨ

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ˆ H

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ˆ H = ˆ T + ˆ V

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−h2

2mi

∇ 2

i

N

∑

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Ceie j

ri − rji< j

N

∑

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

22

)(2

1

2)(ˆ r

rrH Δ+

∂∂

−=Δ μμ

h

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

h

212212211121

22

21

2 111111

2ˆ

ˆˆˆ

ppepepepepeee

e

e

eel

nucleinucleielelel

rC

rrrrrC

mmH

VVTH

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−−−+⎭⎬⎫

⎩⎨⎧ ∇

+∇

−=

++= −

h

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

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Ψ(v r ) = cm ∗Φm (

v r )

m

F

∑

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Φm (v r )

Hydrogen MoleculeHydrogen Molecule

HOMOHOMO

LUMO LUMO

Hydrogen MoleculeHydrogen Molecule

Bond DensityBond Density

Ab Initio/Ab Initio/DFTDFT

Complete Description!Complete Description! Generic!Generic! Major Drawbacks:Major Drawbacks:

Mathematics can be cumbersomeMathematics can be cumbersome Exact solution only for hydrogenExact solution only for hydrogen

InformaticsInformatics Approximate solution time and storage intensiveApproximate solution time and storage intensive

– Acquisition, manipulation and dissemination Acquisition, manipulation and dissemination problemsproblems

Approximate MethodsApproximate Methods SCF (Self Consistent Field) Method (a.ka. SCF (Self Consistent Field) Method (a.ka.

Mean Field or Hartree Fock)Mean Field or Hartree Fock) Pick single electron and average influence of Pick single electron and average influence of

remaining electrons as a single force field (Vremaining electrons as a single force field (V0 external)external)

Then solve Schrodinger equation for single Then solve Schrodinger equation for single electron in presence of field (e.g. H-atom electron in presence of field (e.g. H-atom problem with extra force field)problem with extra force field)

Perform for all electrons in system Perform for all electrons in system Combine to give system wavefunction and Combine to give system wavefunction and

energy (Eenergy (E) Repeat to error tolerance (ERepeat to error tolerance (Ei+1-Ei)

Correcting ApproximationsCorrecting Approximations

Accounting for Electron Accounting for Electron CorrelationsCorrelations DFT(Density Functional Theory)DFT(Density Functional Theory) Moller-Plesset (Perturbation Theory)Moller-Plesset (Perturbation Theory) Configuration Interaction (Coupling Configuration Interaction (Coupling

single electron problems)single electron problems)

Geometry OptimizationGeometry Optimization

First Derivative is ZeroFirst Derivative is Zero

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

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

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dV (v r )

dv r

= 0

Modeling ProgramsModeling Programs

ObservablesObservables Equilibrium bond lengths and anglesEquilibrium bond lengths and angles Vibrational frequencies, UV-VIS, Vibrational frequencies, UV-VIS,

NMR shiftsNMR shifts Solvent Effects (e.g. LogP)Solvent Effects (e.g. LogP) Dipole moments, atomic chargesDipole moments, atomic charges Electron density mapsElectron density maps Reaction energies Reaction energies

Comparison to ExperimentsComparison to Experiments Electronic Schrodinger Equation gives bonding Electronic Schrodinger Equation gives bonding

energies for non-vibrating molecules (nuclei fixed energies for non-vibrating molecules (nuclei fixed at equilibrium geometry) at 0Kat equilibrium geometry) at 0K Can estimate G=Can estimate G= S using frequenciesS using frequencies EEoutout NOT NOT ΔΔHHff!!

Bond separation reactions (simplest 2-heavy atom Bond separation reactions (simplest 2-heavy atom components) provide path to heats of formationcomponents) provide path to heats of formation

334323 CH2CH CH CHCHCH →+

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ΔH fCH3CH2CH3= −ΔEbondseparation - ΔH fCH4

+ 2ΔH fCH 3CH3

ΔEbondseparation = E prodQM − E react

QM

= 2ECH3CH3

QM − (ECH3CH2CH3

QM + ECH 4

QM )

Ab InitioAb Initio Modeling Limits Modeling Limits

Function of basis and method usedFunction of basis and method used AccuracyAccuracy

~.02 angstroms~.02 angstroms ~2-4 kcal~2-4 kcal

NN HF - 50-100 atomsHF - 50-100 atoms DFT - 500-1000 atomsDFT - 500-1000 atoms

Semi-Empirical MethodsSemi-Empirical Methods

Neglect Inner Core ElectronsNeglect Inner Core Electrons Neglect of Diatomic Differential Neglect of Diatomic Differential

Overlap (NDDO)Overlap (NDDO) Atomic orbitals on two different Atomic orbitals on two different

atomic centers do not overlapatomic centers do not overlap Reduces computation time Reduces computation time

dramaticallydramatically

Other MethodsOther Methods

EnergeticsEnergetics Monte CarloMonte Carlo Genetic Genetic

AlgorithmsAlgorithms Maximum Entropy Maximum Entropy

MethodsMethods Simulated Simulated

AnnealingAnnealing

DynamicsDynamics Finite Difference Finite Difference Monte CarloMonte Carlo Fourier AnalysisFourier Analysis

Large Scale Modeling (>1000 Large Scale Modeling (>1000 atoms)atoms)

ChallengesChallenges Many bodies (Avogardo’s number!!)Many bodies (Avogardo’s number!!) Multi-faceted interactions (heterogeneous, Multi-faceted interactions (heterogeneous,

solute-solvent, long and short range solute-solvent, long and short range interactions, multiple time-scales)interactions, multiple time-scales)

Informatics Informatics Split problem into set of smaller problems Split problem into set of smaller problems

(e.g. grid analysis-popular in engineering)(e.g. grid analysis-popular in engineering) Periodic boundary conditionsPeriodic boundary conditions Connection tablesConnection tables

Large Scale ModelingLarge Scale Modeling

Hybrid MethodsHybrid Methods Different Spatial RealmsDifferent Spatial Realms

Treat part of system (Ex. Solvent) as Treat part of system (Ex. Solvent) as classical point particles and remainder classical point particles and remainder (Ex. Solute) as quantum particles(Ex. Solute) as quantum particles

Different Time DomainsDifferent Time Domains Vibrations (pico-femto) vs. sliding (micro)Vibrations (pico-femto) vs. sliding (micro) Classical (Newton’s 2Classical (Newton’s 2nd Law) vs. vs.

Quantum (TDSE)Quantum (TDSE)

Reference MaterialsReference Materials Journal of Molecular Graphics and Journal of Molecular Graphics and

ModelingModeling Journal of Molecular ModellingJournal of Molecular Modelling Journal of Chemical PhysicsJournal of Chemical Physics THEOCHEMTHEOCHEM Molecular Graphics and Modelling SocietyMolecular Graphics and Modelling Society NIH Center for Molecular ModelingNIH Center for Molecular Modeling ““Quantum Mechanics” by McQuarrieQuantum Mechanics” by McQuarrie ““Computer Simulations of Liquids” by Computer Simulations of Liquids” by

Allen and TildesleyAllen and Tildesley

Modeling ProgramsModeling Programs Spartan (www.wavefun.com)Spartan (www.wavefun.com) MacroModel (www.schrodinger.com)MacroModel (www.schrodinger.com) Sybyl (www.tripos.com)Sybyl (www.tripos.com) Gaussian (Gaussian (www.gaussian.comwww.gaussian.com)) Jaguar (Jaguar (www.schrodinger.comwww.schrodinger.com)) Cerius2 and Insight II (Cerius2 and Insight II (www.accelrys.comwww.accelrys.com)) QuantaQuanta CharMMCharMM GAMESSGAMESS PCModelPCModel AmberAmber

SummarySummary

Types of ModelsTypes of Models Tinker ToysTinker Toys Empirical/Classical (Newtonian Physics)Empirical/Classical (Newtonian Physics) Quantal (Schrodinger Equation)Quantal (Schrodinger Equation) Semi-empirical Semi-empirical

Informatic ModelingInformatic Modeling Conformational searching (QSAR, ComFA)Conformational searching (QSAR, ComFA) Generating new potentials Generating new potentials Quantum InformaticsQuantum Informatics

Next TimeNext Time

QSAR (Read Chapter 4)QSAR (Read Chapter 4)

MMFF EnergyMMFF Energy

StretchingStretching

( )⎭⎬⎫

⎩⎨⎧ −+−+−= 202020 )(

12

7)(1*)( ijijijijijijbondbond rrcsrrcsrrKE

MMFF EnergyMMFF Energy

BendingBending

{ })(1*)( 020ijkijkijkijkangle cbKE θθθθθ −+−=

MMFF EnergyMMFF Energy

Stretch-Bend InteractionsStretch-Bend Interactions

{ }( )000 )()( ijkijkkjkjkjiijijijkanglebond rrKrrKE θθ −−+−=−

MMFF EnergyMMFF Energy

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

( ) ( ) ( ){ }Φ++Φ++Φ+= 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 ε

Intermolecular/atomic Intermolecular/atomic modelsmodels

General form:General form:

Lennard-Jones Lennard-Jones

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V = V (r) + V (ri,rj ) + V (ri,rj ,rk ) + .....i< jj<k

N

∑i< j

N

∑

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V (rij ) = 4εσ

r

⎛

⎝ ⎜

⎞

⎠ ⎟

12

1 2 3 −

σ

r

⎛

⎝ ⎜

⎞

⎠ ⎟6 ⎡

⎣ ⎢

⎤

⎦ ⎥

1 2 3

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

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

δ+=