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Molecular Modeling and Molecular Modeling and InformaticsInformatics
C371C371
Introduction to Introduction to CheminformaticsCheminformatics
Kelsey ForsytheKelsey Forsythe
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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
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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
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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
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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
−=
≠
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MMFF PotentialMMFF Potential
E = E = EEbonbondd + + EEangleangle + + EEangleangle
-bond-bond + + EEtorsiontorsion + + EEVDWVDW + + EEelectrostaticelectrostatic
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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
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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
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Schrodinger’s EquationSchrodinger’s Equation
- Hamiltonian operator- Hamiltonian operator
Gravity? Gravity? €
ˆ H Ψ = EΨ
€
ˆ H
€
ˆ H = ˆ T + ˆ V
€
−h2
2mi
∇ 2
i
N
∑
€
Ceie j
ri − rji< j
N
∑
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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
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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
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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)
€
Ψ(v r ) = cm ∗Φm (
v r )
m
F
∑
€
Φm (v r )
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Hydrogen MoleculeHydrogen Molecule
HOMOHOMO
LUMO LUMO
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Hydrogen MoleculeHydrogen Molecule
Bond DensityBond Density
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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
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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)
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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)
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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
€
dV (v r )
dv r
= 0
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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
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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 →+
€
ΔH fCH3CH2CH3= −ΔEbondseparation - ΔH fCH4
+ 2ΔH fCH 3CH3
ΔEbondseparation = E prodQM − E react
QM
= 2ECH3CH3
QM − (ECH3CH2CH3
QM + ECH 4
QM )
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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
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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
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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
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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
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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)
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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
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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
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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
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Next TimeNext Time
QSAR (Read Chapter 4)QSAR (Read Chapter 4)
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MMFF EnergyMMFF Energy
StretchingStretching
( )⎭⎬⎫
⎩⎨⎧ −+−+−= 202020 )(
12
7)(1*)( ijijijijijijbondbond rrcsrrcsrrKE
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MMFF EnergyMMFF Energy
BendingBending
{ })(1*)( 020ijkijkijkijkangle cbKE θθθθθ −+−=
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MMFF EnergyMMFF Energy
Stretch-Bend InteractionsStretch-Bend Interactions
{ }( )000 )()( ijkijkkjkjkjiijijijkanglebond rrKrrKE θθ −−+−=−
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MMFF EnergyMMFF Energy
Torsion (4-atom bending)Torsion (4-atom bending)
( ) ( ) ( ){ }Φ++Φ++Φ+= 3cos12cos1cos15.0 321 VVVEtorsion
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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
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Intermolecular/atomic Intermolecular/atomic modelsmodels
General form:General form:
Lennard-Jones Lennard-Jones
€
V = V (r) + V (ri,rj ) + V (ri,rj ,rk ) + .....i< jj<k
N
∑i< j
N
∑
€
V (rij ) = 4εσ
r
⎛
⎝ ⎜
⎞
⎠ ⎟
12
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σ
r
⎛
⎝ ⎜
⎞
⎠ ⎟6 ⎡
⎣ ⎢
⎤
⎦ ⎥
1 2 3
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
Van derWaals repulsionVan derWaals repulsion London AttractionLondon Attraction
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MMFF EnergyMMFF Energy
Electrostatics (ionic compounds) Electrostatics (ionic compounds) D – Dielectric ConstantD – Dielectric Constant - electrostatic buffering constant- electrostatic buffering constant
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