Molecular Modeling Fundamentals: Modus in Silico
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Transcript of Molecular Modeling Fundamentals: Modus in Silico
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Molecular ModelingMolecular ModelingFundamentals: Fundamentals: Modus in Modus in
SilicoSilico
C372C372
Introduction to Introduction to Cheminformatics IICheminformatics II
Kelsey ForsytheKelsey Forsythe
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"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)
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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!!
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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
VHH D0(1 e a(R R0 ))2
ˆ H E
Valence Valence Bond Bond TheoryTheory
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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
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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
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Atomic UnitsAtomic Units
PE qiq j
40rij
PE qiq j
rij
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ThermodynamicsThermodynamics
How might we compute relevant How might we compute relevant thermodynamic quantities?thermodynamic quantities? Equipartition TheoremEquipartition Theorem Harmonic Oscillator ApproximationHarmonic Oscillator Approximation
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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
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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
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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
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
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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 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
*
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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
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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
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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
rkF
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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
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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
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MMFF EnergyMMFF Energy
Stretch-Bend InteractionsStretch-Bend Interactions
000 )()( ijkijkkjkjkjiijijijkanglebond rrKrrKE
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MMFF EnergyMMFF Energy
Torsion (4-atom bending)Torsion (4-atom bending)
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
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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 jjk
N
i j
N
V (rij )4 r
12
r
6
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
nij
jiticelectrosta
RD
qqE
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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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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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Hydrogen MoleculeHydrogen Molecule
Bond DensityBond Density