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Quantum Chemistry in Drug Quantum Chemistry in Drug Design and Discovery: Where Design and Discovery: Where
We are and Where We are GoingWe are and Where We are Going
MotivationMotivation Linear-Scaling QM Linear-Scaling QM QM based protein/small molecule scoring QM based protein/small molecule scoring functionfunction SpectroscopySpectroscopy
NMRNMR Electron Density and X-RayElectron Density and X-Ray
Status of Theoretical Status of Theoretical Approaches/Problems in Approaches/Problems in
BiologyBiology
Fundamental problems remain unsolved Fundamental problems remain unsolved WaterWater Hydrophobic effectHydrophobic effect Protein folding Protein folding Protein/small molecule interactions (drug design)Protein/small molecule interactions (drug design) etc.etc.
Hence, current theoretical approaches are insufficientHence, current theoretical approaches are insufficient
Current Theoretical Approaches Current Theoretical Approaches to Problems in Biologyto Problems in Biology
Classical Mechanics (standard approach) Classical Mechanics (standard approach) Molecular mechanical potentialsMolecular mechanical potentials Purely empirical potentialsPurely empirical potentials QSAR analysisQSAR analysis
Statistical Mechanics (standard approach)Statistical Mechanics (standard approach) Analyze trajectories (g(r), correlation functions, etc.)Analyze trajectories (g(r), correlation functions, etc.) Free energy methodsFree energy methods
Mathematical tools (standard for all potentials)Mathematical tools (standard for all potentials) Energy minimizationEnergy minimization Molecular dynamicsMolecular dynamics etc.etc.
Quantum Mechanics (less common approach)Quantum Mechanics (less common approach) Cluster models (continuum solvation)Cluster models (continuum solvation) QM/MMQM/MM Linear-scaling QMLinear-scaling QM
Strengths and Weaknesses of Strengths and Weaknesses of Classical and Quantum Classical and Quantum
PotentialsPotentialsClassical Mechanics (standard approach) Classical Mechanics (standard approach)
Highly approximate models (Coulombic electrostatics)Highly approximate models (Coulombic electrostatics) Rapidly evaluatedRapidly evaluated Good approach for ensemble generationGood approach for ensemble generation Quality of potentials highly dependent on parameterizationQuality of potentials highly dependent on parameterization
Quantum Mechanics (less common approach)Quantum Mechanics (less common approach) Fewer approximations (in the limit very accurate models)Fewer approximations (in the limit very accurate models) Expensive calculationsExpensive calculations Good for examining single snapshotsGood for examining single snapshots Quality of potentials are well understoodQuality of potentials are well understood Used to build classical modelsUsed to build classical models Highly successful in organic and inorganic chemistryHighly successful in organic and inorganic chemistry
Hence, applying a QM approach to biological problems is the logical next stepHence, applying a QM approach to biological problems is the logical next step
What are the Hurdles to a QM What are the Hurdles to a QM Model in (Structural) Biology?Model in (Structural) Biology?
Very computationally expensiveVery computationally expensive Linear-scaling algorithmsLinear-scaling algorithms Parallel computingParallel computing
What model to useWhat model to use Exploit model chemistriesExploit model chemistries
Semiempirical HamiltoniansSemiempirical HamiltoniansDensity Functional TheoryDensity Functional TheoryHartree-Fock TheoryHartree-Fock TheoryQuantum Monte-CarloQuantum Monte-Carlo
Ensemble generationEnsemble generation Novel sampling approachesNovel sampling approaches Use classical models to generate ensemblesUse classical models to generate ensembles
SpectroscopySpectroscopy NMRNMR X-rayX-ray etc.etc.
Computational biology approachComputational biology approach Leverage the repetitive nature of biologyLeverage the repetitive nature of biology Bioinformatics databases Bioinformatics databases
Our Vision of Quantum BiologyOur Vision of Quantum Biology
Exploit Exploit Linear-scaling algorithmsLinear-scaling algorithms Parallel computingParallel computing Model chemistriesModel chemistries
Semiempirical HamiltoniansSemiempirical HamiltoniansDensity Functional TheoryDensity Functional TheoryHartree-Fock TheoryHartree-Fock TheoryQuantum Monte-CarloQuantum Monte-Carlo
Exploit ensemble generation protocolsExploit ensemble generation protocolsNovel sampling approachesNovel sampling approachesUse classical models to generate ensemblesUse classical models to generate ensembles
SpectroscopySpectroscopyNMRNMRX-rayX-ray
Exploit statistical approachesExploit statistical approachesLeverage the repetitive nature of biologyLeverage the repetitive nature of biologyBioinformatics databasesBioinformatics databases
Why Can We Think About Why Can We Think About Using Quantum Mechanics?Using Quantum Mechanics?
Divide and ConquerDivide and Conquer Divides QM system into a set of Divides QM system into a set of smaller subsystems.smaller subsystems. “ “Solves” matrix diagonalization Solves” matrix diagonalization problem.problem. Parallelizable.Parallelizable. Uses standard energy expressions. Uses standard energy expressions. Obtain gradients using standard Obtain gradients using standard methods.methods.
S. L.Dixon and K. M. Merz, Jr. J. Chem. Phys. 104, 6643-6649 (1996)S. L. Dixon and K. M. Merz, Jr. J. Chem. Phys. 107, 879-893 (1997)A. van der Vaart, D. Suarez, K. M. Merz, Jr. J. Chem. Phys. 113, 10512-10523 (2000)
Divide and ConquerDivide and Conquer“Onion-Skin” Strategy“Onion-Skin” Strategy
--LYS----ASP----GLY----PRO----CYS----ASN----TRP----GLY----ALA----VAL----GLN
--GLU----ALA----LEU----GLY----CYS----ARG----LYS----SER----ASN----GLU----TYR
Subsystem k Subsystem k+4
CoreRegion
BufferRegion 1
BufferRegion 2
Divide and ConquerDivide and Conquer“Onion-Skin” Strategy“Onion-Skin” Strategy
Pm n
= Dm n
aP
m n
a
a = 1
Nsub
å D m na
=
0 if c m Î Buffer2 or c n Î Buffer2
0 if c m Î Buffer1 and c n Î Buffer1
1 nmn otherwise
ì
í
ï
î ï
P m n
a= n i
ac m i
a( )
*
c n i
a
i
MOs
ån
i
a=
2
1 + exp e i
a- e F( ) kT
[ ]
¥ The global density matrix
¥ Where
¥ With Fermi energy selected to yield occupation #'s that satisfy:
Pm m
= Dm m
a
ni
a
cm i
a2
i
MOs
åa =1
Nsub
åm = 1
N
åm =1
N
å = nelec
Divide & Conquer ("DivCon") vs Standard Calculation
Linear vs. Exponential Scaling
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500 600
Number of Atoms Per Molecule
CP
U R
esou
rces R
eq
uir
ed
(Secon
ds r
eq
uir
ed
to c
om
ple
te
on
e S
CF C
ycle
)
Current StandardScales Exponentially, Rendering
It Unsuitable for Routinely AnalyzingLarge Biomolecules
"Divide & Conquer" Scales Linearly
Drug targetsLarge Biomolecules
(Proteins ~2,500 atoms)
Small molecule drug candidates(50-150 atoms)
Errors in Heat of Formation Using D&C
Implicit Solvation in Implicit Solvation in Biological SystemsBiological Systems
• Use Poisson-Boltzmann Theory in conjunction with Divide and Conquer.• CM1/CM2 charges were key to making this approach sucessful.• Model fit (nonpolar term) to simultaneously reproduce solvation free energies of small molecules and LogP values of a wide range of compounds.
PB: Tannor, Marten, Murphy, Friesner, Sitkoff, Nicholls, Honig, Rignalda, Goddard J. Am. Chem. Soc. 1994, 116(26), 11875-11882.
CM1 and CM2: Li, Zhu, Cramer, Truhlar J. Phys. Chem. 1998, 102, 1820-1831.Storer, Giesen, Cramer, Truhlar J. Computer-Aided Molecular Design 1995, 9, 87-110.
Parameterization: Brothers and Merz to be submitted.
Implicit Solvation in Biological Implicit Solvation in Biological Systems - ProteinsSystems - Proteins
Solvation Free Energies of Proteins in Water Calculated by DivCon-PB Methodology.
Protein Atoms/Res/q GRF Greorg
Gnp Gsol
SCRF iteratsCrambin 642/46/0 -316.7 23.4 19.7 -273.5 11
BPTI 888/58/+6 -1336.3 69.7 26.6 -1239.8 14
CspA 1010/69/0 -1175.5 109.3 28.6 -1073.5 15
Lysozyme 1960/129/+8 -1936.3 129.3 45.3 -1761.7 13
Subtilisin E 3854/275/-2 -1856.3 166.8 74.8 -1614.7 15
Gogonea and Merz J. Phys. Chem. A. 1999, 103, 5171-5188
Do We Understand Do We Understand Intermolecular Interactions Intermolecular Interactions
between Biomolecules?between Biomolecules? Current understanding is at the classical level, but Intermolecular (and intramolecular in biomolecules) interactions are inherently quantum in nature. Can we use quantum chemistry to better understand interactions in biomolecular systems?
Variations in Variations in Point ChargesPoint Charges
• Variation of on polar atoms is +/-0.3e (Mulliken, CM1 or CM2)
• Arises due to variations in thelocal environment of the atoms
Charge Transfer Effects : HIV-1 ProteaseCharge Transfer Effects : HIV-1 Protease
+ve Dq => Charge transferred from Inhibitor to Protease
-ve Dq => Charge transferred from Protease to Inhibitor
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
HIVPA76889
HIVPA76982
HIVPA78791
HIVP Ace-Pep
HIVPIndinavir
HIVPSB203386
HIVPXK263
Inhibitor
Dq (electron)
Mulliken
CM1
CM2
How well do We Understand How well do We Understand Biomolecular Intermolecular Biomolecular Intermolecular
Interactions?Interactions?Current understanding has
limitations due to the neglect of polarization and charge transfer
effects.
Thus, QM models can significantly Thus, QM models can significantly contribute to increasing our contribute to increasing our
understanding of these effectsunderstanding of these effects
QM Based Protein/Ligand QM Based Protein/Ligand Scoring FunctionScoring Function
A quantum mechanics based approach for more fundamental A quantum mechanics based approach for more fundamental understanding of ligand/drug-protein interaction.understanding of ligand/drug-protein interaction.
Score function includes CT and polarization effects which are Score function includes CT and polarization effects which are generally ignored by standard score functions. generally ignored by standard score functions.
Score function can be systematically improved via appropriate Score function can be systematically improved via appropriate parameterization. parameterization.
Pose generation via empirical or classical approaches. Pose generation via empirical or classical approaches.
Primary screen via empirical or classical approaches.Primary screen via empirical or classical approaches.
QM based scoring for final selection of compounds - QM based scoring for final selection of compounds - i.ei.e., ., secondary computational screen.secondary computational screen.
Medicinal Chemistry Feedback: Medicinal Chemistry Feedback:
ValidateValidate ValidateValidate and Validate some moreand Validate some more
Protein-Ligand Binding Protein-Ligand Binding (Docking)(Docking)
L
P
I. The Unbound State II. Ligand Recognition
P
III. The Protein Ligand Complex
L
PL L
L
Methodology: Thermodynamic Cycle to Calculate Methodology: Thermodynamic Cycle to Calculate Free Energy of BindingFree Energy of Binding
Binding Free Energy calculated as:Binding Free Energy calculated as:
DGbs = DGb
g + DGsolvPS - DGsolv
P - DGsolvS
DGbg = DHb
g - TDSbg
DHbg = DH f
g + ( 1R 6 )LJ
DSg = DSAC ,N ,O,S + num(rot _ bonds)
P SPS
+
+
Gas Phase
Solvent
40 Protein-ligand Complexes
56%53% 52% 51%
47%44% 44% 43%
25%23%
17% 17%14%
8%
0%
10%
20%
30%
40%
50%
60%
QMSc
ore
(a)
QMSc
ore
(b)
Xscor
e (a
)
QMSc
ore
(unp
aram
eter
ized)
DrugS
core
(b)
SYBYL
/D-S
core
(b)
SYBYL
/Che
mSc
ore
(b)
SYBYL
/Gsc
ore
(b)
Ceriu
s2/Lig
Scor
e (b
)
Ceriu
s2/P
MF (b
)
Ceriu
s2/P
LP (b
)
Ceriu
s2/LUDI (
b)
SYBYL
/F-S
core
(b)
Autod
ock
(b)
Score Function
R2
(a) parameterized on this data set; (b) parameterized on other data sets
Source: Renxiao Wang, Yipin Lu and Shaomeng Wang, Comparative Evaluation of 11 Scoring Functions for Molecular Docking J.Med.Chem. 2003, 46, 2287-2303. For QMScore date, Kaushik Raha, Merz lab at Pennsylvania State University, unpublished study.
HIV-1 Protease - XK263 (HIV-1 Protease - XK263 (1hvr)1hvr)
0
1
2
3
4
5
TotalScoreCerius2/PLPSYBYL/F-ScoreCerius2/LigScore
DrugScoreCerius2/LUDI
XscoreAutodock
Cerius2/PMFSYBYL/Gscore
SYBYL/ChemScoreSYBYL/D-Score
Score Function
Rank, RMSD
Native Rank Best Rank RMSD
-2400
-2200
-2000
-1800
-1600
-1400
-1200
-1000
0 5 10 15 20
RMSD (Ao)
TotalScore
FKBP - Rapamycin (FKBP - Rapamycin (1fkb)1fkb)
0
2
4
6
8
10
12
TotalScoreCerius2/PLPSYBYL/F-ScoreCerius2/LigScore
DrugScoreCerius2/LUDI
XscoreAutodock
Cerius2/PMFSYBYL/Gscore
SYBYL/ChemScoreSYBYL/D-Score
Score Function
Rank, RMSD
Native Rank Best Rank RMSD
-1200
-1000
-800
-600
-400
-200
0
0 5 10 15 20
RMSD (Ao)
TotalScore
Conclusions and Future Directions• First generation (AM1 based) results are very promising and can be readily refined.
• Explore further parameterization to improve predictive capability.
• QM geometry optimization (ligand only) to further refine structures.
Preliminary Studies of Semiempirical Preliminary Studies of Semiempirical Electron Densities of Biomolecules and Electron Densities of Biomolecules and
Potential ApplicationsPotential Applications
Can we compute reasonable electron densities (EDs) of biomolecules using semiempirical Hamiltonians?How good are they with respect to experimental EDs? Ab initio computed EDs?What are their potential uses in X-ray studies of macromolecules?
Experimental X-Ray CrystallographyExperimental X-Ray Crystallography
X-ray experiments measure the intensities I(h k l) of the diffraction peaks and derive the structure factors F(h k l).
Fourier transformation is used to obtain the electron density distributions r(x y z) in molecule crystals.
Because of the lack of phase angles a(h k l), special techniques have to be applied (heavy-atom methods, anomalous scattering, and molecular replacement, etc.) and structure determination involves an iterative process called refinement.
I(h k l) = F (h k l)2
r(x y z) =1V
F (h k l)l
åk
åh
å exp - 2pi(hx + ky + lz) + ia (h k l)[ ]
A Typical Diffraction Spectrum from an A Typical Diffraction Spectrum from an XRD ExperimentXRD Experiment
Reflections only appear at discrete angles (h k l).
Peak intensities are related to structure factors by:
I(h k l) µ F (h k l)2
Theoretical Studies of Electron Density Theoretical Studies of Electron Density DistributionsDistributions
Ab initio or semiempirical calculation of electron density.
Theoretical structure factors can be simulated by Fourier transformation of theoretical densities. Methods have been described to handle/model temperature factors.
Periodic Hartree-Fock and density functional calculations of small molecules now feasible with, for example, the program CRYSTAL.
With our linear-scaling technologies we can evaluate the ED of macromolecules.
r(r) = Y(r1, r2 ,K ,rn ,s1, s2,K , sn)ò2dr2L drnds1L dsn
= Pmn f m r( )n
åm
å f v r( )
CRYSTAL: de Vries, Feil and Tsirelson Acta. Cryst. 1999, B56, 118-123
QMED Calculations of Macromolecules QMED Calculations of Macromolecules with Semiempirical Hamiltonianswith Semiempirical Hamiltonians
Typical semiempirical models employ the core approximation, but we need the core electron density in order to match with experiment.
Full EDs can be obtained by augmenting the QM-derived valence EDs with spherical core EDs.
The main question remains, though - How good are these EDs?
AM1 EDs: Ho, Schmider, Edgecombe and Smith, Jr. Int. J. Quantum Chem.1994, S28, 215Core model: Cioslowski and Piskorz Chem. Phys. Lett. 1996, 255, 315-319
Quantum Mechanical Electron Quantum Mechanical Electron Densities of p-Nitropyridine-N-OxideDensities of p-Nitropyridine-N-Oxide
AM1 (DIVCON) HF/6-31G* (G98)
Quantum Mechanical Electron Quantum Mechanical Electron Densities of a Protein CrambinDensities of a Protein Crambin
Ultra-high resolution structure (0.54Å, Teeter et al., 2000).
46 residues, 648 atoms.
The QM ED map currently contains only the electron distribution for a static structure as opposed to a time and space average, but otherwise agrees well with the experimental map.
A Small Molecule Test CaseA Small Molecule Test Case
Recent work by Perpetuo et al (Acta Cryst. B55, 70-77, 1999).
3 molecules studied: N-(trifluomethyl) formamide, N-(2,2,2-trifluoethyl) formamide, and 2,2,2-trifluoethyl isocyanide.
1170 independent reflections.
70 parameters used in refinement.
R=0.0498
Preliminary ResultsPreliminary Results
Structure Factors (QM w/ o T fac v.s. Raw)
y = 0.6991x
R2 = 0.8753
0
5
10
15
20
25
30
35
40
45
50
0 10 20 30 40 50 60 70
Preliminary Results Preliminary Results -- Cont’d-- Cont’d
Structure Factors (Atomic v.s. Raw)
y = 0.8213x
R2 = 0.9291
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70
Structure Factors (QM v.s. Raw)
y = 0.5594x
R2 = 0.9221
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40 50 60 70
R=0.196 R=0.173
Current Status and Future DirectionsCurrent Status and Future Directions
Currently further validating computed ED on small molecules.
Application areas we are pursuing by providing aspherical ED descriptions: Aid the macromolecular refinement process by introducing
another constraint. Allow for deconvolution of anisotropic density
distributions from the anisotropic temperature factors. Study macromolecules with the Atoms in Molecules (AIM)
theory. .
SummarySummaryOur Vision of Quantum Our Vision of Quantum
BiologyBiologyExploit Exploit
Linear-scaling algorithmsLinear-scaling algorithms Parallel computingParallel computing Model chemistriesModel chemistries
Semiempirical HamiltoniansSemiempirical HamiltoniansDensity Functional TheoryDensity Functional TheoryHartree-Fock TheoryHartree-Fock TheoryQuantum Monte-CarloQuantum Monte-Carlo
Exploit ensemble generation protocolsExploit ensemble generation protocolsUse classical models to generate ensemblesUse classical models to generate ensemblesNovel sampling approachesNovel sampling approaches
SpectroscopySpectroscopyNMRNMRX-rayX-ray
Exploit statistical approachesExploit statistical approachesLeverage the repetitive nature of biologyLeverage the repetitive nature of biologyBioinformatics databasesBioinformatics databases
General Conclusions General Conclusions • Application of QM to large biomolecular systems are opening up new avenues to aid in our understanding of biomolecular solvation, inhibition, etc.
• QM gives a better account of electrostatic interactions than typical classical models.
• Quantum mechanics and classical mechanics can work synergistically to achieve our desired goal of understanding biomolecular structure, function and inhibition.
AcknowledgementsAcknowledgements• Steve Dixon• Arjan van der Vaart• Dimas Suarez • Lance Westerhoff• Martin Peters• Kaushik Raha• Ed Brothers • Andrew Wollacott• Ken Ayers• Bryan Op’t Holt• Ning Liao• Xiadong Zhang• Bing Wang• Guille Estiu
AcknowledgementsAcknowledgements
• DOE• NIH• NSF • AMBER Development Team• Pharmacopeia, Inc.• QuantumBio Inc.