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.