Quantum Chem In Drug Design & Discovery

Quantum Chemistry in Drug Design and Discovery: Where We are and Where We are Going  Motivation  Linear-Scaling QM  QM based protein/small molecule scoring function  Spectroscopy NMR  Electron Density and X-Ray 

Status of Theoretical Approaches/Problems in Biology Fundamental problems remain unsolved     

Water Hydrophobic effect Protein folding Protein/small molecule interactions (drug design) etc.

Hence, current theoretical approaches are insufficient

Current Theoretical Approaches to Problems in Biology Classical Mechanics (standard approach)   

Molecular mechanical potentials Purely empirical potentials QSAR analysis

Statistical Mechanics (standard approach)  

Analyze trajectories (g(r), correlation functions, etc.) Free energy methods

Mathematical tools (standard for all potentials)   

Energy minimization Molecular dynamics etc.

Quantum Mechanics (less common approach)   

Cluster models (continuum solvation) QM/MM Linear-scaling QM

Strengths and Weaknesses of Classical and Quantum Potentials Classical Mechanics (standard approach)    

Highly approximate models (Coulombic electrostatics) Rapidly evaluated Good approach for ensemble generation Quality of potentials highly dependent on parameterization

Quantum Mechanics (less common approach)      

Fewer approximations (in the limit very accurate models) Expensive calculations Good for examining single snapshots Quality of potentials are well understood Used to build classical models Highly successful in organic and inorganic chemistry

Hence, applying a QM approach to biological problems is the logical next step

What are the Hurdles to a QM Model in (Structural) Biology? Very computationally expensive  

Linear-scaling algorithms Parallel computing

What model to use 

Exploit model chemistries Semiempirical Hamiltonians Density Functional Theory Hartree-Fock Theory Quantum Monte-Carlo

Ensemble generation  

Novel sampling approaches Use classical models to generate ensembles

Spectroscopy   

NMR X-ray etc.

Computational biology approach  

Leverage the repetitive nature of biology Bioinformatics databases

Our Vision of Quantum Biology Exploit   

Linear-scaling algorithms Parallel computing Model chemistries Semiempirical Hamiltonians Density Functional Theory Hartree-Fock Theory Quantum Monte-Carlo



Exploit ensemble generation protocols Novel sampling approaches Use classical models to generate ensembles



Spectroscopy NMR X-ray



Exploit statistical approaches Leverage the repetitive nature of biology Bioinformatics databases

Why Can We Think About Using Quantum Mechanics?

Divide and Conquer  Divides QM system into a set of smaller subsystems.  “Solves” matrix diagonalization problem.  Parallelizable.  Uses standard energy expressions.  Obtain gradients using standard methods. S. L.Dixon and K. M. Merz, Jr. J. Chem. Phys. 104, 6643­6649 (1996) S. L. Dixon and K. M. Merz, Jr.  J. Chem. Phys. 107, 879­893 (1997) A. van der Vaart, D. Suarez, K. M. Merz, Jr. J. Chem. Phys. 113, 10512­10523 (2000)

Divide and Conquer “Onion-Skin” Strategy Buffer Region 1

Buffer Region 2

Core Region

­­LYS­­­­ASP­­­­GLY­­­­PRO­­­­CYS­­­­ASN­­­­TRP­­­­GLY­­­­ALA­­­­VAL­­­­GLN ­­GLU­­­­ALA­­­­LEU­­­­GLY­­­­CYS­­­­ARG­­­­LYS­­­­SER­­­­ASN­­­­GLU­­­­TYR

Subsystem k

Subsystem k+4

Divide and Conquer “Onion-Skin” Strategy  

P

µ ν

=



∑



α



µ ν

α

α

 µ ν =   χ µ ∈  χ ν ∈ 

µ ν

   µν 

α = 

  α

µ ν

=

  χ ∈  χ ∈  µ  ν  

α

∑

α

 



( µ  )



α

ν 

α 

 =  + 



α

[(

ε 

− ε  )



]

 

∑ µ =



 µµ =







∑ ∑ µ =  α =



α µµ

∑ 

α



α

  µ



=  

Divide & Conquer ("DivCon") vs Standard Calculation (Seconds required to complete one SCF Cycle)

Linear vs. Exponential Scaling

CPU Resources Required

3000

2500 Current Standard Scales Exponentially, Rendering It Unsuitable for Routinely Analyzing Large Biomolecules

2000

1500

1000

500

"Divide & Conquer" Scales Linearly

0 0

100 Small molecule drug candidates (50-150 atoms)

200

300

400

Number of Atoms Per Molecule

500

600

Drug targets Large Biomolecules (Proteins ~2,500 atoms)

Errors in Heat of Formation Using D&C

Implicit Solvation in Biological Systems • Use Poisson­Boltzmann 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), 11875­11882. CM1 and CM2: Li, Zhu, Cramer, Truhlar J. Phys. Chem. 1998, 102, 1820­1831. Storer, Giesen, Cramer, Truhlar J. Computer­Aided Molecular Design 1995, 9, 87­110.

Implicit Solvation in Biological Systems - Proteins Solvation Free Energies of Proteins in Water Calculated by DivCon­PB Methodology. Protein           Crambin

Atoms/Res/q  

GRF 

Greorg 

Gnp 

  Gsol 

642/46/0

­316.7 

 

23.4

19.7

­273.5

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, 5171­5188

SCRF iterats 11

   

Do We Understand Intermolecular Interactions 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?

O Glutamic Acid

N

Min Max O (Ð) O (­0.716, ­0.413)

(­0.135, +0.140)

Histidine

N

O N

N O (­0.543, ­0.292)

Glycine

N

O Asparagine

N (­0.574, ­0.332) O N O (­0.566, ­0.337)

Aspartic Acid

N O (Ð)

O O (­0.569, ­0.342)

Alanine

N

Variations in Point Charges • Variation of on polar atoms is  +/­0.3e (Mulliken, CM1 or CM2) • Arises due to variations in the local environment of the atoms

Charge Transfer Effects : HIV-1 Protease 0.1

0.05

0

HIVP A76889

HIVP A76982

HIVP A78791

HIVP AcePep

HIVP Indinavir

HIVP SB203386

HIVP XK263

-0.05   

  ∆ -0.1

-0.15

-0.2

-0.25

+ve ∆q => Charge transferred from Inhibitor to Protease

Inhibitor

-ve ∆q => Charge transferred from Protease to Inhibitor

How well do We Understand Biomolecular Intermolecular Interactions? Current understanding has limitations due to the neglect of polarization and charge transfer effects. Thus, QM models can significantly contribute to increasing our understanding of these effects

QM Based Protein/Ligand Scoring Function A quantum mechanics based approach for more fundamental understanding of ligand/drug-protein interaction. Score function includes CT and polarization effects which are generally ignored by standard score functions. Score function can be systematically improved via appropriate parameterization. Pose generation via empirical or classical approaches. Primary screen via empirical or classical approaches. QM based scoring for final selection of compounds - i.e., secondary computational screen.

• Medicinal Chemistry Feedback: Validate

Validate

and Validate some more

Protein-Ligand Binding (Docking) I. The Unbound State

II. Ligand Recognition

L P

P

L

L

L

III. The Protein Ligand Complex

L P

Methodology: Thermodynamic Cycle to Calculate Free Energy of Binding PS P S Gas Phase +

+

Binding Free Energy calculated as: PS P S ∆Gbs = DGbg + DGsolv - DGsolv - DGsolv

DGbg = DHbg - TDSbg g g DHb = DH f + ( 1 6 )LJ R DSg = DSAC ,N,O,S + num(rot _ bonds)

Solvent

40 Protein-ligand Complexes 60%

56%

53%

50%

52%

51% 47%

44%

44%

43%

R

2

40%

30%

25%

20%

23% 17%

17%

14% 8%

10%

Xs

(u co np re ar (a am ) et er iz D ed ru ) gS SY co BY re L/ (b SY D ) -S BY c L/ or Ch e (b em ) Sc SY or BY e (b L/ ) Ce G sc riu or s2 e /L (b ig ) Sc or Ce e riu (b ) s2 /P M Ce F (b riu ) s2 / P Ce LP riu (b s2 ) /L SY U D BY I L/ (b F) Sc or e (b Au ) to do ck (b )

or e Sc

M Q

M

Sc

or e

Q

Q M

Sc

or e

(a

)

(b )

0%

Score Function

(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 (1hvr) -1000 -1200

0

5

10

15

20

-1400 -1600 -1800

TotalScore -2000 -2200

RMSD (Ao)

-2400 5 4 3 2 Rank, RMSD 1 0

Xscore Autodock DrugScore TotalScore Cerius2/PLP Cerius2/PMF Cerius2/LUDI SYBYL/Gscore SYBYL/F-Score SYBYL/D-Score Cerius2/LigScore Score Function SYBYL/ChemScore Native Rank

Best Rank RMSD

FKBP - Rapamycin (1fkb) 0 0

5

10

15

20

-200 -400 -600 -800 TotalScore -1000 -1200

RMSD (Ao)

12 10 8 6 Rank,4 RMSD 2 0

Xscore Autodock DrugScore TotalScore Cerius2/PLP Cerius2/PMF Cerius2/LUDI SYBYL/Gscore SYBYL/F-Score SYBYL/D-Score Cerius2/LigScore SYBYL/ChemScore Score Function Native Rank

Best Rank RMSD

 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 Electron Densities of Biomolecules and Potential 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 X­ray studies of  macromolecules?

Experimental X-Ray Crystallography X­ray experiments measure the intensities I(h k l) of the diffraction  peaks and derive the structure factors F(h k l).

I(h k l) = F(h k l)

2

Fourier transformation is used to obtain the electron density  distributions ρ(x y z) in molecule crystals.

1 ρ(x y z) = V

ååå h

k

F(h k l) exp [- 2pi(hx + ky + lz) + ia (h k l)]

l

Because of the lack of phase angles α(h k l), special techniques have  to be applied (heavy­atom methods, anomalous scattering, and  molecular replacement, etc.) and structure determination involves an  iterative process called refinement.

A Typical Diffraction Spectrum from an XRD 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 Distributions Ab initio or semiempirical calculation of electron density.

ò Y(r , r ,K ,r ,s , s ,K , s ) = å å P f ( r )f (r )

ρ(r) =

1

mn

m

n

2

m

1

2

n

2

dr2 L drn ds1 L dsn

v

n

Theoretical structure factors can be simulated by Fourier transformation  of theoretical densities. Methods have been described to handle/model  temperature factors. Periodic Hartree­Fock and density functional calculations of small  molecules now feasible with, for example, the program CRYSTAL. With our linear­scaling technologies we can evaluate the ED of  macromolecules. CRYSTAL: de Vries, Feil and Tsirelson  Acta. Cryst. 1999, B56, 118­123

QMED Calculations of Macromolecules with 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, 215 Core model: Cioslowski and Piskorz Chem. Phys. Lett. 1996, 255, 315­319

Quantum Mechanical Electron Densities of p-Nitropyridine-N-Oxide AM1 (DIVCON)

HF/6­31G* (G98)

Quantum Mechanical Electron Densities of a Protein Crambin Ultra­high 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 Case Recent work by Perpetuo et al (Acta Cryst.  B55, 70­77, 1999). 3 molecules studied: N­(trifluomethyl)  formamide, N­(2,2,2­trifluoethyl) formamide,  and 2,2,2­trifluoethyl isocyanide. 1170 independent reflections. 70 parameters used in refinement. R=0.0498

Preliminary Results Structure Factors (QM w/o T fac v.s. Raw) 50 45

y = 0.6991x R 2 = 0.8753

40 35 30 25 20 15 10 5 0 0

10

20

30

40

50

60

70

Preliminary Results -- Cont’d Structure Factors (QM v.s. Raw)

Structure Factors (Atomic v.s. Raw) 70

45

40

60

y = 0.5594x

35

y = 0.8213x

50

R 2 = 0.9221

2

R = 0.9291

30

40

25

20

30

15 20 10 10

5

0

0 0

10

20

30

40

R=0.196

50

60

70

0

10

20

30

40

R=0.173

50

60

70

Current 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.

Summary Our Vision of Quantum Biology Exploit   

Linear-scaling algorithms Parallel computing Model chemistries Semiempirical Hamiltonians Density Functional Theory Hartree-Fock Theory Quantum Monte-Carlo



Exploit ensemble generation protocols Use classical models to generate ensembles Novel sampling approaches



Spectroscopy NMR X-ray



Exploit statistical approaches Leverage the repetitive nature of biology Bioinformatics databases

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.

Acknowledgements • 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

Acknowledgements • DOE • NIH • NSF  • AMBER Development Team • Pharmacopeia, Inc. • QuantumBio Inc.