Computergestützte Biophysik I

Bert de Groot, Udo Schmitt, Helmut Grubmüller

Max Planck-Institut für biophysikalische ChemieTheoretische und Computergestützte BiophysikAm Fassberg 1137077 Göttingen

Tel.: 201-2308 / 2300Email: bgroot@gwdg.de

uschmit1@gwdg.dehgrubmu@gwdg.de

www.mpibpc.gwdg.de/groups/grubmueller

Overview: Computational Biophysics I

L1/P1: Introduction, Protein structure and function, molecular dynamics, approximations, numerical integration, argon

L2/P2: Tertiary structure, force field contributions, efficientalgorithms, electrostatics methods, protonation, periodic boundaries, solvent, ions, NVT/NPT ensembles, analysis

L3/P3: Protein data bank, structure determination by NMR / x-ray; refinement

L4/P4: Monte Carlo, Normal mode analysis, principal components

L5/P5: Aquaporin / ATPase: two examples from current research

L6/P6: Quantenmechanische Methoden: Grundlagen, Hartree-Fock

L7/P7: Dichtefunktionaltheorie, Semiempirische Methoden

Bio-MolecularQuantum Mechanics

Density Functional Theory &QM/MM & Applications

Enzymes: life‘s catalyst

• Proteins that catalyze chemical reactions• Highly specific (for ONE substrate)• Dramatic acceleration of reaction rates

How do enzymes achieve this dramatic accelaration?

-Destabilizing the reactants/productsvs. stabilizing the transition state

- Purely electrostatic effect

- Nuclear Quantum effects (tunneling)

- Special Promoting vibrations

- preorganization of enzyme

catk

E S ES EP E P+ → → +�

Enzymatic activityinvolves „chemistry“Enzymatic activity

involves „chemistry“Breakdown of empirical force fields: no chemical bond

breaking and formation in „harmonic models“

L R

L Rc L c Rψ = +

Summary

• Molecular Hamiltonian (electrons+nuclei)• Born-Oppenheimer approximation• Hartree product + antisymmetry -> Slater Det• Variational principle -> HF equations• LCAO: atomic basis for MO• Basis set (GTO)• Hartree-Fock solution basis for post-Hartree-

Fock methods (pertubation or variational) • Efficient software packages available

Born-Oppenheimer approximation

22 2

,

ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ| |ˆ| | | |

i jiP p

k i i j k lk lk i i j

e Z Ze Z eH T T

r rr R R R< <= + −∑ + ∑ + ∑

−− −

2 2ˆ ˆˆ ˆ2 2

i iP i p i

i e

P pT T

M m

⟨ ⟩ ⟨ ⟩⟨ ⟩ = ∑ ⟨ ⟩ = ∑≪

1836i

e

M

m≥1.)

ˆ ˆˆ( , )elecH r R

Hartree-Fock method

• Mean field ansatz (Hartree)

1 1 2 2( ; ) ( ) ( ) ( ) ( )k k N Nr R r r r rφ φ φΦ = Φ =�� �

…

22 2

,

2 2 2

,

2

ˆ ˆˆ ˆ ˆ| | | | | |

ˆ

ˆ ˆ ˆ2 | | | |

ˆˆ ˆ| |

i jielec p

k i i j k lk i i j k l

k i

k k i k le k i k l

kk k l

k l

e Z Ze Z eH T

r R R R r r

p e Z eC

m r R r r

eh C

r r

< <

<

<

= −∑ + ∑ + ∑− − −

= ∑ −∑ + ∑ +− −

= ∑ + ∑ +−

product ansatzSpatial orbitals

Density Functional Theory (DFT)

• wavefunction <-> electron density

3N dimensional object 3 dimensional

( )1, , , , , ,i j N

x x x xΨ … … … ( )rρ�

2( ) ( )

i

i

r rρ ϕ=∑� �

Density Functional Theory (DFT)

( ) ( )( )

( )

2

1 1 2 1, , , , , ,

0

N electrons

N i j Nr N ds dx dx x x x x

r

r dr N

ρ

ρ

ρ

= ⋅ Ψ

→ ∞ =

=

∫ ∫

∫

�… … … … …

�

� �

History of Electronic Structure Methods

• Thomas,Fermi 1927: first DFT (self consistent field!)• Hartree 1928: Hartree equation • Fock 1930: Hartree-Fock• Roothaan,Hall 1951: LCAO in HF• Hohenberg-Kohn 1964: famous theorem(s)• Kohn-Sham 1965: variational (practical) approach

Hohenberg-Kohn Theorem

2nucleii

i i

e Z( ) = external potential from nuclei =

|r - r |ext

V r ∑�

� �

22 2

,

2 2 2

,

2 2

ˆ ˆˆ ˆ ˆ| | | | | |

ˆ

ˆ ˆ ˆ2 | | | |

ˆ

ˆ ˆ2 | |

i jielec p

k i i j k lk i i j k l

k i

k k i k le k i k l

kext

k k le k l

e Z Ze Z eH T

r R R R r r

p e Z eC

m r R r r

p eV C

m r r

< <

<

<

= −∑ + ∑ + ∑− − −

= ∑ −∑ + ∑ +− −

= ∑ + ∑ + +−

Hohenberg-Kohn Theorem

ext

ext elec

The external potential V (r) is (to within a constant) a unique functional

ˆof (r); since, in turn V (r) fixes H we see that the many particle ground

state is a unique functional of (r)

ρ

ρ

�

� �

�

nucleii

i i

Z( ) = external potential from nuclei =

|r - r |ext

V r ∑�

� �

( ) ( )ext

r V rρ ⇒� � ( ) ( )

extV r rρ⇒� �

Proof “Existence Theorem”

ext extV ( ) V ( ) that give rise to the same (r) associated with r r ρ′≠ Ψ�� �

reduction ad absurdum: proof by contradiction

[ ] [ ]( ) ( ) ( )ext ext

F V r r F V rρ ′= =� � �

0ˆ ˆ ˆ;

extH H V H E= + Ψ = Ψ 0

ˆ ˆ ˆ; ext

H H V H E′ ′ ′ ′ ′ ′= + Ψ = Ψ

ˆ ˆ ˆ ˆ ˆE H H H H H′ ′ ′ ′ ′ ′ ′ ′= Ψ Ψ < Ψ Ψ = Ψ Ψ + Ψ − Ψ

ext ext

E E V V′ ′ ′ ′< + Ψ − Ψ

( ) ( ) ext extE E dr r V Vρ′ ′< + −∫� �

( ) ( ) ext extE E dr r V Vρ′ ′< + −∫� �

E E E E′ ′+ < + [ ] ( )E rρ→ ( ) ( )ext

V r rρ⇒� �

“Variational Theorem”

[ ] [ ] [ ] [ ]int extE T V Vρ ρ ρ ρ= + +

2

ˆ ˆˆ ˆ| |

elec extk l

k l

eH T V C

r r<= + ∑ + +

−

“The ground state energy can be obtained variationally: the density

that minimizes the total energy is the exact ground state density”

[ ] ˆ( )elec

E r Hρ = Ψ Ψ

[ ] [ ]( ) ( )E r E rρ ρ ′<

Kohn-Sham equation

DFT analogon to Hartree-Fock equations:

[ ] [ ] [ ] [ ]0 int extE T V Vρ ρ ρ ρ= + +

( ) ( ) 0elec elecdr r N dr r Nρ ρ= → − =∫ ∫� � � �

[ ] [ ] [ ] [ ]( )0 int ( ( ) ) 0ext elec

E T V V dr r Nδ ρ δ ρ ρ ρ λ ρ= + + − − =∫� �

So far everything is exact: ab initio !!!

But what are the Functionals? [ ] [ ] [ ]int, and ext

T V Vρ ρ ρ

Kohn-Sham approximations

[ ] [ ] [ ]int, and ext

T V Vρ ρ ρ

( ) extdr r Vρ∫� �

1 ( ) ( )[ ( )]

2 | |XC

r rdr dr E r

r r

ρ ρρ

′′ +

′−∫ ∫� �

� � �� �

5

3[ ( )] ( )

Thomas-Fermi

T r dr rρ ρ∫� � �∼

Uniform electron gas:

N electrons in box of volume V with neutralizing background charge

4

3[ ( )] ( ) XCE r dr rρ ρ∫� � �∼

10% error

Kohn-Sham approximations

1 1 1

2 2 2

1 2

( ) ( ) ( )

( ) ( ) ( )1( , , , )

!

( ) ( ) ( )

a b a N

a b a N

N

a N b N a N N

x x x

x x xx x x

N

x x x

ϕ ϕ ϕ

ϕ ϕ ϕ

ϕ ϕ ϕ

+

+

+

Φ =

⋯

⋯…

⋮ ⋮ ⋮

⋯

21

2i i

i i

T t= − ∇ =∑ ∑

Slater determinant

21[ ( )] [ ( )]

2i i i XC

i

T r T rρ ϕ ϕ ρ= − ∇ +∑� �

Kinetic energy part:

XC XC[ ( )] E [ ( )] = E [ ( )]XC

T r r rρ ρ ρ′+� � �

2( ) ( )

i

i

r rρ ϕ=∑� �

Still unkown!!!

[ ]T ρ

Kohn-Sham equation

[ ] [ ] [ ] [ ]

[ ]

0 int

21 1 ( ) ( )= [ ( )]

2 2 | |

ext

i i i XC ext

i

E T V V

r rdr dr E r V

r r

ρ ρ ρ ρ

ρ ρϕ ϕ ρ ρ

= + +

′′− ∇ + + +

′−∑ ∫ ∫

� �� � �

� �

Hartree energy

( )N N

0 ij i

i=1 j=1

[{ }] E [ ] - i j ij

L ϕ ρ λ ϕ ϕ δ= −∑∑2

( ) ( )i

i

r rρ ϕ=∑� �

N N

0 ij i

i=1 j=1

[{ }] = E [{ }] - 0i i j

Lδ ϕ δ ϕ λ δ ϕ ϕ =∑∑

Kohn-Sham equation

[ ]2( )1

( ) ( ) ( ) ( ) 1, 2, ,2 ( )

Kohn-Sham non-linear integro-differential equations

XC

Hartree ext i i i

E rv r v r r r i N

r

δ ρϕ ε ϕ

δρ

− ∇ + + + = =

…

( ) ( ) ( ) 1, 2, ,KS

i i i ih r r r i Nϕ ε ϕ= = …

But can be recast in matrix form-> ideal for computers

HC SCε=

Self-consistent field method (SCF)

""

( ) ( )

LCAO (linear combination

of atomic orb.)

basis set !!!!

atomicorbitals

i i

i

r c rφ χ= ∑

In search for the divine Functional(s)

• Local density approximation (LDA)

• Generalized Gradient Approximation (GGA) 1980is (BLYP,P PBE)

• Hybrid (B3LYP) 1996

[ ( )] ( ) [ ( )] energy densityXC XCE r dr r rρ ρ ε ρ= ∫� � � �

13 1

39 3

[ ( )] ( )8

XC r rε ρ ρπ

= −

� � Thomas,Fermi,Slateretc.

Becke: “everything is legal” approach

Perdew: ”based on first principles”[ ( ), ( )]XC r rε ρ ρ∇� �

Combine Hartree-Fock with DFT !

Success of DFT

• Treat electron correlation based on density

• No rigorous way how to improve results = find better functional ( trial & error )

• Approximate (semi-empirical)

• Modern hybrid functionals B3LYP:Hartree-Fock < B3LYP < MP2 < CI

( )1( ) , , , , , ,i j N

r x x x xρ ↔ Ψ�

… … …

4N∼

4N∼

5 6N

−∼

Ne∼

Basis sets

• Atom centered: • Slater-Type orbitals GaussianTO

( )1

( , , , , , , )

( , )Ar rn m

l

r n l m

r e Yζ

χ θ φ ζ

θ φ− −−

∼

Spherical harmonicsQuantum numbersn,l,m

parameter

2( )

( , , , , , , )

Ar r i j k

x y z i j k

e x y zζ

χ ζ

− −

∼

parameter

Ar

- good approximation

- slow computation

- Pretty bad approximation

- Vary fast

- Take many of them in LCAO!

Basis sets

But in DFT also so-called

“plane waves”

i ( ) ( ) k r

k

r a k eφ ⋅=∑� �

�

��

Electronic structuresoftware packages

• Gaussian,Molpro,Q-Chem,CPMD,Turbomole,Jaguar etc. written in Fortran,C,C++

Provide efficient implementations of electronicstructure methods on modern supercomputersand user-friendly interfaces (use by non-specialist)

Nobel prize Kohn 1998

What can we do with DFT?

DNA crystal

2x12 base pairs

Water + counter ions

periodic boundary cond.

2388 atoms

~ 40 000 basis functions

Towards Larger SystemsQuantum mechanical/Molecular Mechanical

(QM/MM)

• Warshel & Levitt 1976MM regionTotal system

Active site (QM)

5 6~ 10 atoms−

~10-200 atoms only

/ˆ ˆ ˆ

total QM MM QM MMH H H H= + +

ˆelec

HForce field

coupling

12 6

/ˆ ( ) ( )

ˆ

electrons MM nuclei MM nuclei MMj i j ij ij

QM MM

i j i j i j ij iji j i j

eQ eZ QH

R Rr R R R

σ σ= + + −

− −∑ ∑ ∑ ∑ ∑ ∑

QM/MM

• Solution to the length scale “barrier” in biophysics

• Main problem: MM part is purely classical

• Violation of antisymmetry principle of total (QM+MM) wavefunction

• -> MM/QM interaction does not lead to proper electron repulsion and dispersion interaction (= van der Waals) !!!!

• Electronic polarization only treated in QM part

MM region

ab initio dynamics of biological systems

QM/MM molecular dynamics

Total system

Active site (QM)

Classical force fields

(CHARMM, GROMOS96 etc.)

~10-100 atoms only

5 6~ 10 atoms−

- density functional theory (DFT)

- “on the fly” computation of ab initio

forces on the nuclei

- simultaneous integration of electronic and nuclear degrees of freedom

- nuclear dynamics classical (point particle approximation)

Total System

72.5 Å

Total system size:

28 650 atoms

System III

membranewith125 POPCphosphor lipids

System II

6504 SPC/E water molecules

System I

protein (230 amino acids)

QM region:

protonated water network consist-ing of 4 and 6 water molecules, respectively

3 Distance Classes for the Coulombic Force Calculation:

QMρ

i(r

i)

MM

NumericalDiscretization of ρi(ri)

MM

MM

Point ChargeApproximation of ρi(ri)

MultipoleExpansion of ρi(ri)

ab initio (DFT) computation of infrared “fingerprint”

i t

tot tot

0

1I( ) dt e (0) (t)

2

∞− ωω = ⟨µ ⋅µ ⟩

π ∫� �

3

tot i i(t) e Z R (t) e d r (r;R(t)) rµ = − ρ∑ ∫� �� ��

Multi-State Empirical Valence Bond(MS-EVB)

Multi-State Empirical Valence Bond(MS-EVB)

- basic idea: quantum resonance of electronic states representing

stable species

- approximate method to model reactivereactive PES

MS-EVB algorithm

• generate relevant EVB states (dynamical adaptive basis)

• compute EVB energy matrix elements(~N force field calculations)

• diagonalize EVB matrix• compute Hellmann-Feynman forces

• perform MD step

10 ns of dynamics in NVE ensemble shows NO drift in kinetic temperature !!!!

Multistate EmpiricalValence Bond (MS-EVB)

Identify stable

chemical species

Find accurate intra- and

intermolecular force field

Compute ab initio PES (binding energies,

minimum energy paths, barrier heights)

Determine off-diagonal elements

(Functional form + parameters)

ab initio ab initio qualityquality PES PES forfor

reactivereactive systemssystems

TIP3P,H3O+ ff

H3O+, H2O

Schmitt&Voth J. Phys. Chem. 1998

“Proton antenna” in bR

“Proton antenna” in bR

• Direct dynamics of proton transfer in bRusing MSEVB

MethodenspektrumMethodenspektrum

CI, MP

CASPT2

CI, MP

CASPT2Length scale

predictivity

Kontinuumelektrostatik

“Coarse graining”

Kontinuumelektrostatik

“Coarse graining”

Empirische KraftfelderEmpirische Kraftfelder

fsp

sn

s

t i

me

Approximative

Methoden

Approximative

Methoden

HF, DFTHF, DFT

nm

Praktikum 28.01.2008 15.00 Uhr

• hands-on experience with Molpro• DFT calculations on water,amino acids etc.• QM/MM example• Multistate Empirical Valence Bond (MSEVB)

Overview: Computational Biophysics II

Sommersemester 2007

L1/P1: Elektrostatik in Proteinen

L2/P2: Free energy calculations, force probe simulations

L3/P3: Nichtgleichgewichtsthermodynamik

L4/P4: Ratentheorie

L5/P5: Enzymkatalyse: Molekulare Details chemischer Reaktionen

L6/P6: Bioinformatik: Sequence Alignment

L7/P7: Strukturvorhersage, Homology Modeling