Bert de Groot, Udo Schmitt, Helmut Grubmüller

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1 Computergestützte Biophysik I Bert de Groot, Udo Schmitt, Helmut Grubmüller Max Planck-Institut für biophysikalische Chemie Theoretische und Computergestützte Biophysik Am Fassberg Göttingen Tel.: / bgroot@gwdg.de uschmit1@gwdg.de hgrubmu@gwdg.de

2 Overview: Computational Biophysics I L1/P1: Introduction, Protein structure and function, molecular dynamics, approximations, numerical integration, argon L2/P2: Tertiary structure, force field contributions, efficient algorithms, 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

3 Bio-Molecular Quantum Mechanics Density Functional Theory & QM/MM & Applications

4 Enzymes: life s catalyst Proteins that catalyze chemical reactions Highly specific (for ONE substrate) Dramatic acceleration of reaction rates

stabilizing the transition state - Purely electrostatic effect -

5 k cat E + S ES EP E + P How do enzymes achieve this dramatic accelaration? -Destabilizing the reactants/products vs. stabilizing the transition state - Purely electrostatic effect - Nuclear Quantum effects (tunneling) - Special Promoting vibrations - preorganization of enzyme

6 Enzymatic activity involves chemistry Breakdown of empirical force fields: no chemical bond breaking and formation in harmonic models L R ψ = c L + c R L R

7 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

8 Born-Oppenheimer approximation 1.) M i m e 1836 ˆ Tˆ = = P 2 ˆ 2 ˆ p i T i P i p i 2M 2m i e ˆ ˆ ˆ e Z e ZiZ i j e H = TP + Tp + + rˆ Rˆ Rˆ Rˆ rˆ rˆ k, i i< j k< l k i i j Hˆ ˆ ˆ elec ( r, R ) k l

9 Hˆ elec Hartree-Fock method ˆ e Z e ZiZ i j e = Tp + + rˆ R R R rˆ rˆ k, i i< j k< l k i i j k l pˆ e Z e = + + C k i k 2 m k, i ˆ k l ˆ ˆ e rk R < i rk rl 2 ˆ e = hk + + C k k< l rˆ rˆ k l Mean field ansatz (Hartree) Φ ( r; R) = Φ ( r ) = φ ( r ) φ ( r ) φ ( r ) k k N N Spatial orbitals product ansatz

10 Density Functional Theory (DFT) wavefunction <-> electron density Ψ ( x,,,,,, ) 1 xi x j x ( ) N ρ r 3N dimensional object 3 dimensional ρ( r ) ϕ ( r ) = i i 2

11 Density Functional Theory (DFT) ρ,,,,,, ρ ( r ) = 0 ρ r dr = N N electrons ( r ) = N ds dx dx Ψ ( x x x x ) ( ) N i j N 2

12 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

13 Hohenberg-Kohn Theorem Hˆ elec ˆ e Z e ZiZ i j e = Tp + + rˆ R R R rˆ rˆ k, i i< j k< l k i i j k l pˆ e Z e = + + C k i k 2 m k, i ˆ k l ˆ ˆ e rk R < i rk rl 2 2 pˆ k e = + + Vext + C k 2 m k< l rˆ rˆ e k l Vext ( r ) = external potential from nuclei = nuclei i 2 e Zi r - r i

14 Hohenberg-Kohn Theorem The external potential V ext (r) is (to within a constant) a unique functional of ρ(r); since, in turn V ˆ ext (r) fixes H elec we see that the many particle ground state is a unique functional of ρ(r) Vext ( r ) = external potential from nuclei = nuclei i Zi r - r i ρ( r ) V ( r) ext V ( r ) ρ( r ) ext

15 ext ext Proof Existence Theorem reduction ad absurdum: proof by contradiction V ( r ) V ( r ) that give rise to the same ρ(r) associated with H ˆ H ˆ V H ˆ E F V r r F V r [ ( )] = ρ ( ) = [ ( ) ] = 0 + ext; Ψ = Ψ 0 ext ext H ˆ = H ˆ + V ; H ˆ Ψ = E Ψ E = Ψ Hˆ Ψ < Ψ Hˆ Ψ = Ψ Hˆ Ψ + Ψ Hˆ Hˆ Ψ ext Ψ E < E + Ψ V V Ψ ext ext ρ( )( ext ext ) ρ ( ) E < E + dr r V V E < E + dr ( r ) V ext Vext E E E E + < + Vext ( r ) ρ( r ) E [ ρ( r) ]

16 Variational Theorem The ground state energy can be obtained variationally: the density that minimizes the total energy is the exact ground state density 2 ˆ ˆ e Helec = T + + Vext + C k< l rˆ rˆ k l [ ρ] = [ ρ] + [ ρ] + [ ρ] E T V V int ext [ ] ˆ E ρ ( r) = Ψ Helec Ψ [ ρ( )] < [ ρ ( )] E r E r

17 Kohn-Sham equation DFT analogon to Hartree-Fock equations: [ ρ] = [ ρ] + [ ρ] + [ ρ] E T V V 0 int ext dr ρ( r ) = N dr ρ( r ) N = 0 elec [ ] ( δ E [ ] [ ] [ ] ) 0 ρ = δ T ρ + Vint ρ + Vext ρ λ( dr ρ( r ) Nelec ) = 0 So far everything is exact: ab initio!!! But what are the Functionals? T [ ρ] V [ ρ] V [ ρ] elec, and ext int

18 5 3 T[ ρ( r )] dr ρ ( r ) Thomas-Fermi Uniform electron gas: Kohn-Sham approximations N electrons in box of volume V with neutralizing background charge [ ρ] [ ρ] [ ρ] T, V and Vext int 1 ρ( r ) ρ( r ) dr dr + EXC[ ρ( r )] 2 r r 4 3 E [ ( )] ( ) XC ρ r dr ρ r dr ρ( r ) Vext 10% error

19 Kohn-Sham approximations Kinetic energy part: T [ ρ] T 1 = = i t 2 2 i i i Φ ( x, x,, x ) = 1 2 Slater determinant N 1 N! ϕ ( x ) ϕ ( x ) ϕ ( x ) a 1 b 1 a+ N 1 ϕ ( x ) ϕ ( x ) ϕ ( x ) a 2 b 2 a+ N 2 ϕ ( x ) ϕ ( x ) ϕ ( x ) a N b N a+ N N 1 T r = + T r 2 [ ρ( )] ϕi i ϕi XC[ ρ( )] i 2 T [ ( )] XC ρ r + E [ ρ( r )] = E [ ρ( r )] XC XC ρ( r ) ϕ ( r ) = Still unkown!!! i i 2

20 Kohn-Sham equation [ ρ] = [ ρ] + [ ρ] + [ ρ] E0 T Vint Vext 1 1 ρ( r ) ρ( r ) ϕ ϕ dr dr E ρ r V ρ 2 = i i i + + XC[ ( )] + ext i 2 2 r r Hartree energy [ ] N N ( ) L[{ ϕ }] = E [ ρ] - λ ϕ ϕ δ i 0 ij i j ij i=1 j=1 N N δ L[{ ϕ }] = δ E [{ ϕ }] - λ δ ϕ ϕ = 0 i 0 i ij i j i=1 j=1 ρ( r ) ϕ ( r ) = i i 2

21 Kohn-Sham equation [ ρ r ] 1 2 δ EXC ( ) + vhartree ( r) + vext ( r) + ϕi ( r) = εiϕi ( r) i = 1,2,, N 2 δρ( r) Kohn-Sham non-linear integro-differential equations KS h ( r) ϕ ( r) = ε ϕ ( r) i = 1, 2,, N i i i i " atomic orbitals" φ( r) = ci χi ( r) i LCAO (linear combination of atomic orb.) basis set!!!! But can be recast in matrix form -> ideal for computers HC = SCε Self-consistent field method (SCF)

22 In search for the divine Functional(s) E [ ρ( r )] dr ρ( r) ε [ ρ( r)] energy density XC = XC Local density approximation (LDA) r 9 3 ε ρ = ρ XC 8 π r 3 [ ( )] ( ) Thomas,Fermi,Slater etc. Generalized Gradient Approximation (GGA) 1980is (BLYP,P PBE) ε XC[ ρ( r ), ρ( r )] Becke: everything is legal approach Perdew: based on first principles Hybrid (B3LYP) 1996 Combine Hartree-Fock with DFT!

23 Success of DFT Treat electron correlation based on density ρ( r ) Ψ x,, x,, x,, x ( ) No rigorous way how to improve results = find better functional ( trial & error ) 1 i j N Approximate (semi-empirical) Modern hybrid functionals B3LYP: Hartree-Fock < B3LYP < MP2 < CI N 4 N N N e

24 Basis sets Atom centered: Slater-Type orbitals χ( r, θ, φ, n, l, m, ζ ) n 1 ζ ( r r A ) m r e Yl ( θ, φ) r A GaussianTO χ( x, y, z, i, j, k, ζ ) e 2 ζ ( r r ) i j k A x y z Quantum numbers n,l,m parameter - good approximation - slow computation Spherical harmonics parameter - Pretty bad approximation - Vary fast - Take many of them in LCAO!

25 Basis sets But in DFT also so-called plane waves φ( r ) a( k ) e = k i k r

26 Electronic structure software packages Nobel prize Kohn 1998 Gaussian,Molpro,Q-Chem,CPMD, Turbomole,Jaguar etc. written in Fortran,C,C++ Provide efficient implementations of electronic structure methods on modern supercomputers and user-friendly interfaces (use by non-specialist)

27 What can we do with DFT? DNA crystal 2x12 base pairs Water + counter ions periodic boundary cond atoms ~ basis functions

28 Towards Larger Systems Quantum mechanical/molecular Mechanical (QM/MM) 5 6 ~ 10 atoms Warshel & Levitt 1976 Hˆ = Hˆ + H + Hˆ total QM MM QM / MM Hˆ elec Force field coupling Total MM region system Active site (QM) ~ atoms only ˆ eq ez Q σ σ HQM / MM = + + ( ) ( ) rˆ R R R R R electrons MM nuclei MM nuclei MM j i j ij 12 ij 6 i j i j i j i j i j ij ij

30 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

31 ab initio dynamics of biological systems QM/MM molecular dynamics Classical force fields (CHARMM, GROMOS96 etc.) 5 6 ~ 10 atoms Total MM region system - density functional theory (DFT) - on the fly computation of ab initio forces on the nuclei - simultaneous integration of electronic and nuclear degrees of freedom Active site (QM) - nuclear dynamics classical (point particle approximation) ~ atoms only

32 Total System Total system size: atoms 72.5 Å

33 System III membrane with125 POPC phosphor lipids

34 System II 6504 SPC/E water molecules

35 System I protein (230 amino acids) QM region: protonated water network consisting of 4 and 6 water molecules, respectively

36 3 Distance Classes for the Coulombic Force Calculation: Multipole Expansion of ρ i (r i ) Point Charge Approximation of ρ i (r i ) MM Numerical Discretization of ρ i (r i ) MM QM ρ i (r i ) MM

37 ab initio (DFT) computation of infrared fingerprint 1 iωt I( ) dt e tot (0) tot (t) 2π 0 ω = µ µ µ = ρ 3 tot (t) e ZiR i(t) e d r (r;r(t)) r

38 Multi-State Empirical Valence Bond (MS-EVB) - approximate method to model reactive PES - basic idea: quantum resonance of electronic states representing stable species

39 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!!!!

40 Multistate Empirical Valence Bond (MS-EVB) Identify stable chemical species H3O+, H2O Find accurate intra- and intermolecular force field TIP3P,H3O+ ff Compute ab initio PES (binding energies, minimum energy paths, barrier heights) Determine off-diagonal elements (Functional form + parameters) ab initio quality PES for reactive systems

41 Schmitt&Voth J. Phys. Chem. 1998

46 Proton antenna in br

47 Proton antenna in br Direct dynamics of proton transfer in br using MSEVB

48 Methodenspektrum Methodenspektrum time Kontinuumelektrostatik Kontinuumelektrostatik Coarse Coarsegraining graining ns Empirische EmpirischeKraftfelder Kraftfelder ps Approximative Approximative Methoden Methoden fs HF, HF,DFT DFT CI, CI,MP MP CASPT2 CASPT2 predictivity nm Length scale

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

50 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

Computational Methods. Chem 561

Computational Methods. Chem 561 Computational Methods Chem 561 Lecture Outline 1. Ab initio methods a) HF SCF b) Post-HF methods 2. Density Functional Theory 3. Semiempirical methods 4. Molecular Mechanics Computational Chemistry " Computational