C372_L1 - Molecular Modeling Fundamentals: Modus in Silico...

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Unformatted text preview: Molecular Modeling Fundamentals: Modus in Silico C372 Introduction to Cheminformatics II Kelsey Forsythe "Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit in chemistry. If mathematical analysis should ever hold a prominent place in chemistry - an aberration which is happily almost impossible - it would occasion a rapid and widespread degeneration of that science." A. Comte (1830) 1992 Nobel Prize in Chemistry Rudolph Marcus (Theory of Electron Transfer) 1998 Nobel Prize in Chemistry John Pople (ab initio) Walter Kohn (DFTdensity functional theory) Characteristics of Molecular Modeling Representing behavior of molecular systems Visual rendering of molecules Mathematical rendering of molecular interactions Tinker toys Tinker Program (Washington Univ. St. Louis) Newton's Laws Kinetic Theory of Gases Matrix Algebra Quantum Theory Graph Theory? Informatics!! Molecular Modeling + = Valence Bond Theory Underlying equations: empirical (approximate, soluble) -Morse Potential V =0 -a -0) D e( R 2 ( -R ) 1 H H ab initio (exact, insoluble (less hydrogen atom)) ^ -Schrodinger Wave Equation H = E Energy Energy = ? E=KE + PE EH G Depends on underlying equations/assumptions: Energy of all/some of particles? Energy = 0? EMMFF NOT EHF Electrostatics Coulombs Law Permittivity used for vacuum Point particles? Solvent effects qiqj P E= 4 0r ij r r F -( E = ) P Poisson Equation Used to calculate electronic properties = - 2 Atomic Units q j q i P E= E= P 4 r 0 ij r i j q j q i Thermodynamics How might we compute relevant thermodynamic quantities? Equipartition Theorem Harmonic Oscillator Approximation Quantum Mechanics All chemical properties for a system are given by the Schrodinger equation ^ = H E No closed form solutions for systems of more than twobodies (H atom) Number of equations too numerous for computation/storage (informatics problem?) Schrodinger's Equation ^ = H E Hamiltonian operator ^ H ^= + ^ ^ HT V h2 2 - 2mi i N C i< j N eiej ri -rj Gravity? Hydrogen Molecule Hamiltonian ^ ^ ^ H = T +V p2 ^ H =- 2 2 2 2 1 2 2 e1 e 2 p p + + + + m p m p me me 1 1 1 1 1 1 C + - - - - re1e 2 rp1 p 2 rp1e1 rp1e 2 rp 2 e1 rp 2e 2 ^ ^ ^ H =T +V el el el-nuclei +V nuclei BornOppenheimer Approximation h2 2 2 1 1 1 1 1 1 ^ =- e1 + e2 +C H - - - - +C el 2 m m r rp1e1 rp1e2 rp2e1 rp2e2 rp1p2 e e e1e2 123 constant Now Solve Electronic Problem Electronic Schrodinger Equation Solutions: m v , the basis set, are of a known form ( ) mr v v r)=m (r) ( c m F Need to determine coefficients (cm) * Wavefunctions gives probability ( ) of finding electrons in space (e. g. s,p,d and f orbitals) Molecular orbitals are formed by linear combinations of electronic orbitals (LCAO) *^ ^ O= Od Statistical Mechanics Molecular description of thermodynamics Temperature represents average state for system of molecules 1 3 mv2 = k T 2 2 Energy of system is not energy of each molecule distribution wi) e i /k ( E T - Condensed Phase Ideal Gas Law not applicable. Boltzmann averaging Use Monte Carlo for spatial/configurational averaging or molecular dynamics to average a property (ergodic hypothesis) 1 N Geometry Optimization First Derivative is Zero At minimum/maximum As N increases so does dimensionality/complexity/beauty/difficulty v dV(r) v =0 dr Multidimensional (macromolecules, proteins) Conjugate gradient methods Monte Carlo methods Empirical Models Simple/Elegant? p p Intuitive?Vibrations ( ) F = - kr Major Drawbacks: Informatics Does not include quantum mechanical effects No information about bonding ( ) e Not generic (organic inorganic) Interface between parameter data sets and systems of interest Teaching computers to develop new potentials from existing math templates MMFF Potential Merck Molecular Force Field -Common organics/biopolymers E = Ebond + Eangle + Eanglebond + E torsion + EVDW + Eelectrostatic MMFF Energy Stretching Ebond 7 0 2 0 2 = K bond (rij - r ) * 1 + cs(rij - rij ) + cs (rij - rij ) 12 0 2 ij ( ) MMFF Energy Bending Eangle = K ( ijk - ) * 1 + cb( ijk - ) 0 2 ijk 0 ijk { } MMFF Energy StretchBend Interactions 0 0 Ebond - angle = K ijk (rij - rij0 ) + K kji (rkj - rkj ) ijk - ijk { }( ) MMFF Energy Torsion (4atom bending) Etorsion = 0.5{V1 (1 + cos ) + V2 (1 + cos 2 ) + V3 (1 + cos 3 )} MMFF Energy Analogous to LennardJones 612 potential London Dispersion Forces Van der Waals Repulsions EVDW 1.07 R = ij * Rij + 0.07 Rij * ij 7 1.07 R * 7 ij - 2 7 *7 Rij + 0.07 Rij Intermolecular/atomic models General form: V= (r ( i,r )+( i,r,r)+..... V )+ V r j Vr j k i<j i<j j< k N N LennardJones 12 6 V(rij ) = 4 - 23 r 1r 123 Van derWaals repulsion London Attraction MMFF Energy Electrostatics (ionic compounds) D Dielectric Constant electrostatic buffering constant qi q j Eelectrostatic = n D( Rij + ) 8.35E28 8.35E28 8.35E28 8.35E28 8.35E28 1.4E18 8.35E28 8.35E28 1.2E18 8.35E28 8.35E28 1E18 8.35E28 8.35E28 8.35E28 8E19 8.35E28 8.35E28 6E19 8.35E28 8.35E28 4E19 8.35E28 8.35E28 2E19 8.35E28 8.35E28 8.35E28 0 8.35E28 0 8.35E28 8.77567E+14 20568787140 2.03098E18 1.05374E18 8.77567E+14 P tloy g o l 20568787140H1.77569E18 c E ra t i f m l o ar de l u p ic e i n r n ee o M 9.66155E19 8.77567E+14 20568787140 1.54682E18 8.82365E19 8.77567E+14 20568787140 1.34201E18 8.02375E19 8.77567E+14 20568787140 1.15913E18 7.26185E19 8.77567E+14 20568787140 9.96207E19 6.53795E19 8.77567E+14 20568787140 8.51451E19 5.85205E19 8.77567E+14 20568787140 7.23209E19 5.20415E19 8.77567E+14 20568787140 6.09973E19 4.59425E19 8.77567E+14 20568787140 5.10362E19 4.02235E19 8.77567E+14 20568787140 4.2311E19 3.48845E19 8.77567E+14 20568787140 3.47061E19 2.99255E19 8.77567E+14 20568787140 2.81155E19 2.53465E19 8.77567E+14 20568787140 2.24426E19 2.11475E19 8.77567E+14 20568787140 1.75987E19 1.73285E19 8.77567E+14 20568787140 1.35031E19 1.38895E19 8.77567E+14 20568787140 1.0082E19 1.08305E19 8.77567E+14 20568787140 7.26787E20 8.15147E20 8.77567E+14 20568787140 4.99924E20 5.85247E20 8.77567E+14 20568787140 3.22001E20 3.93347E20 8.77567E+14 20568787140 1.87901E20 2.39447E20 8.77567E+14 1 20568787140 1.23547E20 0.5 1.5 2 9.29638E21 2.5 3 3.5 8.77567E+14 20568787140 3.29443E21 4.56475E21 4 8.35E28 8.35E28 8.35E28 8.35E28 1.4E18 8.35E28 8.35E28 8.35E28 1.2E18 8.35E28 8.35E28 1E18 8.35E28 8.35E28 8E19 8.35E28 8.35E28 8.35E28 6E19 8.35E28 8.35E28 4E19 8.35E28 8.35E28 2E19 8.35E28 8.35E28 8.35E28 0 8.35E28 0 8.35E28 Er l oiloy g o l m pa t ar de l u ic e i n r n e 9.66155E19 oM e 8.77567E+14 P t f H1.77569E18 c 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 20568787140 8.77567E+14 1 20568787140 0.5 1.5 8.77567E+14 20568787140 1.54682E18 1.34201E18 1.15913E18 9.96207E19 8.51451E19 7.23209E19 6.09973E19 5.10362E19 4.2311E19 3.47061E19 2.81155E19 2.24426E19 1.75987E19 1.35031E19 1.0082E19 7.26787E20 4.99924E20 3.22001E20 1.87901E20 2 9.29638E21 2.5 3.29443E21 8.82365E19 8.02375E19 7.26185E19 6.53795E19 5.85205E19 5.20415E19 4.59425E19 4.02235E19 3.48845E19 2.99255E19 2.53465E19 2.11475E19 1.73285E19 1.38895E19 1.08305E19 8.15147E20 5.85247E20 3.93347E20 2.39447E20 1.23547E20 3 3.5 4.56475E21 8.77567E+14 20568787140 2.03098E18 1.05374E18 4 Hydrogen Molecule Bond Density ...
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This note was uploaded on 06/28/2011 for the course C 372 taught by Professor Yoonsuplee during the Spring '11 term at Korea Advanced Institute of Science and Technology.

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