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 (DFT-density functional theory)

Characteristics of Molecular Modeling Representing behavior of molecular systems Representing behavior of molecular systems Visual rendering of molecules Visual rendering of molecules Tinker toys Tinker toys Tinker Program (Washington Univ. St. Louis) Tinker Program (Washington Univ. St. Louis) Mathematical rendering of molecular interactions Mathematical rendering of molecular interactions Newton’s Laws - Kinetic Theory of Gases Newton’s Laws - Kinetic Theory of Gases Matrix Algebra - Quantum Theory Matrix Algebra - Quantum Theory Graph Theory? Informatics!!

Molecular Modeling += Underlying equations: empirical (approximate, soluble) - Morse Potential ab initio (exact, insoluble (less hydrogen atom) ) - Schrodinger Wave Equation ValenceBondTheory

Energy Energy = ? E=KE + PE Depends on underlying equations/assumptions: Energy of all/some of particles? Energy = 0? E MMFF NOT E HF

Electrostatics Coulombs Law Coulombs Law Permittivity used for vacuum Permittivity used for vacuum Point particles? Point particles? Solvent effects Solvent effects Poisson Equation Poisson Equation Used to calculate electronic properties Used to calculate electronic properties

Atomic Units

Thermodynamics How might we compute relevant thermodynamic quantities? How might we compute relevant thermodynamic quantities? Equipartition Theorem Equipartition Theorem Harmonic Oscillator Approximation Harmonic Oscillator Approximation

Quantum Mechanics All chemical properties for a system are given by the Schrodinger equation All chemical properties for a system are given by the Schrodinger equation No closed form solutions for systems of more than two-bodies (H- atom) No closed form solutions for systems of more than two-bodies (H- atom) Number of equations too numerous for computation/storage (informatics problem?) Number of equations too numerous for computation/storage (informatics problem?)

Schrodinger’s Equation - Hamiltonian operator - Hamiltonian operator Gravity? Gravity?

Hydrogen Molecule Hamiltonian Born-Oppenheimer Approximation Born-Oppenheimer Approximation Now Solve Electronic Problem Now Solve Electronic Problem

Electronic Schrodinger Equation Solutions: Solutions:, the basis set, are of a known form, the basis set, are of a known form Need to determine coefficients (c Need to determine coefficients (c m ) Wavefunctions gives probability ( ) of finding electrons in space (e. g. s,p,d and f orbitals) 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) Molecular orbitals are formed by linear combinations of electronic orbitals (LCAO)

Statistical Mechanics Molecular description of thermodynamics Molecular description of thermodynamics Temperature represents average state for system of molecules Temperature represents average state for system of molecules Energy of system is not energy of each molecule - distribution Energy of system is not energy of each molecule - distribution Condensed Phase - Ideal Gas Law not applicable. Condensed Phase - Ideal Gas Law not applicable. Boltzmann averaging Boltzmann averaging Use Monte Carlo for spatial/configurational averaging or molecular dynamics to average a property (ergodic hypothesis) Use Monte Carlo for spatial/configurational averaging or molecular dynamics to average a property (ergodic hypothesis)

Geometry Optimization First Derivative is Zero - At minimum/maximum First Derivative is Zero - At minimum/maximum As N increases so does dimensionality/complexity/beauty/difficulty As N increases so does dimensionality/complexity/beauty/difficulty Multi-dimensional (macromolecules, proteins) Multi-dimensional (macromolecules, proteins) Conjugate gradient methods Conjugate gradient methods Monte Carlo methods Monte Carlo methods

Empirical Models Simple/Elegant? Simple/Elegant? Intuitive?-Vibrations ( ) Intuitive?-Vibrations ( ) Major Drawbacks: Major Drawbacks: Does not include quantum mechanical effects Does not include quantum mechanical effects No information about bonding (  No information about bonding (  e ) Not generic (organic inorganic) Not generic (organic inorganic) Informatics Informatics Interface between parameter data sets and systems of interest Interface between parameter data sets and systems of interest Teaching computers to develop new potentials from existing math templates Teaching computers to develop new potentials from existing math templates

MMFF Potential E = E bond + E angle + E angle-bond + E torsion + E VDW + E electrostatic E = E bond + E angle + E angle-bond + E torsion + E VDW + E electrostaticE bondE angleE angle-bond E torsionE VDW E electrostaticE bondE angleE angle-bond E torsionE VDW E electrostatic Merck Molecular Force Field -Common organics/biopolymers

MMFF Energy Stretching Stretching Stretching

MMFF Energy Bending Bending Bending

MMFF Energy Stretch-Bend Interactions Stretch-Bend Interactions Stretch-Bend Interactions Stretch-Bend Interactions

MMFF Energy Torsion (4-atom bending) Torsion (4-atom bending) Torsion (4-atom bending) Torsion (4-atom bending)

MMFF Energy Analogous to Lennard-Jones 6-12 potential Analogous to Lennard-Jones 6-12 potential Analogous to Lennard-Jones 6-12 potential Analogous to Lennard-Jones 6-12 potential London Dispersion Forces London Dispersion Forces Van der Waals Repulsions Van der Waals Repulsions

Intermolecular/atomic models General form: General form: Lennard-Jones Lennard-Jones Van derWaals repulsion London Attraction

MMFF Energy Electrostatics (ionic compounds) Electrostatics (ionic compounds) Electrostatics (ionic compounds) Electrostatics (ionic compounds) D – Dielectric Constant D – Dielectric Constant  - electrostatic buffering constant  - electrostatic buffering constant

Hydrogen Molecule Bond Density Bond Density