Molecular Modeling: Reaction Rates C372 Introduction to Cheminformatics II Kelsey Forsythe

What’s in a rate? Chemical Rate Law Chemical Rate Law Rate depends on: Rate depends on: Anything which changes motion of system Anything which changes motion of system Pressure, temperature Pressure, temperature Number of elements (atoms, molecules etc.) Number of elements (atoms, molecules etc.) Rate  f(P,T)*g(N) Rate  f(P,T)*g(N)

Chemical Intuition Elementary Reaction Steps? Elementary Reaction Steps? Deconstructing reaction in terms of simple one or two component reactions Deconstructing reaction in terms of simple one or two component reactions

Rate Law

General General

Rate Law Typically measure rate as function of temperature at constant pressure Typically measure rate as function of temperature at constant pressure Note: ‘a’ can have ANY value Note: ‘a’ can have ANY value

Rate Laws Zero Order Zero Order First Order in A First Order in A Second Order in A Second Order in A

Integrating Rate Law Oft used approximations: Oft used approximations: Steady state approximation Steady state approximation Pseudo first order reaction Pseudo first order reaction Identifying slow/rate-determining step Identifying slow/rate-determining step Rapid equilibration step(s) Rapid equilibration step(s) Equal concentrations of reactants Equal concentrations of reactants

Connections to Thermodynamics Develop a microscopic picture of how a reaction proceeds (i.e. some wall/barrier must be surmounted) Develop a microscopic picture of how a reaction proceeds (i.e. some wall/barrier must be surmounted)

Arrhenius Rate Theory Based on empirical results Based on empirical results Van’t Hoff plots Van’t Hoff plots Postulated following formulas Postulated following formulas

Transition State Theory A+B  AB ‡  P A+B  AB ‡  P  AB ‡ is intermediate or transition state complex  AB ‡  P fast relative to A+B  AB ‡  ALL AB ‡ reactive

Transition State Theory A+B  AB ‡  P A+B  AB ‡  P Use MM, Semi-Empirical or Ab Initio to calculate frequencies and estimate thermodynamic values Use MM, Semi-Empirical or Ab Initio to calculate frequencies and estimate thermodynamic values k TST (T)>k exact (recrossing effects; MD corrections) k TST (T)>k exact (recrossing effects; MD corrections) k classical <k quantal (tunneling corrections; QTST, Centroid TST) k classical <k quantal (tunneling corrections; QTST, Centroid TST)

Transition State Theory Ex. Michelis-Menton method for enzymatic reactions Ex. Michelis-Menton method for enzymatic reactions E+S  ES  P E+S  ES  P  Assume rate increases linearly w/ E-concentration  Assume S>>E

Michelis-Menton method for enzymatic reactions S approaches infinity S approaches infinity S approaches S<<1 S approaches S<<1

Michelis-Menton method Theory vs. Experiment From R. Lumry, E. L. Smith and R. R. Glantz, 1951, J. Am. Chem. Soc. 73, From R. Lumry, E. L. Smith and R. R. Glantz, 1951, J. Am. Chem. Soc. 73, The hydrolysis of carbobenzoxyglycyl-L-tryptophan using pancreatic carboxypeptidase catalyst The hydrolysis of carbobenzoxyglycyl-L-tryptophan using pancreatic carboxypeptidase catalyst [S](L- tryptoph an)  0 (mMs -1 )

Michelis-Menton method Theory vs. Experiment The hydrolysis of carbobenzoxyglycyl-L- tryptophan using pancreatic carboxypeptidase catalyst The hydrolysis of carbobenzoxyglycyl-L- tryptophan using pancreatic carboxypeptidase catalyst Least Squared analysis displayed agreement with experimental results

Incorporating Dynamics (Recrossings etc.) Dividing surface Dividing surface Reaction Coordinate? Reaction Coordinate? Decomposing full N-D space into a single reaction coordinate or minimum energy path through the Born-Oppenheimer surface Decomposing full N-D space into a single reaction coordinate or minimum energy path through the Born-Oppenheimer surface

Reaction Coordinate?

Reaction Coordinate Minimum Energy Path on Born- Oppenheimer surface Minimum Energy Path on Born- Oppenheimer surface Steepest Descent path Steepest Descent path Passes through saddle point/transition state Passes through saddle point/transition state

Reaction Coordinate SN2 Exchange

Rate Simulations Require knowledge of molecular dynamics Require knowledge of molecular dynamics Position of atoms/molecules Position of atoms/molecules Distribution/partition function of species Distribution/partition function of species Environment (Temperature etc.) Environment (Temperature etc.) Phase (liquid, solid, gas) Phase (liquid, solid, gas)

Molecular Dynamics Solve Newton’s Equations Solve Newton’s Equations Mathematically, if know initial values of forces, momenta and coordinates: Mathematically, if know initial values of forces, momenta and coordinates: Taylor series expansion Taylor series expansion

Molecular Dynamics Taylor series expansions Taylor series expansions Similar equations for the velocity and acceleration Similar equations for the velocity and acceleration

Molecular Dynamics Various numerical approximations Various numerical approximations Predictor-Corrector Predictor-Corrector Gear Gear Verlet Verlet Leap Frog Method Leap Frog Method Runge-Kutta Runge-Kutta Optimal Integrator: Optimal Integrator: Maximize time step Maximize time step Minimize strorage/time Minimize strorage/time Conserve energy Conserve energy

Molecular Dynamics Predictor-Corrector Truncate Taylor Expansions Truncate Taylor Expansions Predict new values for r,v and a Predict new values for r,v and a Calculate “correct” acceleration using equation of motion Calculate “correct” acceleration using equation of motion

Molecular Dynamics Predictor-Corrector Correct predicted values Correct predicted values Modify c’s such that error  (  t) L+1 ) (L th order method) Modify c’s such that error  (  t) L+1 ) (L th order method)

Molecular Dynamics Verlet Solve Newton’s Equation Solve Newton’s Equation Velocities eliminated Velocities eliminated Simpletic (preserves underlying physics) Simpletic (preserves underlying physics) Error  (  t) 4 (vs. (  t) 3 for predictor-corrector at same order) Error  (  t) 4 (vs. (  t) 3 for predictor-corrector at same order) Larger steps possible Larger steps possible Less storage/time required Less storage/time required

MD Method Comparison S. K. Gray, D. W. Noid and B. G. Sumpter, J. Chem. Phys. 101(5) 4062(1994)MD ODE=pc-method Si2 = Position-Verlet Si4 ~ RKNystrom  CH  t max =10ps

MD Method Comparison S. K. Gray, D. W. Noid and B. G. Sumpter, J. Chem. Phys. 101(5) 4062(1994)MD ODE=pc-method Si2 = Position-Verlet Si4 ~ RKNystrom  CH  t max =10ps

Addendum PCModel (Serena Software) PCModel (Serena Software) Utilizies a modified version of Verlet called the Beeman algorithm Utilizies a modified version of Verlet called the Beeman algorithm Gilbert: Often for large molecular systems when one can separate time scales the larger motions can be sampled less often than the faster time scale (bond vibrations, fs) motions thus making such calculations more computationally feasible Gilbert: Often for large molecular systems when one can separate time scales the larger motions can be sampled less often than the faster time scale (bond vibrations, fs) motions thus making such calculations more computationally feasible

Molecular Dynamics Quantum Corrections ZPE ZPE Isotope effects Isotope effects Tunneling Tunneling

Collections of Particles Brownian motion Brownian motion Non-linear behavior Non-linear behavior Characterize Characterize Mean free path Mean free path Avearage # collisons Avearage # collisons Flux Flux

Collections of Particles Brownian motion Brownian motion Condensed phase systems Condensed phase systems  Diffusion constant  friction/viscocity  Diffusion constant  friction/viscocity MD!!!

Quantum Scattering Theory Applicable to gas phase reactions (di/tri atomics) Applicable to gas phase reactions (di/tri atomics) Solve time-dependent schrodinger equation Solve time-dependent schrodinger equation Determine scattering matrix Determine scattering matrix Determine scattering cross section Determine scattering cross section Calculate rate constant Calculate rate constant Use k(T) to get thermodynamic quantities Use k(T) to get thermodynamic quantities

Quantum Rate Theory Rate  Flux through hypersurface Rate  Flux through hypersurface

Other Methods PST (Phase Space Theory) PST (Phase Space Theory) RRK/RRKM theory RRK/RRKM theory TST for unimolecular reactions (e.g. no intrinsic barrier) TST for unimolecular reactions (e.g. no intrinsic barrier) VTST (Variational Transition State Theory) VTST (Variational Transition State Theory) Finds (n-1) surface which minimizes the rate Finds (n-1) surface which minimizes the rate Marcus Theory Marcus Theory Applicable to electron transfer Applicable to electron transfer Oxidation-reduction Oxidation-reduction Photosynthesis Photosynthesis Centroid Theory Centroid Theory Based on Feynman path integrals (quantum particle = centroid of collection of classical particles) Based on Feynman path integrals (quantum particle = centroid of collection of classical particles)

Advanced Simulation Methods Monte Carlo Monte Carlo Applicable to macro-systems Applicable to macro-systems QM/MD QM/MD Use QM for Force evaluation Use QM for Force evaluation Use classical MD to propogate atoms/molecules Use classical MD to propogate atoms/molecules