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Molecular Modeling: Reaction Rates. C372 Introduction to Cheminformatics II Kelsey Forsythe. What’s in a rate?. Chemical Rate Law Rate depends on: Anything which changes motion of system Pressure, temperature Number of elements (atoms, molecules etc.) Rate a f(P,T)*g(N).

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molecular modeling reaction rates

Molecular Modeling:Reaction Rates

C372

Introduction to Cheminformatics II

Kelsey Forsythe

what s in a rate
What’s in a rate?
  • Chemical Rate Law
  • Rate depends on:
    • Anything which changes motion of system
      • Pressure, temperature
      • Number of elements (atoms, molecules etc.)
  • Rate a f(P,T)*g(N)
chemical intuition
Chemical Intuition
  • Elementary Reaction Steps?
    • Deconstructing reaction in terms of simple one or two component reactions
rate law5
Rate Law
  • General
rate law6
Rate Law
  • Typically measure rate as function of temperature at constant pressure
  • Note: ‘a’ can have ANY value
rate laws
Rate Laws
  • Zero Order
  • First Order in A
  • Second Order in A
integrating rate law
Integrating Rate Law
  • Oft used approximations:
    • Steady state approximation
    • Pseudo first order reaction
    • Identifying slow/rate-determining step
    • Rapid equilibration step(s)
    • Equal concentrations of reactants
connections to thermodynamics
Connections to Thermodynamics
  • Develop a microscopic picture of how a reaction proceeds (i.e. some wall/barrier must be surmounted)
arrhenius rate theory
Arrhenius Rate Theory
  • Based on empirical results
    • Van’t Hoff plots
    • Postulated following formulas
transition state theory
Transition State Theory
  • A+B  AB‡ P
    • AB‡ is intermediate or transition state complex
    • AB‡ P fast relative to A+BAB‡
    • ALL AB‡ reactive
transition state theory12
Transition State Theory
  • A+B  AB‡ P
  • Use MM, Semi-Empirical or Ab Initio to calculate frequencies and estimate thermodynamic values
    • kTST(T)>kexact(recrossing effects; MD corrections)
    • kclassical<kquantal (tunneling corrections; QTST, Centroid TST)
transition state theory13
Transition State Theory
  • Ex. Michelis-Menton method for enzymatic reactions
    • E+S  ESP
      • Assume rate increases linearly w/ E-concentration
      • Assume S>>E
michelis menton method for enzymatic reactions
Michelis-Menton method for enzymatic reactions
  • S approaches infinity
  • S approaches S<<1
michelis menton method theory vs experiment
Michelis-Menton method Theory vs. Experiment
  • From R. Lumry, E. L. Smith and R. R. Glantz, 1951, J. Am. Chem. Soc. 73, 4330.
  • The hydrolysis of carbobenzoxyglycyl-L-tryptophan using pancreatic carboxypeptidase catalyst
michelis menton method theory vs experiment16
Michelis-Menton method Theory vs. Experiment
  • The hydrolysis of carbobenzoxyglycyl-L-tryptophan using pancreatic carboxypeptidase catalyst

Least Squared analysis displayed agreement with experimental results

incorporating dynamics recrossings etc
Incorporating Dynamics (Recrossings etc.)
  • Dividing surface
  • Reaction Coordinate?
    • Decomposing full N-D space into a single reaction coordinate or minimum energy path through the Born-Oppenheimer surface
reaction coordinate19
Reaction Coordinate
  • Minimum Energy Path on Born-Oppenheimer surface
    • Steepest Descent path
    • Passes through saddle point/transition state
rate simulations
Rate Simulations
  • Require knowledge of molecular dynamics
    • Position of atoms/molecules
    • Distribution/partition function of species
    • Environment (Temperature etc.)
    • Phase (liquid, solid, gas)
molecular dynamics
Molecular Dynamics
  • Solve Newton’s Equations
  • Mathematically, if know initial values of forces, momenta and coordinates:
    • Taylor series expansion
molecular dynamics23
Molecular Dynamics
  • Taylor series expansions
  • Similar equations for the velocity and acceleration
molecular dynamics24
Molecular Dynamics
  • Various numerical approximations
    • Predictor-Corrector
      • Gear
    • Verlet
      • Leap Frog Method
    • Runge-Kutta
  • Optimal Integrator:
    • Maximize time step
    • Minimize strorage/time
    • Conserve energy
molecular dynamics predictor corrector
Molecular DynamicsPredictor-Corrector
  • Truncate Taylor Expansions
  • Predict new values for r,v and a
  • Calculate “correct” acceleration using equation of motion
molecular dynamics predictor corrector26
Molecular DynamicsPredictor-Corrector
  • Correct predicted values
  • Modify c’s such that error O((dt)L+1) (Lth order method)
molecular dynamics verlet
Molecular DynamicsVerlet
  • Solve Newton’s Equation
    • Velocities eliminated
    • Simpletic (preserves underlying physics)
      • Error a (dt)4 (vs. (dt)3 for predictor-corrector at same order)
      • Larger steps possible
    • 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
MD Method ComparisonS. 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

1000 CH2

tmax=10ps

md method comparison s k gray d w noid and b g sumpter j chem phys 101 5 4062 1994 md29
MD Method ComparisonS. 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

1000 CH2

tmax=10ps

addendum
Addendum
  • PCModel (Serena Software)
    • 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
molecular dynamics quantum corrections
Molecular DynamicsQuantum Corrections
  • ZPE
  • Isotope effects
  • Tunneling
collections of particles
Collections of Particles
  • Brownian motion
    • Non-linear behavior
    • Characterize
      • Mean free path
      • Avearage # collisons
      • Flux
collections of particles33
Collections of Particles
  • Brownian motion
    • Condensed phase systems
    • <r2> a Diffusion constant a friction/viscocity

MD!!!

quantum scattering theory
Applicable to gas phase reactions (di/tri atomics)

Solve time-dependent schrodinger equation

Determine scattering matrix

Determine scattering cross section

Calculate rate constant

Use k(T) to get thermodynamic quantities

Quantum Scattering Theory
quantum rate theory
Quantum Rate Theory
  • Rate a Flux through hypersurface
other methods
Other Methods
  • PST (Phase Space Theory)
  • RRK/RRKM theory
    • TST for unimolecular reactions (e.g. no intrinsic barrier)
  • VTST (Variational Transition State Theory)
    • Finds (n-1) surface which minimizes the rate
  • Marcus Theory
    • Applicable to electron transfer
      • Oxidation-reduction
      • Photosynthesis
  • Centroid Theory
    • Based on Feynman path integrals (quantum particle = centroid of collection of classical particles)
advanced simulation methods
Advanced Simulation Methods
  • Monte Carlo
    • Applicable to macro-systems
  • QM/MD
    • Use QM for Force evaluation
    • Use classical MD to propogate atoms/molecules