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)
Chemical Intuition

Elementary Reaction Steps?

Deconstructing reaction in terms of simple one
or two component reactions
Rate Law
n A A  nB B
1  A 1 B 


 Rate
n A t
n B t
Rate Law

General
# reac tan ts
n R
i
i

# products
m P
j
i
j
j
Rate  appearance of a product =
1 dPi 
m i dt
 
1 d Rj
 -disappear ance of reactant = n j dt
# reactants
= -k(T)
r


R
 i
i
i
 k(T)
# products
p


P
 i
i
i
Rate Law

Typically measure rate as function of
temperature at constant pressure
1 A 1 B 


 Rate
n A t
nB t
1 A
a


Forward Rate   k (T ) * A
n A t

Note: ‘a’ can have ANY value
Rate Laws

Zero Order

First Order in A
 A
0
Forward Rate 
 k(T) * A(t)
t
A(t)  A(t 0)  k(T) * (t  t0 )
Forward Rate 

 A
1
 k(T) * A(t)
t
A(t)  A(t 0)ek(tt0 )


Second Order in A
 A
2
Forward Rate 
 k(T) * A(t)
t
1
1

 k(T) * (t  t 0 )
A(t) A(t 0 )
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

Develop a microscopic picture of how a
reaction proceeds (i.e. some wall/barrier must
be surmounted)
Arrhenius Rate Theory

Based on empirical results

Van’t Hoff plots
1
ln( k(T)) 
T

Postulated following formulas
A0  y  int ercept
Ea
 slope
kb
k(T)  A0e

Ea
kb T
 Arrhenius rate law
Transition State Theory
A+B  AB‡ P




k
AB‡ is intermediate or transition state complex
AB‡ P fast relative to A+BAB‡
ALL AB‡ reactive
TST
G 

k BT
kT

e
h
= (frequency of attempts) * (probabili ty of success)
Transition State Theory
A+B  AB‡ P

k TST  0 e
G 

k BT
G 

k BT
kT

e
h
= (frequency of attempts) * (probabili ty of success)
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 Theory

Ex. Michelis-Menton method for enzymatic
reactions

E+S  ES P


Assume rate increases linearly w/ E-concentration
Assume S>>E
d P 
k 2 E 0
0 

k 1  k 2
dt
1
k1 S 
Michelis-Menton method
for enzymatic reactions

S approaches infinity
k2 E 0
0 
 k2 E 0  Vmax
k k
1 1 2
k1 S 


S approaches S<<1
k2 E 0
k S Vmax
0 
 Vmax * 1

S   k S 

k k
k1  k 2 K m
1 1 2
k1 S 
Km 
k1  k 2
k1
Michelis-Menton method
Theory vs. Experiment


[S](L2.5
tryptoph
an)
5.0
10.0
15.0
20.0
u0(mMs-1)
0.036
0.053
0.060
0.064
0.024
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. Experiment

The hydrolysis of
carbobenzoxyglycyl-Ltryptophan using pancreatic
carboxypeptidase catalyst
Least Squared analysis
displayed agreement
with experimental
results
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 Coordinate?
8.35E-28
8.35E-28
8.35E-28
8.35E-28
8.35E-28
1.4E-18
8.35E-28
8.35E-28
1.2E-18
8.35E-28
8.35E-28
1E-18
8.35E-28
8.35E-28
8.35E-28
8E-19
8.35E-28
8.35E-28
6E-19
8.35E-28
8.35E-28
4E-19
8.35E-28
8.35E-28
2E-19
8.35E-28
8.35E-28
8.35E-28
0
8.35E-28
0
8.35E-28
8.77567E+14
20568787140
2.03098E-18
1.05374E-18
8.77567E+14
20568787140
1.77569E-18
Empirical Potential for Hydrogen Molecule9.66155E-19
8.77567E+14
20568787140
1.54682E-18
8.82365E-19
8.77567E+14
20568787140
1.34201E-18
8.02375E-19
8.77567E+14
20568787140
1.15913E-18
7.26185E-19
8.77567E+14
20568787140
9.96207E-19
6.53795E-19
8.77567E+14
20568787140
8.51451E-19
5.85205E-19
8.77567E+14
20568787140
7.23209E-19
5.20415E-19
8.77567E+14
20568787140
6.09973E-19
4.59425E-19
8.77567E+14
20568787140
5.10362E-19
4.02235E-19
8.77567E+14
20568787140
4.2311E-19
3.48845E-19
8.77567E+14
20568787140
3.47061E-19
2.99255E-19
8.77567E+14
20568787140
2.81155E-19
2.53465E-19
8.77567E+14
20568787140
2.24426E-19
2.11475E-19
8.77567E+14
20568787140
1.75987E-19
1.73285E-19
8.77567E+14
20568787140
1.35031E-19
1.38895E-19
8.77567E+14
20568787140
1.0082E-19
1.08305E-19
8.77567E+14
20568787140
7.26787E-20
8.15147E-20
8.77567E+14
20568787140
4.99924E-20
5.85247E-20
8.77567E+14
20568787140
3.22001E-20
3.93347E-20
8.77567E+14
20568787140
1.87901E-20
2.39447E-20
8.77567E+14
20568787140
9.29638E-21
0.5
1
1.5
2
2.5
31.23547E-20
3.5
8.77567E+14
20568787140
3.29443E-21
4.56475E-21
4
Reaction Coordinate

Minimum Energy Path on BornOppenheimer surface
Steepest Descent path
 Passes through saddle point/transition state

Reaction Coordinate
SN2 Exchange
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

Solve Newton’s Equations
F  mx

Mathematically, if know initial values of forces,
momenta and coordinates:

Taylor series expansion
2
dr
1d r
2
r(t 0  t)  r0 
 t 

(

t)

2
dt t0
2 dt t
0
n
1 d r
n
.... + (-1)

(

t)
n! dt n t
n
0
Molecular Dynamics

Taylor series expansions
2
dr
1d r
2
r(t 0  t)  r0 
 t 

(

t)

2
dt t0
2 dt t
0
n
1 d r
n
.... + (-1)

(

t)
n! dt n t
n
0


Similar equations for the velocity and acceleration
Molecular Dynamics

Various numerical approximations

Predictor-Corrector


Verlet



Gear
Leap Frog Method
Runge-Kutta
Optimal Integrator:



Maximize time step
Minimize strorage/time
Conserve energy
Molecular Dynamics
Predictor-Corrector

Truncate Taylor Expansions

Predict new values for r,v and a
Calculate “correct” acceleration using equation of motion

1
r(t 0  t)  r0  v(t 0 )  t  a(t 0 )  (t) 2
2
v(t  t)  v 0  a(t 0 )  t
a(t  t)  a(t 0 )
F  ma  
dV
dr

a c (t  t ) 
r ( t t )
dV
dr
r ( t t )
m
Molecular Dynamics
Predictor-Corrector

Correct predicted values
a c (t  t )  a (t  t )  a
r c  r  c0  a
v c  v  c1  a
a c  a  c2  a

Modify c’s such that error O((t)L+1) (Lth order method)
Molecular Dynamics
Verlet

Solve Newton’s Equation
d 2r
F  ma  m  2
dt
r(t  t)  2  r(t)  r(t  t)  a(t)  (t) 2



Velocities eliminated
Simpletic (preserves underlying physics)



Error a (t)4 (vs. (t)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
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)MD
ODE=pc-method
Si2 = Position-Verlet
Si4 ~ RKNystrom
1000 CH2
tmax=10ps
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
ZPE
 Isotope effects
 Tunneling

Collections of Particles

Brownian motion


Non-linear behavior
Characterize



Mean free path
Avearage # collisons
Flux
 c 
J  Flux   D 
 x  t
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
(t)
S-matrix
Cross Section
(
k(T)
 S,G
Quantum Rate Theory

Rate a Flux through
hypersurface

ˆ
k  F   (rˆ) (rˆ)
rˆ
Other Methods
PST (Phase Space Theory)
 RRK/RRKM theory



VTST (Variational Transition State Theory)


TST for unimolecular reactions (e.g. no intrinsic barrier)
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

Monte Carlo


Applicable to macro-systems
QM/MD
Use QM for Force evaluation
 Use classical MD to propogate atoms/molecules
