C8855 Advanced Molecular Modelling Methods -1C8855
Advanced Molecular
Modelling Methods
Petr Kulhánek
kulhanek@chemi.muni.cz
National Centre for Biomolecular Research, Faculty of Science
Masaryk University, Kamenice 5, CZ-62500 Brno
JS/2022 Present Form of Teaching: Rev1
Lesson 2
Empirical Valence Bond Theory
C8855 Advanced Molecular Modelling Methods -2Nobel
Laureates in Chemistry 2013
http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2013/
University of Southern
California, Los Angeles,
CA, USA
Stanford University School
of Medicine, Stanford, CA,
USA
Université de Strasbourg,
Strasbourg, France,
Harvard University,
Cambridge, MA, USA
C8855 Advanced Molecular Modelling Methods -3Essential
Works
Lifson, S.; Warshel, A. Consistent Force Field for Calculations of Conformations Vibrational
Spectra and Enthalpies of Cycloalkane and N-Alkane Molecules. J. Chem. Phys. 1968, 49,
5116–&.
Levitt, M.; Lifson, S. Refinement of Protein Conformations Using a Macromolecular Energy
Minimization Procedure. J. Mol. Biol. 1969, 46, 269–&.
Honig, B.; Karplus, M. Implications of Torsional Potential of Retinal Isomers for Visual
Excitation. Nature 1971, 229, 558–&.
Warshel, A.; Karplus, M. Calculation of Ground and Excited-State Potential Surfaces of
Conjugated Molecules .1. Formulation and Parametrization. J. Am. Chem. Soc. 1972, 94,
5612–&.
Levitt, M.; Warshel, A. Computer-Simulation of Protein Folding. Nature 1975, 253, 694–698.
Levitt, M. Simplified Representation of Protein Conformations for Rapid Simulation. J. Mol.
Biol. 1976, 104, 59–107.
Warshel, A.; Levitt, M. Theoretical Studies of Enzymic Reactions - Dielectric, Electrostatic
and Steric Stabilization of Carbonium-Ion in Reaction of Lysozyme. J. Mol. Biol. 1976, 103,
227–249.
C8855 Advanced Molecular Modelling Methods -4Foundation
of Methods
Newtonian (classical) mechanics
Quantum Mechanics (QM) Molecular Mechanics (MM) Coarse-grained Mechanics (CG) Continuum
(polarizable)
environment(water)
coarse-graining level
C8855 Advanced Molecular Modelling Methods -5Multiscale
Methods
Quantum Mechanics (QM) Molecular Mechanics (MM) Coarse-grained Mechanics (CG) Continuum
(polarizable)
environment(water)
coarse-graining level
QM/MM methods
C8855 Advanced Molecular Modelling Methods -6Multiscale
Methods
Quantum Mechanics (QM) Molecular Mechanics (MM) Coarse-grained Mechanics (CG) Continuum
(polarizable)
environment(water)
coarse-graining level
Valence state for EVB
protein or
reference system
and a few solvation layers
rest of
water environment
C8855 Advanced Molecular Modelling Methods -7Enzymatic
Reactions
MM QM active site only
tens to hundreds atoms
QM
MM
rest of enzyme
solvent (water)
Catalyzed reaction – hydrolysis of phosphodiester bond
SN2
C8855 Advanced Molecular Modelling Methods -8Enzymatic
Reactions
Enzyme: ~4,300 atoms
Water: ~42,000 atoms
Total: ~46,500 atoms
MutH
C8855 Advanced Molecular Modelling Methods -9Enzymatic
Reactions
Enzyme: ~4,300 atoms
Water: ~42,000 atoms
Total: ~46,500 atoms
MutH
Beyond any QM method!
C8855 Advanced Molecular Modelling Methods -10Boundary
Problems
Active site of MutH
(hydrogen atoms are not shown
for clarity)
• How to cut covalent bonds?
• There is an unphysical tension at boundary due to incompatible precision of QM and
MM potentials.
• MM atoms cannot be polarized by QM zone
• QM atoms can be (over)polarized by MM atoms
e-
eTwo
electrons formally
contribute to s-bond (single
bond).
C8855 Advanced Molecular Modelling Methods -11Quantum
Mechanics
),()(),(ˆ RrRRr mmme EH =
How to describe wave function (WF) of many-body systems?
➢ Molecular Orbital (MO) Theory
➢ Valence Bond (VB) Theory
MO is now prevalent because MO was simpler to implement than VB in early stages of QM
development.
C8855 Advanced Molecular Modelling Methods -12Molecular
Orbital Theory
)()(
1
i
m
j
jijii c rr =
= 
One-electron functions expressed using basis functions:
Molecular Orbital (MO) theory is about construction of many-body system wavefunction
(WF).
MO-LCAO - molecular orbital as linear combination of atomic orbitals
)}()()...()()()(){()...,,,,...,,( 22221111,2121 nnnn
P
nn rrrPsignrrr ssssss =
One-electron approximation:
C8855 Advanced Molecular Modelling Methods -13Valence
Bond Theory
Shurki, A.; Derat, E.; Barrozo, A.; Kamerlin, S. C. L. How Valence Bond Theory Can Help You
Understand Your (Bio)Chemical Reaction. Chem. Soc. Rev. 2015, 44 (5), 1037–1052.
https://doi.org/10.1039/C4CS00241E.
Valence Bond (VB) theory is about construction of many-body system wavefunction (WF).
C8855 Advanced Molecular Modelling Methods -14EVB
vs hybrid QM/MM
reactants (MM)
products (MM)
EVB (QM)
MM
EVB
QM
Empirical Valence Bond (EVB) Hybrid QM/MM
• no boundary problems
• simulation on mapping potentials
• reference reaction for calibration
Warshel, A.; Weiss, R. An Empirical Valence Bond Approach for Comparing Reactions in
Solutions and in Enzymes. J. Am. Chem. Soc. 1980, 102 (20), 6218–6226.
https://doi.org/10.1021/ja00540a008.
• boundary problems
• any QM theory (DFT, semiempirics, etc.)
C8855 Advanced Molecular Modelling Methods -15Reactants
and products as VB states
𝐻11 = 𝜀1 𝐻22 = 𝜀2 + 𝛼
valence states described by MM
a constant, which is a correction
to different MM reference states
𝐻11 𝐻22
reactant state
product state
C8855 Advanced Molecular Modelling Methods -16Reactants
and products as VB states
𝐻11 = 𝜀1 𝐻22 = 𝜀2 + 𝛼
valence states described by MM
a constant, which is a correction
to different MM reference states
𝐻11
𝐻22
reactant state
product state
what about
transition state?
C8855 Advanced Molecular Modelling Methods -17Empirical
Valence Bond
111 =H  += 222H
𝐻12 = 𝐻21 =?
𝐸 𝐸𝑉𝐵
valence states described by MM
off-diagonal mixing terms
𝐻11
𝐻22
kkk E  =H
Secular problem
Hamiltonian matrix






=
2221
1211
HH
HH
H
Secular problem has two solutions:
E2 > E1 = EEVB
ground state energy = EVB potential
C8855 Advanced Molecular Modelling Methods -18Empirical
Valence Bond
111 =H  += 222H
𝐻12 = 𝐻21 =?
𝐸 𝐸𝑉𝐵
valence states described by MM
off-diagonal mixing terms
𝐻11
𝐻22
kkk E  =H
Secular problem
Hamiltonian matrix






=
2221
1211
HH
HH
H
WF describes contributions of individual valence
states to the total system description.
C8855 Advanced Molecular Modelling Methods -19EVB
Assumption
The empirical valence bond (EVB) model provides an extremely powerful way for modeling
and analyzing chemical reactions in solutions and proteins. However, this model is based
on the unverified assumption that the off diagonal elements of the EVB Hamiltonian do
not change significantly upon transfer of the reacting system from one phase to another.
This ad hoc assumption has been rationalized by its consistency with empirically observed
linear free energy relationships, as well as by other qualitative considerations.
Nevertheless, this assumption has not been rigorously established. The present work
explores the validity of the above EVB key assumption by a rigorous numerical approach.
This is done by exploiting the ability of the frozen density functional theory (FDFT) and the
constrained density functional theory (CDFT) models to generate convenient diabatic
states for QM/MM treatments, and thus to examine the relationship between the diabatic
and adiabatic surfaces, as well as the corresponding effective off diagonal elements. It is
found that, at least for the test case of SN2 reactions, the off diagonal element does not
change significantly upon moving from the gas phase to solutions and thus the EVB
assumption is valid and extremely useful.
Hong, G.; Rosta, E.; Warshel, A. Using the Constrained DFT Approach in Generating
Diabatic Surfaces and Off Diagonal Empirical Valence Bond Terms for Modeling Reactions
in Condensed Phases. The Journal of Physical Chemistry B 2006, 110 (39), 19570–19574.
https://doi.org/10.1021/jp0625199.
C8855 Advanced Molecular Modelling Methods -20EVB
Assumption
Hong, G.; Rosta, E.; Warshel, A. Using the Constrained DFT Approach in Generating
Diabatic Surfaces and Off Diagonal Empirical Valence Bond Terms for Modeling Reactions
in Condensed Phases. The Journal of Physical Chemistry B 2006, 110 (39), 19570–19574.
https://doi.org/10.1021/jp0625199.
Warshel, A.; Weiss, R. An Empirical Valence Bond Approach for Comparing Reactions in
Solutions and in Enzymes. J. Am. Chem. Soc. 1980, 102 (20), 6218–6226.
https://doi.org/10.1021/ja00540a008.
26 years !!!!!
𝐻12 = 𝐻21 = 𝐴𝑒−𝜇(𝑟 𝑖𝑗−𝑟 𝑜)
off-diagonal mixing terms
empirical parameters
𝐻12 = 𝐻21 = 𝐴
off-diagonal mixing terms - typical form
some characteristic geometry
parameter (broaken/formed bond)
C8855 Advanced Molecular Modelling Methods -21EVB
Parameters
111 =H  += 222H 𝐻12 = 𝐻21 = 𝐴
EVB in its simplest form has only two parameters, which are characteristic for given
reaction but independent on environment (water, protein interior).
off-diagonal mixing termsvalence states described by MM
empirical parameters
C8855 Advanced Molecular Modelling Methods -22Reaction
coordinate
How to promote the reaction, i.e., the transition between reactant and product states?
𝐸𝑔𝑎𝑝𝐸𝑔𝑎𝑝
𝜉 = 𝐸 𝑔𝑎𝑝 = 𝐻22 − 𝐻11 = (𝜀2+𝛼) − 𝜀1
Suitable reaction coordinate is the energy gap between two valence states:
𝜉 = 𝐸 𝑔𝑎𝑝
it can be omitted
transition state at
𝐸𝑔𝑎𝑝 = 0
reactants products
C8855 Advanced Molecular Modelling Methods -23Simulations
on EVB potential
➢ It is possible to run MD simulations on EVB potential.
➢ However, the EVB parameters need to be provided before the simulation.
➢ Free energies (reaction and activation free energies) can be calculated using any biasing
techniques:
➢ Metadynamics
➢ Umbrella Sampling
➢ Adaptive Biasing Force method
➢ etc.
kinetics
RT
G
B
e
h
Tk
k


−
=
Eyring equation (theoretical model)
standard activation Gibbs energy
rate constant
(experiment)
C8855 Advanced Molecular Modelling Methods -24Simulations
on EVB potential
→ the EVB parametrization is thus tedious and computationally demanding ….
Reaction in water
(reference
system)
Reaction catalyzed
by enzyme
tune EVB
parameters
biased
simulation
determine
Δ𝐺𝑟 Δ𝐺≠
agree with
exp?
no yes
biased
simulation
determine
Δ𝐺𝑟 Δ𝐺≠
prediction
training
initial
prms
any new EVB parameters require to run a new simulation
1. training/parametrization
model the reaction in water environment (reference system) until the reaction and
activation free energies are the same as obtained by experiments
2. prediction
use the optimized EVB parameters to predict impact of the enzyme on the reaction
and activation free energies (determine the catalytic effect of the enzyme)
C8855 Advanced Molecular Modelling Methods -25Simulations
on EVB potential
→ the EVB parametrization is thus tedious and computationally demanding ….
Reaction in water
(reference
system)
Reaction catalyzed
by enzyme
tune EVB
parameters
biased
simulation
determine
Δ𝐺𝑟 Δ𝐺≠
agree with
exp?
no yes
biased
simulation
determine
Δ𝐺𝑟 Δ𝐺≠
prediction
training
initial
prms
any new EVB parameters require to run a new simulation
1. training/parametrization
model the reaction in water environment (reference system) until the reaction and
activation free energies are the same as obtained by experiments
2. prediction
use the optimized EVB parameters to predict impact of the enzyme on the reaction
and activation free energies (determine the catalytic effect of the enzyme)
C8855 Advanced Molecular Modelling Methods -26Mapping
Potential and Reference Reaction
Reaction in water (reference system) Reaction catalyzed by enzyme
Simulation on mapping potential Simulation on mapping potential
computationally demanding
but no EVB parameters are needed
virtually describe all possible reactions
21)1(  +−=mapE
no EVB parameters are required to run simulations on the mapping potential
both simulations are thus independent; they can be run in a parallel
C8855 Advanced Molecular Modelling Methods -27Mapping
Potential and Reference Reaction
Reaction in water (reference system) Reaction catalyzed by enzyme
Simulation on mapping potential Simulation on mapping potential
computationally demanding
but no parameters are needed
virtually describe all possible reactions
EVB parameters that reproduces
experimental reaction and activation
Gibbs free energies.
computationally cheap
EVB reconstruction via FEP/US (repeat)
21)1(  +−=mapE
FEP/US – Free Energy Perturbation/Umbrella Sampling
training
C8855 Advanced Molecular Modelling Methods -28Mapping
Potential and Reference Reaction
Reaction in water (reference system) Reaction catalyzed by enzyme
Simulation on mapping potential Simulation on mapping potential
EVB parameters that reproduces
experimental reaction and activation
Gibbs free energies.
EVB reconstruction via FEP/US
Predicted reaction and activation Gibbs free
energies for catalyzed reaction.
computationally demanding
but no parameters are needed
virtually describe all possible reactions
computationally cheap
EVB reconstruction via FEP/US (repeat)
21)1(  +−=mapE
FEP/US – Free Energy Perturbation/Umbrella Sampling
prediction
C8855 Advanced Molecular Modelling Methods -29-
Results
Fig. 3 A comparison of schematic EVB free
energy profiles for the reaction catalysed
by haloalkane dehalogenase (DhlA, Fig.
2), obtained in the gas-phase (g),
aqueous solution (w) and in the enzyme
active site (p).
https://doi.org/10.1039/C4CS00241E
C8855 Advanced Molecular Modelling Methods -30Technical
Notes
https://doi.org/10.1039/C4CS00241E
➢ Breaking and forming bonds must be described by a Morse potential? Why?
➢ LJ interaction for atoms involved in breaking and forming bonds must be softened in
the repulsive region.
C8855 Advanced Molecular Modelling Methods -31EVB
Pros & Cons
Pros:
➢ it is computationally cheap in comparison to hybrid QM/MM method (no expensive QM
calculations are preformed)
➢ it can be very accurate (all errors related to models used can be suppressed by tuned
empirical parameters)
➢ Egap as reaction coordinate (non-local but simple to calculate)
➢ Simulations on mapping potential simplify the EVB parametrization and prediction.
Cons:
➢ it cannot be used if mechanism of reaction in reference system and enzyme differs
(empirical parameters are not then transferable)
➢ it cannot go beyond valence state setup (it cannot reveal new reaction pathways)
C8855 Advanced Molecular Modelling Methods -32-
Questions
• Is EVB still used in research?
C8855 Advanced Molecular Modelling Methods -33Further
Readings
Mones, L.; Kulhanek, P.; Simon, I.; Laio, A.; Fuxreiter, M. The Energy Gap as a Universal
Reaction Coordinate for the Simulation of Chemical Reactions. J. Phys. Chem. B 2009, 113
(22), 7867–7873. https://doi.org/10.1021/jp9000576.
Mones, L.; Kulhanek, P.; Florian, J.; Simon, I.; Fuxreiter, M. Probing the Two-Metal Ion
Mechanism in the Restriction Endonuclease BamHI. Biochemistry 2007, 46 (50), 14514–
14523. https://doi.org/10.1021/bi701630s.
Shurki, A.; Derat, E.; Barrozo, A.; Kamerlin, S. C. L. How Valence Bond Theory Can Help You
Understand Your (Bio)Chemical Reaction. Chem. Soc. Rev. 2015, 44 (5), 1037–1052.
https://doi.org/10.1039/C4CS00241E.