Primer on Empirical Valence Bond Calculations

Primer on Empirical Valence Bond
calculations
Jordi Villà-Freixa
jordi.villa@uvic.cat
Universitat de Vic - Universitat Central de Catalunya
COZYME webinars
May 4th, 2023

Why? What? How? Examples Conclusions
1/24
Contents
1 Introduction
2 What was the EVB made for?
3 How is it implemented?
The theoretical framework
Variants
A practical example of implementation
4 The basic method and beyond
5 Conclusions
Jordi VF [FCTE] COZYME webinars 2023-05-05

2/24
Introduction1
1
Romero-Rivera, A., Garcia-Borràs, M. & Osuna, S. Computational tools for the evaluation of
laboratory-engineered biocatalysts. Chem. Commun. 53, (2016)

3/24
Introduction
General QM/MM implementation:
EQM/MM = EQM(A) + Eint(A, B) + EMM(B)
= EQM(A) + Eint,el(A, B)
| {z }
PNB
i=1
R
ρA(r)
qi,B
|r−Ri,B|
dr3
+Eint,ne(A, B) + EMM(B)
= Eemb
QM (A) + Eint,ne(A, B) + EMM(B)
QM/MM are needed to incorporate solvent effects in energy
calculations

4/24
Limitations of the standard QM/MM approach I
q sampling is still very reduced2
q Difficult set up of collective variables to determine reaction
coordinate.

5/24
Limitations of the standard QM/MM approach II
where, if the reaction (ql) and
the complementary (qs)
collective coordinates are
orthogonal:a
a
Kim, B. et al. Reaction Path-Force Matching in
Collective Variables: Determining Ab Initio QM/MM
Free Energy Profiles by Fitting Mean Force. J Chem
Theory Comput 17, 4961–4980 (2021)
2
Xie, L., Cheng, H., Fang, D., Chen, Z.-N. & Yang, M. Enhanced QM/MM sampling for free energy calculation
of chemical reactions: A case study of double proton transfer.The Journal of Chemical Physics 150, 044111 (2019)

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The empirical valence bond (EVB) method I
q In the EVB formalization, chemical reactions are described by
mixing the relevant diabatic states:3
Hii = αgasi + Uintra(r, q) + USs(r, q, r0
, s) + Uss(r0
, q0
)
q The off-diagonal terms can be taken as constant or a simple
expression dependent of some reaction coordinate, e.g.:
Hij = A exp{−µ(r − r0)}
The EVB Hamiltonian is used to calculate the energy of a
system at each point along the PES.

7/24
The empirical valence bond (EVB) method II
q Then, if HEVB =

H11 H12
H12 H22

, the secular equation is soved:
HEVBCg = EgCg

8/24
The empirical valence bond (EVB) method III
q In a typical implementation, only two states are taken:
Reactants and Products
q Free energy perturbation (FEP) is used to move the system
from state 1 to state 2:
εm = λmε1 + (1 − λm)ε2
with λm = 0, . . . , 1, producing
∆G(λn) = ∆G(λ0 → λn) =
n−1
X
i=0
δG(λi → λi+1)
where, taking β = kBT:
δG(λi → λi+1) =

1
β

log hexp{−β}(εi+1 − εi)ii

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The empirical valence bond (EVB) method IV
a)
b)

10/24
The empirical valence bond (EVB) method V
q Finally, to obtain the actual free energy as a function of the
global reaction coordinate Xn = ε2 − ε1:
exp{−β∆G(Xn
)} = exp{−β∆G(Xn
)}hexp{−β (εm(Xn
) − Eg (Xn
))}im
3
Warshel, A. Weiss, R. M. An empirical valence bond approach for comparing reactions in solutions and in
enzymes. J. Am. Chem. Soc. 102, 6218–6226 (1980).; Kamerlin, S. C. L. Warshel, A. The empirical valence
bond model: theory and applications. WIREs Computational Molecular Science 1, 30–45 (2011)

11/24
Some variants I
Vdiab =

V11(Q) V12(Q)
V12(Q) V22(Q)

where, from Chang-Miller’s formalism
V 2
12(Q) = [V11(Q) − V (Q)][V22(Q) − V (Q)]
q Truhlar’s multicofiguration molecular mechanics4 using
Shepard interpolation of quadratic expressions in the PES:
V 2
12(Q) =
M
X
k=1
Wk(Q)V 0
12(Q, k)

12/24
Some variants II
q Schlegel and Sonnenberg’s approach for off-diagonal terms:5
V 2
12(Q) =
X
K
3N−6
X
i≥j≥0
BK:ijg(Q|QK , i, j, αK )
q Voth’s multiscale EVB (MS-EVB)6
4
Kim, Y., Corchado, J. C., Villà, J., Xing, J. Truhlar, D. G. Multiconfiguration molecular mechanics algorithm
for potential energy surfaces of chemical reactions. The Journal of Chemical Physics 112, 2718–2735 (2000)
5
H. B. Schlegel and J. L. Sonnenberg, J. Chem. Theory Comput., 2006, 2, 905–911.
6
1. Maupin, C. M., Wong, K. F., Soudackov, A. V., Kim, S. Voth, G. A. A multistate empirical valence bond
description of protonatable amino acids. J Phys Chem A 110, 631–639 (2006).

13/24
Implementation in Q67
7
Bauer, P. et al. Q6: A comprehensive toolkit for empirical valence bond and related free energy calculations.
SoftwareX 7, 388–395 (2018).

14/24
Applications of EVB8
8
Glennon, T. M., Villà, J. Warshel, A. How Does GAP Catalyze the GTPase Reaction of Ras?: A Computer
Simulation Study. Biochemistry 39, 9641–9651 (2000).

15/24
Applications of EVB9
9
Asadi, M., Oanca, G. Warshel, A. Effect of Environmental Factors on the Catalytic Activity of
Intramembrane Serine Protease. J. Am. Chem. Soc. 144, 1251–1257 (2022

16/24
Applications of EVB

17/24
Applications of MS-EVB10
10
Hu, Y., Wang, S., He, Y. An, L. Evaluation of proton transport and solvation effect in hydrated Nafion
membrane with degradation. Phys. Chem. Chem. Phys. 24, 29024–29033 (2022).

18/24
Activity screening I
11

19/24
Activity screening II
11
Risso, V. A. et al. Enhancing a de novo enzyme activity by computationally-focused ultra-low-throughput
screening †Electronic supplementary information (ESI) available: Additional simulation details and table of the full
list of variants predicted by FuncLib Chem Sci 11, 6134–6148 (2020)

20/24
Extensions: multiscale simulations12
EXPLORATION
REFINEMENT
Potential Energy Surface Coordinates Mapping
Energy
Energy
Internal Coordinates
Coarse-grained
All-Atom
12
Avila, C. L., Drechsel, N. J. D., Alcántara, R. Villà-Freixa, J. Multiscale molecular dynamics of protein
aggregation. https://pubmed.ncbi.nlm.nih.gov/21348836/

21/24
Conclusions
q QM/MM vision is needed to integrate the effect of the
protein/solvent environment in the enzyme reactivity
calculations
q despite the advances in using ab initio QM approaches in
molecular simulations, the sampling is still very limited
q EVB provides a way to obtain free energy profiles:
q with proper sampling
q allowing multiple replicas
q exploring the relevant reaction coordinate

22/24
Nayanika Das Josep Solà Martí Ferriz Jordi Villà-Freixa
-250
-200
-150
-100
-50
0
50
100
150
200
10
20
30
40
50
(ε1+ε2)/2
ε1
ε2
time / ps
Computational Biochemistry and Biophysics Lab

24/24
References
q Warshel, A. (1991). Computer Modeling of Chemical
Reactions in Enzymes and Solutions. John Wiley Sons.
q Kamerlin, S. C., Warshel, A. (2010). Q. Rev. Biophys., 43(4),
305-360.

Primer on Empirical Valence Bond Calculations

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