MASARYKOVA UNIVERZITA
PŘÍRODOVĚDECKÁ FAKULTA
NÁRODNÍ CENTRUM PRO VÝZKUM BIOMOLEKUL
Rigorózní práce
BRNO 2021 ONDŘEJ SCHINDLER
MASARYKOVA
U N I V E R Z I T A
PŘÍRODOVĚDECKÁ FAKULTA
NÁRODNÍ CENTRUM PRO VÝZKUM BIOMOLEKUL
Parciální atomové náboje
peptidů
Rigorózní práce
Ondřej Schindler
Brno 2021
Bibliografický záznam
Autor: Mgr. Ondřej Schindler
Přírodovědecká fakulta, Masarykova univerzita
Národní centrum pro výzkum biomolekul
Název práce: Parciální atomové náboje peptidů
Studijní program: Biochemie
Studijní obor: Biomolekulární chemie a bioinformatika
Akademický rok: 2021/2022
Počet stran: 58
Klíčová slova: parciální atomové náboje, peptidy, empirické metody, EEM,
QEq, EQEq, SQE, SQE+qO, SQE+qp, parametrizace, Optimized
Guided minimization
Bibliographie Entry
Author:
Title of Thesis:
Degree Programme:
Field of Study:
Academic Year:
Number of Pages:
Keywords:
Mgr. Ondfej Schindler
Faculty of Science, Masaryk University
National Centre for Biomolecular Research
Partial atomic charges of peptides
Biochemistry
Biomolecular chemistry and bioinformatics
2021/2022
58
partial atomic charges, peptides, empirical methods, EEM,
QEq, EQEq, SQE, SQE+qO, SQE+qp, parameterization,
Optimized Guided minimization
Abstrakt
Parciálních atomové náboje našly své uplatnění v mnoha odvětvích chemie. Jejich
koncept totiž vhodně aproximuje komplikovanou elektronovou hustotu na sadu reálných
čísel. Své uplatnění náboje našly zejména v počítačové chemii, chemoinformatice, bioinformatice
a nanovědách. Nejpřesnější parciální atomové náboje jsou vypočítány přímo na
základě elektronové hustoty, získané pomocí kvantové mechaniky. Naneštěstí je ale tento
proces extréme výpočetně náročný, zejména pak pro velké molekuly. Z tohoto důvodu
bylo za posledních 50 let vyvinuto mnoho empirických metod, které počítají parciální atomové
náboje za nesrovnatelně kratší čas. Tyto metody využívají pouze souřadnice atomů
a případně informace o vazbách mezi nimi. Empirické metody musí projít procesem
parametrizace, kdy jsou hledány takové parametry, s jejichž použitím se výsledné empirické
parciální náboje co nejvíce podobají těm z kvantové chemie. Problém parametrizace
zejména sofistikovanějších empirických metod na homogenních sadách molekul je ale
taktéž výpočetně náročný a stále ne zcela vyřešený. Z tohoto důvodu bylo jen několik
empirických metod parametrizováno pro peptidy. V této práci je nejdříve vyvinuta
rychlejší univerzální technika pro parametrizaci empirických metod Optimized Guided
Minimization. Následně je zavedena empirická metoda Split-Charge Equilibration with
parameterized initial charges, která v kombinaci s novým atomovým typem Bonded Atoms
reprodukuje kvantově mechanické náboje výrazně lépe než dosud publikované modely.
Abstract
Partial atomic charges have found their application in many fields of chemistry. Their
concept suitably approximates the complicated electron density to a set of real numbers. In
particular, they have found their application in computational chemistry, chemoinformatics,
bioinformatics and nanoscience. The most accurate atomic charges are obtained directly
from the electron density using the apparatus of quantum chemistry. Unfortunately, this
process is extremely computationally expensive, especially for large molecules. For this
reason, many empirical methods have been developed over the last 50 years that calculate
partial atomic charges in much shorter time. These methods use only the coordinates of
the atoms and occasionally information about the bonds between them. Empirical methods
have to go through a process of parameterization, where parameters are searched for
such that the resulting empirical partial charges are as close as possible to those from
quantum chemistry. However, the problem of parameterization, especially in the case of
more sophisticated empirical methods and homogeneous sets of molecules, is also computationally
challenging and still not fully solved. For this reason, only a few empirical
methods have been parameterized for peptides. In this paper, a faster general-purpose technique
developed for parameterizing empirical methods, Optimized Guided Minimization,
is first presented. Subsequently, the empirical method Split-Charge Equilibration with
parameterized initial charges is introduced. This method, in combination with the new
atomic type Bonded Atoms, reproduces quantum mechanical charges significantly better
than previously published models.
Poděkování
Chtěl bych poděkovat své ženě Lucii za její povzbuzující úsměv vždy, když už jsem se
vším chtěl dopravdy praštit. Dík patří také mým dětem Michaelovi a Elišce za to, že mi
neustále nejrůznějšími způsoby připomínají, že existují i jiné neméně důležité záležitosti
než jen věda.
Srdečně také děkuji mým rodičům, kteří mě v mém studiu a v celém mém životě podporují.
Zejména nyníjsem jim vděčný za všechen čas, který tráví se svými vnoučaty (a já tak mohl
psát tuto práci).
Rád bych vyjádřil vřelý dík mé skvělé vedoucí doc. RNDr. Radce Svobodové, Ph.D
za její lidský přístup a hlavně úžasné studijní podmínky, které se snaží pro své studenty
vytvářet. Bez nich by tato práce jen stěží mohla vzniknout. Poděkovat bych chtěl také
mému konzultantovi RNDr. Tomáši Rackovi za jeho ochotu věnovat mi desítky hodin
strávených nejen s Pythonem. Nikdo mi nedal do profesního života tolik, jako Ty.
Dík patří také virtuální organizaci MetaCentrum za poskytnutí výpočetních zdrojů pro
tuto práci.
Na závěr bych chtěl poděkovat Bohu za všechny jeho milosti a skvělé lidi, které mi
posílá do cesty.
Prohlášení
Prohlašuji, žejsem svoji rigorózní práci vypracoval samostatně s využitím informačních
zdrojů, které jsou v práci citovány.
Brno 11. Října 2021
Ondřej Schindler
Contents
Chapter 1. Introduction 12
I Theory 14
Chapter 2. Partial atomic charges 15
2.1 Methods for calculation of charges 16
Chapter 3. Quantum mechanical methods for charge calculation 17
3.1 Quantum mechanics (QM) 17
3.2 Quantum chemistry 18
3.2.1 Quantum chemistry methods 19
3.2.2 Basis sets 20
3.3 Charge distribution schemes 21
Chapter 4. Empirical methods 22
4.1 Atom types definition 23
4.2 Electronegativity equalization method 23
4.3 Charge equilibration 24
4.4 Extended charge equilibration 24
4.5 Split-charge equilibration 25
4.6 Split-charge equilibration with initial formal charges 25
Chapter 5. Parameterization 26
5.1 Types of optimization methods 26
5.2 Quality criteria 27
5.2.1 Root-mean-square deviation (RMSD) 27
5.2.2 Pearson correlation coefficient (R) 27
5.3 Validation 27
II Implementation 28
Chapter 6. Software MACH 29
6.1 Functionality and interface 29
6.1.1 Functionality 29
-8-
Contest 9
6.1.2 Input and output 29
6.1.3 Execution of the program 29
6.2 Main algorithms 30
6.2.1 Optimized guided minimization 30
6.2.2 Split-charge equilibration with initial parameterized charges 31
6.2.3 Bonded atoms 31
6.2.4 Objective function 31
6.3 Design and implementation 32
6.3.1 Programming language 32
6.3.2 Libraries 32
6.3.3 Main classes 33
Chapter 7. Atomic charge calculator II 34
III Methods 35
Chapter 8. Reference data 36
8.1 Sets of molecules 36
8.2 QM charges 37
8.3 Comparison of parameterization approaches GDMIN and optGM - setup . . 38
8.4 Comparison of empirical charge calculation methods - setup 38
IV Results and Discussion 39
Chapter 9. Comparison of parameterization approaches GDMIN and optGM 40
Chapter 10. Comparison of empirical charge calculation methods 42
10.1 Comparison results using atomic type HBO 42
10.2 Comparison results using atomic type BA 45
Chapter 11. Publication & Future plans 47
11.1 Main publication 47
11.1.1 My contribution 47
11.1.2 Related posters 47
11.2 Other publications 48
11.3 Ideas for future work 48
Chapter 12. Conclusion 49
Bibliography 50
Contents of the electronic attachment 58
List of Figures
2.1 Visualisation of the unequally distributed electron density 15
3.1 Scheme of quantum chemistry methods 20
10.1 Correlation plot with constant charges 44
10.2 Correlation plot for SQE+qO/HBO on the PUB_pept set 46
10.3 Correlation plot for SQE+qp/BA on the PUB_pept set 46
List of Tables
4.1 Comparison of CPU times of partial atomic charges calculation using
quantum chemistry and the empirical methods 22
8.1 Summary information about used sets of molecules 37
9.1 Comparison of achieved statistical metrics and CPU times for GDMIN and
optGM 41
10.1 Comparison of statistical metrics obtained for the selected empirical methods
with usage of HBO atomic types definition 43
10.2 Comparison of statistical metrics obtained for empirical methods with
usage of BA atomic types definition 45
-10-
Nomenclature
Atomic types definition
HBO Highest bond order
BA Bonded atoms
BA2 Bonded atoms to the second order
Empirical methods
EEM Electronegativity equalization method
QEq Charge equilibration
EQEq Extended charge equilibration
SQE Split-charge equilibration
SQE+qO Split-charge equilibration with initial formal charges
SQE+qp Split-charge equilibration with parameterized initial charges
Optimization methods
GDMIN Guided minimization
optGM Optimized guided minimization
SLSQP Sequential least squares programming method
Quantum mechanics
DFT Density functional theory
B3LYP Becke, 3-parameter, Lee-Yang-Parr functional
NPA Natural population analysis
Statistics terms
RMSD Root-mean-square deviation
R Pearson correlation coefficient
LHS Latin hypercube sampling
Chapter 1
Introduction
The partial atomic charges are real numbers approximating the complex and hard to interpret
electron density. Each atom in a molecule is assigned a partial charge that represents
how much electron density belongs to this atom.. Partial atomic charges are often used in
computational chemistry [1, 2, 3], chemoinformatics [4, 5, 6], bioinformatics [7, 8], and
nanoscience [9, 10]. It should be said that partial atomic charges are only a non-physical
theoretical concept and therefore cannot be observed or measured in any way. They can be
only calculated and there are two main approaches for it.
The most reliable way is direct derivation of partial atomic charges from the electron
density. First, the distribution of electrons in the orbitals (the so-called electron population)
is calculated and then the electron population is distributed among the atoms of
the molecule via a population analysis (e.g., MPA [11, 12], NPA [13, 14]). However,
obtaining the electron density of a molecule using quantum mechanics is a problem with
high computational complexity, and therefore, a long computational time is required.
For this reason, so-called empirical methods have been developed. They do not work
with the electron density but use only information about the atomic coordinates and, if
need be, the bonds between them [15, 16, 17, 18, 19, 20, 21]. These methods use common
physicochemical laws (e.g., Coulomb's law). Empirical methods can use constants derived
from quantum mechanics and tabular values (e.g., Charge equilibration [22] or Partial
Equalization of Orbital Electronegativity [23]). Or they must go through a parameterization
process during which parameters are sought such that the resulting partial atomic charges
correlate as closely as possible with those obtained using quantum chemistry [24, 25, 26].
The use of empirical methods for homogeneous sets of molecules (i.e., systems composed
from just several types of residues) is still challenging [21, 25]. The main reason is the
small range of values of partial atomic charges of some atomic types (e.g., single-bonded
O). Such atomic types tend to come out constant during parameterization which makes the
resulting parameters unusable.
-12-
Chapter 1. Introduction 13
In this work, problem of partial atomic charges for peptides is solved in four steps:
1. Parameterization technique optGM - a lot of parameterizations was conducted in
this work. A faster and more accurate technique was necessary, because original
optimization technique Guided minimization (GDMIN) [27], which was proved as
the best optimization method for parameterization of empirical Electronegativity
equalization method (EEM), was never tested with more complicated empirical
methods. [15].
2. Empirical method SQE+qp - improved version of Split-Charge Equilibration with
initial formal charges method (SQE+qO) [20]. This empirical method is suitable for
homogeneous systems because of usage of parameterized initial charges instead of
formal charges.
3. Bonded Atoms atomic types definition - atoms are classified for calculation according
to element and bonded atoms, (e.g., C/CCCH for a carbon connected to
three other carbon atoms and one hydrogen,).
4. Comparison of SQE+qp/BA with other empirical methods - in this step, SQElike
methods and frequently used empirical methods are parameterized by optGM
using three sets of molecules containing small organic molecules. The resulting
quality metrics are compared.
Parti
Theory
-14-
Chapter 2
Partial atomic charges
The partial atomic charges [28] are a set of real numbers that describe how much electron
density belongs to each atom in a molecule. The electron density is distributed unevenly in
the molecule due to the different properties of the atoms, especially their electronegativities,
and thus affects its physical properties. The great advantage of partial atomic charges
is their simplicity compared to the rather difficult to interpret electron density. Partial
atomic charges are a theoretical concept and therefore cannot be measured or observed.
We can only calculate them.
Nevertheless, partial atomic charges have found their use in many applications, including
prediction of the dissociation constant pKa [29, 30, 31], various charge descriptors
[32], QSAR and QSPR modeling [33, 34, 35], similarity searches [36, 37], pharmacophore
design [38], virtual screening [39], prediction of bonding sites and reactivity of molecule
[40]. A visual representation of the distribution of electron density and partial atomic
charges can be seen in Figure 2.1.
Figure 2.1: Visualisation of the unequally distributed electron density in the molecule of
ethanol by Avogadro [41].
-15-
Chapter 2. Partial atomic charges 16
2.1 Methods for calculation of charges
In general, there are two approaches to calculate partial atomic charges. The partial atomic
charges approximate the electron density and therefore the most reliable way is to obtain
them directly from the electron density. As we will see in the next chapter, obtaining
the electron density is a computationally challenging problem. Moreover, the quantum
mechanical (QM) charges depend on the chosen method, basis set and charge scheme.
For many purposes, however, QM charges can be approximated by empirical charges.
There are a large number of empirical methods for calculating partial atomic charges.
These empirical methods calculate the charges significantly faster because they work only
with empirical parameters, atomic coordinates and possibly the topology of the molecule
during their calculation. The parameters are most often obtained during parameterization.
However, due to parameterization, the parameters are only transferable to molecules that
were used as a reference set during parameterization.
Chapter 3
Quantum mechanical methods for
charge calculation
This chapter provides a brief introduction to quantum chemistry. In particular, emphasis
is placed on the procedures leading to the acquisition of partial atomic charges. First, the
differences between the macroworld and the microworld are described. Next, the basic
methods for calculating the electronic structure of a molecule and the influence of the
basis set on the result are presented. The end of the chapter is concerned with quantum
mechanical partial atomic charges.
3.1 Quantum mechanics (QM)
The laws of macrosocm differ significantly from the laws of the microworld described by
quantum mechanics. In the particle world we find phenomena that have no analogues in
Newtonian mechanics [42]. For example, depending on the external conditions, particles
sometimes behave like particles and sometimes like waves. We talk about the dual nature
of particles [43]. Another phenomenon, unknown to the world of macro-objects, is
the uncertainty principle, which states that the more precisely we determine one of the
conjugate properties of a system (e.g., position and momentum), the less precisely we
can determine its other conjugate property [44]. Unlike Newtonian mechanics, a particle
is described by a wave function that carries, among other things, information about the
probability of where the particle is. Below we can see the timeless Schrodinger equation,
which describes the particle in non-relativistic terms1
[45].
#*I> = £*I> (3.1)
where *F is the wave function describing the particle, E is the energy of the particle
and H is the Hamiltonian describing interactions in system.
'There are also other equations, e.g., the Dirac equation describing relativistically spin one-half particles.
Such equations will not be further dealt with in the present thesis.
-17-
Chapter 3. Quantum mechanical methodsfor charge calculation 18
The Hamiltonian can by written as
H = fn + fe + Vm + Vee + Vt
en (3.2)
where te and fn represent the kinetic energies of electrons and nuclei, Vnn and Vee
represent nucleus-nucleus and electron-electron repulsions and Ven means electron-nucleus
attraction. Because the wave function itself has no physical interpretation, we must apply
a suitable operator on it to determine the measurable physical quantities. There are three
main conditions that the wave function must satisfy. It must be single-valued, continuous
and quadratically integrable. The third condition is related to the probability of the location
of the particle in space, which is given as the square of the wave function.
The probability of a particle being in an infinitesimal volume of space is infinitesimal.
On the other hand, if we search for the particle in all conceivable space, it is quite certain
that we will find it. So the wave function must satisfy another condition.
3.2 Quantum chemistry
The electronic structure determines most of the chemical properties of atoms and molecules.
The theoretical study of electronic structure is performed by quantum chemistry using the
apparatus of quantum mechanics. Quantum chemistry methods search for the corresponding
form of the Schrodinger equation 3.1, which carries information about the electronic
structure of the system under study. The biggest problem is the fact that the Schrodinger
equation can only be solved analytically for very simple systems. Therefore, its solution by
numerical methods is necessary, which is very computationally demanding. The quantum
chemistry cooperates with experimental chemistry, when the results of quantum chemistry
are verified by experiment and the results of the experiment are explained by quantum
chemistry.
Two important approximations are used during the calculation. The Born-Oppenheimer
approximation uses the different masses of the proton and the electron. The proton, which
is approximately 1800x heavier than the electron, also moves approximately 1800x slower
[46]. In the case of heavier atoms, this difference in speed between the electron and the
nucleus is an order of magnitude greater. This allows us to separate the motions of the
nucleus and electron and describe them separately. As written above, we are unable to
analytically solve the Schrodinger equation for multi-electron systems. Therefore, using the
second approximation, the wave function of the electron is calculated in the temporal meanfield
of the other electrons [47]. The problem with this approach is that the correlation
energy between the electrons is neglected, which results in a higher calculated energy than
the system actually has. Various more sophisticated methods attempt to solve this problem
but at the cost of increased computational time.
f(x)dx= \V(x)\2
dx (3.3)
(3.4)
Chapter 3. Quantum mechanical methodsfor charge calculation 19
3.2.1 Quantum chemistry methods
There are three main groups of quantum chemical methods. Ab initio methods, semiempirical
methods and methods based on Density functional theory. The first group are
ab initio methods, because they use only physical constants in the calculation and thus
calculate everything from the beginning. These methods are generally more accurate than
semiempirical methods. The basic ab initio method is the Hartree-Fock self-consistent
field method [48]. This method, however, neglects the correlation energy of the electrons
and thus converges only to the so-called Hartree-Fock method which is always higher than
the actual energy of the system. More advanced methods try to overcome this problem,
including Configuration interaction [49], various Perturbation methods [50] and currently
the most accurate the method of Coupled clusters [51, 52]. However, calculations using
these methods are already significantly computationally demanding and completely unrealizable
for large molecular systems.
The second group of methods are semiempirical methods, which use parameters obtained
by experiment in addition to the laws of quantum mechanics. These methods are
less demanding on computing time. On the other hand, they are unsuitable for more accurate
calculations. Moreover, they are often limited to the group of molecules on which
the method has been parameterized. Semiempirical methods include, for example, the
extended Hiickel method [53] or the frequently used PM methods [54].
The third group of methods are DFT methods, which states that the electron energy is
the electron density functional [55, 56]. These methods do not try to find the shape of the
wave function as such, instead they try to calculate the electron density, which is a much
simpler problem. This is because from the wave function we can find out the probability
of a particular single electron being in a space. When calculating the electron density,
however, we ask with what probability any electron is in space. The problem with these
methods is that the exact form of the functional describing the energy is not known. Many
functionals have been developed [57], one of the most widely used being B3LYP [58].
The accuracy and computational complexity of DFT methods is approximately between
semiempirical and ab initio methods. In Figure 3.1, we can see the different complexities
of the individual methods and the relationships between them.
Chapter 3. Quantum mechanical methodsfor charge calculation 20
Semi-Empirical
MO-Theory
Perturbation Theory (MBPT)
(e.g. MP2: n \ MP3: n6
, MP4: n7
)
Hartree-Fock (HF), SCF, MO-LCAO
,
P = |(|J,(1)(|)2(2) <p,._,(«-l)<p»|
r,4
Density
Functional
Theory
<n3
-n4
)
Configuration Interaction (CI)
*)=(c| ».)+Ecr| *,;)+! czl'vzh )
(e.g. SDCI: nft
)
Coupled Cluster Methods (CC)
I*)-***. (e.g.CCSD: n 6
,
(f = T, + t + ) CCSD(T):n7
)
X
"Dressed CI Methods"
CPF, ACPF, MCPF, CEFA
Multiconfiguration SCF
(MCSCF, CASSCF, GVB)
Multireference
Perturbation Theory
(CASPT2, MR-MP2, MR-MPn?)
Multireference
Configuration Interaction
(e.g. MR-SDCI, MR-ACPF)
Multireference
Coupled Cluster Theory
Figure 3.1: Scheme of quantum chemistry methods with their computational complexities
depending on the size of the system (n corresponds to the number of electrons in a system).
[59].
3.2.2 Basis sets
An important part of quantum chemical calculations is the so-called basis set, which is a
set of basis functions or also atomic orbitals from which molecular orbitals are formed by
linear combination [60, 61].
n
where <I> represents the molecular orbital, c represents the coefficient and x is the basis
function, n is the number of basis function and i is the number of the molecular orbital.
Analytically calculated atomic orbitals of hydrogen would appear to be the best basis
functions. However, this approach is not suitable for the calculation itself and therefore
Slater-type orbitals (STO) were first developed, which, although they preserved the shape
of the hydrogen atomic orbitals, were still not mathematically ideal. Nowadays, the most
used are the Gaussian-type orbitals [62]. However, since the Gaussian function alone does
not describe the atomic orbitals of hydrogen very well, each basis function is treated as a
combination of several Gaussian functions. In general, the larger the basis set, the better
the result we can achieve. The problem lies in the increasing computational complexity,
and so a trade-off between computational time and achieved accuracy is always chosen for
quantum chemical calculations.
Chapter 3. Quantum mechanical methodsfor charge calculation 21
3.3 Charge distribution schemes
Because the charges are just a theoretical concept and we cannot determine their exact
values by an experiment, many methods for their calculation have been developed. Among
the best known methods is Mulliken population analysis (MPA) [11, 12, 63]. However,
the partial atomic charges according to this approach are strongly influenced by the basis
set used. Moreover, the charges do not converge well with increasing basis set. Therefore,
a better approach is Natural population analysis (NPA) [14, 64], which performs orbital
localization and orthonormalization before assigning electron densities to individual atoms.
Other approaches attempt to map partial atomic charges to a measurable quantity, which can
be compared with an experimental value (e.g., dipole moment). These approaches include
the Merz-Kollman charges (ESP) [65] and its improved version RESP [66]. Mention may
be made of the CM5 charges, the essence of which is to compare the electron density of
the atom itself and that of the atom in the molecule [67]. Atoms in Molecules (AiM)
approaches look for local minima in the electron density, which they take to be boundaries
between atoms, and assign electron densities to individual atoms accordingly [68]. The
numerical value of the partial atomic charges depends on the chosen quantum chemical
method, the used basis set and selected charge distribution scheme. The type of charge is
often described in the form method/basis/charge distribution scheme. An example of this
would be B3LYP/6-311G/NPA.
Chapter 4
Empirical methods
While obtaining partial atomic charges by quantum mechanics is the most accurate way,
since they are derived directly from the electron density, this way is computationally very
demanding. For this reason, many empirical methods have been developed [15,19,20, 22,
69]. Their main advantage is that they are able to calculate partial atomic charges more
quickly. These methods work with empirical parameters and physicochemical tabular
values instead of electron density. The biggest disadvantage of empirical methods is the
necessity of their parameterization in case we want to reproduce quantum mechanical
charges1
. Parameterization is a computationally demanding optimization problem where
we try to find a set of parameters using which the deviation between the empirical and
quantum chemical charges is as small as possible. This implies another problem of
empirical methods. Because of the parameterization, they are limited to the set of molecules
for which they were parameterized. Empirical methods calculate partial atomic charges
from information about the position of the atoms and possibly the bonds between them.
As can be seen in Table 4.1 especially for large structures the difference in computational
time is extreme.
Table 4.1: Comparison of CPU times of partial atomic charges calculation using quantum
chemistry and the empirical methods E E M and SQE. CPU AMD EPYC Processor (with
IBPB) 2.40 GHz used for calculations.
Structure Atoms QM E E M SQE
Cystine 26 285 s 0.002 s 0.003 s
6ATW 538 18 days 0.009 s 0.024 s
lOPS 963 48 days 0.031s 0.098 s
In general, empirical methods can be divided into conformationally independent methods,
which work only with information about the bonds between atoms, and conformationally
dependent methods, which work with the coordinates of the atoms and possibly the
bonds between them. In this paper, I only consider conformationally dependent methods,
which are more accurate, especially if they also use information about the bonds between
atoms [15, 29].
1
There are also empirical methods that do not try to reproduce quantum mechanical charges and thus
create their own another type of partial atomic charges
-22-
Chapter 4. Empirical methods 23
4.1 Atom types definition
Before calculating the partial atomic charges by any empirical method, it is necessary to
divide the atoms into classes according to atomic types. The most trivial and straightforward
division is by element (H, C, N, O, ...). As it turns out, such a division may be too
coarse and therefore several other definitions of atomic types have been published. The
definition of Highest bond order (HBO), where the atomic type is defined as the element
and the highest bond order (HI, CI, C2, C3,...) [24, 70], has proved to be useful. Other
definitions, for example, add information about formal charges or membership to a specific
functional group [71, 72, 73]. In general, the finer the definition of atomic types used,
the better results we can get. The price is more demanding parameterization and higher
risk of overfitting. For simplicity, the combination of the empirical method and the atomic
type definition used is denoted in the form empirical method/atomic type definition (for
example, QEq/HBO).
4.2 Electronegativity equalization method
The oldest empirical conformationally dependent method for calculating partial atomic
charges is Electronegativity equalization method (EEM), which is derived from DFT
EEM uses two empirical parameters for each atomic type, representing electronegativity
and hardness. In addition, it works with one global parameter identical for all atomic types
[15]. The solution of the E E M consists of a system of n linear equations with n unknowns,
where n is the number of atoms in the molecule:
m
K
''1.2
K
1,2
K K
r\,n r2,n
V i i
K
-1
rin
1
- l
\xJ
(~Xi\
-Xi
~Xn
V Q J
(4.1)
where T];- denotes the hardness of atom i, K is the global parameter, T];- denotes the electronegativity
of atom i, rij denotes the distance between atoms i and j, % denotes the
total electronegativity of the molecule, Q represents the total charge of the molecule, and
q\, #2, • • • ) Qn are the unknown partial atomic charges.
Chapter 4. Empirical methods 24
4.3 Charge equilibration
There are many different versions of Charge equilibration (QEq), which differ in the way
they introduce the approximation of the Coulomb integral into the model, which appears
in the off-diagonal elements of the matrix [16]. Apart from this difference, QEq is quite
similar to EEM. In this work, we use a version of QEq [17], as described in the following
equation system:
I m Ji,2
Ji,2 m
J\,n Jl,n
Jl,n - 1 \
h,n " I
r]n -1
f-xA
-X2
Xn
V I 1 ••• 1 0 / \xj \ Q J
where Coulomb integral is aproximed by
1 3 ^ / 3
(4.2)
(4.3)
4.4 Extended charge equilibration
Extended charge equilibration (EQEq) uses parameters for electron affinity (EA) and
ionization potential (IP) instead for electronegativity and hardness. In addition, the method
uses two global parameters [18]. The method is expressed as a system of linear equations:
Jl,2 J°2
J\,n J2,n
\ 1 1
J\,n 1\
J° -1
1 0 / \x)
(~Xi\
-X2
~Xn
\ Q )
(4.4)
where the electronegativity of an atom is expressed as the arithmetic mean of the
ionization potential and the electron affinity
Xi —
the self-Coulomb integral is defined as
IPi+EA,
(4.5)
jy = ipt-EAi
and Coulomb integral is approximated by
XK I 1
JiJ
j2 d 2
K
J
?JRi
>J
K2
R
(4.6)
(4.7)
where X and K are global parameters and / y is the geometric mean of jf and f-.
Chapter 4. Empirical methods 25
4.5 Split-charge equilibration
The Split-charge equilibration (SQE) brings information about the bonds between atoms
into the model. This makes the model significantly more complex. In addition to the
parameters for electronegativity and hardness, SQE uses a parameters for the atomic radius
and a parameters for bond hardness. The term split-charges means the charge transferred
along the bond between two atoms [19]. The SQE method written in the form of a system
of linear equations is expressed by the equation:
(THTT
+ dmg(K))qsp = Tx
where T represents the incidence matrix describing the molecular topology, H is the
hardness matrix that describes the interactions between the atoms, diag(jc) is the diagonal
matrix with bond hardnesses, qsp is the vector with split-charges and % is the vector of
atomic electronegativities. Coulomb integral in hardness matrix is expressed as
erf
'hi —
ILL
JiJ = ^ — (4-8)
v
hJ
where represents the width of atom i. After this calculation, the partial atomic
charges are obtained from the split-charges as
q = TT
qsp (4.9)
4.6 Split-charge equilibration with initial formal charges
Unlike previous methods, SQE cannot work with the total charge of a molecule. Therefore,
this method is not suitable for molecules with charged functional groups. The Split-charge
equilibration with initial formal charges (SQE+qO) method adds information about the
formal charges of individual atoms (and thus the charge of the whole molecule) to the
model [20]. The formalism of the method can then be expressed as follows
(THTT
+ diag(jc)) qsp = T(x-HqQ + 7}* q0) (4.10)
where qo is the vector of initial formal charges, r\ is the vector of atomic hardnesses,
and * is the element-wise product. The calculation of atomic charges from split charges is
then modified to:
q = TT
qsp + q0 (4.11)
Chapter 5
Parameterization
Parameterization [21, 24,26] is an optimization problem in which we try to find parameters
for a method that give the best possible result. The so-called Objective Function tells
us how good the result is [21, 74]. It compares the calculated values with reference
values in a statistical way. The shape of the objective function is therefore critical for the
parametrization. Parameterization is a computationally intensive problem because we have
to search for extremes in N+l dimensional space, where N is the number of parameters.
5.1 Types of optimization methods
There are many optimization methods. These methods can be divided into global and local
optimization methods. Local optimization methods iteratively move down from a given
starting point along a hypersurface of parameters until they find an extremum. Therefore,
the local optimization methods are fast. Their disadvantage is their ability to find only
one local minimum that is closest to the specified starting point along the gradient. In
contrast, global optimizing methods try to find a global extremum in the entire defined
space. Therefore, global optimization methods are much slower. In practice, global and
local optimization methods are often used together. For example when local optimization
is used to refine the results of the global optimization.
Optimization methods can be also divided according to whether they also use first
and possibly second derivatives. In fact, methods that use derivatives are generally faster.
In practice, however, we often do not know the derivative of the Objective Function. In
this case, we can compute the derivatives numerically, but this significantly increases the
computational requirements of the optimization.
There are several other criteria according to which optimization methods can be classified.
Some methods can work with conditions and constraints on parameters. Another
specific type are stochastic methods that work with randomness during optimization. Methods
that try to mimic the laws of biology, chemistry or physics are also a large group.
-26-
Chapter 5. Parameterization 27
5.2 Quality criteria
During the parameterization, the reference charges qR
are compared with the empirical
charges qE
at each step by an objective function. Both simple statistical metrics (e.g.,
average deviation, maximum deviation) and a more advanced approaches can be used for
the comparison.
5.2.1 Root-mean-square deviation (RMSD)
In the case of empirical methods, Root-mean-square deviation (RMSD) [75] is often used,
defined as
RMSD(q
R
,qE
) = J-fd\qf-qf\^ (5.1)
V n
(=i
where qf and qf represent i-th reference and empirical charge. RMSD can take values
from 0 to infinity. The smaller the value, the more similar the sets are. RMSD is sensitive
to outliers due to squaring of the individual differences. The value of RMSD depends on
the units chosen. [75]
5.2.2 Pearson correlation coefficient (R)
Another frequently used statistical metric is the Pearson correlation coefficient [76]. This
coefficient can take values from -1 to 1, where 1 indicates perfect agreement between the
test sets, 0 no correlation and -1 indirect dependence. A value above 0.8 is considered a
strong correlation. R does not depend on the chosen units. Like RMSD, R is sensitive to
outliers. In practice, the square of R2
is often used. R is defined as
s.,(rf_-7)(«r-?) _ ( 5 .2 )
V E L i ( « f - « R
) 2
V S ' - i ( « ? - « E
) 2
where qR
and qE
represent average of reference and empirical charges.
5.3 Validation
Validation is a step after parameterization. Validation verifies the transferability of parameters.
In practice, the set of molecules is usually randomly shuffled and split into two
parts (e.g., 80% and 20%). These parts are called training set and test set. The parameterization
itself is then performed only on the training set. After parameterization, the
partial atomic charges for the test set are calculated using the parameters found. These
calculated charges are then compared with the reference QM charges. If the statistics for
the test set and training set are similar, we can consider the method and its parameters to
be transferable. If the statistics for the training and test set of molecules are significantly
different, the parameters are not transferable. Low transferability is usually due to either
so-called overfitting or a small training set of molecules. Overfitting can occur when there
are many parameters in the model. Indeed, with a large number of parameters, it may
happen that in addition to the phenomenon under study, we start to describe random noise
in the data during parameterization.
Part II
Implementation
-28-
Chapter 6
Software MACH
All parameterizations was conducted via our highly optimized internal software tool M A C H
available freely at GitHub under the MIT licence [77]. This chapter introduces this software
tool.
6.1 Functionality and interface
6.1.1 Functionality
The main function of our M A C H software is a parameterization of empirical methods.
The parameterization can be performed with the atomic types plain, HBO and BA. In
addition to the parameterization mode, it can also calculate the empirical charges by any of
the implemented methods, compare two sets of charges, and output summary information
about a set of molecules.
6.1.2 Input and output
MACH reads molecules in SDF format [78]. Parameters are defined in JSON file. Parameter
files are in the electronic attachment. Input and output charges are represented in
the same way as in [79]. M A C H is controlled via command line. During the calculation,
MACH creates a directory where it stores all results, logs and an HTML file with correlation
plots, statistics and other information about the calculation. All input files and source
code of M A C H are also copied to the directory to make the results fully reproducible.
6.1.3 Execution of the program
The M A C H software runs via the terminal. Command line parameters are also used to set
up M A C H and set the path to input files. For example, parameterization of SQE+qp by
optGM is started with the command:
./mach.py —mode parameterization —chg_method SQEqp
—optimization_method optGM —sdf_file example.sdf
—ats_types_pattern ba —ref_chgs_file example.chg
—params_file parameters.json
-29-
Chapter 6. Software MACH 30
6.2 Main algorithms
MACH was already introduced in my master thesis. For this work it has been extended
with several algorithms and approaches.
6.2.1 Optimized guided minimization
Parameterization via M A C H is performed using our developed algorithm Optimized guided
minimization (optGM) [21]. Algorithm of optGM is based on Guided Minimization
(GDMIN) [27]. Its development was necessary because GDMIN has only been tested on
the simple EEM/HBO method, which has significantly fewer parameters and converges
more easily than, for example, SQE+qp/BA. The main difference between GDMIN and
optGM are:
• optGM only uses a suitable subset (i.e., a subset of molecules containing at least N
atoms of each atomic type present in the original training set) of molecules in several
steps of the parameterization process. Evaluation of the objective function in these
steps is therefore significantly faster.
• The number of initial samples can be substantially higher since they are only evaluated
on a subset. A large number of initial samples is necessary to sufficiently cover
the parameter space in methods with multiple atom and bond parameters.
• The number of local optimizations, which are the most time-demanding part of the
parameterization, is limited to just the best candidate samples.
• Local optimization, unlike GDMIN, is not limited by the number of steps and so can
iterate until it converges.
The parameterization technique optGM is algorithm for a global optimization which
consists of 9 steps1
:
1. Sampling space of parameters by N samples generated by Latin Hypercube Sampling
(LHS) algorithm [80,81].
2. Evaluation of objective function for samples with the usage of a subset^1
).
3. Selection of best M samples.
4. Evaluation of objective function for M selected samples with the usage of the whole
set of molecules.
5. Selection of / best candidates from M samples.
6. Local minimization of / candidates with the usage of a subset(S2
). Local minimization
metod SLSQP [74] is used for this purpose.
'In this work we use the settings N=10,000, Sl
=\, M=300,7=3, 52
=5
Chapter 6. Software MACH 31
7. Evaluation of objective function for minimized I candidates with the usage of the
whole set of molecules.
8. Selection of best candidate from I candidates.
9. Local minimization of the best candidate.
6.2.2 Split-charge equilibration with initial parameterized charges
In addition to all methods described in Chapter 4, the empirical method Split-charge equilibration
with parameterized initial charges (SQE+qp) developed by us is also implemented
in MACH. In principle, this method is similar to SQE+qO. However, SQE+qp uses a
parameterized charges instead of a formal charges. Thus, the initial estimate of the partial
atomic charge is not limited to integer values. The numerical values of initial charges are
obtained in the process of parameterization [21]. However, since there is no guarantee
that the sum of the parameterized initial charges will be equal to the total charge of the
molecule, it is necessary to normalize them before the calculation.
where Q is the total molecular charge, n is the number of atoms in the molecule and qp
is vector with initial parameterized charges.
6.2.3 Bonded atoms
The atomic type Bonded atoms (BA) [21] is also implemented in software MACH. This
atomic type is defined as the element and the elements bonded to it (H/C, O/CH, C/CCHH,
...). Using such a definition of atomic types leads to a large number of parameters and is
therefore only applicable to homogeneous sets (e.g., peptides and proteins).
6.2.4 Objective function
In MACH, the objective function is implemented as in [24] and my diploma thesis. It is
the sum of the average molecular RMSD and the average RMSD for atomic types.
where Nm represents the number of molecules, Nm is the number of atom types, Mi is
the i-th molecule and AT) is the j-th atom type. The average RMSD for atomic types is
included in the objective function because there can be an order of magnitude difference
in the number of atoms of each atomic type in sets of molecules.
F0 =
Y!J=\ RMSDMI L%RMSDATJ
Nm Nat
(6.1)
Chapter 6. Software MACH 32
6.3 Design and implementation
6.3.1 Programming language
MACH is developed in the interpreted object-oriented programming language Python
version 3.8.5. This programming language was chosen because development in it is very
fast. Another advantage is the huge number of available libraries. The computations in
Python itself are relatively slow. However, using appropriate libraries (e.g. Numpy, Scipy,
Numba) will make calculations up to an order of magnitude faster.
6.3.2 Libraries
MACH utilize several well-known Python libraries in the versions provided by the package
manager Anaconda 4.10.3.:
• NumPy [82] is basic library for computing with multidimensional arrays. These
functions are written in compiled languages C/C++ or Fortran, which increases their
performance compared to pure Python. It might be also compiled to use Intel Math
Kernel Library (MKL) [83], a library of optimized math routines for science and
engineering.
• Numba [84] is one of the Python compilers, whose goal is to accelerate the operation
of Python programs. It belongs among the just-in-time compilers, which means the
code is compiled once the program starts running. It uses L L V M compiler [85].
• SciPy [74] is a standard library for scientific purposes. It is built on NumPy, which
is why its basic data structure is a n-dimensional array. It provides many advanced
functions for optimization, integration, interpolation, signal and image processing
and so forth.
• NLopt [86] is a library for nonlinear optimization. It implements numerous global
and local optimization methods.
• Bokeh [87] is a library for visualization. Unlike the better known library Matplotlib,
it supports image interactivity, which is possible thanks to HTML and JavaScript.
Display and interactivity is enabled through a web browser.
Chapter 6. Software MACH 33
6.3.3 Main classes
MACH is implemented using object-oriented programming techniques. Thus, M A C H
consists of classes that describe important objects for parameterization and charge calculation.
During implementation, emphasis was placed on easy extensibility. New atomic
type definitions, empirical methods and optimization methods can be easily added. The
main classes are:
Molecule It carries information about the molecule. In particular, about its atoms and the
bonds between them. It also stores several molecular descriptors.
SetOfMolecules loads and stores molecules.
ChargeMethod is a super class containing functions common to all the empirical methods,
e.g. loading parameters. The individual classes of empirical methods inherit from
this class.
Comparison compares reference and empirical sets of charges, calculate statistics and
writes the results to HTML file.
Parameterization is responsible for the whole process of parameterization.
Objective function compares the degree of correlation between empirical and reference
charges during parameterization.
Chapter 7
Atomic charge calculator II
Atomic charge calculator II (ACCII) [88] is an interactive web application for the calculation
of partial atomic charges via non-QM empirical charge calculation approaches and for
the visualization of these charges. ACC II is composed of a frontend and a backend. The
frontend is a modern web application written in JavaScript using the Bootstrap library. Its
first function is to read the user input that consists of molecular structure(s) and computation
settings (e.g. one of the charge calculation methods that are integrated into the backend).
Its second purpose is to present the output, i.e. calculated charges. These charges are
available as downloadable data files, and can also be visualized via the LiteMol viewer,
which is part of the ACC II frontend. The backend is a Python Flask application. A l l
the computations of charges are carried out by the core C++ application, which integrates
20 non-QM empirical charge calculation approaches along with parameters from literature.
To make our results available to the scientific community, we added empirical methods
SQE, SQE+qO, and SQE+qp to ACC II, including all the parameter sets used in the thesis.
Therefore, all these methods are now accessible for the broader research community for
quick and precise empirical atomistic charge calculation. ACCII is available at
//acc2.ncbr.muni.cz.
-34-
Part III
Methods
-35-
Chapter 8
Reference data
This chapter presents the sets of molecules used to compare the empirical methods. The
reference Q M partial atomic charges of these sets are described here. The chapter also
describes the parameterization setup.
8.1 Sets of molecules
In this work, we utilized three datasets of molecules, described in Table 8.1. The first
two datasets are composed of organic molecules and were also used for the comparison
and parameterization of empirical charge calculation methods in previous publications
[89, 90]. DTP_small is a simple set (a low number of small-sized molecules with low
variability) while CCD.gen is more complex. DTP_small contains organic molecules used
as drugs; CCD_gen includes organic molecules acting as protein ligands. The last dataset,
PUB.pept, was created directly for this publication. It contains small peptides obtained
from the PubChem database [91]. It represents a dataset of molecules with homogeneous
atomic types. All 3D conformers are downloaded as SDF files from the PubChem database,
where they were generated by OEOmega. All structures are additionally manually prepared
in Chimera as zwitterions (charges are assigned at pH 7) by removing hydrogen from the Cterminal
-carboxyl group and adding hydrogens to the N-terminal -amino group. Peptides
were chosen to contain all possible proteinogenic amino acids and to cover the range of
integer charges, from -2, over neutral to +2. Each dataset was divided into two subsets:
a training set and a test set containing 80% and 20% of the molecules, respectively. The
division was done randomly, and the stratification was included during the separation. The
list of molecules that comprised the training and test and the datasets in SDF format are
provided in the eletronic attachment.
-36-
Chapter 8. Reference data 37
Table 8.1: Summary information about used sets of molecules.
Denotation DTPsmall CCDgen PUBpept
Database DTP NCI wwPDB CCD PubChem
Number of
molecules
1,956 4,443 60
Number of atoms 62,977 204,760 2,636
Atomic types
(HBO)
H1,C1,C2,N1,
N2, 01,02, SI
H1,C1,C2, C3,
N1,N2, N3, 01,
02,F1,P2, S1,S2,
C l l , B r l
H1,C1,C2,N1,
N2, O l , 02, SI
Size of molecules 6-176 3-305 20-70
The main type of
molecules
Organic molecules
(drug-like)
Organic molecules
(ligands)
Di- and
tripeptides
Source of 3D
structures
Generated by CORINA Generated by
OEOmega
Publication [90, 92] [90] [21]
QM charges B3LYP/6-311G/NPA B3LYP/
6-31G*/NPA
8.2 QM charges
The Q M charge calculation approach B3LYP/6-311G/NPA was selected for calculating
the QM reference charges (i.e., charges used for the parameterization and evaluation of all
the compared empirical methods) on datasets DTP_small and CCD.gen. These charges
were used because the combination of the B3LYP theory level, the 6-311G basis set, and
NPA proved to be very suitable for parameterizing empirical charge calculation methods
[4, 5, 25, 89]. For the dataset PUB_pept, the QM charge calculation approach B3LYP/6-
31G*/NPA was selected. The method and the population analysis are the same as for the
first two datasets, but the basis set 6-31G* was used. The reason for this is that 6-311G is
too complex and not applicable for proteins, which we will focus on in the future. The basis
set 6-31G* represents a robust enough and feasible replacement, and was also often used
to parameterize empirical charge calculation methods [93, 94, 95]. The QM charges for
all the datasets were calculated with Gaussian 09 [96]. The files with QM partial atomic
charges for molecules from all the datasets are available in the electronic attachment.
Chapter 8. Reference data 38
8.3 Comparison of parameterization approaches GDMIN
and optGM - setup
The comparison was carried out as follows. Both parameterization schemes were used on
all three training sets of molecules. For each calculation, a random seed 1 was used for
reproducibility1
. HBO atomic types definition was used for all parameterizations. The
optGM parameters were set to Sl
=\, M=300, 1=3, S2
=5 (see Section 6.2.1). The most
demanding settings from the publication [27] were used as parameters for GDMIN. We
used different number of initial seeds in each run, starting from 1,000 and doubled that
number in the next run. The procedure ended if more than 8.106
initial samples were
reached, or the individual parameterization took more than 50 hours to finish. From these
results, the best parameters according to the common quality criteria were selected. Then,
we compared the time that each scheme used. Afterward, the parameters computed for
each dataset were used to calculate empirical charges for this dataset (i.e., using its training
subset and also using its test subset).
8.4 Comparison of empirical charge calculation methods
- setup
For comparison of empirical charge calculation methods, a parameterization of all the
methods (i.e., EEM, QEq, EQEq, SQE, SQE+qO, SQE+qp) was performed on all three
sets of molecules. HBO atomic types definition was used for all the datasets. The optGM
parameters were set to N=10,000,Sl
=l, A/=300,1=3, Sz
=5 (see Section 6.2.1). Each combination
was run five times with seeds from 1 to 5. From these results, the best parameters
according to the common quality criteria were selected. Finally, the parameters computed
for each dataset were used to calculate empirical charges for this dataset.
A comparison of the empirical methods using atomic types BA was performed on the
PUB_pept set only. The setting used was the same as for the comparison using HBO atomic
types.
The pseudo-random number generator always generates the same random numbers with the same seed.
Part IV
Results and Discussion
-39-
Chapter 9
Comparison of parameterization
approaches GDMIN and optGM
In this chapter, the optGM parameterization technique with its original version is compared.
For this purpose, a parameterization of the SQE method was performed via GDMIN
and optGM of all three datasets (with HBO atomic types). This parameterization was only
done for SQE, because other empirical methods have a low number of parameters; thus
their parameterization is considerably less time demanding, making GDMIN sufficient for
them. The HBO atomic type was chosen because it is frequently used and only creates a
small number of atomic classes. Thus the calculation of parameters is markedly less time
demanding than for BA atomic types, and can even be done by GDMIN in a reasonable
time (a few days).
The values of statistics metrics and CPU times are summarized in Table 9.1. More
detailed data are in the electronic attachment. Results show that the parameters obtained
by optGM provide charges, which correlate with QM comparably or slightly better than the
charges calculated using the parameters obtained by GDMIN. The metrics for the test set
show the same trend and therefore the parametrization technique can be considered robust
and its results transferable. Moreover, optGM provides results significantly faster than
GDMIN. Therefore, optGM proved to be a more appropriate parameterization approach
and was used for the subsequent experiments presented in this work.
-40-
Chapter 9. Comparison of parameterization approaches GDMIN and optGM 41
Table 9.1: Comparison of achieved statistical metrics and CPU times for GDMIN and
optGM. Empirical method SQE with HBO atomic types definition was used. The better
results are highlighted in blue.
Parameterization DTPsmall
approach R 2
R M S D R M S D A T Param. time [h:m:s]
G D M I N ř r a ř n 0.9941 0.0256 0.0599 19:47:22
optGMf r a ( M 0.9945 0.0249 0.0399 0:40:07
ODMENA 0.9943 0.0255 0.0501
optGMí e í í 0.9949 0.0244 0.0401
Parameterization CCD_gen
approach R 2
R M S D R M S D A T Param. time [h:m:s]
G D M I N ř r a i n 0.9934 0.0334 0.0884 40:03:11
optGMi r a i M 0.9950 0.0292 0.0406 10:27:05
G D M I N t o i 0.9933 0.0331 0.1087
optGMí e í í 0.9949 0.0290 0.0542
Parameterization PUBpep
approach R 2
R M S D R M S D A T Param. time [h:m:s]
G D M I N ř r a ř n 0.9875 0.0519 0.0756 18:37:16
optGMi r a i M 0.9875 0.0518 0.0746 0:04:58
ODMENA 0.9862 0.0552 0.0882
optGMí e í í 0.9860 0.0556 0.0873
Chapter 10
Comparison of empirical charge
calculation methods
As the second step of our study, SQE and SQE+qO, and the newly developed SQE+qp
method was compared with the common approaches (i.e., EEM, QEq, and EQEq). We
used for this comparison the set PUB_pept. Moreover, also sets of organic molecules (i.e.,
DTP_small and CCD_ gen) were chosen for testing, because we plan to use SQE+qp also
for proteins, which often contain various organic ligands.
10.1 Comparison results using atomic type HBO
The statistical values are summarized in Table 10.1. Correlation charts and results for each
seed can be found in the electronic attachment. Although the statistical results may look
good, parameterizations of some combinations of empirical methods and sets of molecules
results in constant partial atomic charges of some atomic types. This usually happens when
some atomic types are significantly underrepresented in the set, or have a too small range
of values of reference charges. Such a case can be seen in Figure 10.1. The results in
which there were constant charges are in Table 10.1 marked by red star.
As we can now see in Table 10.1, almost all methods give good results for the sets of
molecules DTP_small and CCD.gen. However, the partial atomic charges from SQE+qp
correlate best. Moreover, the other methods tend to produce constant charges for the
homogeneous set of PUB_pept and are therefore unsuitable for such sets.
-42-
Chapter 10. Comparison of empirical charge calculation methods 43
Table 10.1: Comparison of statistical metrics obtained for the selected empirical methods.
HBO atomic types were used. Statistical metrics were obtained for the test set. The best
results are highlighted in blue. The results with constant charges are marked by red star.
Method
DTPsmall
Method
R 2
R M S D R M S D A T
E E M 0.9725 0.0569 0.0987
QEq 0.9729 0.0564 0.0987
EQEq 0.9824 0.0451 0.1025
SQE 0.9954 0.0231 0.0399
SQE+qO* 0.9926 0.0293 0.0707
SQE+qp 0.9960 0.0215 0.0412
Method
CCDgen
Method
R 2
R M S D R M S D A T
E E M 0.9791 0.0586 0.1734
QEq 0.9794 0.0582 0.1738
EQEq 0.9836 0.0518 0.1672
SQE 0.9952 0.0279 0.0481
SQE+qO* 0.9926 0.0349 0.0726
SQE+qp 0.9961 0.0253 0.0544
Method
PUBpept
Method
R 2
R M S D R M S D A T
EEM* 0.9795 0.0674 0.0929
QEq* 0.9796 0.0671 0.0990
EQEq 0.9826 0.0620 0.1386
SQE* 0.9862 0.0553 0.0911
SQE+qO* 0.9946 0.0344 0.0587
SQE+qp 0.9973 0.0243 0.0498
Chapter 10. Comparison of empirical charge calculation methods 44
QM charges
Figure 10.1: Correlation plot for the EEM/HBO on set of peptides PUB_pept. The partial
charges of oxygen, represented by the red dots, are constant.
Chapter 10. Comparison of empirical charge calculation methods 45
10.2 Comparison results using atomic type BA
In the previous section, we verified that the SQE+qp method gives high-quality charges
for different types of molecules. However, the SQE+qp method is different from other
methods compared. The other methods have a problem when the charge range is small and
tend to assign a constant charge to such types. SQE+qp, because it works with an initial
estimate of the partial charge, gives the better results the smaller the charge ranges of each
atomic type are. Thus, the ideal atomic type definition for SQE+qp is BA. Unfortunately,
BA can only be used for homogeneous sets of molecules such as PUB_pept. In the case
of heterogeneous sets, using BA would lead to a huge number of atomic types and poor
transferability of the resulting parameters. Thus, the comparison of empirical methods
was repeated on the PUB_pept set using BA. The methodology used was the same as in
the previous chapter. The summarized results can be found in Table 10.2. Detailed results
are in the electronic attachment.
Table 10.2: Comparison of statistical metrics obtained for empirical methods on PUB_pept
with usage of BA atomic types definition. Statistical metrics were obtained for the test set.
The best results are highlighted in blue. The results with constant charges are marked by
red star.
Method R 2
R M S D RMSDAT
EEM* 0.9964 0.0285 0.0641
QEq* 0.9947 0.0342 0.0747
EQEq* 0.9963 0.0286 0.0462
SQE 0.9945 0.0349 0.0608
SQE+qO* 0.9976 0.0228 0.2149
SQE+qp 0.9991 0.0141 0.0342
In all statistics, SQE+qp is again the best. The results show that the definition of BA
atomic types is not suitable for the other methods. If we compare the quality of the partial
atomic charges for the set of PUB_pept molecules before this work (see Section 10.1,
which discusses SQE+qO/HBO) and after this work (see Section 10.2, which discusses
SQE+qp/BA) we find that the RMSD is almost 2.5 times lower. However, if we compare
the RMSD with the EQEq/HBO method, which unlike SQE+qO/HBO does not have constant
charges for some types, the R M S D is reduced more than four times. In both cases,
the R M S D a t value also improved significantly. In addition, we have solved the problem
of constant charges for peptides. This trend is also confirmed in Figures 10.2 and 10.3.
Although the results of the work are optimistic, the problem is still not completely
solved. As we can see in Figure 10.3, the oxygens still do not come out completely
optimally. Research is currently underway to test how much the inclusion of hydrogen
bonds in the model would help to improve the results. It would also be useful to extend
the model to include other charge schemes (e.g., MPA, HIR, CM5).
Chapter 10. Comparison of empirical charge calculation methods 46
too-
X,
-
£
-1
-1
SQE+qO
J *
0
QM charges
Figure 10.2: Correlation plot for SQE+qO/BA on the PUB.pept set.
en
01
OC
u
ri
•iH
>H
s
m
-1
SQE+qp
• c
• H
• N
• O
S
••mr
•
/*
7»
r
1 0
QM charges
Figure 10.3: Correlation plot for SQE+qp/BA on the PUB_pept set.
Chapter 11
Publication & Future plans
11.1 Main publication
The optGM parameterization technique and the empirical SQE+qp method were published
as a paper in the Journal of Cheminformatics (impact factor 5.5). In this article, I share
first authorship with my PhD thesis advisor RNDr. Tomáš Racek.
• Ondřej Schindler, Tomáš Racek, Aleksandra Maršavelski, Jaroslav Koča, Karel
Berka, and Radka Svobodová. "Optimized SQE atomic charges for peptides accessible
via a web application". In: Journal of Cheminformatics 13(1), 1-11 (2021)
11.1.1 My contribution
My contribution to this work is the development of the global optimization technique
optGM, the empirical method SQE+qp and atomic types BA. Furthermore, I performed
all parameterizations and evaluation of the results. I also participated in the revision of the
manuscript.
11.1.2 Related posters
The results of this thesis were presented as a poster at the ELIXIR CZ Annual Conference
in 2019 and 2020.
• Ondřej Schindler, Tomáš Racek, Radka Svobodová, and Jaroslav Koča. "Partial
atomic charges for proteins". In: ELIXIR CZ Annual Conference (2019)
• Ondřej Schindler, Tomáš Racek, Radka Svobodová, Karel Berka, and Jaroslav Koča.
"SQE charge calculation and its applicability for proteins". In: ELIXIR CZ Annual
Conference (2020)
-47-
Chapter 11. Publication & Future plans 48
11.2 Other publications
In addition, I am co-author of two other publications and two posters. All papers are related
to partial atomic charges.
• Tomáš Racek, Ondřej Schindler, Dominik Toušek, Vladimír Horský, Karel Berka,
Jaroslav Koča, and Radka Svobodová. "Atomic Charge Calculator II: web-based
tool for the calculation of partial atomic charges". In: Nucleic Acids Research 48.W1
(2020)
• Tomáš Racek and Ondřej Schindler. "Computer-aided model design of empirical
methods for calculating partial atomic charges". In: Proceedings of the 2021
European Simulation and Modelling Conference (2021). [accepted]
• Tomáš Racek, Ondřej Schindler, Radka Svobodová, and Jaroslav Koča. "Empirical
methods for calculation of partial atomic charges - applicability for proteins?" In:
ENBIK (2018)
• Tomáš Racek, Ondřej Schindler, Vladimír Horský, Jaroslav Koča, Dominik Toušek,
and Radka Svobodová. "AtomicChargeCalculator II". in: ELIXIR CZ Annual Conference
(2019)
11.3 Ideas for future work
I am currently finalizing a methodology for calculating the partial atomic charges of
proteins. Once completed, I need to extend the model to include metalloproteins as well as
ligands. This will require both the creation of new reference sets and modification of the
calculation methodology (for example, the definition of BA atomic types is not applicable
to organic ligands). The final goal of my work is to calculate conformationally dependent
partial atomic charges for all proteins in the Protein Data Bank and make them available
to the scientific community.
Chapter 12
Conclusion
Partial atomic charges find many applications in computational chemistry, chemoinformatics
and bioinformatics. Currently, fast and sophisticated methods for charge calculation
were developed (e.g., EEM, QEq, EQeq). However, even these advanced methods have
limitations (e.g., they require empirical parameters). Moreover, their application for peptides,
proteins, and other macromolecules is problematic. An empirical charge calculation
method that is promising for peptides and other macromolecular systems is the SQE and its
extension SQE+qO. Unfortunately, only one parameter set is available for these methods,
and their implementation is not easily accessible.
In our work, we first developed and tested the optGM parameterization scheme. This
scheme produces parameters comparable to the GDMIN method, but in a significantly
shorter time. Therefore, optGM is also applicable for large datasets and charge calculation
approaches with more parameters (i.e., SQE, SQE+qO, and SQE+qp). An implementation
of optGM is available on GitHub.
Then, we developed the SQE+qp empirical charge calculation method and compared
this method with the empirical methods EEM, QEq, EQeq, SQE, and SQE+qO. We
found that for heterogeneous datasets with drug-like organic molecules, SQE-like methods
performed comparably and improved upon the traditional electronegativity equalization
approaches. For a homogeneous dataset with peptides, SQE+qp provided the best results
and outperformed all other approaches, including SQE+qO. We also introduced a new
atom classification type, BA, tailored to peptides and likely other homogeneous datasets.
The combination of SQE+qp with BA atomic types proved to be an excellent solution for
peptides.
The main contribution of our work is that it makes SQE, SEQ+qO and its extension
SEQ+qp together with their parameter sets accessible to the users via ACC II web
application (https: //acc2 .ncbr .muni . cz/) and also via a command-line application.
Therefore, all these methods are now available for the broad research community for quick
and precise empirical atomistic charge calculation.
This work is summarized in the article:
• Ondřej Schindler, Tomáš Racek, Aleksandra Maršavelski, Jaroslav Koča, Karel
Berka, and Radka Svobodová. "Optimized SQE atomic charges for peptides accessible
via a web application". In: Journal of Cheminformatics 13(1), 1-11 (2021)
-49-
Bibliography
[1] Mikko J Vainio and Mark S Johnson. "Generating Conformer Ensembles Using
a Multiobjective Genetic Algorithm". In: Journal of Chemical Information and
Modeling 47.6 (2007), pp. 2462-2474.
[2] Heloisa S Muniz and Alessandro S Nascimento. "Towards a critical evaluation of
an empirical and volume-based solvation function for ligand docking". In: PloS
one 12.3 (2017), e0174336.
[3] Efstratios Kritikos and Andrea Giusti. "Reactive Molecular Dynamics Investigation
of Toluene Oxidation under Electrostatic Fields: Effect of the Modeling of Local
Charge Distribution". In: The Journal of Physical Chemistry A 124.51 (2020).
[4] Radka Svobodová Vařeková, Stanislav Geidl, Crina-Maria Ionescu, Ondřej Skřehota,
Michal Kudera, David Sehnal, Tomáš Bouchal, Ruben Abagyan, Heinrich J
Huber, and Jaroslav Koča. "Predicting pKa Values of Substituted Phenols from
Atomic Charges: Comparison of Different Quantum Mechanical Methods and
Charge Distribution Schemes". In: Journal of Chemical Information and Modeling
51.S (2011), pp. 1795-1806.
[5] Stanislav Geidl, Radka Svobodová Vařeková, Veronika Bendová, Lukáš Petrusek,
Crina-Maria Ionescu, Zdeněk Jurka, Ruben Abagyan, and Jaroslav Koca. "How
Does the Methodology of 3D Structure Preparation Influence the Quality of pKa
Prediction?" In: Journal of Chemical Information and Modeling 55.6 (2015),
pp.1088-1097.
[6] Sivakumar Prasanth Kumar, Prakash C Jha, Yogesh T Jasrai, and Himanshu A
Pandya. "The effect of various atomic partial charge schemes to elucidate consensus
activity-correlating molecular regions: a test case of diverse QSAR models". In:
Journal of Biomolecular Structure and Dynamics 34.3 (2015), pp. 540-559.
[7] John D Holliday, Stephen P Jelfs, Peter Willett, and Peter Gedeck. "Calculation of
Intersubstituent Similarity Using R-Group Descriptors". In: Journal of Chemical
Information and Computer Sciences 43.2 (2003), pp. 406^-11.
[8] Ann E Cleves, Stephen R Johnson, and Ajay N Jain. "Electrostatic-field and
surface-shape similarity for virtual screening and pose prediction". In: Journal
of Computer-Aided Molecular Design 33.10 (2019), pp. 865-886.
[9] Chi-Hung Chuang, Mintu Porel, Rajib Choudhury, Clemens Burda, and Vaidhyanathan
Ramamurthy. "Ultrafast Electron Transfer across a Nanocapsular Wall:
Coumarins as Donors, Viologen as Acceptor, and Octa Acid Capsule as the Mediator".
In: The Journal of Physical Chemistry B 122.1 (2018), pp. 328-337.
-50-
Bibliography 51
[10] Dan Luo, Feng Wang, Jianfa Chen, Fanghao Zhang, Luo Yu, Dezhi Wang, Richard
C Willson, Zhaozhong Yang, and Zhifeng Ren. "Poly(sodium 4-styrenesulfonate)
Stabilized Janus Nanosheets in Brine with Retained Amphiphilicity". In: Langmuir
34.12 (2018), pp. 3694-3700.
[11] Robert S Mulliken. "Electronic Population Analysis on LCAO-MO Molecular
Wave Functions. I". In: The Journal of Chemical Physics 23.10 (1955), pp. 1833-
1840.
[12] Robert S Mulliken. "Electronic Population Analysis on LCAO-MO Molecular
Wave Functions. II. Overlap Populations, Bond Orders, and Covalent Bond Energies".
In: The Journal of Chemical Physics 23.10 (1955), pp. 1841-1846.
[13] Alan E Reed and Frank Weinhold. "Natural bond orbital analysis of near-HartreeFock
water dimer". In: The Journal of Chemical Physics 78.6 (1983), pp. 4066-
4073.
[14] Alan E Reed, Robert B Weinstock, and Frank Weinhold. "Natural population
analysis". In: The Journal of Chemical Physics 83.2 (1985), pp. 735-746.
[15] Wilfried J Mortier, Swapan K Ghosh, and S Shankar. "Electronegativity-equalization
method for the calculation of atomic charges in molecules". In: Journal of
the American Chemical Society 108.15 (1986), pp. 4315-4320.
[16] Akifumi Oda and Shuichi Hirono. "Geometry-dependent atomic charge calculations
using charge equilibration method with empirical two-center Coulombic
terms". In: Journal of Molecular Structure: THEOCHEM 634.1-3 (2003), pp. 159-
170.
[17] Jaap N Louwen and Eelco TC Vogt. "Semi-empirical atomic charges for use in
computational chemistry of molecular sieves". In: Journal of Molecular Catalysis
A: Chemical 134.1-3 (1998), pp. 63-77.
[18] Christopher E Wilmer, Ki Chul Kim, and Randall Q Snurr. "An extended charge
equilibration method". In: The journal of physical chemistry letters 3.17 (2012),
pp. 2506-2511.
[19] Razvan A Nistor, Jeliazko G Polihronov, Martin H Miiser, and Nicholas J Mosey.
"A generalization of the charge equilibration method for nonmetallic materials".
In: The Journal of Chemical Physics 125.9 (2006), p. 094108.
[20] Toon Verstraelen, Ewald Pauwels, Frank De Proft, Veronique Van Speybroeck,
Paul Geerlings, and Michel Waroquier. "Assessment of Atomic Charge Models for
Gas-Phase Computations on Polypeptides". In: Journal of Chemical Theory and
Computation 8.2 (2012), pp. 661-676.
[21] Ondřej Schindler, Tomáš Racek, Aleksandra Maršavelski, Jaroslav Koča, Karel
Berka, and Radka Svobodová. "Optimized SQE atomic charges for peptides accessible
via a web application", eng. In: Journal of Cheminformatics 13 (2021). ISSN:
1758-2946.
[22] Anthony K Rappe and William A Goddard III. "Charge equilibration for molecular
dynamics simulations". In: The Journal of Physical Chemistry 95.8 (1991),
pp.3358-3363.
Bibliography 52
[23] Johann Gasteiger and Mario Marsili. "Iterative partial equalization of orbital electronegativity—a
rapid access to atomic charges". In: Tetrahedron 36.22 (1980),
pp.3219-3228.
[24] Tomáš Racek, Jana Pazúriková, Radka Svobodová Vařeková, Stanislav Geidl, Aleš
Křenek, Francesco Luca Falginella, Vladimír Horsk, Václav Hejret, and Jaroslav
Koča. "NEEMP: software for validation, accurate calculation and fast parameterization
of E E M charges". In: Journal of cheminformatics 8.1 (2016), p. 57.
[25] Crina-Maria Ionescu, Stanislav Geidl, Radka Svobodová Vařeková, and Jaroslav
Koča. "Rapid Calculation of Accurate Atomic Charges for Proteins via the Electronegativity
Equalization Method". In: Journal of Chemical Information and
Modeling 53.10 (2013), pp. 2548-2558.
[26] Radka Svobodová Vařeková, Zuzanna Jirousková, Jakub Vaněk, Šimon Suchomel,
and Jaroslav Koča. "Electronegativity equalization method: Parameterization and
validation for large sets of organic, organohalogene and organometal molecule".
In: International journal of molecular sciences 8.7 (2007), pp. 572-582.
[27] Jana Pazúriková, Aleš Křenek, and Luděk Matýska. "Guided Optimization Method
for Fast and Accurate Atomic Charges Computation", eng. In: Proceedings of
the 2016 European Simulation and Modelling Conference. Ed. by José EvoraGómez
and José Juan Hernandez-Cabrera. Ghent, Belgium: EUROSIS - ETI,
2016, pp. 267-274. isbn: 978-90-77381-95-3.
[28] Vijay P Gupta. Principles and applications of quantum chemistry. Academic Press,
2015.
[29] Radka Svobodová Vařeková, Stanislav Geidl, Crina-Maria Ionescu, Ondřej Skřehota,
Tomáš Bouchal, David Sehnal, Ruben Abagyan, and Jaroslav Koča. "Predicting
pKa values from EEM atomic charges". In: Journal of cheminformatics 5.1
(2013), p. 18.
[30] Jinhua Zhang, Thomas Kleinoder, and Johann Gasteiger. "Prediction of p K a values
for aliphatic carboxylic acids and alcohols with empirical atomic charge descriptors".
In: Journal of chemical information and modeling 46.6 (2006), pp. 2256-
2266.
[31] Adam C Lee and Gordon M Crippen. "Predicting pKa". In: Journal of chemical
information and modeling 49.9 (2009), pp. 2013-2033.
[32] Roberto Todeschini and Viviana Consonni. Handbook of molecular descriptors.
Vol. 11. John Wiley & Sons, 2008.
[33] Radka Svobodová Vařeková, Stanislav Geidl, Crina-Maria Ionescu, Ondřej Skrehota,
Michal Kudera, David Sehnal, Tomáš Bouchal, Ruben Abagyan, Heinrich J
Huber, and Jaroslav Koca. "Predicting pKa values of substituted phenols from
atomic charges: comparison of different quantum mechanical methods and charge
distribution schemes". In: Journal of chemical information and modeling 51.8
(2011), pp. 1795-1806.
Bibliography 53
[34] Taravat Ghafourian and John C Dearden. "The Use of Atomic Charges and Orbital
Energies as Hydrogen-bonding-donor Parameters for QSAR Studies: Comparison
of MNDO, AMI and PM3 Methods". In: Journal of pharmacy and pharmacology
52.6 (2000), pp. 603-610.
[35] Kevin C Gross, Paul G Seybold, and Christopher M Hadad. "Comparison of
different atomic charge schemes for predicting pKa variations in substituted anilines
and phenols". In: Internationaljournal of quantum chemistry 90.1 (2002), pp. 445-
458.
[36] Dietmar Stalke. "Meaningful structural descriptors from charge density". In: Chemistry-A
European Journal 17'.34 (2011), pp. 9264-9278.
[37] Jorge Galvez, R Garcia, MT Salabert, and Rosa Soler. "Charge indexes. New topological
descriptors". In: Journal of Chemical Information and Computer Sciences
34.3 (1994), pp. 520-525.
[38] Mati Karelson, Victor S Lobanov, and Alan R Katritzky. "Quantum-chemical
descriptors in QSAR/QSPR studies". In: Chemical reviews 96.3 (1996), pp. 1027-
1044.
[39] Paul D Lyne. "Structure-based virtual screening: an overview". In: Drug discovery
today 7.20 (2002), pp. 1047-1055.
[40] Arkadiusz Z Dudek, Tomasz Arodz, and Jorge Gálvez. "Computational methods
in developing quantitative structure-activity relationships (QSAR): a review". In:
Combinatorial chemistry & high throughput screening 9.3 (2006), pp. 213-228.
[41] Marcus D Hanwell, Donald E Curtis, David C Lonie, Tim Vandermeersch, Eva
Zurek, and Geoffrey R Hutchison. "Avogadro: an advanced semantic chemical
editor, visualization, and analysis platform". In: Journal of cheminformatics 4.1
(2012), p. 17.
[42] Walter Greiner. Quantum mechanics: an introduction. Springer Science & Business
Media, 2011.
[43] Peter Atkins. Paula J. Atkins'physical chemistry. 2006.
[44] Jan Celý. "Základy kvantové mechaniky pro chemiky, I". In: Principy. UJEP, Brno
(1981).
[45] Rudolf Polák and Rudolf Zahradník. Obecná chemie: stručný úvod. Academianakladatelství
Akademie věd ČR, 2000.
[46] Max Born and Robert J Openheimer. "Zur Quant en theorie der Molekeln". In:
Ann. Phys 84 (1927), p. 57.
[47] Warren J Hehre. "Ab initio molecular orbital theory". In: Accounts of Chemical
Research 9.11 (1976), pp. 399-406.
[48] Clemens CJ Roothaan. "New developments in molecular orbital theory". In: Reviews
of modern physics 23.2 (1951), p. 69.
[49] John A Pople, Rolf Seeger, and Raghavachari Krishnan. "Variational configuration
interaction methods and comparison with perturbation theory". In: International
Journal of Quantum Chemistry 12.S11 (1977), pp. 149-163.
Bibliography 54
[50] Christian M0ller and Milton S Plesset. "Note on an approximation treatment for
many-electron systems". In: Physical review A6.1 (1934), p. 618.
[51] Rodney J Bartlett and George D Purvis. "Many-body perturbation theory, coupledpair
many-electron theory, and the importance of quadruple excitations for the
correlation problem". In: International Journal of Quantum Chemistry 14.5 (1978),
pp. 561-581.
[52] Daniel Hollas, Vít Svoboda, Ondřej Svoboda, and Petr Slavíček. Kvantová chemie:
První čtení. 2014.
[53] Roald Hoffmann. "An extended Hiickel theory. I. hydrocarbons". In: The Journal
of Chemical Physics 39.6 (1963), pp. 1397-1412.
[54] James JP Stewart. "Optimization of parameters for semiempirical methods II.
Applications". In: Journal of computational chemistry 10.2 (1989), pp. 221-264.
[55] Andrew R Leach and AR Leach. Molecular modelling: principles and applications.
Pearson education, 2001.
[56] Wolfram Koch and Max C Holthausen. A chemist's guide to density functional
theory. John Wiley & Sons, 2015.
[57] Roberto Car. "Density functional theory: Fixing Jacob's ladder". In: Nature chemistry
8.9 (2016), p. 820.
[58] Philip J Stephens, Frank J Devlin, Cary F Chabalowski, and Michael J Frisch.
"Ab initio calculation of vibrational absorption and circular dichroism spectra
using density functional force fields". In: The Journal of physical chemistry 98.45
(1994), pp. 11623-11627.
[59] Hierarchy of ab-initio Post-HF methods. Copied with a kind permission ofprof.
Martin Kaupp. TU Berlin.
[60] Ernest R Davidson and David Feller. "Basis set selection for molecular calculations".
In: Chemical Reviews 86.4 (1986), pp. 681-696.
[61] Roman M Balabin. Communications: Intramolecular basis set superposition error
as a measure of basis set incompleteness: Can one reach the basis set limit without
extrapolation? 2010.
[62] Warren J Hehre, Robert Ditchfield, and John A Pople. "Self—consistent molecular
orbital methods. XII. Further extensions of Gaussian—type basis sets for use in
molecular orbital studies of organic molecules". In: The Journal of Chemical
Physics 56.5 (1972), pp. 2257-2261.
[63] Robert S Mulliken. "Criteria for the Construction of Good Self-Consistent-Field
Molecular Orbital Wave Functions, and the Significance of LCAO-MO Population
Analysis". In: The Journal of Chemical Physics 36.12 (1962), pp. 3428-3439.
[64] Alan E Reed and Frank Weinhold. "Natural bond orbital analysis of near-HartreeFock
water dimer". In: The Journal of chemical physics 78.6 (1983), pp. 4066-
4073.
Bibliography 55
[65] Chandra U Singh and Peter A Kollman. "An approach to computing electrostatic
charges for molecules". In: Journal of Computational Chemistry 5.2 (1984),
pp.129-145.
[66] Christopher I Bayly, Piotr Cieplak, Wendy Cornell, and Peter A Kollman. " A
Well-Behaved Electrostatic Potential Based Method Using Charge Restraints for
Deriving Atomic Charges: The RESP Model". In: The Journal of Physical Chemistry
97.40 (1993), pp. 10269-10280.
[67] Aleksandr V Marenich, Steven V Jerome, Christopher J Cramer, and Donald G
Truhlár. "Charge model 5: An extension of Hirshfeld population analysis for the
accurate description of molecular interactions in gaseous and condensed phases".
In: Journal of chemical theory and computation 8.2 (2012), pp. 527-541.
[68] Richard FW Bader. "Atoms in molecules". In: Accounts of Chemical Research
18.1 (1985), pp. 9-15.
[69] Christopher E Wilmer, Ki Chul Kim, and Randall Q Snurr. "An Extended Charge
Equilibration Method". In: The Journal of Physical Chemistry Letters 3.17 (2012),
pp. 2506-2511.
[70] Crina-Maria Ionescu, Radka Svobodová Vařeková, Jochen H M Prehn, Heinrich
J Huber, and Jaroslav Koča. "Charge profile analysis reveals that activation of
pro-apoptotic regulators Bax and Bak relies on charge transfer mediated allosteric
regulation". In: PLoS computational biology 8.6 (2012), el002565.
[71 ] Kwang-Hwi Cho, Young Kee Kang, Kyoung Tai No, and Harold A Scheraga. "A fast
method for calculating geometry-dependent net atomic charges for polypeptides".
In: The Journal of Physical Chemistry B 105.17 (2001), pp. 3624-3634.
[72] Michael K Gilson, Hillary SR Gilson, and Michael J Potter. "Fast assignment
of accurate partial atomic charges: an electronegativity equalization method that
accounts for alternate resonance forms". In: Journal of chemical information and
computer sciences 43.6 (2003), pp. 1982-1997.
[73] Olexander Yakovenko, Alexander A Oliferenko, Volodymyr G Bdzhola, Vladimir
A Palyulin, and Nikolai S Zefirov. "Kirchhoff atomic charges fitted to multipole
moments: implementation for a virtual screening system". In: Journal of computational
chemistry 29.8 (2008), pp. 1332-1343.
[74] Eric Jones, Travis Oliphant, and Pearu Peterson. "SciPy: Open source scientific
tools for Python". In: (2014).
[75] Vladimir N Maiorov and Gordon M Crippen. "Significance of root-mean-square
deviation in comparing three-dimensional structures of globular proteins". In:
Journal of molecular biology 235.2 (1994), pp. 625-634.
[76] Richard C Graham. Data analysis for the chemical sciences. VCH, 1993.
[77] Ondřej Schindler. dargen3/MACH. URL: https b. com/dargen3/MACH.
[78] CTFile Formats, [cit. 2019-03-15]. URL: http : / /c
ctfile.pdf.
Bibliography 56
[79] Jakub Vaněk. "Parametrizace equalizačních metod [online]". Diplomová práce.
Masarykova univerzita, Fakulta informatiky, Brno, 2009 [cit. 2019-04-03]. URL:
https://is.muni.cz/th/weyqa/.
[80] Jon C Helton and Freddie Joe Davis. "Latin hypercube sampling and the propagation
of uncertainty in analyses of complex systems". In: Reliability Engineering &
System Safety 81.1 (2003), pp. 23-69.
[81] Michael D McKay, Richard J Beckman, and William J Conover. "Comparison of
three methods for selecting values of input variables in the analysis of output from
a computer code". In: Technometrics 21.2 (1979), pp. 239-245.
[82] Stefan Van Der Walt, S Chris Colbert, and Gael Varoquaux. "The NumPy array:
a structure for efficient numerical computation". In: Computing in Science &
Engineering 13.2 (2011), p. 22.
[83] Endong Wang, Qing Zhang, Bo Shen, Guangyong Zhang, Xiaowei Lu, Qing Wu,
and Yajuan Wang. "Intel math kernel library". In: High-Performance Computing
on the Intel® Xeon Phi™. Springer, 2014, pp. 167-188.
[84] Siu Kwan Lam, Antoine Pitrou, and Stanley Seibert. "Numba: A llvm-based python
jit compiler". In: Proceedings of the Second Workshop on the LĽVM Compiler
Infrastructure in HPC. ACM. 2015, p. 7.
[85] Chris Lattner and Vikram Adve. "LLVM: A compilation framework for lifelong
program analysis & transformation". In: Proceedings of the international symposium
on Code generation and optimization: feedback-directed and runtime optimization.
IEEE Computer Society. 2004, p. 75.
[86] Steven G Johnson. "The NLopt nonlinear-optimization package (Version 2.3)". In:
(2012). URL: http://ab-initio.°/„20mit. °/„20edu/nlopt.
[87] Bokeh Development Team. Bokeh: Python library for interactive visualization.
2014. URL: http: //www.bokeh. pydata. org.
[88] Tomáš Racek, Ondřej Schindler, Dominik Toušek, Vladimír Horský, Karel Berka,
Jaroslav Koča, and Radka Svobodová. "Atomic Charge Calculator II: web-based
tool for the calculation of partial atomic charges". In: Nucleic Acids Research
48.W1 (2020).
[89] Stanislav Geidl, Tomáš Bouchal, Tomáš Racek, Radka Svobodová Vařeková, Václav
Hejret, Aleš Křenek, Ruben Abagyan, and Jaroslav Koča. "High-quality and
universal empirical atomic charges for chemoinformatics applications". In: Journal
of Cheminformatics 7.59 (2015).
[90] Tomáš Raček, Jana Pazúriková, Radka Svobodová Vařeková, Stanislav Geidl, Aleš
Křenek, Francesco Luca Falginella, Vladimír Horský, Václav Hejret, and Jaroslav
Koča. "NEEMP: software for validation, accurate calculation and fast parameterization
of E E M charges". In: Journal of Cheminformatics 8.57 (2016).
[91] Sunghwan Kim, Jie Chen, Tiejun Cheng, Asta Gindulyte, Jia He, Siqian He,
Qingliang Li, Benjamin A Shoemaker, Paul A Thiessen, Bo Yu, L Zaslavsky,
J Zhang, and E E Bolton. "PubChem 2019 update: improved access to chemical
data". In: Nucleic Acids Research 47.D1 (2019), pp. D l 102-D2109.
Bibliography 57
[92] Radka Svobodová Vařeková, Zuzana Jiroušková, Jakub Vaněk, Šimon Suchomel,
and Jaroslav Koča. "Electronegativity Equalization Method: Parameterization and
Validation for Large Sets of Organic, Organohalogene and Organometal Molecule".
In: International Journal of Molecular Sciences 8.7 (2007), pp. 572-582.
[93] Yongzhong Ouyang, Fei Ye, and Yizeng Liang. " A modified electronegativity
equalization method for fast and accurate calculation of atomic charges in large
biological molecules". In: Physical Chemistry Chemical Physics 11.29 (2009),
pp.6082-6089.
[94] Patrick Bultinck, Wilfried Langenaeker, Philippe Lahorte, Frank De Proft, Paul
Geerlings, Christian Van Alsenoy, and JP Tollenaere. "The electronegativity equalization
method II: applicability of different atomic charge schemes". In: The Journal
of Physical Chemistry A 106.34 (2002), pp. 7895-7901.
[95] Patrick Bultinck, Ruben Vanholme, PLA Popelier, Frank De Proft, and Paul
Geerlings. "High-speed calculation of AIM charges through the electronegativity
equalization method". In: The Journal of Physical Chemistry A 108.46 (2004),
pp.10359-10366.
[96] Mike J Frisch et al. Gaussian 16 Revision B.01. 2016.
[97] Ondřej Schindler, Tomáš Racek, Aleksandra Maršavelski, Jaroslav Koča, Karel
Berka, and Radka Svobodová. "Optimized SQE atomic charges for peptides accessible
via a web application". In: Journal of Cheminformatics 13(1), 1-11 (2021).
[98] Ondřej Schindler, Tomáš Racek, Radka Svobodová, and Jaroslav Koča. "Partial
atomic charges for proteins". In: ELIXIR CZ Annual Conference (2019).
[99] Ondřej Schindler, Tomáš Racek, Radka Svobodová, Karel Berka, and Jaroslav
Koča. "SQE charge calculation and its applicability for proteins". In: ELIXIR CZ
Annual Conference (2020).
[100] Tomáš Racek and Ondřej Schindler. "Computer-aided model design of empirical
methods for calculating partial atomic charges". In: Proceedings of the 2021
European Simulation and Modelling Conference (2021). [accepted].
[101] Tomáš Racek, Ondřej Schindler, Radka Svobodová, and Jaroslav Koča. "Empirical
methods for calculation of partial atomic charges - applicability for proteins?" In:
ENBIK (2018).
[ 102] Tomáš Racek, Ondřej Schindler, Vladimír Horský, Jaroslav Koča, Dominik Toušek,
and Radka Svobodová. "AtomicChargeCalculator II". In: ELIXIR CZ Annual Conference
(2019).
Contents of the electronic attachment
Additional file 1 Detailed description of parameterization process. Description of optGM
parameterization scheme and preparation of PUB.pept dataset.
Additional file 2 Division of datasets into training and test sets. Lists of IDs of molecules
comprising training and test sets for each dataset.
Additional file 3 Datasets. Molecules for all datasets in SDF format.
Additional file 4 QM charges. Files with QM partial atomic charges for molecules from
all datasets.
Additional file 5 Details of GDMIN and optGM comparison. Description of procedure,
values of quality metrics and correlation graphs.
Additional file 6 Details of empirical charge method comparison. Description of procedure,
values of quality metrics and correlation graphs.
Additional file 7 Parameter sets. All parameter sets obtained during the comparison of
empirical methods.
Additional file 8 Poster from the ELIXIR CZ Annual Conference 2019
Additional file 9 Poster from the ELIXIR CZ Annual Conference 2020
-58-