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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5435_Библиотеки_им_академика_М_И_Перельмана
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sure of the potential of a system to do work at a constant temperature and pressure.
This can be calculated using the equation:
ΔG = ΔH − TΔS (10:1)
where ΔH is the change in enthalpy, T is the temperature in kelvin, and ΔS is the
change in entropy.
10.2.2 Enthalpy
The enthalpy and entropy changes affect the free energy of the system (ΔG). This
change in free energy determines whether the process is favorable or not. If ΔG <0,
then the process is energetically favorable and spontaneous, meaning the two mole-
cules will bind. However, if ΔG > 0, then the process is unfavorable and nonspontane-
ous, meaning the two molecules will not bind [23].
Enthalpy (ΔH) refers to the amount of heat that is absorbed or released during a
reaction. This can be calculated using the equation:
Δ H =
X
H productsðÞ−
X
H reactantsðÞ (10:2)
where
X
H (products) is the sum of the enthalpies of the products and
X
H (reac-
tants) is the sum of the enthalpies of the reactants.
10.2.3 Entropy
The entropy change in a binding process involves any restrictions on the movement
of molecules that occur when two moleculesbindtooneanother.Whentwomole-
cules are separated, they have more translational and rotational degrees of free-
dom compared to when they are bound, which results in an increase in entropy. In
thecaseofbinding,thismovementisrestricted, resulting in a reduction in entropy
[22, 24].
Entropy (ΔS) refers to the degree of disorder in a system. It can be calculated
using the equation:
Δ S = S productsðÞ−
X
S reactantsðÞ (10:3)
where
X
S (products)isthesumoftheentropiesoftheproductsand
X
S (reac-
tants) is the sum of the entropies of the reactants.
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10.2.4 Boltzmann factor
The Boltzmann factor (e–ΔG/RT) is a statistical factor that relates the energy of a sys-
tem to its probability of being in a particular state. The term R in this equation refers
to gas constant (8.314 J/mol
✶
K) and T refers to the temperature in kelvin [25].
10.2.5 Equilibrium constant
The equilibrium constant (K
eq
) is a measure of the balance between the concentra-
tionsofareactantanditsproductatequilibrium.Itcanbecalculatedusingthe
equation:
K
eq
=
Products½
n
Reactants½
m
(10:4)
In the given context, [Products] and [Reactants] represent the concentrations of the
products and reactants, respectively, while n and m denote the stoichiometric coeffi-
cients of the products and reactants, respectively [26, 27].
10.2.6 Binding free energy
The binding free energy (ΔG
bind
) can be calculated using the equation:
ΔG
bind
= − RT ln K
D
ðÞ (10:5)
where R is the gas constant, T is the temperature in kelvin, and K
D
is the dissociation
constant. The dissociation constant i s related to the equilibrium constant by the
equation
K
D
=
1
K
eq
(10:6)
The dissociation constant can also be expressed in terms of the concentrations of the
bound and unbound molecules by the equation:
K
D
=
Unbound½
Bound½
(10:7)
where [Unbound] refers to the concentration of the unbound molecule and [Bound]
refers to the concentration of the bound molecule [28, 29].
The binding free energy quantifies the intensity of the interaction between two
molecules, such as a protein and a ligand, when they bind together. This interaction
can exhibit attraction or repulsion and arises from a combination of electrostatic
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forces, van der Waals interactions, and weak chemical bonds. As a thermodynamic
concept, the binding free energy holds significant significance in drug design, as it
governs the effectiveness of a drug molecule. There are several thermodynamic fac-
tors that contribute to the binding free energy, which can be calculated using various
mathematical equations [28]. One of the most commonly used equations is the Gibbs
free energy equation:
Δ G binding = Δ H binding − TΔ S binding (10:8)
where ΔG binding is the change in Gibbs free energy upon binding, ΔH binding is the
change in enthalpy (heat content) upon binding, T is the temperature in kelvin, and
ΔS binding is the change in entropy (disorder) upon binding. The enthalpy term (ΔH
binding) reflects the energetics of the binding interaction itself, while the entropy
term (ΔS binding) reflects the changes in molecular organization that occur upon
binding [30]. A favorable change in ΔH binding indicates a net decrease in the energy
required for binding, while a favorable change in ΔS binding indicates an increase in
the disorder of the system, making the binding more favorable. The temperature fac-
tor (T) is simply a conversion factor between energy and temperature.
Another commonly used equation is the Van’t Hoff equation:
ln K
d
ðÞ= −Δ H
binding
=R 1=TðÞ+
Δ S
binding
R
(10:9)
where K
d
is the equilibrium dissociation constant, ΔH
binding
is the change in enthalpy,
ΔS
binding
is the change in entropy, R is the gas constant, and T is the temperature in
kelvin. The Van’t Hoff equation expresses the relationship between the binding con-
stants and the temperature and can be used to calculate ΔH
binding
and ΔS
binding
from
experimental data. A favorable change in ΔH
binding
leads to an increase in binding af-
finity (lower K
d
), while a favorable change in ΔS
binding
contributes to an increase in
binding entropy, also leading to a lower K
d
[31, 32].
10.3 Molecular mechanics and quantum mechanics
10.3.1 Molecular mechanics
Molecular mechanics (MM) is basically used to calculate the molecular interactions,
energies as well as nuclear motion [33, 34]. Subatomic particles specially, electrons
are not considered in this study. Also, it was assumed that nuclei are much heavier as
compared to electron and that’s why the movement of the electrons is negligible as
per Born–Oppenheimer approximation [35, 36]. Generally, MM states that the heavy
nuclei are connected with spring which actually represents the bonds which showed
different types of oscillation in different conditions. There are several assumptions in
216 Abhimannu Shome et al.
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the MM, that is, atoms are representing as spherical ball and bonds are represented
as spring, the potential functions and the parameters are evaluating by force fields.
The molecular interactions were determined by the conformation of t he atom-like
particles [37, 38].
10.3.1.1 Traid tool concept
MM is composed of three basic tools and all of them are interconnected with algo-
rithms. Together those tools are called triad tools. The basic components of the x are
force fields, parameter sets, minimizing algorithms, e tc. Before going further, we
must know “what is force field?” Force filed is nothing but a set of functions and con-
stants that used to calculate potential energy as well as the interaction between two
molecules. In terms of potential energy, it is emerged as the sum of all the force field
function in a system, displayed in equation [39, 40]:
E =
X
ij
k
ij
x
i
x
j
+
X
ijk
k
ijk
x
i
x
j
x
k
(10:10)
In this equation k
ij
is a constant depending on bond length and k
ijk
is a constant depend-
ing upon the bond angle between X
i
, X
j
, and X
k
. MM calculates the d ifferent energy
level between two or more conformation, state or level. Now, the parameters in the
traid tool include a set of parameters or you can call it different reference point and
force constant which allows to calculate the different level of potential energy. Next is
minimizing algorithm; this is used to determine stable geometrical position with lesser
electronic repulsion as well as less sterically hindered depicts in fig 10.1 [41, 42].
10.3.1.2 The harmonic oscillator model for molecules
Harmonic oscillator model is a very simple model which is composed of a moving
mass fixed to a wall by a spring. Here, small atoms such as hydrogen which move
faster than the heavier atom is considered as stationary relative to the faster atoms.
An English scientist, Robert Hook, states the mechanism behind elasticity as well as
Figure 10.1: Triad tool diagram.
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the approximation that state at which a certain amount of strain or stress can cause
the deformation of the material body depicts in fig 10.2 [43, 44]:
ν =
1
2π
ffiffiffi
k
μ
s
μ =
m
1
m
2
m
1
+ m
2
(10:11)
10.3.1.3 Energy due to stretching
The Stretching phenomenon of the chemical bonding which alter its proper position
influence potential energy. Also, it was described in Hook’s law [45]:
V
stretching
= 143.88
k
s
2
l − l
0
ðÞ
2
1 − 2 l − l
0
ðÞðÞ (10:12)
where k
s
is the force constant of stretching in mdyn/A, l
0
stands for bond length in A,
and l is the actual bond length.
10.3.1.4 Energy due to bending
The bending of the bond alters the bond angle as well as it increases the poten tial
energy:
E
θ
= ð0.21914k
θ
θ − θ
0
ðÞ
2
Þð1 + 7 × 10
8
θ − θ
0
ðÞ
4
Þ (10:13)
where k
θ
is denoted as force constant due to bending in mdyn/A; similarly, actual
bond angle is θ and θ
0
is the natural bond angle. Here, 0.21914 is the conversion fac-
tor [46].
m
1
m
2
Figure 10.2: Harmonic oscillator model.
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10.3.1.5 Energy due to torsional strain
Intramolecular rotation of the intramolecular dihedral angles such the conversion of
the conformation of chair conformation into boat conformation is given by
E
tor
=
V
1
2
1 + cosωðÞ+
V
2
2
1 + cos2ωðÞ+
V
3
2
1 + cos3ωðÞ (10:14)
Here, V
1
, V
2
, and V
3
are denoted as force constant in Fourier series in kcal/mol and ω
is denoted as the torsion angle ranged between 0° and 180° [46, 47].
10.3.1.6 The ab initio potential
Two-body potential calculation is always crucial. It can be obtained by fitting a bunch
of chose function of data. Rolf Eggenderger’s Ab initial calculation obtained potential
interactions between neon atoms:
VrðÞ= a
1
exp −a
2
r
a
0
2
"#
+ a
3
exp −a
4
r
a
0
2
"#
+ a
5
exp −a
6
r
a
0
2
"#
+ a
7
r
a
0
− 10
+ a
8
r
a
0
− 8
+ a
7
r
a
0
− 6
(10:15)
10.3.1.7 Force fields
The molecular modeling involved in the estimation of intra and intermolecular bind-
ing forces and potential energies. More precisely, based on different functional forms
of parameters which used to calculate potential energies, there are several force
fields, that is, MM2, MM3, MM4, CHARMM, and AMBER [48, 49].
10.3.1.7.1 MM2, MM3, MM4, and MMFF94
Allinger et al. [34] developed a series of force fields based on MM to investigate the dy-
namics of small molecules. These force fields are particularly useful in studying organic
intermediates, radicals, and other molecules containing sp-, sp
2
-, and sp
3
-hybridized car-
bon atoms. The approach involves optimizing the interatomic distances by considering
electron diffractions during the vibrational motion at room temperature. Many of these
force fields follow a point-charge electrostatic model, where dipole charges are assigned
to the molecular bonds. Thus, the force fields simulate the interactions within the mole-
cule based on the distribution of charges. The dipole-dipole interactive energies are
emerged as the total electrostatic energy of the model [34, 50, 51]. The Merck molecular
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force field (MMFF) is another force field reported in 1996 by Halgren. It is designed to
be a transferable force field for the pharmaceutical compounds that can finely adjust
the conformational energy as well as nonbonding interactions. Both the gas phase as
well as condense phase is adequate for the calculation [51].
10.3.1.7.2 CHARMM
Chemistry at Harvard Macromolecular Mechanics (CHARMM) is developed in 1995 by
MacKerell et al. [52]. It is widely used in simulation of the small molecule to solvate
complexes in the cases of macromolecules. It performed a wide range of calculation
regarding the energy, geometry, local minima, time-dependent dynamic behavior,
and barriers to rotation, vibrational frequencies, and free energy [52, 53]:
E
potðÞ
= E
bond
+ E
torsion
+ E
oop
+ E
elect
+ E
vdW
+ E
constraint
+ E
user
(10:16)
Here, out of the plane denoted as the OOP is denoted as improper torsion, the van der
Waals term is derived from rare-gas potentials. Electrostatic term is derived from sol-
vent effect. Here, hydrogen bonding term is not included which is implicated as the
combination of van der Waals and electrostatic term [53].
The key features and principles of the CHARMM force field are as follows:
– Atom types: Each atom in a molecule is assigned a specific atom type in the
CHARMM force field, reflecting its chemical identity and functional groups. These
atom types are based on chemical knowledge, experimental data, and QM calcula-
tions [54].
– The CHARMM force field utilizes harmonic potential functions to represent the
stretching of covalent bonds. Parameters like bond lengths and force constants
are determined based on a combination of experimental data and QM calcula-
tions. This ensures that the bond stretching behavior is accurately captured
within the force field.
– Angle bending: Angle potential terms are included to account for the bending of
bond angles. Parameters for equilibrium angles and force constants are deter-
mined through fitting to experimental and ab initio data [55].
– Torsional rotations: The CHARMM force field incorporates dihedral angle poten-
tials to model the rotation around covalent bonds. It utilizes a combination of har-
monic and periodic functions to accurately describe torsional energy profiles.
Improper torsions are also considered to maintain proper molecular geome-
try [56].
– Van der Waals interactions: The CHARMM force field accounts for van der Waals
interactions between nonbonded atoms using Lennard–Jones potential terms. Pa-
rameters for these interactions are derived from experimental data and QM cal-
culations [57].
– Electrostatic interactions: Electrostatic interactions between charged or polar
groups are d escribed within the CHARMM force field using partial charges as-
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signed to individual atoms. The force field also includes Coulombic potential en-
ergy terms to capture long-range electrostatic effects [54].
10.3.1.7.3 AMBER
AMBER is a widely employed force field that is commonly used to parameterize or-
ganic systems including proteins and nucleic acids. It demonstrates effectiveness, par-
ticularly with atom types found in six- and five-membered rings. AMBER tends to
yield satisfactory results for various aspects such as gas-phase model geometries, sol-
vation free energies, vibrational frequencies, and conformational energies. It is worth
noting that AMBER offers both a united atom representation and an all-atom repre-
sentation. In the united atom representation, nonpolar hydrogen atoms are not explic-
itly represented, but are instead merged with the heavy atoms they are bonded to.
AMBER falls within an intermediate category between force fields that consistently
employ multiple terms for all torsions and those that utilize only a single term. To
maintain proper stereochemistry at c hiral centers, united atom force fields like
AMBER often incorporate improper torsion terms [58, 59].
The key characteristics of the AMBER force field are as follows:
– Atom types: Each atom in a molecule is assigned a specific atom type within the
AMBER force field, representing its chemical identity and functional groups.
These atom types are determined based on chemical knowledge, experimental
data, and QM calculations [60].
– The AMBER force field utilizes harmonic potential functions to represent the
stretching of covalent bonds. The parameters for bond lengths and force con-
stants are determined by combining experimental data and QM calculations. This
approach ensures that the bond stretching behavior is accurately captured within
the AMBER force field.
– Angle bending: Angle potential terms are incorporated to account for the bending
of bond angles. The equilibrium angles and force constants are obtained through
fitting to experimental and ab initio data.
– Torsional rotations: The AMBER force field employs dihedral angle potentials to
model the rotation around covalent bonds. It utilizes a combination of harmonic
and periodic functions to accurately describe torsional energy profiles [61].
– The AMBER force field incorporates van der Waals interactions between non-
bonded atoms by employing Lennard–Jones
potential terms. The parameters associated with these interactions are deter-
mined through a combination of experimental data and QM calculations. This
ensures an accurate representation of van der Waals forces within the AMBER
force field.
– Electrostatic interactions: The AMBER force field considers partial charges as-
signed to individual atoms and Coulombic potential energy terms to describe electro-
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static interactions between charged or polar groups. These charges are determined
through QM calculations or empirical fitting.
10.3.1.7.4 OPLS
The optimized potentials for liquid simulations (OPLS) force field is a widely recog-
nized and extensively utilized empirical force field in molecular dynamic (MD) simula-
tions. It was originally developed by Jorgensen et al. [62, 63] during the 1990s and has
since undergone continuous updates and enhancements. The OPLS force field is partic-
ularly renowned for its exceptional accuracy in characterizing organic molecules and
is frequently employed in various fields including the study of small organic com-
pounds, drug discovery, and biomolecular simulations. The strength of the OPLS force
field lies in its incorporation of bo th atom-based and bond-based parameters. It en-
compasses terms that account for covalent bonds, bond angles, dihedral angles, van
der Waals interactions, and electrostatic interactions [62, 63].
The key principles and features of the OPLS force field are as follows:
– Atom types: Each element in a molecule is assigned specific atom types in the OPLS
force field, capturing the distinct chemical behavior and functional groups. These
atom types are based on both chemical intuition and experimental data [64].
– Bond stretching: The OPLS force field accounts for the stretching of covalent
bonds within a molecule using harmonic potentials. Parameters such as equilib-
rium bond lengths and force constants are derived from experimental data and
QM calculations.
– Angle bending: To describe the bending of bond angles, the fo rce field incorpo-
rates terms for angle potentials. Parameters including equilibrium angles and
force constants are determined through fitting to experimental and ab initio data.
– Torsional rotations: The OPLS force field incorporates dihedral angle potentials to
model the rotation around covalent bonds. It employs a combination of harmonic
and periodic functions to accurately describe torsional energy profiles.
– van der Waals interactions: In order to capture van der Waals interactions be-
tween nonbonded atoms, the OPLS force field includes Lennard–Jones potential
terms. Parameters for these interactions are derived from experimental data and
QM calculations [65].
– Electrostatic interactions: The OPLS force field utilizes a combination of partial
charges assigned to individual atoms and electrostatic potential energy terms to
describe electrostatic interactions between charged or polar groups. Charges are
derived from QM calculations or empirical fitting.
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10.4 Quantum mechanics
Erwin Schrödinger and Werner Heisenberg established the current theory of QM sep-
arately in 1925. The Schröd inger technique uses equations with partial differentials,
although Heisenberg’s approach uses matrices; nonetheless, both approaches were
proven to be mathematically equal over a year afterward. The traditional wave equa-
tion appears to provide an improved practical understanding of the Schrödinger
equation. However, the Schrödinger equation may be seen as a matter-wave version
of the wave equation [66].
10.4.1 The time-independent Schrodinger equation
Let us begin by looking at the one-dimensional traditional wave equation:
d
2
u
dx
2
=
1
v
2
d
2
u
dt
2
(10:17)
where v denotes the velocity.
By adding variable distinction:
ux, tðÞ= ψ xðÞftðÞ (10:18)
we get
ftðÞ
d
2
ψ xðÞ
dx
2
=
1
v
2
xðÞ
xðÞ
d
2
ftðÞ
dt
2
(10:19)
If we choose one of the conventional wave equation responses for f(t), including eiωt
(the constant may be handled subsequently in the normalization), we get
d
2
ψ xðÞ
dx
2
=
−ψ
2
v
2
ψ xðÞ
(10:20)
We now have a normal differential equation that describes the matter wave’s spatial
amplitude as a function of position. A particle’s energy is the sum of its kinetic and
potential parts:
E =
P2
2m
+ VxðÞ (10:21)
which can be solved for the momentum, p, to obtain
p =
2mE− VxðÞ½
fg
2
(10:22)
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