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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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Now we can use the de Broglie formula (eq. (3.4)) to get an expression for the
wavelength:
λ =
h
p
=
h
2mE− VxðÞ½
fg
1
2
(10:23)
The term ω
2
/ν
2
in eq. (10.20) can be rewritten in terms of λ if we recall that ω =2πϑ
and ϑλ= v,whereω is the angular momentum, λ is the wavelength, and ϑ is the
frequency:
ω
2
v
2
=
4π2ϑ
2
v
2
=
4π
2
λ
2
=
2m½E − VxðÞ
v
2
(10:24)
where h = h/2π. When this result is substituted into eq. (10.20) we obtain the famous
time-independent Schrödinger equation [4]:
d
2
ψ xðÞ
dx
2
+
2m
h
2
E − VxðÞ½ψ xðÞ
= 0 (10:25)
which is almost always written in the form:
−
h
2
2m
d
2
ψ xðÞ
dx
2
+ VxðÞψ xðÞ= Eψ xðÞ (10:26)
This single-particle one-dimensional equation can easily be extend ed to the case of
three dimensions, where it becomes
−
h
2
2m
∇
2
ψ rðÞ+ VrðÞψ rðÞ= Eψ rðÞ (10:27)
A two-body problem can also be treated by this equation if the mass m is replaced
with a reduced mass.
It is important to emphasize that the relationship to the conventional wave equation
has its limitations. For instance, it is not possible to directly derive the time-dependent
Schrödinger equation using the same approach. The time-dependent equation includes
incomplete initial derivatives with respect to time rather than partial second derivatives.
In fact, Schrödinger initially proposed his time-independent equation before introducing
the more comprehensive time-dependent equation.
10.4.2 The time-dependent Schrödinger equation
Now let us turn our attention to the time-dependent Schrödinger equation. While we
were able to derive the time-independent Schrödinger equation for a single particle
using fundamental methods based on the wave equation and the de Broglie relation-
224 Abhimannu Shome et al.
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ship, the time-dependent Schrödinger equation cannot be derived using similar funda-
mental techniques. Instead, it is generally treated as a postulate in QM. The time-
dependent Schrödinger equation for a single particle in three dimensions is expressed
as follows:
ih −
dψ r, tðÞ
dt
= −
h
2
2m
∇
2
ψ r, tðÞðÞ−
h
2
2m
+ VrðÞψ r, tðÞ (10:28)
where V is anticipated to have an actual purpose and illustrates the system’s potential for
power. Wave mechanics is a subfield of QM that incorporates dynamical law (eq. (10.25)).
It is important to point out that eq. (10.25) is still unable to take into consideration the
spin or relativistic impacts. The time-dependent equation may, obviously, be employed to
generate the time-independent equation. If the wavefunction is written as a product of
spatial and temporal variables, ψ(r,t) = ψ(r)f(t), then eq. (10.25) becomes
ψ rðÞih −
df tðÞ
dt
= ftðÞ −
h
2
2m
∇
2
+ VrðÞ
ψðrÞ (10:29)
or
−
ih
ftðÞ
−
df
dt
= 1=ψ rðÞ −
1
ψ rðÞ
−
h
2
2m
∇
2
+ VrðÞ
ψ rðÞ (10:30)
Because the left side is simply an expression of t while the right side is just a function
of r, both halves must be equivalent to a constant. If we informally label this constant
E (especially as the right-hand side certainly must have energy levels), we get two or-
dinary differential equations:
1=ftðÞ −
1
ftðÞ
df tðÞ dt
=
Þ −
df tðÞ
dt
= −
iE
h
(10:31)
and
−
h
2
2m
∇
2
ψ rðÞ+ VrðÞψ rðÞ
= Eψ rðÞ (10:32)
−
h
2
2m
∇
2
+ VrðÞ
ψ rðÞ= Eψ rðÞ (10:33)
The expression in the square brackets on the left is known as the Hamiltonian operator.
The latter equation is the time-independent Schrödinger equation once more. The
first equation is simply solved, yielding
ftðÞ=
e
−iEt
h
(10:34)
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The Hamiltonian in eq. (10.30) is a Hermitian operator, and Hermitian operators must
have real eigenvalues, hence E is real. This indicates that the solution f(t) is completely
oscillatory because the magnitude of f(t) never varies (recall Euler’s formula e ± iθ =
cos θ ± i sin θ).
Thus, if
ψ r, tðÞ= ψ rðÞe
−iEtðÞ=h
(10:35)
The entire wavefunction ψ(r,t) varies from ψ(r) solely by a constant magnitude phase
component. This has some fascinating ramifications. To begin, the quantity ψ(r,t)
2
is
time-independent, as we can simply demonstrate
ψ r, tðÞ
2
= ψ ✶ r, tðÞψ r, tðÞ=
eiEt
h
ψ ✶ r, tðÞ
−iEtðÞ=h
ψ rðÞ= ψ ✶ rðÞψ rðÞ (10:36)
Additionally, if ψ(r,t) satisfies eq. (10.31), the desired result for any time-independent
operator is likewise time-independent. Using the same logic as before
A = ψ ✶ r, tðÞAψ r, tðÞ= ψ ✶ rðÞA ψ rðÞ (10:37)
To identify these explanations, wavefunctions of the sort described in eq. (10.31) are
referred to as stationary phases. Although the state ψ(r,t) is quasistationary, the parti-
cle it represents is not.
However, eq. (10.31) is a specific way of solving eq. (10.26). The general solution to
eq. (10.26) is a linear combination of these specific solutions, that is
ψ r, tðÞ=
i cie − iEit
h ψirðÞ
(10:38)
10.5 Binding free energy calculation
via scoring function
10.5.1 Empirical scoring
Empirical scoring in molecular docking is a computational approach utilized to predict
the binding affinity between a ligand and a target protein or receptor. Molecular dock-
ing plays a crucial role in drug discovery and design by analyzing and predicting the
interactions between a ligand and a protein [67]. Empirical scoring functions in molecu-
lar docking rely on empirical knowledge and statistical analysis to estimate the binding
affinity or energy of the ligand–protein complex. These scoring functions consist of var-
ious terms that represent different energetic contributions and interactions occurring
at the ligand–protein interface. Empirical scoring functions are derived from statistical
226 Abhimannu Shome et al.
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analyses of experimentally determined ligand–protein complex structures and their
corresponding binding affinities. Techniques such as machine learning and regression
analysis are often employed to optimize and train the scoring functions using diverse
training datasets. It is important to note that empirical scoring functions provide simpli-
fied representations of the complex binding process. Their accuracy and reliability de-
pend on the quality and diversity of the training data as well as the assumptions and
approximations made during their development [68].
10.5.2 Semiempirical scoring
In semiempirical scoring, the interaction energy is typically calculated using a combi-
nation of empirical force field parameters and QM calculations. Unlike fully empirical
scoring functions, semiempirical methods incorporate some level of QM description,
which allows for a more accurate representation of the electronic structure and ener-
getics of the system. Semiempirical scoring methods offer a compromise between ac-
curacy and computational efficiency [69]. They provide a more detailed treatment of
the electronic structure compared to fully empirical scoring functions, while still
being computationally feasible for large-scale docking studies. It is important to note
that the accuracy of semiempi rical scoring depends on the quality of the empirical
force field parameters and the level of theory employed in the QM calculations. The
development and validation of semiempirical scoring functions require extensive
parameterization and testing against experimental data to ensure their reliability and
applicability to diverse ligand–protein systems [70].
10.5.3 Force field-based scoring
Force field scoring involves the calculation of the potential energy of a system by con-
sidering the interactions between atoms or atom groups present in the system. These
interactions encompass both bonded interactions (such as covalent bonds, bond an-
gles, and torsion angles) and nonbonded interactions (such as van der Waals forces
and electrostatic interactions). Force field scoring finds widespread use in diverse do-
mains including MD simulations, molecular docking, and structure-based drug design
[68]. It provides a practical and computationally efficient way to evaluate the energet-
ics and behavior of molecular systems, aiding in the understanding of chemical pro-
cesses, predicting ligand binding affinities, and designing novel compounds. It is
important to note that force field scoring has its limitations. The accuracy of the results
depends on the quality and applicability of the force field parameters, which may not
fully capture all the nuances of real systems. Additionally, force fields are typically op-
timized for specific classes of molecules, and their transferability to other systems
should be carefully assessed. In summary, force field scoring is a computational ap-
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proach that utilizes a set of mathematical equations and parameters to calculate the
potential energy of molecular systems. It incorporates bonded and nonbonded interac-
tions to evaluate the stability and energetics of molecules and complexes. Force field
scoring plays a fundamental role in molecular modeling and provides valuable insights
into molecular behavior and properties [71].
10.5.4 Consensus scoring
Consensus scoring is a computational approach used in drug discovery and virtual
screening to combine multiple scoring functions or methods to improve the accuracy
and reliability of binding affinity predictions. It aims to mitigate the limitations and
biases inherent in individual scoring methods by considering a consensus or agreement
among multiple approaches. In consensus scoring, a variety of scoring functions or
methods are applied to calculat e th e bin ding affinity or energy of a ligand–protein
complex [72]. These scoring methods can include empirical, semiempirical, or physics-
based force field methods as well as knowledge-based or machine learning-based ap-
proaches. By combining multiple scoring methods, consensus scoring aims to capture
different aspects of ligand–protein interactions and account for their collective contri-
bution to binding affinity. It can provide a more robust and accurate prediction com-
pared to individual scorin g methods alone. Consensus scoring can be beneficial in
virtual screening and lead opt imization, as it helps identify potential ligands with
higher confidence and improves the hit rate in drug discovery campaigns. It can also be
applied to evaluate the performance and reliability of individual scoring methods by
comparing their predictions against the consensus. It is important to note that the suc-
cess of consensus scoring relies on the diversity and quality of the individual scoring
methods incorporated. Careful selection and validation of the individual scoring meth-
ods are crucial to ensure their compatibility and complementary nature [73].
10.5.5 Knowledge-based scoring
Knowledge-based scoring is a computational approach used in drug discovery and mo-
lecular modeling to assess the quality and compatibility of ligand–protein interactions
based on existing knowledge from experimental data or structural databases. It relies
on statistical analysis and empirical observations to derive scoring functions that cap-
ture the preferences and tendencies of known ligand–protein complexes. In knowledge-
based scoring, the scoring function is derived from a database of known ligand–protein
complexes or from statistical analyses of experimentally determined structures [74].
Knowledge-based scoring provides a data-driven and empirical approach to assess
ligand–protein interactions. It captures the preferences and trends observed in experi-
mental data, enabling the prediction of binding affinities and identification of potential
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ligands with high compatibility. The success of knowledge-based scoring depends on the
quality and diversity of the reference database. A larger and more diverse database im-
proves the accuracy and coverage of the scoring function, leading to more reliable pre-
dictions. Additionally, knowledge-based scoring can be combined with other scoring
approaches, such as physics-based force fields or machine learning algorithms, to en-
hance the accuracy and coverage of the predictions [75].
10.6 Binding free energy calculation methods
The calculation of binding free energy is a computational method employed in molecu-
lar modeling and drug discovery to assess the strength of interaction between a ligand
and a target protein. It quantifies the thermodynamic stability of the ligand–protein
complex by evaluating the alteration in free energy during the binding process [76].
Binding free energy is determined by considering multiple energetic factors, such as
changes in enthalpy (ΔH)andentropy(ΔS) upon binding. The principles and methods
employed in the calculation of binding free energy are as follows:
MD simulations: These are often employed to generate an ensemble of ligand–
protein conformations and capture the dynamic behavior of the complex. These
simulations use classical force fields to describe interatomic interactions and simu-
late the motion of the system over time:
– Thermody namic integration: One common method for calculating binding free
energy is thermodynamic integration. This approach involves computationally
transforming the ligand from a nonbound state to a bound state while measuring
the potential energy difference along the transformation pathway. By integrating
the potential energy differences, the free energy change upon binding can be esti-
mated [77].
– FEP: This is another widely used method for binding free energy calculations. FEP
involves computationally mutating the ligand from a reference state to a bound
state while gradually modifying the atomic interactions. The free energy differ-
ence between the two states is then estimated using statistical mechanics and
thermodynamic equations.
– Alchemical calculations: Alchemical calculations are a key component of binding
free energy calculations. These calculations involve modifying the interactions be-
tween the ligand and the protein during the simulation, such as altering the van
der Waals parameters or charges. Alchemical transformations allow for the esti-
mation of the energetic changes associated with ligand binding [78].
– Sampling and convergence: Adequate sampling of the conformational space is
crucial to obtain accurate binding free energy estimates. Enhanced sampling tech-
niques, such as replica exchange MD or accelerated MD, are often employed to
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enhance exploration of the relevant states. Convergence of the calculated free en-
ergy values is assessed through statistical analysis and convergence criteria.
10.6.1 Extra-precision docking
XP docking, also known as empirical scoring function-based docking, is a computa-
tional method used in molecular docking to predict the binding modes and affinities
of ligands to target proteins. It relies on empirical scoring functions that approximate
the binding free energy by considering various types of interactions between the li-
gand and the protein [79].
In XP docking, the key steps involved are as follows:
– Protein preparation: The target protein is prepared by removing water molecules,
adding missing atoms, assigning partial charges, and optimizing the protein struc-
ture if necessary. The protein is typically represented as a rigid entity during the
docking process.
– Ligand preparation: The ligand is prepared by generating its 3D structure, adding
hydrogen atoms, assigning partial charges, and optimizing the ligand conforma-
tion if required. The ligand can be treated as a flexible or rigid entity, depending
on the specific docking program and settings.
– Scoring function: XP docking employs an empirical scoring function to evaluate
the fitness or quality of ligand binding to the protein. The scoring function con-
sists of various terms that account for different types of interactions such as van
der Waals forces, electrostatic interactions, hydrogen bonding, and hydrophobic
effects. These terms are typically derived from experimental data or knowledge-
based potentials and are combined to calculate a total score representing the
binding affinity.
– Search algorithm: XP docking utilizes a search algorithm to explore the conforma-
tional space and find the optimal ligand binding pose. Various search algorithms,
such as genetic algorithms, Monte Carlo methods, or gradient-based optimization,
canbeemployedtosampledifferentligand orientations and conformations
within the binding site.
– Scoring and ranking: During the docking p rocess, each ligand pose is evaluated
using the scoring function, and the poses are ranked based on their calculated
scores. The top-ranked poses are considered as potential binding modes of the li-
gand to the protein.
– XP docking provides a computationally efficient approach for predicting ligand
binding modes and affinities. It takes advantage of empirical scoring functions
that approximate the binding free energy based on known protein-ligand interac-
tions. This approach is particularly useful in virtual screening, lead optimization,
and structure-based drug design, where large compound libraries need to be
screened to identify potential drug candidates [80].
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10.6.2 Molecular mechanics with generalized Born and surface
area solvation (MM-GBSA)
MM-generalized Born (GB) surface area (MM-GBSA) is a computational method used to
estimate the binding free energies of ligand–protein complexes in drug discovery and
molecular modeling. It combines MM calculations to evaluate the molecular interac-
tions and a GBSA model to account for solvation effects [81]. In MM-GBSA calculations,
the binding free energy (ΔG
bind
) is computed as the difference between the free energy
of the bound complex (ΔG
complex
) and the sum of the free energies of the individual
components (ΔG
ligand
+ ΔG
protein
). The key steps involved in MM-GBSA are as follows:
– The initial stage of MM-GBSA involves conducting MM calculations to optimize
the structures of the ligand, protein, and their complex. These calculations utilize
force fields that characterize the interactions between atoms, encompassing as-
pects like bond stretching, angle bending, dihedral rotation, and nonbonded in-
teractions (including van der Waals and electrostatic forces).
– GBSA model: The GBSA model is employed to estimate the solvation free energy
contributions. This model approximates the solvent effect by calculating the electro-
static solvation energy based on the molecular surface area and the GB equation.
The nonpolar solvation energy is usually estimated using empirical parameters or a
solvent-accessible surface area (SASA) model.
– Energy calculations: The energy contributions for the ligand, protein, and complex
are computed by summing the corresponding terms from the force field calcula-
tions. These terms include bonded energy terms (bond stretching, angle bending,
dihedral rotation), nonbonded energy terms (van der Waals and electrostatic in-
teractions), and solvation energy terms (from the GBSA model) [81].
– The calculation of free energy involves determining the binding free energy (ΔG
bind
)
by subtracting the sum of the individual free energies of the ligand (ΔG
ligand
)and
the protein (ΔG
protein
) from the free energy of the complex formed by their binding
(ΔG
complex
). To further analyze the contributions of different molecular interactions,
the binding free energy can be decomposed into various energy terms.
– MM-GBSA offers a computationally efficient and feasible approach for estimating
binding free energies in ligand–protein complexes. It provides a good balance be-
tween accuracy and computational cost, making it a favorable alternative to
more computationally intensive methods like FEP or thermodynamic integration
while still delivering reliable results [82].
10.6.3 Molecular dynamics simulation
MD simulation is a computational technique used to study the dynamics and behavior
of atoms and molecules over time. It is widely employed in various fields, including
chemistry, physics, biology, and materials science, to investigate the properties, interac-
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tions, and structural changes of systems at the atomic level. In MD simulations, the be-
havior of the system is modeled by numerically integrating the equations of motion for
all atoms or particles in the system [77]. The key principles and steps involved in MD
simulations are as follows:
– Force field and initial configuration: A force field is selected to describe the inter-
atomic interactions in the system. The force field consists of mathematical func-
tions and parameters that describe the potential energy associated with bond
stretching, angle bending, dihedral rotation, and nonbonded interactions (van der
Waals forces and electrostatic interactions). An initial configuration is defined,
specifying the positions, velocities, and sometimes the orientations of the atoms or
particles.
– Integration of equations of motion: The equations of motion, typically derived
from classical mechanics (e.g., Newton’s second law), are numerically integrated
using algorithms such as the Verlet algorithm or the leapfrog algorithm. The inte-
gration propagates the positions and velocities of the atoms forward in time, al-
lowing the system to evolve dynamically.
– Time step and ensemble: MD simulations are carried out in discrete time steps.
The choice of the time step is a trade-off between accuracy and computational
efficiency, considering the system’s dynamics and stability. The ensemble, such as
the canonical (NVT) or isothermal–isobaric (NPT), specifies the conditions under
which the simulation is performed including temperature, pressure, and particle
number or volume constraints.
– Boundary conditions and periodicity: To simulate larger systems or avoid edge
effects, periodic boundary conditio ns are often employed. In this approach, the
system is replicated periodically, creating an infinite lattice of repeated units.
This allows the simulation of an effectively larg er system while preserving the
interactions and dynamics of the original system.
– Integration time and equilibration: MD simulations are typically divided into two
phases: equilibration and production. During equilibration, the system is allowed
to reach a stable and representative state by running the simulation for a suffi-
cient duration. This ensu res that the system’s energy, temperature, and other
properties have equilibrated and converged to their desired values.
– Analysis and interpretation: After the equilibration phase, the production phase
of the simulation is carried out to collect data for analysis. Various properties and
quantities of interest can be calculated such as energy, temperature, pressure, ra-
dial distribution functions, diffusion coefficients, or structural changes over time.
Statistical analysis and visualization techniques are used to interpret and under-
stand the simulation results.
– MD simulations provide insights into the behavior and properties of molecular
systems at the atomic level, enabling the exploration of conformational changes,
dynamical processes, thermodyn amics, and interactions. They are particularly
valuable in studying protein folding, protein-ligand binding, enzymatic reactions,
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material properties, and other phenomena that require a detailed understanding
of atomic-level dynamics.
It is essential to acknowledge that MD simulations have certain limitations, which in-
clude the accuracy of force field parameters, the choice of integration algorithm, and
the restricted timescale accessible within the simulation duration. However, to over-
come these challenges and enhance the accuracy of MD simulations, advanced techni-
ques such as enhanced sampling methods (e.g., replica exchange MD or metadynamics)
and QM/MM simulations can be employed. Overall, MD simulation is a computational
approach that enables the modeling of atomic and molecular motion and behavior over
time. It provides valuable insights into the dynamics, interactions, and properties of mo-
lecular systems, allowing scientists to investigate complex processes and phenomena at
the atomic scale [83].
10.6.4 Monte Carlo simulation
Monte Carlo simulation is a computational technique used to model and analyze sys-
tems or processes that involve randomness and uncertainty. It is widely applied in
various fields including physics, statistics, finance, engineering, and computer science.
The Monte Carlo method utilizes random sampling to approximate complex mathe-
matical calculations and make probabilistic predictions [84].
In Monte Carlo simulation, the key principles and steps involved are as follows:
– Problem formulation: The problem or system under investigation is defined,
along with the specific question or objective of the simulation. This includes spec-
ifying the variables, parameters, constraints, and mathematical relationships that
govern the system’s behavior.
– Random sampling: Monte Carlo simulation employs random sampling to generate
a large number of random inputs or scenarios within the defined problem space.
The random values are typically drawn from probability distributions that reflect
the uncertainty or variability associated with the system’s parameters.
– Model evaluation: For each set of random inputs, the model or mathematical re-
presentation of the system is evaluated to calculate the corresponding output or
result. This involves performing calculations, simulations, or mathematical trans-
formations based on the defined relationships and equations.
– Aggregation and analysis: The output results obtained from multiple iterations of
the model evaluation are aggregated and analyzed statistically. This includes cal-
culating summary statistics (such as mean, standard deviation, and percentiles)
and generating probability distributionsorhistogramstorepresenttheuncer-
tainty and variability in the outcomes.
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