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Chapter • Three-Dimensional Structure of Biomolecules
. Fig. 14.19 The DNA base pairs
of cytosine(C) with guanine(G)
and thymine(T) with adenine(A)
on the individual steps are formed
by complementary hydrogen bonds.
Each base carries asugar–phosphate group that is coupled with the
polymer chain. It affords adouble-helical construction with aminor
(green) and major (yellow) groove
(cf. . Fig.14.18). If viewed from
parallel to the steps, four groups
can be seen in the major groove that
possess either hydrogen-bond donor
(blue), hydrogen-bond acceptor (red),
or hydrophobic properties (gray).
Three such groups are aligned in the
minor groove. If an attempt is made
to read the interaction pattern from
this side, an GC or CG pair and an
AT or TA pair will be recognized as
identical. Here the orientation of the
interaction pattern cannot be distinguished. In the major groove, on the
other hand, the pattern of exposed
interaction is unambiguous. Therefore, proteins read information about
the DNA from the major groove
14
magnesium ions. Because of its important role in transmitting genetic information, several important drugs
act on DNA. Two examples are briey mentioned here.
Cisplatin 14.4 is areactive metal complex that can covalently bind to the nitrogen atoms of two nucleobases
on two adjacent steps of the DNA by replacing both
chlorine substituents (. Fig.14.20). This cross-linking
distorts the DNA in such away that the sequence information is no longer readable. Cisplatin and analogues
such as carboplatin are used as potent chemotherapeutic
agents in cancer therapy. Daunorubicin 14.5 is arepresentative with aslightly different mechanism of action,
but it also prevents the reading of DNA base pairs. The
planar molecular moiety of 14.5 slips largely between
two adjacent base pairs, causing astructural distortion
of the DNA (intercalation). This intravenously administered cytostatic drug is used as acombination therapy for the treatment of acute leukemias. Many natural
products also use this intercalation mechanism for their
antibacterial activity spectrum. Other pharmaceutical
research approaches attempt to use segments of DNA
itself for therapy. Such modied oligonucleotide therapeutics are discussed in Sect.32.4.
Every third bond in the polymer chain of aprotein is
-
an amide bond. It is the fundamental building block
in the protein backbone and the mutual spatial ar-
rangement of the sequential planar amide bonds determines the overall architecture of aprotein.
Typical arrangements involving the amide NH and
-
C═O groups in hydrogen bonds lead to α-helical and
β
-pleated sheet structures. Reversal of the polymer
chain in space is achieved in turns that can adopt
avariety of distinct geometries.
Helices, sheets, and turns, the secondary structural
-
elements, assemble into motifs and domains to form
the tertiary and quaternary structure of proteins.
The function of aprotein is not necessarily coupled
-
to aparticular folding pattern; however, the catalytic
and ligand-functional sites within afolding class are
found at the same position.
Nature separates fold-stabilizing residues from func-
-
tion-carrying amino acids to keep the individual steps
of the dual optimization problem separated.
Proteases recognize peptide sequences specically via
-
the binding in well-tailored pockets on both sides of
the cleavage site.
Peptide libraries with an attached photometric or
-
uorescent label that can be cleaved by the protease
reaction help to elucidate the substrate prole of different proteases.
Structural arrangements of molecular portions found
-
in multiple crystal structures can be arranged sequentially in akinematic order to provide an idea of adynamic process.
The spatial arrangement of amino acid residues ex-
-
erting aparticular chemical transformation is highly
Bibliography and further reading
. Fig. 14.20 Crystal structure of an oligomeric DNA segment after
areaction with cisplatin 14.4 (left) or intercalation with daunorubicin 14.5 (right). In both cases, the DNA molecule is severely distorted
and the genetic information on the DNA cannot be read for cell division. Cisplatin reacts with the nitrogen atoms of two nucleobases
(here guanine) of the DNA on neighboring steps with substitution
of both chlorine atoms in achemical reaction. With its planar tetracyclic ring system, daunorubicin intercalates between two neigh-
conserved and can reside on protein architectures
with similar geometry that are constructed from deviating folds.
The DNA molecule encodes our genetic information
-
and forms adouble helix of two opposing strands
wrapped like ahandrail by sugar–phosphate polymer
chains. They carry the complementary base pairs on
the stacked steps of the double helix. The individual
base pairs form atypical H-bonding pattern in the
center of the helix.
The typical H-bonding pattern of bases allows each
-
single strand of DNA to be complementary to the
second strand. When DNA is amplied, the double-stranded DNA can be completed from the single-stranded DNA. Asmall and alarge groove are
formed between the sugar–phosphate backbone. In
the large groove, the coding of the base pairs on each
step can be read from the side. By complexing bases
on adjacent steps with metal ions or by intercalating
planar agents between these steps, the DNA is geometrically distorted which will prevent reading of the
boring base pairs by spreading the DNA along the helix axis. The
compound’s amino sugar accommodates in the DNA minor groove.
(7 https://sn.pub/PMGNV7)
genetic information, which consequently inhibits cell
growth.
General Literature
C. Branden and J. Tooze, Introduction to Protein Structure, 2nd Ed.,
Garland Publ. Inc., New York (1999)
G. A. Jeffrey and W. Saenger, Hydrogen Bonding in Biological Struc-
tures, Springer Verlag, Berlin (1991)
H. B. Bürgi and J. D. Dunitz (Eds.), Structure Correlation, Vol. 1 and
2, VCH, Weinheim (1994)
G. E. Schulz and R. H. Schirmer, Principles of Protein Structure,
Springer Edition, New York (1978)
Special Literature
I. Schechter, A. P. Berger etal., On the size of the active site in prote-
ases, Biochem. Biophys. Res. Commun., 27,157–162 (1967)
P. I. Lario and A. Vrielink, Atomic Resolution Density Maps Reveal
Secondary Structure Dependent Differences in Electronic Distribution, J. Am. Chem. Soc., 125, 12787–12794 (2003)
Chapter • Three-Dimensional Structure of Biomolecules
F. A. Allen, O. Kennard and R. Taylor, Systematic Analysis of Struc-
tural Data as a Research Technique in Organic Chemistry, Acc.
Chem. Res., 16, 146–153 (1983)
C. A. Orengo, D. T. Jones and J. M. Thornton, Protein Superfamilies
and Domain Superfolds, Nature, 372, 631–634 (1994)
K. Vyas, H. Monahar and K. Venkatesan, Thermally Induced O to N
Acyl Migration in Salicylamides. Thermal Motion Analysis of the
Reactants, J. Phys. Chem., 94, 6069–6073 (1990)
G. Klebe, The Use of Composite Crystal-Field Environments in Mo-
lecular Recognition and the De Novo Design of Protein Ligands,
J. Mol. Biol., 237, 212–235 (1994)
W. J. L. Wood, A. W. Patterson, H. Tsuruoka, R. K. Jain, and J. A.
Ellman, Substrate Activity Screening: A Fragment-Based Method
for the Rapid Identication of Nonpeptidic Protease Inhibitors, J.
Am. Chem. Soc., 127, 15521–15527 (2005)
O. Koch and G. Klebe, Turns Revisited: A Uniform and Comprehen-
sive Classication of Normal, Open, and Reverse Turn Families
Minimizing Unassigned Random Chain Portions, Proteins: Struct.,
Funct. Bioinform., 74, 353–367 (2008)
PDB Database: http://www.rcsb.org/pdb/home/home.do (Last accessed
Nov. 17, 2024)
CSD Database: http://www.ccdc.cam.ac.uk/products/csd/ (Last ac-
cessed Nov. 17, 2024)
IsoStar Database: https://www.ccdc.cam.ac.uk/solutions/software/iso-
star/ (Last accessed Nov. 17, 2024)
Mogul, assessment of molecular conformations: https://www.ccdc.cam.
ac.uk/solutions/software/mogul/ (Last accessed Nov. 17, 2024)
14
Molecular Modeling
Contents
15.1 3D Structural Models as Well-Established
Tools in Chemistry – 234
15.2 Strategies in Molecular Modeling – 234
15.3 Knowledge-Based Approaches – 235
15.4 Force Field Methods – 236
15.5 Quantum Chemical Methods – 237
15.6 Computing and Analyzing Molecular Properties – 239
15.7 Molecular Dynamics: Simulation of Molecular Motion – 239
15.8 Dynamics of aFlexible Protein in Water – 242
15.9 Model and Simulation: Where Are the Dierences? – 243
15.10 Synopsis – 244
Bibliography and further reading – 245
© The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2024
G. Klebe, Drug Design, https://doi.org/10.1007/978-3-662-68998-1_15
Chapter • Molecular Modeling
15
Molecules are most commonly communicated in chemistry as two-dimensional molecular representations that
show their topology. This formalism is well established
and has proven enormously fruitful. The ability of
achemist to quickly grasp and intellectually process such
structures should not be underestimated. However, the
notation has its limitations. In particular, the three-dimensional shape of amolecule is not immediately apparent from the chemical formula. However, geometry
is of great importance for the physical, chemical, and
biological properties of drugs and thus for drug design.
Therefore, 3D structure determination (Chap.13) is of
particular importance. Whenever possible, the experimentally determined 3D structure of the compound and
the target protein is used to explain the mode of action
and the structure–activity relationship. Notwithstanding
this, there is often the problem that these structures are
not always available. In these cases, the explanation of
experimental results is limited to the structural consideration of generated models. In some cases where it is even
not possible to determine an experimental structure, an
attempt is made to calculate amodel. Often the properties of molecules can be estimated more easily by calculations than by time-consuming measurements. In many
cases, molecules can adopt several stable geometries.
Through extensive simulations, it is possible to calculate
these geometries and derive ideas about the dynamic
transformations between different states. Computational
models are, therefore, acrucial link to correlate experimental data with design ideas. Computations also allow
scanning the structural property space around agiven geometry and assessing its relevance for agiven molecular
property. The computational methods available for drug
design, the assumptions and approximations on which
they are based, and what can be deduced from them for
the development of anovel drug will be the subject of
this chapter.
15.1 3D Structural Models as Well-
Established Tools in Chemistry
Three-dimensional structure models have been used since
JacobusH. van’t Hoff and Joseph Le Bel. Emil Fischer
reported in his book Aus meinem Leben about avacation
in Italy:
“In the previous winter 1890/91 Iwas busy with the
»
task of clarifying the conguration of sugar, without
entirely achieving my goal. Then the thought came to
me in Bordighera that the decision about the conguration of pentose has to do with its relation to trioxyglutaric acid. Unfortunately for lack of amodel Icould
not tell to what extent such acids are possible according
to theory and Itherefore posed the question to Baeyer.
He picks up such things with great enthusiasm, and di-
rectly constructed carbon atoms from balls of bread and
toothpicks. But after many attempts he gave the cause up,
ostensibly because it was too hard. Later in Würzburg
after considering good models at length, Imanaged to
nd the conclusive solution.”
Linus Pauling was the rst to propose the α-helix as
asecondary structure in proteins.
“The key to Linus’s success was his reliance on the simple
»
laws of structural chemistry. The α-helix had not been
found by only staring at X-ray pictures. The essential
trick, instead, was to ask which atoms like to sit next to
each other. In place of pencil and paper, the main working tools for this work were aset of molecular models
supercially resembling the toys of preschool children.”
These were the words Nobel Laureate James Watson
used to describe Pauling’s approach in his book The Dou-
ble Helix. Pauling’s success was also based on asound
knowledge of theoretical chemistry. For example, Pauling knew that an amide bond is rigid and at, whereas
his competitors, William Bragg, Max Perutz, and John
Kendrew, mistakenly believed that it was exible. James
Watson and Francis Crick followed the same course as
Pauling in their search for the structure of DNA:
“We could thus see no reason why we should not solve the
»
DNA problem in the same way [as Pauling]. All we had
to do was build aset of molecular models and begin to
play—with luck the structure would be aspiral.”
Based on this background, the achievement of Watson and Crick seems even more impressive. They were
awarded the Nobel Prize in 1962 for the elucidation of
the double-helix structure of DNA. This example should
underscore the importance of models in science. To end
with aword from Francis Crick: “Agood model is worth
its weight in gold.”
15.2 Strategies in Molecular Modeling
In contrast to the 1950s and 1960s, today’s computers
have impressive graphical and computational capabilities. Correspondingly, programs for working with molecular models are available. The new eld of molecular
modeling has emerged. This term encompasses the representation and manipulation of realistic three-dimensional molecular structures along with the calculation of
their physicochemical properties. The main methods used
in molecular modeling are summarized in . Table15.1.
In principle, molecular modeling can be approached
from two directions. One approach is to extrapolate the
geometry and physicochemical properties from known
experimental data to the structure under investigation.
. • Knowledge-Based Approaches
The other approach is to try to obtain as accurate acomputational prediction as possible by starting from rst
principles. Quantum chemical methods and force eld
calculations are part of this strategy. In practice, both
approaches are used in parallel and are increasingly coupled. If relevant experimentally determined structures are
available, it would be stupid not to use them for model
building. On the other hand, quantum chemical and molecular mechanical approaches are widely applicable and
provide reliable results. Most importantly, models can be
changed gradually, one parameter at atime. Experiments
usually do not allow this. Instead, experimental investigations typically change several parameters at the same
time, often unavoidably.
The construction of astructural model is achieved
in three steps:
Generation of astarting model,
-
Optimization and analysis, and
-
Work with the model.
-
It is advisable to stay as close as possible to experimental structures when creating the initial starting model.
. Table 15.1 Overview of the most important molecular
modeling approaches in pharmaceutical research
Technique Objective
Interactive
computer
graphics
Modeling small
molecules
Comparing
molecules
Protein modeling
Modeling of
protein–ligand
interactions
Ligand design Searches in 3D databases
Display of 3D structures
3D Structure generation (CONCORD,
CORINA)
Molecular mechanics—force elds
Molecular dynamics
Quantum mechanical techniques
Conformational analysis
Calculation of physicochemical properties
Superimposition of molecules according
to their similarity
Volume comparisons
3D-QSAR (e.g., CoMFA methods)
Sequence comparisons
Protein homology modeling
Protein-folding simulations
Binding constant calculations
Ligand docking
Structure-based ligand design
de novo design
Virtual screening
This can be done by consulting the crystal structure of
acompound. The Cambridge Crystallographic Database, which stores experimentally determined crystal
structures of small molecules, is searched and the geometry of the resulting hits that most closely resembles the
query molecule is used. The next step is to optimize the
molecule using aforce eld calculation.
There are also off-the-shelf programs available for
generating starting models, which translate a2D structural formula into a3D spatial structure according to the
principle of amolecular construction kit. These “elec-
tronic molecular construction kits” store lists of bond
lengths and angles, as well as preferred fragment geometries, and build molecules according to asophisticated
set of rules. In fractions of asecond, they determine the
3D spatial structure for the 2D structural formula. The
rst programs in this eld were CONCORD by Robert Pearlman in Austin, Texas, USA, and CORINA by
Johann Gasteiger and Jens Sadowski at the University
of Erlangen, Germany. Both programs can be used to
generate 3D structures of small molecules. In addition,
CCDC’s program Mogul provides a comprehensive
source of information for comparing and validating
acomputed molecular model with experimental data.
However, these programs cannot build the 3D structure
of aprotein. More sophisticated techniques are required
to predict the spatial structure of aprotein from its sequence (Sect.20.6).
15.3 Knowledge-Based Approaches
Perhaps the most widely used techniques in molecular
modeling are knowledge-based approaches. They attempt to use the vast amount of knowledge accumulated
from experimentally determined molecular structures,
crystal packing, protein structures, protein sequences,
and structure–activity relationships of protein–ligand
complexes, etc. to efciently solve a given problem.
Basically, the computer program mimics the approach
that a conscientious scientist would also take. First, as
much experimental data as possible are collected and
analyzed. Important sources of information are the
Cambridge Crystallographic Database, which contains
more than 1.25 million crystal structures of small molecules, and the Protein Databank (PDB), which contains
more than 227,000 protein and DNA structures. Physicochemical properties are also available in databases.
For example, the Beilstein database of nearly 10million
chemical structures contains pKa values for more than
20,000 compounds. For amino acid residues in proteins,
the PKAD database has been developed (7 http://com-
pbio.clemson.edu/PKAD-2/). The challenge lies in the
extraction of the necessary data for the question at hand
from the enormous plethora of electronically available
information. Furthermore, it must be considered that the
15
12
ij
6
ij
6
ij
Chapter • Molecular Modeling
. Fig. 15.1 E is the total energy of amolecule or complex of mole-
cules. It is made up of several contributions. The rst term describes
the energy change when achemical bond is stretched or compressed.
In this example, it describes the so-called harmonic potential with the
force constant Kb and the equilibrium bond length b0 as aparameter.
The second term describes the energy as afunction of the bond angleΘ. Again, the harmonic potential is used with the force constants
KΘ and the equilibrium constant Θ0. The third contribution describes
the change in energy when the dihedral angle is changed, and the last
term represents noncovalent interactions. The sum of three terms is
used for this last contribution. The rst term
tive and increases rapidly with decreasing distance. It describes the repulsion between atoms that come too close together. The contribution
Cij=r
from
distance rij, but not as fast as the repulsive term. It describes attractive
interactions, which are also called dispersion interactions. Other attractive interactions exist between polar molecules which are also proportional to
. Fig.18.5). The last term q
tions based on Coulomb’s law, which works with apoint charge model.
The dielectric constant isε. The noncovalent contribution to the total
energy, without the electrostatic term, is called van der Waals energy
is always negative and tends to zero with increasing
r
(for adescription of the potentials see Sect.18.12,
/εrij describes the electrostatic interac-
iqj
=r
ij
is always posi-
data come from different sources and could be partially
erroneous or were measured and collected under barely
comparable conditions.
The largest growth in electronically available data
recently has occurred in the area of DNA sequences.
Hundreds of genomes have been sequenced, and new
ones are added weekly. The nearly endless number of sequences can only be conquered with intelligent searching
protocols. Modeling of protein structures is now being
conducted on alarge scale with novel knowledge-based
approaches using machine learning and articial intelligence (Sect.20.6). The generated models are collected in
adatabase of computed structure models and contains
more than amillion models (7 https://www.rcsb.org/
news/6304ee57707ecd4f63b3d3db).
15.4 Force Field Methods
Force eld methods, also known as molecular mechanics,
are empirical techniques for calculating molecular geometries. The goal of aforce eld calculation is to determine
an energetically favorable three-dimensional structure of
amolecule or complex of molecules. The forces acting
between the atoms are described in an analytical form,
and appropriate parameters have to be assigned to the
different terms. In principle, alarge number of such
analytical functions can be imagined. We will consider
asimple form that has been used very often. It considers
covalent and noncovalent forces. The central idea of molecular mechanics is the assumption that the bond lengths
and angles take on values that are close to the standard
values in molecules. Steric interactions, this means the
repulsion of two atoms that are not directly bonded to
each other, can cause some bond lengths and angles not
to take their ideal values. These repulsive interactions are
also known as van der Waals interactions. In the simplest
case, the deviations from the ideal values are described by
aparabolic potential (the so-called “harmonic” potential,
which applies to the motion of amass on afreely suspended spring). However, this form ignores the fact that
very strong distortions lead to bond breaking. To better
describe such strong distortions in the potential prole,
adistance-dependent function with an exponential characteristic (e.g., Morse potential) is often used.
In 1946, three terms, van der Waals interaction, bond
stretching, and angular deformation, were rst proposed
to be sufcient to calculate the structure and energy of
molecules. At that time, however, performing such calculations was extremely difcult. It was not until the
availability of computers increased that molecular mechanics calculations gained importance. In addition to
the three terms originally proposed, atypical force eld
in use today contains at least one additional contribution
that takes into account rotations about the dihedral angles
(. Fig.15.1). In addition, many force elds use terms for
electrostatic interactions. To do this, each atom must be
assigned apartial charge. The sum of these charges gives
the formal charge of the whole molecule. This is usually
set to zero for uncharged particles.
Coulomb’s law is used to describe the forces that occur between charges. This law states that the product of
interacting charges is inversely related to the square of
the distance between them, or, considering the poten
tial, inversely related to the distance. The assignment of
charges and the correct choice of the dielectric constant
are critical to the correct treatment of electrostatic energy contributions. These values are in the denominator
of Coulomb’s law and can take values between ε = 80
-
. • Quantum Chemical Methods
for water and ε = 1 for vacuum. This dampens the electrostatic interactions in water very rapidly, while in
avacuum they tend to reach much further. Choosing
the correct dielectric constant for force eld calculations
in proteins is very difcult. Many values between ε = 4
and ε = 20 have been tried. The constant is sometimes
assumed to be environment dependent, so that larger values are chosen near the surface than for the interior of
the protein. The van der Waals interactions are modeled
by the Lennard–Jones potential. This interaction has an
attractive term falling at arate of 1/r6 and arepulsive
term falling at arate of 1/r12 (. Fig.15.1). The result of
the combination of these terms is agradient that is very
large near the atoms and approaches zero as the distance
increases. In between, it passes through apotential energy minimum (. Fig.18.5). In addition to A/r6–C/r12
as the functional form, alternative distance dependencies
with other potentials or exponential gradients have been
used in force elds.
Aforce eld is derived by calibration to experimental
data or to the results of high-level quantum mechanical
calculations. The 3D structures of small molecules and
force constants derived from infrared and Raman spectroscopy are used. It is clear that different parameters
must be used for asingle bond between two carbon atoms
than for adouble bond. Therefore, several different atom
types per element are dened in aforce eld. The crystal
packing of small organic molecules can be consulted to
parameterize nonbonded interactions. Amino acids and
many functional groups of active compounds can exist
in either aprotonated or deprotonated state depending on
the applied pH conditions (so-called titratable groups).
The strength of the resulting interactions is strongly
dependent on the charge state of the functional groups
involved. The acidity or basicity of agiven functional
group is determined by its pKa value. This indicates how
easily agroup accepts or releases aproton. This property,
in turn, depends heavily on the partial charge that the
group carries and what other charges are in the immediate vicinity of the group. Thus, the pKa will change when
afunctional group is placed in adifferent environment.
For example, carboxylic acids become more acidic when
placed near apositive charge. On the other hand, their
acidic nature will change if apartially negatively charged
group is in the vicinity. This effect must be taken into account in areliable force eld calculation. An attempt can
be made to predict the protonation state in protein–ligand complexes with such calculations. The contribution
to the energy content of the complex is determined by
evaluating all possible combinations of protonated states
of titratable groups. In this way, the shift in pKa values
of functional groups can be estimated. To better describe
the shift of charges in molecules, so-called polarizable
force elds can be applied. They adjust the charge distribution to the local distances and take into account that
charges are displaced or “polarized” in the molecule by
attractive or repulsive effects. Certainly, these force elds
give amore realistic picture. However, the disadvantage
is that they are computationally much more demanding
and require many more parameters to be adjusted.
The importance of water as abinding partner in the
formation of protein–ligand complexes was emphasized
in Sect.4.6. The formation of aprotein–ligand complex
causes achange in the solvation conditions for the molecules involved. This has to be taken into account in the
force eld calculations. In order to do this, aforce eld
is combined with estimates for the contribution of solvation. Approaches such as the MM-PBSA or MM-GBSA
methods try to sum up these contributions over the local environment in asurface-dependent manner. Newer
methods such as the Grid Inhomogeneous Solvation
Theory (GIST) method can calculate thermodynamic
contributions such as solvation enthalpy and entropy
from the interaction contributions collected during amolecular dynamics simulation (Sect.15.7). These values
are then mapped onto agrid for subsequent graphical
analysis.
The choice of arelevant initial starting geometry is
important for any force eld calculation. A force eld
calculation involves energy minimization. If one starts
with an energetically unfavorable geometry, the force eld
will travel “downhill” to the next local minimum on the
multidimensional energy surface (Sect.16.2). If one starts
with two different geometries, the structures obtained at
the end of the minimization can be different, depending
on which local minimum is reached. Many molecules,
especially protein–ligand complexes, can adopt many energetically favorable conformations. It is, therefore, recommended to perform multiple force eld calculations
starting from different geometries. Molecular dynamics
simulations, discussed below, also provide asolution to
this problem of getting trapped in local minima.
15.5 Quantum Chemical Methods
In quantum mechanical approaches, the electronic structure of molecules is calculated using the Schrödinger
equation. However, its mathematically closed solution is
only possible for simple cases such as the hydrogen atom
or the molecular ion of hydrogen, +. For molecules with
more than one electron, approximate methods must be
used to solve the quantum mechanical “many-body
problem.” The most commonly used approximation is
the Hartree–Fock method. Here the many-body prob-
lem is reduced to several single-body problems. The sum
of electron–electron interactions within amolecule is
replaced by an effective eld, which can be iteratively
rened and optimized. This is where the popular name
SCF (self-consistent eld) comes from. In this model,
each electron “sees,” in addition to the potential of the
nuclei, an averaged potential of the remaining electrons.
Chapter • Molecular Modeling
15
The state of each electron in amolecule is, thus, described
by asingle-particle function called the atomic orbital (AO)
or, in amolecule, the molecular orbital (MO). The wave
function of the entire molecule is the antisymmetric
product of the many orbitals considered. The Hartree–
Fock equation is then obtained under the condition that
the optimally chosen orbitals lead to aminimum overall
energy. The main shortcoming of the Hartree–Fock approach, namely the neglect of the electron correlation,
can be corrected by more sophisticated methods, which,
however, increase the computational time considerably.
Quantum mechanical ab initio calculations allow the
calculation of the molecular structure and electron density
distribution as well as molecular properties without the
assumptions necessary for force eld calculations. In many
cases, it is difcult to make apriori predictions based on
the hybridization state of the atoms. For example, in the
case of amines and sulfonamides, it is often impossible
to predict whether the atoms bonded to the nitrogen in
these compounds are all in the same plane or whether the
nitrogen deviates with apyramidal environment. In aforce
eld calculation, this must be specied at the outset of the
calculation by assigning which atom type to which atom
in amolecule (i.e., in the above case, whether the nitrogen
should be in aplanar or pyramidal local geometry). Of
course, if the wrong atom type is chosen, the resulting
structure will be meaningless. As an advantage, quantum
mechanical calculations do not require such assumptions.
Most currently applied force elds use apoint-charge
model to describe the electrostatic interactions. One op-
tion to derive the atomic charges is to calculate the electrostatic potential of asmall molecule containing the group
of interest using quantum mechanical methods. Aset of
partial charges is then assigned to the different nuclei in
order to reproduce the quantum mechanical potential as
accurately as possible. These charges can then be transferred to force eld calculations for use in alarge system.
Another important application of quantum mechanical calculations in drug design is the calculation of
conformational energies of small molecules to calibrate
force elds. The force elds developed for proteins and
peptides are based on conformational energies calculated
quantum-mechanically for small peptides.
In contrast to force eld methods, quantum mechanical techniques are able to take into account the polarization of the electron density caused by the inuence of
neighboring groups. For example, the amide bond dipoles
in an α-helix are all oriented in the same direction, so they
add up to asignicant total dipole moment (Sect.14.2).
As aconsequence, such large and enhanced dipoles can
polarize other groups located at the end of the helix. In
fact, induced dipoles are incompletely described by standard force eld methods (see polarizable force elds). This
is not aproblem for quantum mechanical methods. Another important area of application is chemical reactions,
for which force elds are hardly parameterized, except for
afew special cases. Here, quantum mechanical methods
are the only option for areliable theoretical description.
Quantum mechanical methods are considerably more
elaborate than force eld methods. The most accurate methods, which also consume the most computational time, are
the so-called ab initio methods. However, these techniques
quickly reach their limits for very large systems. Therefore,
other less computationally intensive methods have been
developed. In these so-called semiempirical methods, certain integrals, the determination of which represents the
rate-determining step in ab initio methods, are replaced by
adequate approximations that can be computed quickly.
The resulting drastic reduction in computational time, at
the expense of accuracy, allows the routine application of
semiempirical calculations to active molecules and proteins. Density functional theory is another faster ab initio
technique. In this method, the position-dependent electron
density distribution is calculated in the ground state for
amany-body system, avoiding the complete solution of
the Schrödinger equation for a many-body system. All
interesting properties are then derived from the electron
density. For large protein–ligand systems, techniques have
been developed that treat the interesting regions, such as
the binding site or the catalytic reaction center, quantum
mechanically. The surrounding regions are approximated
with afaster force eld method (QM/MM methods).
Recently, there has been alot of discussion about new
methods using articial intelligence and machine learn-
ing. This is the attempt to transfer the way humans learn
and think to computers. The idea is to give computer
algorithms some intelligence. The goal is to use learning
algorithms that allow the computer to nd answers and
solve problems on its own, without having to develop
anew special program for each case. Neural networks
are an important tool in this area. In 2024, the Nobel
Prize in Physics was jointly awarded to John J. Hopeld
(Princeton Univ., New Jersey, USA) and Geoffrey E.
Hinton (Univ. of Toronto, Canada) for their fundamental discoveries and inventions that enable machine learning with articial neural networks. Such methods are especially powerful for independent data analysis. Today’s
computers have the storage capacity and speed to crunch
through vast amounts of data in avery short time and
“remember” it all in away that no human brain could
capture, correlate, and evaluate in the same amount of
time. In the eld of drug development, such methods
are not new. Data analysis methods and procedures have
always been used to identify correlations in often high-dimensional data spaces. As the data have become larger
and more complex, the algorithms have become better,
and the computers have become faster. We will return
to these quantitative assessments of structure–property
correlations several times in Chaps.18,19, 20. Despite
the excitement of the ever-increasing ood of data to be
evaluated, it is important to remember that every evaluation, and ultimately every insight, is already contained in
. • Molecular Dynamics: Simulation of Molecular Motion
the data. If the data are not reliable, meaning that if they
are too “noisy” due to bad and erroneous information,
even the best machine learning or articial intelligence
algorithm will hardly bring any new insights to light.
But what can articial intelligence contribute to the
use of quantum chemistry in drug design? As mentioned
above, quantum chemistry is characterized by high accuracy. But its computational cost is still prohibitive for
many problems. Today, we are increasingly taking the approach of breaking molecules into small building blocks
and performing sophisticated quantum chemical calculations on the resulting building blocks to determine their
energy and geometry. When alarge dataset of molecules
is processed in this way and care is taken to ensure that
the building blocks of interest are repeatedly embedded in
different molecular environments, it is possible to generate
adataset with inherently redundant information. From
such adataset, one can then use articial intelligence to
infer parameters for agiven force eld. Typically, neural
networks are used for this purpose. Force elds obtained
in this way represent areal advance. They approach the
accuracy of quantum chemical calculations, but can be
computed as quickly as typical empirical force elds.
15.6 Computing and Analyzing Molecular
Properties
The result of amolecular mechanics or quantum chemical calculation is initially aset of atomic coordinates
that dene the three-dimensional shape of the molecule.
What can be done with this? An important application of
the calculations is the determination of conformational
energies: this is the relative energy of one molecular conformation compared to another (Sect.16.1).
Two other molecular properties can be calculated: the
shape and size of amolecule, along with its electronic
characteristics. All current graphics programs have several ways of displaying such properties with the spatial
structure of molecules. The most important ones are
summarized in . Fig.15.2.
The most commonly used representation is aline or
stick representation (Dreiding models), sometimes atoms
are displayed as small spheres. Usually acolor coding
is used to represent the atoms: nitrogen is blue, oxygen
is red, sulfur is yellow, uorine is turquoise, chlorine is
green, bromine is brown, and iodine is purple. Hydrogen
atoms are shown in white, but are usually omitted for
clarity. Carbon atoms are usually shown in black or gray.
In most gures in this book, carbon atoms that belong to
the protein are shown in orange, and carbon atoms that
belong to the ligand are shown in gray, but sometimes in
adifferent color is applied to distinguish them in different molecules. Another display option is the space-lling
model, which shows van der Waals surfaces. In this representation, each atomic nucleus is represented by asphere
whose size corresponds to the van der Waals radius. Values for these radii are derived from crystal packing or
from very accurate ab initio calculations. Such representations are also known as CPK models (named after the
scientists Corey, Pauling, and Koltun).
There are other ways to represent surfaces
(. Fig.15.3). The solvent-accessible surface has proven
particularly valuable for proteins. The most commonly
used way to display a protein in this book is atransparent opaque white surface. The van der Waals surfaces in
. Fig.15.3 (left) give the impression that there is agap at
the location marked by the arrow. However, this gap is so
narrow that no other atom can t into it. Therefore, the
solvent-accessible surface (. Fig.15.3, center) is less mis-
leading. It is created by rolling asphere with aradius of
1.4 Å, the size of awater molecule, over the surface of the
studied molecule. This surface appears much smoother.
The depressions that are still present mean that small molecules—at least one water molecule—can actually t in
there. The Lee–Richards surface is less commonly used,
but very helpful. It is chosen so that ligand atoms that
come into contact with atoms of the protein under investigation will lie directly on this surface (. Fig.15.3, right).
The surface can also be colored. For example, each
atom type can be assigned acolor, and then the color of
the next closest atom is used in that part of the surface.
Similar representations where the surface of the molecule
is colored according to other properties, such as electrostatic or hydrophobic potential, are very instructive and
often used.
15.7 Molecular Dynamics:
Simulation of Molecular Motion
None of the processes we are interested in take place
at 0 K, but rather at body temperature, which is about
310 K. It is, therefore, obvious that not only the potential
energy but also the kinetic energy must be considered.
Molecules also move at room temperature, which is close
to body temperature (about 295 K). They diffuse and
change shape by adopting different conformations. The
exibility and adaptability of both binding partners play
an important role in protein–ligand interactions. Aprerequisite for protein binding is that the ligand can adopt
aconformation with ashape that ts into aprotein–binding pocket. On the other hand, the protein is exible to
some extent. For example, side chains on the surface can
adopt different conformations or entire domains can
move relative to one another. The mutual adaptation of
protein and ligand shapes plays an important role in the
formation of protein–ligand complexes.
Molecular dynamics simulations (MD) are atheoretical method for describing these effects. Molecular
dynamics simulations follow the motion of atoms and
molecules under the inuence of aselected force eld. It