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Chapter • Molecular Modeling
. Fig. 15.2 Different computer graphics representations of dopa-
mine (Sect.1.4, formula 1.13). Carbon atoms are colored gray, hydrogen atoms are white, nitrogen atoms are blue, and oxygen atoms are
15
red. (a) Dreiding models. (b)Ball-and-stick models. (c)Space-lling
models (CPK representation). (d)Solvent-accessible surface. (e)Elec-
is assumed that the interactions between particles obey
the laws of classical mechanics. For this purpose, the
Newtonian equations of motion are solved in parallel and
stepwise for all particles simultaneously. It is usually assumed that the force between two particles is not affected
by the individual spatial positions of the other particles.
In practice, a starting geometry is generated rst
(. Fig.15.4). If an experimentally determined structure,
such as the crystal structure of aprotein–ligand complex,
is available, it is used as the starting point. To account for
the surrounding water shell, the complex is immersed in
a“water bath,” meaning alarge number of water mole-
cules surround it. Furthermore, asufcient number of
ions is added to keep the whole system in an electrically
neutral state. To avoid boundary effects on the “walls,”
atrick called “periodic boundary conditions” is applied
to the water bath. When the simulated protein complex
approaches such awall and wants to leave the water bath,
trostatic potential projected on the surface (positively charged areas
are blue, negatively charged areas are red). (f)Highest-occupied molecular orbitals (HOMO), calculated for the uncharged dopamine molecule. The blue or red areas of the wave function indicate adifferent sign
the computer will treat the process as if the complex had
reentered the water bath from the opposite side. Formally,
this eliminates the boundary regions of the water bath.
At the beginning of the actual simulation, each atom
is assigned arandom starting velocity with an arbitrary
orientation. The velocities are chosen so that, on average,
they correspond to the desired temperature (taken from
aBoltzmann distribution). Then all forces from all surrounding atoms acting on agiven atom are calculated. At
xed time intervals, the next position is calculated using
Newton’s equations of motion, and so on. The step size
is typically one femtosecond (1 fs = 10
−15
s). This small
step size is necessary because many extremely fast processes occur at the molecular level. The evolution of the
motion is followed for several nano- to microseconds
and visualized in the form of atrajectory. Ten nanoseconds are sufcient to follow the movement of side chains
and sometimes smaller movements of protein domains.
. • Molecular Dynamics: Simulation of Molecular Motion
. Fig. 15.3 Denitions of molecular surfaces. The van der Waals
surface is shown on the left. The arrow marks alocation where there is
an irrelevant gap that is too small to accommodate even asingle water
molecule. Center Solvent-accessible surface created by rolling awater
However, this is not sufcient to describe conformational
changes of protein domains or the diffusion of adrug
molecule into the binding pocket. This requires signicantly longer simulation times in the microsecond range.
The folding of aprotein is also difcult to follow with
this technique. The time required for protein folding on
the real time scale is between 20 ms and 1 h. The calculation of one time step (1 fs) still requires seconds of
computing time even on the fastest computers. However,
new algorithms and computers with more specialized
architectures are being developed that will make such
simulations possible in the foreseeable future.
Another important application of MD simulation is
the calculation of binding afnities. In principle, the Gibbs
free energy of binding ∆G can be calculated for agiven
system. Large systems such as protein–ligand complexes
have many molecular degrees of freedom, e.g., groups
can rotate around bonds and the molecules change shape
in the process. Due to Brownian molecular motion, such
asystem is constantly in motion and can take on many
different geometries, or as we say, congurations, over
time. The frequency with which aparticular conguration is assumed, on average over time, depends on its
energy content. In low energy states, we encounter the
system often, while in higher energy states we encounter
it rather rarely. From the point of view of statistical thermodynamics, one now collects the energy contributions
of all the congurations that the system assumes over
time along the trajectory and summarizes them in the
so-called partition function. One automatically obtains
information about the distribution of the system over
the many energetic states. This allows conclusions to be
drawn about the entropy content of the system; this is
how the energy is distributed over the degrees of freedom of the system. In the context of protein–ligand interactions, differences in free energies between protein
complexes formed with different ligands are of particular
interest. From this, it is possible to estimate which ligand
is likely to have better binding properties compared to
an alternative ligand. This is of great importance for the
design of new drug candidates prior to their synthesis.
Methods have been developed to structurally morph one
molecule across the surface. Right The Lee–Richards surface shows
putative contact positions of atoms that will lie directly on the surface
of the molecule being studied
ligand into another during the trajectory calculation. For
example, amethyl group is transformed into achlorine
atom, and the force eld parameters of amethyl group
are gradually transformed into those of achlorine atom
during the course of the calculation. These free energy
calculations provide an estimate of how much the binding mode and thermodynamic binding prole change
during the transition from ligandA to ligandB. Meanwhile, methodological concepts have also been developed
to completely eliminate certain groups at agiven scaffold
position or to create new groups from scratch at these positions. Such calculations are still quite computationally
intensive, so they are used to design individual synthesis
candidates. For broad screening of many thousands of
compounds with docking methods (virtual screening,
Sect.7.6), where very large amounts of data have to be
evaluated, simple empirical energy functions are still used
to estimate binding afnities.
In Sect.15.4, it was mentioned that the search for
local and global minima of amolecule or amolecular
complex can be approached with molecular dynamics
simulations. Many approaches have been developed to
speed up such calculations. One of them is metadynam-
ics. It optimizes the sampling of the potential surface
for minima by cleverly distorting the individual potentials during the calculation. This allows the simulation
to leave local minima quickly. It was once casually expressed as follows: the method lls in potential minima during the simulation with previously calculated
“computational sand.” This “sand” is derived from
arrangements that the simulation had already “visited”
previously on the energy surface. In this way, the metadynamics bypasses arrangements that are visited most
often during the course of acalculation. Thus, it is able
to search alarger area of the energy surface of the system under investigation in amuch shorter time.
Umbrella sampling is used to simulate changes in
amolecular system along apredened coordinate, such
as the path of aligand into the binding pocket of aprotein. The system is forced along apath of interest by
perturbing certain potentials. MD simulations are performed along this path in several time windows. The
15
Chapter • Molecular Modeling
. Fig. 15.4 Schematic of a molecular dynamics simulation. The
starting geometry is either an experimentally determined structure
or ageometry that was optimized with aforce eld. Randomly, each
atom is assigned an appropriate starting velocity taken from aBoltzmann distribution. Then Newtonian equations of motion are stepwise
solved beginning with these starting conditions and subsequently the
coordinates are periodically saved
calculated properties in each time window are compiled
and correlated along the entire path. This gives arough
idea of how and with what energy prole aligand might
bind to aprotein.
15.8 Dynamics of aFlexible Protein
in Water
Finally, an example of the application of molecular dynamics simulations to follow the motion of amolecule
in solution is presented. A protein–ligand complex can
be used to investigate which parts of aprotein–binding
pocket or aligand in the complex remain rigid and which
are exible, and whether the shape of abinding pocket
changes with time.
The enzyme aldose reductase has been shown to be
avery exible protein. It is able to adapt its binding pocket
to the shape of acomplexed ligand in many different ways.
This property is related to the biological function of this
protein. It reduces avery wide range of aldehyde substrates. Its exact function and role as atarget structure
for drug therapy is discussed in Sect.27.4. Highly exible
and adaptable proteins present aspecial challenge to drug
design. From the many crystal structure determinations, it
has become clear that there are several parent conformations for aldose reductase that are most likely in dynamic
equilibrium with one another. Abinding ligand selects
aconformation from this equilibrium that ts, and this
conformation is stabilized upon binding. If abinding ligand selects aconformation from this equilibrium and stabilizes it by binding, this process is called conformational
selection. However, it is also possible that the geometry
of the open pocket is not determined until the ligand has
bound at least in the vicinity of the pocket. This process
is called induced t. These two mechanisms are thought to
play an important role in many exible proteins, and they
are particularly valid for structurally highly dynamic receptors such as GPCRs, which are discussed in Chap.29.
Matthias Zentgraf performed extensive molecular
dynamics simulations on aldose reductase. The resulting prole was consistent with multiple crystallographic
structure determinations with this enzyme. Amino acids,
which are repeatedly found in many protein–ligand complexes with modied geometries, were also shown to be
very exible in the MD simulations. When the trajectory
of such simulations is evaluated, it is apparent that the
protein ips between the above-mentioned parent confor-
mations. In addition, many geometries appear that have
only small but structurally critical variations from these
parent conformations. For example, small areas of the
binding pocket open up to accommodate an additional
methyl group or aphenyl ring on aligand. Such information can be used directly in the design of new inhibitors.
To obtain an overview of the exibility of aprotein,
the variation of atomic positions from one simulation
state to the next is calculated along atrajectory. As with
aphotographic lm, these instantaneous images of the
complex are called “snapshots.” In particular, it becomes
transparent when aprotein uctuates in one conformation for acertain amount of time before ipping to another geometry. As it progresses, it can either return to
the original geometry or ip to ayet different parent geometry. Such an orientation map is shown in . Fig.15.5.
From this map, it can be seen that the protein spends
some time in several parent conformations. Superimposing representative snapshots from these clusters of parent
conformations gives avery instructive picture of which
groups in the binding pocket show increased exibility.
In this example, the side chains of two adjacent phenyl
alanines (Phe 121 and Phe 122, . Fig.15.6) are particularly involved. These can swing out of the way to open
anew, previously closed cavity in the binding pocket.
In the context of drug design, such information can be
translated into the design of new inhibitors that can occupy new binding pockets. In this way, improved afnity
or selectivity for the target protein can be achieved.
. • Model and Simulation: Where Are the Dierences?
15.9 Model and Simulation:
Where Are the Differences?
To conclude this chapter, we will briey compare and
contrast the terms “model” and “simulation.” Molecular
models are used to address questions that are difcult or
impossible to answer experimentally. What different conformations can amolecule assume? This question is currently difcult to resolve experimentally. Will apotential
drug candidate t into the binding pocket of aprotein?
This question is also difcult to answer experimentally in
apredictive way without investing alot of time and effort in actually doing the necessary experiments. The use
of models is afundamental part of every scientic discipline. In chemistry, models have always played acentral
role. Chaps.23–31 show how models based on the crystal
structures of protein–ligand complexes can make an important contribution to drug design, especially in the preselection of possible molecular candidates for synthesis.
The term “simulation” describes calculations with
models. For agiven mathematical model, several options
or variable combinations can be quickly evaluated on the
computer. Such studies can contribute signicantly to
abetter understanding of the system. Along with theory
and experiment, computer simulations have been called
the third pillar of exact science.
However, beware of too high expectations in the eld
of drug design! It should not be overlooked that the performance of areasonable simulation requires that the
fundamental model is accurate and its limitations are
well understood. In many areas of engineering, this requirement is well met, so that simulation plays an important role in the design of automobiles or computer chips.
Unfortunately, chemistry is more complicated. Today’s
molecular models can be used to assemble and prioritize
compounds for synthesis. They can also be used to design ligands with improved binding properties. However,
current models are often not accurate enough to allow
detailed simulations of protein–ligand complexes with
sufcient accuracy to determine abinding afnity. This
is mainly due to the need to correctly account for the behavior of the water molecules involved in the binding. In
addition, polarization effects that alter the protonation
states of functional groups of both the protein and the
bound ligand are major problems that are still largely unresolved. Given the importance of molecular modeling
in the eld of rational drug design, this can only mean
. Fig. 15.5 The development with time of the spatial deviations of
various snapshots along the simulation trajectory are visualized on
this map. Large deviations are color-coded with red, medium-sized
deviations with green, and small deviations are colored blue. Green de-
lineated square areas are recognizable along the main diagonal. There
the complex spends time near aparent conformation. The transition
to the next square represents aip to anew geometry. If sectors out-
side the main diagonal are colored increasingly red, the geometry will
deviate more strongly from the previously adopted conformation. If
an area outside the diagonal is reached that is green, the newly adopted
geometry will not be very different from astate that the system had
once reached. With such amap, it is possible to see which of the many
parent conformations acomplex oscillates between
Chapter • Molecular Modeling
15
. Fig. 15.6 Representative snapshots were taken from the different
square area along the main diagonal in . Fig.15.5 and superimposed
upon one another. It can be seen that above all else, the side chains
of the phenyl alanines Phe 121 and Phe 122 can undergo extensive
motion in the binding pocket. In doing so, they can also adopt confor-
that more effort must be made to collect experimental
evidence in order to develop improved models.
15.10 Synopsis
Models have been and are still used in chemistry in
-
general, but in particular in modern drug design.
Computer graphics is aversatile tool to display structures and models along with various properties assigned and/or geometrically superimposed onto these
molecules.
Structures can be calculated by starting from rst
-
principles and by trying to regard physics as closely as
possible. This is done with quantum mechanical calculations. Because these methods easily become elaborate and computationally intractable, an alternative
is the use of empirical approaches. They are based on
much simpler physics, normally classical mechanics,
and treat molecules as aset of point charges in space
interconnected by springs following harmonic potentials.
mations (e.g., the light-blue geometry) that open anew hydrophobic
cavity in the binding pocket once had. With such amap, it is possible
to see which of the many parent conformations acomplex oscillates
between
Empirical approaches can only be used if enough
-
experimental data are available to parameterize and
calibrate the empirical concepts. Therefore, large databases assembling knowledge about molecular properties have been developed. Meanwhile, attempts are
also being made to quantum-chemically calculate
small building blocks of molecules in order to build
up alarge database, which can then be used to parameterize force elds using machine learning methods.
Molecular mechanics to compute the geometry of
-
molecules are based on empirical force elds. They
comprise multiple energy terms that describe mutual
interactions in the molecules either through bonds
or through space. Particular potentials are used to
describe the torsional barrier to rotations around single bonds. Furthermore, nonbonded interactions are
handled by special potentials.
The accuracy and required computational capacity
-
of quantum chemical approaches depend on the sophistication of the basis sets of atomic or molecular
orbitals used for the calculations. Parameterization of
some parts of the calculations with empirical data can
Bibliography and further reading
signicantly reduce the computational requirements.
Density functional theory is afaster approach and
works with electron density distributions instead of
orbitals. Combinations of quantum chemical methods and force eld approaches have been developed
to handle large systems such as protein–ligand complexes.
Properties such as charges can be displayed on the
-
surface of molecules. Different types of surfaces have
been dened such as the van der Waals surface or the
solvent-accessible surface.
Molecular dynamics simulations are normally based
-
on potentials derived from empirical force elds. They
consider the properties of amolecule under dynamic
conditions by solving Newtonian equations of motion. As aresult, the motion of amolecule can be
evaluated over time by analyzing the so-called molecular trajectory.
Molecular dynamics simulations can be used to study
-
the exibility of aprotein next to its ligand-binding
site. Such simulations can show multiple conformations of the protein that are competent to accommodate different ligands.
Computer simulations allow the possible properties
-
of molecules under different test conditions to be
enumerated. They help to interpret results from experiments or help to predict properties of molecules
to better plan the next experiments.
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Angew. Chem. Int. Ed. Eng., 27, 1009–1020 (1988)
B. Pullman, Molecular Modelling, With or Without Quantum Chem-
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Rivail, Ed., Studies in Physical and Theoretical Chemistry, Vol. 71,
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W. D. Cornell etal. , A Second Generation Force Field for the Simu-
lation of Proteins, Nucleic Acids, and Organic Molecules, J. Am.
Chem. Soc. 117 5179–5197 (1995)
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molecular Systems, ESCOM, Leiden (1989)
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T. Kurtzman, T. Solvation Thermodynamic Mapping of Molecular
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(2016)
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news/6304ee57707ecd4f63b3d3db (Last accessed Nov. 17, 2024)
Bibliography and further reading
General Literature
K. B. Lipkowitz and D. B. Boyd, Eds., Reviews in Computational
Chemistry, VCH, Weinheim (1990)
U. Burkert and N. L. Allinger, Molecular Mechanics, ACS Monograph
177, American Chemical Society, Washington (1982)
J. M. Goodfellow, Ed., Computer Modelling in Molecular Biology,
VCH, Weinheim (1995)
A. Leach, Molecular Modelling: Principles and Applications, 2nd Ed.
Prentice Hall (2001)
A. Hinchliffe, Molecular modelling for beginners. John Wiley & Sons
(2003)
J. Gasteiger and T. Engel, Eds. Chemoinformatics: a textbook. John
Wiley & Sons (2006)
G. Schneider and K.-H. Baringhaus, Molecular design: concepts and
applications. Wiley-VCH (2008)
D. S. Sholl, J. A. Steckel, Density Functional Theory: A Practical In-
troduction, John Wiley & Sons, Inc. (2009)
W. Kauzmann, Quantum chemistry: an introduction. Elsevier (2013)
T. Tsuneda, Density Functional Theory in Quantum Chemistry.
Springer Science & Business Media (2014)
S. J. Jang, Quantum Mechanics for Chemistry. Springer, Berlin (2023)
Special Literature
E. Fischer, Aus meinem Leben, Springer, Berlin, 1922, p.134
J. D. Watson, The double helix, Phoenix, London; originally published
by Weidenfeld & Nicholson (1968)
Conformational Analysis
Contents
16.1 Many Rotatable Bonds Create Large
Conformational Multiplicity – 248
16.2 Conformations Are the Local Energy
Minima of aMolecule – 249
16.3 How to Scan Conformational Space Eciently? – 249
16.4 Is It Necessary to Search the Entire
Conformational Space? – 250
16.5 The Diculty in Finding Local Minima Corresponding
to the Receptor-Bound State – 251
16.6 An Eective Search for Relevant Conformations by
Using aKnowledge-Based Approach – 252
16.7 What Is the Outcome of aConformational Search? – 253
16.8 Synopsis – 253
Bibliography and Further Reading – 253
© 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_16
Chapter • Conformational Analysis
Assembling amolecule with amodeling kit makes it clear
that rotations about single bonds are easy to perform.
The molecule takes on adifferent shape, or as chemists
say, it is transformed into adifferent conformation. In
areal molecule, the rotations around these bonds are
not completely free. They are subject to apotential, and
the molecule adopts certain energetically favorable arrangements as it rotates. n-Butane is the simplest case
(. Fig. 16.1). The central torsional or dihedral angle
determines the relative orientation of the two bonds to
the methyl groups. When n-butane is rotated out of the
arrangement with the two bonds to the methyl groups
in a180° orientation (trans), the methyl group on the
“front” carbon and the hydrogen atom on the “back”
carbon are directly coincident with each other at angles of rotation of 120 and 240°, called “eclipsed.” In
this geometry, they come closer to each other, so this
arrangement is unfavorable for steric reasons. At an angle of rotation of 60 and 300°, the groups are again in
a“staggered” geometry, which is an energetically more
favorable situation. This arrangement is somewhat less
favorable than the staggered trans orientation because
of the spatial proximity of the methyl groups, which are
now said to be “gauche” to each other. Finally, at 0 and
360° along the rotation path, an orientation is adopted in
which both methyl groups are exactly behind each other.
This is an even more unfavorable orientation.
16.1 Many Rotatable Bonds Create Large
Conformational Multiplicity
Several energy maxima and minima can be passed
through in the course of afull 360° rotation, depending
on which atoms and groups are attached to the rotatable bond. They are at different energy levels relative to
one another. The lowest minimum is called the global
minimum, and the energetically higher minima are called
local minima. Knowledge of these minima is important
because molecules adopt geometries that correspond to
such energy minima. To nd these minima, calculations
are necessary. One possible method is to systematically
rotate all rotatable bonds, e.g., in 10° steps. At each step,
the energy of the molecule is calculated using aforce
eld. All minima found correspond to possible conformations of the molecule.
Most drug-like molecules have many single bonds
and, therefore, exhibit more than one rotatable bond.
For these bonds, multiple torsion angle values can be
assumed. These values must be combined for all rotatable
bonds in the molecule. The number of possible combinations increases multiplicatively. The molecule n-hex-
ane has three rotatable bonds. If, analogous to n-butane, three local minima are assumed for each rotatable
bond (±60, 180, and 300°), we can expect 3 × 3 × 3 = 27
minima. However, to systematically search for these
minima in 10° steps, it would be necessary to evaluate
16
. Fig. 16.1 Butane, CH3CH2CH2CH3, is made up of alinear chain
of carbon atoms. If the terminal methyl groups are covering each
other after rotation around the central C–C bond, the torsion angle
about the central bond is 0°. At a60° angle, the “back” methyl group
is half way between the “front” methyl group and ahydrogen atom.
This situation is called a“gauche” orientation. At 120°, amethyl group
and ahydrogen atom are eclipsed to each other. At 180°, the terminal
methyl groups are exactly opposite each other. Here, the energetically
most favorable situation, the trans orientation, is achieved. From now
on, the course of the rotation is mirror symmetrical, and ends in the
starting position at 360°. The orientations at 120 and 240° are energetically less favorable than the 180° orientation by 14.6 kJ/mol. The
gauche orientations at 60 and 300° are the least favorable ones and are
25.5 kJ/mol higher in energy. If aminimization method is applied that
can only run “downhill,” the three minima on the potential curve can
be reached by starting, for example, at angles of 110, 130, and 250°
. • How to Scan Conformational Space Eciently?
36 × 36 × 36 = 46,656 positions. In principle, the energy
must be calculated for each of these positions. However,
not all angle positions will lead to reasonable geometries.
It can happen that parts of the molecule fold back upon
itself and that these parts mutually superimpose. Such
collisions can be identied by computer programs, and
the corresponding geometry is discarded. It is also easy
to imagine that with an increasing number of rotatable
bonds, the number of local minima and adoptable geometries will increase dramatically in asystematic search.
16.2 Conformations Are the Local Energy
Minima of aMolecule
It has been shown in the last chapter that the energy and
geometry of amolecule can be calculated using aforce
eld or aquantum mechanical method. In this way, all
possible combinations of angles around the rotatable
bonds in amolecule can be found that correspond to
energetically favorable states. The mathematical method
that is used to search for such aminimum geometry
can only move downhill on the potential energy surface
(Sect.15.5). For this, the potential of n-butane should
be considered again (. Fig.16.1). If an angle of 130°
is used as astarting value, the minimization ends with
atrans geometry. If an angle of 110° is started with,
which is only 20° distant, the optimization will lead to
agauche orientation. By doing this, two of the three pos-
sibilities are detected. The third minimum that mirrors
the gauche conformation will be reached if, for example,
an angle of 350° is started from. In this way, all three
conformations are found for the simplest possible case.
How are complex molecules to be approached? In
principle, in exactly the same way. Because it is not known
which torsion angles of the individual single bonds will
give access to which potential minima, that is, stable
conformations, the minimization must be started from
numerous angles for each of the single bonds. From these
values the minimization always goes “downhill.” The
minima on the potential surface are found in this way.
The art is to efciently dene the starting points from
which agiven geometry is minimized. This is avery laborious task, particularly with large molecules. It is akin to
ahiker in the mountains searching for the deepest valley.
Adenosine monophosphate 16.1 should serve as
an example (. Fig.16.2). The analysis focuses on the
ve-membered ribose ring, the bond to the nitrogen in
the adenine, and the three bonds of the sugar phosphate
side chain. What conformations can this molecule assume? Rotations around the open chain bonds are performed in 10° steps. In the systematic search for the ribose ring, only orientations that allow the ring to close
are considered. To get arough overview of the hypothetically obtained geometries, the distance between the
center of the adenine scaffold and the phosphorus atom
. Fig. 16.2 Adenosine monophosphate 16.1 exhibits the conforma-
tionally exible ribose ring and four open-chain torsion angles, τ1–τ4.
During the conformational analysis rotations are performed around
these open-chain torsion angles. To get arough description of the attained geometry, the distance between the phosphorus atom in the side
chain and the adenine scaffold (⊗) is measured
is measured in each generated geometry. This distance
falls between 4.5 and 9.3 Å for the more than 300,000
generated geometries. To estimate the energy content of
the molecule in any geometry, its van der Waals energy
(Chap.15) is calculated. Such acalculation is very fast.
The energies of the 300,000 geometries are found between
0 and 64 kJ/mol. The generated structures are not yet in
local potential minima. To achieve this, each initial geometry must be minimized (cf. the potential energy curve
of n-butane in . Fig.16.1). The resulting conformations
are then compared to see whether the same local minima are reached by starting from different points. With
300,000 starting geometries, this is quite atedious task!
It is akin to letting our hiker walk downhill from every
cross-section on the map of the Alps to nd the deepest
valley. Hopefully he will be granted great longevity so
that he lives long enough to see the results of the search!
Can this search be made more effective?
16.3 How to Scan Conformational
Space Efficiently?
Sometimes rolling the dice is better than systematic exploration! The hiker could randomly choose places in the
mountains from which to descend into the next valley.
With alittle luck, he will nd the deepest valley with
much less effort. Such Monte Carlo methods are very
popular in conformational analysis. The starting angles
for the conformation search are chosen at random. The
name of the method, which gives free rein to chance as
in gambling, is an allusion to the casino in Monte Carlo.
Another approach is molecular dynamics. The hiker
would have to get into an airplane that ies between the
mountains at high speed and changes direction at every
obstacle. After acertain amount of time, the hiker jumps
out of the plane and walks to the bottom of the valley.
The higher the plane ies, the fewer mountain peaks it
encounters and the faster it can traverse the mountains.
Chapter • Conformational Analysis
16
Molecular dynamics follows a molecular trajectory
(Sect.15.7) and stores the geometry at predened time
intervals to be used as starting points for energy minimization in aconformational analysis. By increasing
the temperature (meaning ying higher), alarger area
of conformational space can be searched in ashorter
period of time.
16.4 Is It Necessary to Search the Entire
Conformational Space?
So far, molecules have been considered in an isolated
state. How does their exibility change when they are
placed in an environment such as the binding pocket of
aprotein? In principle, their conformational exibility
does not change. It could be that minima are found at
different positions with different relative energies due
to electrostatic and steric interactions with the binding
pocket. This raises the question of whether the torsion
angles for aligand in abinding pocket need to be sought
in all regions. If energy minima occur preferentially at
certain torsion angles, it makes sense to limit the search
to these angles. For example, the hiker might get the impression that the villages are mostly in valleys and hardly
ever on peaks or slopes. Therefore, all villages would be
worthwhile as starting points for his minimum search.
Ligands in the binding pocket of a protein are under the inuence of directional interactions from the
amino acids located there. Similar conditions exist for
molecules in acrystal lattice. There, the environment is
made up of identical copies of neighboring molecules
(Chap.13). These undergo directional interactions with
the molecule, analogous to the amino acids in the binding pocket. Interestingly, the molecular packing density
inside aprotein is similar to that of organic molecules in
acrystal lattice. As mentioned earlier, the crystal structures of many organic molecules are known and stored
in adatabase. Unfortunately, experience has shown that
the conformation of aexible molecule in acrystal structure is often not identical or even similar to the geometry
of the molecule in the binding pocket of aprotein. The
same is true for conformations found in solution.
Accordingly, the receptor-bound conformation of
amolecule cannot be unambiguously deduced from its
small-molecule crystal structure or from that in solution. Nevertheless, much can be learned from crystal
structures. For example, it is not the whole molecule
that should be considered, but rather individual torsion
angles. The potential energy for the central torsion angle
of n-butane is shown in . Fig.16.1. When the angles for
multiple C–CH2–CH2–C fragments are extracted from
adatabase of small-molecule crystal structures, they
tend to cluster in areas where the potential energy curve
shows local minima. Adenosine monophosphate 16.1
has four open-chain torsion angles τ1–τ4 (. Fig.16.2).
The bond between the ribose ring and the adenine backbone forms the torsion angle τ4. Another fragment is
the phosphate group with the oxygen and the attached
carbon in the chain (τ3). This fragment occurs in alarge
number of different structures in the database. Arepresentative picture can be expected because this fragment will occur in very many different environments if
enough crystal structures are considered. The results
of such searches for the four torsion angles τ1–τ4 are
shown in . Fig.16.4 as frequency distributions, socalled histograms. Experience has shown that for many
torsion angles there are clearly preferred values. This is
the case here for τ1, τ2, and τ3. One might ask why this
statistical evaluation is not better performed for ligands
participating in crystallographically studied protein–ligand complexes. Unfortunately, the diversity of these
data is still limited, and the data are usually not precise
enough for the desired evaluation. Nevertheless, comparative studies have shown that the same torsion angles
are preferentially found in protein–ligand complexes and
small–molecule crystal structures (. Fig.16.3). Meanwhile, the Cambridge Crystallographic Data Center has
developed atool called Mogul that provides easy access
to the statistical frequencies found for the distribution
of dihedral angles in torsion angle fragments found in
crystal structures.
The experience that torsion angles prefer certain
values can be used in the conformational search. The
angle τ4 between the ribose ring and the adenine backbone shows abroad distribution over many possible
values (. Fig.16.4). Unfortunately, the search cannot
be narrowed down here. The situation is better for the
other angles τ1–τ3. There are only certain values populated. If the systematic search is restricted to these areas
and asearch is performed in 10° steps around the mean
value, only 6340 geometries would have to be generated.
Almost the same distances between phosphorus and adenine are covered within the 5.9–9.3 Å range as in the
unrestricted search. Avan der Waals energy calculation
on these geometries yields values between 0 and 16.3 kJ/
mol. In contrast to the results of Sect.16.2, all geometries corresponding to the energetically more unfavorable
regions are discarded.
How can it be conrmed that this restricted search
covers the part of the conformational space that includes the receptor-bound conformations? Adenosine
monophosphate 16.1 often occurs as a substructure
of cofactors in protein complexes, so there is enough
information available about receptor-bound conformations for this particular example. They come from crystal structures of proteins with these cofactors bound.
The distance range of 5.9–9.2 Å between the adenine
backbone and the phosphorus in the receptor-bound
structures covers the same range as was found in the
exhaustive systematic search. It can, therefore, be assumed that enough geometries have been generated to