Schlick T. Molecular Modeling and Simulation: An Interdisciplinary Guide
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240 8. Theoretical and Computational Approaches to Biomolecular Structure
4. Successful multidisciplinary collaborations. Notable examples are
collaborations between mathematicians and biologists in relating knot
theory with DNA topology and geometry [246, 613, 1370]; between com-
puter scientists/engineers and biologists regarding DNA computing [245,
pp. 26–38], [40, 117, 162, 779, 929, 1083, 1155]; between biological and
mathematical/physical scientists in propelling biology-information tech-
nology, or bioinformatics; and between material scientists and biomedical
scientists regarding nanomaterials for biological applications, creating the
field of nanotechnology/bionanotechnology [442, 787, 904, 1059, 1202].
1
See [635], for example, for a description of some mathematical challenges
in genomics and molecular biology and [252,530] for biological challenges
in the 21st century.
5. The wealth of readily available Internet and web resources, sequence
and structure databases, and highly-automated analysis tools.
2
8.1.2 The Future of Biocomputations
“One of these days,” believes Nature’s former editor John Maddox, “somebody
will begin a paper by saying, in effect, ‘Here is the Hamiltonian of the DNA mol-
ecule’ and will then, after a little algebra, explain just why it is that the start and
stop codons of the genetic code have their precise functions, or how polymerase
molecules work in the process of transcription”[812].
Some of us might be somewhat skeptical that “a little algebra” will suffice to
explain DNA’s functional secrets, but many remain confident that a large amount
of computation with carefully developed models will bring us closer to that goal
in the not-too-distant future.
8.1.3 Chapter Overview
In this chapter, we introduce molecular mechanics from its quantum-mechanical
roots via the Born-Oppenheimer approximation. Following a brief overview
of current quantum mechanical approaches, we discuss the three underlying
1
Nanotechnology is an emerging science of creating functional materials, devices, and systems
on the basis of matter at the nanometer scale, including macromolecules, and the exploitation of novel
properties and phenomena on this scale for biomedicine, technology, and more. See, for example, gen-
eral principles for a National Nanotechnology Initiative on www.nano.gov; the 24 November 2000
issue of Science, volume 290, highlighting issues in nanotechnology; the September 2001 special
issue of Scientific American devoted to ‘The Science of the Small’; and recent reviews on the
application of nanomaterials to biology and medicine [442] for cell imaging, cell tracking, and can-
cer treatment (diagnosis and therapy) [904], as well as innovative synthetic DNA-based enzymes and
aptamers [787].
2
Caution is certainly warranted regarding the quality of some unreviewed online information.
As stated in the New York Academy of Sciences Newsletter of Oct./Nov. 1996, “There may be debate
about whether the explosive growth in electronic communication has made life better or worse, but
there’s no question that it has made life faster”.
8.2. Quantum Mechanics (QM) Foundations of Molecular Mechanics (MM) 241
principles of molecular mechanics: the thermodynamic hypothesis, additivity, and
transferability. We then describe the choices that must be made in formulating the
potential energy function: configuration space, functional form, and energy pa-
rameters. We end by mentioning some of the current limitations in force fields.
The next chapter discusses details of the force field form and origin.
8.2 Quantum Mechanics (QM) Foundations
of Molecular Mechanics (MM)
Quantum mechanical methods are based on the solution of the Schr¨odinger equa-
tion [742, 978]. This fundamental approach is attractive since 3D structures,
molecular energies, and many associated properties can be calculated on the basis
of fundamental physical principles, namely electronic and nuclear structures of
atoms and molecules. Indeed, the quantum mechanics pioneer Paul Dirac is be-
lieved to have expressed the sentiment that the Schr¨odinger equation reduces
theoretical chemistry to applied mathematics [765]!
Although historically quantum calculations were practical only for very small
systems, exciting developments in both software and hardware (computer speed
as well as memory) have made quantum-mechanical calculations feasible for
larger systems, including biomolecules, with various approximations (see [465],
for example; more below). See the Nobel Lecture address by John Pople for field
perspectives [1011].
The presentation of this important area of modeling is very brief here. For
comprehensive treatments, see textbooks by Warshel and Cramer [270,1342]and
excellent reviews [573, 628, 1163–1165,1348,1442].
8.2.1 The Schr
¨
odinger Wave Equation
The Schr¨odinger wave equation describes the motions of the electrons and nuclei
in a molecular system from first principles. This equation can be written as
Hψ
n
= E
n
ψ
n
, (8.1)
where the Hamiltonian operator
H is the sum of the kinetic (E
k
) and potential
(E
p
) energy of the system:
H(
P,
X)=E
k
(
P )+E
p
(
X), (8.2)
where
P and
X denote the collective momentum and position vectors for all the
nuclei and electrons in the molecule. The potential energy E
p
(
X) originates from
electrostatic interactions among all the variables.
According to this description, the quantum states E
n
(eigenvalues) form a dis-
crete set, corresponding to the eigenfunctions ψ
n
, for the system of electrons
and nuclei. The Schr¨odinger equation thus describes the spatial probability
distributions corresponding to the energy states in a stationary quantum system.
242 8. Theoretical and Computational Approaches to Biomolecular Structure
Traditional electronic structure methods — important more from a historical
perspective — calculate these eigenstates associated with the discrete energy lev-
els by diagonalization of the Hamiltonian matrix, of order of the number of basis
functions. This cubic scaling is prohibitive for large systems.
8.2.2 The Born-Oppenheimer Approximation
In the Born-Oppenheimer approximation to the Schr¨odinger equation, the
motions of the molecule are separated into two levels: electrons and nuclei.
First, only the electrons are considered as independent variables of
H, while
positions of the nuclei are assumed fixed. This is generally a good approximation
because the nuclei — much heavier than the electrons — are typically fixed on the
timescale of electronic vibration. The resulting eigenvalues from this analysis of
electron motion represent electronic energy levels of a molecule as a function of
atomic coordinates. These energy states are known as Born-Oppenheimer energy
surfaces (BOES).
In the second level of the Born-Oppenheimer approximation, the BOES of
the electronic ground state (E
p
(X),whereX is the collective position vector
of the nuclei in the system), are used as the potential energy of the Hamiltonian
(eq. (8.2)) instead of the Coulombic potential. The quantum mechanical behavior
of the nuclei is then investigated.
In theory, quantum mechanical treatments should be the tool of choice for
reliable description of complex chemical processes, since no experimental infor-
mation is needed as input. Yet, an accurate analytic solution of the Schr¨odinger
equation is not feasible except for small molecules. Thus the equation must be
generally solved through standard approximations, which naturally reduce the ac-
curacy obtained. The range of application is further limited by the computational
complexity required by these techniques, but advances are rapidly occurring on
this front.
Two basic quantum-mechanical approaches are used in practice: ab initio and
semi-empirical. Both rely on the Born-Oppenheimer approximation that the nu-
clei remain fixed on the timescale of electronic motion, but different types of
approximations are used. The former is more rigorous than the latter and hence
more computationally demanding. For example, in Hartree-Fock type calcula-
tions, the computational requirements, including computer time and memory,
scale as M
2
F
to M
4
F
where M
F
is the dimension of the Fock matrix. This dimen-
sion corresponds to the size of the computational basis set used to approximate
the wave functions, related to the number of electron orbitals, N
e
.
8.2.3 Ab Initio QM
The name ab initio implies non-empirical solution of the time-independent
Schr¨odinger equation, or solutions based on genuine theory. However, besides
the Born-Oppenheimer approximation, relativistic effects are ignored, and the
concept of molecular orbitals (or wave functions) is introduced.
8.2. Quantum Mechanics (QM) Foundations of Molecular Mechanics (MM) 243
In ab initio methods, molecular orbitals are approximated by a linear combina-
tion of atomic orbitals. These are defined for a certain basis set, often Gaussian
functions. The coefficients describing this linear combination are calculated by a
variational principle, that is, by minimizing the electronic energy of the molecular
system for a given set of chosen orbitals.
This energy (known as Hartree Fock energy) and the associated coefficients are
calculated iteratively by the Self Consistent Field (SCF) procedure with positions
of the nuclei fixed. This calculation is expensive for large systems — as computa-
tion of many integrals is involved — and makes ab initio methods computationally
demanding.
In practice, the basis set for the molecular wave functions is represented in
computer programs by stored sets of exponents and coefficients. The associated
calculated integrals are then used to formulate the Hamiltonian matrix on the
basis of interactions between the wave function of pairs of atoms (off-diagonal
elements) and each atom with itself (diagonal elements) via some potential that
varies from method to method. An initial guess of molecular orbitals is then ob-
tained, and the Schr¨odinger equation is solved explicitly for a minimum state of
electronic energy.
The quality of the molecular orbitals used, and hence the accuracy of the calcu-
lated molecular properties, depends on the number of atomic orbitals and quality
of the basis set. The electronic energy often ignores correlation between the mo-
tion of the electrons, but inclusion of some correlation effects can improve the
quality of the ab initio results.
Density Functional Theory (DFT)
DFT can be formulated as a variant of ab initio methods where exchange/
correlation functionals are used to represent electron correlation energy [962].
DFT methods are based on the use of the electron density function as a ba-
sic descriptor of the electronic system. In the DFT Kohn-Sham formulation, the
electronic wave function is represented by a single ground-state wave function
in which the electron density is represented as the sum of squares of orbital
densities.
DFT schemes differ by their treatment of exchange/correlation energy, but
in general this class of methods offers a good combination of accuracy and
computational requirements, especially for large systems. DFT methods are com-
putationally more efficient than conventional ab initio methods that correct for
electron correlations, but have a similar scaling complexity due to the N
3
e
diago-
nalization cost of the Hamiltonian matrix. With linear-algebra advances, however,
this standard diagonalization expense can be circumvented with approaches that
yield linear scaling by localization of the electronic degrees of freedom (i.e., elec-
tron density, density matrix, and orbital calculations) [282, 465, 1148, 1409,for
example] (see below).
244 8. Theoretical and Computational Approaches to Biomolecular Structure
8.2.4 Semi-Empirical QM
In semi-empirical methods, the matrix elements associated with the wave-
function interactions are not explicitly calculated via integrals but are instead
constructed from a set of predetermined parameters. These parameters define
the forms and energies of the atomic orbitals so as to yield reasonable agree-
ment with experimental data. Thus, most integrals are neglected in semi-empirical
methods, and empirical parameters are used as compensation. Although good pa-
rameterization is a challenging task in these approaches, and parameters are not
automatically transferable from system to system, semi-empirical methods retain
the flavor of quantum approaches (solution of the Schr¨odinger equation) and are
less memory intensive than ab initio methods. As in ab initio methods, the quality
of the results depends critically on the quality of the approximations made.
8.2.5 Recent Advances in Quantum Mechanics
Traditional quantum calculations are dominated by the cost of solving the elec-
tronic wave function expressed in terms of the Hartree-Fock operator. Solution
of the wave functions via inversion of the Fock matrix for a minimal basis set
(roughly the number of electrons, N
e
) requires O(M
4
F
) work. While still much
better than classical orbital calculations that include correlations, this scaling lim-
its applications to large systems. It has been noted, however, that by exploiting the
sparsity of the Fock matrix — zero elements due to the rapid decay with distance
of orbital overlapping — algorithms developed in linear algebra for banded sys-
tems can reduce the cost to linear scaling, i.e., O (M
F
), or to the near-linear cost of
O(M
F
log M
F
) when Ewald forces are included [726,951,1414,1445]. See also
[464] for a physics-community perspective of electronic structure calculations
(rather than the chemistry community).
Linear Scaling
Linear scaling algorithms are also possible by various divide-and-conquer algo-
rithms that localize electronic calculations (for both ab initio and semi-empirical
methods) [465, 1409] and by reformulating the Hamiltonian diagonalization as a
minimization problem. The localization is achieved by using different treatments
for the strong intramolecular interactions and the weak intermolecular interac-
tions [755]. The reformulation as an optimization problem entails construction
of an energy functional, which is minimized with respect to some variational
parameters that represent the electronic ground state [282].
The nonlinear conjugate gradient method (see Chapter 11) exploits the ma-
trix sparsity pattern and is very modest in both computational and storage
requirements. See Daniels and Scuseria [282] for a comparison among several
linear-scaling approaches for semi-empirical calculations, including pseudo-
diagonalization of the Fock matrix via orthogonal rotations of the molecular
8.2. Quantum Mechanics (QM) Foundations of Molecular Mechanics (MM) 245
Figure 8.1. The concept of QM/MM methods, as introduced by Warshel & Levitt [1344],
in which a limited part of the system (reaction region) is treated quantum mechanically,
while the surrounding solvent and remaining biomolecular system is treated classically.
Figure kindly provided by Arieh Warshel.
orbitals, conjugate-gradient density matrix search, and purification of the density
matrix via transformations applied to a function expanded in terms of Chebyshev
polynomials.
Biomolecular Applications
Such advances in complexity reduction are making possible quantum calculations
on biomolecules under various simplifications. Semi-empirical methods have
been applied to protein, DNA, and RNA systems (e.g., [86,638,639,754,1415]),
and ab initio approaches have been applied to smaller systems (e.g., DNA bases
and base pairs [552, 755], cyclic peptides [756], and the reactive centers of
enzymes [144, 1185]). Indeed, hybrid QM/MM (quantum mechanics/molecular
mechanics) methods, as first introduced in 1976 [1344] (see Figure 8.1), are
routinely used for studies of enzyme mechanisms and related aspects of protein
dynamics [776,1442]. See Figures 8.2 and 8.3 and Box 8.1 for examples.
As shown in Figure 8.1, QM/MM methods rely on dividing the biomolecular
systems into two regions: a small region enclosing the active site that is treated
quantum mechanically, and the remaining region (including solvent and biomol-
ecule), which is modeled by classical molecular mechanics force fields. The key
to such QM/MM methods is the coupling between the electric field from the sur-
rounding and the QM Hamiltonian in the active-site region. This requires careful
treatment of the boundary between the QM and MM regions, either by using
hybrid orbitals for the connection or a linked atom approach.
Recent advances in the field have come from using higher-level ab initio QM
Hamiltonians, as first proposed by Singh & Kollman [1194] or, alternatively,
empirical versions, like the valence bond method (EVB) proposed by Warshel
246 8. Theoretical and Computational Approaches to Biomolecular Structure
and co-workers [54, 408, 1342]. However, using ab initio QM Hamiltonians has
introduced new challenges of proper sampling and computing the associated free
energy pathway. The calculation of free energies from QM/MM simulations can
be performed by averaging over the system’s configurations via perturbations
from a reference surface; however, such sampling for accurate free energy evalua-
tions as well as calculations of pK
a
values remain challenging and form an active
area of research. See recent reviews in [441,452,572–574, 628,1165,1442].
Box 8.1: Illustrative QM and QM/MM Applications
Semi-empirical quantum-mechanical applications have been used, for example, to study
aqueous polarization effects on biological macromolecules, by comparing free energies in
the solvated versus gas-phase states [1415]. Figure 8.2 from [638] illustrates the quantum-
mechanically derived electrostatic potential of the polarized electron density of A, B, and
Z-DNA relative to the gas phase density. The maps indicate how the electrostatic potential
changes when the electron density of the DNA is polarized by the reaction field of the
solvent. Another example of semi-empirical applications is the study of the active site of
an enzyme (cytidine deaminase) to delineate a mechanism for ligand attack [754].
In ab-initio applications, electronic and vibrational properties have been determined
from optimized geometries. Ab initio methodologies have also been applied to many
nucleic-acid base systems and their complexes, for example, to study stacking interactions
[20, 552, 1317], hydrogen-bonding [1318], and basepair planarity properties.
Combined ab initio QM/MM approaches can be used to study enzyme catalysis
[54, 1344], for example to deduce the reaction paths and free energy barriers for the
two steps of the reaction catalyzed by enolase [1443, 1444]. This two-step reaction in-
volves abstraction of a proton from the substrate to produce an intermediate, followed by
departure of a hydroxyl group with the assistance of a general acid.
The calculations have identified catalytically important residues and water molecules
at the active site of the enzyme (see Figure 8.3) and have shown that the electrostatic
interactions driven by metal cations at the active site strongly favor the first, but strongly
disfavor the second, step of the reaction. This dilemma appears to be resolved by a tailored
organization of polar and charged groups at the enolase active site, exploiting the two dif-
ferent orientations of charge redistribution involved in each of the two reaction steps. Thus,
the enzyme environment might provide an essential platform for the reaction mechanism.
This finding may ultimately be tested experimentally.
Enzyme catalysis associated with DNA repair and replication is another important
example where QM/MM applications have been insightful. Understanding the fidelity of
DNA repair and replication mechanisms requires tracking the conformational and energetic
pathways associated with these processes, and simulations have the potential to suggest
the transient intermediates that are beyond the capabilities of experiment. In DNA poly-
merases, the repair involves the chemical incorporation of a nucleotide in the DNA by
phosphodiester bond formation [1029]. QM/MM studies of DNA pol β, for example, sug-
gest a Grotthuss hopping mechanism of proton transfer between water molecules and three
8.2. Quantum Mechanics (QM) Foundations of Molecular Mechanics (MM) 247
conserved aspartates in the active site. When a correct unit (i.e., C opposite a template
G) is incorporated, a lower activation energy is involved compared to the incorrect unit
(A opposite G) [1033]. Similar water-assisted mechanisms were later identified in other
polymerase systems such as Dpo4 [1334, 1338] and T7 DNA polymerase [1333]. Other
QM and QM/MM studies [18, 49, 144, 408, 409, 769] also suggest alternative pathways,
depending on the structure of the complex, the protonation states, and the environment
(water and ions), underscoring the versatility of polymerase mechanisms (see Fig. 8.4).
8.2.6 From Quantum to Molecular Mechanics
An alternative approach is known as molecular mechanics, also referred to as the
force-field or potential energy method [24,175,185,764,871,898,938,1335,1359].
The Born-Oppenheimer approximation to the potential energy with respect to the
nuclei, E
p
(X), can be imagined as the target function in molecular mechan-
ics (X represents the collective position vector for the nuclei). The electrons
can be regarded as implicit variables of this potential. However, unlike quantum
mechanics, this potential function must be evaluated empirically.
Mechanical Molecular Representation
An underlying principle in molecular mechanics is that cumulative physical forces
can be used to describe molecular geometries and energies. The spatial conforma-
tion ultimately obtained is then a natural adjustment of geometry to minimize the
total internal energy. A molecule is considered as a collection of masses cen-
tered at the nuclei (atoms) connected by springs (bonds); in response to inter
and intramolecular forces, the molecule stretches, bends, and rotates about those
bonds (Fig. 8.5). This simple description of a molecular system as a mechanical
body is usually associated with a “classical” system. Remarkably, this classical
mechanics description — an appropriate characterization even as the amount
of quantum-mechanical information used to derive force fields increases —
works generally well for describing molecular structures and processes, with the
exception of bond-breaking events.
In practical terms, molecular mechanics involves construction of a potential en-
ergy, a function of atomic positions, from a large body of molecular data (crystal
structure geometries, vibrational and microwave spectroscopy, heats of forma-
tion, etc.). Entropic contributions are either neglected or approximated by various
techniques. Minimization of this function can then be used to compute favor-
able regions in the multidimensional configuration space, and molecular dynamics
simulations can further explain the system’s thermally accessible states.
Early Days
As introduced in Chapter 1, the first molecular mechanics implementations date
to the 1940s, but only in the late 1960s/early 1970s did the availability of digital
248 8. Theoretical and Computational Approaches to Biomolecular Structure
Figure 8.2. The electrostatic potential surface of the electronic response density of
A, B and Z-DNA calculated by the linear-scaling electronic structure and solvation
methods of York and coworkers [638]. The electronic response density is defined as
Δρ(r)=ρ
sol
(r) − ρ
gas
(r) where ρ
gas
(r) is the relaxed electron density in the gas phase
and ρ
sol
(r) is the relaxed electron density in solution (approximated by a continuum
dielectric model).
computers make such calculations tractable. One of the force-field pioneers, the
late Shneior Lifson, who developed the Consistent Force Field approach with
doctoral student Arieh Warshel [766], recalled that “the empirical [force field]
method did not always enjoy a high prestige value among theoretical chemists,
particularly those engaged in quantum chemistry”[765].ButLifsonwentonto
suggest that in the early days of the field the notions of force-field consistency
and transferability were neither always carefully treated nor well understood. The
power of the empirical force-field approach to describe collectively properties of
related molecules — a description not possible by quantum mechanics — was
also not fully appreciated.
A classic example illustrating the early success of molecular modeling com-
putations is the correct interpretation of peculiar experimental results. In 1967,
consistent force-field calculations predicted two stable conformations for a cy-
cloalkane (1,1,5,5–tetramethylcyclodecane). Oddly, neither structure resembled
the crystal form [135]. However, the average of the two computed molecular
mechanics structures matched the experimental data exactly! Thus, empiri-
cal calculations revealed that the regular crystal contained a distribution of
8.2. Quantum Mechanics (QM) Foundations of Molecular Mechanics (MM) 249
Figure 8.3. Catalytically important protein residues and water molecules at the active site
of enolase as identified by ab initio QM/MM calculations by Weitao Yang and coworkers
[778]. The color coding is according to categories of energetic contribution to stabilization
of the transition state in either the first or second step of the reaction (e.g., sky blue, dark
golden, green, orange); oxygens are colored red, nitrogens are dark blue, and hydrogen
atoms of amino acid residues are omitted. Lys345 (K345, bottom center) is the general base
that captures a proton from the substrate (PGA, 2-phospho-D-glycerate) in the first reaction
step. Glu211 (E211, top right) is the general acid that assists departure of a hydroxyl group
in the second reaction step. Residues in sky blue are hydrogen bond donors interacting
with the substrate. Those in dark golden and orange are positioned to counteract the effects
of the two magnesium ions (labeled) in the second reaction step but have small energetic
effects on the first reaction step. Residues colored green are others found to have energetic
effects on the reaction.
two conformers. This was indeed confirmed later by experimentation. Today,
this theme emerges often from molecular dynamics simulations, where the spatial
and temporal conformational-ensemble average corresponds to the experimental
structure [209].