[Interdisciplinary Applied Mathematics] Molecular Modeling and Simulation: An Interdisciplinary...

8Theoretical and ComputationalApproaches to Biomolecular Structure

Chapter 8 Notation

SYMBOL DEFINITION

Vectorsa,b, c interatomic distance vectors (in definition of τ )nab,nbc unit normalsrij interatomic distance vector (xj − xi)xi position vector of vector i, components xi1, xi2, xi3

˜P collective momentum (for nuclei and electrons)˜X collective position (for nuclei and electrons)X collective position (nuclei only), components

x1, . . . ,xN ∈ R3N

V collective velocity (nuclei only)ψn eigenfunctions of the Hamiltonian operatorScalars & Functionsbi bond length iqi Coulomb partial charge for atom i

rij interatomic distance (between atoms i and j)t0 initial timeAij , Bij Lennard-Jones coefficientsH Hamiltonian operator (Ek + Ep)Ebond bond length energyEbang bond angle energyEcoul Coulomb energyEk kinetic energyElocal local (short-range) energyEn eigenvalues of the Hamiltonian operator (quantum states)Enonlocal nonlocal (long-range) energyEp potential energy (also E)

T. Schlick, Molecular Modeling and Simulation: An Interdisciplinary Guide, 237Interdisciplinary Applied Mathematics 21, DOI 10.1007/978-1-4419-6351-2 8,c© Springer Science+Business Media, LLC 2010

238 8. Theoretical and Computational Approaches to Biomolecular Structure

Chapter 8 Notation Table (continued)

SYMBOL DEFINITION

Etor torsional (dihedral) angle energyELJ Lennard-Jones energyKijk bond angle force constantMF dimension of the Fock matrixN number of atomsNe number of electronsRn Euclidean space of dimension nSij bond length force constantSB set of bondsSBA set of bond anglesSDA set of dihedral anglesSNB set of atom pairs computed in the nonbonded energyT temperatureVn torsional barrier height of periodicity nε dielectric functionθi or θijk bond angle (e.g., θdha for donor–hydrogen· · ·acceptor

sequence in a hydrogen bond)τi or τijkl dihedral or torsion angleΔt timestep

Science says the first word on everything, and the last word onnothing.

Victor Hugo (1802–1885).

8.1 The Merging of Theory and Experiment

8.1.1 Exciting Times for Computationalists!

Computational techniques for exploring three-dimensional (3D) structures of nu-cleic acids and proteins are now well recognized as invaluable tools for revealingdetails of molecular conformation, motion, and associated biological function.Over a decade ago, the theoretical chemist Henry Schaefer declared: “It is clearthat theoretical chemistry has entered a new stage . . . with the goal of being noless than full partner with experiment” [1092].

Commenting on the two 1998 Chemistry Nobel Prize awardees in quantumchemistry — Walter Kohn of the University of California, Santa Barbara, andJohn Pople of Northwestern University — a reporter in The Economist wrote: “Inthe real world, this could eventually mean that most chemical experiments are

8.1. The Merging of Theory and Experiment 239

conducted inside the silicon of chips instead of in the glassware of laboratories.Turn off that Bunsen burner; it will not be wanted these ten years” (17 October1998)!

It remains to be seen how effective “in silico biology” will be, but clearly anew enthusiasm is stirring in the molecular biophysics community in the wake ofmany methodological and technological improvements. The following categoriesreflect improvements that lend computation a stronger basis than ever before:

1. Improvements in instrumentational and experimental techniques[1043]. Rapid advances in sequencing and mapping genomes are beingmade, as discussed in Chapter 1, making available an enormous amountof biological data requiring analysis [83]. Many procedures in biomo-lecular NMR and electron spectroscopy [361, 908, 1184], including 4DNMR [1433] and X-ray crystallography [110, 994, 1224], are acceleratingthe rate and improving both the accuracy and scope of structure determi-nation; much of this progress was stimulated by the structural genomicsinitiatives. And newer microscopic techniques such as scanning tunnel-ing and cryo [61, 189, 336, 361, 685, 1077, 1286] are being applied. Withsuch advances, structures that were considered unconquerable a decadeago are being resolved — RNAs, nucleosomes, ribosomes, ion channels,and membrane proteins, for example — although limitations still remainfor macromolecular complexes. Theoreticians can use these structureswith other experimental data and biological information as solid bases forsimulations and computer analyses.

2. New models and algorithms for molecular simulations. Improved forcefields are being developed, some simpler (e.g., coarse-grained [1117])and others more complex (e.g., polarizable, more variables [223]) thatare sophisticated versions of early coarse-grained (e.g., [749]) or po-larizable (e.g., [1344]) protein folding models. Innovative models forprotein and DNA folding and dynamics and for macromolecular com-plexes are shedding insights into important biological processes. Fasterand more advanced dynamic simulations of complex systems with fullaccount of long-range solvation and ionic effects are being used. Quantum-mechanical and classical/quantum hybrid studies of biomolecules arebecoming routine. And many enhanced sampling methods are making pos-sible simulation of a wide range of conformational states of biomolecules,including computation of reaction rates [1116, 1117].

3. The increasing speed and availability of supercomputers, parallelprocessors, and distributed computing. Faster, cheaper, and smaller com-puting platforms and graphics workstations are entering the laboratoryat reduced costs, though computer memory requirements are explod-ing. Supercomputing centers are making available very fast computingplatforms, and specialty hardware like Anton [1169] hold promise foradvancing macromolecular modeling and simulation.


4. Successful multidisciplinary collaborations. Notable examples arecollaborations between mathematicians and biologists in relating knottheory 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 andmathematical/physical scientists in propelling biology-information tech-nology, or bioinformatics; and between material scientists and biomedicalscientists regarding nanomaterials for biological applications, creating thefield of nanotechnology/bionanotechnology [442, 787, 904, 1059, 1202].1

See [635], for example, for a description of some mathematical challengesin genomics and molecular biology and [252,530] for biological challengesin the 21st century.

5. The wealth of readily available Internet and web resources, sequenceand 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, “somebodywill 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 andstop codons of the genetic code have their precise functions, or how polymerasemolecules work in the process of transcription” [812].

Some of us might be somewhat skeptical that “a little algebra” will suffice toexplain DNA’s functional secrets, but many remain confident that a large amountof computation with carefully developed models will bring us closer to that goalin the not-too-distant future.

8.1.3 Chapter Overview

In this chapter, we introduce molecular mechanics from its quantum-mechanicalroots via the Born-Oppenheimer approximation. Following a brief overviewof current quantum mechanical approaches, we discuss the three underlying

1Nanotechnology is an emerging science of creating functional materials, devices, and systemson the basis of matter at the nanometer scale, including macromolecules, and the exploitation of novelproperties 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 2000issue of Science, volume 290, highlighting issues in nanotechnology; the September 2001 specialissue of Scientific American devoted to ‘The Science of the Small’; and recent reviews on theapplication 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 andaptamers [787].

2Caution 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 debateabout whether the explosive growth in electronic communication has made life better or worse, butthere’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, andtransferability. We then describe the choices that must be made in formulating thepotential 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) Foundationsof Molecular Mechanics (MM)

Quantum mechanical methods are based on the solution of the Schrodinger equa-tion [742, 978]. This fundamental approach is attractive since 3D structures,molecular energies, and many associated properties can be calculated on the basisof fundamental physical principles, namely electronic and nuclear structures ofatoms and molecules. Indeed, the quantum mechanics pioneer Paul Dirac is be-lieved to have expressed the sentiment that the Schrodinger equation reducestheoretical chemistry to applied mathematics [765]!

Although historically quantum calculations were practical only for very smallsystems, exciting developments in both software and hardware (computer speedas well as memory) have made quantum-mechanical calculations feasible forlarger systems, including biomolecules, with various approximations (see [465],for example; more below). See the Nobel Lecture address by John Pople for fieldperspectives [1011].

The presentation of this important area of modeling is very brief here. Forcomprehensive treatments, see textbooks by Warshel and Cramer [270,1342] andexcellent reviews [573, 628, 1163–1165, 1348, 1442].

8.2.1 The Schrodinger Wave Equation

The Schrodinger wave equation describes the motions of the electrons and nucleiin a molecular system from first principles. This equation can be written as

Hψn = Enψn , (8.1)

where the Hamiltonian operator H is the sum of the kinetic (Ek) and potential(Ep) energy of the system:

H( ˜P , ˜X) = Ek( ˜P ) + Ep( ˜X), (8.2)

where ˜P and ˜X denote the collective momentum and position vectors for all thenuclei and electrons in the molecule. The potential energyEp( ˜X) originates fromelectrostatic interactions among all the variables.

According to this description, the quantum states En (eigenvalues) form a dis-crete set, corresponding to the eigenfunctions ψn, for the system of electronsand nuclei. The Schrodinger equation thus describes the spatial probabilitydistributions corresponding to the energy states in a stationary quantum system.


Traditional electronic structure methods — important more from a historicalperspective — calculate these eigenstates associated with the discrete energy lev-els by diagonalization of the Hamiltonian matrix, of order of the number of basisfunctions. This cubic scaling is prohibitive for large systems.

8.2.2 The Born-Oppenheimer Approximation

In the Born-Oppenheimer approximation to the Schrodinger equation, themotions of the molecule are separated into two levels: electrons and nuclei.

First, only the electrons are considered as independent variables of H , whilepositions of the nuclei are assumed fixed. This is generally a good approximationbecause the nuclei — much heavier than the electrons — are typically fixed on thetimescale of electronic vibration. The resulting eigenvalues from this analysis ofelectron motion represent electronic energy levels of a molecule as a function ofatomic coordinates. These energy states are known as Born-Oppenheimer energysurfaces (BOES).

In the second level of the Born-Oppenheimer approximation, the BOES ofthe electronic ground state (Ep(X), where X is the collective position vectorof 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 behaviorof the nuclei is then investigated.

In theory, quantum mechanical treatments should be the tool of choice forreliable description of complex chemical processes, since no experimental infor-mation is needed as input. Yet, an accurate analytic solution of the Schrodingerequation is not feasible except for small molecules. Thus the equation must begenerally solved through standard approximations, which naturally reduce the ac-curacy obtained. The range of application is further limited by the computationalcomplexity required by these techniques, but advances are rapidly occurring onthis front.

Two basic quantum-mechanical approaches are used in practice: ab initio andsemi-empirical. Both rely on the Born-Oppenheimer approximation that the nu-clei remain fixed on the timescale of electronic motion, but different types ofapproximations are used. The former is more rigorous than the latter and hencemore computationally demanding. For example, in Hartree-Fock type calcula-tions, the computational requirements, including computer time and memory,scale as M2

F to M4F where MF is the dimension of the Fock matrix. This dimen-

sion corresponds to the size of the computational basis set used to approximatethe wave functions, related to the number of electron orbitals, Ne.

8.2.3 Ab Initio QM

The name ab initio implies non-empirical solution of the time-independentSchrodinger equation, or solutions based on genuine theory. However, besidesthe Born-Oppenheimer approximation, relativistic effects are ignored, and theconcept of molecular orbitals (or wave functions) is introduced.


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 Gaussianfunctions. The coefficients describing this linear combination are calculated by avariational principle, that is, by minimizing the electronic energy of the molecularsystem for a given set of chosen orbitals.

This energy (known as Hartree Fock energy) and the associated coefficients arecalculated iteratively by the Self Consistent Field (SCF) procedure with positionsof the nuclei fixed. This calculation is expensive for large systems — as computa-tion of many integrals is involved — and makes ab initio methods computationallydemanding.

In practice, the basis set for the molecular wave functions is represented incomputer programs by stored sets of exponents and coefficients. The associatedcalculated integrals are then used to formulate the Hamiltonian matrix on thebasis of interactions between the wave function of pairs of atoms (off-diagonalelements) and each atom with itself (diagonal elements) via some potential thatvaries from method to method. An initial guess of molecular orbitals is then ob-tained, and the Schrodinger equation is solved explicitly for a minimum state ofelectronic 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 qualityof the basis set. The electronic energy often ignores correlation between the mo-tion of the electrons, but inclusion of some correlation effects can improve thequality 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, theelectronic wave function is represented by a single ground-state wave functionin which the electron density is represented as the sum of squares of orbitaldensities.

DFT schemes differ by their treatment of exchange/correlation energy, butin general this class of methods offers a good combination of accuracy andcomputational requirements, especially for large systems. DFT methods are com-putationally more efficient than conventional ab initio methods that correct forelectron correlations, but have a similar scaling complexity due to the N3

e diago-nalization cost of the Hamiltonian matrix. With linear-algebra advances, however,this standard diagonalization expense can be circumvented with approaches thatyield linear scaling by localization of the electronic degrees of freedom (i.e., elec-tron density, density matrix, and orbital calculations) [282, 465, 1148, 1409, forexample] (see below).


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 insteadconstructed from a set of predetermined parameters. These parameters definethe forms and energies of the atomic orbitals so as to yield reasonable agree-ment with experimental data. Thus, most integrals are neglected in semi-empiricalmethods, and empirical parameters are used as compensation. Although good pa-rameterization is a challenging task in these approaches, and parameters are notautomatically transferable from system to system, semi-empirical methods retainthe flavor of quantum approaches (solution of the Schrodinger equation) and areless memory intensive than ab initio methods. As in ab initio methods, the qualityof 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. Solutionof the wave functions via inversion of the Fock matrix for a minimal basis set(roughly the number of electrons, Ne) requires O(M4

F ) work. While still muchbetter than classical orbital calculations that include correlations, this scaling lim-its applications to large systems. It has been noted, however, that by exploiting thesparsity of the Fock matrix — zero elements due to the rapid decay with distanceof orbital overlapping — algorithms developed in linear algebra for banded sys-tems can reduce the cost to linear scaling, i.e.,O(MF ), or to the near-linear cost ofO(MF logMF ) 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-empiricalmethods) [465, 1409] and by reformulating the Hamiltonian diagonalization as aminimization problem. The localization is achieved by using different treatmentsfor the strong intramolecular interactions and the weak intermolecular interac-tions [755]. The reformulation as an optimization problem entails constructionof an energy functional, which is minimized with respect to some variationalparameters 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 storagerequirements. See Daniels and Scuseria [282] for a comparison among severallinear-scaling approaches for semi-empirical calculations, including pseudo-diagonalization of the Fock matrix via orthogonal rotations of the molecular


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 densitymatrix via transformations applied to a function expanded in terms of Chebyshevpolynomials.

Biomolecular Applications

Such advances in complexity reduction are making possible quantum calculationson biomolecules under various simplifications. Semi-empirical methods havebeen 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 basesand base pairs [552, 755], cyclic peptides [756], and the reactive centers ofenzymes [144, 1185]). Indeed, hybrid QM/MM (quantum mechanics/molecularmechanics) methods, as first introduced in 1976 [1344] (see Figure 8.1), areroutinely used for studies of enzyme mechanisms and related aspects of proteindynamics [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 biomolecularsystems into two regions: a small region enclosing the active site that is treatedquantum mechanically, and the remaining region (including solvent and biomol-ecule), which is modeled by classical molecular mechanics force fields. The keyto 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 carefultreatment of the boundary between the QM and MM regions, either by usinghybrid orbitals for the connection or a linked atom approach.

Recent advances in the field have come from using higher-level ab initio QMHamiltonians, as first proposed by Singh & Kollman [1194] or, alternatively,empirical versions, like the valence bond method (EVB) proposed by Warshel


and co-workers [54, 408, 1342]. However, using ab initio QM Hamiltonians hasintroduced new challenges of proper sampling and computing the associated freeenergy pathway. The calculation of free energies from QM/MM simulations canbe performed by averaging over the system’s configurations via perturbationsfrom a reference surface; however, such sampling for accurate free energy evalua-tions as well as calculations of pKa values remain challenging and form an activearea 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 studyaqueous polarization effects on biological macromolecules, by comparing free energies inthe 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, andZ-DNA relative to the gas phase density. The maps indicate how the electrostatic potentialchanges when the electron density of the DNA is polarized by the reaction field of thesolvent. Another example of semi-empirical applications is the study of the active site ofan enzyme (cytidine deaminase) to delineate a mechanism for ligand attack [754].

In ab-initio applications, electronic and vibrational properties have been determinedfrom optimized geometries. Ab initio methodologies have also been applied to manynucleic-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 thetwo 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 bydeparture of a hydroxyl group with the assistance of a general acid.

The calculations have identified catalytically important residues and water moleculesat the active site of the enzyme (see Figure 8.3) and have shown that the electrostaticinteractions driven by metal cations at the active site strongly favor the first, but stronglydisfavor the second, step of the reaction. This dilemma appears to be resolved by a tailoredorganization 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 importantexample where QM/MM applications have been insightful. Understanding the fidelity ofDNA repair and replication mechanisms requires tracking the conformational and energeticpathways associated with these processes, and simulations have the potential to suggestthe 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 byphosphodiester 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


conserved aspartates in the active site. When a correct unit (i.e., C opposite a templateG) 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 otherpolymerase systems such as Dpo4 [1334, 1338] and T7 DNA polymerase [1333]. OtherQM 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 theforce-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 thenuclei, Ep(X), can be imagined as the target function in molecular mechan-ics (X represents the collective position vector for the nuclei). The electronscan be regarded as implicit variables of this potential. However, unlike quantummechanics, this potential function must be evaluated empirically.

Mechanical Molecular Representation

An underlying principle in molecular mechanics is that cumulative physical forcescan be used to describe molecular geometries and energies. The spatial conforma-tion ultimately obtained is then a natural adjustment of geometry to minimize thetotal internal energy. A molecule is considered as a collection of masses cen-tered at the nuclei (atoms) connected by springs (bonds); in response to interand intramolecular forces, the molecule stretches, bends, and rotates about thosebonds (Fig. 8.5). This simple description of a molecular system as a mechanicalbody is usually associated with a “classical” system. Remarkably, this classicalmechanics description — an appropriate characterization even as the amountof quantum-mechanical information used to derive force fields increases —works generally well for describing molecular structures and processes, with theexception 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 (crystalstructure geometries, vibrational and microwave spectroscopy, heats of forma-tion, etc.). Entropic contributions are either neglected or approximated by varioustechniques. Minimization of this function can then be used to compute favor-able regions in the multidimensional configuration space, and molecular dynamicssimulations can further explain the system’s thermally accessible states.

Early Days

As introduced in Chapter 1, the first molecular mechanics implementations dateto the 1940s, but only in the late 1960s/early 1970s did the availability of digital


Figure 8.2. The electrostatic potential surface of the electronic response density ofA, B and Z-DNA calculated by the linear-scaling electronic structure and solvationmethods of York and coworkers [638]. The electronic response density is defined asΔρ(r)= ρsol(r) − ρgas(r) where ρgas(r) is the relaxed electron density in the gas phaseand ρsol(r) is the relaxed electron density in solution (approximated by a continuumdielectric model).

computers make such calculations tractable. One of the force-field pioneers, thelate Shneior Lifson, who developed the Consistent Force Field approach withdoctoral 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]. But Lifson went on tosuggest that in the early days of the field the notions of force-field consistencyand transferability were neither always carefully treated nor well understood. Thepower of the empirical force-field approach to describe collectively properties ofrelated molecules — a description not possible by quantum mechanics — wasalso 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 resembledthe crystal form [135]. However, the average of the two computed molecularmechanics structures matched the experimental data exactly! Thus, empiri-cal calculations revealed that the regular crystal contained a distribution of


Figure 8.3. Catalytically important protein residues and water molecules at the active siteof 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 stabilizationof the transition state in either the first or second step of the reaction (e.g., sky blue, darkgolden, green, orange); oxygens are colored red, nitrogens are dark blue, and hydrogenatoms of amino acid residues are omitted. Lys345 (K345, bottom center) is the general basethat captures a proton from the substrate (PGA, 2-phospho-D-glycerate) in the first reactionstep. Glu211 (E211, top right) is the general acid that assists departure of a hydroxyl groupin the second reaction step. Residues in sky blue are hydrogen bond donors interactingwith the substrate. Those in dark golden and orange are positioned to counteract the effectsof the two magnesium ions (labeled) in the second reaction step but have small energeticeffects on the first reaction step. Residues colored green are others found to have energeticeffects on the reaction.

two conformers. This was indeed confirmed later by experimentation. Today,this theme emerges often from molecular dynamics simulations, where the spatialand temporal conformational-ensemble average corresponds to the experimentalstructure [209].


Figure 8.4. Mechanism of Grotthuss hopping chemical reaction along with keyintermediates captured during phosphoryl transfer in pol β common to the G:C and G:Asystems [1033]. Solid arrows in the scheme indicate the migration path of the proton,and the dotted lines represent the nucleophilic attack and formation of pyrophosphate(PPi) [1033]. The reaction intermediates shown are: (a) reaction state of the closed nu-cleotide-bound enzyme state; (b) de-protonation of the the O3′H to water; (c,d) protontransfers to Asp192; (e) proton transfers to Asp190; (f) proton reaches the pyrophosphateunit to obtain the final product. The colors represent: blue (D256), gold (D190), magenta(D192), yellow (dCTP), cyan (CYT: terminal DNA primer), black (the O3′H-proton),green (the O3′ oxygen, attacking nucleophile), dark green (central phosphorus), red (leav-ing O3A oxygen), and light green (Magnesium). The oxygen and hydrogen atoms ofwater molecules are in gold and white, respectively. The key distances A = O3′–Pα andB = P–O3A are shown; in the reaction, A decreases from 2.90 A in the reactant state toA = 1.70 A in the product, and B increases from 1.7 A in the reactant state to 5.8 A inthe product state. The rate limiting step in the chemical pathway is found to be the initialde-protonation of the O3′H. The activation energy was found to be about 17 kcal/mol forG:C and ≥ 21 kcal/mol for G:A. The QM/MM procedure involves a novel protocol usingenergy minimization, dynamics simulations, and quasi-harmonic free-energy calculations.

8.3. Molecular Mechanics: Underlying Principles 251

nonbondedinteraction

τ

θ

b

Figure 8.5. A molecule is considered a mechanical system in which particles are connectedby springs, and where simple physical forces dictate its structure and dynamics.

8.3 Molecular Mechanics: Underlying Principles

In theory, the overall effectiveness of molecular mechanics depends on thevalidity of its three underlying principles: (1) basic thermodynamic assumption,(2) additivity of the effective energy potentials, and (3) transferability of thesepotentials.

8.3.1 The Thermodynamic Hypothesis

Does Sequence Imply Structure?

The thermodynamic hypothesis assumes that many macromolecules are drivennaturally to their folded native structure (i.e., with minimal or no intervention ofcatalysts) largely on the grounds of thermodynamics. This behavior is in contrastto other processes in the central dogma that require the crucial interplay of en-zyme machinery to lower activation barriers. Though it is important to considerthese complex factors when modeling macromolecules, the basic thermodynamichypothesis of the empirical approach appears reasonable for many biomoleculesunder study by molecular mechanics and dynamics.

Indeed, experiments have supported the intrinsic folding/entropy connectionfor many small globular proteins [52]. Thus, a strong thermodynamic force drives‘scrambled’ conformations of high free energy to the native state of low freeenergy. Folding in many cases is reversible (denatured ⇐⇒ native state) andattainable from many different configurations.

Theoretical analyses based on statistical studies of spin glasses further suggestthat the probability that a randomly synthesized protein will exhibit a thermody-namically dominant fold (i.e, the global minimum of the free energy) increasesrapidly as temperature decreases [1167].


As discussed in Chapter 3, the resilience in general of overall protein structuresto small mutations — in some cases even at critical regions, as in the Ropturn ([205] and Figure 3.10) — is another indicator of the strong thermody-namic tendency of proteins to adopt and retain native structure/folds. For higherorganizational forms of DNA, the intrinsic topology also appears to induce afolded configuration determined in large part by base composition and the ionicmedium [269].

The kinetics involved in such folding processes represent another subject ofintense interest (see [861, 1257], the discussion in Chapter 2, and Box 8.2).

Box 8.2: The Thermodynamic Hypothesis as Exemplified by The Levinthal Paradox

The concept introduced by Cyrus Levinthal in 1969 [744], which became termed the“Levinthal paradox”, is that a protein cannot find its native state by a random searchthrough all its possible conformations. This is because such an exhaustive enumerationwould in theory require eons of years, while real proteins fold on the sub-second andsecond timeframe. Levinthal thus hypothesized that well defined pathways must exist forprotein folding in nature [743].

Recently, protein-folding kinetics have been analyzed and interpreted on the basis ofsimple simulations of polypeptide models. The “old view” that emerged from Levinthal’swork is now being merged with the “new view” [313, 314, 707], which promotes manycompeting folding pathways rather than a unique pathway with well-defined intermediates.The two views can be merged when the folding pathways all share a unifying pattern ofnative contacts and can be described within a common multidimensional energy landscape[1385]. (See also discussion in Chapter 2).

While this protein folding problem is generally formulated purely in terms of “sequencedictates structure”, it is clear that in many cases the folding process in vivo is enhancedor facilitated through the external intervention of additional molecules such as chaper-ones. Such agents may stabilize intermediates, prevent aggregates of misfolded proteins,and/or reduce the activation energies effectively [561, 833]. Members of the chaperoneclass that includes the heat-shock protein hsp70 bind to newly-synthesized proteins andshort linear peptides, possibly preventing aggregates. Other chaperones assist in proteintranslocation across membranes. Members of the chaperonin class include the well-studiedGroEL/GroES complex from bacteria [387] that bind to partially-folded polypeptides andassist in the folding. Many questions are now being investigated regarding the foldingkinetics in all these systems (see [519, 1326], for example).

8.3.2 Additivity

The molecular mechanics principle of additivity assumes that the effectivemolecular energy can be expressed as a sum of potentials derived from simplephysical forces: van der Waals, electrostatic, mechanical-like strains arising from


“ideal” bond length and angle deviations, and internal torsion flexibility (rotationof two chemical groups about the bond joining them). The forces can be separatedinto local (bonded) and nonbonded (nonlocal) terms.

Local Terms

The local components for macromolecules can typically be written as:

Elocal(X) =∑

bondsi

Ebond(bi) +∑

bond anglesi

Ebang(θi) +∑

dihedral anglesi

Etor(τi),

(8.3)where summations extend over the sets of bonds {bi}, bond angles {θi}, anddihedral angles {τi} (see Fig. 8.5). Functional forms are typically harmonic (e.g.,S[b − b]2) and trigonometric (e.g., functions of cos(θ)), as discussed in the nextchapter. Note that all internal variables are functions of interatomic distances,rij . Cross terms (like E(b, b′) or E(b, θ)) are not commonly used for proteinsand nucleic acids because the associated force constants (‘off-diagonals’ in theforce-constant matrix) usually have much smaller values than those associatedwith the ‘diagonal’ terms. Such force constants, for example Sbb′ for bond/bondinteractions or Kbθ for bond/bond-angle terms, are associated with potentials ofthe form Sbb′(b − b)(b′ − b′) and Kbθ(b − b)(θ − θ), respectively, which arecommon in small-molecule force fields (e.g., [30]).

Nonlocal Terms

The nonlocal components can be written as:

Enonlocal(X) =∑

nonbonded pairs(i,j), i<j

Er(rij), (8.4)

where the functions are often rational (e.g.,∑Ni

i=1 r−m). These energies are

usually comprised of van der Waals and Coulombic contributions between non-bonded atom pairs. (Nearby atom pairs counted in the local energy may be omittedto avoid double counting).

Benefits of Separability

The natural separability into bonded (local) and nonbonded (nonlocal) terms canbe exploited in the design of minimization algorithms that use function structureto accelerate convergence (see Chapter 11). This separability is also the basis formultiple-timestep protocols for molecular dynamics that update local and nonlo-cal forces at different frequencies (see Chapters 13 and 14). This difference inforce updating is reasonable because the local terms generally change rapidly,while the nonlocal terms change more slowly, with distance and time.

While the number of nonbonded terms grows quadratically O(N2) with thenumber of atoms, the number of local terms — involving pairs, triplets, andquadruplets of bonded atomic sequences — grows only linearly with size.


This computational complexity of the nonbonded terms is especially a burden insimulation protocols for large systems that require updating for many iterations,as in macromolecular dynamics. Fortunately, some work can be reduced since thenonbonded Coulomb forces change more slowly with distance than the bondedterms, and hence can be updated less often than the local forces. Chapter 10 isdevoted to the nonbonded forces.

Multibody Potentials

In this typical energy formulation, nonbonded interactions are “effective pairpotentials” (i.e., additive two-body forces). For electrostatic interactions, for ex-ample, these effective pair potentials only reflect in some average sense the chargedistribution due to molecular polarizability. It is clear that many-body contribu-tions are important for accurate reproduction of certain molecular properties. Forexample, accurate account of dispersion forces for polar molecules (i.e., mole-cules in which the charges are nonuniformly distributed), such as water, requirethree-body interaction potentials:

Eijk(rij , rjk, rik).

These potentials can be expressed as a composite of trigonometric and rationalfunctions in terms of three bond lengths and three bond angles [131]. Indeed, theimportance of water as a solvent for proteins and nucleic acids was realized in the1970s [1344] and has prompted development of ‘polarized’ water potentials [244,1343, for example]. In practice, these models increase the number of interactionsites per water molecule.

8.3.3 Transferability

The principle of transferability assumes that potentials can be developed to in-corporate all experimental data for representative structures and then be appliedsuccessfully to the prediction of large biological molecules composed of the samechemical subgroups. This is basically a reasonable assumption since bond lengthsand bond angles tend to adopt similar values in different molecular species undernormal conditions. However, under special straining forces, such as in cycloalka-nes, these values may vary significantly from ‘ideal’ or average values. Modelingthe structures and properties of complex aromatic or conjugated systems3 alsorequires special treatment, such as using sophisticated quantum-mechanical cal-culations to obtain bond orders of conjugated bonds, which can be related tobond lengths and stretching constants, and in this way reducing the problem tomolecular mechanics ([29, 899, 1210, 1246, 1341], for example).

3Aromatic compounds are benzene-like in structure and properties. Conjugation is a structuralfeature caused by the overlap of the π orbital with other orbitals in the molecule.


Functional Variations in Geometry

In small molecules, the environment-dependence of geometric trends can bemodeled by a function rather than a constant. For example, Allinger andcoworkers devised functions for bond lengths in small-molecule force fields(MM3/MM4) to account for the electronegativity of attached substituents [1210]and for hyperconjugation [26]. The former parameterization [1210] can, for ex-ample, accurately model the shorter C–C bond in fluoroethane (C2H5F) relativeto ethane (C2H6), since the fluorine is electronegative; similarly, it can also ac-count for a longer C–O bond in an alcohol where the electropositive hydrogen isattached to an oxygen (like ethyl alcohol, CH3CH2OH), relative to an analogousmolecule in which a carbon rather than hydrogen is attached (as in dimethyl ei-ther, C2H5OC2H5). A functional dependence for reference bond lengths used forhyperconjugated systems [26] can account for bond-length trends in molecularspecies in which bond orbitals overlap and lead to resonance, like longer C–H orC–C bonds in carbonyl compounds (e.g., H–C–C=O ↔ H++ C=C–O).

Proliferation of Atom Types

An alternative approach for incorporating the environment dependence of geo-metric tendencies is to increase the number of ‘atom types’. Thus, atom typesreflect the molecular environment (e.g., aromatic carbon in a nitrogenous base)and hybridization (e.g., sp2 or sp3) [175, 805, 1359].

There are around 160 atom types, for example, in the CHARMM force fieldfor proteins (version 22) and nucleic acids (version 27) [415, 804–806]: 62 car-bons, 31 hydrogens, 28 nitrogens, 18 oxygens, 4 sulfurs, 3 phosphorus atoms,6 fluorines, one heme iron, and 7 different ions; see Table 8.1 for some examplesof these atom types and Figure 8.6 for illustrations for selected residues [805].Theproliferation of atom types in modern force fields thus attempts to improve com-patibility with experiment. Alternatively, force fields may be restricted to certainfamilies of molecules such as alkanes, amides, and carboxylic acids [185, 764].

I emphasize that individual functions should not be transferred from one forcefield to another, since the entire potential is parameterized as a whole to reproduceconsistently experimental data.

Overall, while the transferability assumption is inherent in this empiricalscience, molecular mechanics has steadily gained recognition through many im-portant contributions. Today’s force fields are excellent for deducing structuresand properties of many molecular systems, especially for small molecules us-ing specialized force fields. One advantage of theoretical calculations is that thethermal motions and lattice effects that influence crystallographically-determinedstructures may not be a problem when accurate force fields are used to predictmolecular structures and properties.


Table 8.1. Examples of atom types defined in CHARMM 22 and 27 [415, 804–806].

Atom Symbol Atom modifier

Carbon C polar (carbonyl, peptide backbone)CA aromaticCC carbonyl (Asn, Asp, Gln, Glu)CPT inter-ring in tryptophanCP1, CP2, CP3 special tetrahedral, in prolineCT1 aliphatic sp3 in CHCT2 aliphatic sp3 in CH2

CT3 aliphatic sp3 in CH3

CN1 nucleic acid carbonyl carbonCN3 nucleic acid aromatic carbon

Oxygen O carbonylOC carboxylateOH1 hydroxylON6 nucleic acid deoxyribose ring oxygenON6B nucleic acid ribose ring oxygen

Nitrogen N prolineNH1 peptideNH2 amideNN1 nucleic acid amide nitrogenNN2 nucleic acid protonated ring nitrogen

Hydrogen H polarHA nonpolarHP aromaticHS thiolHN1 nucleic acid amine protonHN2 nucleic acid ring nitrogen proton

8.4 Molecular Mechanics: Model and EnergyFormulation

In practice, the challenge and success of molecular mechanics rely on botheffective formulation of the potential energy function and the application ofsuitable search algorithms (e.g., multivariate minimization, sampling, dynamicsimulations). With the well known difficulties of large-scale minimization (seeChapter 11) and the timestep problem in dynamics (Chapters 13 and 14), the struc-tural outcome — and hence biological implications of any calculation — dependson the combination of modeling and algorithmic techniques employed. Indeed,

8.4. Molecular Mechanics: Model and Energy Formulation 257

C

CNH

C

C

CH

C

C

HC

CH2

N Ciα

HAH

Ri

H C

O

OH CT1

HB

Ri

C

O

NH1

H

CH (CH3)2

CH3CT3

HA

HA

HA

CT1

CT3

HA

CT3

HA

HA

HA

HA

HA

HA

CH2OHCT2

HA

HA

OH1 H

CH2SH CT2

HA

HA

S HSCH2 C

O

NH2CT2

HA

HA

CC NH2

H

H

O

CP1

N CP3

CP2

CP2HA

HA

C

BH

HA

HA

HAO

HA

H

H

HH

H

CH2CA

CA CA

CA

CACAHP HP

HP

HPHP

CT2

HA

HA

H H

HCY

CANY

CPT

CPT

CACA

CACA

CT2

HA

HA

HP

HP

HP

HP

H

HP

Polypeptide Building Block

Alanine R:

Serine R:

C

HNCH2

CH2

H2C

Proline R:

Valine R:

Phenylalanine R:

H

COO

Asparagine R:

Typical Nomenclature

Tryptophan R:

Cysteine R:

_

Figure 8.6. Examples of atom types as used in polypeptides in the CHARMM program[805, 806].


some strengths and weaknesses of molecular mechanics have been debated openlyin a series of papers [455, 668, 1068, 1069]; see also Homework Assignment 7.Some valid questions are:

• How accurate are quantum-mechanically derived partial charges?

• How do Cartesian and dihedral-angle representations affect results?

• Is it appropriate to introduce arbitrary scale factors in energy coefficients toenhance agreement with experiment?

The cautions and possible pitfalls of force field derivations and parameter choicesare thus important to emphasize, even for state-of-the-art force fields.

In formulation of the energy, three basic and important decisions are involved:(1) representative configuration space, (2) functional form, and (3) numericalvalues for the parameters. We discuss each in turn.

8.4.1 Configuration Space

A Question of Size

The atomic configuration4 space for a molecular system consists of 3N − 6degrees of freedom, where N is the number of atoms. (Six degrees of freedomare removed for rigid-body translation and rotation invariance of the energy).Thus, the number of degrees of freedom for proteins and nucleic acids typicallyranges in order from 103 to 104. With explicit representation of water mole-cules, this number rises by at least an order of magnitude (see, for example,solvated-macromolecular system sizes in Table 1.2 of Chapter 1).

The complete set of 3N − 6 degrees of freedom can be described directly bylists of Cartesian coordinates for all the atoms in the system. Alternatively, internalvariables such as bond lengths, bond angles, and dihedral angles may be used; thisinternal representation has been quite successful for the study of proteins [1068,for example].

Other representations have been used for biomolecules to reduce this numberof variables. Reductions are possible by fixing bond lengths and angles and work-ing in dihedral angle space, or by restricting energetic pathways to approximateformulations. However, these representations do not usually lead to enhancedefficiency unless the work involved in the nonbonded computations — the realcomputational ‘bottleneck’, see Chapter 10 — is reduced significantly.

4Configuration, a more general term than conformation, is often used by mathematicians todescribe the shapes of objects in space. The more chemical term conformation often refers toconfigurations that differ from one another through rotations of groups of atoms about the bond con-necting them (i.e., dihedral angles). Note, however, that in organic chemistry a configuration refersto stereoisomers, molecules with different connectivity arrangements which cannot be converted intoone another through rotations about bonds.


The Pseudorotation Description

An example of a parameter reduction approach is the pseudorotation pathdeveloped for nucleic acids [35,271,750]. This energetic path, initially developedfor the hydrocarbon cyclopentane [766] and later extended to ribose and deoxyri-bose sugars, constrains the energy of five-membered sugar rings to a wavelikemotion from a mean plane. This plane can be defined in various ways by po-sitions of the five skeletal ring atoms. See Chapter 5 for a discussion of thesedescriptions in the section on furanose conformations.

While conceptually simple, the reduction in degrees of freedom from 9 (3·5−6)to 2 is clearly approximate. The pseudorotation approximation was noted to pro-duce anomalies in overall nucleic acid structures [937, 944], inconsistencies withring closure, and mathematical difficulties in expressing the energy derivativesin terms of the independent conformational variables. More generally, while forsome systems simplifications in the representative conformation space may be ac-ceptable, it is usually advantageous to avoid constraints altogether [1101]. (Thepseudorotation concept remains a useful analysis tool, however).

Cartesian Space

Cartesian coordinate space is most convenient for direct differentiation of theenergy in terms of the independent parameters, as realized in the early days offorce field development by Lifson and Warshel [766], and therefore applicationof efficient second-derivative Newton minimization methods (see Chapter 11).Cartesian space is also most natural for implementation of molecular dynamics,since the generated molecular trajectories

{X(t0), X(t0 + Δt), . . . , X(t0 + nΔt), . . .},where t0 is the initial time reference and Δt is the timestep (see molecular dynam-ics chapters), generally rely on recursive expressions. That is, the new positionsand velocities

X(t0 + nΔt) and V (t0 + nΔt)

are explicit functions of the previous positions and velocities

X(t0 + (n− 1)Δt) and V (t0 + (n− 1)Δt) .

With all these considerations, it is usually advantageous to enforce con-straints — if necessary — by means of penalty functions (i.e., soft constraints).Harmonic penalty functions can be used, for example, to keep bond lengths andangles close to their observed values.

8.4.2 Functional Form

Composition

The potential energyE of a molecular model is typically constructed as the sum ofcontributions from the following types of terms: bond length and bond angle strain


terms (Ebond and Ebang), a torsional potential (Etor), a Lennard-Jones potentialto model repulsion at short interatomic separations and attraction at long distances(ELJ), and a Coulombic potential among the pairs of charged particles in thesystem (Ecoul):

E = Ebond + Ebang + Etor + ELJ + Ecoul (8.5)

Ebond =∑

i,j∈SB

Sij

(

rij − rij)2

Ebang =∑

i,j,k∈SBA

Kijk(cos θijk − cos θijk)2

Etor =∑

ijk�∈SDA

∑

n

(

V nijk�

2[

1 ± cos(nτijk�)]

)

ELJ =∑

i,j∈SNB

(−Aij

r6ij+Bij

r12ij

)

Ecoul =∑

i,j∈SNB

(

qiqjε(rij) rij

)

These equations represent first approximations to the physical potentials butare nonetheless reasonable for biomolecules. In these general expressions, thesymbols SB, SBA, and SDA denote the sets of all bonds, bond angles, and dihe-dral angles. The nonbonded set, SNB, typically includes all (i, j), i < j, atompairs separated by three bonds or more. Bond and angle variables capped by barsymbols denote reference values associated with these quantities.

The 6/12 Lennard Jones potential above (i.e., attraction of form −A/r6 andrepulsion of form B/r12) is typical for large-molecule force fields because ofits mathematical convenience; a Buckingham potential (with the same functionalform for attraction but an exponential repulsion term of formB exp(−B′r)) [766]is in principle closer to the electronic structure of the atom and thus provides abetter potential fit over a broader range of interparticle distances [30].

Molecular Geometry

To be more precise, we define the analytic expressions for the geometric quanti-ties in the potential energy.

Positions and distances. For a molecular system of N atoms (possibly inclu-ding atom groups) in Cartesian coordinate space, let

xi = (xi1, xi2, xi3), i = 1, . . . , N ,

denote the position vector of atom i, and

rij ≡ xj − xi


denote the distance vector from atom i to j. Our potential energy functionE thendepends on all Cartesian variables of the atoms:

E = E(X) ≡ E(x1,x2, . . . ,xN )

whereX is the collective vector in the Euclidean space Rn of dimension n = 3N .For any vector a we denote its standard Euclidean magnitude by ‖a‖. For

convenience later in writing the potential energy function, we also denote aninteratomic vector magnitude ‖rij‖ as rij , in nonbold type.

Bond angles. A bond angle θijk formed by a bonded triplet of atoms i–j–k isexpressed as an inner (or dot) product:

cos θijk =(xk − xj) • (xi − xj)

rij rkj. (8.6)

Dihedral angles. A dihedral angle τijk� , defining the rotation of bond i–jaround j–k with respect to k–l, is expressed as (see Fig. 3.14 in Chapter 3and Fig. 8.5):

cos τijk� = nab • nbc. (8.7)

The vectors nab and nbc denote unit normals to planes spanned by vectors {a, b}and {b, c}, respectively, where a = rij , b = rjk , and c = rk�. Denoting θab andθbc as angles θijk and θjk�, respectively, we write:

cos τijk� =a × b

‖a‖ ‖b‖ sin θab• b × c‖b‖ ‖c‖ sin θbc

. (8.8)

The sign of τijk� is set by the sign of the triple scalar product a • (b× c).

Lagrange’s identity. To simplify potential energy equations and differentiation[1103], it is convenient to work with inner product expressions and use Lagrange’sidentity:

(a × b) • (c × d) = (a • c)(b • d) − (b • c)(a • d) . (8.9)

This produces the alternative expression for cos τ :

cos τijk� =(a × b) • (b × c)

[ (a × b) • (a × b) (b × c) • (b × c) ] 1/2

=(a • b)(b • c) − (a • c)(b • b)

{

[ (a • a)(b • b) − (a • b)2 ] [ (b • b)(c • c) − (b • c)2 ]}

1/2.(8.10)

According to this convention, τ = 0o defines a cis coplanar orientation for atomsi–j–k–l, τ = 180o defines a trans coplanar orientation, and a positive sign cor-responds to a clockwise rotation of the far bond with respect to the near bond(when viewed along the j–k bond).


Derivatives. First and second derivatives of these expressions are often neededfor structure minimization, molecular dynamics, and various conformational anal-yses (e.g., normal modes). The derivative expressions are tedious to derive (seeHomework Assignment 8) and care must be used to avoid singularities, but var-ious algorithmic procedures have been developed to simplify the task in practice[1103, for example].

8.4.3 Some Current Limitations

Parameterization details for each potential energy term, including functional vari-ations, are described in the next chapter. We conclude this chapter by mentioningsome general limitations of current molecular force fields:

1. Many Force Field Choices. At present, there is no “universal force field”,nor are the many force fields in use close to converging to one anotherin some sense. Essentially, users around the world develop a preferencefor one force field over another on the basis of their target application andpractical factors such as cost and convenience.

For example, the MM2/3/4 force-field family developed by NormanAllinger and coworkers is a popular choice for small molecular systems[25, 30, 696]. Protein modelers often use the CHARMM package devel-oped by Martin Karplus and coworkers [174]. Nucleic acid modelers mightprefer the AMBER program developed by the late Peter Kollman andcoworkers [204]. Harold Scheraga’s team developed the dihedral-angleECEPP family of force fields for protein modeling [58, 897]. Other forcefields are also available, such as GROMOS [950], OPLS-AA [618, 629],CFF [614], and CVFF [371]; see also [489, p. 76] and [4] for lists ofavailable force fields and associated molecular modeling packages, andTable 11.1 of Chapter 11 for minimization algorithm information for someof these programs.

A current effort is the development of programs which utilize force fieldsthat cover a wider range of chemical systems. This, however, involves atrade-off between force field accuracy and versatility [1123]. The MMFFforce field, for example, developed at Merck in the late 1990s [497–500],was designed to model a wide range of organic molecules and proteins.Still, the recent trend has been to incorporate force field segments forspecialized systems in existing modeling packages, for example to modela complex between small organic and drug-like molecules with biomo-lecular systems. The OPLS-AA force field, in particular, covers a widevariety of organic molecules. Recently, general parameters for organic mol-ecules have been developed for usage in combination with the AMBER andCHARMM force fields through easy extensions [1300, 1331].


2. Variability in Functional Form and Numerical Parameters. As detailedin the next chapter, a great deal of variability exists in the functional formsused for each potential energy term as well as in the numerical values forthe associated parameters.

3. Inclusion of Explicit Hydrogen Bonding. Some macromolecular forcefields use explicit hydrogen-bonding potentials, though these potentialswere more common in older versions that did not consider hydrogensexplicitly. Note that the strength of a hydrogen bond is determined byits geometry, which depends on the distances associated with the Donor-Hydrogen · · · Acceptor sequence and the Donor-Hydrogen · · · Acceptorangle (θdha) formed about the hydrogen atom. For a linear hydrogen bond(usually strongest), θdha has the ideal value of 180o. In the current bio-molecular AMBER and CHARMM force fields, for example, the properdependence of hydrogen bonding on distance and angular orientations isadequately treated with the electrostatic and Lennard-Jones terms.

In theory, classical electrostatic and (quantum) bonding forces can accountfor hydrogen-bonding interactions. Thus much work has gone into for-mulating appropriate hydrogen-bond potentials for small-molecule forcefields.

Hydrogen-bond potentials can be derived from crystal packing studies orquantum-mechanical calculations, or introduced to correct weak interac-tions [898]. A re-optimization of hydrogen-bond potentials for MM3 [768]based on ab initio calculations concluded that hydrogen-bond interactionsare more complex than previously thought. In that work, the hydrogenbond potential was formulated as a complex function of the Hydrogen · · ·Acceptor distance, the Donor-Hydrogen bond length, and the cosine of theθdha angle. While this [768] and earlier works treated the overall nuclearcharges, more recent work has shown that a better representation involvesplacing the center of the electron density where the lone pairs are and notwhere the nucleus is [796].

4. Electrostatic Approximations. In the simple Coulomb potential describedabove, interactions between pairs of atoms often consider only pointcharges for each atom. However, the charge distribution about each nucleusis clearly more complex, and in some cases induced dipole effects (i.e., dis-tortion of the electron distribution) are important to consider. This notion ofmodeling electronic polarizability (roughly a measure of the distortion ofan electronic charge cloud) is a continuing focus in force field design (see[223, 243, 501, 630, 970, 1335, 1343] for example).

5. Limited Use of Quantum-Mechanical Information. In theory, quantum-mechanical theory from molecular orbital techniques can be applied tosmall molecules to improve functional form and assign more accurate


parameters (e.g., to fit parameters to ab initio relative energies or to thequantum-mechanical energy surface curvature). In practice, this fitting isnot easy to perform and depends on the quality of the quantum calculations(e.g., basis set). Yet, quantum-based calculations have been used to assignthe electrostatic partial charges, and the newer generation of force fields re-lies on quantum calculations wherever possible (e.g., [497, 630, 768, 805,842]), in combination with increasingly accurate experimental measure-ments. Thus, using both experimental and quantum calculations — despiteeach technique’s limitations — can provide the best overall results.

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