The venerable deoxyribonucleic acidmolecule has not always held thespotlight. From Gregor Mendel’sfirst careful work in 1865 to the
painstaking evidence Oswald Avery, Alfred Her-shey, and Marsha Chase and colleagues obtainedfrom 1944 through 1954, we have experienceddazzling progress in our appreciation of DNAand our ability to read, interpret, and manipu-late heredity’s master molecule. Just short of ahousehold word, DNA now plays a key role inmedical diagnoses through gene markers, bio-engineering and nanotechnology constructs, his-torical analyses, crime and forensics, and familylineage verifications, to name a few.
As we embark on a new millennium, emerginggenomic research areas seek to characterize geneproducts and relate them among species, and ex-pand our interest beyond a single molecule to in-tegrated cellular structures and functions. At leasttwo features are key to achieving these importantgiant leaps of genome integration in the comingdecade. The first is a better ability to compute the
3D structures of biomolecules from the primarysequence (amino acids in proteins and nucleotidesin nucleic acids). The second is the developmentof efficient computational technologies andstrategies to analyze sequences, structures, andfunctions. We rely crucially on such tools to ex-tract knowledge from the wealth of emergingdatabase information on biomolecules.
Although attention has focused on proteinstructure and folding (how single-strandedpolynucleotides fold back on themselves to formcomplex 3D molecular architectures), analogousproblems in DNA and its cousin ribonucleic acid(RNA) are at least as important and perhaps evenmore challenging. Unlike the relatively compactstructure of globular proteins, DNA has manylevels of structural hierarchy, from length scalesof nanometers for several base pairs to microm-eters for several thousand base pairs. Crucial forrecognition by proteins, DNA’s sequence-de-pendent behavior on the base-pair level must beanalyzed. The study of large-scale DNA foldingon the thousand-base-pair level is also of greatinterest because of its importance to the pack-aging of the genome into chromosomes and theassociated biological regulation processes.
Given these broad goals, studying DNA begsfor multidisciplinary collaborations that involvenot only chemists and biologists but also mathe-maticians and other physical scientists. In this ar-ticle, we describe some of the computational
38 COMPUTING IN SCIENCE & ENGINEERING
COMPUTATIONAL CHALLENGES INSIMULATING LARGE DNA OVERLONG TIMES
Simulating DNA’s dynamics requires a sophisticated array of algorithms appropriate forDNA’s impressive spectrum of spatial and temporal levels. The authors describecomputational challenges, solution approaches, and applications that their group hasperformed in DNA dynamics.
C O M P U T A T I O N A LC H E M I S T R Y
TAMAR SCHLICK
New York University and Howard Hughes Medical Institute DANIEL A. BEARD, JING HUANG, DANIEL A. STRAHS, AND
challenges of simulating DNA’s dynamics, focus-ing on the large-scale and long-time modelingwork in our group. These approaches incorpo-rate chemistry and biology as well as elements ofmathematical topology and geometry, elasticitytheory, mechanics, and scientific computing.
The DNA molecule and its inherentflexibility
The classic DNA double helix that FrancisCrick and James D. Watson described in 1953 isa flexible ladder-like structure of two intertwinedpolynucleotide chains running in anti-parallelfashion. The nucleotide building block consistsof sugar (deoxyribose), phosphate, and baseunits. One strand runs from the C5′–OH groupof the first sugar to the C3′–OH group of thelast, while the complementary strand runs fromC3′–OH of the first sugar group’s partner to thecorresponding C5′–OH end of the last base. Theladder rails comprise alternating sugars andphosphates, and each ladder rung is a nitroge-nous base pair held together by two or three hy-drogen bonds. Adenine (A) often pairs withthymine (T), and guanine (G) frequently pairswith cytosine (C). The spaces formed betweenthe helical backbone and the imaginary cylinder
that encloses the DNA are termed major and mi-nor grooves; they have different dimensions be-cause of the sugar-based linkages’ asymmetrywith respect to the base-pair plane (see Figure1).
We use standard atom and dihedral-angle la-beling schemes for nucleic acids. The sequenceof nitrogenous bases in the 5′ to 3′ strand speci-fies the DNA’s composition; thus, the sequence5′ TATAAAAG 3′ implies the complementarystrand 5′ CTTTTATA 3′. Besides A–T and G–Cbase pairs, researchers have observed many otherhydrogen-bonding patterns for normal and mod-ified bases, especially in RNA molecules. (RNAshave uracil (U) instead of thymine, and ribose in-stead of deoxyribose.) Other references provideexcellent introductions to DNA structure.1,2
DNA’s 3D structure depends on many factors:base composition, environmental conditions(such as relative humidity and salt concentration),and the presence of other molecules that inter-act with DNA (such as proteins or drugs). As inproteins, the DNA sequence contains subtle in-formation on local variations that can becomecollectively pronounced over large spatial scales.Sequence-dependent variations are manifestedby rotational and translational deformations fromideal helical orientations (in which the base pairs
Figure 1. (a) The TATA-box binding protein (TBP), bound to wildtype adenovirus DNA (with central 5’ TATAAAAG 3’sequence), whose coordinates are available from the crystal structure. The distorted DNA element from this co-crystal isrotated to highlight its 90o bend. (b) The molecular-dynamics simulation cell (hexagonal prism) of the AdMLP DNA element. The phosphate-neutralizing sodium ions are yellow and the water molecules are faint red-and-white sticks. Themajor (Mgr.) and minor (mgr.) grooves are also shown. (c) The two computed MD-ensemble averages of the TATA box-containing DNA systems of 14 base pairs, the wildtype sequence (WT), and its single base pair variant 5’ TAAAAAAG 3’(A29, an adenine-rich DNA sequence known as an A-tract). The TATA box is indicated in blue (A29) or red (WT), and theglobal helical axes are illustrated for each system.
TBP/DNA complex
TATA-box DNA
mgr.(minor
groove)
Mgr.(major
groove)
MD 14-bp system Bending in TATA elements
A29 WT
(a) (b) (c)
40 COMPUTING IN SCIENCE & ENGINEERING
are all perpendicular to the global helical axis).Average roll and tilt rotations—deformationsalong the long and short base-pair axes, respec-tively—are generally a few degrees in protein-free DNA but can be more pronounced for DNAbound to proteins (see Figure 1). Twist—the ro-tation along the global helical axis from one basepair to the next—exhibits a range of values belowand above the average of 34° associated with the10.5 base pair/turn repeat of canonical B-DNAin solution. (B-DNA is the classic, most com-monly observed structural form of DNA.) Othertranslational deformations from the idealizedstructure identify the base pairs’ locations with re-spect to the global helical axis.1 Since the two hy-drogen-bonded base pairs may themselves devi-ate from planarity—we also observe a nonzeropropeller twist angle, which can be large as 20° ormore in certain sequence environments.
As an example of sequence effects, considerthe intrinsically curved DNA in Figure 2 that re-sults from adenine-rich sequences in which fiveor six consecutive adenines are phased with thehelical repeat (A-tracts), as Donald Crothers andhis colleagues first discovered in the early1980s.3 The global helical curve’s overall bend-ing is not large on the dodecamer level (11°), butit is pronounced when the sequence pattern, andthus bending propensity, repeats. Figure 2 alsoshows two other lengths of DNA—linear DNAof 1.2 kbp and supercoiled DNA of 12 kbp (kbpis thousands of base pairs). From the figure, wecan view short DNA as a relatively stiff and
straight rod, while large DNA resembles flexi-ble polymers undergoing Brownian motion.
Besides the sequence’s profound effect onDNA structure, the molecule’s architecture as awhole—handedness, helical geometry, and soon—is sensitively affected by the environment.For example, the canonical B-DNA Crick andWatson described was deduced from X-ray dif-fraction analyses of the sodium salt of DNAfibers at 92% relative humidity. Another right-handed form of DNA—now termed A-DNA—emerged from early fiber diffraction data at themuch lower value of 75% relative humidity. Thisalternative helical geometry is prevalent in dou-ble-helical RNA structures. The peculiar left-handed DNA helix, termed Z-DNA for itszigzag design, was discovered in the 1970s inC–G polymers at high salt concentrations. Its bi-ological significance remains uncertain, but evi-dence suggests that the conversion from B to Z-like DNA acts as a genetic regulator.
Beyond these three canonical helical forms,we now recognize numerous variations inpolynucleotide structures—both helical andnonhelical forms—duplexes, triplexes, quadru-plexes, as well as parallel DNA, and hybrids ofRNA, DNA, and other polymers.4 Still, B-DNAis thought to be the dominant form under phys-iological conditions. One reason for its preva-lence is that the B-DNA helix can smoothlybend about itself to form a (left-handed) super-helical structure (plectoneme, or interwoundstructure) with minimal changes in the local
Figure 2. Models of DNA at four different length scales: (a) an A-tract dodecamer with an overall curvature of 11o, (b) amodel of 120 base pairs of a phased A-tract sequence, (c) linear DNA of 1.2 kbp, and (d) supercoiled DNA of 12 kbp. Ourcomputed dodecamer by all-atom molecular dynamics served as the model for constructing the 120-base-pair system;the larger linear and supercoiled structures are representative of the thermal equilibrium ensemble, as generated byBrownian dynamics simulations. The curve for the long DNA represents the double helix.
5’
C
G
C
G
A
A
A
A
A
A
C
G
3’(a) (b) (c) (d)
NOVEMBER/DECEMBER 2000 41
structure (see Figure 2d). This property facili-tates distant interactions in long DNA, thepackaging of long stretches of genomic DNA inthe cell—by promoting volume condensationand protein wrapping—and template-directedprocesses such as replication and transcriptionthat require the DNA to unwind.5 (The genomecontent [in total base pairs] varies from organ-ism to organism but roughly increases with thenumber of different cell types. For example,bacterial genomes have approximately 106 to107 base pairs, but mammals contain approxi-mately 109 base pairs. Because the eukaryoticnucleus size is approximately 5 µm [also the cellsize in prokaryotes], a value much smaller thanthe length associated with that amount ofstretched DNA, five orders of magnitude ofDNA condensation must occur.)
DNA’s two levels of resolution
Two levels of DNA structure form the centralfocus of molecular-simulation research: nu-cleotide (or base pair) level and kilobase pairlevel. The former involves the study of a dozenor so base pairs, focusing on sequence effects andlocal interactions between DNA and proteins orother biomolecules. The latter involves long cir-cular or linear DNA, focusing on global struc-ture and folding kinetics related to biologicalprocesses such as site-specific recombination.6,7
High-resolution methods such as nuclear mag-netic resonance and crystallography for struc-ture determination guide atomic-level models.Lower-resolution techniques such as gel elec-trophoresis and electron microscopy provide in-formation on supercoiled DNA.
MD applicationsAn example in the first application area is in-
trinsically bent, adenine-rich DNA studied withall-atom molecular dynamics (MD). Simulationsproduce insights into the controversial relation-ship between crystallographic and solution dataof DNA A-tracts and the forces that stabilizebending. Specifically, research supports prefer-ential bending of A-tracts into the minorgroove,8,9 and consolidates experimental obser-vations concerning the departure of some crystalmodels from this orientation.10
MD simulations of protein-binding DNA se-quences that vary by a single base pair from eachother have helped interpret experimental data11,12
regarding the relation between the sequence ofthe DNA promoter and the biological transcrip-
tional activity of DNA–protein complexes.Specifically, many groups13–15 have simulatedshort DNA segments called TATA elements thatbind to the TATA box-binding protein (TBP);this binding is a prerequisite for transcription ini-tiation in eukaryotes.11 Significantly, protein bind-ing imposes a large distortion on DNA. However,the protein succeeds in inducing this enormousdeformation because of the DNA’s incisive coop-eration: evolution has apparently selected theTATA box element 5′ TATAAAAG 3′ found inadenovirus because of its inherent flexibility.16
Our recent simulations of 13 single base pairTATA variants16 have revealed several featuresof this sequence-dependent deformability:
• the preferred TATA sequence bends flexiblyinto the DNA’s major groove, commensuratewith the protein deformation (Figure 1);
• optimal backbone shielding by counterionssupports this bending;
• a disordered water–DNA interface further fa-cilitates this motion and thus TBP binding; and
• specific local motions at the TATA ends are as-sociated with high-activity sequences.
Intrinsic curvatureAmong the many questions addressed by com-
putational scientists studying DNA are the ef-fects of intrinsic curvature on DNA conforma-tion,17 DNA site juxtaposition18 (the close spatialapproach of linearly distant regions), and chro-matin folding.19,20 DNA site juxtaposition bringstogether in space linearly distant DNA seg-ments. Many reactions such as site-specific re-combination and transcription depend on sucha spatial approach; in some cases, this interac-tion only occurs if the DNA is supercoiled. Sim-ulations help us understand the reasons for thisrequirement, the mechanisms involved in juxta-position, and the dependence of site juxtaposi-tion on the level of DNA superhelicity, salt con-centration, site separation, and DNA length (seethe “Site juxtaposition kinetics” sidebar).
Modeling chromatin folding involves studyingthe dynamics of the nucleoprotein complex thatcompacts the genomic material in eukaryoticcells. Dynamics of this spool-like complex (madeof DNA wrapped around histone protein cores)plays a key role in regulating basic cellularprocesses such as chromosomal condensation andreplication. The 11-nanometer nucleosome coreparticle’s crystal structure from 199721 was a tourde force of structural biology, but how higher-order forms are organized remains a mystery. In
particular, elucidating the details of the transitionbetween the more open and more compact struc-ture will help us better understand transcriptionalregulation and DNA packaging (see the “Chro-matin folding simulations” sidebar).
Modeling challenges The different nature of these levels and associ-
ated problems requires different computationalapparatuses. Namely, the nucleotide level is usu-ally investigated with all-atom molecular me-chanics and dynamics protocols,6 while the kilo-base pair level is studied by macroscopic modelsinvestigated with Monte Carlo, Brownian, andLangevin dynamics.22 The all-atom approachfaces the challenge of large system sizes in fully
Site juxtaposition kineticsOur recent investigations into supercoiled DNA
dynamics have focused on understanding the jux-taposition mechanism of linearly distant sitesalong the DNA contour and how variations in thesuperhelical density and salt concentration affectthe process. Juxtaposition of linearly distant sites,which occurs on the time scale of milliseconds, isrequired for a variety of processes, including site-specific recombination and certain transcriptionalevents. However, current experimental techniquescannot probe the kinetics involved in greatdetail.1,2 Surprisingly, we find that the site juxta-position mechanism depends critically on the saltconcentration. At low salt, we identify randomcollision as the dominant mechanism, but at highsalt, juxtaposition proceeds by slithering3 (the ran-dom reptational, bidirectional movements of thetwo opposite segments along the superhelicalaxis) coupled to branching rearrangements of theDNA supercoil.
Specifically, our simulations show that at lowsalt concentrations and at low DNA superhelicaldensities, the DNA structure is more irregular.Such loose supercoiling enhances flexibility—theDNA structure undergoes large global superhelicaldistortions. Because supercoiling increases theequilibrium probability of juxtaposition two ordersof magnitude,4 at low salt concentrations we ob-serve an increase in site juxtaposition rates withsuperhelical density commensurate with theincrease in juxtaposition probability.
In contrast, at physiological concentrations (rela-tively high salt), the site juxtaposition rate is deter-mined by the combined effects of slithering,branch creation and deletion, and interbranch col-lisions, and is not sensitive to the changes in thesuperhelicity.5 Here, circular DNAs adopt regular, tightly in-terwound superhelical structures, usually branched for DNAlarger than 3 kbp (see Figure A). In such branched DNAstructures, these three processes combine to accelerate the site juxtaposition process.
Theoretical analyses6 of site juxtaposition, assumingpurely reptational slithering, reveal an average collision
time that scales with L3. More realistic motions that involvebranch creation and deletion along with slithering result7 injuxtaposition times that scale approximately as L2. Our sim-ulations suggest a near-quadratic length dependence ofsite juxtaposition rates at high salt conditions.5 Hence, atphysiological conditions, the juxtaposition rate is not sensi-tive to the changes in the equilibrium juxtaposition proba-
42 COMPUTING IN SCIENCE & ENGINEERING
Figure A. Brownian-dynamics snapshots of 3 kbp circular DNA with superhelicaldensity σ = –0.06 under both (1) low (0.01 M) and (2) high (0.20 M) salt conditions.Discrete 30 base-pair segments model the DNA.
0.0 ms
(1)
0.2 ms 0.4 ms 0.6 ms
0.8 ms1.0 ms 1.2 ms 1.4 ms
1.6 ms 1.8 ms 2.0 ms 2.2 ms
0.0 ms 0.2 ms 0.4 ms0.6 ms
0.8 ms 1.0 ms 1.2 ms 1.4 ms
(2) 1.6 ms 1.8 ms 2.0 ms 2.2 ms
solvated models, sensitivity to force-field and sim-ulation protocol, accurate treatment of long-range electrostatic interactions, and limitation ofsimulation times and hence configurational sam-pling range. The macroscopic representation islimited by model approximations, treatment ofhydrodynamic forces and ionic effects, and prop-agation methods. Both levels are thus challenged
by fundamental model assumptions and largecomputational requirements.
Table 1 shows typical setups and computationalrequirements for these two types of models. Fig-ure 3 shows the percentage of computational workfor different program components. In all-atommolecular and Langevin dynamics protocols, theiterative updating procedure for defining coordi-nates and momenta is relatively simple, even inmultiple time step (MTS) methods, and most ofthe work involves energy and force evaluation ateach time step. The most expensive part of thiscalculation involves the long-range Coulomb po-tentials and associated forces. Although this taskhas largely been accelerated with fast adaptivemultipole or Ewald-type methods that approachnear-linear complexity with size N (typically O(Nlog N)), the time step limitation (femtosecond-order time steps) dictates millions of steps to spana relatively short time in a biomolecule’s life. MTSmethods for both Newtonian and Langevin dy-namics combined with efficient implementationson parallel platforms have also helped alleviate thiscomputational burden,23–26 letting us simulatelarger system sizes over longer times. Recent workon alleviating resonance instabilities by the LN al-gorithm27,28 has extended time step values to wellover 10 fs for the slow forces, with net speedupsas indicated in Table 1 and Figure 3. Still, the com-putational requirements for atomic-level detail re-main large. Currently, we can only accomplishlonger simulation times for small systems withsimplified long-range force treatments and dedi-cated supercomputing time.29
In Brownian-dynamics (BD) simulations ofsupercoiled DNA, the propagation equationsthat dictate each set of coordinates are fairlycomplex when torsional motion and hydrody-namic forces are involved—elaborations on thestandard Ermak and McCammon scheme30 arenecessary.31 Prescribing the motion essentiallyrequires a prediction–correction step becauseeach discrete segment’s rotation is coupled to themovement of the associated bead’s local coordi-nate frame. Incorporating hydrodynamics effectsentails solving a dense linear system that involvesthe configuration-dependent hydrodynamic ten-sor to define the random force at each step. Aswe discuss later, this task is generally accom-plished by a Cholesky factorization, which in-creases as O(N3) with system size. Although theelectrostatic forces dominate the computationaltime for small and moderately sized DNA sys-tems, the work associated with hydrodynamicsdominates for large systems (see Figure 3). Here
bility, as deduced from Monte Carlo work.Figure A illustrates the juxtaposition kinetics
at these two salt conditions. At the lowconcentration (series A1), juxtaposition of twosites (indicated by black and green spheres)proceeds through a rearrangement of theglobal structure. At the higher salt concentra-tion (series A2), the intertwined structure isfairly regular, and juxtaposition proceeds byslithering and branch sliding. In particular, athree-branch structure remains fairly stable athigh salt while thermal motions result in moredrastic rearrangements for the low-salt case. Athigh salt, the highlighted beads graduallyslither toward one another and remain in closeproximity from 1.4 ms to 2.0 ms, while at lowsalt the juxtaposition event (occurring at 2.0ms) is short-lived.
References1. C.N. Parker and S.E. Halford, “Dynamics of Long-Range
Interactions on DNA: The Speed of Synapsis during Site-
Specific Recombination of Resolvase,” Cell, Vol. 66, No.
4, Aug. 1991, pp. 781–791.
2. R.B. Sessions et al., “Random Walk Models for DNA
Synapsis by Resolvase,” J. Molecular Biology, Vol. 270,
No. 3, July 1997, pp. 413–425.
3. K.R. Benjamin et al., “Contributions of Supercoiling to Tn3
Resolvase and Phage Mu Gin Site-Specific Recombination,”
J. Molecular Biology, Vol. 256, No. 1, Feb. 1996, pp. 50–65.
4. A.V. Vologodskii and N.R. Cozzarelli, “Effect of Super-
coiling on the Juxtaposition and Relative Orientation of
DNA Sites,” Biophysical J., Vol. 70, No. 6, June 1996, pp.
2548–2556.
5. J. Huang, T. Schlick, and A. Vologodskii, “Dynamics of
Site Juxtaposition in Supercoiled DNA,” to be published
in Proc. Nat’l Academy of Science USA, 2000; schlick@
nyu.edu.
6. J.F. Marko and E.D. Siggia, “Statistical Mechanics of Su-
percoiled DNA,” Physical Rev. E, Vol. 52, No. 3, Sept.
1995, pp. 2912–2938.
7. J.F. Marko, “The Internal ‘Slithering’ Dynamics of Super-
coiled DNA,” Physica A, Vol. 244, 1997, pp. 263–277.
matin, the nucleoprotein complex that compacts the ge-nomic material in eukaryotic cells. The chromatin fiber iscomposed of a chain of globular histone protein octamersconnected by linker DNA segments. Continuous with thelinker DNA is a 150-base-pair left-handed supercoil of DNAthat is wrapped around each octamer. The entire repeatingunit of a core particle (octamer plus wrapped DNA) andlinker DNA is denoted the nucleosome. Chromatin con-
denses into a compact form, which is a critical regulator oftranscription and replication.
This system’s size demands a biophysical description inthe spirit of polymer-level models of DNA.1–3 The core pro-tein complex, however, is much less regular in terms ofshape and charge distribution than simple DNA. To modelthe electrostatic interactions in this complex, we developedan algorithm for optimizing a discrete N-body Debye-Hückel potential to match the electric field predicted by thenonlinear Poisson-Boltzmann equation.4 The nucleosome
Figure B. Chromatin modeling based on our dinucleosome model of two electrostatically charged core particles connected by an 18-nmlinker DNA modeled as an elastic wormlike chain (top left corner). The dinucleosome folding trajectory in part (1) reveals spontaneousfolding into a condensed structure in a few nanoseconds. We refined the 30-nm fiber (48 nucleosome units) constructed as a solenoid fromthe dinucleosome fold motif using Monte Carlo methods to obtain the structure in part (2).
0 ns
(1)
Nucleosome cores
1 ns 2 ns
Linker DNA
H3 tail
(2)
3 ns 4 ns 5 ns
6 ns 7 ns 8 ns
30 n
m
Cha
rge
[e]
10
0
–10
NOVEMBER/DECEMBER 2000 45
we show that an alternative algorithm (whichMarshall Fixman proposed over a decade ago32)dramatically reduces computational times forBD simulations of long DNA. Other recent ap-plications are described elsewhere.33,34
The elastic model for long DNA
The elastic-rod approximation has provenvaluable for studying superhelical DNA’s globalfeatures (such as long range and time flexibility).Using ideas from polymer physics, we can char-acterize long DNA by its contour length L and abending rigidity A. We can relate the DNA’smean square displacement ⟨R2⟩ to the persistencelength pb, which is essentially the length scale onwhich the polymer directionality is maintained:
⟨R2⟩ = 2 pb L . (1)
Thus, for lengths � pb, we can consider theDNA to be straight, but for lengths � pb, a bet-ter description is a bent random coil undergo-ing Brownian motion. This length-dependentflexibility is apparent from Figure 2, whichshows DNA on length scales much smaller andmuch greater than pb. The persistence length ofDNA in vivo is approximately 50 nm, or ap-proximately 150 base pairs at physiologicalmonovalent salt concentrations. The persistencelength is also related to the bending force con-stant A as
A = pb kB T (2)
where kB is Boltzmann’s constant and T is thetemperature. Thus, the floppy polymer writhesthrough space as a wormlike chain, with thebending rigidity—which tries to keep the DNAstraight—balanced by thermal forces—whichtend to bend it in all directions.
We can write the elastic-deformation energyas a sum of bending and twisting potentials, withbending and torsional-rigidity constants (A andC) deduced from experimental measurements ofDNA bending and twisting.35 Similar to Equa-tion 2, the torsional rigidity C is related to thetwisting persistence length ptw by
C = (ptw kB T) /2. (3)
The bending constant does not have the 1/2 fac-tor because bending involves two axial compo-nents of the deformation (roll and tilt) perpen-dicular to the global helical axis.
particles in Figure B use the 277-point chargemodel that we incorporate into a macrolevelpolynucleosome model. In this way, a biomole-cule’s atomic-level details are efficiently inte-grated into an accurate biophysical descriptionof a system too large to treat on the atomicscale. Energy parameters for the DNA (chargedensity and elasticity constants) are adoptedfrom studies of DNA supercoiling. We havetested the resulting model parameters againstavailable experimental data, such as trans-lational diffusion constants from chicken ery-throcyte polynuclesomes under varying saltconcentrations.5
In Figure B1 we plot a 4-ns trajectory repre-senting the folding of a two-nucleosomesystem at monovalent salt concentration of Cs
= 0.05 M. The N-terminal H3 tail is positivelycharged and associates with the negativelycharged linker DNA. The linker DNA adopts abent configuration. Based on the observed foldmotif for this system, we can construct largersystems, such as the 48-nucleosome fiber inFigure B2. The predicted fiber is a right-handedsolenoid with a diameter of approximately 30nm, in agreement with experimental observa-tions on chromatin. Our work continues to ex-plore the internal structure of the 30-nm fiberand to interpret folding and unfolding proc-esses associated with acetylation and phospho-rylation of the histone proteins.5
References1. W.K. Olson and V.B. Zhurkin, “Modeling DNA Defor-
mations,” Current Opinion in Structural Biology, Vol. 10,
No. 3, June 2000, pp. 286–297.
2. J.A. Martino, V. Katritch, and W.K. Olson, “Influence of
Nucleosome Structure on the Three-Dimensional Fold-
ing of Idealized Minichromosomes,” Structure with Fold-
ing & Design, Vol. 7, No. 8, Aug. 1999, pp. 1009–1022.
3. L. Ehrlich et al., “A Brownian Dynamics Model for the
Chromatin Fiber,” Computer Applications in the
Biosciences, Vol. 13, No. 3, June 1997, pp. 271–279.
4. D. Beard and T. Schlick, “Modeling Salt-Mediated Elec-
trostatics of Macromolecules: The Algorithm DiSCO
(Discrete Charge Surface Charge Optimization) and Its
Application to the Nucleosome,” Biopolymers, Vol. 58,
The bending term is proportional to thesquare of the curvature κ and the twisting en-ergy is proportional to the twist deformation:
(4)
In these equations, s denotes arc length, and the in-tegrals are computed over the entire closed DNAcurve of length L0. The DNA’s intrinsic twist rate isω0 (such as 2π/10.5 radians between successive basepairs). In addition to these bending and twistingdeformations, other components account forstretching interactions, electrostatic (screenedCoulomb in the form of Debye-Hückel), and hy-drodynamic interactions (see the sidebar “A com-putational model for supercoiled DNA”).
BD propagation algorithm andhydrodynamics
To simulate long-time trajectories of DNAmotion,36 researchers commonly use Donald Er-mak and J. Andrew McCammon’s30 BD algo-rithm. The algorithm updates particle positionsaccording to
(5)
where Xn denotes the collective position vectorfor the N particles at the nth time step (timen∆t), f n is the systematic force (negative gradi-
ent of the potential energy), D(Xn) is the config-uration-dependent diffusion tensor, and Dij is theijth entry of D(Xn). The allowable time step ∆tfor BD is typically in the range of 100 picosec-onds, orders of magnitude greater than the sub-femtosecond time steps used in all-atom MD.
The random-displacement vector Rn is in-cluded to mimic thermal interactions with thesolvent. It is a Gaussian white noise process re-lated to D with covariance structure given by
⟨Rn⟩ = 0, ⟨(Rn)(Rn)T⟩ = 2∆t D(Xn). (6)
(⟨(Rn)(Rm)T⟩ = 0 for m ≠ n.) In the BD algorithm, thediffusion tensor D defines the hydrodynamic inter-actions among the particles, as well as the correla-tion structure of the random motions. A reasonablechoice for the mathematical form of D is theRotne-Prager hydrodynamic tensor,37 which rep-resents a second-order approximation for two beadsdiffusing in a Stokes fluid. (The termin Equation 5 is also zero for this tensor.)
Obtaining this random force Rn in BD algo-rithms turns out to be the computational bottle-neck. We can compute a vector Rn with covari-ance specified by Equation 6 according to
Rn = 2(∆t)1/2Lz (7)
where z is a vector of uncorrelated random num-bers chosen from a Gaussian distribution withzero mean and unit variance (that is, ⟨zn(zm)T⟩ =δnmI, ⟨zn⟩ = 0). The matrix L comes from aCholesky factorization of Dn:
D = LLT. (8)
∂ ∂∑ D Xij ij/
Xn 1 Xn tDijX j rn
k T(Xn) f n Rn
j
B
+ = +∂
∂
+
⋅ +
∑∆
∆tD
E E E s s sB T= + = + −∫ ∫Ad
Cd
2 22
02κ ω ω( ) ( ) .
Table 1. The complexity of DNA dynamics simulations.
Resolution System and size (N) Technique and protocol Simulation range CPU performance
Macroscopic DNA, Linear DNA, 1.2 kbp Second-order BD, 10 ms 10.9 days8 base pairs per bead (150 beads) hydrodynamics, ∆t: 600 ps
Macroscopic DNA/ 48 nucleosomes Monte Carlo 1 million steps 60 daysprotein, 8 base pairs & linker DNA (240 DNA per bead & protein core beads, 48 core beads)
MD stands for molecular dynamics, LD for Langevin dynamics, and BD for Brownian dynamics. All computations are reported on anSGI Origin 2000 with 300-MHz R12000 processors. The LN scheme, named for its origin in a Langevin normal-mode approach, com-bines force splitting by extrapolation with Langevin dynamics to alleviate severe resonances and allow large outer time steps.27
NOVEMBER/DECEMBER 2000 47
The above factorization of D requires O(N3)floating-point operations and consumes most ofthe CPU’s time for BD simulations of large sys-tems. Increasing the efficiency of these hydro-dynamics calculations is the key to alleviating thecurrent limitations on the size of DNA systemsand the time scale of trajectories that we can sim-ulate using BD.
The Fixman alternative to the Cholesky fac-torization of D involves calculating y, the vectorof correlated random numbers, as
y = Sz (9)
instead of Equation 7. Here S is the square rootmatrix of D (D = S2) and not the Cholesky factorL. Fixman’s idea was to expand the vector y as a
Hydro
MD with Verlet integrator
(a)
3,000
100
75
50
25
0
4,000
3,000
2,000
1,000
N
A
BC D
E
F
10,000 40,000
CP
U (
%)
Day
s
Nonbond forceand bookkeeping
CPU/10ns
MD with LN integrator
(b)
3,000
100
75
50
25
0
4,000
3,000
2,000
1,000
N
A B C D EF
10,000 40,000
CP
U (
%)
Day
s
Nonbond forceand bookkeeping
CPU/10ns
BD with Cholesky factorization
(c)
1003 kbp
100
75
50
25
0
120
90
60
30
0
N
Force
Hydro
2006 kbp
400123 kbp
% C
PU
wor
k
Day
s
CPU/10ms
BD with polynomial expansion
(d)
1003 kbp
100
75
50
25
0
120
90
60
30
0
N
Force
2006 kbp
400123 kbp
% C
PU
wor
k
Day
s
CPU/10ms
Figure 3. The computational complexity of DNA simulations on the all-atom (top) and macroscopic (bottom) levels. The upperplots correspond to molecular-dynamics calculations, with the fraction of CPU time devoted to calculating the nonbonded energy and forces (blue line) plotted against the number of atoms, N. The plot in (a) corresponds to the standard Verlet integrator with ∆t = 1.0 fs, while the plot in (b) corresponds to the LN integrator with the time step protocol of 1/2/120 fs.The various all-atom systems correspond to lysozyme (A, 2,857 atoms, in vacuo), 12-bp A29 TATA box element (B, 11,013atoms, solvated in a hexagonal prism), 14-bp A29 TATA box element (C, 15,320 atoms, solvated in a hexagonal prism), triosephosphate isomerase (TIM) (D, 18,733 atoms, solvated in a truncated octahedral prism), TIM (E, 23,635 atoms, solvated in arectangular prism), and the wildtype TBP/WT DNA complex (F, 37,703 atoms, solvated in a rectangular prism). For the plots in(c) and (d), the fraction of CPU time associated with the hydrodynamics calculations (green lines) and with force calculations(blue line) in Brownian-dynamics simulations of large DNA is plotted versus the system size in kbp. The red curves in all plotscorrespond to the right-hand axis, the total number of days required to compute a trajectory of 10 ns for all-atom MD, and 10ms for macroscopic BD. All computations were performed on one R12000 processor of an SGI Origin 2000 with 300-MHzprocessors.
48 COMPUTING IN SCIENCE & ENGINEERING
series of Chebyshev polynomials, a calculationthat requires O(N2) operations, compared to thestandard method’s O(N3) operations.
The sidebar “Polynomial expansion for Brow-nian random force” describes computing the ex-pansion for y. The procedure requires deter-mining bounds on the maximum and minimumeigenvalues of D(Xn). (In practice, we can ap-proximate these bounds—which are required toscale the matrix for the Chebyshev expansion—by computing the eigenvalues of D(X0) and as-suming that the magnitudes do not change dras-tically.) Then, once we have determined theorder M of the expansion according to some er-
ror criterion, we expand y in terms of polynomi-als with coefficients determined for the square-root function.
The computational work required for thestandard Cholesky treatment of hydrodynamicsdominates BD simulations for large systems, asin Figure 3c. When we apply the vector polyno-mial expansion described earlier, the hydrody-namics calculations consume approximately 33%of the CPU time, regardless of system size. For a 12-kbp system, BD using the standardCholesky factorization requires twice the CPUtime as our implementation of the vector poly-nomial expansion. This acceleration is not large,
A computational model forsupercoiled DNA
Following Stuart Allison’s pioneering work,1 we can rep-resent wormlike DNA as a series of N virtual objects (orbeads) connected in a closed loop. The centers of thebeads, denoted by ri, represent a discrete polymer chain’svertices, and local coordinate unit vectors {ai, bi, ci} associ-ated with each bead describe the DNA molecule’s internalconfiguration. For circular DNA, the index i = N + 1 incideswith the first index i = 1. Euler angles {α i, βi, γi} specify therotation of the (i – 1)th to the ith coordinate system.2
The configuration-dependent potential energy ismodeled as the sum of stretching, bending, twisting, andelectrostatic interactions:
E = ES + EB + ET + EC. (A)
We compute the stretching energy ES from the sum ofsquared deviations in segment length:
(B)
where lo is the resting length of each interbead segment and lo= Lo/N, where Lo is the DNA molecule’s target length. Settingthe stretching constant to results in devia-tions in realized segment lengths of less than 1% from lo.
3,4
We calculate Equation 4 in the main text’s discrete analogfrom the set of Euler angles:
(C)
where φo is the equilibrium excess twist due to superhelical
winding: φo = 2πσ(lo/lh). Here σ is the superhelical density ofDNA, ∆Lk/Lko is a normalized linking number difference(typically around –0.05), and lh is the DNA helical repeatlength of about 3.55 nm.
Following Dirk Stigter’s work,5 we approximate the elec-trostatic energy by the Debye-Hückel potential associatedwith point charges at the centers of the beads:
(D)
where ν is the effective linear-charge density along thechain, ε is the dielectric constant of water, 1/κ is the Debyelength, and rij is the scalar distance between beads i and j.(We do not consider the j = i + 1 term here, because it iscounted in the stretching term.) For a monovalent salt con-centration of 40 mM, 1/κ = 1.52 nm and ν = –3.92 e • nm–1.(This screening parameter κ should not be confused withthe curvature symbol introduced earlier.)
References1. S.A. Allison, “Brownian Dynamics Simulation of Wormlike Chains: Flu-
orescence Depolarization and Depolarized Light Scattering,” Macro-
molecules, Vol. 19, No. 1, Jan. 1986, pp. 118–124.
2. S.A. Allison, R. Austin, and M. Hogan, “Bending and Twisting Dynam-
ics of Short Linear DNAs: Analysis of the Triplet Anisotropy Decay of a
209 Base Pair Fragment by Brownian Simulation,” J. Chemical Physics,
Vol. 90, No. 7, Apr. 1989, pp. 3843–3854.
3. D. Beard and T. Schlick, “Inertial Stochastic Dynamics: I. Long-Time
Step Methods for Langevin Dynamics,” J. Chemical Physics, Vol. 112,
No. 17, May 2000, pp. 7313–7322.
4. D. Beard and T. Schlick, “Inertial Stochastic Dynamics: II. Influence of
Inertia on Slow Kinetic Properties of Supercoiled DNA,” J. Chemical
Physics, Vol. 112, No. 17, May 2000, pp. 7323–7338.
5. D. Stigter, “Interactions of Highly Charged Colloidal Cylinders with
Applications to Double-Stranded DNA,” Biopolymers, Vol. 16, No. 7,
July 1977, pp. 1435–1448.
EC
lorijj i
ijr=
( )> +∑
−( )ν
ε
κ2
1
exp
E EAl
Cl
B To
ii
N
oi
i
N
i o
+ =
+ + −
=
=
∑
∑
2
2
2
1
1
2
β
α γ φ( ) .
h k T lB o= 1500 2, /
Eh
r r lS i ii
N
o= − −+=∑2 1
1
2( )
NOVEMBER/DECEMBER 2000 49
because the system size in terms of beads is notlarge (several hundred, see Table 1). However,we can realize greater CPU gains for larger sys-tems. The Chebyshev alternative to theCholesky factorization also opens the door toother BD protocols (such as our recent inertialBD idea)38,39 and is crucial to BD studies of finermodels, such as those that are base-pair-basedrather than bead-based. Now that the BD com-putational bottleneck is reduced to electrostaticsand hydrodynamics (O(N2) for both), fast electro-static methods help accelerate computation fur-ther, especially for the chromatin system,31 wherethe number of charges is much greater than thenumber of hydrodynamic variables or beads.
We have witnessed considerableprogress over the past twodecades in simulating the dy-namics of DNA, both on the
all-atom and macroscopic levels.6,7 It was only inthe early 1990s that we could simulate stable, fullysolvated models of DNA oligonucleotides withtraditional MD methods. Both improved forcefields and longer-range electrostatics modelingcreated these advances. Although such improve-ments continue, this success has opened the doorto investigating many of DNA’s subtle sequence-dependent properties that are key to regulatorybiological processes. The notion of DNA as a pas-sive partner to protein interactions has largely
Polynomial expansion for Brownianrandom force
To expand the vector y = Sz, we consider Chebyshevpolynomials defined over the interval [-1, 1]. The scalingfactors k1 and k2 are introduced, where
G = k1D + k2I (A)
so that the eigenvalues of G have magnitudes less than 1.We define the order-M Chebyshev expansion of the square-root matrix as
(B)
where the {am} are scalar coefficients and the {Cm} are theChebyshev polynomial functions G. The expansion for y =Sz has a similar form:
(C)
where zm is the vector Cm(G)z. We found M = 10 suitablefor our applications to achieve errors of less than 0.1% for3-kbp systems. (We use a double-precision algorithm, withmachine epsilon 10–15.)
We define the Chebyshev polynomials for the matrix ex-pansion according to the formula
Cm+1 = 2GCm – Cm–1; C0 = I; C1 = G (D)
From this, we obtain the polynomials defining the vectorexpansion for zm = Cm(G)z as
Although calculation of SM according to Equation B requires
a series of matrix–matrix multiplications of complexityO(N3), the expansion of yM defined by Equations C and E in-volves only matrix–vector multiplications, an O(N2) process.
The Chebyshev coefficients for the expansion of a func-tion g(λ) are given by
(F)
where the λj are distributed according to
(G)
In Equation F, cm(λj) represents the mth Chebyshev poly-nomial for the scalar case:
The function g(λj) is the square root function, scaled by thefactors introduced in Equation A:
. (I)
We determine the scaling factors k1 and k2 so that1
k1λmax + k2 = 1k1λmin + k2 = –1
(J)
where λmax and λmin are reasonable upper and lowerbounds on the eigenvalues of D.
Reference1. M. Fixman, “Construction of Langevin Forces in the Simulation of Hy-
drodynamic Interaction,” Macromolecules, Vol. 19, No. 4, 1986, pp.
1204–1207.
gk
kjj( )
/
λλ
=−
2
1
1 2
λ πj
jM
= ++
⋅
cos .2 1
1 2
a g c cm jj
M
m j m==∑ ( ) ( ) /λ λ
0
2
y C GM mm
M
m mm
M
ma z a z= == =
∑ ∑0 0
( ) ,
S C GM m mm
M
a==
∑0
( )
50 COMPUTING IN SCIENCE & ENGINEERING
been discarded in favor of the view of DNA as animportant influencing factor on these processes.Ongoing advances in time-step integration (seeTable 1), configurational sampling, and efficientimplementation of MD programs on parallel ar-chitectures will continue to push the capabilitiesof DNA and DNA–protein modeling toward ex-perimental time frames. (Of course, these meth-ods are general and also applicable to proteins).
The parallel studies focusing on DNA’s struc-ture and kinetics on scales much greater than itspersistence length require different algorithmictools to capture DNA’s inherent floppiness andstrong dependence on the ionic concentrationand solvation. Researchers have applied MonteCarlo and Langevin and Brownian dynamics tothese problems, but they encounter computa-tional bottlenecks too. To study kinetic processesof supercoiled DNA, which are largely unre-solvable by traditional experimental techniques,these algorithms must be accelerated and broad-ened in scope. For example, we can replace thetraditional O(N3) treatment of the random forcein BD simulation with a more economical O(N2)procedure involving Chebyshev polynomials toallow the study of much larger DNA systems ormore refined models where each bead representsa specific base pair. This finer resolution is im-portant for modeling sequence-dependent bend-ing and twisting deformations as observed ex-perimentally (hence appropriate elastic constantscan be derived). This enhanced resolution willundoubtedly develop significantly in the nextdecade. A related review of collective-variablemodeling for nucleic acids appears elsewhere.40
Ultimately, we must bridge the all-atom andpolymer-level representations, but this mergingis technically challenging. Hybrid approachessuch as those that eliminate the explicit repre-sentation of the solvent molecules through theuse of generalized Born potentials hold greatpromise.41,42 At the spectrum’s other end, intro-ducing quantum degrees of freedom through hy-brid molecular mechanics–quantum mechanicsshould broaden the scope of problems that wecan study.43 To be sure, in all these exciting stud-ies, computational scientists will continue to playa key role in advancing our understanding ofmacromolecular structure and function.
AcknowledgmentsThe work on DNA supercoiling started with Wilma Olson,and the recent work on site juxtaposition is in collaborationwith Alex Vologodskii. We gratefully acknowledge supportfrom the National Science Foundation (ASC-9157582,
ASC-9704681, BIR-9318159), the National Institutes ofHealth (R01 GM55164), and a John Simon Guggenheimfellowship. Tamar Schlick is an investigator at the HowardHughes Medical Institute. (See group papers at http://monod.biomath.nyu.edu/.)
References1. R.R. Sinden, DNA Structure and Function, Academic Press, San
Diego, Calif., 1994.
2. A.D. Bates and A. Maxwell, “DNA Topology,” In Focus, OxfordUniv. Press, New York, 1993.
3. D. Crothers, T.E. Haran, and J.G. Nadeau, “Intrinsically BentDNA,” J. Biological Chemistry, Vol. 265, No. 13, May 1990, pp.7093–7096.
4. N.B. Leontis and E. Westhof, “Conserved Geometrical Base-Pair-ing Patterns in RNA,” Quarterly Rev. Biophysics, Vol. 31, No. 4,Nov. 1998, pp. 399–455.
5. A.V. Vologodskii and N.R. Cozzarelli, “Conformational and Ther-modynamic Properties of Supercoiled DNA,” Ann. Rev. BiophysicsBiomolecular Structure, Vol. 23, 1994, pp. 609–643.
6. D.L. Beveridge and K.J. McConnell, “Nucleic Acids: Theory andComputer Simulation, Y2K,” Current Opinion in Structural Biol-ogy, Vol. 10, No. 2, Apr. 2000, pp. 182–196.
7. W.K. Olson and V.B. Zhurkin, “Modeling DNA Deformations,”Current Opinion in Structural Biology, Vol. 10, No. 3, June 2000,pp. 286–297.
8. M.A. Young and D.L. Beveridge, “Molecular Dynamics Simula-tions of an Oligonucleotide Duplex with Adenine Tracts Phasedby a Full Helix Turn,” J. Molecular Biology, Vol. 281, No. 4, Aug.1998, pp. 675–687.
9. D. Sprous, M.A. Young, and D.L. Beveridge, “Molecular Dy-namics Studies of Axis Bending in d(G5-(GA4T4C)2-C5) and d(G5-(GT4A4C)2-C5): Effects of Sequence Polarity on DNA Curvature,”J. Molecular Biology, Vol. 285, No. 4, Jan. 1999, pp. 1623–1632.
10. D. Strahs and T. Schlick, “A-Tract Bending: Insights into Experi-mental Structures by Computational Models,” J. Molecular Biol-ogy, Vol. 301, No. 3, Aug. 2000, pp. 643–663.
11. G.A. Patikoglou et al., “TATA Element Recognition by the TATABox-Binding Protein Has Been Conserved Throughout Evolu-tion,” Genes & Development, Vol. 13, No. 24, Dec. 1999, pp.3217–3230.
12. G. Guzikevich-Guerstein and Z. Shakked, “A Novel Form of theDNA Double Helix Imposed on the TATA-Box by the TATA-Bind-ing Protein,” Nature Structural Biology, Vol. 3, No. 1, Jan. 1996,pp. 32–37.
13. L. Pardo et al., “Binding Mechanisms of TATA Box-Binding Pro-teins: DNA Kinking Is Stabilized by Specific Hydrogen Bonds,”Biophysical J., Vol. 78, No. 4, Apr. 2000, pp. 1988–1996.
14. O. Norberto de Souza and R.L. Ornstein, “Inherent DNA Curva-ture and Flexibility Correlate with TATA Box Functionality,”Biopolymers, Vol. 46, No. 6, Nov. 1998, pp. 403–441.
15. A. Lebrun, Z. Shakked, and R. Lavery, “Local DNA StretchingMimics the Distortion Caused by the TATA Box-Binding Protein,”Proc. Nat’l Academy of Science USA, Vol. 94, No. 4, Apr., 1997,pp. 2993–2998.
16. X. Qian, D. Strahs, and T. Schlick, “Sequence-Dependent Struc-ture and Flexibility of TATA Elements Has Been Selected by theTATA-Box Binding Protein (TBP),” to appear in 2000; [email protected].
17. G. Chirico and J. Langowski, “Brownian Dynamics Simulationsof Supercoiled DNA with Bent Sequences,” Biophysical J., Vol.71, No. 2, Aug. 1996, pp. 955–971.
18. D. Sprous and S.C. Harvey, “Action at a Distance in Supercoiled
DNA: Effects of Sequences on Slither, Branching and Intermole-cular Concentration,” Biophysical J., Vol. 70, No. 4, Apr. 1996,pp. 1893–1908.
19. J.A. Martino, V. Katritch, and W.K. Olson, “Influence of Nucleo-some Structure on the Three-Dimensional Folding of IdealizedMinichromosomes,” Structure with Folding & Design, Vol. 7, No.8, Aug. 1999, pp. 1009–1022.
20. L. Ehrlich et al., “A Brownian Dynamics Model for the ChromatinFiber,” Computer Applications in the Biosciences, Vol. 13, No. 3,June 1997, pp. 271–279.
21. K. Luger et al., “Crystal Structure of the Nucleosome Core Parti-cle at 2.8 Å Resolution,” Nature, Vol. 389, No. 6648, Sept. 1997,pp. 251–260.
22. T. Schlick, “Modeling Superhelical DNA: Recent Analytical andDynamic Approaches,” Current Opinion in Structural Biology, Vol.5, No. 2, Apr. 1995, pp. 245–262.
23. T. Schlick, E. Barth, and M. Mandziuk, “Biomolecular Dynamicsat Long Time Steps: Bridging the Timescale Gap between Simu-lation and Experimentation,” Ann. Rev. Biophysics BiomolecularStructure, Vol. 26, 1997, pp. 179–220.
24. T. Schlick et al., “Algorithmic Challenges in Computational Mol-ecular Biophysics,” J. Computational Physics, Vol. 151, No. 1, May1999, pp. 9–48.
25. P. Koehl and M. Levitt, “Theory and Simulation: Can TheoryChallenge Experiment?” Current Opinion in Structural Biology, Vol.9, No. 2, Apr. 1999, pp. 155–156.
26. S. Doniach and P. Eastman, “Protein Dynamics Simulations fromNanoseconds to Microseconds,” Current Opinion in Structural Bi-ology, Vol. 9, No. 2, Apr. 1999, pp. 157–163.
27. E. Barth and T. Schlick, “Overcoming Stability Limitations in Bio-molecular Dynamics: I. Combining Force Splitting via Extrapo-lation with Langevin Dynamics in LN,” J. Chemical Physics, Vol.109, No. 5, Aug. 1998, pp. 1617–1632.
28. E. Barth and T. Schlick, “Extrapolation versus Impulse in Multi-ple-Time Stepping Schemes: II. Linear Analysis and Applicationsto Newtonian and Langevin Dynamics,” J. Chemical Physics, Vol.109, No. 5, Aug. 1998, pp. 1632–1642.
29. Y. Duan and P.A. Kollman, “Pathways to a Protein Folding In-termediate Observed in a 1-Microsecond Simulation in Aque-ous Solution,” Science, Vol. 282, No. 5389, 23 Oct. 1998, pp.740–744.
30. D.L. Ermak and J.A. McCammon, “Brownian Dynamics with Hy-drodynamic Interactions,” J. Chemical Physics, Vol. 69, No. 4,Aug. 1978, pp. 1352–1360.
31. D. Beard and T. Schlick, “Computational Modeling Predictsthe Structure and Dynamics of the Chromatin Fiber,” sub-mitted to Structure with Folding & Design, 2000; [email protected].
32. M. Fixman, “Construction of Langevin Forces in the Simulationof Hydrodynamic Interaction,” Macromolecules, Vol. 19, No. 41986, pp. 1204–1207.
33. R.M. Jendrejack, M.D. Graham, and J.J. de Pablo, “Hydrody-namic Interactions in Long Chain Polymers: Application of theChebyshev Polynomial Approximation in Stochastic Simula-tions,” J. Chemical Physics, Vol. 113, No. 7, Aug. 2000, pp.2894–2900.
34. M. Kröger et al., “Variance Reduced Brownian Simulation of aBead-Spring Chain under Steady Shear Flow Considering Hy-drodynamic Interaction Effects,” J. Chemical Physics, Vol. 113,No. 11, Sept. 2000, pp. 4767–4773.
35. P.J. Hagerman, “Flexibility of DNA,” Ann. Rev. Biophysics Bio-physical Chemistry, Vol. 17, 1988, pp. 265–286.
36. H. Jian, T. Schlick, and A. Vologodskii, “Internal Motion of Su-percoiled DNA: Brownian Dynamics Simulations of Site Juxta-position,” J. Molecular Biology, Vol. 284, No. 2, Nov. 1998, pp.287–296.
37. J. Rotne and S. Prager, “Variational Treatment of Hydrodynamic
Interaction in Polymers,” J. Chemical Physics, Vol. 50, 1969, pp.4831–4837.
38. D. Beard and T. Schlick, “Inertial Stochastic Dynamics: I. Long-Time Step Methods for Langevin Dynamics,” J. Chemical Physics,Vol. 112, No. 17, May 2000, pp. 7313–7322.
39. D. Beard and T. Schlick, “Inertial Stochastic Dynamics: II. Influ-ence of Inertia on Slow Kinetic Properties of Supercoiled DNA,”J. Chemical Physics, Vol. 112, No. 17, May 2000, pp. 7323–7338.
40. I. Lafontaine and R. Lavery, “Collective Variable Modeling of Nu-cleic Acids,” Current Opinion in Structural Biology, Vol. 9, No. 2,Apr. 1999, pp. 170–176.
41. B.N. Dominy and C.L. Brooks, III, “Development of a General-ized Born Model Parameterization for Proteins and NucleicAcids,” J. Physical Chemistry B, Vol. 103, No. 18, May 1999, pp.3765–3773.
42. D. Bashford and D.A. Case, “Generalized Born Models of Macro-molecular Solvation Effects,” Ann. Rev. Physical Chemistry, Vol.51, 2000, pp. 129–152.
43. J. Gao, “Methods and Applications of Combined Quantum Me-chanical and Molecular Mechanical Potentials,” Reviews in Com-putational Chemistry, Vol. 7, K.B. Lipkowitz and D.B. Boyd, eds.,VCH Publishers, New York, 1996, pp. 119–185.
NOVEMBER/DECEMBER2000 51
Tamar Schlick is a professor of chemistry, mathematics, and computerscience at New York University. She is also an associate investigator at theHoward Hughes Medical Institute. Her technical interests include compu-tational and structural biology, specifically on algorithms for biomolecularmodeling and simulations and their application to proteins and nucleicacids. She received her PhD in mathematics from the Courant Institute ofMathematical Sciences. Contact her at the Dept. of Chemistry andCourant Inst. of Mathematical Sciences, New York Univ., 251 Mercer St.,New York, NY 10012; [email protected].
Daniel A. Beard is a postdoctoral fellow at New York University. He receivedhis PhD in bioengineering from the University of Washington. Contact him atthe Dept. of Chemistry, 31 Washington Place, 1021 Main, New York Univ.,New York, NY 10003; [email protected].
Jing Huang is a fourth-year graduate student in chemistry at New York Uni-versity. Contact her at the Dept. of Chemistry, 31 Washington Place, 1021Main, New York Univ., New York, NY 10003; [email protected].
Daniel A. Strahs is a Howard Hughes Medical Institute research specialistat New York University. He received his PhD in biochemistry from the Al-bert Einstein College of Medicine. Contact him at the Dept. of Chemistry,31 Washington Place, 1021 Main, New York Univ., New York, NY 10003;[email protected].
Xiaoliang Qian is a fifth-year graduate student in chemistry at New YorkUniversity. Contact him at the Dept. of Chemistry, 31 Washington Place,1021 Main, New York Univ., New York, NY 10003; [email protected].
