Large-Scale Conformational Transitions in Supercoiled DNA ...stanford.edu/~ajspakow/publications/KS-supercoil.pdf · Article Large-Scale Conformational Transitions in Supercoiled

Article

Large-Scale Conformational Transitions inSupercoiled DNA Revealed by Coarse-GrainedSimulation

Brad A. Krajina1 and Andrew J. Spakowitz1,2,3,4,*1Department of Chemical Engineering, 2Department of Applied Physics, 3Department of Materials Science and Engineering, and 4BiophysicsProgram, Stanford University, Stanford, California

ABSTRACT Topological constraints, such as those associated with DNA supercoiling, play an integral role in genomic regu-lation and organization in living systems. However, physical understanding of the principles that underlie DNA organization atbiologically relevant length scales remains a formidable challenge. We develop a coarse-grained simulation approach for pre-dicting equilibrium conformations of supercoiled DNA. Our methodology enables the study of supercoiled DNA molecules atgreater length scales and supercoiling densities than previously explored by simulation. With this approach, we study the confor-mational transitions that arise due to supercoiling across the full range of supercoiling densities that are commonly explored byliving systems. Simulations of ring DNA molecules with lengths at the scale of topological domains in the Escherichia colichromosome (~10 kilobases) reveal large-scale conformational transitions elicited by supercoiling. The conformational transi-tions result in three supercoiling conformational regimes that are governed by a competition among chiral coils, extended plec-tonemes, and branched hyper-supercoils. These results capture the nonmonotonic relationship of size versus degree ofsupercoiling observed in experimental sedimentation studies of supercoiled DNA, and our results provide a physical explanationof the conformational transitions underlying this behavior. The length scales and supercoiling regimes investigated here coincidewith those relevant to transcription-coupled remodeling of supercoiled topological domains, and we discuss possible implicationsof these findings in terms of the interplay between transcription and topology in bacterial chromosome organization.

INTRODUCTION

In living organisms, an incredible amount of genetic mate-rial must be compacted into a small cellular space whilemaintaining the ability to orchestrate complex genetic pro-cesses that occur over a broad hierarchy of length scales.As a consequence of the multiscale nature of genetic pro-cesses and the highly crowded molecular environment inwhich they occur, biological systems face a complex setof topological constraints that must be managed for survival(1). To this end, all living organisms rely on a diverse familyof enzymes—topoisomerases—that elicit a rich display oftopological transformations that play a prominent role ingenomic compaction, replication, segregation, transcription,and genetic recombination (2,3).

DNA supercoiling, the winding of the DNA-double helixabout itself under torsional strain, is a powerful example ofthe role of topological control over genomic organization. In

Submitted March 21, 2016, and accepted for publication July 28, 2016.

*Correspondence: [email protected]

Editor: Tamar Schlick.

http://dx.doi.org/10.1016/j.bpj.2016.07.045

� 2016 Biophysical Society.

Escherichia coli and other bacteria, the circular DNA chro-mosome is regulated in an underwound topological linkingstate by topoisomerases, which gives rise to supercoiling ofthe chromosome. The degree of supercoiling is dynamicallyadapted to environmental cues (4), and the chromosomeis further dynamically partitioned into looped topologicaldomains with average sizes of ~10 kilobases (5–7). Inboth eukaryotes and prokaryotes, supercoiling facilitatescompaction of the genome into the cell (1) and is believedto be intimately connected to regulation of gene expres-sion (7–9).

Although supercoiled DNA has been extensively studiedtheoretically (10–18) as well as experimentally (19–26),both simulation and experimental studies that directly inves-tigate the large-scale organization of supercoiled DNA aretypically limited to supercoiling densities up to the averagesupercoiling density in E. coli (s ~ �0.06). However, thetopological state of the bacterial chromosome is highly dy-namic (27), and both DNA gyrase and RNA polymerase arecapable of generating negative supercoiling densities far inexcess of average supercoiling levels (28–30).

Biophysical Journal 111, 1339–1349, October 4, 2016 1339

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Krajina and Spakowitz

Moreover, multiple lines of evidence indicate that RNApolymerase plays an active role, along with topoisomerases,in defining and remodeling topological domains in bacterialchromosomes (6,31–33). Supercoiling, in turn, regulatestranscription, leading to a complex interplay between chro-mosome topology and gene expression that is not fullyunderstood. Thus, a theoretical understanding of DNA orga-nization that spans the full range of supercoiling that isaccessible to DNA motor proteins may provide crucial in-sights into the physical principles that underlie genomic or-ganization in living systems.

In this work, we investigate the large-scale conforma-tional transitions that emerge as a function of the intramo-lecular topological linking state in DNA rings with lengthsup to 104 basepairs, which corresponds to the typical sizeof topological domains in E. coli. We develop a coarse-grained model for supercoiled DNA, and conduct simula-tions across the full range of supercoiling levels accessibleto DNA gyrase. Our simulations reveal conformationaltransitions in DNA that give rise to three distinct super-coiling regimes as DNA is increasingly underwound: achiral random coil regime, an extended plectoneme regime,and a branched hyper-supercoil regime. The transition tobranched hyper-supercoils at large supercoiling densitiesis predicted for the first time, to our knowledge, in oursimulations, which explore a broader range of physiologi-cally relevant supercoiling densities than have been consid-ered in previous simulation work at this length scale. Theextended plectoneme to branched hyper-supercoil transitionemerges at length scales coincident with topological do-mains in bacteria, and linking deficits that are within therange of those generated in the wake of actively transcribingRNA polymerase. Our results capture the nonmonotonicdependence of hydrodynamic size versus supercoilingexperimentally measured by dynamic light scattering andsedimentation and provide an explanation for the conforma-tional transitions underlying these experimental results. Wediscuss possible implications of these findings in terms ofthe interplay between transcription and topology in genomicorganization.

Despite decades of theoretical and computational inves-tigation on supercoiled DNA (10–18,34–36), the behaviorof supercoiled DNA rings in free solution at the lengthscale of topological domains in bacteria with supercoil-ing <s ~ �0.07 has remained underexplored. However,these larger linking deficits represent an important biolog-ical regime, because DNA gyrase is capable of generatingsupercoiling densities up to s ~ �0.12, and RNA polymer-ase has been observed to generate similar or greater linkingdeficits (28–30). Although simulation work on the scaleof topological domains of E. coli has been previously re-ported, these studies were limited to supercoiling densitieswith jsj < 0.07 (11). Previous simulation work at larger de-grees of unwinding has been restricted to smaller rings(25,37), where branching is limited, or to the presence of

1340 Biophysical Journal 111, 1339–1349, October 4, 2016

extensional forces that suppress plectonemic supercoilformation (38). In contrast, our simulations explore thebehavior of DNA rings in solution, without extensionalforces, and extend predictions for s < �0.06 up to thelength scale of topological domains in E. coli, where weobserve previously unreported structural transitions associ-ated with branching statistics.

Analytical free-energy models have been previouslyleveraged to predict the force-torque phase behavior ofDNA held under extensional forces in the regime of super-coiling densities studied in our current work and even largerdegrees of unwinding (36), but this analytical theory entailsa number of top-down simplifying assumptions regardingthe geometry and energetics of supercoiled DNA. Further-more, although complementary experimental single-mole-cule mechanics studies have reported the behavior ofunderwound DNA over a broad range of linking deficits(24,26,39–41), experimental data at degrees of underwind-ing with s< �0.07 have been restricted to sufficiently largeextensional forces where supercoiled plectoneme formationis suppressed. At lower tensions (<0.4 pN), where plecto-neme formation is favored, the experimental phase diagramof supercoiled DNA cannot be extracted from the extensionversus winding behavior for s < �0.07 (40). Thus, the pre-dictions of free energy models at very low extensional forcesand large degrees of unwinding have not been directlyexperimentally verified. In contrast, our simulations pro-vide bottom-up predictions for the geometry of supercoilswithout relying on the simplifying assumptions of freeenergy models, and our hydrodynamic radius calculationsare compared directly to experimental data in free solution.The simulations provided in this work thus provide new in-sights, to our knowledge, into the behavior of supercoiledDNA at length scales and supercoiling densities of directbiological significance.

Furthermore, the systematic coarse-graining approach forsimulating supercoiled DNA that we describe in this worksignificantly relieves the computational burden associatedwith simulating large DNA under topological constraints(~10-fold compared to levels of coarse-graining requiredfor the discrete wormlike chain model (34)). This enablesus to explore the conformation statistics of larger DNA ringsthan have been previously simulated, up to the scale ofthe l-phage genome (48.5 kilobases). In addition, thereduced computational expense associated with simulatingthe DNA chain at genomically relevant length scalesopens the opportunity to incorporate additional levels ofcomplexity in the future, such as protein interactions andcrowding, which is essential to understanding the physicsthat underlies the hierarchical organization of genomes.Thus, the systematic coarse-graining approach that weleverage in this work presents, to our knowledge, newpossibilities for exploring the physical behavior of topolog-ically constrained DNA at genomically relevant lengthscales.

Conformational Changes in Supercoiled DNA

MATERIALS AND METHODS

Coarse-grained model of supercoiled DNA

The DNA molecule is modeled as a thermally fluctuating isotropic elastic

filament whose conformations are dictated by three energetic contributions:

a chain deformation energy Echain, a twist energy Etwist associated with

winding or underwinding the double-helix, and a steric repulsive energy

Erepulsion between DNA segments along the chain. Our coarse-grained

model for supercoiled DNA (shown schematically in Fig. 1) extends our

discrete shearable stretchable wormlike chain (dssWLC) (42,43) to incor-

porate twist and topological constraints.

In the dssWLC model, the detailed structure of the DNA double-helix is

replaced by a string of discrete beads of total contour length L with spatial

bead positions~ri, separated by a discretization length D. Thus, a ring chain

consisting of m beads separated by a discretization length D possesses a to-

tal contour length L¼mD. The dssWLC defines a universal coarse-graining

procedure for mapping the bending persistence length of a semiflexible

polymer lp onto a set of effective elastic parameters that are able to

accurately capture chain statistics at all length scales above the discretiza-

tion D (42,43). The polymer configuration is additionally represented by an

auxiliary set of orientation vectors ~ui, which capture the coarse-grained

tangent of the chain configuration. The chain conformational free energy

is given by

Echain ¼Xi¼ 1

m

264

eb

2Dj~ui �~ui�1 � h~R

t j2þek2D

�~Ri$~ui�1 � Dg

�2 þ et

2Dj~Rt

i j2

375; (1)

where ~Ri ¼~ri �~ri�1, and~Rt

i ¼ ~Ri �~Ri$~ui�1. The effective elastic param-

eters eB, ek, g, et, and h are determined by a choice of lp andD, and define a

set of chain elastic modes: bending, stretching, equilibrium length,

shearing, and bend-shear coupling, respectively. The coarse-grained

bending modulus eB describes the tendency of the tangent vectors along

the chain to remain aligned. The stretch modulus ek describes the entropiccost of stretching or compressing the chain along the tangent vector away

from the ground-state distance g. The shear modulus et describes the resis-

tance of the bead displacement vector~Ri to shearing away from the coarse-

grained tangent ~ui, and the bend-shear coupling term h characterizes the

tendency of the chain to bend in the direction that it is sheared. In this

work, we take the bending persistence length of DNA to be 50 nm and

FIGURE 1 Schematic of the discrete-stretchable-shearable wormlike

chain model. The DNA double-helix is coarse-grained into a polymer of

discrete beads separated by a contour length D. Each bead is associated

with an interbead displacement vector R!

i and a coarse-grained orientation

u!i. To see this figure in color, go online.

obtain eB, ek, et, h, and g for a given D using the dssWLC coarse-graining

procedure that is described in detail elsewhere (42,43).

We incorporate twist elasticity of the DNA double-helix by assuming that

every point along the contour length of the double-helical axis can be

represented by a local rate of twist u, which represents the rate at which

the two DNA strands twist around the double-helical axis per unit con-

tour length. In the absence of torsional strain, u corresponds to the

natural helical pitch of B-DNA of 1 turn per 10.5 basepairs, i.e.,

u0 ¼ ð2p=10:5 bpÞ. We assume a quadratic free-energy penalty for the

local twist rate uðsÞ, deviating from the natural helical twist rate u0,

Etwist

kBT¼ lt

2#dsðu� u0Þ2 ¼ ð2pÞ2ðTw� Tw0Þ2lt

2L; (2)

where lt is the twist persistence length. In this work, we take the twist

persistence length to be 70 nm, which agrees with experimental measure-

ments based on topoisomer distributions generated by ligation (44), but is

slightly smaller than the value of z100 nm measured in single-molecule

mechanics studies (24). The second form of Eq. 2 arises from integration

over all possible local twist fluctuations consistent with a given global twist

Tw ¼ ð1=2pÞ#dsuðsÞ, which gives the total number of times that the two

DNA strands wind around the double-helical axis.

A DNA ring can be topologically characterized by its linking number, Lk,

which specifies the number of times the DNA backbone strands wind

around one another (45), and can be decomposed into two geometric prop-

erties of the conformation—the total sum of local twists in the DNA double-

helix Tw and the writhe of the DNA double-helix about itself, Wr (45,46)

(i.e., Lk ¼ Tw þ Wr). The writhe Wr can be computed from the chain

conformation (47,48):

Wr ¼ 14p

R L

0ds1

R L

0ds2

ð~rðs1Þ�~rðs2ÞÞj~rðs1Þ�~rðs2Þj3$ð~uðs1Þ � ~uðs2ÞÞ: (3)

Equation 3 intuitively represents how many times the DNA double-helical

axis~rðsÞ winds about itself, and can be interpreted as the average number of

signed crossings that can be counted when the conformation of the double-

helical axis is projected onto a two-dimensional plane. Thus, the total num-

ber of times that the two DNA strands wind around one another, Lk, is

the sum of the number of times they wind about the double-helical axis,

Tw and the number of times the double-helical axis winds about itself,

Wr. By specifying a constant linking number constraint, the total twist asso-

ciated with any spatial conformation of a ring DNA in our model can be

determined implicitly by calculating the writhe from Eq. 3 and applying

Tw ¼ Lk � Wr (34).

To prohibit unphysical overlap of the DNA chain with itself, we incor-

porate repulsion between DNA segments in close proximity by a steric

Lennard-Jones repulsion of the form,

Eijrepulsion

kBT¼ VHC

12

"�LHC

Dij

�12

� 2

�LHC

Dij

�6

þ 1

#; (4)

for Dij < LHC, and Eijrepulsion ¼ 0 for DijRLHC, where VHC is the strength of

the repulsive potential, Dij is the closest distance of approach between seg-

ments i and j, and LHC is the steric diameter, which we take to be the bare

diameter of the B-form DNA (2 nm) unless otherwise noted. DNA is a

densely charged polyelectrolyte, and thus, exhibits ionic strength-depen-

dent electrostatic repulsions beyond the bare steric repulsion, except in

highly screening salt solutions. It has been shown that the salt-dependent

repulsion of the DNA double-helix can be approximated by an effective

steric diameter that varies with ionic strength. Where noted, to account

for electrostatic repulsions in <3 M monovalent salt, we take our value

of LHC to be the effective steric diameter of DNA that has been determined

as a function of salt concentration from knotting and sedimentation exper-

iments (15,49). Our implementation of the repulsive potential thus accounts



for both the bare steric prohibition of chain crossing as well as the electro-

static repulsion between DNA double-helices at larger length scales.

Calculations of the hydrodynamic radius RH are performed incorporating

hydrodynamic interactions approximated by the Rotne-Prager-Yamakawa

tensor (50,51). RH values represent the average over all configurations in

equilibrium Monte Carlo simulations. In our coarse-graining procedure,

the effective hydrodynamic radius of each bead is approximated as the

bead radius that preserves the hydrodynamic radius of a discretized rod

characterized by the same D, L, and bare steric diameter as the DNA ring.

Monte Carlo simulation procedure

We generate equilibrium conformations of supercoiled DNA molecules us-

ing a Metropolis Monte Carlo approach that combines local chain Monte

Carlo moves (crankshaft moves and random subchain displacements) and

a replica exchange (52) algorithm based on exchange between topological

replicas. We find that at high supercoiling densities jsj > 0.06, where

conformational frustration presents a considerable challenge, the topologi-

cal replica exchange algorithm leads to significant improvement in sam-

pling compared to local moves alone. In each Monte Carlo trajectory, 103

local moves are performed between each replica exchange trial, and

trajectories are collected for at least 4 � 107 Monte Carlo steps. All

ensemble-averaged quantities are taken over at least 6 � 103 realizations

of the polymer conformation, which are obtained from at least three inde-

pendent Monte Carlo simulations. Histograms for the radius of gyration for

the 10-kilobase plasmid are generated using 1.3 � 104 realizations, which

are collected from seven replicate simulations.

Because the local Monte Carlo moves allow for the possibility of forming

knots through moves that involve passing the chain through itself, we reject

any moves that lead to nontrivial knot topology. The Alexander polynomial

is a topological invariant that can be computed directly from a given chain

conformation, and is constant for all realizations of a given knot topology.

Knot topology of the polymers in our Monte Carlo simulations is enforced

by rejecting any Monte Carlo steps that produce conformations with values

of the Alexander polynomial not equal to that of the trivial knot (53). We

note that during local moves that involve chain crossing, the linking topol-

ogy is strictly enforced, because the twist Tw associated with any conforma-

tion consistent with the linking number Lk is obtained implicitly from the

writhe Wr from Tw ¼ Lk � Wr.

FIGURE 2 Systematic coarse-graining of supercoiled DNA. The left figure p

with supercoiling density s ¼ �0.04 and discretization lengths of 26, 39, 40, an

coarse-grained Monte Carlo simulations of the hydrodynamic radius RH as a

compared against experimental measurements from dynamic light scattering (bl

ence, the predicted theoretical scaling behaviors for a slender rigid rod (76) and a

bars in the simulations (defined as 90% confidence intervals for the mean based

trajectories) are smaller than the size of the symbols. To see this figure in color


Simulations of 1-, 2.686-, 4.3-, 5.8-, 10-, and 48.5-kilobase plasmids

are performed using 100, 100, 120,150, 250, and 485 beads, respectively.

Using our systematic coarse-graining procedure, we find that in a 1- and

2.686-kilobase plasmid, all simulations across a range of discretization

lengths D up to 54 basepairs produce a dependence of the radius of gyration

on supercoiling density s that falls on a single universal curve (Fig. S1

in the Supporting Material). We find that for modestly supercoiled DNA

s ¼ (Lk – Lk0)/Lk0 ~ �0.04, the coarse-graining length scale can be

increased to D¼ 100 bp without loss of accuracy in prediction of the radius

of gyration and hydrodynamic radius, while significantly reducing compu-

tation time.

RESULTS AND DISCUSSION

Systematic coarse-graining of supercoiled DNA

Our systematic coarse-graining procedure enables compu-tationally efficient simulation of large-scale conformationalfeatures of significantly longer DNA rings than have beenpreviously simulated. Fig. 2 illustrates the process of sys-tematically increasing the coarse-graining length scale Din supercoiled DNA rings with sizes ranging from 2.686kilobases up to the size of the l-phage genome (48.5 kilo-bases). This coarse-graining process enables more detailedfeatures of the superhelix to be resolved for small rings,while allowing us to capture the large-scale conformationalfeatures of long rings without the computational burdenof finely discretizing the chain. This coarse-graining pro-cedure achieves a 10-fold reduction in computation timewhen the discretization length D for the 48.5-kilobaseDNA ring is coarse-grained from the value required fordiscrete wormlike chain model (~30 bp) (11,14,34) upto the length scale of coarse-graining (100 bp) used togenerate the data in Fig. 2. We emphasize that thiscomputational speedup is achieved when implementing

rovides snapshots from Monte Carlo simulations of supercoiled DNA rings

d 100 bp (in order of increasing total DNA length). The right plot provides

function of contour length L of supercoiled DNA molecules (red circles),

ue squares) (74) and fluorescence microscopy (blue circles) (75). For refer-

self-avoiding branched lattice animal are provided (13) (dashed lines). Error

on boot-strap resampling the average values from independent Monte Carlo

, go online.


our software with different coarse-graining levels on iden-tical hardware, and is not a consequence of hardwareadvances that have occurred following previous workthat utilized the discrete wormlike chain. As a validationof our approach, the right plot of Fig. 2 compares ourpredicted scaling behavior for the hydrodynamic radius ofsupercoiled DNA rings with lengths from 1 kilobase to48.5 kilobases to experimental diffusivity measurementsfrom dynamic light scattering and fluorescence microscopy.Although the experimental supercoiling densities were notreported, the simulated value of s ¼ �0.04 is withinthe range of supercoiling densities typically reported forplasmid preparations from E. coli (8,54–56). The simulatedscaling behavior of the hydrodynamic radius with length isin good agreement with experimental measurements acrossthe full range of lengths simulated, and extends computa-tional predictions for the hydrodynamic radius to previ-ously unexplored lengths (15). We compare our results totwo limiting scaling behaviors: a slender rigid rod, whichis a possible model for a perfectly rigid unbranched plecto-neme; and a self-avoiding hyperbranched polymer withnondraining hydrodynamic interactions, which has beenproposed as a model for large supercoiled DNA mole-cules (13). We find a scaling behavior that is intermediatebetween these two limiting behaviors over the full rangeof lengths considered. Thus, our coarse-graining approachcasts new insights, to our knowledge, into the behaviorof supercoiled DNA at a broad hierarchy of length scalesand opens new opportunities for studying large-scale prop-erties of supercoiled DNA.

FIGURE 3 The top plot provides a visualization of conformational tran-

sitions in circular DNA as a function of supercoiling density s and length L

in kilobases. Conformations represent typical snapshots from Monte Carlo

simulations. Coloring denotes the implicit torsional twist experienced by

the DNA double-helix, according to the color bar provided. The bottom

plot provides the corresponding values of the asphericity for each DNA

length as a function of supercoiling density s. Shading represents 90% con-

fidence intervals for the mean obtained by boot-strap resampling average

results from independent Monte Carlo trajectories. To see this figure in

color, go online.

Monte Carlo simulations of large-scaleconformational transitions in supercoiled DNA

Monte Carlo simulations with lengths ranging from 1 to 10kilobases reveal a rich spectrum of conformational transi-tions that emerge as DNA is negatively supercoiled acrossthe full range of linking deficits that are accessible toDNA gyrase. The top image in Fig. 3 illustrates typicalconformations observed at supercoiling densities s ¼(Lk – Lk0)/Lk0 that coincide with distinct supercoiling re-gimes in sufficiently long DNA molecules. We define theasphericity A as

A ¼ ðl1 � l2Þ2 þ ðl1 � l3Þ2 þ ðl2 � l3Þ22ðl1 þ l2 þ l3Þ2

; (5)

where l1, l2, and l3 are the square roots of the principal mo-ments of gyration, ordered as l1 > l2 > l3. The asphericityis a scale-independent shape descriptor that measures theextent to which the principal moments of gyration of a poly-mer deviate from one another, and possesses a value of 0 foran isotropic polymer conformation, and a value of 1 for aperfect rod (57).

The bottom figure in Fig. 3 provides the correspondingdependence of the asphericity on supercoiling density andlength. The top-left plot of Fig. 4 provides the corre-sponding radii of gyration of DNA rings with increasinglynegative supercoiling densities, and the remaining plots ofFig. 4 (marked B, C, and D) provide radius of gyration his-tograms for the three labeled supercoiling densities on the10-kilobase dataset.

In relatively short DNA molecules (<3 kilobases),increasing jsj results in the formation of unbranched,rodlike plectonemic conformations as the DNA writhes toaccommodate torsional stress due to underwinding. Fig. 3together with Fig. 4 indicates the presence of two distinctcoiling regimes. Below jsj < 0.03, the DNA forms aloose, highly fluctuating chiral coil in which increasingjsj results in overall compaction of the chain as reflectedby decreasing RG to a local minimum, and a subtle increasein the asphericity as the DNA winds into increasingly


A B

C D

FIGURE 4 (A–D) Dependence of the average radius of gyration RG on the supercoiling density s for DNA rings generated by Monte Carlo simulations for

four chain lengths. The three supercoiling densities marked B,C, andD on the 10-kilobase dataset are used to generate the radius of gyration histograms in the

corresponding plots. Histograms are constructed from 1.3 � 104 realizations of the chain conformation. Representative Monte Carlo snapshots within the

histograms indicate the prevailing conformational states. Shading in (A) represents 90% confidence intervals for the mean obtained by boot-strap resampling

average results from independent Monte Carlo trajectories. To see this figure in color, go online.


anisotropic chiral coils. Above jsj ¼ 0.03, the superhelixbecomes increasingly tightened into a regular plectonemicconformation, which results in increasing RG and aspher-icity as the plectoneme becomes extended into a rodlikeconfiguration in which the superhelical axis behaves likean unbranched semiflexible polymer. Beyond jsj ¼ 0.06,which is close to the typical linking deficit in E. coli, amonotonic increase in RG and asphericity is observed inthe 1-kilobase ring, whereas the 2.686-kilobase ring ex-hibits a modest decrease in both RG and asphericity dueto buckling of the plectonemic axis into more solenoidalconfigurations.

For sufficiently long supercoiled DNA rings, thermal-induced branching of the interwound plectonemic confor-mation arises. The onset of branching in DNA superheliceswith increasing DNA length is dictated by the balance be-tween the finite bending energy associated with extrudinga branch and the entropic benefit of branching, which isexpected to grow with the length as ~kBlog L (13). Theemergence of branching, which occurs only rarely in a2.686-kilobase plasmid but is observed in many conforma-


tions of plasmids above ~5 kilobases, results in a richsequence of conformational transitions as the DNA dou-ble-helix is underwound.

The simple conformational progression with supercoilingdensity s observed in smaller plasmids is not observed inlonger DNA rings. The 5.8- and 10-kilobase plasmidsinitially exhibit a compaction of the chain as the DNA iswound into loose, fluctuating chiral coils that are progres-sively tightened. Above jsj ¼ 0.03, the superhelix tightensand stiffens, leading to an increasing prevalence ofextended, tightly interwound plectonemic conformationswith occasional branching, and the RG and asphericity riseto a local maximum near �s z 0.08.

The branched 5.8-kilobase and 10-kilobase plasmidsexhibit sharply contrasting behavior beyond this point.Beyond �s z 0.08, the RG and asphericity in the 5.8-kilo-base DNA rise sharply due to emerging prevalence of elon-gated plectonemic conformations in which branching issuppressed, which compete with more branched supercoils.Unlike the smaller plasmids, the 10-kilobase DNA compen-sates for the increasing torsional strain by buckling into

FIGURE 5 Dependence of the hydrodynamic radius (normalized to RH at

s ¼ 0.0) ~RH on the supercoiling density s for a 10-kilobase and a 2.686-

kilobase (inset) plasmid. Monte Carlo simulations are represented by red

circles and experimental measurements (23,59) are represented by blue

circles. Shading represents 90% confidence intervals for the mean obtained

by boot-strap resampling average results from independent Monte Carlo

trajectories. To see this figure in color, go online.


highly branched configurations, leading to the hyper-super-coiling regime. The transition is marked by a progressivecollapse in the radius of gyration with increasing jsjabove jsj z 0.08, during which branched hyper-supercoilsbecome increasing prominent in the heterogeneous confor-mational population. This final supercoiling regime coin-cides with the maximum degree of supercoiling that isaccessible by DNA gyrase (28) and occurs within the rangeof supercoiling densities generated upstream of translocat-ing RNA polymerase in in vitro and in vivo transcriptionexperiments (29,30,58).

The RG distributions for the 10-kilobase dataset in Fig. 4indicate the emergence of heterogeneous, multimodal sizedistributions in molecules that are supercoiled beyond thechiral coil regime. The corresponding conformational fluc-tuations involve a competition between highly extended,mostly unbranched plectonemes and branched hyper-super-coiled conformations. The multimodal distributions indicatethe existence of free energy barriers between extended plec-tonemes and hyper-supercoiled conformations. This mayhave important consequences for the dynamics of chromo-somal reorganization, which we will explore in future work.

Comparison to experimental hydrodynamicradius measurements

We compare the simulated predictions described hereto experimental measurements of conformational transi-tions in supercoiled plasmids. A number of studies haveinvestigated the conformations of supercoiled plasmidsusing atomic force microscopy (20) or electron micro-scopy (19,25), which reveal conformations that are qualita-tively in agreement with the configurations realized in oursimulations.

Dynamic light scattering and sedimentation studies,which yield the hydrodynamic radius RH, provide statisticalaverages over full thermodynamic ensembles in solution.Fig. 5 provides comparisons between our simulations andexperiments of the dependence of RH on the supercoilingdensity for a 2.686 kilobase plasmid (23) and a 10-kilobaseplasmid (59). The experimental s for the 10-kilobaseplasmid from Wang (59) has been corrected for an ethidiumbromide unwinding angle of 23�, which is within therange of experimental estimates (60,61). The 2.686-kilobaseplasmid is simulated with an effective repulsive diameter of5 nm, which is consistent with the effective repulsive diam-eter of the DNA double-helix (49) under the ionic conditions(100 mM monovalent salt) in the dynamic light scatteringexperiments. The 10-kilobase plasmid was simulated usingthe bare steric diameter of the DNA double helix (2 nm) forcomparison to the highly screening electrostatic conditions(3 M CsCl) in the sedimentation experiments.

The mostly unbranched 2.686-kilobase plasmid exhibitsa simple monotonic progression of RH as the molecule isincreasingly supercoiled, with a value that nearly saturates

at jsj above the chiral coil to plectoneme transitions z �0.03. Beyond this linking deficit, no further confor-mational transitions are observed, and the modest decreasein RH with increasing jsj reflects the tightening of the un-branched superhelix.

In contrast, the 10-kilobase plasmid demonstrates a non-monotonic dependence of RH on supercoiling that is quali-tatively consistent with that of RG, possessing two distinctlocal extrema. These local extrema, observed in both theexperimental and simulation studies, demarcate the bound-aries of the three distinct supercoiling regimes describedabove: the chiral coil regime for jsj < 0.03, the extendedplectoneme regime for 0.03 < jsj < 0.08, and the branchedhyper-supercoil regime for jsj > 0.08.

The dependence of RH on supercoiling in the 10-kilobaseplasmid shows reasonable agreement between simulationsand experiments. However, the simplemodel used in our sim-ulations provides a slight overestimate of the hydrodynamicradius at high jsj. This may be partially due to our simpletreatment of intersegment repulsions in highly screeningelectrostatic conditions, which are based exclusively on ste-ric effects, and neglect other short-range forces that will playa role in intersegment repulsions at high supercoiling density.Although the experiments were conducted in very high salt(3 M CsCl) and thus exhibit strong screening of electrostaticeffects, additional forces may play an important role inthis regime. For instance, osmotic pressure measurementsof DNA repulsion in crowded solutions demonstrate thatfor short separations between the surfaces of two double he-lices (z1 nm), fluctuation-enhanced hydration forces andelectrostatic double-layer effects lead to short-range repul-sion (62,63). Also, the simple treatment of hydrodynamicinteractions based on the Rotne-Prager-Yamakawa tensordoes not fully capture hydrodynamic effects that will bepresent for very closely spaced DNA segments.



The dependence of RH and other structural featureson supercoiling density has been considered in detail inprevious theoretical and simulation work (11,13–18). How-ever, previous simulation work on plasmids with lengthsexceeding six kilobases has been restricted to supercoilingdensities up to s ¼ �0.06, which is only one-half the super-coiling density that is accessible to DNA gyrase (28) andwell below the supercoiling density accessible to RNA po-lymerase (29,30). Although recent simulation work wasconducted for supercoiled DNA rings with jsj as large as0.086, this work was limited to plasmid sizes up to sixkilobases (17), which is below the length scale where weobserve the extended plectoneme to branched hyper-super-coil transition to occur. The simulations provided here forthe 10-kilobase plasmid thus provide new predictions atphysiologically relevant supercoiling densities. In partic-ular, the experimentally observed nonmonotonic sedimenta-tion behavior for the 10-kilobase plasmid for jsj > 0.06 waspreviously speculated to be due to secondary structural tran-sitions in the double-helix (11), and a satisfying model re-producing this effect has not been put forth by simulationor analytical theory. However, we show here that the non-monotonic behavior at higher jsj can be explained by a sim-ple elastic model without invoking secondary transitions.

Possible implications for transcriptional controlof chromosome organization

We now cast these findings in the context of currentunderstanding of the role of supercoiling and transcription ingenomic organization in living systems. A growing body ofwork reveals that regulation of DNA supercoiling in bacterialchromosomes entails an interplay between topoisomerasesand transcription by RNA polymerase (6,10,31–33). Ac-cording to the twin supercoiled domain model (64), activelytranscribing RNA polymerase must locally unwind the DNAdouble helix to provide access to the antisense strand. Thisrotation generates positively supercoiled DNA in front ofthe translocating transcription machinery, and leaves hyper-negatively supercoiled DNA in its wake.

Consistent with this model, both in vitro and in vivostudies indicate that RNA polymerase is a powerful torquegenerator, and linking deficits below s ¼ �0.10 have beenmeasured as a result of transcription at strong promotersboth in the presence of functional Topo I (DNA relaxingenzyme) as well as in Topo I-deficient mutants (28–30).Studies in E. coli and Salmonella have shown that RNApolymerase is capable of actively creating and remodelingdynamic supercoiled topological domains (6,33). Simi-larly, recent experimental work in Caulobacter revealsthat highly transcribed genes appear to demarcate topolog-ical domains in the chromosome (7), and in vivo measure-ments of interloci distances in Caulobacter resemble thosetheoretically predicted for a branched supercoiled confor-mation with densely packed plectonemes (65). In all three


of these organisms, topological domains are measured topossess average sizes of ~10 kilobases (7,33) and theimportance of transcription in regulating the size of thesetopological domains is suggested by the high conservationof genomic interspacing of highly expressed genes inthe chromosomes of phylogenetically distant bacteria(33). Thus, the linking deficits and length scales simulatedin this work are of direct relevance to the length scalesand linking deficit regimes that are implicated in tran-scriptional regulation of chromosomal topology, and mayhave important implications for chromosomal organizationand dynamics.

Relevance of noncanonical secondary structuraltransitions in highly supercoiled DNA to this work

It has been suggested that elastic models for supercoiledDNA, such as the model used here, provide limited applica-bility for jsj < �0.06, due to the formation of noncanonicalsecondary structures, such as melted regions (L-DNA),Z-DNA, cruciforms, and H-form DNA (11). It is true thatsingle-molecule mechanics studies demonstrate the forma-tion of noncanonical secondary transitions when DNA ishighly underwound (24,26,66). However, these experimentsinvolve holding the DNA molecule under tension, wherewrithe fluctuations are strongly suppressed compared toDNA rings in free solution. A recent single-molecule me-chanics study on torsionally induced melting of DNAobserved the onset of melting in DNA held under large ten-sion (up to 36 pN) at linking deficits near sz �0.05. Com-plete conversion of B-DNA to L-DNA is not observed untils z �1.6 (66). Similarly, in two other recent single-mole-cule experiments, at tensions where no plectoneme forma-tion occurs (R1 pN), the onset of DNA melting was notobserved until s z �0.02 (26,40). However, at these ten-sions, most of the linking deficit will be stored in twist,because the DNA was held at end-to-end extension nearlyequal to its contour length, and buckling due to plectonemeformation was not observed. In contrast, in our simulations,at large values of jsj, only ~17% of the linking deficit isstored in twist in the 10-kilobase DNA, and the remainderis stored in writhe (Fig. S2). Thus, even at the largest linkingdeficit studied here s z �0.12, the twist deficit density isonly ðDTw=DLKÞz0:02, which is within the range of twistdensities where linear elastic torque behavior is observed inrandom DNA sequences (26,40,66).

Moreover, noncanonical structures, in particular H-form,Z-form, and cruciform DNA, require specific DNA se-quences that are susceptible to their formation undertorsional strain (67–69). To our knowledge, studies thatdemonstrate the formation of Z-DNA at the twist densitiesconsidered in this work typically utilize sequences thathave been specifically engineered to form Z-DNA inresponse to supercoiling, such as pyridine-pyrimidine tan-dem repeats (26,70,71).


Local denaturation and melting are also sequence-dependent. For instance, supercoiled pUC19 plasmids(s z �0.06) have been experimentally observed by atomicforce microscopy to exhibit local denaturation bubbles dueto supercoiling, but the formation of these bubbles wasfound to be restricted to a particular AT-rich portion ofthe plasmid sequence under physiological salt conditions(21,72). Similarly, recent single-molecule fluorescence mi-croscopy experiments of l-DNA under torsion demon-strated the onset of localized melting (~15% probability)under moderate supercoiling s z �0.07, but even up tosupercoiling densities of sz �0.10, melting was restrictedto a particular AT-rich portion within the origin of replica-tion (73). These experiments also involved an extensionalforce, in which a significant portion of the DNA contourlength was partitioned into the extended, rather than plecto-nemic, state.

In addition, recent electron microscopy and molecular dy-namics studies demonstrated that in very small DNA rings(336 bp), where DNA supercoiling requires the presenceof sharply bent loops at the apices of the plectoneme, localdenaturation may occur to accommodate kinks in the DNA(25). In this case, the short length of the DNA minicirclesimposes a geometric constraint for writhing that likely en-hances the formation of denatured regions that enable sharpbending. These sequence- and geometry-dependent nonca-nonical effects may certainly play an important role in bio-logical systems. However, the extent to which these localeffects contribute to the large-scale conformation statisticsof supercoiled DNA in free solution has remained underex-plored. Our current predictions thus provide important inputfor elucidating this question, and we will further explorethe consequences of local regions of melting and kinkingsusceptibility on large-scale DNA conformations in futurework.

CONCLUSIONS

In this work, we investigate the large-scale conformationaltransitions that emerge due to underwinding of DNA ringsat length scales up to the size of typical topological domainsin bacterial chromosomes. We explore the full range ofsupercoiling densities that are accessible to DNA gyrase,using a coarse-grained simulation approach for predict-ing organization of supercoiled DNA. This coarse-grainedapproach enables the study of supercoiled DNA moleculesat length scales up to the genome of l-phage, and supercoil-ing densities significantly larger than those that have beenconsidered in previous simulations at the length scale ofthe topological domains of E. coli. Our simulations predicta rich display of conformational transitions that arise asDNA rings are underwound. In rings with lengths commen-surate with the size of topological domains in bacterialchromosomes, we observe the emergence of three distinctconformational regimes as DNA is increasingly under-

wound, which are regulated by a competition among chiralcoils, extended plectonemes, and branched hyper-super-coils. The extended plectoneme to hyper-supercoil transi-tion was found to occur at supercoiling densities in therange of those that have been measured upstream ofactively transcribing RNA polymerase, which may haveimplications for transcriptional control of chromosomalorganization.

Despite decades of active experimental, theoretical, andsimulation work exploring the physical behavior of super-coiled DNA, the conformation statistics of supercoiledDNA at the length scale of topological domains in bacteria(~10 kilobases) and across the full range of linking deficitsaccessible to DNA gyrase (up to s ¼ �0.12) has remainedunderinvestigated. Previous simulation and theoreticalwork investigating supercoiled DNA over this range ofsupercoiling densities has been restricted to smaller plas-mids (25,37) or the presence of extensional forces thatsuppress plectoneme formation (38), or has relied uponsimplifying top-down assumptions about the geometry ofthe superhelix (13,36). By comparison, our bottom-up pre-dictions for the conformations of supercoiled DNA extendpredictions for highly underwound DNA up to the scaleof topological domains in E. coli, and we observe distinctstructural transitions in the highly underwound regime(s < �0.07) that are absent in smaller rings. Moreover,the systematic coarse-graining approach developed heresignificantly relieves the computational burden of simu-lating large, moderately supercoiled DNA compared toprevious approaches, as we demonstrate with our coarse-grained simulations of the l-phage genome. Thus, thiswork provides new insights into the behavior of supercoiledDNA at biologically relevant length scales and across super-coiling densities accessible to both DNA gyrase and RNApolymerase, and opens new opportunities for unveiling thephysical principles of genomic organization.

In summary, the simulations performed here provide newpredictions for the organization of supercoiled DNA atlength scales relevant to topological domains in bacterialchromosomes and topological states accessible to DNA mo-tor proteins that remodel topological domains. Despite thesimplicity of the model, our simulations recapitulate exper-imental measurements of the dependence of the size ofsupercoiled DNA at supercoiling densities previously unex-plored by simulation at these length scales. These findingsprovide important input for future work in understandingthe physical principles that underlie genomic organization,and we hope that our theoretical predictions will motivatefurther experimental studies that explore this importantbiophysical problem.

SUPPORTING MATERIAL

Two figures are available at http://www.biophysj.org/biophysj/supplemental/

S0006-3495(16)30663-4.


http://www.biophysj.org/biophysj/supplemental/S0006-3495(16)30663-4

http://www.biophysj.org/biophysj/supplemental/S0006-3495(16)30663-4


AUTHOR CONTRIBUTIONS

B.A.K. designed research, performed research, contributed analytic tools,

analyzed data, and wrote the article; and A.J.S. designed research and wrote

the article.

ACKNOWLEDGMENTS

We thank Stanford University and the Stanford Research Computing Center

for providing computational resources and support that have contributed to

these research results. The authors gratefully acknowledge helpful discus-

sions with Sarah Heilshorn, Sebastian Doniach, and Nick Melosh.

B.A.K. was supported by the Stanford BioX Fellowship Program. This

work was also supported by the National Science Foundation, Physics of

Living Systems Program (grant No. PHY-1305516).

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