Polymer-and-salt induced toroids of hexagonal DNA

Polymer-

and

Salt-induced

Toroids of

Hexagonal

DNA

Job Ubbink and Theo Odijk

Department of Polymer Technology, FacultyofChemical EngineeringandMaterialsScience, DelftUniversityofTechnology, 2600 GA, Delft,The Netherlands

ABSTRACT Amodel is proposed for polymer- and salt-induced toroidal condensates of DNA,basedonarecenttheoryofthe undulation enhancement of the electrostatic interaction in the bulk hexagonal phase of semiflexible polyions. Inacontinuum

approximation, the thermodynamic potential ofa monomoleculartoroid maybe splitup in bulk, surface, and curvature con-tributions. With thehelp ofanapproximate analytical minimization procedure, the optimaltorus dimensionsarecalculated as afunction of the concentrations of inert polymer and added salt. Thestabilityofthe torus isanalyzedinterms of its surfacetension andabulk meltingcriterion. Thetheoryshould beapplicableto

qp-toroids

thatarenot too thick.

INTRODUCTION

Underawide variety of

conditions,

randomly coiled DNA in dilute aqueoussolution may be forcedto condense into

globulesofvaryingdegrees oforder. This collapsemay be induced by neutral polymers like polyethylene oxide with added monovalent salt

(Lerman,

1971), orby a numberof polyvalent cations like polylysine

(Laemmli,

1975), the poly-aminesspermidine (3+) (GosuleandSchellman, 1976)and

spermine (4+)

(Chattoraj

et al., 1978; Gosule and Schell-man,1978), cobalt hexammine

(3+)

(Widom and Baldwin, 1980), or magnesium(2+) inwater/ethanol mixtures(Post

andZimm, 1982).

Thepolymer-and-salt-inducedori+-condensation ofDNA was first studied by Lerman (1971) who discovered that

the rate ofsedimentationof DNAinhighly dilute solutions increased stronglywhen the concentrationsofpolyethylene oxide and NaCl were above certain critical levels. He was abletoestimateanaveragehydrodynamic radius ofthe sedi-menting particles, which indicated that within theglobules

the DNA strands were tightly packed, the volume of a globulebeing of theorder ofthe self-volume ofoneDNA

molecule.

In additional investigations, the spatial structure of the

condensateswasresolved to some extent. From x-raystudies it was concluded that the DNA strands were stacked in a

lattice,thehexagonalorder ofwhich increasedwith increas-ingpolymer concentration(Maniatisetal.,1974; Evdokimov et al., 1972). Spectroscopic studies indicated that the DNA

double helix did on the whole retain its normal B form (Jordanetal.,1972).Inelectronmicrographsalarge fraction oftheglobules adoptedmore orlesssimple geometries, like loosely wound toroidal globules (Evdokimov et al., 1972;

Evdokimovetal., 1976).Thepackingofthe DNAstrands in the globules tended to be tighter the greater the polymer concentration.

qi-Condensation

appears to be fully

revers-Receivedfor publication 3 May 1994 andinfinal form 17 October 1994. AddressreprintrequeststoDr.Theo Odijk,Department of Polymer

Tech-nology, Delft UniversityofTechnology,P.O. Box 5045, 2600 GA, Delft, TheNetherlands. Tel.: 31-71-145346; Fax: 31-71-274397.

i 1995by the Biophysical Society 0006-3495/95/01/54/08 $2.00

ible; ondilutingthe solution to concentrations ofpolymeror salt lower than the critical concentrations needed for con-densation to occur, the solution becomes isotropic again (Lerman, 1973).

Severalyears after the discovery of+'-condensation, it was observed that low concentrations of spermidine or spermine, polyamines that are found in vivo, could also induce a collapse of DNA. Again, regularmorphologies of the globules were observed. Bloomfield and co-workers reported rather well defined toroids making up the larger fraction of theglobules, which are often also rodlike structures(Arscott et al., 1990;Plum etal., 1990).

Inelectronmicroscopy studiesitwasshown that theDNA strands were circumferentially wrapped around the cen-tersof the toroids(MarxandRuben, 1983,1986), thereby

allowing a closepacking of the DNA strands without the need for sharp kinks orbends.

Inboth types ofcondensation, small globules of well de-finedgeometry are observed only at low concentrations of DNA. AthigherDNA concentrations, aggregation sets in, giving rise to muchlarger and less compact structures. For

example, in the case of thecondensation induced by mag-nesium it was shown thatsometimesthemonomolecular

con-densatesareincoexistence withtherandomDNAcoils(Post

and Zimm, 1982).

Quantitatively, thecondensationphenomenaare notwell

understood, althoughawidevarietyofmodelshas been

pro-posed.Forinstance,

qi-condensation

has beeninterpreted in

termsof thecoil-to-globule transition (FrischandFesciyan,

1979; PostandZimm, 1979) that is observedin the case of

flexible polymersin a poor solvent(Lifshitzet al., 1978). A

persistencesegmentof DNA hasalargeaspectratioimplying orientational order within a tightly packed globule as has been stressedbyGrosbergandcolleagues(Grosberg, 1979;

Grosberg and Khokhlov, 1981; Grosberg et al., 1982; Grosbergand Zhestkov, 1986). The free energy of toroidal globules has been estimated assuming the DNA phase is nematic (Grosberg and Khokhlov, 1981; Grosberg and

Zhestkov, 1986).InarecentpaperbyBloomfield (1991)an overviewis givenofthefree energy ofhexagonallypacked

The general statistical problemof DNA condensation is greatlycomplicated bythestiffness ofDNAcausing the ori-entational andtranslationaldegreesof freedomtobestrongly coupled. Here,wefocus ontherelativelymodestproblemof

+i-condensation of toroidal DNAbasedon a recent, appar-ently successful theory of the hexagonal phase (Odijk,

1993a,b, 1994).Thecollapse of theDNAcoil iscausedby

theosmoticpressure of thesurrounding semidilute polymer

solution(Lifshitzetal.,1978).Wedonotdescribe theinitial stagesof thecompression of the coil butratherapproach the problem from the opposite point ofview, i.e., the hexago-nallyorderedglobule and itsmelting behavior.Inparticular,

we do notjustify why theglobule should adopt a toroidal shape; this is merely a supposition, although we note that

somejustification fornematically ordered toroidswas

pre-sentedby GrosbergandZhestkov

(1986).

Wedogiveanovel

treatmentof theeffectof undulations andsurfaceterms,the onset of deviations from a toroidal shape and the possible import ofthe interaction energy of curvature.

Theoutline of thepaperisasfollows.Wefirstneglect undulations as it turns out that their effect is easily

in-cludedattheend of theanalysis.Wehavetoevaluate the electric potentialofcurvedDNA soas toderive the elec-trostaticfree energy ofDNAhexagonally packed within

a toroid, which is split up in volume, surface, and cur-vature terms. The bulkpolymer solution is supposed se-midilute; otherwise the osmotic pressure is too weak to

induce condensation. Thefree energyis again separated

involumeandsurface(depletion)terms.Theoptimal di-mensions ofa torusarecalculatedbyminimizing thetotal

potential of the whole system.

ELECTRIC POTENTIAL EXERTED

BY

CIRCULAR DNA

The DNA double helix may beapproximated byauniformly charged cylinder ofdiameterD, bent intoacircleofradius Rmuchlarger than the Debye lengthK-1,sothatthepolyion

may be considered to be slightly curved on the scale of a

Debye length. For a 1-1 electrolyte the Debye length is

given by K2

87rQn.,

with Q

q2/EkBT

the Bjerrum length,q the

efementary

charge, Ethedielectricconstant ofthesolvent, and

ns

theexcess salt concentration. To a goodapproximation,theelectrostatic potentialaround the

polyion may be obtained by solving the nonlinear Poisson-Boltzmann equation (Fixman, 1979). However,

as under excess salt conditionsthe distance between

ad-jacentDNAwinds inthehexagonal lattice is larger than

D +

2K-1,

we areinterestedonlyin the outer double layer of the potential, which decays essentially as a

super-positionofDebye-Huckel potentials.

If thechargedcylinder isstraight,various workers have

arguedthat it may bereplacedbyaline charge of effective

linearcharge density adjusted so as to letthetails of the

respective potentials match (Brenner and Parsegian, 1974). A convenient means ofcalculating

veff

is via the

Boltzmann equation (Philip and Wooding, 1970;

Stroobants et al., 1986). Upon bending thecylinder and the concomitant line charge, veffwill decrease a bit be-causemorecounterions areattracted tothe bentcylinder in view of the enhancement of the bare potential.

Nev-ertheless, we neglect this renormalization as it has the

following form:

veff(R) = veff[1 - constant(KR)2]

(1)

for it will turn out that only the thin double layer

ap-proximation (DK > 2) is of practical interest. (Note that

Eq.1mustbeanexpansion in thesquareofthecurvature.)

Furthermore, effects of image charges may be disre-garded at high ionic strengths. Ultimately, our analysis will retain solely the leading terms in an expansion in

termsof the small parameter (aK)-1 wherea is the

dis-tance between the centerlines of adjacent DNA winds inthe torus (in the case of interest here, aK > 5).A sec-ond smallparameter will be aIR; therefore, in addition, we expand all results systematically in terms of the curvature.

Next,we choose a Cartesian coordinate system withthe origin fixed at a point on the line charge, with the z axis aligned tangentially along the line charge, and the x axis pointing alongits outward normal(seeFig. 1).Thedistance

rbetweenapointP inthexyplaneandapoint Sonthe line charge is givenby:

r2= r0 sin2,y + (rocos y-R(cos 0- 1))2+R2sin20

=

ro

R

cos-12R2

S

R3)

(2)

where yis theanglebetween theline OPandthexaxis,

and 0 = (sIR),with sthelengthof the arcOS. Equation

2 represents the deviation from the straight line in

Philip-Wooding solution to the nonlinear

Poisson-I,Y

termsof thecurvature.As

argued

above,

the electrostatic

potential, scaled

by

-q/kBT,

at

point

Pisasumof

Debye-Huckel potentials:

ITR

exp[-Kr]

tp

=

tp(ro,

y,

R)

21 ds r

weobtain theasymptoticform of the

potential

of the circular DNA orits equivalentcircular line

charge:

4 -

Irod(ro)

+

it(ro,

y,

R)

itrod(rO)

=

I()

Kr

exp[-Kro]

(3)

where IF Q

veff.

The bestwayof

expanding

this

integral

asymptotically

is by

introducing

the variable t = r

-ro:

rdt=(1 + cosy)sds-6R2 ds

(4)

Theintegral is transformed to:

g ~~exp[-Kt]

-2F

exp[-Kro]

dt

+

)p

K

s

3

(5)

o0 + -ORcs s-6R2

Because KR >> 1, the errors introduced as we let the upper limit of

integration

goto

infinity

are

exponentially

small.

Expanding the denominator to

O(R-2)

gives:

- 2F

exp[-Kro]

r dt

exp[-Kt]

0 (6)

X

[1-R

cos y

Rj2cos2,y

+ +

R21

Todisposeofsinthe

integrand,

it is convenienttointroduce:

W

2ro

t+t2[1+

cosyJ

52-12R2

(7)

Equation

6 may then be written as:

P00

Uexp[-Kt]

t - 2F

exp[-Kr0]

J dt [I

0 (8)

X

[1-2Rcos

+

3R2

+

8R21

With the

help

of the

leading

terms:

f

dt

exp[-Kt]

(2r0)-12

fdt

t-"'2

exp[-Kt]

0

0~~~~~

(9) I 1 1/2

t2Kro

J dt

exp[

-Kt]W"

2(2r0)ln

dt

tl'2exp[-

Kt]

(10) (Tro 1/2 2K

3rO

(11)

(12)

where c is the correction to the electrostatic

potential

for

slight curvature:

Ic =0P'oirod(rO)[2Rcrd2R8R2cosycosR3r2 2 8K?21] (13) The coefficient of the last term is notvery

meaningful

as

termsof4(1/(Kr0))have been deleted in

Eqs.

9 and 10.

FREE

ENERGY OF A HEXAGONALLY

PACKED

TORUS

WeconsideraDNA

globule,

which isassumedtobetoroidal inshape andthe size ofwhich isdetermined

by

thetworadii

RC

and

Rt

(Fig.

2).

One DNA molecule is

wrapped

circum-ferentiallyaround thecenterof thetorus.In its

bulk,

the DNA exhibitsperfect

hexagonal

order withalattice

spacing

a.We

momentarily suppose the DNAundulations are

negligible.

Hence, the

hexagonal

packing

mustcontinueall thewayto the surface. Provided the DNAis

long

enough,

we

adopt

a continuum approximation so that the DNA strands are alignedwithanaverage

spacing

aonthetoroidal

surface,

the cross section of which is circular

(Fig.

2).

The totalfree energy of theDNAtorusmaybe writtenas:

ft

= F+A =

Felo

+

Fejic

+

Fb+ AelO

+

Aelc

(14)

whereFisthe total bulk free energy of the torus,Ais its

surface free energy,

Felo

and

AelO

are,

respectively,

the bulk and surface electrostatic free

energies

of the hex-agonal lattice as ifit were

straight,

Fe,c

and

Ael

care the

t/

(a)1~~~~~~

respectivetorrectionterms tothe electrostatic free energy ofinteractiondue to the slighttoroidal curvature, and Fb

is the bending free energy.

To obtain the electrostaticfree energy per unit length

ofstrand in the bulk of the toroidwesimply multiplythe

effective charge density veff onthe test strandbythe

elec-trostatic potential atits axis. Asonly thefar field ofthe Poisson-Boltzmann potential is relevant and the

super-position approximation holds, we may obtain the total

potential atthe test strandbyjust summingoverthe

hex-agonal lattice. Only the nearest neighbors in the lattice (i.e., three-pairinteractions) contribute significantly

be-cause the electrostatic potential falls off exponentially

and Ka >> 1. The total bulk electrostatic free energy is

thenobtainedbyintegrating the freeenergydensity, i.e.,

thefree energy per unit length of strand divided by the surface area of ahexagonal unit cell

(3112a2/2),

over the torus volume V = 2i r2

R,

Fel = 4 31/2

,7.2Veff

IIrod(a)R2 (15)

kBTa

Thetotalelectrostaticfree energyof thetorusis lowerthan

its total bulkelectrostatic freeenergy,becauseonthe surface

twoof the sixnearestneighborsassumedtobe presentinthe bulkcalculationaremissing.We thushave anegative

elec-trostatic surfacefree energy. This is consistentwithour ne-glectof entropy orundulations.A DNAstrand isrepelled by

thetoroidal bulk. The surface freeenergyasif the latticewere straight is obtained by multiplyingthefreeenergydensity by the surface area S =

4wl2RtRc

ofthe torus:

Ael,

_4T2vef

rod(a)'c

kBT

fft,daa

written as:

Fei,c

7 [ 21 =-

X1

3

Tveff

qr.d(a)

[1

+ 11I kBT 2[ 7K.aJ Fb 4uP kBT 31/2a2' where the integral Iis given by:

Rt+Rc[R2

-(R

-Rt)2]/2

I -=R R

-dR

Rt -Rc

=

rR2 = irR-

i'R2

-R

2)1/2

;= t 7( t c 2 R (19) (20) (21) for

Rc

<<

Rt

Notethatthepotentialdivergencehasdisappeared,which

justifiesourcontinuumapproximation. The calculationof Iis effected by the substitution R =

R,

+

Rc

cos y.

Similar considerationsallowus toevaluate theanalogous surfaceterms. The density of the electrostatic surface free energyofcurvature is:

Wel,c

kBT

(22)

Vefftrod

3112cos

0 7a

cos2

0 7a 1 1

a L 2R

16R

32

8K]

Rj

whereR =

Rt

+

Rc

cos0isthelocalradiusofcurvature.The surface free energy of curvatureis obtained byanintegration

overthe surface:

Aelc

_

(16)

kBT

I

2iR

Sie1

C 16 aRc

R

c(0)

e"

d0'-

7~Tveff

tp

.(a)

kBT

(23)

Next,the freeenergydensity of interactionand curvatureis

given by (see Eq. 13):

;elc

2veff Ft W 3+12a) ( IW

kBT

31/2a2 L

6~~,R

2

)2

i~,~

+ 3'12a )]

(17)

7 312veff

tkd(a)

[ 2

8R2 1+7KaJ

where Ris the local radiusofcurvature. Bysymmetry,the term of order R-1 disappears as it should. The elastic free energy density is given by:

_b P

kB T 3112a2R2 (18)

where P is the persistence length of DNA.

We again adopt a continuum approximation: a << RC9

a <<

Rt.

A summation over the strands amounts to an integrationoverthevolumeeventhough Rmaybe small, as we shall see. The free energies of curvature may be

whereonlythefirsttermhasbeen retained.c(o) =

2Tru

R(O) isthelocal circumference of thetorus. Eq. 23 issimply the surfaceequivalent of Eq. 17.

LOCAL

EQUILIBRIUM

IN A

SEMIDILUTE

POLYMER SOLUTION

WemayassumethattheDNAandpolymer phasesare

com-pletely immiscible sothat thepolymer isnot present in the DNAphase,orvice versa. Theinterfacebetween the DNA

globule and the polymersolution is considered to be well

defined, inthesensethat itisstable, relativelysmooth, and

permeable only to water and salt.

The polymer is neutral, flexible, highly soluble, and

sufficiently highin concentration to ensure that we are in the semidilute regime. According to scaling theory, the

propertiesof the solution are governed by the correlation

length (,which is ameasure of the average mesh size in the polymer network. The correlation length decreases

quite strongly with increasing polymer volume fractionv

(de

Gennes, 1979):

where A is theKuhnlength. Eq.24 is valid if thepolymer chain is long enough.

Inthe stabilization of the DNA globule by the polymer solution,twoeffects havetobediscerned.

First,

wehave the

osmoticpressure that scales as(de Gennes, 1979):

HP =C-3

kBT

(25)

where C3 is a constant. The second effect thatis of

impor-tancehere is the depletion ofthepolymerat thesurfaceof

theglobule(Joannyetal., 1979).Ifwesupposethepolymer is inert withrespect tobothDNAandsalt, thepolymer con-centration in alayerof characteristic thickness (surrounding

the toroidal surface is lower than in the bulk. The surface

tension that results from thisdepletion layer scales as:

and curvature free energy terms determines the aspect ratio

R,/RC

of the globule. Positive surface free energy contribu-tions will tend to keep the torus short and thick; the free energy of curvature in general favors a more extended configuration.

Atthis stage it isconvenientto consider the ratio of the surface terms:

Aei*

l VC

Cvff

tI=

d(a)(_

E 32

,yp-

S

C2a

C2Qa

(30)

where

Ael,c

may beneglected. If

g,

> 1,the total surface free energy is negative and the torusis unstable, for it tends to increase its surface areawithout bound. We therefore have to adopt the condition

g,

' 1 as a stringent criterion for

stability. The ratio of the curvature termsbecomes:

F_lC 21Ea2 7

3"12C2a3

92

'Fb

8QP

8KPf3

Arenormalizationtheoryof 'yPdoes notexistatpresent; the numerical coefficients C2andC3areclosetounityinEqs. 25 and 26.

Thethermodynamic potential of the DNAtorus in equi-librium with the polymer solutionmaythus be writtenasthe sum of the total free energy of the torus and the work of

expanding it against the polymer stress (Kirkwood and

Oppenheim, 1961):

Q1

=

Qt

+

HlpV

+ yps

The thermodynamic potential may be written as (see Eqs. 14-16, 19-21, 23,and 25-26): 2k V k2V 41T4R6v121

fQ(Rc

V) R + 272R2 -1-1 c RC2c X + constant X V (32) with: (27) Here, the polymer solutionis avery large reservoirsothat

thepolymer concentrationremainsvirtually constantunder variationsin volume andsurface area of the torus.

The numericalminimization offl doesnotyielda great deal ofinsight. Hence,we apply anapproximate analytical procedure. Inthe continuumlimit (Rc/a =

0(10)

ormore),

thetwovolumetermsinEqs.15 and 27overwhelmthe other terms in view of the estimates

RC

= (3(R) = (3(P) and

Veffl'rod =

C(Q-1).

In addition,we note thatthe potential decays exponentially sothattheequilibrium lattice spacing

mustbe very close to the hypothetically straightened hex-agonal phaseinequilibriumwiththesamepolymer solution.

As the total amount of DNA in a globule is constrained, knowledge of the lattice spacing also fixes the volume of the globule, V =

3'12La2/2,

where L is the length of the DNA

molecule. Therefore,we have: dQf

aFel,1

dV &V +

fp

= °

C2

E 2 Qa 7

312r2E

4Tr2P k = + 3112a2

(33)

(34)

Accordingly,

afk/aRc

= 0yields:

k1V5

_ 2R2 [1 2R2 R2)-1/2

One useful limit isathighsalt andfairly low polymer

con-centrations(k1 C2 -2andk2

44wXP/3112a2)

foraspectratios

Rt/Rc

> 2. 3'12C2a2R1 2PRc 2 L

p2V4

1/5

tRV

3(1/2r2C2a2

-33/2C2L2a6

)1/5 c V327r4pg2) (28) (36)

(37)

(38) which, together with Eq. 16, leads to:

31/2EK

C3 - (29)

where E is defined by E =- Qvefflrod(a) = F2((2/Ka)"2

exp[-Ka]. InEq. 29termsof

((llKa))

havebeen deleted.

At a fixed volume of the torus, we have one remaining degree offreedom,

RC

for instance. The balance of surface

DISCUSSION

Inourcalculationof the freeenergyofthetorus,undulations

of the DNAchainwereneglected. Potentially, however,even

chainundulationsmuchsmaller thanthe lattice spacingmay strongly enhance the electrostatic interaction between the winds, leading to a significant increase in the equilibrium

latticespacing. Moreover, if the undulations become larger

'P= C2 -2

than acertain fraction of the latticespacing,the lattice may undergo a melting transition (Odijk, 1993b).

We nowshow that the incorporation of the polyion

un-dulations into the previous analysis is straightforward. The total free energy ofa straight lattice ofweakly undulating semiflexible polyions (undulatory amplitude d << a),

bal-ancingthe electrostatic interactionagainstentropic confine-ment, maybecalculatedbyassumingaGaussian distribution

of theundulations. Fordetailsofthe undulationtheory, see Odijk 1993a(inTable I the entry for R = 4nmshould read 0.19 (0.011); the theory iscompared with osmotic

experi-ments on DNA (Podgornik et al., 1989), muscle filament (Millman,

1986),

and tobacco mosaic virus

(Millman

et

al.,

1984)). The viewpoint that the decay length ought to be renormalizedinastraightforward way(Podgornik and

Par-segian, 1990) has been criticized in the review by Odijk (1994). The statusofthesetheories has also been discussed

inthe context of field theorybydeVries (1994).

Minimization of the total free energy leadsto an

asymp-totic relation betweena andd(Odijk, 1993a):

d8'3

E= 2cQ2

(39)

9P13K2 (9

where thecoefficientcis estimatedtobe

2'.

Eis defined

by - UE,where Uisanundulatory factor: exp[½/2K2d2]

U 1 +

Kd2I2a

(40)

Theosmotic pressure isgiven by:

eIel 2c

3'/2,K

(41)

kBT

331/2Kad813p

1/3 Qa

Theequilibriumlatticespacingaandtheundulation

ampli-tuded maybe obtainednumerically from Eqs.39-41. Wesee

that the effect ismostpronouncedathigh concentrations of addedsalt, because the undulationenhancement ofthe

elec-trostaticpotentialbecomes significantwhen theundulatory amplituded is

(K-1).

InEq. 41,the bare interaction E(see

Eq.29) is replacedwith theundulation-enhanced interaction

E. In a new analysis of the electrostatics of a DNAtorus

incorporating undulations,wewouldhavetosubstituteEfor Ewherever the latter occurs, at least to within afactor of unity. Hence, our main result, Eq. 36, remains legitimate evenwhen thereareundulations. In thecontinuumlimit,the

undulations and spacing aregiven by Eqs. 28, 39, and41. Recently,a melting criterion for the hexagonal phase of DNAwasformulatedbycombining the theoryof undulation enhancement with the Lindemann melting rule (Odijk, 1993b). The latter states that on the melting curve of a

po-sitionallyordered system, the ratio 1 of the undulation

am-plitudeand thelatticespacing is a constant of order 0.1. For

macroscopic hexagonal phases of DNA, I

turns

out to be -0.13

(Odijk,

1993b). Forahexagonal toroid,we expecta melting transitionataboutthesamevalueof the Lindemann ratioI=

dia

',0.13,

atleast inthecontinuumlimit(a<<

RC

andC(Rc)=

C(Rt)

= Y(P)), where the bulk free energy terms aremuchlarger than the surface and curvature free energies. Asthe critical deflection length of the toroidal lattice is muchlarger than the lattice deflectionlengthA =

P113'&,

the undulations will not be suppressed by curvature (Odijk,

1993c). The higher order term discussed inOdijk'spaperis much smaller than those discussed here.

Ourtheoryfor DNA toroids isapplicablefor aspect ratios

R!Rc

>2, i.e., for toroids that are not too thick. Hence, we

imposeanupperlimit on thesize of theglobule becausethe aspect ratio decreases with increasing length of the DNA molecule(as L-l/5for

R,Rc

>2).Of course, this limitdepends

onthe salt andpolymer concentrations. Thedimensions of the torus are thus confined within rather narrow bounds in

viewof ourinsistenceonthe continuum limit(which restricts

the minimum size of the DNA).

If the aspect ratio becomeslargerthan2,thesimplemodel presented heredoesnotsufficequantitatively.As

Rc

tendsto

Rt,

the local density ofcurvature free energy will increase verystrongly, leadingto adeformation of the toroidalshape.

The surfacewill tend to flatten nearthe hole in the torus,

therebyincreasingthesurface free energy, butreducingthe

otherwisestrongincreaseincurvaturefree energy. Wehope

to returnto this issue.

Fortheglobuletobe stable thesurface tensionshouldbe positive; otherwise, an unlimited increase in surface area wouldbefavorableand the toruswouldcease toexist.

There-fore,

g,

2 1, so that the salt and polymer concentrations requiringstability arerelated by the following inequality:

v c43113-T23C23A4'3Q213n23 (42)

This condition neglects entropic contributions (first postulated forcrystals of hard particles by Kirkwood(1950)

long ago) that might enforce absolute stability ofthe

hex-agonalphase.

Finally, althoughwewould like tocompare Eq. 36with experiments, theyseemtolie outside thelimits of validityof the presentanalysis.Laemmli's nontoroidalglobules studied atveryhighsalt(Laemmli, 1975)appear to bein the regime where

RC

isveryclose to

Rt

andwherethe toroid model must

failas wehavealready pointed out.Toroidshave been

wit-nessed in the t+-condensationof DNAby Evdokimov et al.

(1976). The dimensions of the torus photographed in their

Fig.

3 agree well with

Eq.

36:

RC

-25nm,

Rt

-50nm, P 50 nm, a 3.7 nm, = 0

((a).

However,

the

globule

consists of many DNAmoleculesso the agreementwith

our single molecule theory may be fortuitous. We hope

thesimple rule Eq. 36 mayinciteexperimentaliststo

re-investigate

ti-toroids.

CONCLUSION

Weagainstressthe limited scope of this work. The DNA macromolecule isviewedas auniformly charged cylinder the undulationsofwhich arestrictlyelastic orwormlike. Thesurrounding ionicdistributionis evaluated withinthe

TABLE 1 Approximations usedtoderiveEq. 36 Polymer scaling theory v' 0.1

Double layer superposition a2 D + K-1 Continuumapproximation RC 5 a

Idealtorus Rt 2 RC

Leading orderapproximations Ka >>1

Poisson-Boltzmann approximation (Fixman, 1979). The

approximations leading to our main result are listed in Table 1.

A referee hasremarked on the difficulty of thecharged worm model for DNA providing an explanation for the

isotropic-precholesteric-cholesteric transition

(Livolant,

1987; Strzelecka andRill, 1991; Merchant andRill, 1994).

Wemayaddtothat thepeculiaritiesfoundrecentlyin the be-havior ofsemidiluteisotropicsolutions of DNA(Strzelecka

andRill, 1992; Wissenburgetal., 1994).On theotherhand,

neutron scattering work oncholesteric DNA (Groot et al., 1994),extended to study the ionicstrengthdependenceofthe

orientationalorder,is wellrepresented by electrostatic theory (K. Kassapidou andJ.vanderMaarel,UniversityofLeiden, manuscript in preparation). Itis important to note that the

isotropic-cholesterictransition is very sensitiveto

perturba-tiveinfluences(e.g.,weak attractiveforces).Inisotropicand

cholesteric solutions long-range positional order is absent

(disregarding the helicoidalorganization of thecholesteric)

and the DNA molecules sample the entire configurational

space beyond the effective diameter of each polyion (Stroobantsetal.,1986).By contrast, in thehexagonalphase

the molecules arepinnedwithinasharply definedpotential

trough, which isconsiderablylesssensitive toperturbations.

Hence, difficulties in explaining the isotropic-cholesteric

transition neednotfalsifytheusefulness of theelectrostatic

model for the hexagonal phase.

REFERENCES

Arscott, P. G., A. Li, and V. A. Bloomfield. 1990. Condensation ofDNAby trivalent cations. I.Effects ofDNAlength and topology

on the size and shape of condensed particles. Biopolymers. 30:

619-630.

Bloomfield, V. A. 1991. Condensation ofDNAby multivalent cations: considerationsonmechanism.Biopolymers. 31:1471-1481.

Bloomfield,V.A., R. W. Wilson, andD. C. Rau. 1980.Polyelectrolyte effects in DNA condensation by polyamines. Biophys. Chem. 11:

339-343.

Brenner, S. L., andV.A.Parsegian.1974. Aphysical methodfor deriving

the electrostatic interaction between rod-like polyions at all mutual angles. Biophys.J. 14:327-334.

Chattoraj,D.K.,L. C.Gosule, and J. A.Schellman.1978. DNA

conden-sation withpolyamines. II.Electronmicroscopic studies.J. Mol. Biol. 121:327-337.

Evdokimov, A. L., Platonov, A. S. Tikhonenko, and Y. M. Varshavsky. 1972. Acompact form of double-stranded DNA in solution. FEBS Lett.

23:180-184.

Evdokimov,Y.M.,T. L.Pyatigorskaya,0.F.Polyvtsev, N. M. Akimenko, V. A.Kadykov,D. Y.Tsvankin, and Y. M. Varshavsky. 1976. A

com-parativex-raydiffractionandcircular dichroismstudy of DNA compact particlesformed in water-salt solutions, containing poly(ethylene glycol). Nucleic AcidsRes. 3:2353-2366.

Fixman, M. 1979. ThePoisson-Boltzmann equation and its application to polyelectrolytes. J. Chem. Phys. 70:4995-5005.

Frisch, H. L., and S. Fesciyan. 1979. DNA phase transitions: the +i-transition

ofsinglecoils. J.PolymerSci. PolymerLett. Ed. 17:309-315.

de Gennes,P.G. 1979. Scaling Concepts in Polymer Physics. Cornell Uni-versity Press, Ithaca, NY.

Gosule,L.C., and J. A. Schellman. 1976. Compact form of DNA induced byspermidine.Nature. 259:333-335.

Gosule, L. C., and J. A. Schellman. 1978. Condensation with polyamines. I. Spectroscopic studies. J.Mol. Biol. 121:311-326.

Groot, L.C. A., M.E. Kuil,J. C. Leyte, J. R. C. van derMaarel, R. K. Heenan,S.M.King, and G. Jannink. 1994. Neutronscattering

experi-ments onmagnetically aligned liquid crystallineDNAfragmentsolutions.

Liquid Crystals. 17:263-276.

Grosberg, A. Y. 1979. Certain possibleconformational states of a uniform

elasticpolymer chain. Biophysics. 24:30-36.

Grosberg,A.Y.,I.Y.Erukhimovitch,and E. I. Shakhnovitch. 1982. On the

theory of+i-condensation. Biopolymers.21:2413-2432.

Grosberg,A.Y.,and A.R. Khokhlov. 1981. Statisticaltheory ofpolymeric lyotropicliquid crystals.Adv. Polymer Sci. 41:53-97.

Grosberg,A.Yu.,and A. V. Zhestkov. 1986. On the compact form of linear

duplexDNA: globularstatesof the uniformelastic(persistent)

macro-molecule. J. Biomol. Struct. Dyn. 3:859-872.

Joanny, J.F., L.Leibler,and P.G. de Gennes.1979. Effects ofpolymersolutions oncolloidstability.J.PolymerSci.PolymerPhys.Ed. 17:1073-1084. Jordan, C. F.,L.S.Lerman,and J. H. Venable. 1972. Structure and circular

dichroism ofDNA inconcentrated polymersolutions. Nat. New Biol.

236:67-70.

Kirkwood,J.G. 1950.Critique of free-volumetheoriesof theliquidstate. J. Chem.Phys. 18:380-382.

Kirkwood, J. G., and I. Oppenheim. 1961. Chemical Thermodynamics. McGraw-Hill,NewYork.

Laemmli,U.K. 1975. Characterization of DNA condensates inducedby poly(ethyleneoxide) and polylysine. Proc. Natl.Acad. Sci. USA. 72: 4288-4292.

Lerman, L.S. 1971. Atransitionto acompactformofDNAinpolymer solutions. Proc.Natl.Acad. Sci. USA. 68:1886-1890.

Lerman, L.S. 1973. Thepolymer- and salt-induced condensation ofDNA. InPhysico-ChemicalProperties of the Nucleic Acids. J. Duchesne, editor. Academic Press, London. 59-76.

Lifshitz,I.M.,A. Y.Grosberg, andA. R.Khokhlov. 1978. Someproblems of thestatisticalphysics of polymer chains with volume interaction.Rev. Mod.Phys. 50:683-713.

Livolant,F.Precholesteric liquidcrystallinestatesofDNA. 1987.J.Phys. 48:1051-1066.

Maniatis, T., J. H. Venable, andL. S. Lerman. 1974. The structure of

+i-DNA.J. Mol. Biol. 84:37-64.

Marx,K.A., and G. C.Ruben.1983.Evidence forhydratedspermidine-calf thymus DNA toruses organized by circumferential DNA wrapping. Nucleic AcidsRes. 11:1839-1854.

Marx,K.A.,andG. C. Ruben. 1986.AStudy of4X-174DNA torusand

lambdaDNAtorustertiarystructureand the implicationsfor DNA

self-assembly.J.Biomol. Struct. Dyn. 4:23-39.

Merchant, K., andRill,R. L.1994. Isotropictoanisotropicphase transition

ofextremely longDNAinanaqueoussaline solution. Macromolecules. 27:2365-2370.

Millman,B. M. 1986.Long-rangeelectrostaticforces in cylindrical systems:

muscleand virusgels.InElectricalDoubleLayersin Biology.M.Blank,

editor. PlenumPress,NewYork. 301-312.

Millman,B.M.,T.C. Irving,B.G.Nickel, andM. E.Loosley-Millman. 1984. Interrod forces inaqueousgelsof tobaccomosaicvirus. Biophys. J. 45:551-556.

Odijk, T. 1993a. Undulation-enhanced electrostatic forces in hexagonal

polyelectrolyte gels. Biophys. Chem. 46:69-75.

Odijk,T.1993b.Evidence for undulation-enhanced electrostatic forces from themeltingcurvesofalamellar phaseand a hexagonalpolyelectrolyte gel. Europhys. Lett.24:177-182.

Odijk, T. 1993c.Physics oftightly curvedsemiflexible polymerchains. Macromolecules. 26:6897-6902.

Odijk, T. 1994. Undulation-Enhanced forces inhexagonal gels of

semi-flexiblepolyelectrolytes. In Macroion Characterization. ACS Symposium Series. 458. K. S. Schmitz, editor. 86-97.

Philip, J. R., and R. A. Wooding. 1970. Solution of the Poisson-Boltzmann equation about acylindrical particle. J. Chem. Phys. 52:

953-959.

Podgornik,R.,andV. A.Parsegian. 1990. Molecularfluctuationsinthe

packing of polymeric liquid crystals. Macromolecules. 23:

2265-2269.

Podgornik, R., D. C. Rau, and V. A. Parsegian. 1989. The action of interhelical forces on the organization of DNA double helices:fluctuation-enhanced decay of electrostatic double-layer and hydration forces. Macromolecules. 22:1780-1786.

Plum, G. E.,P.G. Arscott, and V. A. Bloomfield. 1990. Condensation of DNAby trivalent cations. II. Effects of cation structure. Biopolymers. 30:631-643.

Post, C. B., and B. H. Zimm. 1979.Internal condensation of a single DNA molecule.Biopolymers. 18:1487-1501.

Post, C. B., and B. H. Zimm. 1982.Light-Scattering study of DNA con-densation: competition between collapse and aggregation. Biopolymers 21:2139-2160.

Stroobants, A.,H. N.W.Lekkerkerker,andT.Odijk. 1986. Effect of

elec-trostaticinteraction on the liquid crystal phase transition in solutions of rodlikepolyectrolytes. Macromolecules. 19:2232-2238.

Strzelecka,T.E., and R. L. Rill. 1991. Phasetransitions in concentrated DNA solutions:ionic strength dependence. Macromolecules. 24:5124-5133. Strzelecka, T. E., andR. L.Rill. 1992. 23Na NMR of concentrated DNA

solutions: salt concentration and temperature effects. J. Phys.Chem. 96:

7796-7807.

de Vries, R. 1994. Undulation-enhanced electrostatic forces in lamellar phases of fluid membranes. J. Phys. IIFrance. 4:1541-1555. Widom, J., andR. L.Baldwin.1980.Cation-induced toroidal condensation

of DNA: studieswithCo3'(NH3)6. J. Mol. Biol. 144:431-453. Wilson, R. W., and V. A. Bloomfield. 1979. Counterion-Induced

conden-sation ofdeoxyribonucleic acid: alight-scattering study. Biochemistry. 18:2192-2196.

Wissenburg, P.,T.Odijk,P.Cirkel, andM.Mandel. 1994.Multimolecular aggregation in concentrated solutions of mononucleosomalDNAin1M sodium chloride.Macromolecules. 27:306-308.