john e. hearst and nathaniel g. hunt- statistical mechanical theory for the plectonemic dna...

8/3/2019 John E. Hearst and Nathaniel G. Hunt- Statistical mechanical theory for the plectonemic DNA supercoil

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Statistical mechanical theory for the plectonemic DNA supercoilJohn E. HearstDepartment of Chemistry, University of California, Division of Chemical Biodynamics, Lawrence BerkeleyLaboratory, Berkeley, California.94720Nathaniel G. HuntDepartment of Physics, University of California, Division of Chemical Biodynamics, Lawrence BerkeleyLaboratory, Berkeley, California 94720(Received 5 August 199 1; accepted 6 September 199 1)The eigenfunctions for circular boundary conditions of the differential equation fust used byHarris and Hearst in 1966 to represent the dynamic properties of the wormlike coil have nowbeen applied to the closed circular coils of high writhe. In order to avoid problems of knottingand excluded volume, the discussion here has been restricted to three-dimensionaleigenfunctions with near plectonemic symmetry, i.e., eigenfunctions which cross each constantz plane only twice. It is concluded that at the natural levels of superhelical density that arefound for DNA in vim, the DNA free of protein must be in a highly reduced configurationalentropy state. The impact of this conclusion on issues of entanglement, chromosomereplication and segregation, and chromosome organization are discussed.

I. INTRODUCTIONDNA supercoiling and packaging are essential featuresof cellular architecture and must be understood before generegulation and the dynamics of cell division can be properlyaddressed. Many points of view have been presented in ef-forts to provide enlightenment on these problems. First, theDNA alone condensation has been studied both experimen-tally1~2 and theoretically.3,4 Chromatin and the nucleosomeproblems, both with respect to structure56 and with respect

to the dynamics of this structure in relation to gene regula-tion and expression, -lo have been studied. DNA supercoil-ing has received extensive study since its discovery, tist ex-perimentally as a static property,-I6 and then dynamically,primarily as linked to transcription.,* Mathematical rulesfor calculating linking number (Lk) and writhe ( Wr) havedeveloped slo~ly~~-~~ and efforts at the determination ofbending2js2 and torsiona128-31 orce constants have led tosome computer modeling of DNA molecules with knownwrithe densities.32In all of these studies what has been lacking has been atractable mathematics which al lows for the representation(albeit approximate) for the various quarternary foldingstructures which DNA is hypothesized to form. Further-more, while the importance of writhe to the issue of foldingand packaging has been obvious to most, and the mathema-tics for the calculation of writhe for any space curve has beenreasonably well described in the literature, there have beenno applications to continuous space-curve representations ofDNA.We were first attracted to this problem by a suspicionthat the oxygen regulation of the photosynthetic gene clusterof Rhodobacter capsulatus (a 46 kilobase-pair region of thisbacterial genome committed to photosynthesis) would berelated to DNA torsional tension.33 After several rounds ofcareful experimentation, there is no definitive evidence thatsignificant changes in torsional tension occur during this cel-

lular process,34 suggesting that this may be a conserved pro-perty in vivo.In an effort to understand the significance of linkingnumber deficit or excess to the polymer configurations avai-lable to DNA in phase space, we have used the dynamictheory of Harris and Hearst3 for the worm-like coil. Wehave concluded that at the high-excess linking-number den-sities (pALk) found in nature ( - 0.05 turns per turns DNAduplex), a DNA molecule is effective in a single conforma-tional state with very low polymer configurational entropy,and we believe that this low-entropy state is likely to be theevolutionary reason for DNA supercoiling. Such a state mayavoid problems of entanglement during DNA replicationand during chromosome segregation. c

II. THE NORMAL COORDINATESThe Harris-Hearst theory for stiff-chain polymers- (thewormlike coil) is a dynamic representation of a polymer interms of four parameters: p, the mass per unit length; J; thefriction factor/unit length; ohend,a bending force constant or

flexibility parameter; and L, the contour length of the chain.The principal assumptions are that the potential energy ofthe chain be given by the continuous integral,Bv=? (1)and s is a contour position coordinate having a range- L /2


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J. E. Hearst and N. G. Hunt: The plectonemic DNA supercoil 9323

(3)(4)

This procedure symmetrizes the bending motions inthree dimensions, leading to a longitudinal degree of free-dom as well as the two transverse degrees of freedom asso-ciated with bending. In addition, a fluctuation in contourlength is artificially introd.uced into the motion of the po-lymer. This approach converts an otherwise nonl inear prob-lem into one requiring the solution of a simple linear differ-ential equation. The theolry has been demonstrated to behighly successful in predici.ing quadratic moments of the po-lymer motion such as the pair-correlation function betweencontour positions on the chain, both in the limit of the rigidrod where the positions are very close to each other in con-tour and in the limit of the random coil where the positionsare many statistical lengths away from each other on thecontour.The time-independent eigenfunctions for circular boun-dary condi tions are simple sines and cosines satisfying thecircular condition. These functions represent a completeorthonormal set, capable of describing all configurationsaccessible to circular stiff-chain molecules. In particular,they are capable of describing configurations with writhe:

Y, = (2/L)12 sin[ (2ms/L) + 51, (5)where S is an arbitrary phase angle between 0 and 2a.Assessment of what is required for functions of definitewrithe leads to some intcdtive simplifications which limitphase space to a certain class of molecular shapes, perhapsignoring other classes which also have writhe. The logic formaking these simplifications is that otherwise phase spacewould include highly knotted configurations, and configura-tions in which the polymer, as a line, touches itself (coinci-dence). Both such classes of structures must be eliminatedfor physical reality. We therefore initially limit ourselves toconfigurations of symmetry similar to those of plectonemicsupercoils, an example provided in Fig. 1. The lowest-energyconfiguration for space curves with stiffness and writhe willtypically have a plectonemic structure.21An eigenfunction with definite writhe must be three di-mensional. Thus, the constraint of writhe requires that theeigenfunctions for the x, y, and z directions are no longerindependent. We arbitrarily assign the long axis (the screwaxis) of the plectonemic helix to the z direction. In order toavoid knotting and coincidence, the sine function in the zdirection experiences only one period, n, = 1, over the con-tour length L of the closed circle. The values of n, and nY arenow allowed to take on any values, but the three-dimen-sional wave function is made up of only one simple sine or

831P3 d.P$wFIG. 1. A DNA plectonemic superhelix. The approxi-mate properties of this positive superhelical structure arethe foll owing: superhelical pitch = 52; Wr = 3.95;Tw = 47.05; Lk = 51; Two = 46.10; Lk,, = 46.10;ATw,, = 0.95; ALk, = 4.90; c = + 0.106 turns/turnDNA helix; d = 90 A.

cosine function along each direction. While this procedure isnot the only one possible, more complex functions can begenerated by taking li near combinations of these simplifiedthree-dimensional functions.We have selected orthonormal eigenfunctions which re-tain the terminology of Harris and Hearst.26 In representingthe approximate plectonemic helix, there is a necessaryphase relationship between the trigonometric functions inthe three directions. The only positions where both x and yequal zero simultaneously are at the ends of the helical struc-ture where z is at one of its two extrema which occur ats = - L /2,O,L /2. This selection for the phasing of s on thecontinuous circular molecule is one of many possibilities, butthis choice is sufficiently general.The vector eigenfunction X has three vector compo-nents X = xi + yj + zk, where the components are

z== E,(2/L)/2COS(2rrS/L),.x=ex(2/L) 12 sin(2m,s/L), (6)y=~,(2/L)sin(25-+7/L),

and s, the contour parameter, has a range - L /2


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x,, = X(s,1 X(s,1. (9)In the context of closed circular DNA molecules as shown inFig. 1, the two phosphodiester backbones of the single DNAstrands may be thought to be the edges of a ribbon. Thelinking number of this closed ribbon is established by wrap-ping the ribbon about a cylinder so that its center axis lies in aplane perpendicular to the cylindrical axis. If one starts withthe ribbon flush with the cylindrical surface (untwisted),the linking number of a closed ribbon is the number of 360Qrotations the ribbon must unde rgo at a single cut relative tothe other end of the cut in order to achieve the final state ofthe ribbon. Because the DNA strands have different polari-ties, 180 otations, leading to moebius strips, do not occur inthe DNA. The linking number, Lk, is always, therefore, apositive or negative integer depending on the direction ofrotation. Once the center axis of the ribbon is no longer con-strained to a single plane, the linking number can be accom-modated by writhe Wr as well as twist Tw. Whitem has pro-ven that

Lk=Tw+Wr. (10)In the DNA world the DNA duplex is assumed to havea definite pitch or helical repeat, h, of 10.4 basepairs per turnor 35.4 A/turn of right-handed helix and a 3.4 8, spacing perbasepair along the helix axis. By convention, the linkingnumber is replaced by a linking number difference, ALk,between the hypothetical linking number it would have ifunconstrained, Lk,, and its actual value. Lk, equals the to-tal number of basepairs N divided by 10.4. Therefore,ALk = Lk - Lk, = Lk - N/10.4.

Because N/10.4 need not be an integer, ALk need not be aninteger either. Changes in ALk still occur in integral steps,however. Twist, Tw for DN A is also measured as a differ-ence from its expected unconstrained value, N/10.4:

ATw = Tw - Tw, = Tw - N/10.4.Therefore,

ALk = ATw + Wr. (11)For any definite circular DNA molecule, ALk is fixed,and the linking number deficit or excess in the DNA willdistribute itselfby modifying the twist of the DNA duplex orby configurational modifications of the axis of the duplex togive its space-curve writhe. The relative importance ofchanges in twist, ATw, and writhe, Wr, in this process hasbeen determined experimentally,16 and has been predictedby Monte Carlo simulations,32 however, no prediction is yetavailable from statistical mechanics. The following theorypresented here achieves that goal for a restricted class ofpolymer configurations, those approximating the plectone-mic helix and, therefore, having cylindrical symmetry aboutan axis in space. The dependencies of configurational poten-tial energy on twist and writhe arc presumed to be uncoupledexcept by virtue of the constraint that ALk remains constant.Numerical evaluation of writhe by Eq. (7) by computeranalysis for any space curve having a parametric representa-tion in a contour parameter s is straightforward as long as

care is taken with the limits of integration. The numeratorand denominator of the argument of integration,f(s, ,s, ), in

Eq. (81, both approach zero as s1 approaches s2. The argu-ment, however, remains finite.IV. IMPORTANT ElGENSTATES AT THERMALEQUILIBFWM

Upon calculating the writhe for each of our three-di-mensional eigenfunctions, a striking fact is observed. Takingsome arbitrary ratio of e, to l X = ev which is far greater than1, say 10, leads to the conclusion that the highest writhe isalways found on the + 1 or - 1 off diagonals of a matrix inthe variables n, and izg This conclusion becomes even moreconvincing as one goes o large n. We conclude, using this setof eigenfunctions, the highest writhe for a given energy mustbe obtained from those functions for which n, = nY + 1 ornp - 1.We are, however, required to determine the correctvalue for QE,, the ratio of the major to minor axis of theplectonemic helix. It is here that Harris-Hearst theoryproves useful. In terms of the eigenfunction expansion coeffi-cients and the eigennumbers, the average length (L ) of thecircular DNA molecule can be written as

(L ) = j-;s (9. $) ds,

9324 J. E. Hearst and N. G. Hunt: The plectonemic DNA supercoil

(12)(L> m7m2(n3~2,) + n;cq + (2)). (13)The time-average potential energy ( V) of the molecule is

(V> = Wbend/2) Gwl5~W(~) + $! 6) + (6)>.(14)

The eigenvalues for the X, y, and z directions are/JCL, (2%-/Jc)[n + ( l/2%q2(hL )n], (15)

where A = y&-d. From the discussion which will follow,for molecul es of large numbers of statistical lengths, the rz2term dominates for low values of ri and the n4 term domi-nates at large values of 12.Small values of IZ relate to therandom-walk characteristics of the distribution of elements;large values of n relate to the local bending of the stiff rod.As in all normal mode analyses, the distribution func-tion for the expansi on coefficients is ust a statement of equi-partitioning of thermal ene rgy to each normal mode. Fromthe distribution function of Harris and Hearst [Eq. (22) ofRef. 35 1 the mean-square average of each expansi on coeffi-cient is

65 hE = (kT%,, I( 1/P, 1, (16)where the subscript TE refers to a thermal equilibrium aver-age according to Harris and Hearst. This equation is validfor a space curve with no cross-sectional area. The bendingforce constant, fibend, s related to the statistical length of theDNA. Again, retaining old terminology,26 the Kuhn statis-tical length is 1/;1 and the persistence length is l/u. Therelationship between the force constant and the parameter ilis27,37

Bbend= kT/W. (171For the z expansion coefficient and large AL , since

n, = 1,(


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The parameter AL 2 is discussed in detail by Harris andHearst.35 It may be approximated by the expressionAL2=(2U)2/(1-{[1-exp( -WL)]/WL)).

(19)For the limit of a very large :number of statistical lengths, theonly case considered in this work, AL = 4(n~C)~, or fourtimes the square of the nurnber of statistical lengths in thecircular DNA.In the random coil limit, where one ignores the n4 termin Eq. (15), ( ( 0.03 turns per turn helix. For the x and y expan-sion coefficients, thermal equilibrium requires that, in thelimit of large n which is the useful limit for the high linkingnumber density found for DNA in vivo,

TABLE I. Writhe matrix for the random coil.nx

..I

2 3 4 5 6 7 8 9 10 11 12 73 74 15 16 17 18 79 2::j:i::~:~$.~.:.:.:.:.m,,_.~.~~~~~.:,:,:.:, - 2 1 -1. 1 -1 1 -1 1.:.:!!.::?.-g; i::s:j:::.:r:..jfii

123456789

b 10

I1121314151677181920

-4-1:.:.:.:::~:~:t:~:~:F$>?T;pz;:awa?1 - 3 2 2 -1 :$i$@::+:.:.:.;:.:; ::::::::::-.p$ 0.A......... .%a% Q+..:..:::.:-1 -2 2 1 ?z;a$$:! j:s;p>>.,:.:r _ !$z&:.:.:::::::.:.~~:::::i::::::$& !+k?:-2 1 -4 3 1 p& $j@;$.~:;.::a:$: :. 1$Xggj.-.....:.:iI 1 6 2 -2 :$.c--,--

g$g3 1 4 g#$. ,:.:,:.:.fg-1 3 -5 4>..>. _ .:.:,:.!.:.:.. ^

2 -1 7. 1 -1 - -2 .l g$g

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these two expansion coefficients equal to each other. Theseexpansion coefficients are now considered to be constants intime.

9326 J. E. Hearst and N. G. Hunt: The plectonemic DNA supercoil(E,& ==( 1/2;,4(4/lL/3)L 3( l/n,)4. (20)A quick calculation of the root-mean-square value of x,

(x2) 12, or the conditions in which DNA is usually found,leads to a value ~20 A. Since the diameter of the DNAcylinder is approxi mately 20 A, this result is unrealistic, in-dicating that the configuration of the plectonemic helix athigh writhe is sterically determined and that the z axis ismaximized to generate a helix with the tightest radius steri-tally allowed. Thermal equilibrium in the sense of Harrisand Hearst can never be achieved for DNA molecules ofhigh writhe density. Within this approximation, thermalfluctuations have only small fractional impact on the dimen-sions of the plectonemic helix and the helix may ,be repre-sented by the eigenstates coming closest to the requiredwrithe. There is thus a reduction of configurational phasespace from 19,~~) he total number of degrees of freedom ofthe stiff chain circle which is of the order of magnitude of 3times the number of basepairs, to a very small number oforder 1.

Using Eq. (2 1)) the x and y axis expansion coefficientsare determined by a choice of the root-mean-square distanced of closest approach of the centers of the two helices..Be-cause of the negative charge density on the DNA backbone,this number is not likely to get smaller than 50 A, althoughan experimental determination of its value is necessary andfeasible. Thus,ex = ey = d( L ,8) 12.From the constraint on L [ Eq. ( 13 ) 1, setting (L ) = L,and from the high writhe requirement that n = n, = n, * 1,the square of the z expansion coefficient becomes< = [L~/(~T)~] - [ni- (n i- 1)21(L~8)(d2).

(23)

Since the thermally predicted values of the expansioncoefficients are not accessible , it is necessary to relate theexpansion coefficients, Ed and eu, to a distance of closestapproach between DNA duplex regions. In the xy plane atypical plot of the distance between the centers of the DNAduplex in a selected eigenstate is shown in Fig. 2 as a functionof s. The mean-square distance for a given eigenfunction is

(d2) = (4/L)(< + 6). (21)It can further be shown that this value is a maximum ifex = 6,. We have thus established the rationale for setting

For the very small values of d relative to L which we areconsidering, the first term is dominant even at large n. Ne-vertheless, this is the expression for the reduction in the zaxis of the helix as the number of turns increases and its pitchdecreases.Table II shows data calculated for DN A where the fol-lowing parameters were used: L is in number of basepairs(bp) and, therefore, equals N, d is 15 bp, l/ii = 300 bp, andwrithe = writhe densityx (3.4)N. The parameter IpWrl isthe absolute value of the writhe density in units of writhe perunit length and the parameter 35.4 lpWr/ is the absolutevalue of the writhe density in turns writhe per turn DNAhelix.

(22)

VI. POTENTIAL ENERGY PER UNlT LENGTHINCLUDING BOTH WRlTHE AND TWISTFor the conditions in Table II and writhe density perturn helix values greater than 0.03 and less than 0.05, it ispossible to derive the following equation for the potentialenergy in A, pE,,, , of the plectonemic helix. The experimen-tal approximation, that the force constants&,,, for bendingand fltwist for torsion are equal, is often applied:P&t =Pbend2r4d2@Wr)41Ptwist~~-YPATw)~I,(24)

FIG. 2. The distance between the center of the DNA duplex in constant xplanes. The abscissaxis the contour par ameters with range from - L /2 to0 left to right and/or from 0 to L 12 right to left. The ordinatey is a relativedistance with range from 0 to approximately 1.4d. The v) in this figureequals d. The particular normal coordinate this plot applies to n, = 9, andn, = 10. It should be noted that while v) is large enough to satisfy thesteric constraints of a cylindrical structure like DNA of finite diameter d,this normal coordinate has~regions which clearly violate the steric con-straint. To properly satisfy the steric contraint uniformly along the contour,linear combinations of the normal coordinates will have to be used. Thisissue will be addressed in detail in a future paper.

TABLE II. Representations for DNA plectonemic helices at physiologicalwrithe densities. Notice that for DNA at these writhe densities, the integerindex n and total writhe are nearly identical.

Ubp) %/CTotal writhe and35.41pWr( = 0.03

in)(integer index)35.4)pWrl = 0.05

(n)2000 60 5.8(7) 9.6( 10)6000 180 17( 18) 29(30)10 000 300 29(30) 47(49)50 000 1500 143( 145) 235(241)

100000 3ooo 286(289) 472(482)



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where writhe density pWr and twist difference densitypATw are defined as the turns of writhe and twist per unitlength of DNA.The equation shows deviations at higher writhe densi-ties because n does not predict the value of writhe at veryhigh writhe densities where terms of order higher than thefourth power in writhe density become important. The highwrithe density domain, equivalent to lower pitch helices, isdiscussed in some detail in the following paper.38 At lowerwrithe densities, thermal equilibrium begins to play a roleand the number of configurational states that must be consi-dered for the DNA grows rapidly.Equation (24)) for energy density, plus Eq. ( 11)) whichprovides a relationship between writhe density and twistdensity, allows one to predict the energy of a coil of constantlinking number density but varying writhe and twist. Thelowest-energy state is also predictable. Since Eq. (24) is es-sentially identical to a more genera l equation, derived in Ref.38 by a different method, conclusions involving numericalpredictions are presented there.

VII. CONCLUSIONBy application of the principles of statistical mechanicsin the form of Harris-Hearst theory to supercoiled closedcircular wormlike coils, we have been able to predict someimportant properties of such coils in their plectonemicforms. ( 1) Above a minimum writhe density, the number ofpolymer configurational states available to the coil drops to anumber near 1. This statement relates only to the configura-tional states within the context o f the continuous model pre-

sented here. The internal vibrational and electronic statesassociated with the atomic structure of DNA have not beenconsidered. (2) The writhe density that DNA has in vivo islarger than this minimum, indicating that in cells DNA hasvery little configurational entropy. (3) In these low-entropystates, the potential energy of the coil depends on writhedensity to the fourth power and higher-order terms.It is of interest to speculate on the evolutionary advan-tage of low-entropy states for DNA in vivo, for the cell mustexpend considerable energy in order to sustain these low-entropy states. First, since DNA is compacted in volume bya factor greater than 10 in living cells relative to the volumeit occupies when free in solution, its entropic freedom mustbe greatly reduced in the cell anyway. Second, writhe densityprovides a possible systematic mechanism for the compac-tion, especially since DNA in viva is known to exist in topolo-gically constrained domams.Finally, knotting of highly compacted DNA could pre-sent major difficulties to the processes of replication, tran-scription, and chromosome segregation. While enzymes ex-ist which can unknot DNA in very dilute solutions whereentropy drives knotted objects apart, the unknotting of high-ly compacted DNA is likely to prove far more difficult un-less a local structural distinction can differentiate a knotfrom two DNA duplexes which happen to be close to eachother. High writhe density can help solve this dilemma intwo ways. It can make knotting very unlikely to start with

because the structural backbone of the DNA is thicker andstiffer, making the threading of one section of DNA into theloop of another section more unlikely. It may also providethe structural distinction by which a knot is distinguish froman ordinary supercoil. In the low-entropy limit, a supercoilhas very uniform characteristics, making it readily distingui-shable from a knot.It is being proposed here that the living cell is obliged tomaintain a state of least DNA entanglement, and this princi-ple of least entanglement requires that the DNA sustain ahigh writhe density in vivo.

ACKNOWLEDGMENTSWe wish to extend our thanks to David Cook, RobertHarris, Vaughan Jones, and Paul Selvin for discussions andsuggestions during the development of this theory. Thiswork was supported in part by the National Institutes ofHealth Grant No. GM 41911 and by the Office of EnergyResearch, Office of Health and Environmental Research ofthe Department of Energy, under Contract No. DE AC03-76SFOOO98.

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john e. hearst and nathaniel g. hunt- statistical mechanical theory for the plectonemic dna...

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