GLOSSARY

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a	DNA hard-core radius
A	Hamaker constant, scaled by kBT
b	coupling parameter of harmonic potential
c	concentration of monovalent salt
cr	coefficient of confinement free energy, confinement in r
cp	coefficient of confinement free energy, confinement in p
dp	root mean square undulation in p
dr	root mean square undulation in r
f	perturbation per unit length of strand
	total free energy of plectoneme per unit length of strand
conf	confinement free energy per unit length of strand
el	undulation-enhanced electrostatic free energy per unit length of strand
el,0	electrostatic free energy of the nonfluctuating configuration per unit length of strand
VdW	van der Waals free energy per unit length of strand
g	generalized bending constant
Gp	Gaussian distribution of undulations in p
Gr	Gaussian distribution of undulations in r
h	helical repeat DNA relaxed state
c	elastic Hamiltonian
kB	Boltzmann's constant
Lk	linking number
Lk0	linking number relaxed state
Lk	excess linking number
m1, m2	fitting coefficients of the approximation of the electrostatic potential
ns	number concentration monovalent salt
p	pitch/2 of plectonemic superhelix
Pb	DNA bending persistence length
Pt	DNA torsional persistence length
q	elementary charge
QB	Bjerrum length = q2/(kBT)
r	radius of plectonemic superhelix
Rc	radius of curvature in plectonemic configuration
s	contour distance
T	absolute temperature
Tw	twisting number
u	angle of plectonemic rotation
ur	amplitude of undulation in r
up	amplitude of undulation in p
w	dimensionless parameter = 2r
Tw	excess twisting number
Wr	writhing number
r	writhe per unit length of strand of the plectonemic helix
Z	function defined by Eq. 19

Greek symbols

	plectonemic opening angle;   2
	gamma function
	dielectric permittivity of solvent
	constant in the undulatory entropy accounting for non-Gaussianity
1	Debye length
c	curvature classical plectonemic configuration
	deflection length
µ	dimensionless parameter = p2/4r2
eff	effective linear charge density of DNA
	Poisson-Boltzmann charge parameter
	distance between two points on the superhelical contour
	specific linking difference
	inverse plectonemic parameter = h0/(4r||)
	dimensionless distance = /2r
	electrostatic potential, scaled by q/kBT
1	electrostatic potential, scaled by q/kBT
	renormalized potential, scaled by q/kBT
0	twist rate relaxed DNA
	excess twist

	INTRODUCTION

TOP ABSTRACT GLOSSARY INTRODUCTION TOPOLOGY ELECTROSTATIC POTENTIAL OF... UNDULATION ENHANCEMENT OF THE... ENTROPY FREE ENERGY DISCUSSION CONCLUDING REMARKS APPENDIX I APPENDIX II REFERENCES

Both the global conformation and the local structure of the DNA double helix depend subtly on applied forces. Entropy, interactions, topological constraints, and external forces are nonlinearly intermingled to various degrees, giving rise to the remarkable structural and functional versatility of the DNA molecule (Bloomfield et al., 1974; Sinden, 1994).

When put under sufficient torsional stress, a closed double-helical chain of DNA will respond by forming superhelical structures that are more or less regular and interwound. In the plectonemic helix (Fig. 1), two strands of the double helix are intertwined, each superhelical strand displaced with respect to the other by half the superhelical pitch. At least two end loops are present, but there may be more loops if branching defects occur.

View larger version (8K): [in this window] [in a new window]   FIGURE 1   A configuration of the plectonemic helix. r is the radius and 2p the pitch of the plectoneme, and  is its opening angle.

In the supercoiling of DNA, topology and twist are intimately related. The topology of a complex molecule like DNA, however, gives rise to multifarious phenomena, whose relevance extends well beyond supercoiling alone (Wasserman and Cozzarelli, 1986; Cozzarelli and Wang, 1990; Bates and Maxwell, 1993; Stasiak, 1996). It may bear on both isolated molecules and those in congested states, on the formation of knots (Liu et al., 1981), and on the catenation of rings (Martin and Wang, 1970). Topological constraints may be permanent or may manifest themselves only transiently when obstructions or entanglements diffuse away.

Our understanding of the biological implications of supercoiling is still incomplete, although many qualitative arguments and models supporting either passive or active roles of supercoiling have been advanced (Sinden, 1987; Cozzarelli and Wang, 1990; Stasiak, 1996). At present it is thought that supercoiling may be functional with respect to the compaction of DNA, in this way enhancing the rate of certain recombination reactions by bringing together distant segments of DNA (Wasserman and Cozzarelli, 1986; Gellert and Nash, 1987) and the regulation of DNA-specific enzymatic activity by a partial unwinding of the double helix, which facilitates a local unstacking of base pairs (Drew et al., 1985).

In other cases, however, supercoiling or the formation of supercoiled domains within a very long DNA molecule may potentially interfere with the proper functioning of the cell. For instance, if the cell were not able to relax excess supercoiling density, both DNA transcription (Liu and Wang, 1987) and the wrapping of DNA into nucleosome core particles (Wolffe, 1992) would be hampered by the accumulation of positive supercoils in the remaining free loops.

In dealing with the myriad topological impediments that occur during normal cell operation, with or without associated elastic stresses, the living cell has at its disposal a complex enzymatic machine, of which the topoisomerases form the center (Wang, 1971, 1991, 1996; Gellert, 1981). Various members of this class of enzymes are able to manipulate the torsional state of the double helix either actively, by introducing twist into the double helix at the expense of the consumption of ATP, or passively, by relaxing the excess twist in the circular DNA. In the latter case the release of excess twist may be the sole driving force of the topological reaction.

The supercoiling of DNA was revealed by electron microscopy after hints of its anomalous behavior in sedimentation experiments (Vinograd et al., 1965). The topology and physical structure of supercoiled DNA have since been studied by a wide variety of techniques, including dynamic light scattering (Langowski et al., 1990), x-ray diffraction (Brady et al., 1987), site-specific recombination and transposition (Boles et al., 1990), microcalorimetry (Seidl and Hinz, 1984), gel electrophoresis (Keller and Wendel, 1974; Keller, 1975; Depew and Wang, 1975; Pulleyblank et al., 1975), dialysis studies of intercalating agents (Bauer and Vinograd, 1970; Hsieh and Wang, 1975), ring closure probabilities (Shore et al., 1981; Shore and Baldwin, 1983), and single-molecule stretching experiments (Strick et al., 1996). Many of these experiments were directed mainly at the elucidation of the topological state itself. Unfortunately, most of the common physical chemical techniques do not allow a precise and unambiguous assignment of supercoil structure because the resolution in the experiments is too weak.

In recent years, however, modern (cryo-) electron microscopic techniques have been applied, aiming at a deeper reassessment of supercoil structure (Boles et al., 1990; Adrian et al., 1990; Bednar et al., 1994). The supercoil parameters are thus becoming better known, and with greater accuracy. Of course, microscopy remains a technique that is never without some ambiguity.

The correct topological relations governing closed DNA were determined merely a few years after the experimental discovery of DNA supercoiling (White, 1969; Fuller, 1971, 1978; Bauer et al., 1978). The conformations of DNA rings and coils under torsion have been studied primarily within the elastic limit (Fuller, 1971; Camerini-Otero and Felsenfeld, 1978; LeBret, 1979, 1984; Benham, 1979, 1983; Tanaka and Takahashi, 1985; Wadati and Tsuru, 1986; Tsuru and Wadati, 1986; Hao and Olson, 1989; Hunt and Hearst, 1991; Shi and Hearst, 1994; Westcott et al., 1997). An analytical study that goes some way in explaining plectonemic structure is the elastic theory by Hunt and Hearst (1991). They calculated the bulk plectonemic parameters as a function of the excluded-volume radius of the DNA.

The thermally averaged properties of supercoiled DNA have been probed extensively by computer simulations (Vologodskii et al., 1979, 1992; Klenin et al., 1991; Olson and Zhang, 1991; Chirico et al., 1993; Rybenkov et al., 1997; Delrow et al., 1997). The simulations differ widely in their degree of sophistication, but the results are, in general, mutually consistent, and the agreement with experiment is satisfactory in most cases.

The analytical development of the statistical mechanics of supercoiling is hampered considerably by the topological constraints (Shimada and Yamakawa, 1984, 1985; Tanaka and Takahashi, 1985; Benham, 1990; Hearst and Hunt, 1991; Guitter and Leibler, 1992; Marko and Siggia, 1994, 1995; Odijk, 1996). Quantitative understanding was first achieved in the consideration of the ring closure probabilities of short stiff chains with torsion (Shimada and Yamakawa, 1984, 1985). The similarity between a superhelical strand undulating within a supercoil and a wormlike chain confined within a harmonic potential was noted by Marko and Siggia (1994), who advanced a simple scaling picture of supercoil structure in the limit of fairly large fluctuations.

Even for tight bending, it has been argued that the entropy and bending of a wormlike chain are superposable to a good approximation (Marko and Siggia, 1995; Odijk, 1996). This introduces a major shortcut to theoretical work. In fact, a semiclassical treatment of supercoil structure may be put forward. Exploiting the special symmetry inherent in the classical elastic Hamiltonian of the plectoneme, we have recently shown that some of the peculiarities of plectonemic DNA observed both in experiment and in computer simulation may be understood in fairly simple terms (Odijk and Ubbink, 1998).

Besides topology, bending, and entropy, there is a fourth problem that needs to be analyzed, namely the interaction of superhelical DNA with itself. Under physiological conditions, the behavior of DNA is strongly influenced by the screened Coulomb forces exerted by its negative phosphate charges. The electrostatic interaction in supercoiled DNA immersed in a monovalent salt solution has been taken into account via the use of an effective diameter, both in simulations (Vologodskii et al., 1992) and in analytical theory (Marko and Siggia, 1994). The effective diameter depends on the ionic strength of the solution (Onsager, 1949; Stigter, 1977), but it was introduced as a statistical concept pertaining to the isotropic interaction between two straight charged rods. The statistical averaging and Boltzmann weighting are, in principle, entirely different in a theory of supercoils. In recent work the use of an effective diameter was circumvented. A soft, exponentially decaying electrostatic potential was taken into account in computer simulations (Fenley et al., 1994; Vologodskii and Cozzarelli, 1995) and, albeit within a bare, unrenormalized approximation, in analytical theory (Marko and Siggia, 1995). In positionally ordered systems, however, we recall that the bare electrostatic interaction is strongly enhanced by even small undulations of the chains around their equilibrium conformation (Odijk, 1993a). Entropy and electrostatics conspire to give rise to an electrostatic-undulatory interaction.

Here we would like to go beyond previous theoretical work in the following ways: 1) The electrostatics is dealt with by summing all interactions in a far-field Poisson-Boltzmann approximation. Closed analytical approximations for the electrostatic potential at all values of the plectonemic parameters are given, which may also be useful outside the context of this paper. 2) The potentially powerful enhancement of the potential by thermal undulations is computed within a Gaussian ansatz for the undulatory confinement. 3) The pitch and radius are two scales determining a plectonemic supercoil. It will turn out that they cannot be treated on the same footing at all. 4) Analytical procedures are employed to handle the total free energy of the plectoneme (i.e., the sum of electrostatics, entropy, bending, and twisting), so that we attain a tractable theory for supercoiling that is of practical use and yields physical insight at the same time.

The outline of the paper is as follows. First, we recapitulate the main topological relations governing covalently closed circular DNA. We calculate, both numerically and asymptotically, the electrostatic potential exerted by the plectonemic configuration to evaluate the free energy of electrostatic interaction. We next discuss the entropic mechanism by which small undulations of the strands within the supercoil couple nonlinearly with the electrostatic potential and present an approximate calculation of this effect. Then the total free energy of the supercoil is cast in the scheme previously proposed by us (Odijk and Ubbink, 1998). We concentrate on the limit of tight supercoiling, for it is then possible to postulate the existence of semiclassical configurations, in which the undulations are small. The free energy consists of an elastic contribution and a perturbative term, the electrostatic-undulatory interaction. We self-consistently minimize the total plectonemic free energy with the help of the iterative procedure derived by us earlier (Odijk and Ubbink, 1998). Our results are compared with the available quantitative data. Finally, in the Appendices, we give a detailed analysis of an entropic coefficient and briefly consider the effect of attractive interactions on plectonemic structure.

	TOPOLOGY

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Even when we disregard the probability of knot formation in the double helix itself, the closure of a double-stranded molecule like DNA can take place in many topologically distinct ways. Either strand closes on itself because the two strands run in opposite directions along the double helix, and the ends of the sugar-phosphate backbones are of a different chemical nature. The number of turns of the strands of the double helix around one another characterizes a specific topological state. For a covalently closed DNA molecule, the appropriate topological invariant is the linking number Lk (Fuller, 1971). Normal B-DNA in the relaxed state forms a right-handed helix characterized by a helical repeat h of ~3.5 nm (or, equivalently, 10.5 bp) (Bates and Maxwell, 1993), so to measure the degree of supercoiling, which may manifest itself in either under- or overwinding of the double helix, it is convenient to introduce the linking number in the relaxed state Lk₀. This number is defined in such way that for B-DNA it is positive (Bauer et al., 1978; Cozzarelli et al., 1990; Bates and Maxwell, 1993).

In 1969 White derived a relation between the linking number and two configurational quantities, one bearing on the local twist of the chain and the other reflecting the global shape of the molecule (White, 1969):

(1)

where Tw is the twisting number, defined by

(2)

The integration is performed along the contour of the axis of the double helix, ₀ is the intrinsic rate of twist of the relaxed double helix, and  is the excess twist. The second quantity introduced in Eq. 1 is the writhing number Wr, which, for an arbitrary space curve, is given by the Gauss integral (White, 1969; Clugreanu, 1959). The writhe is a functional of the configuration of the axis of the double helix only. The energy of a supercoil depends on the twist that can be eliminated via Eq. 1 in favor of the writhe. In this way, the energy conveniently becomes a functional of the configuration vector.

Analytical evaluation of the writhing number is generally cumbersome; simple analytical approximations have been derived in several cases, including that of the regular interwound configuration (Fuller, 1971; White and Bauer, 1986). We will need the writhe per unit length of strand of a plectonemic superhelix,

(3)

In Eq. 3 it is assumed that end loops may be neglected. The plus and minus signs hold for left- and right-handed plectonemic helices, respectively. r is the radius, and 2p is the pitch of the superhelix (see Fig. 1); the two variables are assumed to be uniform. The plectonemic opening angle  is defined by tan  = p/r.

Deviations from the relaxed state are measured by the excess linking number Lk = Lk  Lk₀ and the excess twisting number Tw = Tw  Lk₀. The writhe is taken to be zero in the relaxed state. Furthermore, Lk₀ = 2₀l/h, where l is the DNA contour length, so we can write

(4)

Both excess quantities may be positive or negative, pertaining either to over- or underwinding of the double helix.

By dividing the excess linking number Lk by Lk₀, we obtain the specific linking difference :

(5)

For a homogeneously supercoiled molecule, the degree of supercoiling is determined completely by the intensive quantity .

	ELECTROSTATIC POTENTIAL OF PLECTONEMIC DNA

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We view the double-stranded DNA molecule as a closed circular curve of cylindrical cross section. Its body is a uniform dielectric with a permittivity much lower than that of water, and its surface is assumed to bear a uniform charge density. In aqueous solution, the electrostatic potential of the supercoil is often screened by excess 1:1 salt, so we address its electrostatics within the nonlinear Poisson-Boltzmann approximation. This has been established to be quite accurate (Fixman, 1979).

The difficult problem of solving the Poisson-Boltzmann equation for the charged plectoneme may be replaced by a much simpler one, however. Because the distances between adjacent winds in the plectonemic helix are typically much larger than about twice the sum of the DNA hard-core radius a and the Debye screening length ¹ (owing to Boltzmann weighting), we are interested in the far-field asymptotic solution to the Poisson-Boltzmann equation only. This solution is essentially a linear superposition of effective Debye-Hückel potentials arising from all of the phosphate charges on the DNA supercoil. In the case of a straight polyion, the charged cylinder may be replaced by a line charge coinciding with the axis of the cylinder (Brenner and Parsegian, 1974). The nonlinear screening in the inner double layers of the charged cylinder is taken into account by adjusting the effective charge density _eff (i.e., the number of charges per unit length along the helical axis) in such a way that the outer double layers of the respective potentials coincide (Stroobants et al., 1986).

Here we consider the potential exerted by a polyion of plectonemic shape, which is again characterized by a radius r and a pitch 2p (Fig. 1). Corrections to the effective charge density _eff due to the superhelical curvature of the polyions may be neglected, for they are of order (R_c)² (Fixman, 1982) when the characteristic radius of curvature R_c  (p² + r²)/r of the plectoneme is much larger than the Debye length.

Next, we superpose Debye-Hückel potentials exerted by the uniformly charged superhelix, whose charge density along the helical axis is _eff. We choose a Cartesian coordinate system (x, y, z) in such a way that the z axis coincides with the central axis of the plectonemic helix (Fig. 2). M: (r, 0, 0) is a point on one strand of the plectonemic helix, and N: (r cos u, ±r sin u, pu) is a point on the opposing strand; the plus and minus signs hold for left- and right-handed superhelices, respectively. u is the parameterization along the plectonemic axis. The distance  between M and N may be written as

(6)

The total Debye-Hückel potential exerted by the opposing strand on point M of the test strand is then given by

(7)

where s is the arclength from (r, 0, 0) to N along the opposing strand, ds = [(p² + r²)^1/2/p]dz.   Q_Beff is an effective charge parameter that may be calculated within the Poisson-Boltzmann approximation (Stroobants et al., 1986; Philip and Wooding, 1970), ¹ is the Debye length defined by ²  8Q_Bn_s, Q_B  q²/k_BT is the Bjerrum length, q is the elementary charge, and n_s is the number concentration of monovalent salt.  is the permittivity of the solvent, k_B is Boltzmann's constant, and T is absolute temperature. In the integrand of Eq. 7 one recognizes the Debye-Hückel potential exerted by an element of arclength, i.e., a Coulomb potential screened by a decaying exponential. The potential has been multiplied by the elementary charge and divided by k_BT to render it dimensionless, for convenience. The electrostatic self-energy of the DNA helix itself will be assumed to be constant.

View larger version (11K): [in this window] [in a new window]   FIGURE 2   The plectonemic coordinate system.

The potential may be usefully expressed as a function of the two dimensionless variables w  2r and µ  p²/4r², so that Eq. 7 is transformed into

(8)

with

(9)

To investigate the physical behavior of the potential, we here anticipate that w  1 and 4µw²  1, for the inner double layers of the strands are unlikely to interpenetrate. We also do not expect twisting forces within the DNA helix to compete with electrostatic forces in the event they become unduly high ( k_BT/nm) upon such interpenetration.

It is seen from the behavior near u = 0 of the integrand in Eq. 8 that the construction of asymptotic expansions for large w that are uniformly valid for all µ > 0 is not standard. Bleistein's method (Olver, 1974) could be used in this case, but the presence of cos u in (u) proves to be awkward. Therefore, we have opted for the usual Laplace method (Olver, 1974; Bender and Orszag, 1978), albeit as it is applied in various regimes, for it does not yield a uniformly valid approximation for integrals of the type in Eq. 8.

For w 1, the integrand in Eq. 8 decays exponentially fast away from some minimum u = u_m of the function (u). A major contribution to the integral comes from the neighborhood of u_m, so we expand (u) around u_m:

(10)

Here we retain only the first nonvanishing term, which is positive. The leading asymptotic contribution to Eq. 8 is then given by (Olver, 1974; Bender and Orszag, 1978)

(11)

where c₁ = 1 if the minimum is at u_m = u₀ = 0 and c₁ = 2 if u_m > 0 (m = 1, 2, ...).

The case 1/4w² < µ < 0.2

We have to distinguish among a number of cases, depending on the value of µ. If µ < 1/4, we have either one or a multiple of local minima, which are to be determined from sin u_m/u_m = 4µ. In view of our lower bound µ > 1/4w², the minima beyond the first may be neglected: this is easily proved by noting that the first minimum u₁ <  and u_m > 2 (m = 2, 3, ...) and w(u_m)  u_m/2. If we approximate sin u₁ by the polynomial (4/²)u₁² + (4/)u₁ (which is reasonable for µ < 0.2), we determine the first minimum to be u₁    ²µ. The lead term for the potential is then approximately given by

(12)

If we now let µ become very small by increasing the radius r while keeping the pitch 2p constant (u₁  ), we ultimately obtain the limiting form for the potential, which is independent of r:

(13)

This is interpreted as the potential at the test strand due to two neighboring line charges, each at a distance of half the superhelical pitch. The line charges are effectively straight on the scale of p.

The case µ = 1/4

For µ larger than ~0.2, u₁ starts to approach zero, and this causes problems. In fact, as µ increases to 1/4, Laplace's method fails because the second-order derivative at the minimum becomes small compared to the value of the next nonvanishing derivative, which is of fourth-order. For µ  1/4, (u) attains only one minimum at u₀ = 0. The case µ = 1/4 is peculiar, for the first nonvanishing derivative at u₀ = 0 is fourth-order. Upon using Eq. 11, we may write for the potential

(14)

The case µ > 1/4

For µ > 1/4, the second-order derivative again comes into play and dominates the contribution from the fourth-order derivative for large enough µ (the third-order derivative vanishes at u = 0 for any µ). For µ somewhat larger than unity, we may again use the Laplace method so as to obtain

(15)

If we let µ   by increasing the pitch 2p while keeping the radius r constant, Eq. 15 reduces to a limiting form independent of p:

(16)

This is interpreted as the potential at a test strand exerted by a straight line charge at a distance 2r. Note the formal equivalence of Eqs. 13 and 16.

We have derived the asymptotic forms of the potential in several regimes to gain physical insight into its dependence on the superhelical pitch angle. Interacting charged rods exert an electric torque on each other, forcing them toward a perpendicular orientation, an effect with measurable impact on various phenomena (Stroobants et al., 1986). In the present analysis (Eqs. 12-16), the influence of twist might appear to be less severe. The simplest uniform approximationa superposition of the two limiting forms given by Eqs. 13 and 16seems not such a bad zeroth-order expression at first sight:

(17)

This was already proposed by Marko and Siggia (1995). See Table 1 for the accuracy of this simple form. Equation 17 becomes fairly poor whenever w  4 and 0.1  µ  1, whereas the asymptotic formulas Eqs. 12, 14, and 15 fare much better.

However, the magnitude of the plectonemic potential is, in itself, not such a serious issue. The two major problems with Eq. 17 are, in fact, as follows: 1) We ultimately need to minimize the total free energy of the supercoil, so derivatives are important; the derivative of ₀ with respect to p is often a vast underestimate of the actual derivative (see Table 1). 2) Undulation enhancement (as we explain below) of any weak exponential-like term one would inadvertently introduce could lead to a huge (fictive) contribution to the undulatory electrostatic energy. We are therefore forced to devise a bare plectonemic energy considerably more accurate than Eq. 17.

Now it so happens that in practice the superhelical pitch angle is rarely smaller than 45°, i.e.,   45° or p  r or µ  1/4. Accordingly, we focus only on regime c as defined above, and the asymptotic form (Eq. 15) suggests an approximation that does not have the unphysical divergence at µ = 1/4:

(18)

(19)

We have adjusted the coefficients m₁ = 0.207 and m₂ = 0.054 to let ₁ agree closely with the numerical evaluation _num of Eq. 8 (see Table 2). Clearly, the function ₁ is accurate enough to circumvent both major difficulties quoted above. Moreover, Eqs. 18 and 19 show that the pitch and radius are definitely not independent variables, as in the superposition formula (Eq. 17). Thus there is a twisting torque of electrostatic origin.

View this table: [in this window] [in a new window]   TABLE 2   Ratio of the approximation 1 (Eq. 18) to the plectonemic electrostatic potential and the numerical calculation nm of Eq. 8

	UNDULATION ENHANCEMENT OF THE ELECTROSTATIC INTERACTION

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If we were to neglect undulations of the strands, the electrostatic free energy per unit length of strand in the plectonemic supercoil would be calculated by multiplying the effective linear charge density _eff  /Q_B of the test strand by the electrostatic potential exerted by its neighbor:

(20)

The factor 1/2 has been introduced to avoid double counting.

However, as is already discernible in electron micrographs, the plectonemic helix is definitely perturbed by thermal undulations, which in some cases may be so wild that it becomes impossible to speak of a regular interwound state. Here we restrict ourselves to plectonemic supercoiling at moderate to high values of the specific linking difference so that the superhelix may be viewed as tightly wound. In this limit, the strands in the plectonemic helix are pinned in a deep potential trough, causing the undulations of the strands within the supercoil to remain fairly weak. The slopes of the free energy well in which the strands are undulating are dominated by the electrostatic interaction favoring some optimal pitch and optimal radius, and by the torsional free energy, coming into play via White's relation, which favors an increasing pitch and decreasing radius.

The strands of the plectonemic superhelix are ordered positionally with respect to one another, so we expect undulation enhancement of the interactions to occur, in a manner similar to that conceived earlier for hexagonal phases of semiflexible polyions (Odijk, 1993a). In particular, owing to the exponentially screened form of the electrostatic interaction, we anticipate a strong enhancement of the bare electrostatic interaction by the undulations.

Now, a rigorous analytical treatment of the statistical mechanics of a plectonemic worm interacting with itself is anything but trivial. The typical radius of curvature is much smaller than the persistence length, so we are in the semiclassical limit (Odijk, 1996), where fairly weak undulations of the chains are defined with respect to a (local) state of minimum energy. The latter may be called a classical limit. The configurational statistics of such tightly curved worms has been dealt with by several methods (Shimada and Yamakawa, 1984; Marko and Siggia, 1995; Odijk, 1996). The general conclusion is that a stiff chain undulates virtually independently of its degree of tight bending. We simply assume that this holds true in our case with electrostatics included, despite the lack of a rigorous mathematical proof. Nevertheless, from a physical point of view, switching on repulsive forces does not increase the import of bending; rather the reverse is true. On the whole, we expect the electrostatics to be balanced by entropy as far as the undulations of the plectoneme are concerned. Next, we know the plectoneme fluctuates about some equilibrium configuration. Clearly positional order exists that is similar but not identical to that of a linear polyion undulating within a hexagonal lattice (Odijk, 1993a). One obvious difference is that a plectonemic strand does not undulate within a potential of simple symmetry. At this stage we simply posit a two-variable description (r and p independent) to introduce coarse-grained undulatory electrostatics. Marko and Siggia (1995) have presented arguments based on pseudopotentials that this is a useful approximation. In this paper we disregard all end effects, including branching.

We now first presuppose that the undulations in both r and p are small. Below, we shall see that we will be forced to modify this hypothesis, but we need to investigate this case first. It is then reasonable to postulate a Gaussian distribution for the undulations in the two-dimensional (r, p)-space:

(21)

where u_r and u_p and d_r/2^1/2 and d_p/2^1/2 are the undulatory amplitudes in r and p and their root mean squares, respectively (d_r r, d_p p). Orientational fluctuations of neighboring polymer segments may be neglected in this limit (Odijk, 1993a).

The two strands in the plectonemic helix are presumed to undulate independently. Let us choose one of the strands and average its potential over all of the undulations (Fig. 3 a):

(22)

The bare potential ₁ given by Eq. 18 is smeared out by the undulations of the test strand exerting the potential. The renormalization of the radial undulation is potentially strong, for it is exponential. But the renormalization of the longitudinal undulation is slight because the dependence of ₁ on p is weak. Equation 22 has been derived using the fact that d_p p.

View larger version (10K): [in this window] [in a new window]   FIGURE 3   The bare electrostatic potential within the plectonemic configuration (r, p) is averaged over all undulations of (a) the strand adjacent to the test strand and (b) the test strand itself.

The strand adjacent to the test strand is also undulating (Fig. 3 b). Because of symmetry, averaging the renormalized potential (Eq. 22) over the undulations of the adjacent strand is equivalent to averaging again over the undulations of its neighbor:

(23)

_el is the free energy of electrostatic interaction per unit length of strand. In Eq. 23 relative terms of (1/r) and (d_p⁴/p⁴) have been consistently deleted.

	ENTROPY

TOP ABSTRACT GLOSSARY INTRODUCTION TOPOLOGY ELECTROSTATIC POTENTIAL OF... UNDULATION ENHANCEMENT OF THE... ENTROPY FREE ENERGY DISCUSSION CONCLUDING REMARKS APPENDIX I APPENDIX II REFERENCES

In the previous section we discussed the mechanism by which small undulations of magnitudes d_r and d_p of the strands within the plectonemic superhelix give rise to an amplification of the bare electrostatic interaction that is weighted unevenly. Now, the reduction of entropy of a worm upon confinement to the close neighborhood of its classical paththe typical transverse wanderings being of average amplitude dis generally expressed in terms of a deflection length   P_b^1/3 d^2/3, which replaces the persistence length as an independent length scale (Odijk, 1983). The free energy of entropic confinement per unit length of the strand in the plectonemic supercoil may then be written approximately as (Odijk, 1983, 1993a; Helfrich and Harbich, 1985; Marko and Siggia, 1994)

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The coefficients c_r and c_p are here 3/2^8/3 (we have reexamined them in Appendix I).

The undulation-enhanced free energy per unit length of strand, which is of electrostatic origin and here scaled by k_BT, is thus expressed as

(25)

All terms contributing to the free energy should be averaged over the relevant semiclassical paths. The undulatory degrees of freedom of a confined worm are weighted via the free energy of confinement; the confinement due to the electrostatic interaction is evaluated within a Gaussian approximation. In principle, the torsional forces, restraining the strands in the supercoil via White's relation (Eq. 4), should also be renormalized over all undulations. An approximate evaluation of the torsional energy, allowing for undulations that do not disrupt the plectonemic symmetry, is straightforward, but the undulatory effect turns out to be negligible. Some advances toward a more rigorous approach have been forwarded by Shimada and Yamakawa (1984, 1985), by Marko and Siggia (1995), and by Odijk (1996). However, simple quantitative approximation schemes for the coupling between torsional deformation and entropy in confined or topologically constrained systems have yet to be proposed. Moreover, the backbone of the plectoneme is not perfectly straight as assumed here, but fluctuates on length scales that are possibly on the order of the superhelical pitch, so that we may expect some influence of these fluctuations on the internal structure of the superhelix. We do not address these topics here, but simply assume that torsional effects are adequately taken into account by considering only the classical plectonemic path.

	FREE ENERGY

TOP ABSTRACT GLOSSARY INTRODUCTION TOPOLOGY ELECTROSTATIC POTENTIAL OF... UNDULATION ENHANCEMENT OF THE... ENTROPY FREE ENERGY DISCUSSION CONCLUDING REMARKS APPENDIX I APPENDIX II REFERENCES

In line with an approximate minimization procedure proposed by us (Odijk and Ubbink, 1998), we write the total free energy per unit length of strand of the plectonemic helix in the following form:

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