INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

Closed circular DNA usually exists in a supercoiled, plectonemic configuration in which the DNA duplex is wound around another part of the same molecule to form a higher order helix. The plectonemic negative supercoiling, with a right-handed interwinding, is the most likely conformation present in vivo. The excess free energy associated with the supercoiling is used in many cellular mechanisms. Examples include the replication and transcription of DNA, the formation of nucleosomes and other protein complexes on DNA, and the formation of altered DNA structures such as cruciforms (Bates and Maxwell, 1993). Apart from the importance in understanding the mechanisms involved in vivo, the interplay between conformation and interactions in topologically constrained polymers is of interest in its own right from a biophysical point of view.

With the advent of high-resolution (cryo-) electron microscopy (Boles et al., 1990; Bednar et al., 1994) and atomic force microscopy (Rippe et al., 1997; Lyubchenko and Shlyakhtenko, 1997; Cherny and Jovin, 2001), the supercoils can be directly visualized. In these studies the plectonemic parameters, i.e., radius, pitch, opening angle, and length projected on the superhelical axis were obtained as a function of superhelical density and/or supporting electrolyte concentration. The main observation is that the shape of the interwound superhelix is generally quite irregular, but the average radius is inversely proportional with the superhelical density and decreases with increasing ionic strength of the supporting medium. However, these visualization techniques are never without some ambiguity. First, it is difficult to preserve the ionic conditions during the elaborate sample preparation procedures and there is always a possible effect of the spreading interface on the molecular conformation (Fujimoto and Schurr, 2002). Furthermore, apart from cryo-electron microscopy, these interfacial techniques are not well adopted to investigate typical three-dimensional solution properties such as plasmid excluded volume effects and crowding. Accordingly, it is desirable to perform bulk radiation scattering experiments to infer the structure of the supercoils close to their native state.

Scattering work on supercoiled DNA is scarce. Early work using low angle x-ray scattering shows rather featureless structure factors, which were interpreted with a toroidal molecular configuration (Brady et al., 1983, 1987). This observation is at odds with the images obtained with modern high-resolution microscopy techniques, which show clearly plectonemic, interwound molecular conformations. The interwound structure has also been observed in more recent small angle neutron scattering (SANS) work on supercoiled DNA either in dilute solution (Hammermann et al., 1998) or in liquid-crystalline environment (Torbet and DiCapua, 1989). From the scaling of the interaction peak position (Bragg spacing) with plasmid concentration, the latter authors obtained a pitch angle 36 ± 5° for concentrations above 45 g of DNA/dm³. With the linking number deficit determined with gel electrophoresis, they further derived a radius r = 8 ± 1 nm and a pitch 2p = 36 ± 4 nm. With a similar procedure as in the present contribution, Hammermann et al. (1998) obtained significantly smaller values for the radius of the superhelix of pUC18 bacterial plasmid (2686 bp). In these previous scattering works, attempts to observe interference over the superhelical pitch were unsuccessful. As we will see shortly, this is mainly because of the existence of a rather broad distribution of topoisomers in a typical DNA preparation.

Supercoiling is also of paramount importance in controlling the formation of a liquid crystal, through the effect of the plectonemic dimensions and possible branching of the superhelix on the excluded volume. The key issue how spatial confinement controls the radius, pitch, and associated excess free energy has not been addressed yet. Accordingly, we thought it useful to focus on the dimensions of the supercoil as a function of plasmid concentration in a range prior and covering the transition to the liquid crystalline phase. For this purpose, we will first derive a mathematical expression for the total scattering function, including interactions among supercoils in the second virial approximation. This expression should include both longitudinal and transverse interference over the pitch and radius, respectively. Furthermore, it will be necessary to include a variation in the molecular shape due to the presence of a distribution in linking number deficit, thermal fluctuations, and/or, in the case of biphasic samples, partitioning over coexisting anisotropic and isotropic phases. From a comparison with our SANS data from pUC18 plasmid in saline solutions, we will estimate the regularity of the superhelical structure and the extent to which the configuration follows locally the classical path. The derived results for the radius, pitch, and opening angle will than be compared with the corresponding literature values obtained with microscopic visualization techniques. Finally, we will discuss the observed concentration dependence of the number of superhelical turns in terms of spatial confinement effects.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

Topology

For a description of the topology, it is assumed that the double-stranded DNA coil takes a helical configuration that is well defined locally at a length scale on the order of the radius r and the pitch 2p (Fig. 1). We let the helix be right-handed with a negative superhelical density  = Lk /Lk₀, with excess linking number deficit Lk and Lk₀ being the linking number if the coil is fully relaxed. The excess linking number deficit is related to the writhing number Wr and the excess twist Tw according to

(1)

(White, 1969). For a right-handed, regular supercoil without end loops, the writhing number is simply proportional to the number of crossings n when viewed perpendicular to the superhelical axis Wr = nsin , with the plectonemic pitch angle  as in Fig. 1 (Bloomfield et al., 2000). L denotes the total length of the supercoil projected on the superhelical axis. It is convenient to define the normalized length 2L/l, with l being the DNA length measured along the contour. From integration along the contour, it follows that

(2)

if the end loops are neglected. The pitch angle  is given by

(3)

and the writhing number reads

(4)

The local structure of the plectoneme is fully characterized by the lengths p and r. These two parameters determine the opening angle , the writhe Wr, and the normalized length of the superhelical axis 2L/l.

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

For an interpretation of the data, it is necessary to calculate the contribution to the scattering from a single DNA molecule. This single chain contribution, which is commonly referred to as the form factor, is completely defined by the topology of the supercoil and can be expressed in terms of p, r, and L. Once the form factor is known, interactions among different supercoils can be included in the description of the total structure factor in the random phase approximation as detailed below.

Form factor

At a larger length scale on the order of the plectonemic length L, the supercoil can branch. In the interpretation of our scattering experiments, branching effects are unimportant however. This is because the momentum transfer range q (defined by the wavelength  of the radiation and the angle  between the incident and scattered beam according to q = 4/sin(/2)) exceeds L¹ by at least an order of magnitude and the scattering is particularly sensitive to interference over a spatial extent on the order of the plectonemic radius and pitch (qL 1). We will accordingly neglect branching in the calculation of the form factor. Furthermore, we will also ignore the scattering contributions due to the end loops and the effects of overall flexibility of the superhelical axis. The error in the form factor introduced by the neglect of the end loops becomes less significant for supercoils with higher superhelical density. The effects of flexibility are minimal for qL_s 1 with L_s the bending persistence length of the supercoil (~100 nm, i.e., being twice the value of the persistence length of the duplex L_p). For the time being, we will also neglect the effect of the radius of gyration r_p of the cross-section of the double-stranded DNA duplex. The latter effect is important only for very high values of momentum transfer where qr_p  1 and can easily be taken into account once the form factor of the superhelix is known.

Before engaging in any detailed calculation, it is interesting to gauge the limiting behavior of the single chain scattering function for low and high values of momentum transfer, respectively. In the low q range, with qr 1 and qp 1, the details of the plectonemic structure (i.e., radius and pitch) are beyond observation. At this rather extended distance scale, but at lengths significantly smaller than L (qL 1), the supercoil is seen as a rodlike object with form factor P(q) = /(qL) (overall flexibility of the superhelical axis is ignored). At higher values of momentum transfer in the range qr 1 and qp 1, the scattering is essentially given by a single strand of the superhelix (not to be confused by a single strand of the DNA duplex). The high q limiting form of the form factor of a single DNA duplex is P(q) = /(ql), with l the DNA contour length (here, the finite cross section of the duplex is ignored). In both regimes, the form factor shows the characteristic q¹ scaling for rodlike particles (Higgins and Benoit, 1994). With increasing value of momentum transfer, the prefactor drops, however, from the DNA density per unit plectonemic length L¹ to the density per unit contour length l¹. The ratio of the prefactors gives, hence, l/L, which is particularly promising from an experimental point of view. It is clear that a full description should agree with these two limiting situations. As we will see shortly, the form factor is particularly rich in information concerning the plectonemic structure in the intermediate q range with qr  1 and qp  1.

The exact form factor pertaining to a regular plectonemic structure is derived in the Appendix and reads

(5)

with J_k being the Bessel functions of integer order. The restriction to the even 2k terms comes from the fact that the contributions of the two opposing strands in the superhelix have been averaged. The form factor is normalized to unity at q = 0. As anticipated, the structure is fully described by the lengths p, r, and L. In the relevant momentum transfer range qL 1, the integration over orientation variable µ can be done analytically. In this range of momentum transfer, the form factor takes the relatively simple approximate form

(6)

in which the summation is restricted to those terms with index k < qp/2. The high q approximate solution of the single chain scattering function is sensitive to the DNA density per unit length projected on the superhelical axis L¹. In the derivation of these expressions, the effect of the radius of gyration r_p of the cross-section of the DNA duplex has been ignored. Because of the difference in length scales, this effect can be taken into account by multiplication of the form factor of the superhelix Eqs. 5 or 6 with the one pertaining to the cross-section

(7)

(van der Maarel, 1999; Zakharova et al., 1999).

The exact and approximate expressions Eqs. 5 and 6, respectively, are compared in Fig. 2. The parameters are typical for pUC18 plasmid with a normalized plectonemic length 2L/l = 0.8, which corresponds with an opening angle  = 53°. The experimental value of the normalized length, deduced from electron microscopy, is found to be 0.82 ( = 55°) and is constant within a margin 0.05 (Boles et al., 1990). For qL exceeding, say, 10, the exact and approximate expressions are indiscernible. Furthermore, the approximate solution has the right low and high q limiting behavior as already anticipated from our scaling arguments. In the intermediate q range, the form factor shows strong oscillatory behavior, which is due to intrasuperhelix interference both in the radial and longitudinal direction over lengths r and p, respectively. The intrasuperhelix interference is more clearly demonstrated if the form factor is normalized in a way that it goes to unity at high q.

View larger version (14K): [in this window] [in a new window]   FIGURE 2   Form factor of a superhelix according to the exact (solid line) and approximate (dashed line) Eq. 5 and 6, respectively. The low and high q limiting behavior is given by /(qL) and /(ql) (dashed-dotted line), respectively. The parameters are the radius r = 7 nm, pitch p = 9.3 nm, and contour length l = 920 nm.

Fig. 3 displays the normalized form factor qlP/ versus qr. The form factor is calculated without as well as with fluctuations in molecular shape as detailed below. In the limit q  0, qlP/ extrapolates to two times the inverse normalized plectonemic length l/L. Apart from their obvious dependence on the average radius (r), the positions of the minimums and maximums in the normalized form factor are rather sensitive to the value of the normalized length (or, according to Eq. 3, the opening angle ). This sensitivity of the periodicity to the normalized length comes from longitudinal interference over the plectonemic pitch through the (pitch dependent) series of Bessel functions in Eqs. 5 or 6. The corresponding terms also ensure the correct high q limiting behavior. The first leading term J(qr) is the form factor of a pair of point scatterers at a constant distance 2r and averaged over a circle. Restriction to this first term in the analysis of the scattering from a plectonemic structure, as has been done in previous scattering works (Brady et al., 1983, 1987; Hammermann et al., 1998), is clearly not sufficient for a precise estimate of the diameter of the superhelix.

View larger version (15K): [in this window] [in a new window]   FIGURE 3   Normalized form factor qlP/ versus qr without (A) and with (B) 20% variation in the radius r (rm = 2 nm). The lines refer to a normalized plectonemic length 2L/l = 0.7 (dashed-dotted), 0.8 (solid), and 0.9 (dashed). The other parameters are as in Fig. 2. Except the dotted line in B, all curves are calculated with a fixed opening angle  = 44°, 53°, and 64° for 2L/l = 0.7, 0.8, and 0.9, respectively. The dotted line includes the effect of an additional 25% fluctuation in pitch about the average value set by the average opening angle  = 53° (2L/l = 0.8).

As we will see shortly, our experimental results do not show such strong oscillatory behavior, because in reality there are fluctuations in the radius r, pitch p, and opening angle . To include these fluctuations in our model, we assume a Gaussian distribution in r with a standard deviation _r and a distance of closest approach of the two opposing DNA strands in the superhelix 2r_m. Here, and in subsequent calculations, the cut off radius r_m was set to 2 nm, being approximately the sum of the DNA duplex outer radius and a condensed counterion layer thickness. As a first approximation, we will also assume that  is constant. The latter assumption is supported by the electron microscopy observation of a constant normalized plectonemic length over a large range in superhelical density 0.12    0.02 (Boles et al., 1990). The pitch p is hence proportional to r, with a proportionality factor given by the normalized length according to Eq. 3. Such a distribution in radius and pitch with constant opening angle can be envisioned due to the presence of a distribution of topoisomers with different linking number deficit in our DNA preparation (see below). As displayed in Fig. 3, a 20% fluctuation in r (_r/r = 0.2) is already sufficient to damp the oscillations to a significant degree. We have also allowed for an additional Gaussian broadening in pitch, which relates to a fluctuation in the opening angle . As can be seen in Fig. 3, an additional 25% fluctuation in p (_p/p = 0.25) about the average value set by the average opening angle  = 53° (2L/l = 0.8) gives a minor effect only.

Total structure factor

Of course, the scattering does not only depend on the properties of a single DNA molecule but is also sensitive to interference among different supercoils. Doing experiments at low solute volume fractions minimizes the latter interference, but practical reasons, such as neutron beam intensity and scattering cross section, set a lower bound to the concentration range. In polyelectrolytes, including DNA, interchain structure at wavelengths on the order of the electrostatic screening length is notoriously difficult to describe (Nierlich et al., 1979; van der Maarel et al., 1992; Yethiraj and Shew, 1996; van der Maarel and Kassapidou, 1998). If the superhelix is modeled as a uniformly charged cylinder with radius r immersed in a medium with ionic strength I, its effective diameter can be calculated in the Debye-Hückel approximation and takes the form (Stigter, 1977; Fixman and Skolnick, 1978)

(8)

Here,  denotes Euler's constant and the screening length ¹ given by ² = 8QI, with Bjerrum length Q = e²/(kT). The constant A = 2Q¹ depends on the effective number of charges per unit length along the superhelical axis _eff. If the charge is renormalized according to the Manning's (1969) condensation concept, _eff = [QrK₁(r)]¹, with K₁ the first order modified Bessel function of the second kind. In 0.05 M monovalent salt and with a bare diameter 20 nm (see below), the effective diameter of the superhelix amounts 21 nm. We have, hence, a sufficient amount of supporting low molecular weight electrolyte, so that the electrostatic interactions are screened within the range of the diameter of a single supercoil. Accordingly, the DNA molecules do not interact very strongly and the scattering behavior is anticipated to agree with virial theory. We will use the most simple random phase approximation (de Gennes, 1979; des Cloiseaux and Jannink, 1990) for the total structure factor

(9)

with N the number of nucleic acid monomers per DNA molecule (i.e., two times the number of base pairs), A₂ the second virial coefficient, and  denotes the DNA concentration in number of plasmids per unit volume. Theoretical expressions for the second virial coefficient are available for, e.g., rigid rods including electrostatic interactions and end effects (Onsager, 1949; Odijk, 1990). However, due to various spurious effects such as branching and overall flexibility, we consider interpretation of the second virial coefficient of DNA supercoils with a rigid rod model not feasible and we will treat A₂ as an adjustable parameter.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

Isolation of pUC18 clone vector DNA

pUC18 plasmid (2686 bp) was prepared from Escherichia coli DH5 (Bhikhabhai et al., 2000). A colony was transformed with pUC18 and grown on a Luria Broth agar plate with ampicillin (100 µg/mL). A single colony was taken to grow a culture in Terrific Broth medium (12 g of tryptone, 24 g of yeast extract, 4 mL of glycerol, 2.3 g of KH₂PO₄, and 12.5 g of K₂HPO₄ per dm³) and ampicillin at 37°C. After 7 h, this culture was put into a fermentor, which contained 30 dm³ Terrific Broth-medium, ampicillin, and an antifoam agent. The bacteria were cultured for 17 h at 37°C under continuous shaking, aeration, and, subsequently, harvested and stored at 20°C. The cells were suspended in TEG buffer (25 mM Tris, 10 mM EDTA, 50 mM glucose, pH 8) and lysed with an alkaline solution (0.2 M NaOH, 1% sodium dodecyl sulfate) at room temperature. Bacterial genomic DNA, cellular debris, and proteins were precipitated by the addition of 3 M potassium acetate at 4°C. After centrifugation, the supernatant was treated with 5 M ammonium acetate to precipitate any residual contaminants (Sun et al., 1994). RNA and protein were removed with Rnase (20 µg/mL, 37°C, 1 h) and proteinase K (100 µg/mL, 55°C, 1 h) treatments, respectively. After precipitation with cold isopropanol, the DNA pellet was dried for a short period and dissolved in TE-buffer (10 mM Tris, 1 mM EDTA, pH 8) for storage at 4°C. The advantage of this procedure is that it has a high yield of typically 75 mg of plasmid DNA per isolation using 4 of 30 dm³ cell culture.

Plasmid characterization, purification, and sample preparation

The integrity of the plasmid was checked with 1% agarose gel electrophoresis in Tris-acetate buffer (40 mM Tris-acetate, 2 mM EDTA, pH 8.3) at 75 V for 1 h (Backendorf et al., 1989). The linking number deficit and the percentage of open circular DNA were determined by 1.4% agarose gel electrophoresis in the same buffer but with 10 µg/mL chloroquine at 50 V for 17 h (van Workum et al., 1996). The results are displayed in Figs. 4 and 5, respectively. Apart from supercoiled DNA, an isolated batch contained some chromosomal DNA and a small amount of aggregated plasmid DNA. The density profile in Fig. 5 shows a rather broad topoisomer distribution with predominant linking number deficit 7 and deviation ±3 (Lk = 7 ± 3), which corresponds with a superhelical density  = 0.03 (for chloroquine-free pUC18 Lk₀ = 262). Furthermore, gel electrophoresis indicates that less than 5% of plasmids are nicked and open circular. The plasmids were further purified with fast performance liquid chromatography on a 40-mL bed volume Q-sepharose column (XK 26/20, Pharmacia BioTech, Uppsala, Sweden) (Prazeres et al., 1998). Gel electrophoresis shows that the purified material is now essentially free of open circular and/or linear plasmid, aggregated plasmid, and chromosomal DNA (Fig. 4). Furthermore, the ratio of the optical absorbencies A₂₆₀/A₂₈₀ = 1.83 indicates that the material is also free of protein. The hypochromic effect at 260 nm (>35%) confirms the integrity of the duplex. For sample preparation, the purified plasmid was precipitated in ethanol and the 10% residual water content in the gently dried pellet was determined with infrared spectroscopy. The pellet was subsequently dissolved in 0.05 M NaCl, and the DNA concentration was determined by weight and checked with ultraviolet spectroscopy. Standard quartz cuvettes with 0.1-cm path length were used.

View larger version (26K): [in this window] [in a new window]   FIGURE 4   Analysis by agarose gel electrophoresis. (Lane 1) Fast performance liquid chromatography purified supercoiled material; (lane 2) nonpurified batch; (lane 3) open circular plasmid.

View larger version (27K): [in this window] [in a new window]   FIGURE 5   Topoisomer distribution in the nonpurified batch by agarose gel electrophoresis in the presence of chloroquine. The density profile shows <5% open circular plasmid and linking number deficit Lk = 7 ± 3. Linking number deficits less than 9 are not resolved.

Scattering

Small angle neutron scattering experiments were done with the PAXY (test experiments only) and D22 diffractometers, situated on the cold sources of the high neutron flux reactors at the Laboratoire Léon Brillouin, CEN de Saclay, and Institute Max von Lauc - Paul Langevin respectively. The temperature was kept at 298 K. The reported data were collected with the D22 instrument in two different configurations. A wavelength of 0.7 nm was selected, and the effective distances between the sample and the planar square multidetector (sample-detector, (S-D) distance) were 2 and 8 m, respectively, with a 0.4-m detector offset for the 2-m S-D distance only. This allows for a momentum transfer range of 0.05 to 4 nm¹. The counting times were approximately 2 h/sample, irrespective S-D distance. Data reduction allowed for sample transmission and detector efficiency. The efficiency of the detector was taken into account with the scattering of H₂O. Absolute intensities were obtained by reference to the attenuated direct beam and the scattering of the solvent H₂O was subtracted. Finally, the intensities were corrected for a small solute incoherent scattering contribution. In the present solutions, the scattering is dominated by the DNA structure, because the nucleotide scattering length contrast exceeds the corresponding values of the small ions by two orders of magnitude. The coherent part of the solvent subtracted SANS intensity gives hence the DNA structure factor according to I(q) = _m ²S(q) with _m the DNA concentration in number of nucleotides per unit volume (_m = N). The contrast  = 11.4 × 10¹² cm has been calculated according to the pUC18 base composition A:G:C:T = 0.245:0.252:0.255:0.248 (NCBI database, accession number L08752) and the nucleotide scattering lengths reported by Jacrot (1976).

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

Scattering experiments were done on samples with plasmid concentration in the range 2 to 27 g of DNA/dm³ in 0.05 M NaCl. This concentration range is just below and covering the transition to a liquid crystal (Torbet and DiCapua, 1989; Reich et al., 1994). The experimental results are displayed in Fig. 6, together with the theoretical structure factors of a plectonemic model and interactions in the second virial approximation. The result obtained from 2 g of DNA/dm³ is not shown, because it is very similar to the one with 3 g/dm³ plasmid concentration. This agreement shows that the effect of interactions among supercoils on the structure factor is vanishing small for DNA concentrations less than, say, 3 g/dm³ (i.e., A₂  0). For the samples with higher plasmid concentration, progressive interactions among supercoils results in a suppression of the intensity at longer wavelengths (smaller q values). Furthermore, the samples with 11 and 27 g/dm³ plasmid concentration show an upturn for very low values of momentum transfer q < 0.1 nm¹, which is not accounted for by the theoretical predictions. If the plasmid concentration exceeds 3 g/dm³, the translucent samples exhibit birefringent domains when observed through crossed polarizers and a solid-like broadening of the ³¹P resonance in the nuclear magnetic resonance spectrum. These observations comply with the progressive growth of liquid crystalline germs in an otherwise isotropic medium. The samples with 2 and 3 g of DNA/dm³ are completely isotropic; the ones with 6 and 11 g of DNA/dm³ are in the biphasic regime, and the sample with 27 g of DNA/dm³ is completely liquid crystalline. The critical boundaries and the characteristics of the liquid-crystalline phase are, however, beyond the scope of the present paper and will be reported in an accompanying paper. Here, we focus on the concomitant effects of the compaction on the dimensions of the supercoil by a full analysis of the structure factor.

View larger version (16K): [in this window] [in a new window]   FIGURE 6   Structure factor S versus momentum transfer q in the double logarithmic representation. The DNA concentration is 3 (), 6 (), 11 () and 27 () g/dm3. The solid lines are calculated according to the theoretical structure factor with the adjusted parameters in Table 1. To avoid overlap, the data are shifted along the y axis with a multiplicative constant.

As displayed in Fig. 6, there is nice agreement with the theoretical predictions according to Eq. 9. We have used the approximate form factor expression Eq. 6, because qL > 20 in our experimental range of momentum transfer. The structure factors do not exhibit a significant intermolecular interaction peak. The agreement with second virial theory and the absence of an interaction peak indicate that the supercoils do not interact too strongly and that the ionic strength is sufficiently high to screen the electrostatic interactions over a relatively short range on the order of the plectonemic diameter. Due to the presence of a significant distribution in plectonemic radius and pitch, the structure factors do not exhibit strong oscillatory behavior. They do show, however, the anticipated q¹ scaling and the drop in prefactor from the inverse plectonemic length L¹ to the inverse contour length l¹ with increasing value of momentum transfer. The determination of the plectonemic dimensions will be further discussed below, once the structure factors are normalized to their high q limiting behavior. For q exceeding, say, 1 nm¹, the structure factors show the characteristic deviation from q¹ scaling related to the finite cross-section of the DNA duplex. A fit of the relevant form factor Eq. 7 gives a radius of gyration r_p = 0.8 nm. This value agrees with the cross-section of the DNA duplex in the B-form. It is a little less than the outer radius 1 nm, due to the relatively open duplex structure related to the existence of grooves (van der Maarel et al., 1992; Zakharova et al., 1999).

The plectonemic structure is most clearly demonstrated in Fig. 7, where the structure factors are normalized in a way that they go to unity at high q. For this purpose, the structure factors are multiplied with q times the contour length l and divided by  times the number of nucleotides per plasmid N. For the sake of completeness, the data are also divided by the form factor pertaining to the cross-section of the duplex P_c, although the latter factor deviates little from unity in the relevant range q < 1 nm¹. The normalized structure factor extrapolates to two times the inverse normalized plectonemic length l/L for q  0 and in the absence of interactions. Due to the distribution in radius and pitch, the higher order oscillations are damped and the first oscillation takes the form of a shoulder (see Fig. 3). With increasing DNA concentration, this shoulder shifts to higher values of momentum transfer with a concomitant suppression at longer wavelengths due to progressive interactions among the supercoils. The deviations observed at very low q values (q < 0.1 nm¹) comply with the existence of a long-range inhomogeneity in DNA density, possibly related to the existence of aggregated plasmid and/or the formation of crystalline spherulites (see accompanying paper). The shift of the shoulder and subsequent minimum toward higher values of momentum transfer with increasing plasmid concentration is related to a concomitant decrease in the average plectonemic pitch and radius. These observations will be further substantiated with the results of the fit procedure of the theoretical structure factor.

View larger version (12K): [in this window] [in a new window]   FIGURE 7   Normalized structure factor qlS/(NPc) versus momentum transfer q. The concentrations and solid lines are as in Fig. 6. To avoid overlap, the data are incrementally shifted along the y axis with 0.5 units.

As discussed in the theoretical section, in the data analysis we neglect the contribution originating from the end loops as well as branching. Flexibility effects on the scattering function of a wormlike chain are less than 5% for our lower bound qL_s  2.5 (with the bending persistence length of the supercoil L_s  100 nm) and become negligible as q is increased (Norisuye et al., 1978). Accordingly, we expect that overall flexibility effects on the scattering from a supercoil are modest in the present q range.

The main contribution to the distribution in radius comes from the presence of a distribution in topoisomers with different linking number deficit as shown by gel electrophoresis (Fig. 5). Another source for a variation in radius is thermal fluctuations, as suggested by, e.g., analytical theory (Marko and Siggia, 1995; Ubbink and Odijk, 1999) and computer simulations (Gebe et al., 1996; Hammermann et al., 1998; Klenin et al., 1998). Furthermore, the partitioning of DNA over coexisting isotropic and liquid crystalline phases with a concomitant change in plectonemic dimensions contributes to a broadening of the distribution in radius. Of course, the opening angle can also fluctuate, but we assume that it takes a constant value. This assumption is supported by the observation of a constant normalized plectonemic length over a large range in superhelical density 0.12    0.02 (Boles et al., 1990). Alternatively, the variation in pitch may be assumed uncorrelated with the variation in radius, but it was checked that such model gives an inferior fit (results not shown). The input parameters for our model are, hence, the plectonemic radius r, its distribution width _r, the opening angle , and the second virial coefficient A₂ (A₂ was set to zero for the 3 g of DNA/dm³ sample). The pitch p, the writhing number Wr, and the normalized plectonemic length 2L/l can subsequently be derived with Eq. 3. Estimation of the distribution widths in the latter parameters was done with standard variance propagation of the optimized variation in radius. The cut off radius pertaining to the distance of closest approach of the two DNA strands was set to r_m = 2 nm, being approximately the sum of the DNA duplex radius and the condensed counterion layer thickness. It was checked that a 25% variation in the latter value has no significant effect on the fitted parameters. The results are collected in Table 1. Note that the margins do not refer to error in the physical measurements, but rather to variation in the molecular shape resulting from the distribution in linking number deficit and/or thermal fluctuations.

In the diluted regime, the plectonemic radius takes a relatively large value r = 10 ± 4 nm, and the supercoils are rather expanded with a solvent and small ions filled core spanning ~10 times the duplex diameter. This value, which is derived under solution conditions, is in good agreement with the electron microscopy result for DNA with a superhelical density   0.03 in 0.105 M monovalent salt (Boles et al., 1990). The spreading of the supercoils on the electron microscopy grid has apparently no significant effect on the derived dimensions. There is also nice agreement with the SANS result of Torbet and DiCapua (1989), who obtained a 8-nm radius from the scaling of the position of the interaction peak in the liquid-crystalline regime above 45 g of DNA/dm³ (  0.05). With a similar experimental procedure as described in the present work, Hammermann et al. (1998) reported a significantly smaller radius 5.5 nm for pUC18 in 0.05 M NaCl. However, the latter authors did not take into account the longitudinal interference over the pitch (and, hence, they did not derive the opening angle), which has an important effect on the structure factor. From the results in Fig. 3 b and the position of the first minimum in the term J(qr), we can estimate that the position of the minimum in the normalized structure factor is shifted by a factor, say, 1.7 towards higher values of momentum transfer once the longitudinal interference is included. If we tentatively scale Hammermann et al.'s result with this factor, we obtain a 9-nm, plectonemic radius which is in reasonable agreement with our result. There is also nice agreement with the value predicted by electrostatic theory within the Poisson-Boltzmann approximation (Marko and Siggia, 1995; Ubbink and Odijk, 1999).

For diluted supercoils with vanishing mutual interaction, the derived values of the normalized length and the opening angle are also close to the electron microscopy values reported by Boles et al. (1990). These authors found an average normalized length 0.82 with a margin 0.05, which corresponds to an opening angle  = 55°. For DNA in solution, we obtain 2L/l = 0.91 and  = 65°. The opening angle is very close to 63 ± 20°, reported for the superhelical regions in p1868 plasmids in air or hydrated in water and obtained with scanning force microscopy in the presence of MgCl₂ (Rippe et al., 1997). There is also nice agreement with the experimental value 62 ± 5°, obtained with the same technique for pPGM1 (2987 bp,   0.06) plasmid in 0.05 M NaCl (Cherny and Jovin, 2001). With the optimized radius and pitch, the writhing number takes the value Wr = 6 ± 3. With linking number deficit Lk = 7 ± 3 (see Materials and Methods section), we derive a writhe per added link Wr/Lk = 0.85. Boles et al. (1990) reported a slightly smaller value Wr/Lk. = 0.72, but in view of the variation set by the distribution in radius and the neglect of end loops in our calculation of the writhe, we consider the agreement gratifying. Furthermore, the variation in writhe agrees with the variation in linking number deficit as observed by gel electrophoresis. This agreement indicates that the main contribution to the variation in adjusted parameters comes from the presence of a distribution of topoisomers rather than thermal fluctuations. To gauge the relative weight of the latter contribution, however, it is necessary to prepare plasmid with a narrow distribution in (preset) linking number deficit in sufficient quantities.

With increasing plasmid concentration, both the radius and pitch are seen to decrease significantly. Although the experiments were done with a constant concentration of added salt, the ionic strength increases from 0.05 to, say, 0.06 M, due to the uncondensed fraction 0.24 of counterions coming from the DNA (Manning, 1969). A decrease in radius with increasing ionic strength was previously observed with cryoelectron microscopy (Bednar et al., 1994), atomic force microscopy in situ (Lyubchenko and Shlyakhtenko, 1997), sedimentation and catenation experiments (Rybenkov et al., 1997a,b), SANS (Hammermann et al., 1998), as well as computer simulations (Gebe et al., 1996; Klenin et al., 1998). However, our increase in ionic strength comes from free counterions only and is so small that we consider spatial confinement effects of entropic origin of greater importance. This is also supported by the fact that the electrostatic interactions are screened over a distance on the order of 8 nm (as derived from the Poisson-Boltzmann equation for cylindrical polyelectrolytes in excess 0.05 M NaCl), which is well within the range of the superhelical diameter (10-20 nm).

As will be reported in an accompanying paper, in the present concentration range a first-order phase transition occurs to a liquid crystal. Except for the isotropic solutions with 2 and 3 g DNA/dm³, our samples are microphase separated with a progressive growth of the liquid crystalline domains with increasing concentration. In the liquid crystalline 27 g of DNA/dm³ sample, the spacing R between the molecules can be obtained from the hexagonal unit cell volume R²L/2 = ¹ and takes the value 13 nm. This spacing is on the same order of magnitude as the effective diameter of the supercoil 11.5 nm, as derived from the linearized Poisson-Boltzmann equation according to Eq. 8 with bare diameter 10 nm (Table 1) and in 0.05 M monovalent salt. To accommodate the molecules in the confined state, the radius of the supercoil has to decrease at the cost of a significant elastic bending and twisting energy of the duplex.

As anticipated, the normalized length and opening angle do not change much, although there is a tendency to smaller values for higher DNA concentrations. In this context, it is of interest to note that Torbet and DiCapua (1989) reported a pitch angle 36 ± 5° in the liquid crystalline regime for plasmid concentrations above 45 g/dm³. This decrease in pitch angle and normalized length shows that, apart from a decrease in diameter, spatial confinement results in a slight compression of the supercoil in the longitudinal direction. On the other hand, the values averaged over the various DNA concentrations in Table 1, 2L/l = 0.85 ± 0.05 and  = 58 ± 5°, are very close to the ones obtained from electron microscopy: 2L/l = 0.82 ± 0.05 and  = 55° (Boles et al. 1990) and theoretical analysis 2L/l = 0.81 and  = 54° (Ubbink and Odijk, 1999).

Because of the (near) constancy of the opening angle, the writhe decreases significantly with increasing packing fraction through the phase transition. According to the fact that through White's Eq. 1 the sum of the excess twist and the writhe are conserved (White, 1969), this decrease in writhe should be compensated by a positive twist exerted on the DNA duplex if the linking number deficit is fixed. With the results in Table 1, the excess twist should increase by approximately eight turns if the concentration is increased from 3 to 27 g of DNA/dm³. This increase in Tw corresponds to a twist of ~1° per base pair, and with a torsional persistence length L_t  75 nm (Bloomfield et al., 2000), the associated free energy amounts ~0.04 kT. If the twist exceeds a certain critical value, the torsion may be (partially) relaxed with the phosphate backbone turned inside and the unpaired bases exposed on the outside (Strick et al., 1998). We found, however, no evidence for this effect, because the cross-sectional radius of gyration of the duplex agrees with the B form for all investigated plasmid concentrations. It is our conjecture that the free energy associated with the excess twist is of paramount importance in controlling the critical boundaries pertaining to the transition to the liquid crystalline phase.

Another possibility for an increase in number of superhelical turns is an ionic strength induced decrease in helical repeat distance of the double-stranded DNA duplex. Cherny and Jovin (2001) observed a change of configuration of negatively supercoiled pPGM1 plasmid (2987 bp) from a plectonemic form with one or two nodes to 14 ± 1 nodes, if the concentration of monovalent salt is increased from 0.001 to 0.05 M. For high salt concentration, the number of nodes approached the expected value for a 3000 bp plasmid with a superhelical density  ~ 0.06. In our situation, the number of nodes (which is on the order of minus the writhe) increases beyond the expected value ~7 ( ~ 0.03) to ~18 with an increase in ionic strength from 0.05 to 0.06 M (the increase in ionic strength comes from uncondensed counterions only). Because such a significant effect from such a small increase in ionic strength is unlikely, we consider a change in helical repeat distance of the duplex irrelevant in explaining the observed change in overall DNA geometry.

For a rigid rod with length L and diameter D, the second virial coefficient including end effects reads A₂ = /4 L² D(1 + 4 D/L) (Onsager, 1949; Odijk, 1990). With the normalized lengths and second virial coefficients in Table 1, we derive an effective diameter pertaining to the excluded volume D = 4, 9, and 15 nm for 6, 11, and 27 g of DNA/dm³, respectively. The latter values are on the same order of magnitude as twice the plectonemic radius (Table 1), which shows that the supercoils take indeed a rather extended configuration. However, because no good models are available to describe the excluded volume of supercoiled DNA, including the effects of branching and overall flexibility, we consider further interpretation of A₂ not feasible.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

With a view to determine the configuration and regularity of plectonemically supercoiled DNA, we have measured the small angle neutron scattering from pUC18 plasmid in 0.05 M NaCl. In the present q range, interference over a spatial extent on the order of the radius and pitch of the supercoil is sampled and, accordingly, effects of overall flexibility and/or branching of the superhelix are beyond observation. We have derived the mathematical expression for the single chain scattering function (form factor) of an interwound structure. In particular, for a precise estimate of the superhelical dimensions and to ensure the correct high q limiting behavior of the structure factor, it appeared to be necessary to include the longitudinal interference over the plectonemic pitch. Interactions among supercoils can be taken into account in the second virial approximation, because at the present ionic strength electrostatic interactions are screened over a spatial extent less than the superhelical diameter. It was found that a local plectonemic structure describes the scattering data well, provided a significant distribution in radius and pitch is included. This distribution comes from a variation in molecular shape resulting from a distribution in linking number deficit and/or thermal fluctuations. Furthermore, the partitioning of DNA over coexisting isotropic and liquid crystalline phases with a concomitant change in plectonemic dimensions contributes to a broadening of the distribution in radius. We have assumed that the plectonemic opening angle is constant, so that the variation in pitch is correlated with the one in radius. A model in which the pitch and radius are allowed to fluctuate independently gives an inferior fit to the data. Furthermore, we would not be able to detect a modest fluctuation in opening angle, because theoretical analysis shows that this gives a minor additional damping of the normalized structure factor only.

For diluted supercoils with vanishing mutual interaction, the derived results for the radius, pitch, opening angle, and normalized length agree with independent electron microscopy measurements by Boles et al. (1990), scanning force microscopy (Rippe et al., 1997), and previous SANS work (Torbet and DiCapua, 1989; Hammermann et al., 1998). Furthermore, the estimated number of superhelical turns (proportional to minus the writhing number), as well as the variation, agrees with the linking number deficit as determined by gel electrophoresis. This suggests that the main contribution to the variation in adjusted parameters comes from the presence of a distribution in topoisomers rather than thermal fluctuations. However, to gauge the relative importance of the fluctuation contribution it is necessary to do experiments with a narrow distribution in preset linking number deficit. With increasing plasmid concentration, prior and covering the transition to a liquid-crystalline phase, the radius and pitch are seen to decrease significantly. The opening angle and normalized length show a slight tendency to decrease as well, but they remain relatively close to the average values 58° and 0.85°, respectively. The latter observation implies that compaction of negatively supercoiled DNA by spatial confinement results in a decrease in writhing number at the cost of a positive twist exerted on the DNA duplex. It is our conjecture that the free energy associated with this excess twist is of paramount importance in controlling the critical boundaries pertaining to the transition to the anisotropic, liquid-crystalline phase.