Theory of sequence-dependent DNA elasticity
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8.In engineering practice the term “o-ring” is commonly used for elastic rings that are circular when stress-free.
9.See also the discussion given in Refs. 323334353637383940.
10.In our numbering of base pairs the choice of the direction of increasing n is arbitrary. Although it is common to number the base pairs so that increasing n corresponds to advance in the direction of the strand that (in discussions of transcription) is called the sequence strand, we do not impose such a convention here. Because we study the implications of the requirement that the energy be independent of the choice of the direction of increasing n, it is necessary for us to consider transformations that change that direction while leaving invariant the orientation and spatial position of the base pairs.
11.The superposed symbols, etc., here serve to emphasize the distinction between a value of a quantity and a function relating the quantity to other quantities.
12.In general, the center of the rectangle is not the barycenter (center of mass) of the corresponding base pair. Here, as in Ref. 64, and are obtained by the following construction: Let be the straight line connecting the C1′ atoms of the deoxyribose groups corresponding to the nth pair base, and let be the line that passes at a right angle through the midpoint of and intersects the straight line containing the C8 atom of the purine base and the C6 atom of the complementary pyrimidine base. The point is taken to lie at the intersection of and the vector is parallel to and points in the direction of the major groove; is perpendicular to both and with
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15.We use the Einstein summation convention for subscripts (but not superscripts).
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19.Such an external force and moment can arise when the nth base pair is in contact with a protein or with a base pair sequentially distant from it. When electrostatic forces are taken into explicit account, P in Eq. (2.10) will include the potential energy of the interaction of charged residues.
20.The configurations discussed in Sec. IV may be considered appropriate to cases in which the ionic strength of the solution is sufficiently high (⩾1 M) that the Debye length is significantly less than the distance between charged sites and the employed values of elastic moduli and intrinsic kinematical parameters have been adjusted to account for the electrostatic interaction of adjacent base pairs.
21.Packer, Dauncy, and Hunter (Ref. 65) have used a computational model to calculate the potential energy surface for base-pair steps as a function of two principal degrees of freedom, slide, and shift, and obtained results that suggest that the presence of close local minima of energy (of the type observed in crystal structures of DNA oligomers in Ref. 5) can limit the range of applicability of quadratic approximations for Their calculations for tetramer steps (Ref. 66) further suggest that one should not blandly assume that interactions of nonadjacent base pairs need not be taken into account. However, there are not as yet a sufficient number of high-resolution crystallographic structures available to treat all 136 tetramer steps in the analysis of the sequence-dependent properties of DNA and to answer the question, posed in Ref. 67, of whether the base-pair neighbors adjacent to a dimer step have a strong influence on its elastic properties.
22.O’Hern et al. (Ref. 68) developed a model of DNA elasticity in which base pairs are assumed to be rigid plates separated by a homogeneous isotropic elastic material, and showed that their model implies that, for each n, is a quadratic function obeying the relations seen in Eqs. (2.23). We here show that the Eqs. (2.23) hold in our more general model when The available experimental data (Refs. 5 and 6) strongly indicate that Eqs. (2.23) hold when and only when A DNA sequence for which at every base-pair step must be either of the type ⋯ATATAT⋯ or ⋯GCGCGC⋯.
23.The values and ranges we give for the moduli and parameters are based on data reported in Ref. 6.
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26. is the permutation symbol of Levi-Civita.
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42.Base-pair level modes of DNA deformability constructed to account for results of measurements of cyclization kinetics and gel electrophoretic mobility of short DNA segments suggest that highly curved DNA is made up of two types of duplex structure within each helical repeat length: the canonical B-type duplex and a perturbed form in which the base pairs are inclined with respect to the duplex axis (Refs. 69 and 70) as they are in an A-type or C-type duplex.
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46.Electron micrographs of a 219 bp segment with such a sequence found in C. fasiculata indicate that the segment has the form of a complete circle (Ref. 71) and DNA segments with higher intrinsic curvature have been synthesized. Koo et al. (Ref. 72) have given strong evidence, based on a study of cyclization kinetics, to the effect that a 210 bp DNA segment with the sequence can form a complete circle when stress-free. Previously, Ulanovsky et al. (Ref. 73) had showed that DNA segments with sequences can be closed to form minicircles with 105, 126, 147, and 168 bp, i.e., with 6, 7, 8. They found that the maximum value of the ratio of concentrations at equilibrium of circular and noncircular products occurred when the sequence length was either 126 or 147 bp, and they suggested that such an observation indicates that it may be possible to obtain stress-free minicircles with as few as 137 bp. See also the review of Crothers et al. (Ref. 74).
47.So as to obtain an o-ring of small size, i.e., with bp, while making the simplifying assumption that the intrinsic shears and are everywhere zero, we chose in Eq. (4.2) to be larger than the available estimates of the upper bound for stress-free values of the roll.
48.The values of and were chosen so that the bending rigidity of each minicircle equals that of naturally straight DNA with a persistence length of 500 Å. The choice ω=0.7 is then compatible with measurements of fluorescence anisotropy of dyes intercalated in open segments of DNA (see Ref. 75). Measurements of topoisomer distributions for miniplasmids (Ref. 76) suggest that ω should be close to 1.4. [See also Bouchiat and Mezard (Ref. 77), who state that single molecule stretching experiments (Ref. 78) yield a value of 1.6 for ω.]
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50.The (approximate) analysis of Ref. 36 yielded the value ω=0.9 as the critical value of ω for loss of stability.
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53.We say “essentially independent,” for the intercalative binding of EtBr changes not only the twist but also the rise (Ref. 51).
54.The present theory is not intended to provide an interpretation of an observed preference of EtBr for binding to pyrimidine-purine base-pair steps (Ref. 82).
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58.As the o-ring we are considering here is composed of periodically repeating units of length 10 bp, the minimum energy configuration for a given distribution of sites of intercalation would not be affected if each such site were shifted by 10 bp in the direction of increasing n. Although a shift in the distribution by less than 10 bp can affect the minimum energy configuration, our calculations show that for Minicircle I the corresponding change in Ψ does not exceed 0.15
59.It follows from the theory of ideal elastic rods that if the minicircle were formed by closing naturally straight DNA, i.e., if Eq. (4.2) were replaced by Eq. (4.1), then the untwisting at a single base-pair step by an angle of magnitude less than 360°, as it corresponds to a change in excess link less than 1, would not affect the shape of the minicircle, i.e., the axial curve would remain circular.
60.There are protein structures that can induce a comparable change in the magnitude of twist, but the changes they induce are generally distributed over several base-pair steps. See, e.g., Ref. 83 reporting a case in which an 8 bp long region is untwisted by 147° in a crystal structure of the human TFIIB-TBP protein complex bound to duplex DNA.
61.See also Ref. 32.
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