Possible Origin of the Increased Torsion Elastic Constant of Small

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Possible Origin of the Increased Torsion Elastic Constant of Small Circular DNAs: Bending-Induced Axial Tension J. Michael Schurr J. Phys. Chem. B, Just Accepted Manuscript • Publication Date (Web): 02 May 2017 Downloaded from http://pubs.acs.org on May 4, 2017

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The Journal of Physical Chemistry

Possible Origin of the Increased Torsion Elastic Constant of Small Circular DNAs: Bending-Induced Axial Tension

J. Michael Schurr Department of Chemistry University of Washington Box 351700 Seattle, WA 98195 E-mail: [email protected] Tel. 206 522 7583

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Abstract Deforming an intrinsically straight elastic rod into a circle is shown to introduce an axial tension that acts to extend the rod, much like an externally applied force. The response of a circular DNA to such axial tension was reckoned using a previously suggested model of a force-dependent cooperative transition between a shorter torsionally softer a conformation and a longer torsionally stiffer b conformation. Each of three earlier reported optimal sets of parameters, that well-fitted both relative extension vs. force and torsion elastic constant vs. force data on single DNAs under tension, was applied here to predict torsion elastic constants of the effective springs between base-pairs for both linear and circular 181 bp and ~210.8 bp DNAs under their respective conditions. Predicted values for both linear and circular species agreed well with their corresponding experimental values, which strongly suggests that the observed 1.4- to 1.5-fold enhancement of the torsion elastic constants upon circularization arises from such axial tension. Experimental torsion elastic constants lie in the range, (6.4 to 6.6)x10−19 J for these linear DNAs, and in the range (9.1 to 9.9)x10−19 J for the corresponding circles, significantly below the limiting value ~12x10−19 J at tensions exceeding 4 pN.

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Introduction Over twenty years ago it was shown that circularization of a 181 base-pair linear DNA caused a ~1.5-fold increase in the torsion elastic constant of the effective springs between base-pairs from α = (6.4 ± 0.5) x 10−19 J to (9.1 ± 0.7 and 9.9±0.8) x 10−19 J for two differently prepared samples of the circular DNA.1 These values were obtained by fitting appropriate theoretical expressions for the linear and circular species, which had been well tested on Brownian dynamics simulations,2 to time-resolved fluorescence polarization anisotropy (FPA) data. FPA was then, and still is, the only method capable of providing such a direct comparison between the torsion elastic constants of unstrained linear and circular forms of the same DNA in this size range. The values obtained for the unstrained linear species were typical of those measured by the same method for many linearized plasmid DNAs of various lengths under similar conditions, namely ~0.1 M univalent cations at 293 K. The values prevailing in the 181 base-pair (bp) circles under such conditions practically matched those, α = (9.12 to 9.53) x 10−19 J, obtained by analyzing topoisomer ratios of circular DNAs of comparable size (205-217 bp) in ~43 mM univalent cations plus 10 mM Mg2+ at 310 K.3-5 In the case of circular 181 bp DNAs, the structural deformation responsible for the increased torsion elastic constant was originally presumed to be bending of the DNA.1 However, extension of the DNA is another possible deformation that was overlooked at the time.

The contour length, or mean rise per bp, of a given DNA was traditionally assumed to be practically constant, largely invariant to applied stresses. This assumption was, and still is,

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commonly invoked in analyses of all kinds of experimental data, including extension vs. force data on single DNAs. Important contradictions of this constant length assumption came from detailed analyses of X-ray solution scattering measurements on 10 to 35 bp DNAs specifically labeled with gold colloids.6-8. The results strongly suggested the existence of a cooperative equilibrium between two distinct conformations with modestly different mean rises per bp. Another contradiction came from an analysis9 of relative extension vs. force data for single DNAs with zero excess twist10, in a paper hereinafter denoted as paper I. By assuming a constant, but unknown, contour length in order to determine the persistence length, this analysis yielded values that increased systematically from, P~ 41 nm at low forces (0.18 to 0.42 pN) up to P ~ 92 nm at 3.9 pN.9 This extreme variation of P from below accepted values at low forces to far above accepted values at higher forces clearly failed to converge to the canonical value, P ~ 50 nm, obtained by numerous other methods, including experiments at forces ≥ 2 pN, and was therefore eschewed. An alternative analysis of the same zero twist data with an assumed constant persistence length, P = 50 nm, yielded rises per bp that increased with increasing tension above 0.25 pN and rose with ever decreasing slope toward a plateau at 3.5 to 3.9 pN (c.f. Figure 2 of ref 9). This observation suggested that tension induced a saturable shift in population of base-pairs from a shorter conformation to a modestly longer conformation with a comparable persistence length and helix repeat (bp/turn).9 Could this shift with increasing tension from a shorter to a longer conformation be the actual “deformation” responsible for the rise in torsion elastic constant upon circularization?

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Torque measurements by different methods on single twisted DNAs held at ≥ 4 pN tension in ~0.1 M univalent cations at 293 K yielded torsion elastic constants, α = (12.1 to 13.0) x 10−19 J,10-13 which are about 2-fold greater than those prevailing in unstrained linear DNAs, and about 1.33-fold greater than those prevailing in the circular 181 and 205 to 217 bp circular DNAs noted above. However, analyses of torque measurements at lower tensions in the range, 0.25 to 3.9 pN,12,13 by using Moroz-Nelson torque theory14,15 yielded an intrinsic torsion elastic constant, α ≈ 6.4 x 10−19 J, at the lowest tensions (0.25 and 0.29 pN), which then rose with increasing tension to reach (9.2 to 10.2) x 10−19 J at 0.74 to 0.91 pN, and finally a plateau in the range, (11.3 to 12.6) x 10−19 J, at 3.5 to 4.0 pN.9 Testing of the Moroz-Nelson torque theory by extensive Monte Carlo simulations indicated that it remained at least moderately, and possibly very, accurate down to tensions as low as 0.25 pN at the superhelix densities examined.16 In paper I, a model was formulated for a simple force-dependent cooperative transition between two conformations with different mean rises per bp. Statistically acceptable sets of the relevant parameters that exhibited similar minimum chi-squared values, were obtained by fitting the model to the “experimental” mean rises per bp at different forces.9 Acceptable cooperativity lengths lay in the range, 224 to 441 bp, indicating a highly cooperative transition. With the four relevant model parameters (P = 50 nm; δ = difference in rise per bp between the longer and shorter states; B0 = intrinsic equilibrium constant for single isolated base-pairs; and J = cooperativity parameter) held fixed at one or another set

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of acceptable values, the α-values of each of the two states were adjusted so as to optimally fit the model with each set of fixed parameters to the experimental inverse α-values for the various tensions. Plausible and equivalent fits were obtained for three different choices of the model parameters spanning the statistically acceptable range, and the torsion elastic constants found for the longer conformer fell in the range, (α = 11.98 to 12.05 x 10−19 J, while the corresponding values for the shorter conformer fell in the range (α = 5.44 to 4.13) x 10−19 J.9

These force-dependent two-state models of a cooperative transition between a shorter torsionally softer and a longer torsionally stiffer conformation accounted reasonably well for the observed variations in both DNA length and torsion elastic constant with tension for all tensions from 3.9 down to 0.25 pN, as shown in Figures 1 and 2 of paper I.9 It also predicted torsion elastic constants at zero tension in the range, α = (5.92 to 6.22) x 10−19 J, well within the range of values, (5.4 to 6.5) x 10−19 J, measured by FPA for various linearized plasmid DNAs 0.1 M univalent ionic strength at 293 K. The acceptable range of δ-values extended from a bit below 0.030 nm to a bit above 0.047 nm. The 0.030 nm value is close to the 0.029 nm estimated by Shi et al. 7 for a 1:1 ratio of populations in the a and b states, and the 0.047 nm value is close to the more recent 0.050 nm estimated by Zettl et al.8 also for a 1:1 ratio of populations. On intuitive grounds it seemed that bending an elastic cylinder into a circle might induce an internal tension along the local rod axis, and if it did, then extension of the DNA caused by that tension, rather than bending per se, could conceivably be the deformation that is

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primarily responsible for the elevated, but still intermediate, values of the torsion elastic constants in the ~ 181 to 217 bp circular DNAs noted above. The possibility that circularization of small ~200 bp DNAs might induce a strain equivalent to an effective tension was also suggested in the legend of Fig. 12 of the supplementary material of ref. 12. The primary objectives of this communication are (1) to demonstrate the existence of such an internal tension and consequent lengthening of a circularly bent elastic cylinder; and (2) to apply the force-dependent cooperative two-state model to circular DNAs containing 181 and 210.8 bp under their respective conditions in order to self-consistently determine the prevailing internal tension, mean rise per bp, the fraction of bp in each state, and the mean inverse torsion elastic constant for each of the previously determined three sets of optimal parameters for that model. As will be seen, the “predicted” torsion elastic constants of the circular DNAs agree surprisingly well with the measured values.

The tension in circularly bent elastic rods The potential energy of a deformed elastic body is the work that must be expended by an external agent in order to effect the particular deformation, and is associated with displaced internal coordinates within that body that resist the deformation. When an intrinsically straight filament with radius r0, length L, and flexural (bending) rigidity A is bent into a (planar) circle and the ends “welded”, so that its axis forms a perfect circle of radius, R = L/(2π), its total bending potential energy is given by

= (A/2)(2π)2/L

Ubend = (A/2)

! |𝑑𝒕/𝑑𝑠|! !

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d𝑠

(1)



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where s is distance along the filament axis, t = dr/ds is the tangent vector at s, r denotes the position of the axis at s in the lab frame, and |dt/ds| is the absolute value of the curvature, which for a (planar) circle is just 1/R = (2π/L). For an isotropic Hookean elastic cylinder, it is well known that the bending rigidity is, A = EI, where E is the Young modulus and I = πr04/2 is the so-called “area moment of inertia” around the local bending axis, which at each point s is perpendicular to the plane of the circle and intersects the axis of the cylinder at that point. It is obvious from equation (1) that increasing L has the effect of decreasing the bending energy, and that is the origin of the tensile force acting to lengthen the DNA. Now consider a small extensional strain, ΔL/L

Possible Origin of the Increased Torsion Elastic Constant of Small

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