Torsional directed walks, entropic elasticity, and DNA twist stiffness

Torsional directed walks, entropic elasticity, and DNA twist stiffness

J. David Moroz and
Philip Nelson*

Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104

Communicated by Sidney R. Coleman, Harvard University, Cambridge, MA (received for review September 4, 1997)

Abstract

DNA and other biopolymers differ from classical polymers because of their torsional stiffness. This property changes the statistical character of their conformations under tension from a classical random walk to a problem we call the “torsional directed walk.” Motivated by a recent experiment on single lambda-DNA molecules [Strick, T. R., Allemand, J.-F., Bensimon, D., Bensimon, A. & Croquette, V. (1996) Science 271, 1835–1837], we formulate the torsional directed walk problem and solve it analytically in the appropriate force regime. Our technique affords a direct physical determination of the microscopic twist stiffness C and twist-stretch coupling D relevant for DNA functionality. The theory quantitatively fits existing experimental data for relative extension as a function of overtwist over a wide range of applied force; fitting to the experimental data yields the numerical values C = 120 nm and D = 50 nm. Future experiments will refine these values. We also predict that the phenomenon of reduction of effective twist stiffness by bend fluctuations should be testable in future single-molecule experiments, and we give its analytic form.

Footnotes

↵ * To whom reprint requests should be addressed. e-mail: nelson{at}dept.physics.upenn.edu.
↵ † Our notation is similar to ref. 16. Throughout we will neglect sequence dependence. In the force regime in question we expect linear rod elasticity to be a good approximation. Higher-order bend elasticity effects are expected to be suppressed by powers of the rod radius (1 nm), which is much smaller than any other length scale. Indeed a linear-elastic model, the “extensible worm-like chain,” describes accurately the extension of torsionally unconstrained DNA up to forces greater than those considered here, with an intrinsic stretch modulus more than 100 times greater than the forces of interest to us (9).
↵ ‡ Certainly the omitted anisotropic couplings also will renormalize the constants A, C in Eq. 1 (unpublished work). The model (Eq. 1) is to be regarded as coarse-grained to the scale of the helix pitch.
↵ § The leading-order perturbative formulas can be obtained directly, without appeal to White’s formula (11). A more elegant approach takes the configuration variables to be a 3 × 3 rotation matrix; the torque term then takes the form −τ ∫ ds[Ω₃ + Ω̌₃]/(1 + t⋅z), where Ω̌_i are the space-fixed angular velocities of a rigid body (unpublished work).
↵ ¶ To justify perturbation theory itself we note that it gives an excellent approximation to the exact solution of the worm-like chain (6) when K > 1, as may be expected from the form of the leading anharmonic correction below. Nonperturbative effects in the variational approach to the worm-like chain (6) are also small when K > 1. Note that our data cuts also eliminate the region σ > 2π/Cω₀ where plectoneme formation (and hence large self-avoidance effects) is expected (32). Raising our threshold on K selected fewer points with little effect on our result.
↵ ‖ The variance of the data from our curves is σ_Z/L = 0.013, comparable to the visible scatter in the data. The formal covariances for A, C, D correspond to very small errors; in practice the fit is visibly worse for C outside the range 70 < C < 150. Omitting either the f/γ or the −Dσ/γ terms makes a poorer fit, as does replacing Eq. 5 by the naïve τ = ω₀σ/C _eff with constant C _eff.
↵ ** An independent check of our measured value of the twist-stretch coupling D is not available, but it is interesting to compare to the situation with RecA. Binding to RecA unwinds it to σ = −0.52 and stretches it by 0.54 times its natural length. Unwinding DNA slightly at constant tension lengthens it by Dσk _B Tω₀ ²/γ (14, 15). Although σ = −0.52 is outside the domain of linear elasticity, applying the formula gives an extension of 0.33; much of the extension is explained by the unwinding without requiring additional tension. This argument at best confirms that our value of D is not unreasonable.