Twist-bend coupling, twist waves, and the shape of DNA loops

S. K. Nomidis, M. Caraglio, M. Laleman, K. Phillips, E. Skoruppa, and E. Carlon

Phys. Rev. E 100, 022402 – Published 5 August 2019

Abstract

By combining analytical and numerical calculations, we investigate the minimal-energy shape of short DNA loops of approximately 100 base pairs (bp). We show that in these loops the excess twist density oscillates as a response to an imposed bending stress, as recently found in DNA minicircles and observed in nucleosomal DNA. These twist oscillations, here referred to as twist waves, are due to the coupling between twist and bending deformations, which in turn originates from the asymmetry between DNA major and minor grooves. We introduce a simple analytical variational shape that reproduces the exact loop energy up to the fourth significant digit and is in very good agreement with shapes obtained from coarse-grained simulations. We, finally, analyze the loop dynamics at room temperature, and show that the twist waves are robust against thermal fluctuations. They perform a normal diffusive motion, whose origin is briefly discussed.

Received 13 April 2019

DOI:https://doi.org/10.1103/PhysRevE.100.022402

Physics Subject Headings (PhySH)

DNA

Polymers & Soft MatterPhysics of Living SystemsStatistical Physics & Thermodynamics

Authors & Affiliations

S. K. Nomidis^1,2, M. Caraglio^1,3, M. Laleman¹, K. Phillips¹, E. Skoruppa¹, and E. Carlon ¹

¹Laboratory for Soft Matter and Biophysics, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
²Flemish Institute for Technological Research (VITO), Boeretang 200, B-2400 Mol, Belgium
³Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, A-6020 Innsbruck, Austria

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Vol. 100, Iss. 2 — August 2019

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Images

Figure 1
Minimal-energy loop shape (left) and curvature as a function of the rescaled arc-length coordinate $s / L$ (right). Blue solid lines are obtained from the exact solution [Eq. (A5)] [21], while red dashed lines show in (a) and (b) the first- and in (c) and (d) third-harmonic solutions, corresponding to Eqs. (10) and (12), respectively.
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Figure 2
Plots of the bending ( $Ω_{1}$ and $Ω_{2}$ ) and twist ( $Ω_{3}$ ) densities of a closed triad-model loop, as functions of the reduced arc-length $s / L$ . The black circles are low-temperature Monte Carlo simulations of the triad model (with 104 bp), while the red and blue solid lines are the first- [Eq. (17)] and third-harmonic [Eq. (18)] Ansätze, respectively. The inset zooms into the one end of the loop, and reveals that the third-harmonic Ansatz more accurately reproduces the data. The stiffness constants were chosen as $A_{1} = 81 nm, A_{2} = 39 nm, C = 105 nm$ and $G = 30 nm$ , calculated in Ref. [17] from the oxDNA2 model. The global phase $ϕ = 4.02$ rad is the only fitting parameter.
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Figure 3
(a) Similar to Fig. 2, but with the data obtained from low-temperature oxDNA2 simulations (black points). For the harmonic approximations [Eqs. (17) and (18), shown with red and blue lines, respectively] we used the same values as in Fig. 2, extracted from oxDNA2 simulations [17]. Once more, the global phase $ϕ = 1.70$ rad is the only fitting parameter. (b) Actual configuration of the loop from which the plots in (a) were produced. The yellow points indicate the bp that were tethered to two fixed points in space by means of harmonic springs, effectively fixing the end distance (center-of-mass distance between the two pairs of yellow beads) at $d = L / 10$ .
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Figure 4
Kymogram plots of the time evolution of the deformation densities $Ω_{i}$ along an oxDNA loop (top three panels), together with the corresponding global phase (bottom panel), obtained from room-temperature simulations. The right subplots show a zoom in the region marked by red lines, revealing that the correlations predicted by Eqs. (17) and (18) survive the effect of thermal fluctuations. In the kymographs, the blue and white regions correspond to positive and negative values of $Ω_{i}$ , respectively.
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Figure 5
Time evolution of the mean-squared displacement of the phase $ϕ$ (blue points), averaged over six independent oxDNA2 runs, which reveals regular diffusion. Fitting to $〈 Δ ϕ^{2} 〉 = 2 D t$ (solid red line), we obtain the value $D = 121 {rad}^{2} / μ s$ for the diffusion coefficient.
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Figure 6
Comparison of the shape among the one-harmonic approximation (10) (dotted green lines), two-harmonics approximation (12) (dashed red lines), and the exact solution (solid blue lines), with the two ends of the loop being fixed at a distance $L / 10$ (a) and $L / 5$ (b).
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