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Statistical mechanics of elastica for the shape of supercoiled DNA: hyperelliptic elastica of genus three

Shigeki Matsutani

Abstract

This article studies the statistical mechanics of elastica as a model of the shapes of the supercoiled DNA, and shows that its excited states can be characterized by the focusing modified KdV (MKdV) equation due to thermal fluctuation. Following the previous paper (Matsutani and Previato, Physica D 430 (2022) 133073), the hyperelliptic solutions of the focusing modified KdV (MKdV) equation of genus three are considered. There appears a pattern as a repetition of the modulation of figure-eight and the inverse 'S' as a thermal fluctuation of elastica, called the S-eight mode. Our model states that the excited states of elastica due to the thermal effect have the S-eight mode, which reproduces the shapes of the AFM image of the supercoiled DNAs observed by Japaridze et al. (Nano Lett. 17 3, (2017) 1938).

Statistical mechanics of elastica for the shape of supercoiled DNA: hyperelliptic elastica of genus three

Abstract

This article studies the statistical mechanics of elastica as a model of the shapes of the supercoiled DNA, and shows that its excited states can be characterized by the focusing modified KdV (MKdV) equation due to thermal fluctuation. Following the previous paper (Matsutani and Previato, Physica D 430 (2022) 133073), the hyperelliptic solutions of the focusing modified KdV (MKdV) equation of genus three are considered. There appears a pattern as a repetition of the modulation of figure-eight and the inverse 'S' as a thermal fluctuation of elastica, called the S-eight mode. Our model states that the excited states of elastica due to the thermal effect have the S-eight mode, which reproduces the shapes of the AFM image of the supercoiled DNAs observed by Japaridze et al. (Nano Lett. 17 3, (2017) 1938).
Paper Structure (6 sections, 5 theorems, 21 equations, 4 figures)

This paper contains 6 sections, 5 theorems, 21 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statistical Mechanics of elastica
  3. Hyperelliptic solutions of the focusing modified KdV equation over ${\mathbb C}$
  4. Hyperelliptic Curves of Genus Three
  5. Numerical results and discussion
  6. Conclusion

Key Result

Theorem 3.1

Mat02b For $((x_1,y_1),\cdots,(x_g,y_g)) \in S^g X_g$, a fixed branch point $b_a$$(a=1, 2, \ldots, 2g+1)$, and $u:= v( (x_1,y_1),$$\cdots,(x_g,y_g) )$, satisfies the MKdV equation over ${\mathbb C}$, where $\partial_{u_i}:= \partial/\partial u_i$ as an differential identity in $S^g X_g$ and ${\mathbb C}^g$.

Figures (4)

  • Figure 1: The orbits of each $\varphi_i$ in the quadrature: (a): $k_1> k_2 > k_3>1.0$. (b): $k_3>k_2>k_1>1.0$.
  • Figure 2: An open excited state of elastica: $(k_1, k_2, k_3) = (1.04, 1.0392, 1.010)$, and the initial condition is $(\varphi_1, \varphi_2, \varphi_3) = (\varphi_{\mathfrak b}, -0.90, -0.90)$. (a): $\psi_{\mathrm r}$ and $\psi_{\mathrm i}$, and (b): a shape of the excited state of elastica.
  • Figure 3: An open excited state of elastica: $(k_1, k_2, k_3) = (1.0260, 1.0259, 1.0008)$$(\varphi_1, \varphi_2, \varphi_3) = (\varphi_{\mathfrak b}, 0.0, 0.0)$ (a) is the profile of $\psi_{\mathrm r}$ and $\psi_{\mathrm i}$ and (b) is the shape of the excited state of elastica.
  • Figure 4: An open curve of excited state of elastica and shape of supercoiled DNA: $(k_1, k_2, k_3) = (6.00, 7.00, 8.00)$, and the initial condition is $(\varphi_1, \varphi_2, \varphi_3) = (\pi-\varphi_{\mathfrak b}, 1.4, 1.4)$. (a) is the profile of $\psi_{\mathrm r}$ and $\psi_{\mathrm i}$, (b) is a shape of the excited state of elastica, and (c) is the shape of a supercoiled DNA, which is a part of the AFM images in JMB.

Theorems & Definitions (6)

  • Theorem 3.1
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • Theorem 4.4