# Adsorption of a Helical Filament Subject to Thermal Fluctuations

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## Abstract

**:**

## 1. Introduction

## 2. Simulations

#### 2.1. Model

#### 2.2. Simulation Results: H-Filaments Confined by a Harmonic Potential

#### 2.3. Simulation Results: H-Filaments Adsorbed in a Localized Surface Potential

#### 2.3.1. Adsorption of H-Filaments, $\gamma >1$

#### 2.3.2. Adsorption of H-Filaments, $\gamma <1$

## 3. Theory

#### 3.1. Instability of the 2D Configuration

- -
- For $B{\omega}_{1}^{2}<C{\omega}_{3}^{2}$, only the largest root ${q}_{1}^{\star}$ is positive. All long wavelength $q<{q}_{1}^{\star}$ are unstable. This corresponds to the case where wavy shapes are favored over circular ones in 2D.
- -
- For $B{\omega}_{1}^{2}>C{\omega}_{3}^{2}$, the two roots ${q}_{1}^{\star},{q}_{2}^{\star}$ are positive. The low q-modes are stable. Intermediate wavelength ${q}_{2}^{\star}<q<{q}_{1}^{\star}$ are unstable. This suggests (it is only a linear stability study) that, when the helical chain goes on the surface, it does so by forming loops of intermediate length. This corresponds to the case where circular shapes are favored over wavy ones in 2D.

#### 3.2. H-Filaments Maintained by a Harmonic Surface Potential

#### 3.3. Stability of a Finite Helical Strand

#### 3.4. H-Filaments Adsorbed in a Localized Surface Potential

## 4. Conclusions

- (a)
- Wavy shapes or 3D loops which avoid self-crossing of the chain, where the frustration of the configuration is mainly localized at some spots along the chain.
- (b)
- Spiral shapes, where the frustration is smeared out and terminal sections escape into 3D tails.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Simulation Method

**Figure A1.**(

**a**) DOS $g(E,\overline{{z}^{2}})$ of WLCT under various strength of harmonic potential $V\left(z\right)=\frac{1}{2}k{z}^{2}$, $k=0,0.05$ (weak confinement) and $k=0.5$ (strong confinement). (

**b**) Loops of WLCT at various k. (

**c**) $\langle {z}^{2}\rangle $ of WLCT in harmonic potential confinement in log-log scale for $k=0.05,0.1,0.2,0.4$, and $1.0$ from top to bottom. The line is used to guide the eye and represents $\langle {z}^{2}\rangle \propto {k}^{-3/4}$.

**Figure A2.**(

**a**) DOS $g(E,\overline{{z}^{2}})$ of WLCT filament interacting with short range attractive potential. (

**b**) Representative conformation of WLCT at weak ($\u03f5=0.1$) and strong ($\u03f5=1.2$) adsorption regime. (

**c**) Loop length distribution $P\left(s\right)$. The line represent the scale of adsorption loop length distribution $P\left(s\right)\propto {s}^{-5/2}$.

## Appendix B. Derivation of Partition Sum

## References and Notes

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**Figure 1.**Various shapes of super-Helical filaments. (

**a**) The helical shape of the ground state conformation of H-filaments (${\omega}_{1}=0.81{\sigma}^{-1}$, ${\omega}_{2}=0$, and ${\omega}_{3}=0.10{\sigma}^{-1}$). The local twist and curvature are represented by a set of orthogonal vectors $\{\overrightarrow{u},\overrightarrow{f},\overrightarrow{v}\}$ along the filament backbone. $\overrightarrow{f},\overrightarrow{v}$ are defined in the material frame orthogonal to the local tangent vector $\overrightarrow{u}$. Top panels in (

**b**–

**d**) show H-filaments with $\gamma >1$, (${\omega}_{1}=0.81{\sigma}^{-1}$, ${\omega}_{2}=0$, and ${\omega}_{3}=0.10{\sigma}^{-1}$), where circular shapes are ground states when squeezed. Bottom panels in (

**b**–

**d**) show H-filaments with $\gamma <1$, (${\omega}_{1}=0.09{\sigma}^{-1}$, ${\omega}_{2}=0$, and ${\omega}_{3}=0.19{\sigma}^{-1}$), where wavy shapes are ground states when squeezed. The localized parts of H-filaments ($\left|z\right|<0.2\sigma $) are shown in red. (

**b**) Bulk conformations subject to thermal fluctuations. (

**c**) Typical conformations of H-filaments confined by harmonic potential $V\left(z\right)=\frac{1}{2}k{z}^{2}$ and (

**d**) adsorbed under a localized surface potential ${U}_{ads}\left(z\right)=4\u03f5[{(\sigma /z)}^{12}-{(\sigma /z)}^{6}]$. The numbers are the strength of harmonic potential k and the surface potential $\u03f5$. The figures with horizontal lines indicating the line of $z=0$ are side views. The flexural modulus $B=50\sigma $k${}_{B}$T and twist modulus $C=B/2$.

**Figure 2.**Various measured properties of H-filaments, consisting of N = 90 monomers, confined by a harmonic potential $V\left(z\right)=\frac{1}{2}k{z}^{2}$. One end is anchored at $z=0$. Symbols ∘, filled ⋄, and *, respectively, stand for: (i) $\gamma >1$; (ii) $\gamma <1$; and WLCT (${\omega}_{1}={\omega}_{2}={\omega}_{3}=0$). (

**a**) The height fluctuation $\langle {z}^{2}\rangle $ as the function of the stiffness k of the harmonic potential. The solid line represents analytical calculation in weak fluctuation limit (Equation (11)) for Case (i), ${\omega}_{1}=0.18{\sigma}^{-1}$, and ${\omega}_{3}=0.10{\sigma}^{-1}$. The inset shows mean squared height $\langle {z}^{2}\rangle $ as a function k on log-log scale. The solid line is used to guide the eye for the exponent in relation $\langle {z}^{2}\rangle \propto {k}^{-3/4}$ expected for the WLC. (

**b**) Average tail length $\langle {l}_{\mathrm{tail}}\rangle $; (

**c**) average loop length $\langle {s}_{\mathrm{loop}}\rangle $; and (

**d**) mean number of confined monomers $\langle {n}_{\mathrm{c}}\rangle $.

**Figure 3.**The loop length distributions for: (

**a**) $\gamma >1$ (∘); and (

**b**) $\gamma <1$ (⋄). Loop length is defined as the segment length that consecutively belongs to $\left|z\right|>0.2$. Two representative regimes, weak confinement regime k = 0.05 and strong confinement regime $k=1.0$, are shown by filled and empty symbols, respectively. The loop distribution with $k=0$ is shown as gray symbols (+) for comparison. For $\gamma >1$, the loop length making half of the 3D helical period is most prevalent at weak confinement. Smaller loops are equally common for $\gamma <1$, where the H-filament is more confined (see Figure 1a). Strongly confined segments show almost flat distributions for $\gamma >1$ and two sub-populations for $\gamma <1$.

**Figure 4.**H-filaments adsorbed in a localized surface potential ${U}_{ads}$. Various properties are measured for a H-filament of length $S=90\sigma $ with one of its ends anchored at $z=0$. Symbols ∘ and filled ⋄ stand for the case of $\gamma >1$ and $\gamma <1$, respectively. WLCT case at strong/weak adsorption regimes are represented as *. (

**a**) $\langle {z}^{2}\rangle $ at various strengths of surface potential $\u03f5$. The inset shows the average number of the adsorbed monomers ($\left|z\right|<0.2$). (

**b**) Average tail length $\langle {l}_{\mathrm{tail}}\rangle $; and (

**c**) average loop length $\langle {s}_{\mathrm{loop}}\rangle $ at various $\u03f5$.

**Figure 5.**The loop length distributions of H-filament $\gamma >1$ (

**a**,

**b**) and $\gamma <1$ (

**c**,

**d**) for the two representative values of $\u03f5$: weak adsorption regime ($\u03f5$ = 0.1) (

**a**,

**c**); and strong adsorption regime ($\u03f5$ = 1.0) (

**b**,

**d**). The envelop of distributions in general follows WLC statistics ${s}^{-5/2}$ (with an offset $\lesssim 1$ in s), which is represented as solid line.

**Figure 6.**Three representative shapes of adsorbed H-filaments ($\gamma >1$) at: (

**a**) $\u03f5$ = 0.4; (

**b**) $\u03f5$ = 0.8; and (

**c**) $\u03f5$ =1.6. The localized parts are shown in red. The middle panels show measured local curvatures $\kappa \left(s\right)$ relative to the preferred curvature ${\omega}_{1}=0.181$. Local curvatures $\kappa \left(s\right)=\sqrt{{\mathsf{\Omega}}_{1}^{2}+{\mathsf{\Omega}}_{2}^{2}}$ (∘) are averaged over several similar conformations along the contour s. Lower panels show twist energy density (green ▵), bending energy density (blue *), and the sum of these two (red ∘).

**Figure 7.**(

**a**) The bare inverse structure factor ${S}_{0}^{-1}\left(q\right)$ defined in Equation (10) (i.e., $k=0$) for ${\omega}_{1}=0.18$ and ${\omega}_{3}=0.10$. Boundary conditions impose flat circular shapes outside of the section of interest. Note the region of instability indicated by negative ${S}_{0}^{-1}$. The harmonic potential of strength k would shift the inverse structure factor upwards by k, ${S}_{k}^{-1}\left(q\right)={S}_{0}^{-1}+k$. Hence, $k>0.005$ stabilizes the flat helical chain against small amplitude fluctuations. In practice, larger k is required to ensure the validity of the quadratic expansion. (

**b**) The height fluctuation $\langle {z}^{2}\rangle $ in weak fluctuation limits as a function of the stiffness fk of the harmonic potential for helical filament (i), ${\omega}_{1}=0.18{\sigma}^{-1}$, ${\omega}_{3}=0.10{\sigma}^{-1}$. Analytical calculation following Equation (11) is shown as solid line and the simulation data for filaments with one end anchored are shown in dots together with typical conformations at some k-values. Flat configurations at k = 0.5 and 1.0 are captured from the top view.

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**MDPI and ACS Style**

Chae, M.-K.; Kim, Y.; Johner, A.; Lee, N.-K.
Adsorption of a Helical Filament Subject to Thermal Fluctuations. *Polymers* **2020**, *12*, 192.
https://doi.org/10.3390/polym12010192

**AMA Style**

Chae M-K, Kim Y, Johner A, Lee N-K.
Adsorption of a Helical Filament Subject to Thermal Fluctuations. *Polymers*. 2020; 12(1):192.
https://doi.org/10.3390/polym12010192

**Chicago/Turabian Style**

Chae, M.-K., Y. Kim, A. Johner, and N.-K. Lee.
2020. "Adsorption of a Helical Filament Subject to Thermal Fluctuations" *Polymers* 12, no. 1: 192.
https://doi.org/10.3390/polym12010192