# Role of Bending Energy and Knot Chirality in Knot Distribution and Their Effective Interaction along Stretched Semiflexible Polymers

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### Model and Simulation Methodology

## 3. Results

#### 3.1. Knots Sizes

#### 3.2. Knots Free Energy

#### 3.3. Relative Orientation of the Knots

## 4. Discussion

- When the knots along the chain are clearly separated and sufficiently far from the hard walls, the free energy is dominated by an entropic interaction between the knots, dependent on the absolute linear distance between them as well as the length of the knots.
- As the knots get closer to one another, but can still be considered as two simple separate knots, a repulsive interaction starts to dominate the free energy, stemming primarily from the steric hindrance of the proximal loops of the knots.
- Finally, as the knots become intertwined at yet smaller effective separations, the absolute magnitude of the free energy—And consequently, the stability of the knotted chain configuration—Is dependent both on the bending stiffness of the polymer chain as well as on the relative chirality of the two knots.

#### 4.1. Elastic Energy Model for the Size of Two Separate, Non-Interacting Knots

#### 4.2. Elastic Energy Model for Chirality Effects in Knot–Knot Interaction

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) An example of a configuration for the ($++$) system, with both knots intertwined. In this configuration, only one prime component(marked in yellow) is identified as “isolated” by our knot identification scheme based on the Minimally Interfering Closure [41]; (

**b**) A small portion of an MD trajectory for the ($++$) system. Yellow shaded regions indicate the portions of the chain occupied by the isolated prime knots, and red and blue lines indicate their first and last bead, respectively. The portion of the chain taken up by the whole composite knot is reported as a gray shaded area. When the knots are intertwined, our algorithm identifies only one isolated knot, which can be seen in the trajectory as a single yellow region surrounded by gray boundaries. Note that the intertwined composite knot can still travel from one side of the chain to the other.

**Figure 2.**(

**a**) Size of separated trefoil knots, plotted as a function of ${\kappa}_{b}$; (

**b**) Size of a nested trefoil knot (circles) as well as of the composite knot (squares) when the two trefoils are intertwined, plotted as a function of ${\kappa}_{b}$.

**Figure 3.**Schematics of the collective order parameter $\left|D\right|$ measuring the linear distance between two prime knots. (

**a**) When both prime components are isolated , the order parameter is given by $\left|D\right|=|{d}_{1,2}|=|{c}_{1}-{c}_{2}|$, where ${c}_{i}=({e}_{i}+{s}_{i})/2$ is the center of knot i on the chain. Here ${e}_{i}$ and ${s}_{i}$ stand for the last and the first bead of the i-th isolated prime knot; (

**b**) When the two knots are intertwined, the center of the hosting knot is taken to coincide with the center of the composite knot ${c}_{1,2}$.

**Figure 4.**Free energy F as a function of the collective order parameter $\left|D\right|$, i.e., the absolute linear distance between the centers of the two knots. $F\left(\right|D\left|\right)$ for different values of the chain bending rigidity ${\kappa}_{b}$ is reported for (

**a**) two trefoil knots with the same relative chirality and (

**b**) two trefoil knots with opposite relative chiralities. In (

**c**,

**d**) we show the same free energies, but with subtracted entropic contribution $-TS\left(\right|D\left|\right)$, as defined in Equation (8).

**Figure 5.**(

**a**) Graphical representation of three quantities discussed in the text: the height of the barrier, ${\u03f5}_{b}$, the depth of the minimum, ${\u03f5}_{d}$, and the interaction distance, ${D}_{\mathrm{int}}$; (

**b**) Dependence of the height of the barrier and the depth of the free energy minimum on the polymer bending rigidity, ${\kappa}_{b}$.

**Figure 6.**(

**a**) Interaction distances of separated knots as a function of ${\kappa}_{b}$. For comparison, we also report the knot size averaged between the ($++$) and ($+-$) systems, $\langle {l}_{k}\rangle $. Note that the interaction distance is always larger than $\langle {l}_{k}\rangle $; (

**b**) Two different rescaling of ${D}_{\mathrm{int}}$: ${D}_{\mathrm{int}}/\langle {l}_{k}\rangle $ and $({D}_{\mathrm{int}}-\langle {l}_{k}\rangle )/{L}_{p}$, with ${L}_{p}$ being the persistence length of the chain.

**Figure 7.**Left-handed ${3}_{1}$ knot configuration obtained by minimizing the energy viewed from the side (

**a**) and from the top (

**b**); (

**c**) Right-handed ${3}_{1}$ knot viewed from above. X indicates the pulling direction. In (

**a**), the direction of the arc length s is indicated by an arrow, and in all panels by the coloring of the knots, from green to blue for increasing s. In panels (

**b**,

**c**), we reported the direction of the vector ${\mathbf{U}}_{k}$ defined in Equation (9). Note that ${\mathbf{U}}_{k}\xb7x$ changes sign with the handedness of the knot; (

**d**) A ($+-$) intertwined composite knot, with the nested knot highlighted by a red shading. Note that when the loop of the nested knot lies inside the loop of the outer knot, both orientation directors point out of the page. On the other hand, when the loop of the two knots form an eight, the two directors point in opposite directions.

**Figure 8.**Free energy of two separated trefoil knots projected along (

**a**) θ and (

**b**) ${\theta}_{\perp}$ for four different values of polymer bending rigidity.

**Figure 9.**Free energy of two intertwined trefoil knots with the same handedness (red curves) and opposite handedness (blue curves) as a function of θ (

**a**,

**c**,

**e**,

**g**) and ${\theta}^{\perp}$ (

**b**,

**d**,

**f**,

**h**), for four different values of polymer bending rigidity: ${\kappa}_{b}=5$, ${\kappa}_{b}=10$, ${\kappa}_{b}=15$, and ${\kappa}_{b}=20{k}_{B}T$. The free energies have been shifted so that their maximum values correspond to zero.

**Figure 10.**(

**a**) A trefoil knot can be decomposed into two loops and a braid. The two enantiomers are defined by the braid. The circles indicate the orthogonal components of the bending force vector. Following the usual conventions, crossed circles indicate chain directors going into the page, while dotted circles indicate chain directors coming out of the page; (

**b**,

**c**): Composing the intertwined knots of equal and/or different chiralities. In an intertwined ($+-$) knot (

**b**), the director joins the junction between the braid and the loop in the same sense, while in an intertwined ($++$) knot (

**c**), it points in opposite sense at the two junctions; (

**b**,

**c**) bottom: Schematic representation of the axial projection (in the direction of the braid axes, assumed to coincide) for the ($+-$) and ($++$) composite knots, respectively.

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**MDPI and ACS Style**

Najafi, S.; Podgornik, R.; Potestio, R.; Tubiana, L.
Role of Bending Energy and Knot Chirality in Knot Distribution and Their Effective Interaction along Stretched Semiflexible Polymers. *Polymers* **2016**, *8*, 347.
https://doi.org/10.3390/polym8100347

**AMA Style**

Najafi S, Podgornik R, Potestio R, Tubiana L.
Role of Bending Energy and Knot Chirality in Knot Distribution and Their Effective Interaction along Stretched Semiflexible Polymers. *Polymers*. 2016; 8(10):347.
https://doi.org/10.3390/polym8100347

**Chicago/Turabian Style**

Najafi, Saeed, Rudolf Podgornik, Raffaello Potestio, and Luca Tubiana.
2016. "Role of Bending Energy and Knot Chirality in Knot Distribution and Their Effective Interaction along Stretched Semiflexible Polymers" *Polymers* 8, no. 10: 347.
https://doi.org/10.3390/polym8100347