# A Simple Analytical Model for Predicting the Collapsed State of Self-Attractive Semiflexible Polymers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Analytical Model

_{1}, Equation (5c) reduces to:

#### 2.2. Simulation Details

_{B}) and the three fundamental units, namely mass (m), distance (σ), and energy (ε), were set to be unity. A Langevin thermostat [20] was used in all simulations, with the damping coefficient set to 10, which leads to an overdamped simulation, and reduced temperature (T*) to be unity (k

_{B}T* = 1). The time scale τ was defined to be $\sqrt{m{\mathsf{\sigma}}^{2}/\mathsf{\epsilon}}$ , and a typical time step of 0.005τ was used. Simulations were carried out for at least ${10}^{8}$ steps. We tracked the radius of gyration (R

_{g}) to monitor the collapse behavior of the chain. The polymer chain was modeled using beads and stiff springs, with each bead representing a generic monomer in the polymer chain. The CG interaction potential (Equation (1a–c), Equation (8a–c)) included both bonded and non-bonded interactions. Harmonic bond and interactions were used to model the bonded interactions, while the Lennard-Jones 12-6 potential was used to model the non-bonded interaction.

## 3. Results and Discussion

_{sim}through V

_{sim}). Specifically, for ${\mathsf{\gamma}}_{\mathrm{e}}/{\mathsf{\gamma}}_{\mathrm{s}}$ values less than 1.3, we observe a globule, and for ${\mathsf{\gamma}}_{\mathrm{e}}/{\mathsf{\gamma}}_{\mathrm{s}}$ between 1.3 and 2.7, we observe bundles with various aspect ratios. For ${\mathsf{\gamma}}_{\mathrm{e}}/{\mathsf{\gamma}}_{\mathrm{s}}$ greater than 6.5, tori with various aspect ratios form. In between 2.7 and 6.5, we observe fluctuations between bundle and torus. As expected, simulations show random coils form at large ${\mathsf{\gamma}}_{\mathrm{e}}/{\mathsf{\gamma}}_{\mathrm{s}}$, here found to be a value greater than 13.

## 4. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Illustrative phase diagram of the parameter space that has been covered in previous simulations and theoretical models in the literature, and in this work. The x-axis is the Kuhn length (${N}_{\mathrm{k}}$) of the polymer chain, and the y-axis is the dimensionless temperature, defined as the k

_{B}T divided by the attractive energy per monomer bead used in each study.

**Figure 2.**Schematics of torus (

**a**) and bundle (

**b**); (

**c**) Schematic of a generic end fold we consider in this work, which has three exposed monomers (colored in gray). The angles in the end fold, ${\vartheta}_{1-3}$, are formed among three consecutive monomer beads. For simplicity, these angles are taken to be 120$\xb0$.

**Figure 3.**Phase diagrams for polymer chain with dimensionless chain length L* = L/σ = 600. Each symbol on the phase diagram marks the parameter values at which a simulation was conducted. Specifically, open circles, crosses, and asterisk symbols are the simulation runs using stiff, harmonic, and cosine bending potentials respectively. A vertical bar connects simulations data producing the same type of structure. The color of the vertical bars, of the dashed lines, and of the regions of the diagram producing this structure coded as follows: blue—random coil (RC), pink—fluctuating between random coil and torus (RC&T), red—torus (T), green—fluctuating between torus and bundle (T&B), yellow—bundle (B), and black- globule (G). Note the pink and blue regions only occur in simulations because we do not model random coil phase. We regard the simulated T&RC phase simply as a torus phase (T) because we do not model the random coil (RC) phase analytically. The dashed lines connecting the vertical bars mark the upper boundaries of the phases obtained from the simulations (e.g., simulation results between yellow and grey dashed lines resulted in a bundle as the final collapsed structure). Exact solutions for the boundaries between the three collapsed phases are shown on the phase diagram (black dash-dotted lines). We have selected representative conformations in each phase, pointed by the arrows. We have calculated the theoretical aspect ratios for these conformations, depicted in the insets I–V. We have also conducted simulations using harmonic bending potential under the same conditions, and the final snapshots from the simulations are supplied in additional insets I–V with subscript “sim”.

**Figure 4.**Phase diagrams for polymer chains with dimensionless chain lengths L* of 80. We overlay the simulation results from this study (solid lines) onto all three phase diagrams and the upper boundaries of the phases obtained from the simulations (colored dashed lines) onto the phase diagrams. The color coding and the definitions for the solid and dashed lines are the same as in Figure 3. We also overlay the simulation results from Kong et al. [10] onto the phase diagram (dotted line).

**Table 1.**Comparison of conformations and their corresponding aspect ratios between simulation using harmonic bending potential and theory for chain with L* = 600. Each simulation is repeated three times and the error bar is taken as the standard deviation among the three aspect ratios. The aspect ratios l*/r* and R*/r* are only shown for the bundle and torus conformations.

${K}_{\mathsf{\theta}}$ | ε | ${\mathsf{\gamma}}_{\mathrm{b}}/{\mathsf{\gamma}}_{\mathrm{e}}{\mathsf{\sigma}}^{3}$ | ${\mathsf{\gamma}}_{\mathrm{e}}/{\mathsf{\gamma}}_{\mathrm{s}}$ | Config. (Sim.) | Asp. ratio (Sim.) | Config. (Theory) | Asp. ratio (Theory) |
---|---|---|---|---|---|---|---|

1.5 | 1 | 0.65 | 0.91 | G | G | ||

3.0 | 1 | 0.65 | 1.30 | G | G | ||

7.5 | 1 | 0.65 | 3.23 | B | 4.3$\text{}\pm \text{}$1.4 | B | 6.0 |

10.0 | 1 | 0.65 | 2.77 | T&B | T&B | ||

13.0 | 1 | 0.65 | 4.31 | T&B | T&B | ||

15.0 | 1 | 0.65 | 6.46 | T | 4.4$\text{}\pm \text{}$1.1 | T | 2.6 |

30.0 | 1 | 0.65 | 12.92 | T | 9.2$\text{}\pm \text{}$1.2 | T | 4.2 |

150.0 | 1 | 0.65 | 44.60 | RC | T | 10.0 |

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**MDPI and ACS Style**

Huang, W.; Huang, M.; Lei, Q.; Larson, R.G.
A Simple Analytical Model for Predicting the Collapsed State of Self-Attractive Semiflexible Polymers. *Polymers* **2016**, *8*, 264.
https://doi.org/10.3390/polym8070264

**AMA Style**

Huang W, Huang M, Lei Q, Larson RG.
A Simple Analytical Model for Predicting the Collapsed State of Self-Attractive Semiflexible Polymers. *Polymers*. 2016; 8(7):264.
https://doi.org/10.3390/polym8070264

**Chicago/Turabian Style**

Huang, Wenjun, Ming Huang, Qi Lei, and Ronald G. Larson.
2016. "A Simple Analytical Model for Predicting the Collapsed State of Self-Attractive Semiflexible Polymers" *Polymers* 8, no. 7: 264.
https://doi.org/10.3390/polym8070264