# Semiflexible Chains at Surfaces: Worm-Like Chains and beyond

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## Abstract

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## 1. Introduction

## 2. The WLC Model: Some Results in 3D and 2D

#### 2.1. Single Chain Properties

#### 2.2. Melt of Worm-Like Chains

#### 2.2.1. Simulations of WLC Melts and Lattice Artifacts

#### Generalities

#### Caveats

#### 2.2.2. Two-Dimensional Polymer Melts: Isotropic-Nematic Phase Transition

#### Flexible Chains

#### Weak Persistence Length Effects

#### Strong Persistence Length Effects

#### 2.2.3. Three-Dimensional Polymer Melts: Corrections to Chain Ideality

## 3. Adsorption of WLC

#### 3.1. Loop and Tail Partition Function

#### 3.2. Reversible Adsorption of an Ideal WLC from Dilute Solution

#### 3.3. Irreversible Chemisorption of a WLC from Dilute Solution

## 4. Filaments in 2D beyond WLC

#### 4.1. The Helical Filament Squeezed in 2D

#### 4.1.1. A Bit of Mechanics

**Ω**are [123]:

**σ**is proportional to the strain, i.e., $\mathbf{\sigma}\left(s\right)=\delta E/\delta \mathbf{\Omega}\left(s\right)\propto $ $\mathbf{\Omega}\left(s\right)$, the expansion stops at the quadratic order in

**Ω**. We limit ourselves to this regime for the moment. See the Section 4.2.2 on filament crunching for nonlinear contributions.

#### 4.1.2. Thermally-Injected Twist-Kinks and Hyper Flexibility: The Arc/Arc Toy Model

#### 4.1.3. The Free Energy Map

#### 4.2. Emergence of Bistability

#### 4.2.1. Long Range Elasticity and Switchability

#### 4.2.2. Bistability and Cooperativity

#### 4.3. Polymorphic Model of Microtubules

## 5. Miscellaneous Topics and Outlook

- We restricted our study to adsorption from dilute solution and mentioned that nematic order is expected in very dense solutions or melts. Even if the nematic order is not thermodynamically stable in the bulk, there may be a thin layer of thickness ℓ closest to the wall where orientational order prevails as described in work by Milner [144]. Strong adsorption of WLC in the melt has been simulated recently [145] with a focus on the distribution of loops and tails, layering effects closest to the wall and local mobility.
- We only considered adsorption on undeformable planar surfaces. Biofilaments can deform soft surfaces like membranes to optimize their surface binding [146]; for example, when the orientation for adsorption is not compatible with the in-plane orientation of the preferred curvature. WLCs have to adapt to the curvature of undeformable shells to adsorb, and this can lead to special arrangements as found in [147]. Surfaces bearing fixed obstacles alter the 2D dynamics of a WLC [148,149].
- Another way to fix polymers on a surface is end-grafting, earlier mentioned in the latest stage of irreversible adsorption. One recent simulation work compares single end graft and middle graft WLC in great detail [153], which is related to the distributions of loops and tails discussed above (Section 3.1). The role of stiffness in densely-grafted brushes was considered early by Pickett and Witten [154]. More recently, simulations in the Binder group discuss grafted WLC in the brush regime and the crossover to the dilute surface regime (so-called mushrooms) [155].
- Anisotropic filaments (tapes) with two flexural moduli [156,157] do occur; examples are synthetic ladder polymers or short (rigid) associated polypeptides. Helical polypeptide tapes [158] can reassemble in a hierarchy of structures. Similar associations play a role in amyloid diseases [158]. Mesini and coworkers studied a family of organogelators and report various self-assembled structures, including microtubes [159]. Recently, the adsorption of helical tapes on rigid surfaces was considered by Quint [160].
- It is sometimes argued that the reptation tube can be represented by some transverse (in a first approximation harmonic) potential [161]. The motion of a WLC confined to an effective tube by a transverse harmonic potential was considered by several authors [21,162]. The loosely-related topic of polymers in random media was considered in [163].
- Some larger scale man-made chains can be considered as semiflexible. The mechanics of semiflexible chains formed by poly(ethylene glycol)-linked paramagnetic particles was studied by Gast and coworkers [164] as a function of the length of the poly(ethylene glycol) spacer.
- Concerning the helical filaments squeezed in 2D discussed in Section 4.1, many open questions remain in the specific biological contexts. Open questions pertain also to the nonequilibrium behavior of squeelices: e.g., when they are transported along molecular motor-covered substrates, where they display circular, spiraling or wavy trajectories [165] that are also observed experimentally [112,166].
- The investigation of cooperativity and switchability discussed in Section 4.2 and Section 4.3 has only just began. For instance, there are indications for cooperative binding of molecular motors on microtubules [167], for which cooperative conformational changes in the tubulins may be responsible [131]. The polymorphic switching model discussed in Section 4.3 has so far only been employed to taxol-stabilized microtubules, for which there is the most experimental evidence (as taxol is the main microtubule stabilizer preventing further polymerization/depolymerization). In fact, many other molecules, especially microtubule-associated proteins, may also induce conformational changes of tubulin upon binding with different associated energy gains $\Delta G$. This subject deserves further study as a new route for microtubule regulation that goes beyond the regulation of spatial distribution and length (as typically discussed in biology) towards a regulation of their mechanical behavior and response.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Stiffness effects and lattice artifacts for bond-fluctuation model (BFM) data obtained for $M=200$ chains of length $N=20$ at a volume fraction $\varphi =8NM/V=0.5$ of occupied lattice sites at a temperature $T=1/\beta =1$ [41]. Panel (

**a**) shows the effective bond length ${b}_{\text{e}}={R}_{\text{e}}\left(N\right)/{(N-1)}^{1/2}$, obtained from the root-mean-squared chain end-to-end distance ${R}_{\text{e}}\left(N\right)$ and the (rescaled) center-of-mass self-diffusion coefficient ${D}_{\mathrm{cm}}$ as a function of the dimensionless parameter $x=\beta \u03f5$. A snapshot of a configuration at $\u03f5=10$ is given in Panel (

**b**). The chains are seen to align along the three principal lattice axes.

**Figure 2.**Snapshots of semiflexible 2D polymers of length $N=256$ obtained by means of molecular dynamics simulation of a Kremer–Grest bead-spring model [56,79]. We show data for four concentrations $c=NM/{L}^{2}=0.125$ ($M=192$ chains, linear box size $L\approx 627$), $c=0.250$ ($M=192$, $L\approx 443$), $c=0.500$ ($M=192$, $L\approx 313$) and $c=0.750$ ($M=384$, $L\approx 362$) and five bending penalties $\u03f5=0$, 2, 4, 8 and 16 (from the bottom to the top). Only small subvolumes of much larger boxes are represented. The configurations have been sampled by increasing ϵ starting with flexible and compact chain systems ($\u03f5=0$) [68,76,80]. While the chains remain compact and segregated at low densities and stiffnesses below the dashed line, they are seen in the opposite limit to align (at least) locally, forming bundles of chains with hairpins, which are extremely difficult to equilibrate.

**Figure 3.**Rescaled sub-chain size $y={R}_{\text{e}}^{2}\left(s\right)/(s-1)$ with s being the arc-length (sub-chain length) for four concentrations and three stiffness penalties corresponding to systems below and around the dashed line in Figure 2. The symbols refer to flexible systems ($\u03f5=0$); the line width for $\u03f5=2$ and 4 increases with density. The dash-dotted line corresponds to the asymptotic slope for perfectly rigid chains. Note that $y\left(s\right)$ becomes strongly non-monotonous with increasing ϵ. However, for a given density and $s\to N$, all $y\left(s\right)$ become similar as long as the system remains isotropic, i.e., ϵ is not too large. Independent of the rigidity, the overall chain size is thus ruled by the (persistence length independent) distance ${d}_{\text{cm}}\approx {(N/c)}^{1/D}$ between chains.

**Figure 4.**Log-log plot of the tangent/tangent correlation function $C\left(s\right)=\langle \mathrm{cos}\theta \left(s\right)\rangle $ versus arc-length s for a 3D polymer melt with chains of length $N=1024$. The symbols show data from MD simulations for a Kremer–Grest-like bead-spring model [10] with three bending penalties: $\u03f5=0,1,2$. The model is similar to the one shown in Figure 2. Thus, the melt with $\u03f5\le 2$ is isotropic, the value $\u03f5=0$ corresponding to fully-flexible chains. The abscissa is scaled by the persistence length ℓ obtained from a fit of the initial decay of $C\left(s\right)$ to Equation (1); see the dashed line in the figure. The solid lines indicate the power law, $C\left(s\right)\sim {s}^{-3/2}$, expected from corrections to chain ideality [10,11].

**Figure 5.**Loop distribution as simulated by molecular dynamics for two chain lengths ($S=30$ b and $S=250$ b). The persistence length is $\ell =10.1$ b throughout. (left) Distribution of the internal loop size upon first adsorption of a loop of size S; only the smallest of the generated internal loops is taken into account; the full distribution is symmetric about $S/2$. We show the power law fit by the single loop partition function (dashed lines) for the stiff loop and for the flexible loop, which apply where they should. Note the rather narrow crossover around $s=2\ell $. The product of loop flexible partition functions nicely accounts for the flattening near $s=S/2$ required by symmetry. (right) Distribution of the size of the loop generated by first re-adsorption of a tail of length $S/2$. Again, the single loop partition functions fit where they should; the narrow crossover is now located around $s=\ell $. The product of the flexible loop and tail partition functions accounts for the upturn near $s=125$ b where the small tail partition function dominates.

**Figure 6.**(

**a**) Schematic squeelix with the angle ϕ that is slaved to the twist angle ψ given by the line in black; (

**b**) typical shape of a squeelix for $\gamma =1$ (see Equation (19)) with a single twist-kink; (

**c**) a squeelix in the dilute regime of twist-kinks with $\gamma =0.997$; (

**d**) a squeelix in the dense regime of twist-kinks with $\gamma =0.52$. The ground states (b), (c) and (d) are from [121].

**Figure 7.**$P\left(\theta \right)$ for $S=100b$, $\omega =0.01/b$ and $\ell =1000b$ obtained by convolution according to Equation (23) using Equation (22). The Boltzmann weight of a twist kink is ${e}^{-E}$. The lines correspond to $E=$ 4 (green), 6 (blue) and 8 (black). The thick dashed line corresponds to $E=-log\left(b\omega \right)\approx 4.6$.

**Figure 8.**${R}_{g}^{2}\left(S\right)$ for $\omega =0.01/b$ and ${l}_{p}=1000b$. The Boltzmann weight of a twist kink is ${e}^{-E}$. The thin lines are for $E=2,4,6,8$ and 10 from top to bottom. The thick line indicate ${R}_{g}^{2}$ with $E=-log\left(b\omega \right)\approx 4.6$. Undulations appear for $E>-log\left(b\omega \right)$.

**Figure 9.**(

**a**) Free energy landscapes without applied force ($f=0$) and (

**b, c**) under increasing force ($f=0.25{k}_{B}T/b$, $f=0.50{k}_{B}T/b$). To fix the force scale: for $b=0.5$ nm, given ${k}_{B}T\approx 4$ pNnm, ${k}_{B}T/b\approx 8$ pN.

**Figure 10.**(

**a**) Two filaments are coupled with elastic springs forming a simple two-chain bundle; (

**b**) the shear and bending degrees of freedom become strongly coupled, leading to long-range deformation effects. An arc formed in the region of length l (blue line) around the origin induces two counter-arcs with opposite curvature in its next proximity. The deformations are screened and vanish only at length scales longer than the elastic screening length λ.

**Figure 11.**A semiflexible filament with elastic tails that cross-link points along its backbone becomes bistable. If the cross-linking point intervals overlap in addition (red and black chains), the curvature switching becomes cooperative. The single arc state in the middle is the energetically most stable (indicated by bold and dashed arrows).

**Figure 12.**Polymorphic crunching: (

**a**) Nonlinear bendable units are coupled in-plane of bending by a ring closure constraint (

**b**). The constraint modifies their effective free energy and gives rise to a bistable monomer potential. (

**c**) Two out of exponentially many energetically-equivalent ground state realizations. Blue and light-blue indicate the regions of opposite curvature.

**Figure 13.**The polymorphic model of microtubules. (

**a**) The switchability of tubulin dimers leads to a competition of three states of the microtubule’s cross-section: straight and long (L), curved (C) and straight and short (S). (

**b**) “Phase diagram” of the polymorphic microtubule model as a function of generalized force f vs. torque m. The existing states in the respective regions are ordered by their polymorphic energies, the one at the bottom having minimum energy. (

**c**) An intermittent buckling event, for a microtubule transported along a motor-covered surface, in case of $f\simeq 0.7$ (corresponding to small switching energies $\Delta G\simeq 0$). The behavior observed is the one of a regular WLC chain. (

**d**) An intermittent buckling event in case of $f\simeq 0.4$ (corresponding $\Delta G\simeq 5\phantom{\rule{0.166667em}{0ex}}{\mathrm{k}}_{\mathrm{B}}\mathrm{T})$. The microtubule curls up.

**Table 1.**Monomer size b along the backbone and persistence length ℓ for various polymers (in water if not specified otherwise) assuming worm-like-chain (WLC) statistics: polyethylene (PE) in dodecanol1, polyisobutylene (PIB) in benzene, polydimethylsiloxane (PDMS) in hexane, atactic polystyrene (PS) in cyclohexane, poly(sodium styrene sulfonate) (NaPSS), poly(diallyl-dimethyl ammonium chloride) (PDADMAC), hyaluronan (HA), duplex-DNA (d-DNA), intermediate filament vimentin (IF) and F-actin. The polymers marked by ⋆ are polyelectrolytes measured in water at high salt; for HA, two sets of values for ℓ are found in the literature possibly linked to association phenomena. The last three polymers are biofilaments measured in physiological conditions. It is questionable whether the simple WLC model is directly applicable to them (see the last section), and the persistence length may be only indicative.

ℓ (nm) | b (nm) | $\ell /b$ | |
---|---|---|---|

PE (Dodecanol1) | 0.59 | 0.13 | 4.5 |

PIB (Benzene) | 0.59 | 0.26 | 2.3 |

PS (Cyclohexane) | 0.86 | 0.26 | 3.3 |

PDMS (Hexane) | 0.57 | 0.29 | 2 |

${}^{\star}$NaPSS (high salt) | 1.0 | 0.25 | 4 |

${}^{\star}$PDADMAC (high salt) | 3.0 | 0.47 | 5.3 |

${}^{\star}$HA (high salt) | 4–5; 7–10 | 1.0 | 4–5; 7–10 |

d-DNA | 50 | 0.34 | 150 |

IF | 10${}^{3}$ | ∼10 | ∼100 |

F-actin | 17 × 10${}^{3}$ | 5 | 3400 |

**Table 2.**Characteristic length scales for chemisorption of a single worm-like chain. The typical zipping loop size is ${s}_{0}$. For long chains $S>{S}^{\star}$, zipping occurs from multiple nucleation points distant along the chain.

ℓ (nm) | b (nm) | ${s}_{o}$ (nm) | ${S}^{\star}$ (nm) | |
---|---|---|---|---|

PE (dodecanol1) | 0.59 | 0.13 | 0.21 | 1.6 |

PIB (benzene) | 0.59 | 0.26 | 0.34 | 1 |

PS (cyclohexane) | 0.86 | 0.26 | 0.38 | 1.9 |

PDMS (hexane) | 0.57 | 0.29 | 0.36 | 0.90 |

NAPSS (high salt) | 1.0 | 0.25 | 0.39 | 2.5 |

PDADMAC (high salt) | 3.0 | 0.47 | 0.80 | 7.6 |

HA (high salt) | 4–5; 7–10 | 1.0 | 1.6–2 | 12–35 |

d-DNA | 50 | 0.34 | ∼1.8 | ∼ 1.4 × 10${}^{3}$ |

IF | 10${}^{3}$ | ∼ 10 | ∼46 | ∼ 2 × 10${}^{4}$ |

F-actin | 17 × 10${}^{3}$ | 5 | ∼75 | 3 × 10${}^{6}$ |

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Baschnagel, J.; Meyer, H.; Wittmer, J.; Kulić, I.; Mohrbach, H.; Ziebert, F.; Nam, G.-M.; Lee, N.-K.; Johner, A.
Semiflexible Chains at Surfaces: Worm-Like Chains and beyond. *Polymers* **2016**, *8*, 286.
https://doi.org/10.3390/polym8080286

**AMA Style**

Baschnagel J, Meyer H, Wittmer J, Kulić I, Mohrbach H, Ziebert F, Nam G-M, Lee N-K, Johner A.
Semiflexible Chains at Surfaces: Worm-Like Chains and beyond. *Polymers*. 2016; 8(8):286.
https://doi.org/10.3390/polym8080286

**Chicago/Turabian Style**

Baschnagel, Jörg, Hendrik Meyer, Joachim Wittmer, Igor Kulić, Hervé Mohrbach, Falko Ziebert, Gi-Moon Nam, Nam-Kyung Lee, and Albert Johner.
2016. "Semiflexible Chains at Surfaces: Worm-Like Chains and beyond" *Polymers* 8, no. 8: 286.
https://doi.org/10.3390/polym8080286