Conformation of a semiflexible filament in a quenched random potential

Motivated by the observation of the storage of excess elastic free energy, prestress, in cross-linked semiflexible networks, we consider the problem of the conformational statistics of a single semiflexible polymer in a quenched random potential. The random potential, which represents the effect of cross-linking to other filaments, is assumed to have a finite correlation length $\xi$ and mean strength $V_0$. We examine statistical distribution of curvature in filament with thermal persistence length ${\ell }_P$ and length $L_0$ in the limit in which ${\ell }_P\gg L_0$. We compare our theoretical predictions to finite-element Brownian dynamics simulations. Finally, we comment on the validity of replica field techniques in addressing these questions.

Figure 1

Three tensed filaments interacting with the pinning potential, shown as a heat map with brighter colors representing higher potential energies. The lowest filament traverses a saddle between local potential maxima. On the right of that saddle point it curves into a deep potential minimum (dark). Similar features may be seen in the other filaments. This is a snapshot from our Brownian dynamics simulations, discussed in Sec. 5.

Figure 2

Examples of the random potentials $V(x,y)$ (shown as a heat map with contour lines) selected from different distributions: (a) energy-controlled distribution, (b) force-controlled distribution, and (c) exponential suppression of high modes. The correlation length is fixed in all three so that $L_x/\xi =20$.

Figure 3

Detail views of the force fields resulting due to the random pinning potentials shown in Fig. 2: (a) force-controlled distribution and (b) exponential suppression of high modes.

Figure 4

Excess arc length $\mathrm{\Delta }L$ [see Eq. (19)] of a semiflexible filament in the quenched pinning potential (with persistence length ${\ell }_P\approx 14\mu \text{m}$) as a function of tension $\tau$. At high tension (orange dashed line) the filament cannot track the bottom of potential valleys, while at low tension (green solid line) or small bending modulus the filament does track the potential valleys with higher fidelity. The blue dots with error bars represent simulation results and the errors show the standard deviations of about 500 filaments. The pinning potential is defined by $V_0=0.175\text{pN}$ and $\xi \approx 1.6\mu \text{m}$.

Figure 5

(a) Plot of $E^{\mathrm{bend}}\left(q\right)$, the energy per mode of a tensed pinned filament (in units of $T$), as a function of $q=\xi z_n$. The pinning potential strength is set by dimensionless $\tilde{E}=V_0^2\xi /T\tau$ using the exponential potential distribution. The small-$q$ modes typically have more bending energy than expected for a thermalized filament without the pinning. The high-$q$ modes are effectively unpinned. (b) Effect of changing tension on bending energy $\tau =0.1,1,5{\tau }_0$, where ${\tau }_0{V}_0^2\xi /T$. We set $\kappa ={\xi }^2{\tau }_0$.

Figure 6

For a given value of the pinning strength and tension, there is a transition at $q^{★}$ between pinned modes $q<q^{★}$, which trap a significant excess energy as compared to the free filament and free modes $q<q^{★}$, which do not. We examine this transition by plotting the ratio of the excess bending energy resulting from the pinning potential $E^{\mathrm{pin}}$ to the energy of that mode without the pinning potential $E^{\mathrm{T}}$. The pinning potential and the tension are $T/2\xi$ and $\sqrt{2\kappa V_0}/\xi$, respectively. The figure is qualitatively the same for other values of these parameters.

Figure 7

The maximum value for the energy stored in the potential is reached at $\xi =1/z_n$, which resembles the resonance absorption spectrum. Here $\xi$ is measured in units of $1/z_n$, $V_0=20(\tau +\kappa )$ in these units.

Figure 8

(a) Simulation snapshot of the initial setup. An initially straight, stress-free filament is constrained to the $xy$ plane and simply supported at its end points. It interacts with a random potential $V(x,y)$ that is shown as a heat map with contour lines. (b) Simulation snapshot of a deformed configuration showing the forces on the filament resulting from the pinning potential.

Figure 9

Weak tension persistence length.

Figure 10

Strong tension correlation function.

Figure 11

Prestress.