Equilibrium fluctuations of a semiflexible filament cross linked into a network

We examine the equilibrium fluctuation spectrum of a semiflexible filament segment in a network. The effect of this cross linking is to modify the mechanical boundary conditions at the end of the filament. We consider the effect of both tensile stress in the network and its elastic compliance. Most significantly, the network's compliance introduces a nonlinear term into the filament Hamiltonian even in the small-bending approximation. We analyze the effect of this nonlinearity upon the filament's fluctuation profile. We also find that there are three principal fluctuation regimes dominated by one of the following: (i) network tension, (ii) filament bending stiffness, or (iii) network compliance. This work provides the theoretical framework necessary to analyze activity microrheology, which uses the observed filament fluctuations as a noninvasive probe of tension in the network.

Figure 1

A schematic diagram of a particular filament (blue) cross linked into a network of similar filaments (red). The cross links are represented by rings.

Figure 2

Above: Schematic diagram of a single semiflexible filament with the left endpoint pinned. Both endpoints are subject to torque-free boundary conditions. The right endpoint is attached to a combination of a longitudinal spring of rest length $x_0$ with spring constant $k_{\parallel }$, and a transverse spring of zero rest length with spring constant $k_{\perp }$. These represent the elastic compliance of the network. Below: A schematic diagram showing the connection of the longitudinal spring and the various lengths defined in the text. The spring's anchor point $x_0$ is positive when that spring applies tension to the perfectly stretched filament, $\ell ={\ell }_{\infty }$. The change in projected length $\mathrm{\Delta }\ell$ is defined oppositely, becoming positive as the filament contracts.

Figure 3

The free propagator and nonlocal vertex. Slashes denote multiplication by momentum squared. The four field interaction depends on two independent momenta.

Figure 4

The first three diagrams in the perturbative expansion of Green's function. No loops are possible with dressed propagators in the connected diagrams, allowing the series to be resummed.

Figure 5

Parameter space spanned by the ratios of the tension length and the longitudinal spring length to the filament's length. The three regions are defined by the type of boundary term that dominates the fluctuation profile: tension, network compliance (spring), or filament bending. The three regions coincide at the point $[{(2/z)}^{1/3}{\pi }^{-4/3},\pi ]$. The persistence length is set to be ${\ell }_p=\ell$ so that $z=1$.

Figure 6

Log-log plot of the two-point function $G\left(p\right)$ with respect to wave number for parameters $\ell =1\mu m,\kappa =0.0413\left(pN\mu m\right),\tau =4.133\left(pN\right),$ and $k_{\parallel }=5(pN/\mu m)$. There are two scaling regimes. The dashed lines illustrate their slopes in the two regimes. At low wave number, the fluctuations are dominated by a combination of tension and network compliance, while at high wave number, they are controlled by the filament's bending stiffness. Network compliance shifts the transition $p_{*}\approx {\left({\ell }_k^3\ell \right)}^{-1/4}$ to the higher wave numbers, provided we are in the spring-dominated regime ${\ell }_t\gtrsim {\left({\ell }_k^3\ell \right)}^{1/4}$.

Figure 7

Plots of the exact root mean-square height-height fluctuations for parameters $\ell =1\mu \text{m},{\ell }_p=4/3\ell ,{\ell }_t=0.1\ell$. The average tensions of $0.1,1,$ and $10(\text{pN}/\mu )$ correspond to lengths ${\ell }_k=0.1123,.0498,$ and $.01937\left(\mu \text{m}\right)$, respectively. The mean tension $\langle \tau \rangle$ is calculated using Eq. (37).

Figure 8

The MFT root-mean-square fluctuations for various values of the transverse spring constant $k_{\perp }$. The inset provides a log plot of the root-mean-square fluctuations at the spring (right-hand side), as a function of $k_{\perp }$. For large large $k_{\perp }$, the endpoint fluctuation scales as $\sqrt{u^2\left(\ell \right)}\sim k_{\perp }^{-1}$, as expected for an ideal spring. Parameter values are $\kappa =0.0413\left(\text{pN}\mu \text{m}\right),\tau =4.133\left(\text{pN}\right),$ and $k_{\parallel }=5(\text{pN}/\mu \text{m})$.