Tension dynamics in semiflexible polymers. I. Coarse-grained equations of motion

Based on the wormlike chain model, a coarse-grained description of the nonlinear dynamics of a weakly bending semiflexible polymer is developed. By means of a multiple-scale perturbation analysis, a length-scale separation inherent to the weakly bending limit is exploited to reveal the deterministic nature of the spatio temporal relaxation of the backbone tension and to deduce the corresponding coarse-grained equation of motion. From this partial integro-differential equation, some detailed analytical predictions for the nonlinear response of a weakly bending polymer are derived in an accompanying paper [O. Hallatschek et al., following paper, Phys. Rev. E 75, 031906 (2007)].

Figure 1

(Color online) Two basic examples of dynamic force-extension experiments. Pulling: a weakly bending polymer in equilibrium is suddenly pulled longitudinally by two external forces $\mathfrak{f}$. Release: a prestretched polymer is suddenly released.

Figure 2

Typical conformations of a freely fluctuating stiff polymer with suppressed rotations. In order to generate these conformations, which serve illustrative purposes only, we have represented each conformation as a linear superposition of the first ten modes that solve the linearized equation of motion. For each realization, the mode amplitudes were then drawn randomly according to their Gaussian distribution in equilibrium.

Figure 3

The absolute value $\mid \mathrm{\Delta }{\Sigma }_{h∕c}^{\gtrless }\mid$ of the spatially varying part $\mathrm{\Delta }{\Sigma }_{h∕c}^{\gtrless }\equiv {\Sigma }_{h∕c}^{\gtrless }\left(\xi \right)-{\Sigma }_{h∕c}^{\gtrless }(\infty )$ of the scaling functions ${\Sigma }_{h∕c}^{\gtrless }$ defined in Eqs. (51, 52), which happens to be the same for hinged and clamped boundary conditions.

Figure 4

Tensile forces acting on a polymer subsection of size $\mathrm{\Delta }s$. Balancing these forces with the drag that arises from the longitudinal velocity of magnitude ${\partial }_t{r}_{\Vert }(s,t)$, one estimates $\mathrm{\Delta }s{\zeta }_{\Vert }{\partial }_t{r}_{\Vert }(s,t)\approx f(s-\mathrm{\Delta }s∕2)-f(s+\mathrm{\Delta }s∕2)$ in the weakly bending limit. For infinitesimal $\mathrm{\Delta }s$, this becomes ${\zeta }_{\Vert }{\partial }_t{r}_{\Vert }(s,t)\approx -{\partial }_{s}f(s,t)$. A further spatial derivative yields ${\zeta }_{\Vert }{\partial }_t{\partial }_s{r}_{\Vert }(s,t)\approx -{\partial }_s^{2}f(s,t)$, which is the time derivative of Eq. (72) (in original units) up to a spatial and ensemble average on the left-hand side, which correspond to an adiabatic and equilibrium approximation, respectively.