Coupling of Transverse and Longitudinal Response in Stiff Polymers

Abstract

The time-dependent transverse response of stiff inextensible polymers is well understood on the linear level, where transverse and longitudinal displacements evolve independently. We show that for times beyond a characteristic time t f , longitudinal friction considerably weakens the response compared to the widely used linear response predictions. The corresponding feedback mechanism is explained by scaling arguments and quantified by a systematic theory. Our scaling laws and exact solutions for the transverse response apply to cytoskeletal filaments as well as DNA under tension. PACS numbers: 61.41.+e, 87.15.La, 87.15.He, 98.75.Da In tracing back the viscoelasticity of the cell to properties of its constituents, a detailed understanding of the mechanical response of single cytoskeletal filaments is indispensable. Due to their large bending stiffness, these filaments exhibit highly anisotropic static [1] and dynamic [2,3,4] features, such as the anomalous t 3 /4 -growth of fluctuation amplitudes in the transverse direction [5,6], i.e., perpendicular to the local tangent. The related response to a localized transverse driving force has so far been examined only by neglecting longitudinal degrees of freedom [6,7], although these polymers are virtually inextensible, and transverse and longitudinal contour deformations therefore coupled. In this Letter we show that longitudinal motion strongly affects the transverse response even for weakly-bending filaments and leads to relevant nonlinearities beyond a characteristic time t f .The physical key factors controlling the transverse response may be understood from Fig. 1, which shows a weakly-bending polymer (bending undulations are exaggerated for visualization) shortly after a transverse driving force f ⊥ has been applied in the bulk. In response to this force, the contour develops a bulge. Due to the backbone inextensibility, this bulge can continue growing only by pulling in contour length from the filament's tails. This effectively reduces the thermal roughness of the contour [8,9,10], at a rate substantially limited by longitudinal solvent friction. The resulting coupling to the longitudinal response tends to slow down the bulge growth. In order to describe this feedback mechanism, we start with a scaling analysis and treat the simpler athermal case first. To connect to the biologically important situations of prestressed actin networks [11] and prestretched DNA [12], we then extend a recent theory of tension dynamics [13] to calculate the nonlinear response for unstretched and prestretched initial conditions. Consider the overdamped dynamics of an initially straight stiff rod of total length L. Suddenly applying a transverse pulling force f ⊥ , for simplicity in the center of the rod, leads to the growth of a bulge deformation. The generated friction in the transverse and longitudinal direction needs to be balanced by corresponding driving forces. Viscous solvent friction is modeled via anisotropic friction coefficients (per length) ζ ⊥ and ζ = ζζ ⊥ with ζ ≈ ...

Tension dynamics in semiflexible polymers. II. Scaling solutions and applications

“…Inclusion of hydrodynamic interactions merely produce logarithmic corrections but would give rise to more interesting effects for polymerized membranes to which our discussion could be generalized with otherwise little change. Other natural generalizations including the transverse nonlinear response of polymers [7], quenches in the persistence length [12] and more complex force protocols [34] are currently also under investigation.…”

“…( 4) is the basis not only for discussing the tension dynamics itself, but also the starting point for analytical and numerical calculations of the longitudinal and transverse nonlinear response of a weakly bending polymer. It is the purpose of the present Part II to treat the former case in detail, while the latter, somewhat more complex case is reserved for a future communication [7]. In Sec.…”

Viscoelastic Behavior of a Single Semiflexible Polymer Chain

“…They consider a weakly bend situation and use a kind of multi-scale perturbation methods. The theory has been applied to the relaxation of an elongated chain after removing an external force [35,36].…”

Longitudinal response of confined semiflexible polymers

“…These early results were later confirmed and generalized by means of a systematic formalism based on a multiple scale perturbation theory [43][44][45]. Recent extensions of these results include prestressed filaments [46], transverse forces [47], extensible backbones [48], and oscillatory forces [49]. A common observation of these studies is that the essential nonlinearities contained in the nontrivial tension profile lead to a mixing of short-and long-wavelength fluctuations, such that experimentally changing the equilibrium mode spectrum of contour fluctuations can have far more drastic consequences as usually suspected, for instance on the scenario-specific relaxation dynamics observed for polymers that were prepared in initially straight conformations by different means [50].…”