For an inextensible polymer, the backbone cannot actually be stretched, but the ther- mally excited transverse excursions discussed above cause the contour to exhibit bending undulations. These thermal fluctuations can be straightened by applying external forces, and the tension f, introduced as constraint force, is a measure of the force necessary to ensure that only fluctuations are straightened but the backbone itself remains at constant length; accordingly, the tension diverges as the extension approaches the contour length (cf. Eq. (1.3) with f as tension). Hence, a reciprocal relation between contour length “stored” in thermal undulations (“stored length”), and tension arises: external forces can
2.2 Tension dynamics for wormlike chains 19
pull out stored length and create tension; once the thermal undulations are re-introduced, the tension relaxes.
By Eq. (2.10a), the absence of longitudinal friction and bulk forces means that the leading-order tension is just a constant and stored length is thus homogeneously distributed along the contour. If a sudden stretching force is applied, the above introduced correlation length`k(t) gains another, much more intuitive interpretation: stored length is pulled out,
but limited by longitudinal friction this happens at first only near the boundary, over distances of length `k(t). Likewise, the increased tension induced by the applied force accordingly penetrates the contour only within a growing region of size`k(t). In this sense,
the physics missed within the linear theory is entirely contained in a proper description of the propagation and relaxation of backbone tension.
Realizing these important features, a theory of tension dynamics on the linear level has been developed by Morse et al. [211, 238], while the case of large force has been discussed both based on a “taut-string” approximation by Seifert et al. [236], as well as using the somewhat complementary “quasi-static” approximation by Brochard-Wyart et al. [25]. Together with accordingly refined mean-field models [153] and other studies based on scaling arguments [2,60,236], “adiabatic” approximations [219,236] or computer simu- lations [190, 269], these approaches lead to an incomplete and, due to partly contradicting assumptions, also somewhat inconsistent picture of tension propagation under different con- ditions. Aiming to provide a unified and systematic theoretical description, Hallatschek et al. [97, 98,99] developed a general theory of tension dynamics in the weakly-bending limit based on a rigorous multiple-scale perturbation theory (for an introduction to this method see the book [108]).
2.2.1. Multiple-scale perturbation theory
This approach, explained in detail in Ref. [98], exploits the previously observed scale sepa- ration `⊥/`k ∝ε1/2 to introduce two different small- and large-scale arclength coordinates
s and ¯sε1/2. Writing dynamic variables such asr(s) andf(s) as an expansion in functions of these two formally independent variables gives a system of equations of motion in each power ofε1/2. In order to obtain a uniformly convergent expansion, the coefficients in each
order should be bounded in the formal limit s → ∞. Most importantly, this condition yields that the tension is a function of the large-scale variable ¯sonly. This finding suggests to spatially average the longitudinal part over small-scale fluctuations, which causes most terms in Eq. (2.9b) to vanish and results in the simple equations of motion:
∂tr⊥ =−r0000⊥ + ¯fr 00
⊥+ξ⊥, (2.17a)
ˆ
ζ∂tr0k =−∂s2¯f ,¯ (2.17b)
where the prime is ans-derivative and the overbar denotes the spatial average. The latter is performed over many uncorrelated segments of length `⊥ and thus effectively produces an ensemble average [98]. Hence, the longitudinal part Eq. (2.17b) can be written as
where % = 12r02
⊥ ≈ r0k is the local density of contour length stored in thermal undulations,
i.e., the stored length density. The transverse equation of motion Eq. (2.17a) for small- scale transverse fluctuations contains a locally constant tension, whose large-scale arclength dependence follows from Eq. (2.18), which describes the deviations of the tension profile from its constant equilibrium value. Integrating over arclength suggests the interpretation that changes in stored length produce tension gradients against longitudinal friction.
2.2.2. Coarse-grained equation of motion for the tension
Using that Eq. (2.17a) is linear, because the large-scale arclength dependence of the tension enters only adiabatically, a mode decomposition of r⊥(s, t) in terms of eigenmodes wq(s)
to the eigenvalues −q2[q2+ ¯f(¯s, t)] provides a solution via the response function
χ⊥(q;t, t0) = e−2q2[q2(t−t0)+ Rt
t0dτf¯(¯s,τ)]Θ(t−t0). (2.19) It is the appropriate extension of the function χ0⊥(q;t, t0) = e−q4(t−t0)Θ(t − t0) used for solving the linear force-free case Eq. (2.10b) in Sec. 2.1.3. Hence, the expectation value
h%i=12r02 ⊥ is given by h%i= * 1 2 " X q Z ∞ −∞ dt0ξ⊥,q(t0)χ⊥(q;t, t0)w0q(s) #2+ . (2.20)
In order to proceed from here, we have to specify the scenarios we will be interested in. Let us assume that the polymer is for t < 0 equilibrated under a possibly inhomogeneous tension profile ¯f0(¯s) and with a possibly different persistence length θ`p. Further, we
assume that the boundary conditions on the contour are such (e.g., hinged) that a spatial average over the squared first derivatives of the eigenmodes gives a constant (this point will be discussed further in Sec. 2.5):
w02 q(s)≈
q2
L. (2.21)
In this case, we can evaluate Eq. (2.20) by using the noise correlation as for Eq. (2.13) to find the spatially averaged stored length h%i¯ as
h%i¯ =X q χ2⊥(q;t,0) Lθ`p[q2+ ¯f0(¯s)] + 2q 2 L`p Z t 0 dt0χ2⊥(q;t, t0) . (2.22)
To obtain an explicit coarse-grained equation of motion for the backbone tension from Eq. (2.18), we switch to the time-integrated tension
¯
F(¯s, t) =
Z t
0
2.3 Longitudinal response 21