Monomer dynamics of a wormlike chain
Jakob Tómas Bullerjahn, Sebastian Sturm, Lars Wolff, Klaus Kroy
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Monomer dynamics of a wormlike chain
J. T. Bullerjahn, S. Sturm, L. Wolff and
K. Kroy
Institut für Theoretische Physik, Universität Leipzig – PF 100920, 04009 Leipzig, Germany
PACS – Dynamics of biomolecules
PACS – Self-diffusion in polymers
PACS – Single-molecule techniques
Abstract – We derive the stochastic equations of motion for a tracer that is tightly attached to asemiflexible polymer and confined or agitated by an externally controlled potential. The generalisedLangevin equation, the power spectrum, and the mean-square displacement for the tracer dynamicsare explicitly constructed from the microscopic equations of motion for a weakly bending wormlikechain by a systematic coarse-graining procedure. Our accurate analytical expressions shouldprovide a convenient starting point for further theoretical developments and for the analysis ofvarious single-molecule experiments and of protein shape fluctuations.
Introduction. –
Many of the tools available to anexperimental biophysicist probe either the fluctuations ofsemiflexible polymers or their response to external forces.This includes dynamic light scattering [1, 2], active andpassive microrheology of polymer networks [3] or cells [4],magnetic bead twisting cytometry [5], DNA relaxationand stretching [6], single-molecule force spectroscopy [7]and electron transfer techniques [8]. From a theoreticalpoint of view, these methods expose different aspects ofthe wormlike chain (WLC) model. It provides a minimaldescription of semiflexible polymer physics in terms ofan inextensible, thermally fluctuating elastic beam, andhas found broad acceptance on grounds of its excellentagreement with experimental data.Deformations of the polymer contour excite a broad spec-trum of bending modes with widely disparate relaxationtimes, thus resulting in anomalous, subdiffusive dynamics.In the linear regime, valid for equilibrium fluctuations orsmall external forces transverse to the polymer backbone,the dynamic mean-square displacement of a tagged (butmechanically unaltered) monomer obeys MSD ⊥ ( t ) ∝ t / [9,10]. For fluctuations parallel to the polymer axis, the ad-ditional longitudinal solvent friction induces tension forceswhich in turn stiffen the polymer and give rise to a dif-ferent scaling behaviour, MSD (cid:107) ( t ) ∝ t / [11]. Tensionalso dominates the response to strong point forces andexternally imposed solvent flows; the resulting equations ofmotion are highly nonlinear and can produce a multitudeof different dynamical regimes even in the course of a singleexperiment [12–16]. In many cases of practical interest, a full evaluation ofthe dynamics would be needlessly complicated and waste-ful, since experimental manipulation and data acquisitionare strongly localised, say, to an attached tracer particle,or a tagged monomer, in the following simply referred toas “the tracer”. Farther parts of the polymer matter onlyinsofar as they contribute to the force on the tracer. It canthen be preferable to integrate out the polymeric degreesof freedom beforehand and subsume them under an effec-tive equation of motion describing the tracer coordinateonly. Such a reduced description is for example knownfor the important special case of a tracer subjected to anexternally prescribed deterministic force protocol [16]. Itcannot, however, easily be extended to accommodate forthe fluctuating forces exerted onto the tracer by an exter-nally controlled confinement potential. Practical examplesthat involve such a potential are provided by various single-molecule manipulation techniques (think e.g. of an actinfilament labelled with a gold nanoparticle that is trappedby optical tweezers). The analysis of high-frequency shapefluctuations of globular proteins, as measured by electrontransfer techniques [8, 17] provides another important ex-ample. Indeed, the WLC has been proposed as one possiblemodel of protein fluctuations, but to date only numericalevaluations of the corresponding noise and friction func-tions are available within a mean-field approximation tothe WLC [20].In the following, we systematically derive the sought-after reduced equation of motion for the tracer coordinate x ( t ) , starting from the WLC in the weakly-bending limit,p-1 a r X i v : . [ c ond - m a t . s o f t ] S e p . T. Bullerjahn et al. which is asymptotically exact for large bending rigidity orstrong stretching force. The resulting GLE, (cid:90) t d τ K ( t − τ ) ˙ x ( τ ) = F ( x, t ) + Ξ( t ) , (1)is not necessarily linear, as it may include an arbitraryexternal potential U ( x, t ) = − (cid:82) d x (cid:48) F ( x (cid:48) , t ) . In the infinite-length limit L → ∞ , the GLE (1) can be worked out explic-itly in terms of the microscopic parameters, which comprisethe length L of the polymer, its bending rigidity κ , andtension f . It is validated by comparison with numerical so-lutions of the exact equations of motion. We moreover givea simple interpolation formula (9) that provides a universaldescription for polymer-bound tracer particles in stronglylocalised externally controlled potentials and possibly alsofor the mentioned protein shape fluctuations. We expectit to become a valuable and convenient tool for analysinga wealth of experimental data and for future theoreticaldevelopments. By providing a physically transparent andconcise parametrisation of measured tracer movements,it will moreover be helpful in the mutual comparison ofdata obtained with diverse experimental techniques. Toexemplify our approach, we calculate various observablescharacterising the time-dependent spatial correlations ofthe tracer motion, such as its power spectrum and mean-square displacement in presence of a harmonic trap. Langevin description of a WLC. –
In the WLCmodel, a semiflexible polymer is mechanically representedas an inextensible elastic beam of length L and bendingrigidity κ . It follows that thermal forces can only inducesignificant bending on length scales larger than the per-sistence length (cid:96) p = κ/ ( k B T ) (in 3 dimensions). In theweakly-bending limit, valid for large persistence length (cid:96) p (cid:29) L or strong external stretching force f (cid:29) k B T /(cid:96) p ,the polymer is essentially straight and can thus be treatedin terms of its small excursions r ⊥ ( s, t ) from the straight-rod ground state. The elastic bending energy in a givenconfiguration r ⊥ ( s, t ) reads [9] H = (cid:90) L d s (cid:20) κ r (cid:48)(cid:48)⊥ ) + f r (cid:48)⊥ ) (cid:21) . Shape fluctuations of the polymer then obey a Langevinequation obtained by balancing the corresponding bendingforces with friction and thermal (Gaussian white) noise [9], ζ ⊥ ˙ r ⊥ = − δ H δr ⊥ + ξ ⊥ = − κr (cid:48)(cid:48)(cid:48)(cid:48)⊥ + f r (cid:48)(cid:48)⊥ + ξ ⊥ (2a) (cid:104) ξ ⊥ (cid:105) = 0 (2b) (cid:104) ξ ⊥ ( s, t ) ξ ⊥ ( s (cid:48) , t (cid:48) ) (cid:105) = 2 ζ ⊥ k B T δ ( t − t (cid:48) ) δ ( s − s (cid:48) ) . (2c)In the vein of a similar treatment for a Rouse chainmonomer [21], we now introduce a tracer at s = s and let xU ( x, t ) − γ ( t ) F ( x, t )+ γ ( t ) f f Figure 1: Force diagram for the combined system of a tracer(red) and an attached WLC (blue). The tracer is displaced byan external potential U ( x, t ) = − (cid:82) F ( x, t ) d x . The constrain-ing force pair ± γ ( t ) fixes the polymer backbone to the tracerposition x . it absorb the external driving force F ( x, t ) , ζ tr ˙ x ( t ) = F ( x, t ) − γ ( t ) + ξ tr ( t ) (3a) ζ ⊥ r ⊥ ( s, t ) = − κr (cid:48)(cid:48)(cid:48)(cid:48)⊥ ( s, t ) + f r (cid:48)(cid:48)⊥ ( s, t )+ γ ( t ) δ ( s − s ) + ξ ⊥ ( s, t ) (3b) r ⊥ ( s , t ) ! = x ( t ) . (3c)Here ζ tr and ξ tr denote an optional friction coefficient andGaussian white noise source for the tracer, respectively.Both may be set equal to zero in the tagged-monomercase. The (Lagrange) forces ± γ ( t ) serve to rigidly tie thetracer to the polymer contour at s , as required by theconstraint (3c). The latter is simplified by transferring to acomoving reference frame, in which the tracer is always atrest, ∆( s, t ) ≡ r ⊥ ( s, t ) − x ( t ) . The corresponding equationof motion acquires a spatially constant friction term dueto the drift between the comoving reference frame and thesolvent, ζ ⊥ ˙∆( s, t ) = − κ ∆ (cid:48)(cid:48)(cid:48)(cid:48) ( s, t ) + f ∆ (cid:48)(cid:48) ( s, t ) + ξ ⊥ ( s, t )+ γ ( t ) δ ( s − s ) − ζ ⊥ ˙ x ( t ) (4a) ∆( s , t ) = 0 . (4b)The formal solution for γ can immediately be written downin the frequency domain, γ ω = − (cid:82) d σ ξ ⊥ ,ω ( σ ) G ω ( s , σ ) G ω ( s , s ) (cid:124) (cid:123)(cid:122) (cid:125) Ξ ω − iωx ω (cid:82) d σ G ω ( s , σ ) G ω ( s , s ) (cid:124) (cid:123)(cid:122) (cid:125) − K ω , where G ( s, s (cid:48) , t ) denotes the transverse Green’s functionof a WLC, i.e. its transverse deformation r ⊥ ( s, t ) in re-sponse to a unit force impulse δ ( s − s (cid:48) ) δ ( t ) . This explicitlyestablishes the link between the microscopic equations ofmotion (2) and the coarse-grained equation for the tracer(1). A practical way of evaluating γ ( t ) numerically consistsin the decomposition of ∆( s, t ) into its normal coordinates[22]. This procedure is discussed for arbitrary boundaryconditions and monomer positions s in the appendix. Analytical solution. –
To proceed analytically, wenow consider the centre monomer only, s = 0 . Insteadp-2onomer dynamics of a wormlike chainof explicitly including the “adhesion force” γ ( t ) δ ( s ) , whichinduces a coupling of different eigenmodes and thus rendersthe dynamics nondiagonal, we can then easily take care ofthe constraint ∆(0 , t ) = 0 by restricting the function spaceaccordingly. Using only those eigenmodes W n ( s ) satisfying W n (0) = 0 , the singular force γ can be read off as follows, κ ∆ (cid:48)(cid:48)(cid:48)(cid:48) (0 , t ) = γ ( t ) δ ( s ) κ (∆ (cid:48)(cid:48)(cid:48) (0 + , t ) − ∆ (cid:48)(cid:48)(cid:48) (0 − , t )) = γ ( t ) . Since the above expression vanishes for odd eigenmodes,only even modes W n ( s ) = W n ( − s ) contribute to γ . Re-quiring further that ∆ (cid:48)(cid:48)(cid:48)(cid:48) constitutes the highest (and only)singularity, we thus find ∆ (cid:48) (0 ± ) = 0 and so obtain thefollowing friction kernel, K ( t ) = 2 (cid:88) n W (cid:48)(cid:48)(cid:48) n (0) (cid:82) L/ W n ( s ) d s (cid:82) L/ W n ( s ) d s e −E n t/ζ ⊥ , where κ W (cid:48)(cid:48)(cid:48)(cid:48) n ( s ) − f W (cid:48)(cid:48) n ( s ) = E n W n ( s ) W n (0) = W (cid:48) n (0) = 0 , and the outer boundary conditions are dictated by thephysical situation. In the infinite-length limit L → ∞ ,valid for t (cid:28) τ , K ( t ) becomes independent both of s andof the choice of outer boundary conditions. Using torquedends W (cid:48) n ( L/
2) = W (cid:48)(cid:48)(cid:48) n ( L/
2) = 0 for convenience, we find K ( t ) ∼ ∞ (cid:88) n =0 (cid:20) n π κL + fL (cid:21) e − t/τ n . (5)with a relaxation time spectrum τ n ∼ ζ ⊥ L κπ n ) + (2 n + 1) f /f L . Here, f L = κπ /L denotes the Euler buckling force. Forsemiflexible polymers, f L is usually small in comparison toexternally applied stretching forces. Mode numbers belowa critical value n c = ( f /f L ) / are then tension-dominated,whereas shorter-wavelength modes exhibit force-free relax-ation and therefore τ n ∝ n − . This divides the frictionalresponse K ( t ) to a transverse force into three differentasymptotic regimes. At short times t (cid:28) τ n c = ζ ⊥ κ/ f ,the backbone tension can be neglected so that K ( t ) sim-plifies to K ( t (cid:28) τ n c ) ∼ √ κ / Γ(1 /
4) ( ζ ⊥ /t ) / . (6)Between τ n c and the terminal relaxation time τ , we have K ( τ n c (cid:28) t (cid:28) τ ) ∼ / (cid:112) f ζ ⊥ /t , (7)and at very long times t (cid:29) τ , the exponential relaxation K ( t (cid:29) τ ) ∝ (cid:40) e − t/τ f = 0 e − t/τ f > (8) K ( t ) [ κ / L ] − − − − t [ L ζ ⊥ /κ ] Figure 2: The analytical interpolation formula (9) compared tothe exact numerical solutions for f = 0 ( (cid:13) ) and f = 10 f L ( (cid:3) ). of the lowest mode provides a natural physical cutoff ofthe scale-free intermediate asymptotic dynamics. Usinga time-dependent mode cutoff at τ n = t , we arrive at theapproximate interpolation formula K ( t ) ≈ √ π √ κ (cid:113)(cid:112) f + 4 ζ ⊥ κ/t − f × (cid:104)(cid:112) f + 4 ζ ⊥ κ/t + 2 f (cid:105) e − t/τ ∗ , (9)which faithfully reproduces the general solution describedin the appendix, see fig.(2).An exact treatment of the binding constraint (3c) renor-malises the terminal relaxation time, which is why weconsider τ ∗ as a free (fit) parameter. In the long-polymerlimit ( L, τ ∗ → ∞ , e − t/τ ∗ → ), eq. (9) reduces to a two-parameter formula.The associated colored noise term Ξ( t ) of eq. (1) canbe determined from the fluctuation-dissipation theorem(FDT), (cid:104) Ξ( t )Ξ( t (cid:48) ) (cid:105) = 2 k B T K ( | t − t (cid:48) | ) . Example applications. –
As an example applicationof our single-coordinate equation of motion eq. (1), we firstrederive the time-dependent MSD of a monomer in thebending-dominated regime. We then include an externalharmonic potential to compute both the time-dependentMSD and the power spectrum of a polymer-bound tracerparticle held in a harmonic trap.For the free polymer in solution, we set F ( x, t ) = 0 andhold the tagged monomer fixed until t = 0 . Using theearly-time asymptote to K ( t ) , eq. (6), its trajectory thenfollows as x ( t ) − x (0) ∼ √ / κ / ζ / ⊥ (cid:90) t Ξ( τ ) d τ ( t − τ ) / , (10)The MSD follows from eq. (10),MSD F =0 ( t (cid:28) τ n c ) ∼ / √ k B Tκ / (cid:20) tζ ⊥ (cid:21) / . p-3. T. Bullerjahn et al. − − − S ( ω ) [ n m s ] ω [Hz]10 − − − S ( ω ) [ n m s ] Figure 3:
Top panel : power spectrum of an unmodified taggedmonomer (no excess friction) with (cid:96) p = 10 µ m , ζ ⊥ = 10 mPa · s , k = 1 pN / nm , T = 300 K , L → ∞ ( i.e. ωτ ∗ → ∞ ) and f = 5 pN ( (cid:3) ), f = 0 ( (cid:13) ). Bottom panel : power spectrum of an at-tached tracer particle that has a perceptible friction coefficient ζ tr = 6 πη water r , r = 10 nm , other parameters as above. TheLorentzian power spectrum of the same bead without the at-tached polymer is shown for comparison (dashed line). This subdiffusive behaviour coincides with previous the-oretical predictions [2] and has been measured directlyand indirectly in networks of polymerized actin [1, 10] andmicrotubuli [5]. Including a stationary harmonic trap ofstiffness k , i.e. , F ( x, t ) ≡ F ( x ) = − kx , equation (10) turnsinto x ( t ) ∼ x (0) E (cid:20) − k ( t/ζ ⊥ ) / √ κ / (cid:21) − k (cid:90) t d τ Ξ( τ ) ∂∂t (cid:48) E (cid:20) − k ( t (cid:48) /ζ ⊥ ) / √ κ / (cid:21) t (cid:48) = t − τ , (11)where E α ( z ) = ∞ (cid:88) n =0 z αn Γ(1 + αn ) denotes the Mittag–Leffler function, which can be regardedas a generalised exponential function. The resulting MSDfor x (0) ≡ readsMSD F (cid:54) =0 ( t (cid:28) τ n c ) ∼ k B Tk (cid:34) − E (cid:20) − k ( t/ζ ⊥ ) / √ κ / (cid:21)(cid:35) . A deterministic external force f ext ( t ) or a dynamicallymoving optical trap, represented by f ext ( t ) ≡ kx ( t ) , canbe included along the same lines by setting F ( x, t ) = − kx + f ext ( t ) . Finally, we calculate the power spectrumdensity for a polymer-bound bead in harmonic confinement, S ω = (cid:90) ∞−∞ (cid:104) x ( t ) x (0) (cid:105) e − iωt d t , as measured by video microscopy or other in-plane tech-niques. (Note that for d position tracking x becomesa d vector transverse to the polymer backbone, and S ω = 2 S ω .)The somewhat lengthy explicit result has a simple struc-ture, suitable for fitting experimental data. Specialising tothe infinite-length limit ( ωτ ∗ → ∞ ), we find S ω k B T = 4 ζ ⊥ R ω + 2 ζ tr [ k − ωζ ⊥ I ω ] + ω [ ζ tr + ζ ⊥ R ω ] (12)where R ω and I ω denote the real and imaginary parts (witha dimension of length) of √ κ (cid:2) ( f − (cid:112) f + 4 iωζ ⊥ κ ) − / + ( f + (cid:112) f + 4 iωζ ⊥ κ ) − / (cid:3) , respectively. For a tracer that causes a perceptible friction( ζ tr = 6 πηr tr > ), the ultimate high-frequency limit is ofthe usual Lorentzian form S ( ω → ∞ ) ∼ ω − . If ζ tr van-ishes or is imperceptibly small compared to the monomerfriction, the decay of the power spectrum is slightly weakerat large frequencies, S ∼ ω − / . This can intuitively beunderstood as a consequence of the frequency-dependent“apparent bead size”, given by the subsection (of length (cid:96) ⊥ , where (cid:96) − ⊥ (cid:39) ζ ⊥ ω/κ ) of the attached polymer thatequilibrates with the bead within one period ω − . Seefig. 3 for a graphical representation.In the absence of backbone tension ( f = 0 ), the powerspectrum (12) simplifies to ( s = sin( π/ , c = cos( π/ ) S f =0 ω k B T = 8 ζ / ⊥ ( κ/ω ) / ( s + c ) + 2 ζ tr (cid:2) k + 2 κ / ( ζ ⊥ ω ) / ( s − c ) (cid:3) + ω (cid:104) ζ tr + 2 ζ / ⊥ ( κ/ω ) / ( s + c ) (cid:105) . Conclusions. –
Starting from the formally exactWLC equation of motion in the weakly-bending rod approx-imation, we have derived a generalised Langevin equationdescribing the dynamics of a tagged monomer of (or “tracer”attached to) a semiflexible polymer. The tracer was al-lowed to be under the influence of an arbitrary externalpotential. Our method is simple, direct and analyticallysolvable. We have furthermore derived a uniformly validanalytic interpolation formula which may serve as a com-pact (two- or three-parameter) parametrisation of, e.g. , themotion of a tracer attached to a semiflexible polymer andmanipulated by an optical trap, or of conformational fluc-tuations of protein domains. With regard to quantitativeapplications using metallic tracer particles in combinationwith optical traps, it might be worthwhile to extend ourp-4onomer dynamics of a wormlike chainresults along the lines of [23] to take the heating of thetracer into account. ∗ ∗ ∗
We acknowledge financial support from the DeutscheForschungsgemeinschaft (DFG) through FOR 877 andthe Leipzig School of Natural Sciences – Building withMolecules and Nano-objects (BuildMoNa).
Appendix. –
For the evaluation of γ ( t ) , we decom-pose ∆( s, t ) into bending eigenmodes W n ( s ) and modeamplitudes a n ( t ) , such that ∆( s, t ) = N (cid:88) n =0 W n ( s ) a n ( t ) κ W (cid:48)(cid:48)(cid:48)(cid:48) n ( s ) − f W (cid:48)(cid:48) n ( s ) = ζ ⊥ τ n W n ( s ) (cid:90) L d σ W n ( σ ) W m ( σ ) = δ nm . The actual physical boundary conditions at the polymerends dictate the detailed functional form of the W n . Afree polymer in solution requires [24] W (cid:48)(cid:48) n ( − L/
2) = W (cid:48)(cid:48)(cid:48) n ( − L/
2) = W (cid:48)(cid:48) n ( L/
2) = W (cid:48)(cid:48)(cid:48) n ( L/
2) = 0 , but the precise choice of boundary conditions is irrelevantto the following discussion. Projecting eq. (4b) onto eachof the W n , we find N + 1 distinct equations of motion forthe a n , ˙ a n ( t ) = − a n ( t ) (cid:18) τ n + ˙ x ( t ) A n (cid:122) (cid:125)(cid:124) (cid:123) (cid:104)W n ( s ) | (cid:105)− (cid:104)W n ( s ) | ξ ⊥ ( s, t ) (cid:105) ζ ⊥ (cid:124) (cid:123)(cid:122) (cid:125) ξ n ( t ) /ζ ⊥ + γ ( t ) ζ ⊥ W n ( s ) (cid:19) , (A.1)where τ n is the relaxation time of the n th eigenmode.The singular force term can be eliminated by choosinga complete set of allowed displacements in mode space, (cid:80) W n ( s ) a n ( t ) = 0 , leading to δ = (cid:0) W ( s ) , −W ( s ) , , . . . , (cid:1) δ = (cid:0) , W ( s ) , −W ( s ) , , . . . , (cid:1) ... δ N = (cid:0) , . . . , , W N ( s ) , −W N − ( s ) (cid:1) . (A.2)Combining the eqs. (A.1) of motion with the correspondingconstraint equations (A.2), the dynamics of the a n aredetermined completely, reading M ∂ t ( a , . . . , a n ) (cid:62) = M ( a , . . . , a N ) (cid:62) + ˙ x ( t ) V x + V ξ ( t ) , where M , and V x,ξ are given by M = W ( s ) −W ( s ) W ( s ) −W ( s ) ... ... W N ( s ) −W N − ( s ) W ( s ) W ( s ) ··· ··· W N ( s ) M = − W s τ W s τ − W s τ W s τ ... ... − W N ( s τN − W N − s τN V x = A W ( s ) − W ( s ) A ...... A N W N − ( s ) − W N ( s ) A N − V ξ ( t ) = 1 ζ ⊥ ξ ( t ) W ( s ) − W ( s ) ξ ( t ) ...... ξ N − ( t ) W N ( s ) − W N − ( s ) ξ N ( t )0 . The solution to eq. (13) then reads ( a , . . . , a N ) (cid:62) ( t ) = e M − M ( t − t ) ( a , . . . , a N ) (cid:62) ( t )+ (cid:90) tt d τ e M − M ( t − τ ) M − (cid:0) ˙ x ( τ ) V x + V ξ ( τ ) (cid:1) . Inserting the above solution into the original equation ofmotion (A.1), we obtain the force of constraint in modespace, ( γ ( t ) δ ( s − s )) n = ζ ⊥ ∂ t ˙ a n ( t ) + ζ ⊥ a n ( t ) /τ n − ξ n ( t ) + ˙ x ( t ) A n ζ ⊥ = ζ ⊥ (cid:90) tt d τ (cid:0) M − M + diag( τ − n ) (cid:1) e M − M ( t − τ ) × M − (cid:0) ˙ x ( τ ) V x + V ξ ( τ ) (cid:1) + ζ ⊥ ˙ x ( t )( M − V x + A n )+ (cid:0) ζ ⊥ M − V ξ ( t ) − ξ n ( t ) (cid:1) . (A.3)The part outside the integral vanishes, except for the N thentry, which is an artefact introduced by the mode cutoffand can safely be ignored: having a finite minimum bendingmode wavelength implies that even at arbitrarily shorttimes, a finite part of the polymer will be dragged alongwith the monomer. We thus obtain γ ( t ) as follows, γ ( t ) = lim N →∞ (cid:20) N (cid:88) n =0 W n ( s ) (cid:21) − W (cid:62) (cid:90) tt d τ (cid:0) M − M + diag( τ − n ) (cid:1) e M − M ( t − τ ) M − (cid:0) ˙ x ( τ ) V x + V ξ ( τ ) (cid:1) , p-5. T. Bullerjahn et al. where W = ζ ⊥ (cid:0) W ( s ) , . . . , W N ( s ) (cid:1) . Identifying the ran-dom and the ˙ x -dependent terms with ξ ( t ) and K ( t ) , re-spectively, we find K ( t ) = W (cid:62) (cid:0) M − M + diag( τ − n ) (cid:1) e M − M t M − V x ,ξ ( t ) = W (cid:62) (cid:90) tt d t (cid:48) (cid:0) M − M + diag( τ − n ) (cid:1) × e M − M ( t − t (cid:48) ) M − V ξ ( t ) . Note that if s coincides with the center of the polymer,antisymmetric modes will not contribute to γ ( t ) ; the calcu-lation then has to be restricted to the symmetric component ∆ s ( s, t ) , otherwise M would be degenerate. The aboveprocedure trivially extends to an inhomogeneous stationaryforce profile f = f ( s ) . In that case, the differential oper-ator κ∂ s − f ∂ s turns into κ∂ s − f (cid:48) ( s ) ∂ s − f ( s ) ∂ s , whichchanges both the eigenmodes W n ( s ) and their respectiveeigenvalues, but the calculation in mode space remainsunaffected. References[1]
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