Active Brownian filaments with hydrodynamic interactions: conformations and dynamics
Aitor Martin-Gomez, Thomas Eisenstecken, Gerhard Gompper, Roland G. Winkler
AActive Brownian Filaments with Hydrodynamic Inter-actions: Conformations and Dynamics
Aitor Martín-Gómez, Thomas Eisenstecken, Gerhard Gompper, and Roland G. Winkler
The conformational and dynamical properties of active self-propelled filaments/polymers are in-vestigated in the presence of hydrodynamic interactions by both, Brownian dynamics simulationsand analytical theory. Numerically, a discrete linear chain composed of active Brownian particlesis considered, analytically, a continuous linear semiflexible polymer with active velocities chang-ing diffusively. The force-free nature of active monomers is accounted for—no Stokeslet fluidflow induced by active forces—and higher order hydrodynamic multipole moments are neglected.Hence, fluid-mediated interactions are assumed to arise solely due to intramolecular forces. Thehydrodynamic interactions (HI) are taken into account analytically by the preaveraged Oseen ten-sor, and numerically by the Rotne-Prager-Yamakawa tensor. The nonequilibrium character of theactive process implies a dependence of the stationary-state properties on HI via the polymer re-laxation times. In particular, at moderate activities, HI lead to a substantial shrinkage of flexibleand semiflexible polymers to an extent far beyond shrinkage of comparable free-draining poly-mers; even flexible HI-polymers shrink, while active free-draining polymers swell monotonically.Large activities imply a reswelling, however, to a less extent than for non-HI polymers, causedby the shorter polymer relaxation times due to hydrodynamic interactions. The polymer meansquare displacement is enhanced, and an activity-determined ballistic regime appears. Over awide range of time scales, flexible active polymers exhibit a hydrodynamically governed subdif-fusive regime, with an exponent significantly smaller than that of the Rouse and Zimm modelsof passive polymers. Compared to simulations, the approximate analytical approach predicts aweaker hydrodynamic effect. Overall, hydrodynamic interactions modify the conformational anddynamical properties of active polymers substantially.
The perpetual conversion of either internal chemical energy, orutilization of energy from the environment, into directed mo-tion is a key feature of active matter . Its respective out-of-equilibrium nature is the origin of intriguing emerging structuraland dynamical properties, which are absent in passive systems.This particularly applies to soft matter systems, e.g., comprised offilaments or polymers, which renders active soft matter a promis-ing class of new materials . Nature provides various exam-ples of filamentous, polymer-like active agents or phenomenawhere activity governs the nonequilibrium dynamics of passivemolecules. Propelled biological polar semiflexible filaments are
Theoretical Soft Matter and Biophysics, Institute for Advanced Simulation and In-stitute of Complex Systems, Forschungszentrum Jülich, D-52425 Jülich, Germany;E-mail: [email protected]; [email protected] ubiquitous, e.g., filamentous actin or microtubules in the cell cy-toskeleton due to tread-milling and motor proteins . In motilityassays, filaments are propelled on carpets of motor proteins an-chored on a substrate, which results in a directed motion andthe appearance of self-organized dynamical patters.
A char-acteristic feature of biological cells is the intrinsic mixture of ac-tive and passive components; specifically the active cytoskeletonand a large variety of passive colloidal and polymeric objects.Here, activity implies an enhanced random motion of tracer par-ticles . Furthermore, the active dynamics of microtubules oractin-filaments leads to an accelerated motion of chromoso-mal loci and chromatin . In addition, ATP-dependent en-zymatic activity-induced mechanical fluctuations drive molecularmotion in the bacterial cytoplasm and the nucleus of eukaryoticcells . Self-propelled rodlike or semiflexible polymer-like objectsare formed via self-assembly, e.g., by dinoflagellates , or growin bacterial biofilms, such as Proteus mirabilis . Synthetic ac- Journal Name, [year], [vol.] ,,
A char-acteristic feature of biological cells is the intrinsic mixture of ac-tive and passive components; specifically the active cytoskeletonand a large variety of passive colloidal and polymeric objects.Here, activity implies an enhanced random motion of tracer par-ticles . Furthermore, the active dynamics of microtubules oractin-filaments leads to an accelerated motion of chromoso-mal loci and chromatin . In addition, ATP-dependent en-zymatic activity-induced mechanical fluctuations drive molecularmotion in the bacterial cytoplasm and the nucleus of eukaryoticcells . Self-propelled rodlike or semiflexible polymer-like objectsare formed via self-assembly, e.g., by dinoflagellates , or growin bacterial biofilms, such as Proteus mirabilis . Synthetic ac- Journal Name, [year], [vol.] ,, a r X i v : . [ c ond - m a t . s o f t ] A ug ive or activated colloidal polymers are nowadays synthesizedin various ways. Assembly of active chains of metal-dielectricJanus colloids (monomers) can be achieved by imbalanced inter-actions, where simultaneously the motility and the colloid inter-actions are controlled by an AC electric field . Electrohydro-dynamic convection rolls lead to self-assembled colloidal chains ina nematic liquid crystal matrix and directed movement . More-over, chains of linked colloids, which are uniformly coated withcatalytic nanoparticles, have been synthesizes . Hydrogen per-oxide decomposition on the surfaces of the colloidal monomersgenerates phoretic flows, and active hydrodynamic interactionsbetween monomers results in an enhanced diffusive motion .Valuable insight into the properties of self-propelled filamentsand polymers, or their passive counterparts embedded in an ac-tive environment, is obtained by computer simulations and ana-lytical theory. Thereby, typically active Brownian polymers (AB-POs), neglecting hydrodynamic interactions (HI) (in the follow-ing, we will denote such polymers as ABPOs-HI), have been con-sidered , but also particular aspects of fluid-mediate interac-tions have been studied . Filaments are modeled as semi-flexible polymers, with an implementation of activity adaptedto the particular propulsion mechanism. Polar polymers, rep-resenting actin filaments or microtubules driven by molecularmotors, are typically propelled by forces tangential to the poly-mer contour . Here, a sufficiently high activityleads to shrinkage and compactification . ABPOs, where ev-ery monomer experiences an independent active force whose ori-entation changes in a diffusive manner , or passive poly-mers embedded in an environment of active Brownian particles(ABPs), exhibit a different behavior. Flexible ABPOs-HI swellwith increasing activity due to local active forces overpoweringthermal noise . Semiflexible ABPOs-HI shrink first atmoderate activities owing to active intramolecular stresses com-peting with bending forces, and swell for higher activities similarto flexible ABPOs-HI . In all cases, a faster dynamics is ob-tained .Hydrodynamics changes the properties of active systems in var-ious ways. Since an individual self-propelled particle—an isolatedmonomer in the case of a colloidal-type polymer —is force andtorque free, it creates a flow field lacking a Stokeslet, but in-cludes higher multipole contributions . Conformationalchanges and the interference of the monomer flow fields leadto autonomous filament/polymer motion even when individualmonomers are non-motile . The conformational and dy-namical properties of polar (actively) driven filaments, which arenot force free, are also strongly affected by hydrodynamic inter-actions . In particular, hydrodynamic coupling between twofilaments leads to cooperative effects .In this article, we analyze the influence of hydrodynamic inter-actions on the conformational and dynamical properties of AB-POs, denoted as ABPOs+HI in the following, by computer sim-ulations and an analytical approach. In simulations, we em-ploy a bead-spring linear phantom or self-avoiding polymer withABP monomers (cf. Fig. 1), where the ABP propulsion direc-tion changes diffusively, and hydrodynamic interactions aretaken into account Rotne-Prager-Yamakawa hydrodynamic ten- sor. For the analytical calculations, we consider a Gaus-sian semiflexible polymer, with active sites modeled byan Ornstein-Uhlenbeck process (active Ornstein-Uhlenbeck par-ticle, AOUP), where the active velocity vector changes ina diffusive manner; here, HI is included via the preaveragred Os-een tensor.
The main purpose of our study is to resolve theinfluence of hydrodynamics on the properties of self-propelledpolymers, respecting the force-free nature of an individual activeagent. Hence, no Stokeslet due to self-propulsion is present. OnlyStokeslets arising from bond, bending, and excluded-volume in-teractions between monomers, as well as thermal forces are con-sidered. Moreover, we neglect higher order multipole contribu-tions of the active monomers, especially the force dipole. Sincewe consider point particles, source multipoles are also absent. Allthese multipoles decay faster than a Stokeslet. Hence, we cap-ture the long-range character of HI in polymers of a broad classof active monomers. As far as near-field hydrodynamic effectsare concerned, our model closest resembles a polymer composedof neutral squirmers , where particular effects by highermultipole interactions between monomers are not resolved.
Our studies reveal a decisive influence of hydrodynamic inter-actions on the polymer conformations and dynamics. In particu-lar, even flexible ABPOs+HI shrink at moderate activities, whereABPOs-HI swell monotonically. At high activities, ABPOs+HIswell, but to an extent, which is considerably smaller than thatof ABPOs-HI. This indicates a dependence of the stationary-statedistribution function on hydrodynamics, an effect absent for pas-sive systems. The reason is the violation of the fluctuation-dissipation theorem of the active processes, which leads to the de-pendence of stationary-state properties on the hydrodynamicallymodified relaxation times. The shrinkage is then a consequenceof the time-scale separation between the thermal process, dom-inating for zero or very weak activities, and the active processwith hydrodynamically accelerated relaxation times. The modi-fied, activity-dependent relaxation times also affect the transla-tion motion, and a subdiffusive time regime appears, where themean square displacement (MSD) exhibits a power-law depen-dence with the exponent α (cid:48) = / , significant smaller than theZimm value, α (cid:48) = / , of a passive polymer.The manuscript is organized as follows. Section 2 describes thediscrete model of the ABPO along with the simulation approach,and presents simulation results. Section 3 describes the contin-uum model of an active polymer, its analytical solution, and dis-cusses conformational and dynamial properties. Section 4 sum-marizes our findings. A semiflexible active polymer is composed of N m active Brownianparticles (ABPs) ( i = ,..., N m , cf. Fig. 1), which obey the Journal Name, [year], [vol.] ,
Illustration of the activity-induced flow by the motion of anABPO+HI. Several ABPs moving by chance together in a certaindirection, indicated by the velocity arrow vvv , drag other connected ABPs,which in turn exert a force,
FFF , indicated by the red arrow, on the fluidinducing Stokes flow. The small arrows vvv n display the direction of theactive velocity. No flow field is generated by the active motion of anindividual ABP. An animation of the dynamics of a discrete ABPO+HI isprovided in the ESI. equations of motion ˙ rrr i ( t ) = v eee i ( t ) + N m ∑ j = H i j [ FFF i ( t ) + ΓΓΓ i ( t )] , (1) ˙ eee i ( t ) = ˆ ηηη i ( t ) × eee i ( t ) . (2)Here, rrr i ( t ) and ˙ rrr i ( t ) denote the position and velocity of particle i , respectively, and vvv i ( t ) = v eee i ( t ) is the active velocity with thepropulsion direction eee i ( | eee i | = ), which changes in a diffusivemanner according to Eq. (2). The forces FFF i ( t ) = − ∇∇∇ rrr i ( U l + U b + U LJ ) following from the bond ( U l ), bending ( U b ), and volumeexclusion ( U LJ ) potentials, U l = κ l N m ∑ i = ( | RRR i | − l ) , (3) U b = κ b N m − ∑ i = ( RRR i + − RRR i ) , (4) U LJ = ε ∑ i < j (cid:34)(cid:18) σ r i j (cid:19) − (cid:18) σ r i j (cid:19) + (cid:35) , r i j < √ σ , r i j > √ σ , (5)where RRR i + = rrr i + − rrr i is the bond vector, rrr i j = rrr i − rrr j the vectorbetween monomers i and j , and r i j = | rrr i j | . The energy ε mea-sures the stength of the purely repulsive potential, and σ is thediameter of a monomer. ΓΓΓ i and ˆ ηηη i are Gaussian and Markovianstochastic processes with zero mean and the second moments (cid:68) ΓΓΓ i ( t ) ΓΓΓ Tj ( t (cid:48) ) (cid:69) = k B T H − i j δ ( t − t (cid:48) ) , (6) (cid:10) ˆ η i α ( t ) ˆ η j β ( t (cid:48) ) (cid:11) = D R δ αβ δ i j δ ( t − t (cid:48) )) , (7)where ΓΓΓ Ti denotes the transpose of ΓΓΓ i , and H − i j the inverse of H i j ; T is the temperature, k B the Boltzmann constant, and D R the rotational diffusion coefficient of a spherical colloid. The tensor H i j ( rrr i j ) = δ i j I / πη l + ( − δ i j ) ΩΩΩ ( rrr i j ) accounts for hydrodynamicinteractions, with the first term including local friction, and theRotne-Prager-Yamakawa tensor ΩΩΩ (cid:0) rrr i j (cid:1) = πη r i j (cid:34) I + rrr i j rrr Ti j r i j + l r i j (cid:32) I − rrr i j rrr Ti j r i j (cid:33)(cid:35) , r i j > l πη l (cid:34)(cid:18) − r i j l (cid:19) I + r i j l rrr i j rrr Ti j r i j (cid:35) , r i j < l , (8)with the solvent viscosity η and the unit matrix I . We assume atouching bead model of spherical colloids, hence, the monomerhydrodynamic radius is half of the bond length l . The Rotne-Prager-Yamakawa tensor insures the positive definiteness of thehydrodynamic tensor even at small distances.The translational equations of motion (1) are solved via theErmark-McCammon algorithm. The procedure to solve theequations of motion (2) for the orientation vectors is described inSec. S-IV of the ESI. We characterize activity by the Péclet number Pe and the ra-tio ∆ between translational, D T = k B T / πη l , and rotational, D R ,diffusion coefficient of an isolated monomer, where Pe = v lD R , ∆ = D T l D R . (9)The coefficient κ l (Eq. (3)) for the bond strength is adjusted ac-cording to the applied Péclet number, in order to avoid bondstretching with increasing activity. By choosing κ l l / k B T = ( + Pe ) , bond-length variations are smaller than of the equi-librium value l . Furthermore, the scaled bending force coeffi-cient ˜ κ b = κ b l / k B T (Eq. (4)) is related to the polymer persistencelength, l p = / ( p ) , by pL = N m ˜ κ b ( − coth ( ˜ κ b )) + κ b ( + coth ( ˜ κ b )) − . (10)The parameters of the truncated and shifted Lennard-Jones po-tential are σ = . l and ε = k B T . We characterize the polymer conformations by the mean squareend-to-end distance. Results for phantom polymers of length L = ( N m − ) l = l and L = l are presented in Fig. 2. Evidently,ABPOs in the presence of hydrodynamic interactions exhibit apronounced shrinkage for (cid:46) Pe (cid:46) , where shrinkage dependson polymer length and is substantially stronger for longer poly-mers. Semiflexible ABPOs+HI shrink stronger than ABPOs-HI,but the effect vanishes gradually as pL → . This is a consequenceof the reduced influence of hydrodynamic interactions for ratherstiff polymers. Yet, the asymptotic swollen value for Pe → ∞ ofABPOs+HI is smaller than the value for ABPOs-HI, for which the-ory predicts L / and simulations yield approximately L / . Hence, hydrodynamic interactions affect the swelling behavior offlexible and semiflexible polymers for all Pe > . In particular, theasymptotic size (cid:104) rrr e (cid:105) ≈ L / for Pe → ∞ , which is independent of Journal Name, [year], [vol.] ,,
FFF , indicated by the red arrow, on the fluidinducing Stokes flow. The small arrows vvv n display the direction of theactive velocity. No flow field is generated by the active motion of anindividual ABP. An animation of the dynamics of a discrete ABPO+HI isprovided in the ESI. equations of motion ˙ rrr i ( t ) = v eee i ( t ) + N m ∑ j = H i j [ FFF i ( t ) + ΓΓΓ i ( t )] , (1) ˙ eee i ( t ) = ˆ ηηη i ( t ) × eee i ( t ) . (2)Here, rrr i ( t ) and ˙ rrr i ( t ) denote the position and velocity of particle i , respectively, and vvv i ( t ) = v eee i ( t ) is the active velocity with thepropulsion direction eee i ( | eee i | = ), which changes in a diffusivemanner according to Eq. (2). The forces FFF i ( t ) = − ∇∇∇ rrr i ( U l + U b + U LJ ) following from the bond ( U l ), bending ( U b ), and volumeexclusion ( U LJ ) potentials, U l = κ l N m ∑ i = ( | RRR i | − l ) , (3) U b = κ b N m − ∑ i = ( RRR i + − RRR i ) , (4) U LJ = ε ∑ i < j (cid:34)(cid:18) σ r i j (cid:19) − (cid:18) σ r i j (cid:19) + (cid:35) , r i j < √ σ , r i j > √ σ , (5)where RRR i + = rrr i + − rrr i is the bond vector, rrr i j = rrr i − rrr j the vectorbetween monomers i and j , and r i j = | rrr i j | . The energy ε mea-sures the stength of the purely repulsive potential, and σ is thediameter of a monomer. ΓΓΓ i and ˆ ηηη i are Gaussian and Markovianstochastic processes with zero mean and the second moments (cid:68) ΓΓΓ i ( t ) ΓΓΓ Tj ( t (cid:48) ) (cid:69) = k B T H − i j δ ( t − t (cid:48) ) , (6) (cid:10) ˆ η i α ( t ) ˆ η j β ( t (cid:48) ) (cid:11) = D R δ αβ δ i j δ ( t − t (cid:48) )) , (7)where ΓΓΓ Ti denotes the transpose of ΓΓΓ i , and H − i j the inverse of H i j ; T is the temperature, k B the Boltzmann constant, and D R the rotational diffusion coefficient of a spherical colloid. The tensor H i j ( rrr i j ) = δ i j I / πη l + ( − δ i j ) ΩΩΩ ( rrr i j ) accounts for hydrodynamicinteractions, with the first term including local friction, and theRotne-Prager-Yamakawa tensor ΩΩΩ (cid:0) rrr i j (cid:1) = πη r i j (cid:34) I + rrr i j rrr Ti j r i j + l r i j (cid:32) I − rrr i j rrr Ti j r i j (cid:33)(cid:35) , r i j > l πη l (cid:34)(cid:18) − r i j l (cid:19) I + r i j l rrr i j rrr Ti j r i j (cid:35) , r i j < l , (8)with the solvent viscosity η and the unit matrix I . We assume atouching bead model of spherical colloids, hence, the monomerhydrodynamic radius is half of the bond length l . The Rotne-Prager-Yamakawa tensor insures the positive definiteness of thehydrodynamic tensor even at small distances.The translational equations of motion (1) are solved via theErmark-McCammon algorithm. The procedure to solve theequations of motion (2) for the orientation vectors is described inSec. S-IV of the ESI. We characterize activity by the Péclet number Pe and the ra-tio ∆ between translational, D T = k B T / πη l , and rotational, D R ,diffusion coefficient of an isolated monomer, where Pe = v lD R , ∆ = D T l D R . (9)The coefficient κ l (Eq. (3)) for the bond strength is adjusted ac-cording to the applied Péclet number, in order to avoid bondstretching with increasing activity. By choosing κ l l / k B T = ( + Pe ) , bond-length variations are smaller than of the equi-librium value l . Furthermore, the scaled bending force coeffi-cient ˜ κ b = κ b l / k B T (Eq. (4)) is related to the polymer persistencelength, l p = / ( p ) , by pL = N m ˜ κ b ( − coth ( ˜ κ b )) + κ b ( + coth ( ˜ κ b )) − . (10)The parameters of the truncated and shifted Lennard-Jones po-tential are σ = . l and ε = k B T . We characterize the polymer conformations by the mean squareend-to-end distance. Results for phantom polymers of length L = ( N m − ) l = l and L = l are presented in Fig. 2. Evidently,ABPOs in the presence of hydrodynamic interactions exhibit apronounced shrinkage for (cid:46) Pe (cid:46) , where shrinkage dependson polymer length and is substantially stronger for longer poly-mers. Semiflexible ABPOs+HI shrink stronger than ABPOs-HI,but the effect vanishes gradually as pL → . This is a consequenceof the reduced influence of hydrodynamic interactions for ratherstiff polymers. Yet, the asymptotic swollen value for Pe → ∞ ofABPOs+HI is smaller than the value for ABPOs-HI, for which the-ory predicts L / and simulations yield approximately L / . Hence, hydrodynamic interactions affect the swelling behavior offlexible and semiflexible polymers for all Pe > . In particular, theasymptotic size (cid:104) rrr e (cid:105) ≈ L / for Pe → ∞ , which is independent of Journal Name, [year], [vol.] ,, -1 -2 -1 -1 -3 -2 -1 Fig. 2
Mean square end-to-end distance (simulations) as a function ofthe Péclet number for semiflexible polymers with (a) N m = ( L = l )monomers (bullets) and pL = × (blue), . × (green), . (red), . × − (cyan), and . × − (purple), and (b) N m = ( L = l )monomers (squares) for pL = × (blue), × (green), (red), (cyan), and − (purple). In (a), the solid lines are theoretical results forABPOs-HI, and bullets are for phantom polymers. In (b) filled squarescorrespond to phantom and open squares to self-avoiding polymers.The dashed lines are guides for the eye. See Fig. 3 for snapshots andthe ESI for a movie. stiffness, is smaller than the value for an ABPO-HI.Self-avoidance reduces the extent of shrinkage, specifically offlexible polymers. This is illustrated in Fig. 2(b). For pL (cid:38) ,the equilibrium value (cid:104) rrr e (cid:105) of a self-avoiding polymer is swollencompared to a phantom polymer. Such an ABPO+HI exhibitsa less pronounced shrinkage for all polymer lengths. Natu-rally, excluded-volume effects vanish with decreasing pL , and for pL < there is hardly any difference between a phantom and aself-avoiding polymer. Moreover, the swelling behavior with andwithout excluded-volume interactions is rather similar in the limit Pe (cid:29) . Interestingly, phantom and self-avoiding polymers showa universal dependence on Pe as they start to swell. Here, ac-tive forces exceed both, excluded-volume interactions and bend-ing forces. As predicted by theory (cf. Sec. 3.3), the internaldynamics is determined by the modes of a flexible polymer, i.e.,intermolecular tension, in this regime. Snapshots of conforma-tions with and without HI are presented in Fig. 3. Fig. 3
Configurations of flexible phantom ABPOs of length N m = forthe Péclet numbers Pe = (top) and (bottom) in the presence(green) and absence (red) of hydrodynamic interactions. A movie of anABPO+HI is available at the ESI. The dynamics of ABPO+HI is characterized by the monomermean square displacement (MSD) averaged over all monomers (cid:104) ∆ rrr ( t ) (cid:105) = ∑ i (cid:104) ( rrr i ( t ) − rrr i ( )) (cid:105) / N m . Figure 4 shows MSDs of a poly-mer with N m = monomers for various Péclet numbers. A pas-sive polymer exhibits the well-known Zimm behavior, with thetime dependence t / of the MSD in the center-of-mass referenceframe for t / τ Z (cid:28) . At long times t / ˜ τ (cid:29) , the center-of-massdisplacement dominates the monomer MSD for all Péclet num-bers. Here, we find the HI-independent MSD (cid:104) rrr cm (cid:105) = v lt / γ R L following from Eq. (1) for Pe (cid:29) (see also Eq. (46)). For Pécletnumbers Pe > , the active ballistic regime, (cid:104) ∆ rrr (cid:105) ∼ t , is presentat short times ( γ R t , t / ˜ τ < ) . Moreover, for t / ˜ τ (cid:38) and moder-ate Péclet numbers, Pe ≈ , activity implies a polymer-specificregime, where the monomer MSD exhibits a power-law depen-dence (cid:104) ∆ rrr ( t ) (cid:105) ∼ t α (cid:48) , with an exponent of α (cid:48) ≈ / , a value smallerthan the exponent α (cid:48) = / of the Zimm dynamics. The reductionof the exponent is a clear consequence of the coupling betweenhydrodynamics and activity, since ABPOs-HI always display slopes α (cid:48) (cid:38) / , where α (cid:48) = / is the value of a passive flexible polymer(Rouse model). However, this regime appears as a crossoverfrom the ballistic to the diffusive regime. Nevertheless, it is aconsequence of hydrodynamics with a sub-diffusive motion. Thepolymer-specific regime vanishes gradually with increasing Pe . Asdiscussed in Sec. 3.3, this is a consequence of the decreasing poly-mer relaxation times with increasing activity. The simulations of the previous section yield a surprising shrink-age of even flexible polymers by hydrodynamic interactions. Inorder to shed light on the underlying mechanisms, we study anmean-field analytical model, where an active polymer is describedby a continuous Gaussian semiflexible polymer model.
Thisapproach has been applied successfully in the analysis the prop-erties of ABPO-HI in close quantitative agreement with sim-ulations. Journal Name, [year], [vol.] , -1 -4 -3 -2 -1 Fig. 4
Mean square displacement of a flexible phantom polymer with N m = ( pL = ) monomers for the Péclet numbers Pe = (blue), (green), (red), and (cyan). The time is scaled by the factor γ R = D R . The solid lines indicate the monomer MSD and the dashedlines the MSD in the polymer center-of-mass reference frame. The blacklines are guides for the eye correspond to a power-law fit of the data inthe respective regime. The polymer is considered as a differentiable space curve rrr ( s , t ) oflength L , with contour coordinate s ( − L / (cid:54) s (cid:54) L / ) and time t .Activity is introduced in analogy to an active Ornstein-Uhlenbeckparticle (AOUP) by assigning a propulsion velocity vvv ( s , t ) toevery point rrr ( s , t ) (cf. Fig. 1) , which changes in an inde-pendent manner. The equation of motion is then given by theLangevin equation ∂ rrr ( s , t ) ∂ t = vvv ( s , t ) + (cid:90) L / − L / ds (cid:48) H ( rrr ( s ) , rrr ( s (cid:48) )) (11) × (cid:20) ν k B T ∂ rrr ( s (cid:48) , t ) ∂ s (cid:48) − ε k B T ∂ rrr ( s (cid:48) , t ) ∂ s (cid:48) + ΓΓΓ ( s (cid:48) , t ) (cid:21) , with boundary conditions for free ends as specified in Refs.42,44,79. The tensor H ( rrr ( s ) , rrr ( s (cid:48) )) accounts for hydrodynamic in-teractions; it is defined as H ( rrr ( s ) , rrr ( s (cid:48) )) = ΩΩΩ ( rrr ( s ) − rrr ( s (cid:48) )) + I δ ( s − s (cid:48) ) / πη , where the second term on the right hand side describesthe local friction, and ΩΩΩ ( ∆ rrr ) = πη | ∆ rrr | (cid:18) I + ∆ rrr ⊗ ∆ rrr | ∆ rrr | (cid:19) (12)is the Oseen tensor . The terms in Eq. (11) with the secondand forth derivative capture chain flexibility, i.e., chain entropy,and bending forces, respectively. The Lagrangian multipliers ν ( s ) and ν = ν ( ± L / ) account for the inextensibility of the polymer(we will denote ν as stretching coefficient in the following), and ε characterizes the bending stiffness . For a polymer in threedimensions, previous studies yield ε = / p and ν = / , where p = / l p and l p is the persistence length . Adopting a mean-field approach, the stretching coefficient ν is independent of s and is determined by the global constraint (cid:90) L / − L / (cid:42)(cid:18) ∂ rrr ( s ) ∂ s (cid:19) (cid:43) ds = L . (13)The stochastic force ΓΓΓ ( s , t ) is assumed to be stationary, Marko-vian, and Gaussian. Within the AOUP description of the analytical calculations, theactive velocity vvv ( s , t ) is a non-Markovian but Gaussian stochasticprocess with zero mean and the correlation function (cid:10) vvv ( s , t ) · vvv ( s (cid:48) , t (cid:48) ) (cid:11) = v le − γ R | t − t (cid:48) | δ ( s − s (cid:48) ) . (14)Here, v is the constant propulsion velocity and γ R character-izes the decay of the velocity correlation function. For a spher-ical colloid in solution, the relation γ R = D R applies, where D R is the rotational diffusion coefficient. The correlation function(14) emerges due to a diffusive motion of either the Ornstein-Uhlenbeck process for the active velocity, or by the change of thepropulsion direction (unit vector) of an ABP. . Since only firstand second moments of the active velocity are required for thecurrent analytical studies, the results are independent of the un-derlying active velocity dynamics of an active site—either AOUPor ABP. Further details on the derivation of the equations of mo-tion are presented in Ref. 42, including a discussion of the factor l in Eq. (14).Self-propelled systems are force and torque free . Hence,only conservative and random forces give rise to Stockeslet-typehydrodynamic interactions in Eq. (11). However, we neglectforce-dipole, source-dipole, and higher multipole flow field con-tributions, as they decay as O ( r − ) with distance compared to a / r decay of the Stokeslet flow field . The hydrodynamic tensor renders Eq. (11) a nonlinear and non-local equation of motion. In order to obtain an (approximate) an-alytical solution, we apply the preaveraging approximation orig-inally proposed by Zimm , where the hydrodynamic tensoris replaced by its average over the stationary state distributionfunction, i.e., H ( rrr ( s ) − rrr ( s (cid:48) )) → (cid:104) H ( rrr ( s ) − rrr ( s (cid:48) )) (cid:105) = H ( s , s (cid:48) ) . Hence,Eq. (11) turns into a linear equation—an Ornstein-Uhlenbeckprocess—with a Gaussian stationary-state distribution functionfor the distance ∆ rrr ( s , s (cid:48) ) = rrr ( s ) − rrr ( s (cid:48) ) of the form Ψ ( ∆ rrr ) = (cid:18) π a ( s , s (cid:48) ) (cid:19) / exp (cid:18) − ∆ rrr a ( s , s (cid:48) ) (cid:19) , (15)with a ( s , s (cid:48) ) = (cid:10) ( rrr ( s ) − rrr ( s (cid:48) )) (cid:11) . Note that a ( s , s (cid:48) ) is not necessar-ily a function of s − s (cid:48) only. The explicit form of a ( s , s ) will bediscussed later. Preaveraging yields ΩΩΩ ( s , s (cid:48) ) = Θ ( | s − s (cid:48) | − l πη (cid:115) π a ( s , s (cid:48) ) I = Ω ( s , s (cid:48) ) I , (16) Journal Name, [year], [vol.] ,,
Mean square displacement of a flexible phantom polymer with N m = ( pL = ) monomers for the Péclet numbers Pe = (blue), (green), (red), and (cyan). The time is scaled by the factor γ R = D R . The solid lines indicate the monomer MSD and the dashedlines the MSD in the polymer center-of-mass reference frame. The blacklines are guides for the eye correspond to a power-law fit of the data inthe respective regime. The polymer is considered as a differentiable space curve rrr ( s , t ) oflength L , with contour coordinate s ( − L / (cid:54) s (cid:54) L / ) and time t .Activity is introduced in analogy to an active Ornstein-Uhlenbeckparticle (AOUP) by assigning a propulsion velocity vvv ( s , t ) toevery point rrr ( s , t ) (cf. Fig. 1) , which changes in an inde-pendent manner. The equation of motion is then given by theLangevin equation ∂ rrr ( s , t ) ∂ t = vvv ( s , t ) + (cid:90) L / − L / ds (cid:48) H ( rrr ( s ) , rrr ( s (cid:48) )) (11) × (cid:20) ν k B T ∂ rrr ( s (cid:48) , t ) ∂ s (cid:48) − ε k B T ∂ rrr ( s (cid:48) , t ) ∂ s (cid:48) + ΓΓΓ ( s (cid:48) , t ) (cid:21) , with boundary conditions for free ends as specified in Refs.42,44,79. The tensor H ( rrr ( s ) , rrr ( s (cid:48) )) accounts for hydrodynamic in-teractions; it is defined as H ( rrr ( s ) , rrr ( s (cid:48) )) = ΩΩΩ ( rrr ( s ) − rrr ( s (cid:48) )) + I δ ( s − s (cid:48) ) / πη , where the second term on the right hand side describesthe local friction, and ΩΩΩ ( ∆ rrr ) = πη | ∆ rrr | (cid:18) I + ∆ rrr ⊗ ∆ rrr | ∆ rrr | (cid:19) (12)is the Oseen tensor . The terms in Eq. (11) with the secondand forth derivative capture chain flexibility, i.e., chain entropy,and bending forces, respectively. The Lagrangian multipliers ν ( s ) and ν = ν ( ± L / ) account for the inextensibility of the polymer(we will denote ν as stretching coefficient in the following), and ε characterizes the bending stiffness . For a polymer in threedimensions, previous studies yield ε = / p and ν = / , where p = / l p and l p is the persistence length . Adopting a mean-field approach, the stretching coefficient ν is independent of s and is determined by the global constraint (cid:90) L / − L / (cid:42)(cid:18) ∂ rrr ( s ) ∂ s (cid:19) (cid:43) ds = L . (13)The stochastic force ΓΓΓ ( s , t ) is assumed to be stationary, Marko-vian, and Gaussian. Within the AOUP description of the analytical calculations, theactive velocity vvv ( s , t ) is a non-Markovian but Gaussian stochasticprocess with zero mean and the correlation function (cid:10) vvv ( s , t ) · vvv ( s (cid:48) , t (cid:48) ) (cid:11) = v le − γ R | t − t (cid:48) | δ ( s − s (cid:48) ) . (14)Here, v is the constant propulsion velocity and γ R character-izes the decay of the velocity correlation function. For a spher-ical colloid in solution, the relation γ R = D R applies, where D R is the rotational diffusion coefficient. The correlation function(14) emerges due to a diffusive motion of either the Ornstein-Uhlenbeck process for the active velocity, or by the change of thepropulsion direction (unit vector) of an ABP. . Since only firstand second moments of the active velocity are required for thecurrent analytical studies, the results are independent of the un-derlying active velocity dynamics of an active site—either AOUPor ABP. Further details on the derivation of the equations of mo-tion are presented in Ref. 42, including a discussion of the factor l in Eq. (14).Self-propelled systems are force and torque free . Hence,only conservative and random forces give rise to Stockeslet-typehydrodynamic interactions in Eq. (11). However, we neglectforce-dipole, source-dipole, and higher multipole flow field con-tributions, as they decay as O ( r − ) with distance compared to a / r decay of the Stokeslet flow field . The hydrodynamic tensor renders Eq. (11) a nonlinear and non-local equation of motion. In order to obtain an (approximate) an-alytical solution, we apply the preaveraging approximation orig-inally proposed by Zimm , where the hydrodynamic tensoris replaced by its average over the stationary state distributionfunction, i.e., H ( rrr ( s ) − rrr ( s (cid:48) )) → (cid:104) H ( rrr ( s ) − rrr ( s (cid:48) )) (cid:105) = H ( s , s (cid:48) ) . Hence,Eq. (11) turns into a linear equation—an Ornstein-Uhlenbeckprocess—with a Gaussian stationary-state distribution functionfor the distance ∆ rrr ( s , s (cid:48) ) = rrr ( s ) − rrr ( s (cid:48) ) of the form Ψ ( ∆ rrr ) = (cid:18) π a ( s , s (cid:48) ) (cid:19) / exp (cid:18) − ∆ rrr a ( s , s (cid:48) ) (cid:19) , (15)with a ( s , s (cid:48) ) = (cid:10) ( rrr ( s ) − rrr ( s (cid:48) )) (cid:11) . Note that a ( s , s (cid:48) ) is not necessar-ily a function of s − s (cid:48) only. The explicit form of a ( s , s ) will bediscussed later. Preaveraging yields ΩΩΩ ( s , s (cid:48) ) = Θ ( | s − s (cid:48) | − l πη (cid:115) π a ( s , s (cid:48) ) I = Ω ( s , s (cid:48) ) I , (16) Journal Name, [year], [vol.] ,, nd the hydrodynamic tensor becomes H ( s , s (cid:48) ) = (cid:20) δ ( s − s (cid:48) ) πη + Ω ( s , s (cid:48) ) (cid:21) I = H ( s , s (cid:48) ) I . (17)The Heaviside step function, Θ ( x ) , in Eq. (16) introduces a lowercut-off, which we choose as l . In a touching bead model of apolymer, l is the bead diameter and, hence, the polymer thickness.The preaveraging approximation has very successfully been ap-plied to describe the dynamics of DNA and semiflexible poly-mers. Even quantitative agreement between analytical theoryand simulations of the full hydrodynamic contribution of ratherstiff polymers is achieved, as well as with measurements onDNA. This demonstrates that preaveraging is also suitable forrather stretched polymers, it, however, fails for rodlike objects. The final linear equation is solved by the eigenfunction expansion rrr ( s , t ) = ∞ ∑ n = χχχ n ( t ) ϕ n ( s ) (18)in terms of the eigenfunctions ϕ n of the equation ε k B T d ds ϕ n ( s ) − ν k B T d ds ϕ n ( s ) = ξ n ϕ n ( s ) , (19)with the eigenvalues ( n ∈ N ) ξ n = k B T (cid:16) εζ n + νζ n (cid:17) . (20)The mode numbers ζ n follow from the boundary conditions. Therespective eigenfunctions and eigenvalues are explicitly presentedin Refs. 42,44,79. Specifically, in the limit of a flexible polymer, pL (cid:29) , the eigenfunctions are ϕ = (cid:114) L , (21) ϕ n ( s ) = (cid:114) L sin (cid:16) n π sL (cid:17) , ∀ n odd (22) ϕ n ( s ) = (cid:114) L cos (cid:16) n π sL (cid:17) , ∀ n even , (23)with the wave numbers ζ n = n π / L and the eigenvalues ξ n = ν k B T π n / L .The equation of motion (11) yields the Langevin equation forthe mode amplitudes, χχχ n ( t ) , d χχχ m ( t ) dt = ∞ ∑ n = H mn [ ΓΓΓ n ( t ) − ξ n χχχ n ( t )] + vvv m ( t ) . (24)The mode representation of the hydrodynamic tensor is H nm =( δ nm + πη Ω nm ) / πη , with the preaveraged Oseen tensor Ω nm . The second moments of the stochastic-force amplitudes
ΓΓΓ n aregiven by (cid:104) Γ n α ( t ) Γ m β ( t (cid:48) ) (cid:105) = k B T δ αβ δ ( t − t (cid:48) ) H − nm , (25)with α , β ∈ { x , y , z } . The mode representation of the correlation function (14) of the active velocity is (cid:104) vvv n ( t ) · vvv m ( t (cid:48) ) (cid:105) = v le − γ R | t − t (cid:48) | δ nm . (26)In Eq. (24), all modes are coupled in general and the set ofequations can only be solved numerically. To arrive at an ana-lytical solution, we neglect the off-diagonal terms of the hydro-dynamic mode tensor H nm , which leads to the decoupled equa-tions d χχχ n ( t ) dt = − τ n χχχ n + H nn ΓΓΓ n ( t ) + vvv n ( t ) , (27)with the relaxation times ˜ τ n = H nn ξ n = τ n + πη Ω nn , (28)and τ n = πη / ξ n , the relaxation times in absenceof hydrodynamic interactions; for flexible polymers τ n = η L / ( ν k B T π n ) . The stationary-state solution of Eq. (24) is χχχ n ( t ) = (cid:90) t − ∞ dt (cid:48) e − ( t − t (cid:48) ) / ˜ τ n (cid:2) vvv n ( t (cid:48) ) + H nn ΓΓΓ n ( t (cid:48) ) (cid:3) (29)for n > , and for n = the solution is χχχ ( t ) = χχχ ( ) + (cid:90) t dt (cid:48) (cid:104) vvv ( ) ( t (cid:48) ) + H ΓΓΓ ( t (cid:48) ) (cid:105) . (30) With the correlation functions of the stochastic forces (25) andvelocities (26), the correlation functions of the mode amplitudesbecome ( t ≥ t (cid:48) ) (cid:10) χχχ n ( t ) · χχχ m ( t (cid:48) ) (cid:11) = δ nm (cid:18) k B T τ n πη e −| t − t (cid:48) | / ˜ τ n (31) + v l ˜ τ n − ( γ R ˜ τ n ) (cid:104) e − γ R | t − t (cid:48) | − γ R ˜ τ n e −| t − t (cid:48) | / ˜ τ n (cid:105)(cid:33) , (cid:10) χχχ ( t ) · χχχ ( t (cid:48) ) (cid:11) = (cid:68) χχχ ( ) (cid:69) + k B T H t (cid:48) (32) + v l γ R (cid:104) γ R t (cid:48) − − e γ R ( t (cid:48) − t ) + e − γ R t + e − γ R t (cid:48) (cid:105) . The eigenfunction expansion (18) and the correlation functions(31) permit us to calculate the mean square distance a ( s , s (cid:48) ) . Ex-plicitly, we find a ( s , s (cid:48) ) = ∞ ∑ n = (cid:68) χχχ n (cid:69) ( ϕ n ( s ) − ϕ n ( s (cid:48) )) , (33)with the stationary-state average (cid:68) χχχ n (cid:69) = k B T τ n πη + v l ˜ τ n + γ R ˜ τ n . (34)As for passive polymers, the relaxation behavior (31) is de-termined by hydrodynamics. Remarkably, however, the mode-amplitude correlation functions (34) depend on the hydrody-namic interactions via the relaxation times (28). Thus, HI Journal Name, [year], [vol.] , hanges, additionally to the dynamics, also the stationary-stateconformational properties of an active polymer. In order to determine the relaxation times ˜ τ n , the double integral Ω nn = (cid:115) π η (cid:90) L / − L / (cid:90) L / − L / Θ ( | s − s (cid:48) | − l c ) ϕ n ( s ) ϕ n ( s (cid:48) ) (cid:112) a ( s , s (cid:48) ) ds (cid:48) ds (35)needs to be evaluated, which itself depends via a ( s , s (cid:48) ) on theOseen tensor Ω nn . Hence, the equation has to be solved in an it-erative and self-consistent manner, where the double integration,combined with the summation of Eq. (33), constitutes a majorcomputational challenge. To arrive at a more easily tractableexpression with a single integral, we apply standard approxi-mations for the integrals over the functions ϕ n in Eq. (35) as,e.g., described in Ref. 69 for a flexible polymer. For a pas-sive semiflexible polymer, a ( s , s (cid:48) ) is only a function of the dif-ference | s − s (cid:48) | . This is no longer the case in the presenceof activity, where a ( s , s (cid:48) ) depends on s − s (cid:48) and s + s (cid:48) in gen-eral. In fact, an analytical expression of a for a flexible ABPO-HI can be calculated by performing the sum in Eq. (33). To ob-tain an approximate expression, which depends on the difference | s − s (cid:48) | only, we replace the difference of the eigenfunctions inEq. (33) by the expression approximately valid for a passive poly-mer, namely ϕ n ( s ) − ϕ n ( s (cid:48) ) = ( n π ( s − s (cid:48) ) / L ) for n odd, and ϕ n ( s ) − ϕ n ( s (cid:48) ) = for n even. As a result, we obtain a ( s ) = L ∑ n , odd (cid:32) k B T τ n πη + v l ˜ τ n + γ R ˜ τ n (cid:33) sin (cid:16) n π L s (cid:17) , (36)and, hence, Ω nn is given by Ω nn = (cid:114) π η L (cid:90) Ll c L − s (cid:112) a ( s ) cos (cid:16) n π L s (cid:17) ds . (37)Aside from the distance a ( s − s (cid:48) ) , which depends on activity viathe relaxation times, this expression is identical with that of apassive polymer. As shown in Sec. S-I of the ESI, the approxi-mations of Eqs. (36) capture the dependence of a on the contourcoordinate well, the better the larger the Péclet number.Assuming a linear dependence of a ( s − s (cid:48) ) on | s − s (cid:48) | , i.e., a ( s ) = a | s | L , as for a passive flexible polymer , we obtain theanalytical solution of Eq. (37), Ω nn = √ π η a √ n , (38)in analogy to the Zimm approach . Choosing for a theresult of a flexible ABPO-HI, namely a = / µ pL + Pe / µ pL ∆ ,where the Péclet number Pe and ∆ are defined in Eq. 9, and µ isgiven by µ = ν / ( p ) , we obtain Ω nn ∼ √ pL µ Pe √ n (39)for Pe (cid:29) .In the following, when not indicated otherwise, the approxi- -1 Fig. 5
Normalized stretching coefficient µ = ν / ( p ) , solution ofEq. (40) , as function of the Péclet number Pe for flexible polymers with pL = , , and without HI (green) and with HI (blue). The resultsare independent of polymer length. mate expressions (36) and (37) are used for the calculation ofthe Oseen tensor. Moreover, we use ∆ = / , the value of a spher-ical colloid of diameter l in solution. The stretching coefficient and relaxation times are interdepen-dent and need to be determined simultaneously. Due to nonlin-earities, specifically in Ω nn , the respective quantities can only bedetermined numerically.We focus here on flexible polymers, where pL (cid:29) , and we set L / l = pL . Then, in terms of the eigenfunction expansion (18),Eq. (13) for the stretching coefficient ν , respectively µ = ν / p ,becomes ∞ ∑ n = (cid:34) k B T τ n πη + v l ˜ τ n + γ R ˜ τ n (cid:35) ζ n = L , (40)with the relaxation times (Eq. (28)) ˜ τ n = τ R µ n ( + πη Ω nn ) , (41)where τ R ≡ η L / ( π k B T p ) is the Rouse relaxation time . Inthe non-hydrodynamic case, i.e., Ω nn = , we find the asymptoticsolution µ = Pe / / ∆ of Eq. (40), independent of pL . For therelaxation times, we recover the Zimm behavior ˜ τ n = τ Z / n / at Pe = , with the longest relaxation time τ Z = η ( L / p ) / / ( √ π k B T ) (Zimm relaxation time).The scaled stretching coefficient, µ , is presented in Fig. 5 as afunction of the Péclet number. For the considered polymer lengthsand stiffness, µ is independent of pL . Moreover, it increases ap-proximately linearly with Péclet number for Pe (cid:38) , somewhatweaker than µ of comparable passive polymers. Hence, hydrody-namics modifies the stretching coefficient.Figure 6 displays longest relaxations times, ˜ τ , as function of Pe . For < Pe (cid:46) , hydrodynamic interactions enhance the de- Journal Name, [year], [vol.] ,,
Normalized stretching coefficient µ = ν / ( p ) , solution ofEq. (40) , as function of the Péclet number Pe for flexible polymers with pL = , , and without HI (green) and with HI (blue). The resultsare independent of polymer length. mate expressions (36) and (37) are used for the calculation ofthe Oseen tensor. Moreover, we use ∆ = / , the value of a spher-ical colloid of diameter l in solution. The stretching coefficient and relaxation times are interdepen-dent and need to be determined simultaneously. Due to nonlin-earities, specifically in Ω nn , the respective quantities can only bedetermined numerically.We focus here on flexible polymers, where pL (cid:29) , and we set L / l = pL . Then, in terms of the eigenfunction expansion (18),Eq. (13) for the stretching coefficient ν , respectively µ = ν / p ,becomes ∞ ∑ n = (cid:34) k B T τ n πη + v l ˜ τ n + γ R ˜ τ n (cid:35) ζ n = L , (40)with the relaxation times (Eq. (28)) ˜ τ n = τ R µ n ( + πη Ω nn ) , (41)where τ R ≡ η L / ( π k B T p ) is the Rouse relaxation time . Inthe non-hydrodynamic case, i.e., Ω nn = , we find the asymptoticsolution µ = Pe / / ∆ of Eq. (40), independent of pL . For therelaxation times, we recover the Zimm behavior ˜ τ n = τ Z / n / at Pe = , with the longest relaxation time τ Z = η ( L / p ) / / ( √ π k B T ) (Zimm relaxation time).The scaled stretching coefficient, µ , is presented in Fig. 5 as afunction of the Péclet number. For the considered polymer lengthsand stiffness, µ is independent of pL . Moreover, it increases ap-proximately linearly with Péclet number for Pe (cid:38) , somewhatweaker than µ of comparable passive polymers. Hence, hydrody-namics modifies the stretching coefficient.Figure 6 displays longest relaxations times, ˜ τ , as function of Pe . For < Pe (cid:46) , hydrodynamic interactions enhance the de- Journal Name, [year], [vol.] ,, -3 -2 -1 Fig. 6
The longest polymer relaxation time ˜ τ , Eq. (41) , normalized bythe corresponding passive value ˜ τ as function of the Péclet number Pe for flexible polymers with pL = (solid), (dashed), and (dotted). The green line shows the result of an active polymer inabsence of HI, where τ ∼ Pe − / . cay of the relaxation time with increasing Pe compared to the non-hydrodynamic case, specifically for pL > . Note that µ ∼ Pe , in-dependent of the polymer length in the considered length regime.This is a consequence of an increase of Ω nn with increasing Pe (cf.Sec. S-I. of ESI). In contrast, a slower decay of ˜ τ is obtainedfor Pe > . Here, we find a strong polymer-length dependence,which is related to particular values of Ω nn (cf. Fig. S.2). The ap-proximation (39) yields the relation ˜ τ ∼ / √ Pe for πη Ω nn (cid:29) ,which describes the Péclet number dependence well in the inter-val < Pe < for pL = . As shown in Fig. S.2 for shorterpolymers, Ω nn varies more slowly with Pe and, hence, ˜ τ decaysfaster with increasing Pe . In the limit Pe → ∞ , Ω nn becomes verysmall and the contribution to the relaxation times vanishes gradu-ally. Hence, ˜ τ approaches the asymptotic dependence ˜ τ ∼ / Pe ,determined by µ .Results on the mode-number dependence of the relaxationstimes for various Pe are presented in Sec. S-III of the ESI. Theintricate dependence of Ω nn on the relaxation times poses a ma-jor challenge for an (approximate) analytical solution, a problemwe were not able to overcome so far. The conformational properties of a polymer are charac-terized by the mean square end-to-end distance (cid:10) rrr e (cid:11) = (cid:10) ( rrr ( L / ) − rrr ( − L / )) (cid:11) , which is given by (cid:68) rrr e (cid:69) = L ∑ n , odd (cid:32) k B T τ n πη + v l ˜ τ n + γ R ˜ τ n (cid:33) (42)in terms of the mode amplitudes of Eq. (34). Numerical resultsfor (cid:10) rrr e (cid:11) of flexible polymers ( pL (cid:29) ) are shown in Fig. 7 forvarious polymer lengths. Starting from the equilibrium value (cid:10) rrr e (cid:11) = L / p at Pe = , ABPOs+HI first shrink with increasing activ-ity and then swell for higher Pe (solid lines), in qualitative agree- -1 -2 -1 Fig. 7
Polymer mean square end-to-end distance (cid:104) rrr e (cid:105) , Eq. (42) , as afunction of the Péclet number Pe for flexible ABPOs+HI of length pL = × (orange), (green), (yellow), and (magenta). Theblue lines correspond to a free-draining flexible polymer with pL = .The dotted curves represent the contribution with the relaxation times τ n and the dashed-dotted curves that with v of Eq. (42) , respectively. ment with the simulation results of Sec. 2.2. In the asymptoticlimit Pe → ∞ , a limiting value (cid:10) rrr e (cid:11) < L is assumed. Thereby,the shrinkage strongly depends on the polymer length and ismore pronounced for longer polymers. As shown in Fig. 7, flex-ible ABPO-HI exhibit a drastically different behavior and swellmonotonically with increasing activity. The reason for the quali-tatively different conformational properties rests on the differentpolymer-length dependence of the Rouse and Zimm relaxationtimes, where τ R / τ Z ≈ √ pL . Hence, in the presence of hydro-dynamic interactions, relaxation times are shorter by the factor / √ pL , which can be orders of magnitude for long flexible poly-mers.Hydrodynamic interactions lead to a polymer-length depen-dence of the swelling with increasing Péclet number ( Pe (cid:28) ∞ ), asshown in Fig. 7. For polymer lengths in the range pL ≈ − , Ω nn depends only weakly on the mode number (cf. Fig. S.2),hence, replacement of the relaxation times ˜ τ n by the relaxationtimes τ Z / n / yields (cid:68) rrr e (cid:69) = Lp µ (cid:20) + Pe ( π ) / ∆ √ pL (cid:18) − √ ζ ( / ) (cid:19)(cid:21) , (43)in the limit γ R ˜ τ n (cid:29) , where ζ ( x ) is Riemann’s zeta function( ζ ( / ) ≈ . ). In the limit of very large pL and Pe (cid:29) , at leastin the vicinity of pL = , ˜ τ n can be approximated by Ω nn ofEq. (39), which yields (cid:68) rrr e (cid:69) ∼ Lp µ Pe √ µ pL . (44)Thus, we find the same dependence on pL for both, small andlarge pL , and (cid:104) rrr e (cid:105) / L decreases as / ( pL ) / . The dependenceon Pe changes from (cid:10) rrr e (cid:11) ∼ Pe for pL ≈ − to (cid:104) rrr e (cid:105) ∼ Pe / for pL ≈ , because µ ≈ Pe . For an ABPO-HI, we found instead (cid:104) rrr e (cid:105) ∼ LPe / / p , since for such a polymer µ ∼ Pe / . Hence, Journal Name, [year], [vol.] , ydrodynamic interactions lead to a qualitative different Pe de-pendence.Figure 7 shows the individual contributions to (cid:10) rrr e (cid:11) —the termwith the relaxation times τ n (dotted lines) and that with v (dashed-dotted lines) in Eq. (42), respectively. The initial shrink-age of (cid:104) rrr e (cid:105) with increasing Pe is caused by the decreasing re-laxation times τ n ∼ / µ with increasing activity. In the ther-mal contribution of Eq. (43), (cid:104) rrr e (cid:105) ∼ L / ( p µ ) , the stretching co-efficient µ increases with increasing activity, which formally im-plies a decreasing persistence length below the value of a passivepolymer, corresponding to more compact conformations than ofthe passive case. The v -dependent term causes a swelling ofthe polymer. For an ABPO-HI, the competing effects lead to anoverall swelling, since swelling exceeds shrinkage. In case of anABPO+HI, swelling is weaker due to fluid-induced collective mo-tion (cf. Fig. 1) compared to the random motion of an ABPO-HI,and (cid:104) rrr e (cid:105) assumes a minimum. Mathematically, this is reflectedby the shorter relaxation times ˜ τ n compared to τ n . Hydrodynamicinteractions accelerate the polymer dynamics and higher Pe arerequired to achieve a significant swelling of an ABPO+HI.The exponents of the (approximate) power-law regimes for thevarious pL values approximately exhibit the above predicted scal-ing relations with respect to Pe ( Pe > ). The shift of the dashed-dotted curves in Fig. 7 to smaller (cid:10) rrr e (cid:11) with increasing polymerlength, pL , reflects the discussed decrease in relaxation times byhydrodynamics.Figure 8 shows a comparison of analytical and simulation re-sults. We find good agreement for short polymers ( N m = ), buttheory yields a less pronounced shrinkage for the longer poly-mers. We like to emphasize that for an ABPO-HI the theoreticalapproach reproduces the simulation data very well. The rea-son of the discrepancy is not evident, but is related to the appliedapproximations, which seem to underestimate hydrodynamic ef-fects. We speculate that the preaveraging approximation may fail,because active fluctuations could be large and the replacement of H ( rrr − rrr (cid:48) ) by (cid:104) H ( rrr − rrr (cid:48) ) (cid:105) no longer be justified. Yet, the analyti-cal expression captures the qualitative behavior, and even morequantitatively the swelling behavior at large Pe is reasonably wellreproduced, although the asymptotic value for Pe → ∞ is some-what overestimated due to the applied mean-field approximationof the bond-length constraint. The dynamics of the polymers is characterized by the site meansquare displacement (MSD) averaged over the polymer contour, (cid:104) ∆ rrr ( t ) (cid:105) = (cid:82) (cid:104) ( rrr ( s , t ) − rrr ( s , )) (cid:105) ds / L , which yields (cid:68) ∆ rrr ( t ) (cid:69) = (cid:68) ∆ rrr cm ( t ) (cid:69) + (cid:68) ∆ rrr ( t ) (cid:69) + (cid:68) ∆ rrr a ( t ) (cid:69) , (45)with the center-of-mass mean square displacement (cid:68) ∆ rrr cm ( t ) (cid:69) = k B TL H t + v l γ R L (cid:0) γ R t − + e − γ R t (cid:1) , (46) -1 Fig. 8
Mean square end-to-end distance as a function of the Pécletnumber of flexible ABPOs+HI for the monomer number N m = ( pL = ) (blue) and N m = ( pL = ) (green). Solid lines correspondto analytical and symbols to simulation results. H = ( + πη Ω ) / ( πη ) , the activity-modified equilibriuminternal-dynamics contribution (cid:68) ∆ rrr ( t ) (cid:69) = L ∞ ∑ n = k B T τ n γ (cid:16) − e − t / ˜ τ n (cid:17) , (47)and the active contribution (cid:68) ∆ rrr a ( t ) (cid:69) = L ∞ ∑ n = v l ˜ τ n + γ R ˜ τ n (cid:32) − e − γ R t − γ R ˜ τ n e − t / ˜ τ n − γ R ˜ τ n (cid:33) . (48)Remarkably, in the center-of-mass MSD only the thermal con-tribution includes hydrodynamics, via H , which depends onactivity through µ , whereas the active term is identical withthat of an ABPO-HI . The reason is that swimming isforce free and no Stokeslet is present. Within the approximation a ( s , s (cid:48) ) ≈ a ( s − s (cid:48) ) , Eq. (37) yields Ω = √ π η a . (49)Hence, for πη Ω (cid:29) , the thermal center-of-mass diffusion co-efficient D = k B T Ω / L increases somewhat due to activity in therange (cid:46) Pe (cid:46) , and decreases for higher Pe (cf. Fig. S.2).In the asymptotic limit Pe → ∞ , the active polymer is stretched,and the hydrodynamic contribution to thermal diffusion de-creases (cf. Fig. S.2 of ESI). As for a passive, rodlike poly-mer , asymptotically hydrodynamic interactions yield onlysmall corrections with respect to the polymer-length dependenceof a non-hydrodynamic (free-draining) polymer.Figure 9 displays the average site mean square displacementfor various Péclet numbers. For a passive flexible polymer, werecover the well-known Zimm behavior, with (cid:104) ∆ rrr ( t ) (cid:105) ∼ t / for t / τ Z (cid:28) , and a crossover to free diffusion for t / τ Z (cid:29) . Inthe presence of activity, four time regimes can be identified • t → ∞ — The MSD is dominated by the linear time dependence Journal Name, [year], [vol.] ,,
Mean square end-to-end distance as a function of the Pécletnumber of flexible ABPOs+HI for the monomer number N m = ( pL = ) (blue) and N m = ( pL = ) (green). Solid lines correspondto analytical and symbols to simulation results. H = ( + πη Ω ) / ( πη ) , the activity-modified equilibriuminternal-dynamics contribution (cid:68) ∆ rrr ( t ) (cid:69) = L ∞ ∑ n = k B T τ n γ (cid:16) − e − t / ˜ τ n (cid:17) , (47)and the active contribution (cid:68) ∆ rrr a ( t ) (cid:69) = L ∞ ∑ n = v l ˜ τ n + γ R ˜ τ n (cid:32) − e − γ R t − γ R ˜ τ n e − t / ˜ τ n − γ R ˜ τ n (cid:33) . (48)Remarkably, in the center-of-mass MSD only the thermal con-tribution includes hydrodynamics, via H , which depends onactivity through µ , whereas the active term is identical withthat of an ABPO-HI . The reason is that swimming isforce free and no Stokeslet is present. Within the approximation a ( s , s (cid:48) ) ≈ a ( s − s (cid:48) ) , Eq. (37) yields Ω = √ π η a . (49)Hence, for πη Ω (cid:29) , the thermal center-of-mass diffusion co-efficient D = k B T Ω / L increases somewhat due to activity in therange (cid:46) Pe (cid:46) , and decreases for higher Pe (cf. Fig. S.2).In the asymptotic limit Pe → ∞ , the active polymer is stretched,and the hydrodynamic contribution to thermal diffusion de-creases (cf. Fig. S.2 of ESI). As for a passive, rodlike poly-mer , asymptotically hydrodynamic interactions yield onlysmall corrections with respect to the polymer-length dependenceof a non-hydrodynamic (free-draining) polymer.Figure 9 displays the average site mean square displacementfor various Péclet numbers. For a passive flexible polymer, werecover the well-known Zimm behavior, with (cid:104) ∆ rrr ( t ) (cid:105) ∼ t / for t / τ Z (cid:28) , and a crossover to free diffusion for t / τ Z (cid:29) . Inthe presence of activity, four time regimes can be identified • t → ∞ — The MSD is dominated by the linear time dependence Journal Name, [year], [vol.] ,, -6 -4 -2 -9 -7 -5 -3 -1 -4 -3 -2 -1 -6 -4 -2 Fig. 9
Mean square displacement of flexible ABPO+HI, Eq. (45) . (a)MSDs for the Péclet numbers Pe = − (blue), Pe = × (orange), Pe = . × (yellow), and Pe = (purple); the polymer length is pL = . The time is scaled by the Zimm time τ Z of a passive polymer.(b) MSDs for the polymer lengths pL = × (blue), × (green), (red), (cyan), and (purple) and Pe = . (cid:104) ∆ rrr ∞ (cid:105) = (cid:104) rrr g (cid:105) denotesthe asymptotic value of the MSD in the center-of-mass reference. Thedashed lines correspond to the MSD in the polymer center-of-massreference frame, Eqs. (47) + (48) , and the solid lines to the overall MSD,Eq. (45) . The black lines indicate power laws in the respective regimes. of the center-of-mass dynamics, with the diffusion coefficient D = k B T H L + v l γ R L . (50)The other terms approach a constant value equal to (cid:104) rrr g (cid:105) ,where (cid:104) rrr g (cid:105) is the active-polymer radius of gyration. The simu-lations results are in agreement with the active contribution to D . • t / ˜ τ (cid:28) (cid:28) γ R t — The active contribution to the MSD is domi-nated by ( γ R ˜ τ (cid:29) ) (cid:68) ∆ rrr a ( t ) (cid:69) = v l γ R L ∞ ∑ n = ˜ τ n (cid:16) − e − t / ˜ τ n (cid:17) . (51)With the power-law dependence ˜ τ n = ˜ τ / n ˜ α of the relaxation time and by replacing the sum by an integral, we find (cid:68) ∆ rrr a ( t ) (cid:69) = v l ˜ τ γ R L (cid:18) t ˜ τ (cid:19) − / ˜ α (cid:90) ∞ dx − e − x ˜ α x ˜ α . (52)With the assumption πη Ω (cid:29) and Eq. (38), which corre-sponds to the exponent ˜ α = / , Eq. (52) yields (cid:68) ∆ rrr a ( t ) (cid:69) ∼ L Pe / ( pL ) t / , (53)i.e., a site sub-diffusive MSD dominated by the internal poly-mer dynamics. The exponent α (cid:48) = − / ˜ α = / of Eq. (53)approximately agrees with the full numerical result / (cf.Fig. 9). Note that the exponent ˜ α ( ˜ τ n ∼ / n ˜ α ) is in fact largerthan ˜ α = / for most Pe (cf. mode-number dependence of re-laxations in Fig. S.3), which leads to an exponent α (cid:48) somewhatlarger than α (cid:48) = / , consistent with the results of Fig. 9, wherewe find α (cid:48) ≈ / . Figure 9(b) emphasizes the universality ofthe internal dynamics with increasing pL . The various curves,especially for the MSD in the center-of-mass reference frame,asymptotically approach a power-law regime with an exponentclose to the predicted value. This polymer specific regime is ev-idently only pronounced for pL (cid:38) . Hence, it is not clearlyvisible in Fig. 4 of the MSD obtained from simulations. • t / ˜ τ , γ R t (cid:28) — Taylor expansion of the exponential functionsin Eq. (48) yields (cid:68) ∆ rrr a ( t ) (cid:69) = v l γ R L ∞ ∑ n = ˜ τ n + γ R ˜ τ n t , (54)consistent with the observed ballistic regime in Fig. 9. Thisregime and its dependence on activity and polymer proper-ties is in quantitative agreement with the simulation results ofFig. 4. • t → — The MSD is dominated by Eq. (47), and all modescontribute. Setting ˜ τ n = ˜ τ / n / and replacing the sum by anintegral yields (cid:68) ∆ rrr ( t ) (cid:69) = L π p µ (cid:18) t ˜ τ (cid:19) / (cid:90) ∞ dx − e − x / x . (55)This is the same relation as obtained for a passive system, ex-cept that µ and ˜ τ depend on activity. With Eq. (39), we findthe Péclet-number dependence (cid:104) ∆ rrr ( t ) (cid:105) ∼ Pe − / ( t / τ Z ) / for Pe (cid:38) and pL (cid:29) . We have presented analytical, numerical, and computer simula-tion results for the conformational and dynamical properties ofactive semiflexible polymers in the presence of hydrodynamicinteractions. In the simulations, the overdamped dynamics ofa bead-spring polymer composed of ABP monomers is studied,with hydrodynamic interactions captured by the Rotne-Prager-Yamakawa hydrodynamic tensor. For the analytical treatment, theGaussian semiflexible polymer model is adopted, which takes intoaccount the polymer inextensibility in a mean-field manner by a
10 | 1–13
Journal Name, [year], [vol.] , onstraint for the contour length. Here, activity is modeled asa Gaussian colored noise process with an exponential temporalcorrelation. Hydrodynamic interactions are taken into account bythe preaveraged Oseen tensor. The linearity of the equation ofmotion allows for its analytical solution. In any case, our activeforces do not generate a Stokeslet, higher-order active multipoleflow fields, which decay spatial as / r or faster with distancefrom the self-propelled active site, are neglected, and only thepresumably dominant Stokeslet field resulting from intramolecu-lar forces are taken into account.Most remarkably, we find a strong influence of hydrodynam-ics on the polymer conformational properties. In absence of hy-drodynamics, active flexible polymers (ABPOs-HI), with pL > ,monotonically swell with increasing activity, Pe , whereas semi-flexible polymers, with pL < , shrink at moderate Pe (the actual Pe -range depends on the polymer length) and swell for higher Pe ,similar to flexible polymers . In contrast, active polymers inthe presence of hydrodynamic interactions (ABPOs+HI) alwaysshrink for moderate Pe independent of stiffness, and swell againfor high activities, where the asymptotic extension for Pe → ∞ of an ABPO+HI is significantly smaller than that of an ABPO-HI. The observed strong influence of hydrodynamics appears overa Péclet-number range well covered by synthetic active colloids,where typically Pe (cid:46) . Two stochastic processes determine the size and shape of anABPO+HI—thermal and active fluctuations. Due to the linear-ity of the analytical equations of motion and the additivity of thenoise, the fluctuations lead to additive contributions to the meansquare end-to-end distance, which are, however, coupled by theinextensibility of the polymer. The active fluctuations yield a con-tribution quadratic in the propulsion velocity (or Péclet number),similar to the quadratic dependence of the MSD of an ABP ,which leads to a swelling of the polymer, however, with a Pe dependence smaller than quadratic due to the increase of thestretching coefficient with increasing Pe . The polymer inexten-sibility implies enhanced fluctuations of the thermal part of (cid:104) rrr e (cid:105) by the active noise—expressed by the factor µ —corresponding toa decreasing persistence length with increasing activity associatedwith a shrinkage of the polymer size.Qualitatively, the behavior can be understood as follows. Anincreasing activity yields an increasing persistent displacement l m / l = v / γ R l = Pe / of a monomer before it changes its propul-sion direction. Hence, any disparity in the propulsion direction isamplified by an increasing Pe and leads to, in average, divergentmonomer trajectories and an increasing intramolecular tensionreflected in the increasing stretching coefficient µ . For an ABPO-HI, the competing shrinkage of the thermal part and swelling ofthe active contribution leads to an overall swelling, since swellingexceeds shrinkage. In case of an ABPO+HI, the reduced swellingcan descriptively be understood by fluid-induced collective mo-tions compared to random motions in absence of HI (cf. Fig. 1).Mathematically, this is reflected by the shorter relaxation times ˜ τ n compared to τ n . To achieve a swelling of an ABPO+HI compara-ble to that of an ABPO-HI requires larger Péclet numbers. As aconsequence, (cid:104) rrr e (cid:105) of an ABPO+HI assumes a minimum at inter-mediate Pe . As mentioned several times, we do not take into accountswimmer-specific flow fields of individual monomers. Nonethe-less, the intramolecular forces create complex flow fields—fromsingle monomers to the full filament. Already for a pair ofmonomers, forces along their bond vector constitutes a forcedipole, aside from a potential Stokeslet. In fact, such a force-dipole field could also exist for a passive polymer, but the strongerforces of active monomers increase the dipole field and its rele-vance for the polymer dynamics significantly. Hence, on largerlength scales, embracing more monomers, the overall flow field israther complex and a large number of hydrodynamic multipolescontribute. This dynamically emerging multipoles are a particu-lar feature of ABPOs+HI and, in their sum, lead to the observedpolymer shrinkage.The polymer dynamics is determined by two relaxation pro-cesses, the orientational relaxation of an active site/monomer,and the polymer relaxation. This leads to distinct time regimes inthe polymer mean square displacement. At short times t / ˜ τ , γ R t (cid:28) , activity leads to a ballistic regime, with an enhanced dynamicscompared to a passive polymer. For t / ˜ τ (cid:28) (cid:28) γ R t , the MSD isdominated by the internal dynamics, and a polymer-characteristicsubdiffusive regime appears. Again, activity and hydrodynamicsplay a decisive role, leading to a power-law dependence with anexponent, α (cid:48) ≈ / , smaller than that of a passive hydrodynamicpolymer. In the asymptotic limit of long times, the enhanced dif-fusive dynamics is no longer affected by the fluid motion, butrather becomes identical to that of an ABPO-HI.Our studies predict a substantial effect of hydrodynamic inter-actions on the properties of active polymers. The shrinkage, evenin the presence of excluded-volume interactions, results in an en-hanced packing, which might be important for DNA organizationwithin the cell nucleus . The actual mechanism of DNA pack-ing is unresolved so far, however, DNA transcription or other lo-cal enzymatic processes, e.g., active-loop extrusion , provide acontinuous local energy influx, and, hence, a source of nonther-mal active noise. Moreover, hydrodynamic interactions could beinvolved in the observed subdiffusive dynamics of chromosomalloci , which is typically related to a viscoelastic or a frac-tal environment . Further experimental studies are necessaryto resolve the relevance of the various possible mechanisms af-fecting the dynamics, such as hydrodynamics, confinement, andviscoelasticity. Acknowledgments
This research was funded by the European Union’s Horizon2020 research and innovation programme under Grant agree-ment No. 674979-NANOTRANS. Financial support by theDeutsche Forschungsgemeinschaft (DFG) within the priority pro-gram SPP 1726 “Microswimmers-from Single Particle Motion toCollective Behaviour” is also gratefully acknowledged. More-over, the authors gratefully acknowledge the computing timegranted through JARA-HPC on the supercomputer JURECA atForschungszentrum Jülich.
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