The physics of active polymers and filaments
TThe physics of active polymers and filaments
Roland G. Winkler and Gerhard Gompper a) (Dated: 25 June 2020) Active matter agents consume internal energy or extract energy from the environment for locomotion andforce generation. Already rather generic models, such as ensembles of active Brownian particles, exhibitphenomena, which are absent at equilibrium, in particular motility-induced phase separation and collectivemotion. Further intriguing nonequilibrium effects emerge in assemblies of bound active agents as in linearpolymers or filaments. The interplay of activity and conformational degrees of freedom gives rise to novelstructural and dynamical features of individual polymers as well as in interacting ensembles. Such out-of-equilibrium polymers are an integral part of living matter, ranging from biological cells with filamentspropelled by motor proteins in the cytoskeleton, and RNA/DNA in the transcription process, to long swarm-ing bacteria and worms such as
Proteus mirabilis and
Caenorhabditis elegans , respectively. Even artificialactive polymers have been synthesized. The emergent properties of active polymers or filaments depend onthe coupling of the active process to their conformational degrees of freedom, aspects which are addressedin this article. The theoretical models for tangentially and isotropically self-propelled or active-bath drivenpolymers are presented, both in presence and absence of hydrodynamic interactions. The consequences fortheir conformational and dynamical properties are examined, emphasizing the strong influence of the cou-pling between activity and hydrodynamic interactions. Particular features of emerging phenomena, inducedby steric and hydrodynamic interactions, are highlighted. Various important, yet theoretically unexplored,aspects are featured and future challenges are discussed.
I. INTRODUCTION
Living matter is characterized by a multitude ofcomplex dynamical processes maintaining its out-of-equilibrium nature.
Molecular machines such as (mo-tor) proteins and ribosomes undergo conformationalchanges fueled by Adenosine Triphosphate (ATP), whichdrive and stir the cell interior.
This triggers a hier-archy of dynamical processes, movements, and trans-port, resulting in a nonequilibrium state of the cell—from the molecular to the whole-cell level—, with in-triguing collective phenomena emerging by migration andlocomotion also on scales much larger than individualcells. The nature of living and active matter systemsimplies nonthermal fluctuations, broken detailed balance,and a violation of the dissipation-fluctuation relation,which renders their theoretical description particularlychallenging. Filaments and polymers are an integral part of bio-logical systems, and their conformational and dynamicalproperties are substantially affected by the active pro-cesses to a yet unresolved extent.
Biological active polymers and filaments — Enzymaticconformational changes are considered to induce fluctu-ating hydrodynamic flows in the cytoplasm, which lead toan enhanced diffusion of dissolved colloidal and polymericobjects.
In addition, kinesin motors walking alongmicrotubule filaments generate forces that affect the dy-namics of the cytoskeletal network, the transport proper- a) Theoretical Physics of Living Matter, Institute of BiologicalInformation Processing and Institute for Advanced Simulation,Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany; Electronicmail: [email protected], [email protected] ties of species in the cell, and the organization of the cellinterior.
Even more, molecular motors give rise tononequilibrium conformational fluctuations of actin fila-ments and microtubules.
Within the nucleus, ATPases such as DNA or RNApolymerase (RNAP, DNAP) are involved in DNA tran-scription, where the information coded in the base-pairsequence of DNA is transcribed into DNA or RNA. Thisproceeds in several steps, where the ATPases locally un-zip the two DNA strands, nucleotides are added to thesynthesized molecules, and the ATPases move along theDNA. Hence, every RNAP/DNAP translocation stepis a complex process, which generates nonthermal fluctu-ations for both, RNAP/DNAP as well as the transcribedDNA.
Various of these active process are considered to beinvolved in the spatial arrangement of the eukaryoticgenome, controlling their dynamical properties, and be-ing essential for the cell function. In particular, ATP-dependent processes affect the dynamics of chromosomalloci and chromatin. Moreover, spatial segregationof active (euchromatin) and passive (heterochromatin)chromatin has been found.
The detailed mechanismneeds to be unraveled, possibly taking active processesinto account.
Microswimmers are a particular class of active matter,where individual agents move autonomously by eitherconverting internal chemical energy into directed motion,or utilizing energy from the environment.
Bio-logical microswimmers, like algae, sperm, and bacteria,are omnipresent.
Numerous biological microswim-mers are rather elongated and polymer- or filament-like,undergoing shape changes during migration. A par-ticular example are elongated swarming bacteria, pro-pelled by flagella, such as
Proteus mirabilis , Vibrio a r X i v : . [ c ond - m a t . s o f t ] J un parahaemolyticus , or Serratia marcescens , whichexhibit an intriguing collective behavior. Other mi-croswimmers organize in chain-like structures, for exam-ple planktonic dinoflagellates or Bacillus subtilis bac-teria during biofilm formation. On a more macroscopicscale, nematodes such as
Caenorhabditis elegans swim and collectively organize into dynamical networks. Similarly, biological polar filaments like actin and mi-crotubules driven by molecular motors exhibit filamentbundling and the emergence of active turbulence, whichis characterized by high vorticity and presence of motiletopological defects.
These diverse aspects illustrate the relevance of activ-ity for the function of cells and other biological polymer-like active objects. However, a cell is a very complexsystem with a multitude of concurrent processes, hence,a classification or the separation of individual active pro-cesses is difficult. Here, a systematic study of emergentphenomena due to activity by synthetic model systemsmay be useful. More importantly, the understanding ofactive processes and their suitable implementation maybe essential in the rational design of synthetic cells.
Synthetic active polymers — Synthetic active or ac-tivated colloidal molecules or polymers are nowa-days obtained in several ways. Various conceptshave been put forward for the design of synthetic ac-tive particles, which can serve as monomers. Typi-cally, propulsion is based on phoretic effects, relyingon local gradients of, e.g., electric fields (electrophore-sis), concentration (diffusiophoresis), and temperature(thermophoresis).
Synthetic active systems ex-hibit a wide spectrum of novel and fascinating phenom-ena, specifically activity-driven phase separation or large-scale collective motion.
Catalytic Janus particles have been shown to spon-taneously self-assembly into autonomously swimmingdimers with a wide variety of morphologies.
Assem-bly of metaldielectric Janus colloids (monomers) intoactive chains can be achieved by imbalanced interac-tions, where the motility and the colloid interactions aresimultaneously controlled by an AC electric field.
Through the application of strong AC electric fields, lin-ear chains of bound Janus particles can be achieved ei-ther by van-der-Waals forces or polymer linkers. Elec-trohydrodynamic convection rolls lead to self-assembledcolloidal chains in a nematic liquid crystal matrix anddirected movement. Linear self-assembly of dielectriccolloidal particles is achieved by alternating magneticfields, where the chain length can be controlled by theexternal field.
Moreover, chains of linked colloids,which are uniformly coated with catalytic nanoparticles,have been synthesizes. Hydrogen peroxide decomposi-tion on the surfaces of the colloidal monomers generatesphoretic flows, and activity induced hydrodynamic inter-actions between monomers results in an enhanced diffu-sive motion. Objectives —Filamentous and polymeric structuresplay a major role in biological systems and are heav- ily involved in nonequilibrium processes. However, weare far from understanding the interplay between out-of-equilibrium fluctuations, corresponding polymer con-formations, and emerging activity-driven self-organizedstructures. Studies of molecular/polymeric active mat-ter will reveal original physical phenomena, and promotethe development of novel smart devices and materials.In this perspective article, we address the distinctivefeatures emerging from the coupling of activity and con-formational degrees of freedom of filamentous and poly-meric structures. Assemblies of active colloidal particleshave been denoted as active colloidal molecules . Sincewe focus not only on synthetic colloidal systems, but alsoon polymers and filaments in biological systems, we willuse the notion active polymers . Different concepts for the active dynamics of polymerscan be imagined or realized. For polymers composedof linearly connected monomers, the latter can be as-sumed to be themselves active, i.e., self-propelling, orto be activated, i.e., externally driven by a nonthermalforce. Self-propulsion typically emerges by the interac-tion of a microswimmer with the embedding fluid. Here,hydrodynamics plays a major role and momentum isconserved. This is denoted as wet active matter. Incontrast, dry active matter is characterized by the ab-sence of local momentum conservation and, hence, hy-drodynamic interactions. Here, other effects, such asstrong particle-particle interactions or contact with amomentum-absorbing medium, as in bacteria gliding orgranular beads vibrating on frictional surfaces, dominateover fluid-mediated interactions. Moreover, externally-driven polymers are subject to forces, which, when theyare dissolved in a fluid, give rise to Stokeslet flows andhydrodynamic interactions. Depending on the nature ofthe polymer environment, the emerging properties by thecoupling of activity and the internal degrees of freedomcan be rather different.
Moreover, the nature ofthe active force determines the driving of a monomer.The forces on filaments driven by molecular motors aretypically assumed to push them along the local filamenttangent.
Another realization are spatially indepen-dent, but time-correlated, active forces on monomers.We will highlight the impact of the various aspects—self-propelled vs. actuated, dry vs. wet, tangentially vs. ran-domly driven—on the polymer conformations, dynamics,and collective effects.The article is organized as follows. Section II de-scribes realizations of dry and wet active forces on thelevel of monomers and bonds. The properties of drypolymers composed of active Brownian particles are dis-cussed in Sec. III. Effects of hydrodynamic interactionsare included in Sec. IV. Section V is devoted to prop-erties of tangentially driven dry polymers, and Sec. VIpresents aspects of hydrodynamically self-propelled poly-mers. Collective effects are discussed in Sec. VII. Finally,Sec. VIII summarizes the major aspects and indicatespossible future research directions.
II. ACTIVE FORCES
In models of active polymers, forces can directly beassigned to individual monomers, or to the bond con-necting two monomers. This defines the minimal activeunits—either a monomer or a dumbbell of two connectedmonomers.
A. Active Brownian particles
A generic model of a self-propelled particle, espe-cially suitable for dry active matter, is the well-know ac-tive Brownian particle (ABP).
Its over-damped equations of motion are given by˙ r ( t ) = 1 γ T F a + 1 γ T ( F ( t ) + Γ ( t )) , (1)˙ e ( t ) = Θ ( t ) × e ( t ) (2)for the translational and rotational motion of the posi-tion r and the orientation e in two or three dimensions(2D, 3D). The active force is F a = γ T v e , with v theself-propulsion velocity and γ T the translational frictioncoefficient; F accounts for all non-active external forces, Γ is the translational thermal noise, and Θ the rotationalnoise. The latter are Gaussian and Markovian randomprocesses with zero mean and the second moments (cid:104) Γ α ( t ) Γ β ( t (cid:48) ) (cid:105) = 2 k B T γ T δ αβ δ ( t − t (cid:48) ) , (3) (cid:104) Θ α ( t ) Θ β ( t (cid:48) ) (cid:105) = 2 D R δ αβ δ ( t − t (cid:48) ) , (4)with k B the Boltzmann factor, T the temperature, D R the rotational diffusion coefficient, and α, β ∈ { x, y, z } .Equation (2) yields the correlation function (cid:104) e ( t ) · e (0) (cid:105) = e − ( d − D R t (5)in d dimensions. Hence, the particle [Eq. (1)] can beconsidered as exposed to thermal and colored noise withthe correlation function (5). B. Active dumbbells and rods
The combination of active monomers into linear as-semblies provides a wide spectrum of possible combina-tions of active forces. This is illustrated for two boundactive particles forming a dumbbell. A bond, not re-stricting the orientational motion of two ABPs, leads toan active Brownian dumbbell. The dynamical proper-ties and the phase behavior of such dumbbells have beenstudied.
Since active dumbbells are a special caseof active Brownian polymers, we refer to the general dis-cussion is Sec. III for results. Specific features by hydro-dynamic interactions are addressed in Sec. VI A.Another extreme case is propulsion along the bond vec-tor only. The equations of motion of the monomers for a (a)(c) (b)(d)
FIG. 1. Illustration of possible combinations of propulsiondirections of active dumbbells. (a) The two ABPs rotateindependently, (b) the beads are propelled along thebond, (c) correlated propulsion in a common directionoblique to the bond, where the propulsion direction maychange in a diffusive manner, and (d) propulsion in non-parallel directions with fixed angles. such a dumbbell are ( i ∈ { , } ) γ T ˙ r i = F a ( t ) + F i ( t ) + Γ i ( t ) , (6)with the active force F a = F a u on every monomer in thedirection of the (unit) bond vector u = ( r − r ) / | r − r | .Note that F i contains the bond force. In the absence ofany external force, thermal noise leads to a rotational dif-fusive motion of u . A generalization allows the activeforce to vary within a certain angle with respect to thebond vector.
Oblique arrangement of the propulsiondirection leads to spiral trajectories in two dimensions, ashas been demonstrated for Janus particles.
Naturallymany other options are possible, and may, in modeling,be chosen according to experimental needs.As indicated, the particular propulsion mechanismleads to a distinct dynamics and also effects on the emer-gent collective behavior can be expect, e.g., motility-induced phase separation (MIPS). In fact, MIPS has beenfound for dumbbells propelled along their bond as wellas those propelled oblique with respect to the bond, where spontaneously formed aggregates break chiral sym-metry and rotate.
Such a rotation is also obtained forsystems of self-propelled particles or rods movingin two dimensions on circles as well as for bend rigidself-propelled filaments.
We focus on uniform systems of active particles of thesame size, not addressing effects appearing in systemsof dumbbells with different radii with correspondinglyasymmetric flow fields.
C. Self-propelled force-free and torque-free monomers influids
Synthetic and biological microswimmers are typicallyimmersed in a fluid and hydrodynamic interactions (HI)are an integral part of their propulsion. Self-propelledparticles move autonomously, with no external force ortorque applied, and hence the total force/torque of theswimmer on the fluid and vice versa vanishes.
Theflow field generated by the microswimmer can be rep-resented in terms of a multipole expansion.
Thefar field is dominated by the force dipole (FD), sourcedipole (SD), force quadrupole (FQ), source quadrupole(SQ), and rotlet dipole (RD) contributions.
In theory and simulations, such an expansion can beexploited to calculate the flow field of individual swim-mers. Alternatively, the squirmer model can be ap-plied, which was originally designed to model ciliatedmicroswimmers.
It is a rather generic model, whichcaptures the essential swimming aspects and is nowadaysapplied to a broad class of microswimmers, rang-ing from diffusiophoretic particles to biological cells.
A linear arrangement of such monomers leads to anintricate coupling of their flow fields and novel emergentconformational and dynamical features.
1. Squirmers
A squirmer is an axisymmetric rigid colloid with a pre-scribed surface fluid (slip) velocity.
For a purelytangential fluid displacement, the surface slip velocity ofa sphere can be described as v sq = ∞ (cid:88) n =1 n ( n + 1) B n sin ϑP (cid:48) n (cos ϑ ) e ϑ (7)in terms of derivatives of the n -th order Legendre poly-nomial, P n (cos ϑ ). Here, ϑ is the angle between thebody-fixed propulsion direction e and the consideredpoint on the colloid surface with tangent vector e ϑ ,and B n is the amplitude of the respective mode. Typ-ically only two modes are considered, i.e., B n = 0 for n ≥ Explicitly, the leading contributionsyield the slip velocity v sq = B sin ϑ (1 + β cos ϑ ) e ϑ . (8)The parameter B = 2 v / β = B /B characterizes the nature of theswimmer, namely a pusher ( β < β > β = 0), corresponding, e.g., to E. coli , Chlamydomonas , and
Volvox , respectively. The far fieldof a squirmer is well described by the flow fields of aforce dipole (FD), a source dipole (SD), and a sourcequadrupole (SQ).
Various extensions to spheroidalsquirmers have been proposed, where some al-low for an analytical calculation of the flow field.
The propulsion direction is not affected by thesquirmer flow field. In case of several squirmers, butin absence of thermal fluctuations, interference of theirflow fields leads to particular hydrodynamic collective ef-fects and structure formation.
In simulation ap-proaches, which account for thermal fluctuations, often different structures are observed, and in dilute solutionthe propulsion orientational correlation function agreeswith Eq. (5) with the diffusion coefficient determined bythe fluid viscosity.
2. Non-axisymmetric swimmers
For a more general description and an extension tonon-axisymmetric swimmers, the swimmer equations ofmotion can be expressed in terms of mobilities.
Such an extension captures the many-body nature of thesurface-force density in a suspension of many colloids, whereas the original Stokes law applies to infinite dilu-tion only. The center-of-mass translational velocity andthe rotational frequency of a single force- and torque-freespherical particle in an unbounded fluid are˙ r = v a ( t ) , ˙ ω = Ω a ( t ) , (9)where the active velocity and active torque are given by v a = − πR (cid:73) v s ( r ) d r, (10) Ω a = − πR (cid:73) r × v s ( r ) d r, (11)with the surface slip velocity v s , e.g., the squirmer veloc-ity of Eq. (8). More general expressions for ellip-soidal particles are presented in Ref. 145. This is a ratherobvious result, but it shows that a colloid translates androtates independently in response to the active surfacevelocity. The case of linearly connected active spheres isdiscussed in Sec. VI B.
III. DRY ACTIVE BROWNIAN POLYMERS (D-ABPO)
The conformational and dynamical properties of dry(free-draining) active Brownian polymers, which will bedenoted as D-ABPO, are typically studied analytically bythe well-known Rouse model of equilibrium polymerphysics.
However, the deficiencies of the model,in particular the extensibility of bonds in the standardformulation, leads to inadequate predictions of activityeffects. In contrast, valuable quantitative predictionshave been obtained by computer simulations and more adequate analytical models, which will be in-troduced in the following.
A. Discrete model of active polymers
Dry semiflexible active polymers can be modeled as alinear chain of N m linked active Brownian particles, withtheir dynamics described by the overdamped equationsof motion (1) and (2). The force F i on particle i ( i =1 , . . . , N m ) includes bond, bending, and excluded-volumecontributions. The two independent parameters v and D R , characterizing activity, are combined in thedimensionless quantities P e = v lD R , ∆ = D T d H D R , (12)with l the equilibrium bond length. The P´eclet number P e compares the time for the reorientation of an ABPmonomer with that for its translation with velocity v over the monomer radius, and ∆ is the ratio between thetranslational, D T = k B T / πηd H , and rotational, D R ,diffusion coefficient of an individual monomer, where d H denotes the monomer hydrodynamic diameter. For a tangent hard-sphere-type polymer, d H = l and ∆ = 1 /
3. In order to avoid artifacts in the polymer struc-tures by activity, the force constant for the bond, κ l , andthat of the excluded-volume Lennard-Jones potential, ε ,can be adjusted according to κ l l /k B T = (10+2 P e ) × and ε/ ( k B T P e ) = 1 as a function of
P e . This ensures afinite bond length within 3% of the equilibrium value fora harmonic bond potential, and an activity-independentoverlap between monomers.
B. Continuum model of active polymers
An analytical description of D-ABPO propertiesis achieved by a mean-field model for semiflexiblepolymers augmented by the active velocity v ( s, t ),which yields the Langevin equation ∂∂t r ( s, t ) = v ( s, t ) (13)+ 1 γ (cid:18) λk B T ∂ ∂s r ( s, t ) − (cid:15)k B T ∂ ∂s r ( s, t ) + Γ ( s, t ) (cid:19) , with suitable boundary conditions for the free ends of lin-ear polymers or periodic boundary conditionsfor ring polymers. Here, γ is the translational frictioncoefficient per unit length and (cid:15) = 3 / p for polymers inthree dimensions, where p is related to the persis-tence length l p via p = 1 / (2 l p ). The terms withthe second and fourth derivative in Eq. (13) account forthe entropic degrees of freedom and bending elasticity,respectively. The Lagrangian multiplier λ is determinedin a mean-field manner by the global constraint of a finitecontour length, L , (cid:90) L/ − L/ (cid:42)(cid:18) ∂ r ( s, t ) ∂s (cid:19) (cid:43) ds = L, (14)From Eq. (5), the correlation function of the velocity v ( s, t ) follows as (cid:104) v ( s, t ) · v ( s (cid:48) , t (cid:48) ) (cid:105) = v le − γ R ( t − t (cid:48) ) δ ( s − s (cid:48) ) , (15)with γ R = 2 D R the damping factor of the rotationalmotion. For the analytical solution, only the first andsecond moment of the distribution of the active velocityare needed. Note that the parameter l in Eq. (15) defines the num-ber of active sites L/l along the polymer. -3 -2 -1 FIG. 2. Mean square end-to-end distances as a function of thePclet number for semiflexible D-ABPO with pL = L/ l p =10 , , , , − , − (bottom to top at P e = 10 − ),and ∆ = 1 / The dashed line is the flexible polymer limitaccording to Eq. (18). “T. Eisenstecken, G. Gompper, and R.G. Winkler, Polymers , 304 (2016); licensed under a CreativeCommons Attribution (CC BY) license.”FIG. 3. Configurations of flexible phantom D-ABPO (red)and active polymers with self-propelled monomers in presenceof HI (S-ABPO) (green) (cf. Sec. IV A) of length N m = 50 forthe P´eclet numbers P e = 1 (top) and
P e = 10 (bottom). “Reproduced with permission from Soft Matter , 8316(2016). Copyright 2016 The Royal Society of Chemistry.” C. Results1. Conformational properties
The conformational properties of linear polymersare characterized by their mean square end-to-enddistance (cid:10) r e (cid:11) = 4 ∞ (cid:88) n =1 (cid:10) χ n − (cid:11) ϕ n − ( L/ , (16)in terms of the eigenfunctions ϕ n of the differential op-erator on the rhs of Eq. (13) and the fluctuations of thenormal-mode amplitudes (cid:10) χ n (cid:11) = 3 k B Tγ τ n + v p (1 + γ R τ n ) τ n . (17)The relaxation times τ n as well as the eigenfunctions de-pend on the activity. The closed expression (cid:10) r e (cid:11) = Lpµ (18)+
P e L pµ∆ (cid:34) − (cid:112) µ ∆pL √ µ tanh (cid:32) pL √ µ (cid:112) µ ∆ (cid:33)(cid:35) is obtained in the limit of flexible polymers, pL (cid:29)
1, withfree ends, with the relaxation times τ n = τ R µn , (19)where τ R = γL / (3 π k B T p ) is the (passive) Rouse relax-ation time.
The activity-depend factor µ accountsfor the constraint (14). A detailed discussion of µ ( P e )for linear and ring polymers is presented in Refs. 95, 155,and 156, respectivelyThe effect of activity on the conformational proper-ties of active polymers is illustrated in Fig. 2. Figure 3shows polymer conformations for various P´eclet numbers.For flexible polymers, activity causes polymer swellingwith increasing
P e , which saturates at L / pL → ∞ as a consequence of the finite contour length.The reason for the swelling is an increase of the per-sistence length L p of the active motion with increasing P e , where L p is defined as the distance displaced by ac-tivity with velocity v in the time 1 / D R of the decayof the correlation function (15), i.e., L p /l = P e/
2. For
P e (cid:29)
1, two groups of monomers moving in oppositedirections will move far before changing their directionsubstantially, which leads to stretching of the in-betweenpart of the polymer. In contrast, semiflexible polymersshrink at weak
P e and swell for large
P e similarly toflexible polymers. Shrinkage is caused by enhanced fluc-tuations transverse to the polymer contour by activity.The bonds prevent fluctuations along the contour, whichleads to an apparent softening of the semiflexible poly-mer. At large P´eclet numbers, tension in the polymercontour, which increases with activity, dominates overthe energetic contribution of bending, so that the lat-ter can be neglected and a semiflexible polymer appearsflexible. The comparison of the theoretical predictions ofthe polymer conformations with simulation results yieldsexcellent agreement.
In general, ring polymers exhibit similar features aslinear polymers.
Specifically, active and thermal fluc-tuations attempt to shrink and crumple a ring-like struc-ture. However, this is opposed by a negative internaltension, whereas the tension in always positive for linearpolymers. In general, activity implies enhanced fluctua-tions of the normal-mode amplitudes.
The fluctuation -1 -8 -7 -6 -5 -4 -3 -2 -1 FIG. 4. Longest relaxation times of semiflexible D-ABPOas a function of the P´eclet number for pL = L/ lp =10 , , , , − , and 10 − (bottom to top). Inset: Mode-number dependence of the relaxation times of active polymerswith pL = 10 − for the P´eclet numbers P e = 10 , 3 × ,10 , and 5 × (bottom to top). The black squares (top)show the mode-number dependence of flexible polymers with pL = 10 . The solid lines indicate the relations for flexible( ∼ n − ) and semiflexible ( ∼ (2 n − − ) polymers, respec-tively. τ is the longest relaxation time. “T. Eisenstecken,G. Gompper, and R. G. Winkler, Polymers , 304 (2016);licensed under a Creative Commons Attribution (CC BY) li-cense.” spectrum is dominated by activity already for moderateP´eclet numbers ( P e (cid:38) v inEq. (17)—and thermal fluctuations matter only for largemode numbers, i.e., at very small length scales. Notably,the major part of the spectrum is determined by tension,with a crossover from a 1 /n to a 1 /n power-law withincreasing mode number (cf. Eq. (17) for τ n ∼ /n ).Hence, conclusions from the exponent on the underlyingfluctuation mechanisms have to be drawn with care, anda 1 /n dependence is not necessarily a sign of dominatingbending modes.Qualitatively, the obtained fluctuation spectrum forrings agrees with that of fluctuating membranes, wherealso an increase of the fluctuations at small wave vectorsby activity compared to an equilibrium system has beenobserved experimentally, in simulations, anddescribed theoretically.
2. Relaxation times
The polymer relaxation times strongly depend on theactivity, as displayed in Fig. 4. The longest time, τ ,decreases with increasing P´eclet number for P e (cid:38) τ ∼ /µ is solely determined by the stretching coefficient µ , andits decrease is a consequence of the finite polymer con- -2 -1 -8 -6 -4 -2 FIG. 5. Mean-square displacements (MSD) of flexible D-ABPO with pL = 10 . The P´eclet numbers are P e = 0, 3, 20,10 , and 5 × (bottom to top). The time is scaled by thefactor γ R = 2 D R of the rotational diffusion. The dashed linescorrespond to the MSD in the polymer center-of-mass refer-ence frame. “Reproduced from T. Eisenstecken, G. Gomp-per, and R. G. Winkler, J. Chem. Phys. , 154903 (2017),with the permission of AIP Publishing” tour length. With increasing stiffness, the initial dropis stronger, but the same asymptotic dependence is ob-tained for P e → ∞ .The inset of Fig. 4 shows the dependence of the re-laxation times τ n of stiff polymers on the mode number.At low P e , we find the well-know dependence τ n /τ ∼ (2 n − − valid for semiflexible polymers. Withincreasing
P e , the ratio τ n /τ increases, and for P e (cid:38) τ n /τ ∼ n − of flexible polymers. At larger n ,the relaxation times cross over to the semiflexible behav-ior again. However, the crossover point shifts to largermode numbers with increasing activity. Hence, activepolymers at large P´eclet numbers appear flexible on largelength and long time scales and only exhibit semiflexiblebehavior on small length scales.The theoretically predicted dependence of τ on P e hasnot been directly confirmed by simulations yet. How-ever, the very good quantitative agreement between the-oretical and simulations results of polymer mean-squaredisplacements supports the reliability of the theoreti-cally obtained activity dependence.
3. Mean-square displacement
The contour-length averaged mean-square displace-ment (MSD) of active polymers is (cid:104) ∆ r ( t ) (cid:105) = (cid:10) ∆ r cm ( t ) (cid:11) + 1 L ∞ (cid:88) n =1 (cid:20) k B T τ n γ (cid:16) − e − t/τ n (cid:17) + 2 v lτ n γ R τ n (cid:18) − e − γ R t − γ R τ n e − t/τ n − γ R τ n (cid:19)(cid:21) , (20)with the center-of-mass mean-squaredisplacement (cid:10) ∆ r cm ( t ) (cid:11) = 6 k B TγL t + 2 v lγ R L (cid:0) γ R t − e − γ R t (cid:1) . (21)Equation (21) resembles the MSD of a single activeBrownian particle, with a ballistic time regime forshort times and a diffusive regime for long times (cf.Fig. 5) with the diffusion coefficient D = k B T [1 +3 P e / (2 ∆ )] / ( γL ). The polymer nature is re-flected in the total friction coefficient γL , in the Brownianmotion (first term on rhs of Eq. (21)), and the number ofactive sites L/l , in the active term.
The center-of-massmotion of polymers is free of internal forces, hence, the
L/l
ABPs contribute independently to the MSD. In thelimit t → ∞ , the center-of-mass MSD is proportional to v l/L . For an isotropic and homogeneous orientation ofthe propulsion directions, we expect no contribution ofthe active force to the MSD. However, by the Gaussiannature of the stochastic process, the orientation fluctua-tions are proportional to (cid:112) l/L , which vanish in the limit L → ∞ , but lead to a finite contribution to the diffusioncoefficient for L < ∞ .The site MSD (monomer MSD), cf. Fig. 5, is stronglyaffected by activity. It increases with increasing P e , andexhibits up to four different time regimes.
In the limit t →
0, the second term on the right hand side of Eq. (20)dominates and (cid:104) ∆ r ( t ) (cid:105) ∼ ( t/τ ) / , exhibiting Rousedynamics, however, with activity-dependent relaxationtime τ . For t/τ (cid:28) γ R t (cid:28)
1, and
P e (cid:29)
1, the ac-tive contribution dominates with a quadratic time depen-dence (cid:104) ∆ r ( t ) (cid:105) ∼ t . On time scales 1 /γ R (cid:28) t (cid:28) τ , theinternal polymer dynamics is most important, with theRouse-like time dependence (cid:104) ∆ r ( t ) (cid:105) ∼ γ R P e ( τ t ) / in the center-of-mass reference frame. Since τ decreasewith increasing activity, this regime shortens with in-creasing P e . At longer times, the center-of-mass MSD(21) dominates.
IV. WET ACTIVE BROWNIAN POLYMERS
Hydrodynamic interactions lead to a qualitative dif-ferent polymer dynamics, as is well established for pas-sive polymers, and first studies on active polymersimmersed in a fluid indicate even pronounced effects ontheir stationary-state conformational properties.
Theinfluence of hydrodynamics depends on the nature of theactive force, i.e., self-propelled or bath-driven monomers.Hydrodynamic interactions between monomers embed-ded in a fluid can be taken into account implicitly by theOseen hydrodynamic tensor for point particles and theRotne-Prager-Yamakawa (RPY) tensor for a sphere ofdiameter l . Alternatively, hydrodynamics can explicitly be takeninto account by mesoscale hydrodynamics simula-tion techniques, such as the lattice Boltzmannmethod (LB), the dissipative particle dynamics(DPD), and the multiparticle collision dynamics(MPC) approach.
Externally driven active poly-mers have been implemented in MPC. As mentioned above, the active, nonthermal processaffecting the monomer dynamics can originate from in-ternal sources or be imposed externally. In the first case,the monomer is force and torque free, whereas in the sec-ond case it is not. This leads to different equations ofmotion and consequently different behaviors.
A. Discrete model of self-propelled active polymers(S-ABPO)
For active polymers with self-propelled monomers (S-ABPO), the equations of motion are ( i = 1 , . . . , N m ) ˙ r i ( t ) = v e i ( t ) + N m (cid:88) j =1 H ij [ F j ( t ) + Γ j ( t )] , (22)where F i comprises all inter- and intramolecular forces asfor D-ABPO. The second moment of the Gaussian andMarkovian random force is now given by (cid:10) Γ i ( t ) Γ Tj ( t (cid:48) ) (cid:11) = 2 k B T H − ij δ ( t − t (cid:48) ) , (23)where H − ij is the inverse of the hydrodynamic tensor H ij ( r ij ) = δ ij πηl I + (1 − δ ij ) G ( r ij ) , (24)and G ( r ) the Oseen or RPY tensor. The rotational mo-tion of the monomers, Eq. (2), is not affected by hydro-dynamics. No Stokeslet due to self-propulsion is takeninto account, only Stokeslets arising from bond, bending,and excluded-volume interactions between monomers, aswell as thermal forces are considered in this descrip-tion. Higher-order multipole contributions of the ac-tive monomers are neglect, especially the force dipole.Since point particles are considered, source multipolesare absent. All these multipoles decay faster than aStokeslet. Hence, the long-range character of HI inpolymers of a broad class of active monomers are cap-tured. As far as near-field hydrodynamic effects are con-cerned, this model is closest to polymers composed of -1 -3 -2 -1 FIG. 6. Mean square end-to-end distance as a functionof P´eclet number for semiflexible S-ABPO with N m = 200( L = 199 l ) monomers for pL = 2 × (blue), 10 (red),and 10 − (green). The bullets correspond to phantom andopen squares to self-avoiding polymers. The dashed lines areguides for the eye. The solid lines are analytical results forsemiflexible free-draining polymers (D-ABPO). “Reproducedwith permission from Soft Matter , 8316 (2016). Copyright2016 The Royal Society of Chemistry.” neutral squirmers, where particular effectsby higher multipole interactions between monomers arenot resolved. Section VI A discusses orientationalcorrelations of the propulsion directions of squirmermonomers of a dumbbell by higher-order multipoles.The equations of motion (22) are solved by the Er-makMcCammon algorithm.
1. Conformational properties
Figure 6 depicts the confirmations of the active poly-mers as a function of
P e , which are strongly influence byHI. Compared to free-draining polymers (D-ABPO), hy-drodynamics leads to a substantial shrinkage of the poly-mers in the range 1 < P e (cid:46) and a reduced swellingfor larger P e . This qualitative difference is illustratedby the snapshots of Fig. 3. The shrinkage depends onpolymer length and is substantially stronger for longerpolymers. Also semiflexible polymers in presence of HIshrink stronger than those in its absence, but the effectvanishes gradually as pL →
0. This is a consequence ofthe reduced influence of hydrodynamic interactions forrather stiff polymers.
Yet, the asymptotic value for
P e → ∞ is smaller than that for D-ABPO, because theconformational properties are determined by polymer en-tropy rather than stiffness in this limit, with a substantialhydrodynamic effect.Self-avoidance reduces the extent of shrinkage, specif-ically for flexible polymers, but the excluded-volume ef-fects vanish with decreasing pL , and for pL < -1 -2 -1 FIG. 7. Polymer mean square end-to-end distance (cid:104) r e (cid:105) , Eq.(26), as a function of the P´eclet number P e for flexible S-ABPOs of length pL = 2 × (orange), 10 (green), 10 (yellow), and 10 (magenta). The blue lines correspond toa free-draining flexible polymers (D-ABPO) with pL = 50.The dotted curves represent the contribution with the relax-ation times τ n and the dashed-dotted curves that with v ofEq. (26), respectively. “Reproduced with permission fromSoft Matter , 8316 (2016). Copyright 2016 The Royal So-ciety of Chemistry.” hardly any difference between phantom and self-avoidingpolymers. Moreover, the swelling behavior with andwithout excluded-volume interactions is rather similarin the limit P e (cid:29)
1. Interestingly, phantom and self-avoiding polymers show a different but universal depen-dence on
P e as they start to swell. Here, active forcesexceed both, excluded-volume interactions and bendingforces.Analytical theory predicts a stronger increase of (cid:104) r e (cid:105) with increasing P e in the swelling regime compared toD-ABPO (cf. Fig. 7). The dependence on
P e changesfrom (cid:104) r e (cid:105) ∼ P e for pL ≈ − to (cid:104) r e (cid:105) ∼ P e / for pL ≈ . Hence, hydrodynamic interactions lead to aqualitative different P e dependence.The polymer collapse is a consequence of the time-scale separation of the thermal and the active contri-bution to the mean square end-to-end distance. Hy-drodynamic interactions enhance the polymer dynamicsand shorten the relaxation times, as is well-know for theZimm model of flexible polymers and also forsemiflexible polymers.
Within the preaveragingapproximation, the relaxation times are given by˜ τ n = τ n πηG nn , (25)with τ n the relaxation times in absence of hydrodynamicinteractions, and G nn the matrix elements of the Os-een tensor in terms of eigenfunctions of the semiflexiblepolymers. Since G nn (cid:62)
0, ˜ τ n (cid:54) τ n . Analytical theory yields the mean square end-to-end distance (cid:10) r e (cid:11) = 8 L (cid:88) n, odd (cid:18) k B T τ n πη + v l ˜ τ n γ R ˜ τ n (cid:19) , (26)where the first term in the brackets arises from thermalfluctuations and the second from activity. However, alsothe relaxation time τ n and the elements G nn depend onactivity. Figure 7 displays the different contributions to (cid:104) r e (cid:105) for various polymer lengths. The initial shrinkageof (cid:104) r e (cid:105) with increasing P e is caused by the decreasingrelaxation times τ n ∼ /P e with increasing activity. The v -dependent term causes a swelling of the polymers.For D-ABPO, the competing effects lead to an overallswelling, since swelling exceeds shrinkage. For S-ABPO,swelling is weaker due to fluid-induced collective motioncompared to the random motion of D-ABPO, and (cid:104) r e (cid:105) assumes a minimum. This is a consequence of ˜ τ n (cid:54) τ n .Here, we like to mention that a certain amount ofshrinkage has also been observed for self-avoiding D-ABPO over a certain range of P´eclet numbers, whichis attributed to specific monomer packing.
2. Dynamical properties
The mean-square displacement of flexible and semi-flexible polymers exhibits the Zimm behavior t / or thedependence t / ( t/ ˜ τ (cid:28) P e <
As forthe D-ABPO, activity yields a ballistic time regime for t/ ˜ τ (cid:28) γ R t (cid:28)
1, and a regime dominated by inter-nal polymer dynamics at times 1 /γ R (cid:28) t (cid:28) ˜ τ for longpolymers, pL (cid:29)
1, and
P e (cid:29)
1. Interestingly, in thelatter regime analytical calculations yield the power-lawbehavior (cid:104) r e (cid:105) ∼ t / , i.e., an exponent even smaller thanthat of D-ABPO. For long times, t/ ˜ τ (cid:29)
1, the diffu-sive regime (21) of D-ABPO is assumed, with the sameactive diffusion coefficient. B. Discrete model of externally actuated polymers(E-ABPO)
In case of externally actuated monomers (E-ABPO),the polymer equations of motion are given by ˙ r i ( t ) = N m (cid:88) j =1 H ij [ γ T v e j ( t ) + F j ( t ) + Γ j ( t )] . (27)Here, also the external active force generates a Stokesletfor each monomer.A possible realization of E-ABPO is a passive polymerembedded in an active bath. Experimentally, an exter-nally driven polymer can in principle be realized by forc-ing a chain of colloidal particles by optical tweezers. Optical forces are very well suited to manipulate objectsas small as 5 nm and up to hundreds of micrometers.
1. Conformational properties
The mean square end-to-end distance of flexible E-ABPO increases monotonically with increasing
P e , incontrast to S-ABPO. As for D-ABPO and S-ABPO,semiflexible polymers shrink initially with increasing
P e and swell again for large P´eclet numbers in a universalmanner. The comparison with the mean square end-to-end distance curves for D-ABPO reveals a stronger im-pact of activity on the conformations of E-ABPO. In par-ticular, flexible polymers swell and semiflexible polymersshrink already for smaller P´eclet numbers. However, thesame asymptotic limit is assumed for
P e → ∞ , whichfollows from Eq. (28) for γ R ˜ τ n (cid:28)
1. Hence, S-ABPOexhibit the more compact structures compared to both,D-ABPO and E-ABPO. In the universal, stiffness- andexcluded-volume-independent high-
P e regime, the meansquare end-to-end distance exhibits the power-law in-crease (cid:104) r e (cid:105) ∼ P e / with increasing P e . This increaseis weaker than that of D-ABPO, with the dependence (cid:104) r e (cid:105) ∼ P e / (cf. Fig. 5), which emphasizes the strongeffect of the internal dynamics on the active polymer con-formational properties. As for the shrinkage of the S-ABPO, enhanced swellingis related to a particular dependence of the mean squareend-to-end distance on the relaxation times, which isgiven by (cid:10) r e (cid:11) = 8 L (cid:88) n, odd (cid:18) k B T τ n πη + v lτ n γ R ˜ τ n (cid:19) . (28)Since τ n γ R ˜ τ n ≥ τ n γ R τ n ≥ ˜ τ n γ R ˜ τ n , (29)the active contribution in Eq. (28) is larger than that ofD-ABPO and S-ABPO, which implies a stronger overallswelling.
2. Dynamical properties
The distinct coupling between activity and hydrody-namics changes the activity-dependence of the relaxationtimes substantially, as illustrated in Fig. 8. Over a widerange of P´eclet numbers, ˜ τ of E-ABPO is larger thanthe relaxation time of D-ABPO and S-ABPO, which isreflected in the strong increase of the mean square end-to-end distance. The (longest) relaxation time decreaseswith in increasing P e , hence, γ R ˜ τ n (cid:28) P e → ∞ . Moreover, ˜ τ exhibits the same dependenceon P e as τ in absence of HI in that limit, which yields -3 -2 -1 -1 FIG. 8. Longest polymer relaxation time ˜ τ normalized bythe corresponding passive value ˜ τ as a function of the P´ecletnumber P e for flexible D-ABPO (green), S-ABPO (blue), andE-ABPO (red) with pL = 10 . the same asymptotic value for (cid:104) r e (cid:105) . We like to empha-size that ˜ τ of semiflexible polymers assumes the same P e dependence and scaling for
P e (cid:29) /γ R (cid:28) t (cid:28) ˜ τ dominated by their internal dynamics,where analytical theory predicts the power-law time de-pendence t / , which is close to t / . V. TANGENTIALLY DRIVEN FILAMENTS
Cytoskeleton polymers such as microtubules are ratherstiff and are typically denoted as filaments. They ex-hibit active motion fueled by molecular motors in vivo and in vitro . By the nature of theirpropulsion mechanism, such filaments are typically mod-eled as tangentially driven rodlike or semiflexi-ble polymers in molecular approaches.
A. Continuum model
The theoretical description of Sec. III B can be adaptedto the case of tangential driving semiflexible polymers,which yields the equation of motion γ ∂∂t r ( s, t ) = f a ∂∂s r ( s, t ) (30)+ 2 λk B T ∂ ∂s r ( s, t ) − (cid:15)k B T ∂ ∂s r ( s, t ) + Γ ( s, t ) , with the active force f a per unit length. This linear par-tial differential equation can be solved by an eigenfunc-tion expansion in terms of the eigenfunctions ψ n ( s ) of1the equation O ψ n ( s ) = − ξ n ψ n ( s ) with the operator O = f a ∂∂s + 2 λk B T ∂ ∂s − (cid:15)k B T ∂ ∂s (31)and the eigenvalues ξ n . However, the operator O is non-Hermitian, hence, an expansion into a biorthogonal ba-sis set has to be applied, with the adjoint eigenvalueequation O † ψ † n ( s ) = − ξ n ψ † n ( s ) and O † = − f a ∂∂s + 2 λk B T ∂ ∂s − (cid:15)k B T ∂ ∂s . (32)In case of a flexible polymers ( (cid:15) = 0), the eigenfunctionsare ψ n ( s ) = (cid:114) L (cid:112) k n + g e − gs [ k n cos( k n s ) + g sin( k n s )] , (33)with k n = nπ/L , g = f a / (4 λk B T ), and the eigenvalues ξ n = 2 λk B T ( k n + g ) for the boundary condition ∂ r /∂s =0 at s = 0 , L , which implies ∂ψ n /∂s = 0 at s = 0 , L .The adjunct eigenfunctions are ψ † n ( s ) = e gs ψ n ( s ), andobey the boundary condition dψ † n ( s ) /ds − gψ † n ( s ) = 0at s = 0 , L . The eigenfunction for the eigenvalue ξ = 0is ψ = (2 g/ ( e gL − / . B. Discrete model
The dynamics of discrete polar active filaments is de-scribed by Eq. (6), with bond, bending, and excluded-volume forces. However, for the active force on particle i , various nonequivalent representations have been ap-plied. In most cases, the active force is assumed to betangential to the chain at a monomer, F ai = f a r i +1 − r i − | r i +1 − r i − | = f a t i , (34)with the unit tangent vector t i . Other simulation studiesutilized push-pull type forces F ai = f a l [( r i +1 − r i ) + ( r i − r i − )] = f a l ( r i +1 − r i − ) , (35)or F ai = f a (cid:20) r i +1 − r i | r i +1 − r i | + r i − r i − | r i − r i − | (cid:21) . (36)For strong bond potentials and small bond-length vari-ations, the latter two variants are essentially identical,but differ from Eq. (34). All these forces yield the samecontinuum limit (Eq. 30). In simulations, we can expectresults to be independent of the adopted discretizationfor not too strong activities. However, deviations willemerge above a certain activity and, in particular, fora small number of monomers. This is reflected by thedifferences in the conformational properties of the con-tinuous filaments (Eq. (30)) and of discrete models (cf.Sec. V C). C. Results
The theoretical analysis of Sec. V A suggest that theconformational properties of flexible polar polymers areunaffected by activity. In particular, the mean squareend-to-end distance is equal to the value of the passivecounterpart.
This is in contrast to simulation re-sults for a discrete model, which indicate a shrinkage offlexible polymers at large active forces f a , for both,the active forces (34) as well as (36), although shrinkageis more pronounced for the force (34). The difference isa consequence of the particular discrete representation ofthe continuous polymers (cf. Sec. V B).So far, very little further analytical results for indi-vidual polar filaments have been derived, specifically forsemiflexible polymers, which might be related to the non-Hermitian character of the equations of motion.Confinement in two dimensions enhances the influ-ence of excluded-volume interactions on the emergingstructures. Here, semiflexible filaments exhibit a tran-sition to a spiral phase with increasing activity at P e ∼ l p /L . Tethered filaments start beating in thepresence of a polar force, or assume spiral shapes whenthe fixed end is able to rotate freely.
The filament dynamics depends on the composition ofits ends. As an example, a load at the leading end leadsto distinct locomotion patterns such as beating and rota-tion, depending on stiffness and propulsion strength.
Simulations predict a substantial influence of hydrody-namic interactions on the rotational dynamics of singleand multiple filaments as well on their collective dynam-ics and structure formation in 3D.
According to thedefinition of Sec. IV B, the latter polymers are externallyactuated by a tangential force of the type of Eq. (36).
VI. POLYMERS WITH SELF-PROPELLED COLLOIDALMONOMERS
Polymers can be composed of monomers with theirself-propulsion modeled by the concepts described inSec. II C, i.e., as squirmers or a more general expan-sion of the monomer flow field. Although various studieshave already been performed based on mobilities, only squirmer dumbbells have been considered so far.In contrast to the Brownian monomers of Secs. III andIV, the hydrodynamically self-propelled monomers in-clude higher order multipoles such as force dipole, sourcedipole, etc., which, by monomer-monomer interac-tions, give rise to additional features.
A. Linear squirmer assemblies
The swimming behavior of athermal squirmer dumb-bells has been investigated by the boundary integralmethod, where, in this context, swimming refers toballistic motion. The calculations show a strong effect of2
FIG. 9. Illustration of the flow field of a dumbbell composed of two spherical squirmers for their preferred propulsion directions.For (a) neutral squirmers ( β = 0) and (b) pullers ( β = 5), the propulsion direction is preferentially antiparallel. In the case of(c) a weak pusher ( β = − β = −
5) exhibit preferentially an arrangement as in (c) as well as an orthogonal arrangement as in (d). Note that the flowfields are superpositions of the flow fields of the individual squirmers. the interfering swimmer flow fields on their orientationand locomotion. In particular, a flow-induced torque pre-vents stable forward swimming of individually freely ro-tating spherical squirmer in a dumbbell. By introducinga (short) bond, which restricts an independent rotation,circular trajectories for pushers and stable side-by-sideswimming for pullers can be obtained.
Athermal squirmer lack rotational diffusion, thus, ab-sence of locomotion of freely rotating spheres in a dumb-bell is a consequence of the missing thermal fluctuationsof the fluid. Such fluctuations can be taken into accountby mesoscale hydrodynamic simulations, e.g., the mul-tiparticle collision dynamics (MPC) method, andlocomotion is obtained. At infinite dilution, rotation isan activity-independent degree of freedom of a squirmer(cf. Sec. II C 1), which, however, strongly affects its long-time diffusion behavior, in the same way as for an ABP(see Eq. (21) for comparison)Simulations of squirmer dumbbells yield preferred ori-entations of the propulsion directions, e i , with respectto each other and the bond connecting their centers, asillustrated in the snapshots of Fig. 9. This prefer-ence can be quantified by the average relative orientationof the propulsion direction, i.e., (cid:104) e · e (cid:105) . As expected forthe active stresses, in dumbbells of neutral squirmers,alignment is very weak, and only for high P e a slightlynegative value is obtained. Puller exhibit very strongorientational correlations, with a preferential antiparallelalignment (cf. Fig. 9 (a), (b)). Strong correlations arealso present for pushers, however, with a parallel align-ment of the e i orthogonal with respect to the bond be-tween the squirmers (Fig. 9(c)), and a near orthogonalarrangement as show in Fig. 9(d).The preference in orientation is reflected in the dumb-bell dynamics. The locomotion of the pusher dumbbellis closest to the dynamics of an ABP, specifically for β ≈ −
1. Significant deviations to an ABP dumbbell ap-pear for smaller and large β . In particular, the preferredantiparallel alignment for neutral squirmers and pullers leads to a pronounced slow-down of the dumbbell dynam-ics with a rather abrupt change at β (cid:38)
0. Notably, thelong-time diffusion coefficient of puller dumbbells at largeactive stresses, β = 5, is reduced by orders of magnitudecompared to that of ABP dumbbells. This is inline withthe theoretical studies, however, thermal fluctuationsare responsible for a finite diffusion coefficient at longtimes.Hydrodynamic interactions between the monomer flowfields strongly affect the conformational properties ofsquirmer polymers. Simulations of dodecamers yieldstrongest swelling for neutral squirmers, and essentiallyno swelling for pullers ( β = 5), with a mean square end-to-end distance close to the passive value. B. Active filaments of tangentially propelled monomers1. Colloidal monomers
The translational and rotational motion of chains oflinearly connected monomers can be described by a setof Langevin equations.
The equations are based onthe integral solution of Stokes flow, which is evaluatedanalytically by an expansion of the boundary fields intensorial spherical harmonics. The omission of hydrody-namic many-body effects by the assumption of pair-wiseadditivity yields a computationally tractable approxima-tion in terms of the Oseen tensor and its derivatives.
For tangentially driven filaments, the orientation of amonomer is no longer a dynamical degree of freedom,and the equations of motion reduce to translation only.In contrast to the Brownian semiflexible polymers andfilaments of Secs. III, IV, and V, where activity originatesfrom an active velocity or force, the presence of higher or-der hydrodynamic multipoles yields active motion evenin absence of a propulsion velocity. This is due to thegeneration of flows perpendicular to the tangent vectorfor a curved contour (cf. Fig. 10) This gives rise to partic-3 -12 -11 -10 -9 -8 -7 -6 -5 -4
FIG. 10. Nonequilibrium stationary states of a single-endtethered filament. Increasing activity (from left to right) leadsto particular stationary-state conformations and rotation (g,i), or filament beating (h). The background color indicatesthe logarithm of the magnitude of the fluid velocity. Repro-duced from Ref. 120 with permission from the Royal Societyof Chemistry. “Reproduced with permission from Soft Mat-ter , 9073 (2015). Copyright 2015 The Royal Society ofChemistry.” ular stationary-state filament conformations and dynam-ical states, such as rotation of free filaments or beatingand rotation of tethered filaments (cf. Fig. 10). Thermal noise will modify this behavior, to an extentwhich remains to be investigated.
2. Pointlike monomers
A somewhat simpler approach for hydrodynami-cally self-propelled, pointlike ”monomers” has beensuggested, where Eq. (1) is employed for the monomerdynamics, however, with constraint rather than harmonicbond forces, and the force-dipole-type active force for abond/link u i +1 , F ai = − F u i +1 , F ai +1 = F u i +1 , (37)where F is magnitude of the force, and its sign impliesextensile ( F >
0) or contractile ( F <
0) dipoles. Theactive forces are assumed to arise from stresses exertedby molecular motors temporarily attached to a bond.The time interval of the attached and detached statesare taken from an exponential distribution.
The ac-tive velocity in the attached state is then v ai = (cid:88) j G ( r i − r j ) F aj ( t ) . (38)Considering a single, long, and densely packed polymer ina spherical cavity, simulations show that hydrodynamicinteractions between extensile dipoles can lead to large-scale coherent motion (cf. Sec. VII). VII. COLLECTIVE BEHAVIOR
Already ABPs and active dumbbells exhibit an intrigu-ing collective behavior, especially motility-induced phase separation.
Steric interactionsby the extended shape of filaments and polymers, andtheir conformational degrees of freedom lead to furthernovel phenomena and behaviors.Assays of microtubules mixed with molecular mo-tors are paramount examples for collective effectsemerging in out-of-equilibrium systems includingpolymers/filaments.
Simplified systems com-prising microtubules and kinesin motor constructsare able to self-organize into stable structures. Themotors dynamically crosslink the microtubules andtheir directed motion along the polar filaments leads toformation of vortex-like structures and asters with themicrotubules arranged radially outward (Fig. 11(a)). Along the same line, highly concentrated actin filamentspropelled by molecular motors in a motility assaydisplay emergent collective motion.
Above a criticaldensity, the filaments self-organize into coherentlymoving structures with persistent density modulations,such as clusters, swirls, and interconnected bands(Fig. 11(b)). Furthermore, addition of a depletionagent to a concentrated microtubule-kinesin mixtureleads to microtubule assemblance into bundles, hundredsof microns long.
Kinesin clusters in bundlesof microtubules of different polarity induce filamentsliding and trigger their extension. At high enoughconcentration, the microtubules form a percolatingactive network characterized by internally driven chaoticflows, hydrodynamic instabilities, enhanced transport,and fluid mixing. Activity destroys long-range nematicordering and leads to active turbulence, with short-rangenematic order and dynamically creation and annihilationof topological defects (Fig. 11(c)).
The instability of the nematic phase of long polar fil-aments is usually attributed to the pusher-like hydrody-namics of systems with extensile force dipoles. A sinu-soidal perturbation of the nematic order, with a wavevector along the filament direction, is amplified by thedipole flow field.
However, a similar type of undulationinstability is observed in simulations of semiflexible po-lar filaments, which are temporarily connected by motorproteins sliding them against each other—without anyhydrodynamic interactions (cf. Fig. 12). This systemis extensile, because motors mainly exert forces on an-tiparallel filaments. For an initially nematic phase withrandom filament orientation, first a polarity-sorting intobands occurs, followed by buckling of the bands, anda transition to an isotropic phase with polar domains(cf. Fig. 12).
The isotropic phase is characterizedby nematic-like +1 / − / Here, it is interesting to note that (i) ac-tive nematics are actually quite difficult to model on thefilament level, because filaments move actively, but can-not have a preferred direction, and (ii) the filaments areactually polar , so that the system has both polar and ne-matic characteristics at the same time. This system hastherefore been termed a polar active nematic in Ref. 239.At surfaces, various flagellated bacteria alter their mor-4
FIG. 11. Large-scale patterns formed by self-organization of biological filaments and molecular motors. (a) Lattice of astersand vortices formed by microtubules. (b) Motility assay with fronts of polar actin clusters moving in the same direction as thefilaments. (c) Network of nematically ordered microtubules exhibiting turbulent motion with a +1 / − / (a) “Reproduced with permission from Nature , 305 (1997). Copyright 1997 Macmillan PublishersLtd.” (b) “Reproduced with permission from Science , 255 (2018). Copyright 2018 AAAS.” (c) “A. Doostmohammadi, J.Ign´es-Mullol, J. M. Yeomans, and F. Sag´ues, Nat. Commun. , 3246 (2018); licensed under a Creative Commons Attribution(CC BY) license.”FIG. 12. Euler-like buckling instability of polar bands by semiflexible, propelled filaments. The snapshots depict a timesequence from (a) the development of polarity-sorted bands, (b), (c) progressive bending and breaking of bands, and finally(d) the formation of the long-time disordered structure. “G. Vliegenthart, A. Ravichandran, M. Ripoll, T. Auth, and G.Gompper, Sci. Adv. in press (2020); licensed under a Creative Commons Attribution (CC BY) license.” phologies as they become more elongated by suppressionof cell division. These so called swarmer cells migratecollectively over surfaces and are able to form stable ag-gregates, which can be highly motile (swarming). Some cells, such as
Proteus mirabilis or Vibrio para-haemolyticus become rather elongated ( > − µm )and polymer-like. Since they are propelled by flagella,which are shorter than the body length and distributedall over the body, locomotion of individual cell is mainlytangential. Figure 13 shows that such cells can un-dergo large conformational changes from nearly rodliketo strongly bend structures while moving collectively ina film with other cells. As actin and microtubule fila-ments in dense suspensions, the swarmer cells exhibit lo-cal nematic order with defects and large-scale collectivemotion.The collective effects of tangentially driven filamentshave been studied by simulations in 2D, and new phases depending on density, stiffness, activity, and aspect ra-tio have been identified.
At moderate densitiesand activities, stiff filaments organize into mobile clus-ters, structures which are reminiscent to those formedby self-propelled rods.
High activities andlow bending rigidity yield spiral formation of individ-ual filaments, which then translate as compact disc-like objects.
With increasing density, spiral collisionsyield again motile clusters.
At moderate densities, areentrant behavior is observed with increasing activity,from small clusters to giant clusters and back to smallclusters.
Here, the disintegration of the giant clusteroccurs when
P e ∼ l p /L , i.e. when active forces be-come comparable to bending forces. A similar thresh-old of P e ∼ l p /L is seen for the transition from open(equilibrium-polymer-like) conformations to spirals inthe dilute case, due to the same force competition. Athigher densities, a rich collective behavior is observed,5 FIG. 13. Image of
P. mirabilis swarmer cells in a colony ac-tively moving across a surface. The green, generally straightcell illustrates the ability of swarmer cells to bend substan-tially. “G. K. Auer, P. M. Oliver, M. Rajendram, T.-Y. Lin,Q. Yao, G. J. Jensen, and D. B. Weibel, mBio , e00210(2019); licensed under a Creative Commons Attribution (CCBY) license.” such as a melt-like structures with topological defects. Even higher densities lead to jammed states of semiflexi-ble filaments at small P´eclet numbers, followed by laningand active turbulence for higher activities.
It is impor-tant to emphasize that this active turbulence occurs in asystem without momentum conservation, and thereforeat zero Reynolds number. Nevertheless, the turbulentstate is characterized by a power-law decay of the kinetic-energy spectrum as a function of wave vector, reminiscentof Kolmogorov turbulence. The turbulent phase exhibitstopological defects and a dynamics similar to the behav-ior seen theoretically in active nematics and experimen-tally in solutions of microtubules in presence of molecularmotors.
A flexible polymer composed of hydrodynamicallyself-propelled, pointlike, and extensile force dipoles (cf.Sec. VI B 2), confined in a sphere, as a model of chromatinorganized in the cell nucleus, shows large-scale coherentmotion (cf. Fig. 15). The extensile force, mimicking a lo-cal molecular motor, reorganizes the polymer by stretch-ing it into long and mutually aligned segments. This pro-cess is driven by the long-range flows generated along thechain by motor activity, which tends to both straightenthe polymer locally and to align nearby regions.
Asa result, large patches with high nematic order appear.Studies of passive and contractile polymers show that thistransition to a highly coherent state is linked with theextensile dipole, which emphasizes the interplay betweenconnectivity of dipoles in a chain and hydrodynamic in-teractions mediated by the embedding fluid.
FIG. 14. (a) Phase diagram of tangentially driven semiflexi-ble filaments as a function of l p /L = 1 / (2 pL ) and P e , wherethe P´eclet numbers is defined as
P e = f a L / ( k B T ). Cyantriangles indicate the melt phase, magenta circles the gas ofclusters, and the red squares the giant clusters phase. Greenfilled symbols depict the simulations with high aspect ratiofilaments, L/l = 100, while the other points correspond to
L/l = 25. (b) Snapshot of a high density turbulent phase at
P e = 90. The filament colors are chosen randomly. “Re-produced with permission from Soft Matter , 4483 (2018).Copyright 2018 The Royal Society of Chemistry.” VIII. CONCLUSIONS AND OUTLOOK
We have demonstrated that the coupling of polymerdegrees of freedom and active driving leads to manynovel phenomena with respect to polymer conformationaland dynamical properties. Depending on the natureof the active process—random vs. tangential driving—polymers swell or collapse, and their dynamics is typ-ically enhanced. Moreover, hydrodynamic coupling ofthe flow field by the active monomeric units leads tolong-range correlations and coherent motion. In ensem-6
FIG. 15. Simulations of active chains with extensile dipoles. (a) Filament configurations at different times; the red segmentsindicate instantaneous dipole locations. (b) The nematic structure of the filaments in a shell adjacent to the surface of theconfining sphere. The color code indicates the nematic order tensor. (c) Chromatin displacement map calculated in a planeacross the spherical domain over a time interval ∆t = 0 .
2. “Reproduced with permission from , 1144263 (2018). Copyright2018 National Academy of Sciences.” bles of active polymers, steric and hydrodynamic interac-tions imply novel long-range collective turbulent motionwith large-scale patterns. The various aspects emphasizethe uniqueness of systems comprised of active polymersand filaments, and renders active soft matter a promisingclass of new materials.
We only begin to understand and to unravel the prop-erties of polymeric active-matter systems. The currentperspective article describes theoretical and modeling ap-proaches and major results of mainly individual polymersand filaments. These approaches establish the basis forfurther studies in various directions.
Phase behavior —As mentioned, active particles andfilaments exhibit novel collective phenomena and newphases such as MIPS. Typically two dimensional (2D)systems are considered, much less attention has beenpaid to three dimensional (3D) realizations. However,we can expect the appearance of novel structures andphases in 3D. First studies of active nematics consist-ing of microtubules and kinesin molecular motor yielda chaotic dynamics of the entire system.
The analy-sis of spatial gradients of the director field show regionswith large elastic distortions, which mainly form curvi-linear structures as either isolated loops or belong to acomplex network of system-spanning lines. These distor-tions are topological disclination lines characteristic of3D nematics.
Far less is know about the phase behavior of activeBrownian polymers, both in 2D and 3D. Comparison ofresults for ABPs and active dumbbells in 2D indicate ashift of the critical activity for the onset of the MIPSto higher values.
We can expect a further shift of thecritical point for longer polymers, and a possible suppres-sion of the phase transition beyond a particular polymerlength.
Confinement —Confinement will alter the propertiesof the active polymers substantially. Already sphericalABPs accumulate at walls. We expect an even strongeraccumulation for active polymers. As an example, sim-ulations of D-ABPO confined in a slit yield deviations in the scaling behavior of the wall force established forpassive polymers.
Active-passive mixtures —New phenomena appear inmixtures of active and passive components. Alreadysystems of ABPs mixed with passive colloids exhibitphase separation and a collective interfacedynamics.
Passive semiflexible polymers embedded inan active bath of ABPs exhibit novel transient states in2D, where an activity-induced bending of thepolymers implies an asymmetric exposure to active par-ticles, with ABPs accumulating in regions of highest cur-vature, as has been observed for ABPs in confinement.
This leads to particular polymer conformations such ashairpins, structures which are only temporarily stableand dissolve and rebuild in the course of time.The presence of both active and passive componentsis a hallmark of living systems. So far, the impactof activity on the properties of the passive componentsis unclear. Simulations of two-component mixtures ofpolymers at different temperatures, which is an alter-native approach to establish a nonequilibrium state, yield phase separation of the two components. Here,the two temperatures account for example for the ac-tivity of hetero- and euchromatin, which could play arole in chromation separation in the cell nucleus.
Al-though temperature control is an acceptable way of im-plementing activity differences, other out-of-equilibriumprocesses should be considered as well. Cells ex-hibit coherent structures—so-called membraneless or-ganelles or condensates—encompassing and concentrat-ing specific molecules such as proteins and RNA in thecytoplasm.
In-vivo experiments suggest that activeprocesses, which occur constantly within such organelles,play a role in their formation.
An understanding ofthe interplay between equilibrium thermodynamic driv-ing forces and nonequilibrium activity in organelle for-mation is fundamental in the strive to elucidate theirfunctional properties, and their contribution to cell phys-iology and diseases.
Active-passive copolymers —Passive copolymer systems7self-assemble into ordered and tuneable structures in awide range of morphologies, including spheres, cylinders,bicontinuous phases, lamellae, vesicles, and many othercomplex or hierarchical assemblies, driven by the pre-ferred attractive and repulsive interactions between thedifferent constituents of the copolymers.
Copoly-mers composed of monomers of different activity willshow an even richer phase and dynamical behavior, andwill provide additional means to control structure forma-tion and transport. Since biological macromolecules arerather intrinsically disorder than homogeneous, studies ofcopolymers will improve our understanding of structureformation in biological cells, for active polymers as well asfor passive copolymers in an active bath. Specifically forpolymeric assemblies of hydrodynamically self-propelledmonomers, a strong hydrodynamical monomer-monomercoupling can be expected, e.g., by monomers with differ-ent active stress or diameter, leading to a rich dynamicsand particular self-organized structures.
Assemblingseveral active/passive monomers into an unit may evenbe a route to design intelligent active particles, whichsense and respond autonomously to other units and showswarm-intelligent behavior as schools of fish or swarms ofbirds.
Active turbulence —As mentioned several times, activenematics and bacteria exhibit chaotic, turbulent behav-ior. Up to now, similarities or differences to Kolmogorov-type turbulence of large Reynolds number fluids havenot been resolved satisfactorily. In particular, the impactof polymer degrees of freedom needs to be investigated.
Hydrodynamic interactions —Various of the discussedexamples highlight the relevance of hydrodynamic inter-actions for the conformational, dynamical, and collectiveeffects of active polymers. Here, a broad range of stud-ies are required to unravel hydrodynamics effects on thephase behavior of active polymers, their properties underconfinement and under flow.
Rheology —Experiments show that the viscosity of a di-lute suspension of swimming bacteria is lower than thatof the fluid medium.
This is a consequence of theinterplay between bacteria (pushers) alignment with theflow and stress generation by the microswimmers, whichenhances the applied shear stress and leads to an appar-ent viscosity reduction that increases with increasing vol-ume fraction of cells.
Even “superfluidity” is obtainedin bacterial suspensions at the onset of nonlinear flow andbacteria collective motion.
Certainly, various of theobserved aspects are of hydrodynamic origin and dependcrucially on the flow field of the microswimmer. Analyti-cal calculations of dilute dumbbell and D-ABPO systemsalready reveal an influence of activity on the zero-shearviscosity and on the shear thinning behavior.
On amore macroscopic scale, shear experiments on long, slen-der, and entangled living worms (
Tubifex tubifex ) showthat shear thinning is reduced by activity and the con-centration dependence of the low-shear viscosity exhibitsa different scaling from that of regular polymers.
Thishighlights the wide range of rheological phenomena in ac- tive matter. Clearly, systematic studies of various kindsof active polymers are needed to shed light onto theirrheological behavior. Ultimately, such polymers mightbe useful as rheological modifiers.In summary, there is a wide range of interesting as-pects of active polymers, ranging from individual poly-mers with their different driving mechanisms, to theircollective properties as a function of concentration, torheological aspects. Simulations and analytical theories,which are based on the described models and approaches,will certainly play a decisive role in the elucidation of themany novel and unexpected nonequilibrium features ofthese systems.
DATA AVAILABILITY
The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest.
ACKNOWLEDGMENTS
We thank C. Abaurrea-Velasco, T. Auth, J. Clopes,O. Duman, T. Eisenstecken, J. Elgeti, D. A. Fedosov,A. Ghavami, R. Isele-Holder, A. Martin Gomez, C.Philipps, A. Ravichandran, M. Ripoll, and G. A. Vliegen-thart, for enjoyable collaborations and stimulating dis-cussions. We gratefully acknowledge partial supportfrom the DFG within the priority program SPP 1726on Microswimmers–from Single Particle Motion to Col-lective Behaviour. A computing-time grant on the su-percomputer JURECA at J¨ulich Supercomputing Centre(JSC) is thankfully acknowledged.
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