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Single Active Ring Model † Emanuel F. Teixeira, ∗ a Heitor C. M. Fernandes, a ‡ and Leonardo G. Brunnet a Cellular tissue behavior is a multiscale problem. At the cell level, out of equilibrium, biochemi-cal reactions drive physical cell-cell interactions in a typical active matter process. Cell modelingcomputer simulations are a robust tool to explore the countless possibilities and test hypotheses.Here, we introduce a two dimensional, extended active matter model for biological cells. A ring ofinterconnected self-propelled particles represents the cell. Translational modes, rotational modes,and mixtures of these appear as collective states. Using analytic results derived from active Brow-nian particles, we identify effective characteristic time scales for ballistic and diffusive movements.Finite-size scale investigation shows that the ring diffusion increases linearly with its size when incollective movement. A study on the ring shape reveals that all collective states are present evenwhen bending forces are weak. In that case, when in translational mode, the collective velocity alignswith the largest ring’s direction in a spontaneous polarization emergence.
Active matter systems are constructed based on interacting el-ements that move using energy or mass fluxes, resulting in anemerging complex behavior . Cells in living tissues are physi-cally active elements playing the role prescribed by the underlyingbiochemical system. Wound healing, morphogenesis, and tumorevolution are essential processes in living organisms and motivateresearch on phenomena related to multi-cellular organization .Computational modeling may identify essential physical ingredi-ents responsible for tissue regenerative behavior . Hypothesisconcerning cell segregation, such as Differential Adhesion andDifferent Velocities , were simulated based on simple point-like,active matter models . However, more sophisticated hypothe-ses taking into account cell cortex tension, such as SuperficialContraction , can not be explored using these models.Cell movement depends on an internal actin fiber structure,which polymerizes or depolymerizes as different membrane partsreach substrate regions with fluctuating rigidity or experimentchemical gradients . To describe the physical forces undersuch a fine structure, extended cell models come into play. Monte-Carlo based model, such as GGH , or more recently, the vertexmodel and the phase-field approach , came in to fill thisgap. These models use energy fluctuations or field equations intheir description. We present here a complementary approach a Instituto de Física, Universidade Federal do Rio Grande do Sul, CP 15051,CEP 91501-970 Porto Alegre - RS, Brazil; E-mail: [email protected],[email protected], [email protected] † Electronic Supplementary Information (ESI) available: [details of anysupplementary information available should be included here]. See DOI:10.1039/cXsm00000x/ based on active molecular dynamics.In this work, we present an extended cell model for active sys-tems able to contemplate several features of other models whilekeeping its simplicity and physical appeal. A model cell is con-structed based on a set of active particles connected by springsand subject to a bending potential, forming a ring. Here we showthe different dynamical states a single ring may assume usingwell-known order parameters to identify collective translation .We also calculate a slightly modified version of the group angu-lar momentum as an order parameter to characterize the singlering collective rotation. We study the ring diffusion and frame itin the context of active Brownian particles using known analyti-cal solution limits and experimental observations . Finally, weinvestigate its shape and size change under different parametersusing the gyration tensor.The paper is structured as follows: In Sec. 2, we present themodel and simulation details; in Sec. 3 results for quantities usedto characterize the behavior collective motion, Sec. 3.1, meansquare displacement and its effective parameters, Sec. 3.2, andring’s morphology, Sec. 3.3; In Section 4, we present our conclu-sions and summarize the results. We model the cell as a ring formed by N active particles heldtogether by N bonds and subject to bending forces (see Fig. 1).This last interaction plays two roles: prevent ring collapse anddetermine its shape, in the absence of other forces, while allow-ing membrane fluctuations. Our two-dimensional system lies ina square box with periodic boundary conditions. We neglect in-ertial effects supposing a low-Reynolds-number regime . The Journal Name, [year], [vol.] , a r X i v : . [ c ond - m a t . s o f t ] F e b verdamped equations governing each particle dynamics are dd t (cid:126) r i ( t ) = v ˆ n i − µ ∑ i ∼ j ∇ U ( (cid:126) r i j ) + (cid:112) D T (cid:126) χ i ( t ) (1) dd t θ i ( t ) = τ (cid:48) arcsin (cid:18) ˆ n i ( t ) × (cid:126) v i ( t ) | (cid:126) v i ( t ) | · ˆ e z (cid:19) + (cid:112) D R ξ i ( t ) (2)where (cid:126) r i ( t ) = ( x i ( t ) , y i ( t )) denotes the i -th particle’s position attime t , µ its mobility, and v the magnitude of the self-propellingvelocity along with its orientation, ˆ n i ( t ) = ( cos θ i ( t ) , sin θ i ( t )) . Thedirection of the self-propelling velocity described by the angle θ i ( t ) , relaxes towards (cid:126) v i ≡ d (cid:126) r i / dt within a characteristic time τ (cid:48) ,while also experiencing angular Gaussian white noise ξ i withzero-mean and second moment (cid:10) ξ i ( t ) ξ j ( t ) (cid:11) = δ i j δ ( t − t ) in-dependently for each particle at each time-step. D R is the rota-tional diffusion coefficient and defines a typical timescale, τ R ≡ / D R , for changes due to angular noise. When translationalnoise is present, each self-propelled particle position is subjectto Gaussian noise with zero-mean and variance (cid:10) (cid:126) χ i ( t ) .(cid:126) χ j ( t ) (cid:11) = δ i j δ ( t − t ) . D T is the translational diffusion coefficient and de-fines a characteristic timescale, τ T ≡ σ / D T , for a particle to dif-fuse a length of the order of its size, σ .The derivatives of the inter-particle bond, bending, andexcluded-volume (EV) potentials generate the forces on each par-ticle, U = U bond + U bend + U EV . (3)For the bond term we use a harmonic potential, U bond = k i + ∑ j = i ( | (cid:126) d j | − r ) , (4)where (cid:126) d j = (cid:126) r j − (cid:126) r j − is the bond vector connecting consecutiveparticles in the ring (see Fig. 1), k is the spring constant and r isthe equilibrium bond length. We introduce a bending potential tocontrol the ring rigidity , U bend = k b i + ∑ j = i (cid:126) d j .(cid:126) d j − | (cid:126) d j || (cid:126) d j − | , (5) k b is the bending rigidity. We model the excluded-volume interac-tion among particles using a Weeks–Chandler–Anderson potential(WCA), U EV = ε (cid:40)(cid:0) σ / r i j (cid:1) − (cid:0) σ / r i j (cid:1) , if r i j < / σ , if r i j ≥ / σ (6) r i j = | (cid:126) r i ( t ) − (cid:126) r j ( t ) | , ε , σ are the distance between particles i and j ,characteristic exclusion volume energy and effective diameter ofa given particle, respectively. It is useful to identify dimensionless parameters to control systemdynamics. From Eq. 1 and Eq. 2, we define B ≡ τ R τ T , (7) Fig. 1
Sketch of a ring segment illustrating vectors, angles and distancesused in the model definition. See text for details. when τ T → ∞ , translational noise is irrelevant and B → . Pa-rameter B measures the relative importance between angular andtranslational noise. Another dimensionless parameter is the rota-tional Péclet number , Pe ≡ τ R τ , (8)with τ ≡ σ / v . Pe relates rotational diffusion and movement’spersistence time.Also, we follow the work by Duman and collaborators , anddefine the flexure number as Fn ≡ N σ ( τ − + τ − T ) µ k b , (9)parameters τ − and τ − T play the role of deforming forces, whilethe bending force, k b , tends to restore the circular shape.Initial conditions specification follows equations, (cid:126) r i ( t = ) = R cos φ i ˆ e x + R sin φ i ˆ e y , (10) ˆ n i ( t = ) = [( − β ) + β sin φ i ] ˆ e x − β cos φ i ˆ e y (cid:113) [( − β ) + β sin φ i ] + β cos φ i , (11)where R = Nr / π , r = / σ , φ i = ( i − ) r / R and i = , ,... N . Weinitialize the ring with a circular shape. Parameter β defines theinitial polarization, ˆ n i , for each particle, β = implies a circularpolarization and β = a parallel one, see Fig. 2. β = 0 β = 1 Fig. 2
Initialization of positions and velocities used in simulation. Par-ticles are disposed in a circle with radius R . Self-propelled velocitiesdirection, ˆ n i , initialization is determined through parameter β ; β = setstranslational motion configuration and β = sets a rotational one. We integrate the basic dynamic equations, Eq. 1 and Eq. 2, us-ing Euler method with a time step in the range ∆ t / τ = [ − : Journal Name, [year], [vol.][vol.]
Initialization of positions and velocities used in simulation. Par-ticles are disposed in a circle with radius R . Self-propelled velocitiesdirection, ˆ n i , initialization is determined through parameter β ; β = setstranslational motion configuration and β = sets a rotational one. We integrate the basic dynamic equations, Eq. 1 and Eq. 2, us-ing Euler method with a time step in the range ∆ t / τ = [ − : Journal Name, [year], [vol.][vol.] , . − ] . Along this work, we use k = ε / σ , v = . , ε = , µ = and σ = . We also define τ ≡ τ (cid:48) / τ . This choice of param-eters guarantees k sufficiently large to render bond length close r . Through the rest of paper, time is in units of τ and we referto reduced time, t / τ , just as t for sake of simplification. A well defined velocity correlation is the signature for collectivemotion . Here we identify both translational and rotationalorders. To quantify translation we use ϕ ( t ) = N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N ∑ i (cid:126) v i ( t ) | (cid:126) v i ( t ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (12)which measures whether self-propelled velocities are aligned pro-moting translational collective movement. To quantify rotationwe use Γ ( t ) ≡ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N ∑ i (cid:126) r i , CM ( t ) × (cid:126) v i ( t ) | (cid:126) r i , CM ( t ) || (cid:126) v i ( t ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (13)where (cid:126) r i , CM ( t ) = (cid:126) r i ( t ) − (cid:126) R CM ( t ) and (cid:126) R CM ( t ) is the center of mass(CM) position. We use here a normalized sum of particles’ angu-lar momentum, a definition close to the one introduced by Erd-mann et all . In collective translation ϕ ( t ) → and Γ ( t ) → . Theopposite happens in collective rotation, ϕ ( t ) → and Γ ( t ) → . We start varying the dimensionless parameters Pe , τ and β at con-stant particle number, flexure number and null translational noise( B = ). We measure order parameters associated with the statesof motion , Eqs. 12 and 13.For N = , τ = . and Pe = , the ring reaches a stationarystate induced by its initial configuration. That is, if β = , thesystem enters a translational collective (RUN) motion. Figure 3adisplays the evolution of the order parameters ϕ and Γ . Figure 6cillustrates the center of mass typical trajectory. When β = , itenters a rotational (ROT) state. Figures 3b and 6d show the or-der parameters and the center of mass trajectory, respectively. For Pe = and τ = , the center of mass performs a persistent ran-dom walk (PRW). Fig. 3c illustrates this observation, and Fig. 6apresents a center of mass trajectory for the same state, but for dif-ferent parameters. In an intermediary parameter region, τ = . and Pe = . , the ring switches between translation and rotationin a run and rotate mode (RRM). This happens independently ofthe initial configurations. Figure 3d shows the time series and theprobability distribution function correspondent to the state. InFig. 6b, we plot a typical center of mass trajectory.To characterize these different modes, we define an order pa-rameter to identify collective motion regardless of its type: O ( t ) ≡ ϕ ( t ) + Γ ( t ) . (14)Collective motion is present when O ( t ) → . Our systematic sim-ulations resulted in the state diagram ( Pe × τ ) for two differentinitializations ( β = and β = ). Fig. 4 shows the case β = ,case β = is similar (not shown). So, independently of initialization, we observe in Fig. 4a-b aregion where collective motion is settled ( (cid:104) O (cid:105) → , yellow region)and another one with small values of (cid:104) O (cid:105) (purple region) sepa-rated by an intermediary region (orange and red). Symbols (cid:104) . (cid:105) indicate time averages.When τ / Pe (cid:28) , we find (cid:104) O (cid:105) → . Meaning that, when the self-propelled velocity orientation relaxation time, τ , is much smallerthan the rotational noise time scale, τ R , particles quickly aligntheir self-propelled velocities, ˆ n , in the direction of the scatteredvelocity. In this limit, and noting that each particle is always inter-acting with at least two neighboring particles, velocity alignmentwill occur according to the mechanism described in Ref. andthe initial condition determines whether the system will be in ro-tational or translational collective motion, see Figs. 3a and 3b.On the other side, when τ / Pe (cid:29) , angular noise destroys col-lective motion implying that (cid:104) O (cid:105) → . Self-propelled velocities ofdifferent particles become uncorrelated, the center of mass mo-tion resulting from a sum of random displacements. This limitcorresponds to the case of active Brownian particles (ABP), aprototypical model to study competition between noise and self-propulsion effects . In a ring with a small number of particles(Fig. 4a and Fig. 4c), we observe a mean value above zero (purplecolor) for O ( t ) in the disordered state, this happens because fluc-tuations δ φ and δ Γ of both order parameters scale with / √ N , sofluctuations in O ( t ) scale with / N (Fig. 5a). In the Supp. Mate-rial C, we analyze the dependence of δ ϕ with √ N in the ABP limit( τ / Pe (cid:29) ). For small number of particles, fluctuations are highin the RW region. For large particle numbers, the system reachesa disordered state with small fluctuations, as shown in Fig. 4b,Fig. 4d and Fig. 5a.In the intermediary region which separates disordered and or-dered states, τ / Pe ∼ , fluctuations of both order parameter, ϕ ( t ) and Γ ( t ) , increase indicating the emergence of a distinct motionstate where the ring switches between rotation and translation.We call this run and rotate motion (RRM) and show a typicaltime series in Fig. 3d. Since parameters ϕ and Γ are complemen-tary in the RRM state, we use their fluctuations δ ϕ and δ Γ , tostudy it. When both fluctuations are close to zero, the system isout of the RRM state, being either in collective or in persistentrandom walk states. Comparing Fig. 4c and Fig. 4d, we observethe shrinking of the RRM region as the ring particles’ number in-creases from N = to N = . The decay in the fluctuations δ φ and δ Γ ( Fig. 5b) with N confirms this tendency: they decrease upto N = , remaining nearly constant for larger N values. Finally,it is interesting to note in Figures 5c and 5d that the fluctuationsscale as N . In both cases, the system presents collective motion.For comparison, we use the same scales of Figures 5a and 5b. In this section, we characterize how the ring’s center of mass be-haves in the different motion states. To illustrate, in Fig. 6 weshow typical center of mass trajectories for each motion state.Note the difference in scales in each case. The active ring has alonger reach when in RUN state (Fig. 6c). The ring diffusion ischaracterized by the center of mass mean-square displacement,
Journal Name, [year], [vol.] , t PDF (a) t PDF (b) t PDF (c) ϕ (t) Γ (t) t PDF (d)
Fig. 3
Steady state time series of the translational order parameter ϕ ( t ) (purple) and the rotational one Γ ( t ) (blue) for a ring composed of N = particles, no translational noise, B = , and flexure number Fn = . Corresponding probability distribution functions (PDF) for whole time series arealso shown. Panels illustrate motions state observed in simulation for set of parameters ( Pe , τ , β ) . (a) RUN state with (1,0.1,0), where fast relaxationof self-propelled velocity in the direction of velocity ensures a permanent translational collective motion with high persistence time. In this case, ϕ ( t ) just fluctuates close to unity for all times. (b) ROT state with (1,0.1,1) in this case, Γ ( t ) just fluctuate close to unity value for all times. (c) PRWstate with (1,10,0), where rotational noise dominates angular dynamics causing all particles to behave as independent ones. (d) RRM state with(0.3,0.1,1); this case presents a dynamics where system alternates between RUN and ROT states; both order parameters alternate high and low valuesand a double PDF peak is observed. See Movies1-4 in Suplem. Material. See text for more details. −3 −2 −1 τ (a) < O > N o C o ll ec ti v e Collective (b) −3 −2 −1 −2 −1 τ Pe R R M R U N o r R O T P R W (c) −2 −1 Pe δ ϕ (d) Fig. 4
State diagram ( Pe × τ ) for collective motion order parameter, (cid:104) O (cid:105) , upper panels, and fluctuation of translational collective motion orderparameter, δϕ , bottom panels. Left panels show results for systems with N = and right ones for N = , both use Fn = and are initialize with β = . The PRW behavior of CM is found in the ABP limit, at high τ and no collective motion. Single state collective motion settles for lowvalues of τ and higher values of Pe , where both contribute to increasepersistent motion of individual particles in such way that a mechanismsimilar to those of Ref. is observed. ROT or RUN states are obtaineddepending on initialization parameter β . At intermediary values of τ and Pe parameters, we observe RRM motion: during a simulation systemalternates between RUN and ROT states. This region, characterized byhigh fluctuation values, shrinks as ring’s size increases and disappears forlarge enough N . Note that parameter axes are in log scale but the colorbar is in linear scale. MSD , obtained by the sliding windowing method. In addition,we take averages over 40 trajectories . The correspondent cen-ter of mass MSD (Figure 7a, red curve) shows a long time in-terval in the ballistic ( ∝ t ) regime. In Fig. 7b we find a similarbehavior, but a short time diffusive regime appears due to transla-tional noise. In both cases, the behavior is diffusive for asymptot-ically large times, as expected. RRM state (Figure 6b and yellowcurves in Figures 7a,b), PRW state (Figure 6a and blue curves in −1 −1 −1 −1 ∝ N −1/2 ∝ N −1 (a) PRW RRM ∝ N (b) −2 −2 −2 −2
20 80 140 200 260 320 380 N RUN ∝ N (c)
20 80 140 200 260 320 380 N δϕδΓδ O ROT ∝ N (d) Fig. 5
Fluctuation of different collective motion order parameters asfunctions of N for a set of parameters ( Pe , τ , β ) with fixed B = and Fn = : (a) Region of absence of collective motion in the ABP limit, τ / Pe (cid:29) ,for parameters (1,100,0). In this region, the system presents a persistentrandom walk for the CM movement, resulting from the summation ofrandom variables, where fluctuations are expect to decrease with N − / .(b) Region RRM, (0.25,0.1,0), where after an initial increase with N itgets more difficult to system to change between RUN and ROT states.(c) and (d) regions of translational, (4,0.1,0), and rotational collectivemotion, (4,0.1,1), respectively. In both regions, fluctuations are almostindependent of N since all particles spend most time completely alignedin a translational motion or rotating around CM. Figures 7a,b) and ROT state (Figure 6d and green curves in Fig-ures 7a,b) present similar trends, but successively smaller ballisticregimes, implying smaller asymptotic diffusion constants.In the limit τ / Pe (cid:29) , each particle behaves as anABP , and the ring as a whole executes a PRW. In thislimit, there is an analytical solution for the MSD , which wedetail in Supp. Material A and B. In Fig. 8a we show the
MSD obtained for the active ring for different relaxation time values, τ , fixed Pe = and β = . In Fig. 8b we use the same parametersand include a small translational noise ( B = . ). In both cases, Journal Name, [year], [vol.][vol.]
MSD obtained for the active ring for different relaxation time values, τ , fixed Pe = and β = . In Fig. 8b we use the same parametersand include a small translational noise ( B = . ). In both cases, Journal Name, [year], [vol.][vol.] , ig. 6 Typical CM evolution in a xy plot for a set of parameters ( Pe , τ , β ) ,with N = , B = and Fn = kept fixed: (a) PRW motion, (0.1,0.3,0);(b) RRM motion, (0.3,0.1,0); (c) RUN motion, (2,0.1,0); and (d) ROTmotion, (2,0.1,1). Note that all panels have different xy ranges and thatRUN motion presents the wider displacement region due to persistentcollective motion of the CM. ROT state show very small displacementsince particles rotate around CM. RRM presents intermediate values ofdisplacements since it depends, essentially, in the fraction of time spendin RUN state. RW motion presents the expected behaviour of persistentmotion of ABP particles where there is no collective motion. Colorsindicate time (see color bar). −10 −8 −6 −4 −2 −3 −2 −1 M S D / σ t ∝ t ∝ t ∝ t (a) −10 −8 −6 −4 −2 −5 −4 −3 −2 −1 M S D / σ t RUNROTRRMPRW ∝ t ∝ t ∝ t (b) Fig. 7
Center of Mass Mean Square Displacement (
MSD ) for the sameparameters of Fig. 6 without (panel (a)), and with (panel (b)) transla-tional noise. Note, however, that for
MSD estimations, we also performaverages over repeated simulations with different noise. In both panels,solid lines correspond to fit of Eq. 15, while the dots indicate numericalresults. Fitted parameters are shown in Table 1. RUN state exhibitshighest values of persistent time and diffusion coefficient, followed byRRM state and RW. In ROT state, particles spend most of time rotat-ing around the CM implying no significant displacement of it. Whentranslational noise is present, B = . , ballistic regime is preceded by adiffusive one. increase in τ reduces the ballistic region extension resulting ina smaller long time diffusion. We note that behavior shown inFigs. 7 and 8 qualitatively resembles the exact solution for ABPs.This observation suggests the possibility of fitting effective param-eters combining translational and rotational noise time scales, MSD ( t ) = σ N τ T t + ( v e τ e ) (cid:20) t τ e + ( e − t / τ e − ) (cid:21) , (15) where τ e and v e are the effective persistence time and the effectiveself-propelled speed, respectively. In Supp. Mat. B, we detail therelation among v e and the center of mass mean square velocity.At long times, t (cid:29) τ e , we find D e f f = lim t (cid:29) τ e MSD ( t ) t = σ N τ T + v e τ e . (16)Table 1 displays fitted parameters for the center of mass MSD ,Eq. 15, of displacements of Figs. 6. Note the large increase in theeffective persistence time in the RUN state. In Fig. 8c and Fig. 8d, −8 −6 −4 −2 −3 −2 −1 M S D / σ t τ = 0.11210100 ∝ t ∝ t (a) PRWRUN −10 −8 −6 −4 −2 −5 −4 −3 −2 −1 M S D / σ t ∝ t ∝ t ∝ t (b) −1 v e / v τ (c) −1 −1 D e ff / v σ τ B = 0B = 0.001 (d)
Fig. 8
Center of Mass Mean Square Displacement (
MSD ) for systemswithout (panel (a)), and with translational noise ( B = . in Panel (b)).Parameters N = , Pe = , β = and Fn = are kept fixed. Both figuresshow different values of parameter τ from high values of it, correspond-ing to RW state (ABP limit and absence of collective motion), to lowvalues, corresponding to RUN state, where all particles move aligned tothe CM velocity. Solid lines correspond to fit of Eq. 15, while dots indi-cate numerical results. (c) Dependence of fitted effective self-propelledspeed v e with τ : in RUN state, all particles are aligned and move withvelocities close to v ; in PRW state, it approaches theoretical predictedvalue (dashed line). (d) Fitted effective constant diffusion as a functionof τ : note that the transition at low τ values. Note, also, that even asmall translational noise (panels (c) and (d)) is sufficient to displace theposition of transition to collective motion by an order of magnitude in τ . we also show the effective values for velocities and diffusion coef-ficients obtained fitting simulation data with that equation. Notethe transition between the low- τ / Pe , collective motion state, andhigh- τ / Pe , ABP state. τ / Pe limit In the ABP limit, or high- τ / Pe limit, the first term in Eq. 2 isnegligible and the rotational dynamics is dominated by rotationalnoise, that is, for N = , τ e → τ R , v e v = √ N ∼ . , Journal Name, [year], [vol.] , able 1 Parameters describing curves of Fig. 7 for fixed parameters N = and Fn = are shown. Effective persistence times and self-propelledvelocities are obtained by fitting Eq. [15] to the MSD simulation’s data. For comparison we show several ratios of these fitted quantities with othersystem parameters. Fig. 6 shows the trajectories of systems in the absence of translational noise. State of Motion Pe τ β B τ e / τ τ e / τ R v e / v D e f f / v σ Pers. Random Walk 0.1 0.3 0 0 0.41 2.05 0.32 0.02Pers. Random Walk 0.3 0.1 0 0.001 1.04 1.73 0.275 0.034Run and Rotate 0.3 0.1 0 0 10.6 17.68 0.786 3.15Run and Rotate 0.7 0.1 0 0.001 2.6 1.86 0.74 0.76Run 2 0.1 0 0 99.32 24.83 0.98 47.6Run 2.5 0.1 0 0.001 73.05 14.61 0.96 33.02Rotate 2 0.1 1 0 0.144 0.036 0.054 0.000169Rotate 2.5 0.1 1 0.001 0.55 0.11 0.089 0.00228as seen in Fig 8c. In this limit, Eq. 16 results in D e f f v σ = B NPe + PeN ∼ . , (17)as can be checked in Fig. 8d for N = . The first term in Eq. 17is negligible, since B = . . As τ decreases, both v e and D e f f depart from the ABP behavior (Figure 8c). Nevertheless, notethat even a small translational noise shifts this departure by oneorder of magnitude (Figure 8d). τ / Pe limit The ballistic regime is extended in this limit, resulting in largeeffective persistence times (Figures 8a and 8b). Also, the effectivevelocity reaches v (Figure 8c), so the whole ring achieves thefree single-particle speed. The consequence is a high diffusionconstant, as can be checked in Fig. 8d. We address the relationship between the ring’s size, N , andthe diffusion coefficient for the different motion states. Fig-ures 9a(RUN), 9b (threshold) and 9c (PRW) show the MSD fordifferent ring sizes (N=20,50,100) using parameter τ to controlthe system’s motion state at constant Pe . We exclude the ROTstate in this analysis. At low τ values we find the RUN state, thetransition to collective motion is set at τ ∼ , and above it wereach the PRW state. All curves present the form described byEq. 15, allowing us to fit the effective parameters.When in RUN state (Fig. 9a), all curves in the ballistic intervalcollapse, indicating the same effective self-propelling speed, v e ,with value close to v (Fig. 8c). We also observe in the inset ofFig. 9a the effective persistence time increasing as N increases.This is also clear in Fig. 9e where τ e is divided by N , and thecurves for the different system sizes coincide at small τ values.The overall result (Eq. 16), in this case, is that the diffusion coef-ficient scales with N , as can be checked in Fig. 9f at low Pe values.The MSD for systems in the threshold to collective motion,Fig. 9b, show curves slightly shifted downward as N increases,indicating a correspondent decrease in v e . On the other side, inthis region, τ e starts to grow with N ( Fig. 9e), resulting that D e remains independent of N on the transition, as indicated by thevertical bar in Fig. 9f. Beatrici previously found this result inthe context of cell segregation.When in PRW state, Fig. 9c, systems present the expected be- havior: for increasing N values, the CM moves less since particlesare uncorrelated. In Fig. 9d we multiply the effective velocitysquared by N to show that curves for different sizes collapse athigh- τ values. Since τ e is close to its single particle value (Table1), following Eq. 16, we expect the ring diffusion to scale withthe inverse of N . This is observed in Fig. 9f at large τ values.Fig. 9f summarizes the dependency of the diffusion constantwith the ring size in the different motion states. There is an evi-dent change in behavior with N : from a decrease in RW state toan increase in RUN state, crossing a region without dependence(marked with a vertical line) at the onset of collective motion. To characterize morphological changes in the shape of the activering, we use the gyration tensor, ℜ ( t ) , defined as ℜ ( t ) = N N ∑ i (cid:126) r i , CM ( t ) ⊗ (cid:126) r i , CM ( t ) , (18)where (cid:126) r i , CM = (cid:126) r i ( t ) − (cid:126) R CM ( t ) and ⊗ is the tensor product. In matrixform, ℜ ( t ) = (cid:34) R xx ( t ) R xy ( t ) R yx ( t ) R yy ( t ) (cid:35) . (19)To quantify the ring extension at time t , we measure the gyration’ssquared radius R g ( t ) ≡ Tr ( ℜ ( t )) = λ ( t ) + λ ( t ) , (20)where λ and λ are the gyration tensor eigenvalues. Anothershape measure is the asphericity , defined as A ( t ) = ( λ ( t ) − λ ( t )) ( λ ( t ) + λ ( t )) . (21)The limiting cases where A ( t ) = and A ( t ) = corresponds to acircle and to a rod, respectively. We use the stationary average ofthose quantities, (cid:10) R g (cid:11) , (cid:104) A (cid:105) and the asphericity fluctuation, δ A , tocharacterize the ring format.Fig. 10 shows the phase diagram for asphericity and its fluctua-tion. As in our previous diagram, Fig. 4, we fix the flexure number( F n = ) and vary parameters τ and Pe . On top of that, we sketchillustrative ring formats. We choose N = because the previ-ously used ring’s size, N = , shows small shape variations for Journal Name, [year], [vol.][vol.]
MSD ) for systemswithout (panel (a)), and with translational noise ( B = . in Panel (b)).Parameters N = , Pe = , β = and Fn = are kept fixed. Both figuresshow different values of parameter τ from high values of it, correspond-ing to RW state (ABP limit and absence of collective motion), to lowvalues, corresponding to RUN state, where all particles move aligned tothe CM velocity. Solid lines correspond to fit of Eq. 15, while dots indi-cate numerical results. (c) Dependence of fitted effective self-propelledspeed v e with τ : in RUN state, all particles are aligned and move withvelocities close to v ; in PRW state, it approaches theoretical predictedvalue (dashed line). (d) Fitted effective constant diffusion as a functionof τ : note that the transition at low τ values. Note, also, that even asmall translational noise (panels (c) and (d)) is sufficient to displace theposition of transition to collective motion by an order of magnitude in τ . we also show the effective values for velocities and diffusion coef-ficients obtained fitting simulation data with that equation. Notethe transition between the low- τ / Pe , collective motion state, andhigh- τ / Pe , ABP state. τ / Pe limit In the ABP limit, or high- τ / Pe limit, the first term in Eq. 2 isnegligible and the rotational dynamics is dominated by rotationalnoise, that is, for N = , τ e → τ R , v e v = √ N ∼ . , Journal Name, [year], [vol.] , able 1 Parameters describing curves of Fig. 7 for fixed parameters N = and Fn = are shown. Effective persistence times and self-propelledvelocities are obtained by fitting Eq. [15] to the MSD simulation’s data. For comparison we show several ratios of these fitted quantities with othersystem parameters. Fig. 6 shows the trajectories of systems in the absence of translational noise. State of Motion Pe τ β B τ e / τ τ e / τ R v e / v D e f f / v σ Pers. Random Walk 0.1 0.3 0 0 0.41 2.05 0.32 0.02Pers. Random Walk 0.3 0.1 0 0.001 1.04 1.73 0.275 0.034Run and Rotate 0.3 0.1 0 0 10.6 17.68 0.786 3.15Run and Rotate 0.7 0.1 0 0.001 2.6 1.86 0.74 0.76Run 2 0.1 0 0 99.32 24.83 0.98 47.6Run 2.5 0.1 0 0.001 73.05 14.61 0.96 33.02Rotate 2 0.1 1 0 0.144 0.036 0.054 0.000169Rotate 2.5 0.1 1 0.001 0.55 0.11 0.089 0.00228as seen in Fig 8c. In this limit, Eq. 16 results in D e f f v σ = B NPe + PeN ∼ . , (17)as can be checked in Fig. 8d for N = . The first term in Eq. 17is negligible, since B = . . As τ decreases, both v e and D e f f depart from the ABP behavior (Figure 8c). Nevertheless, notethat even a small translational noise shifts this departure by oneorder of magnitude (Figure 8d). τ / Pe limit The ballistic regime is extended in this limit, resulting in largeeffective persistence times (Figures 8a and 8b). Also, the effectivevelocity reaches v (Figure 8c), so the whole ring achieves thefree single-particle speed. The consequence is a high diffusionconstant, as can be checked in Fig. 8d. We address the relationship between the ring’s size, N , andthe diffusion coefficient for the different motion states. Fig-ures 9a(RUN), 9b (threshold) and 9c (PRW) show the MSD fordifferent ring sizes (N=20,50,100) using parameter τ to controlthe system’s motion state at constant Pe . We exclude the ROTstate in this analysis. At low τ values we find the RUN state, thetransition to collective motion is set at τ ∼ , and above it wereach the PRW state. All curves present the form described byEq. 15, allowing us to fit the effective parameters.When in RUN state (Fig. 9a), all curves in the ballistic intervalcollapse, indicating the same effective self-propelling speed, v e ,with value close to v (Fig. 8c). We also observe in the inset ofFig. 9a the effective persistence time increasing as N increases.This is also clear in Fig. 9e where τ e is divided by N , and thecurves for the different system sizes coincide at small τ values.The overall result (Eq. 16), in this case, is that the diffusion coef-ficient scales with N , as can be checked in Fig. 9f at low Pe values.The MSD for systems in the threshold to collective motion,Fig. 9b, show curves slightly shifted downward as N increases,indicating a correspondent decrease in v e . On the other side, inthis region, τ e starts to grow with N ( Fig. 9e), resulting that D e remains independent of N on the transition, as indicated by thevertical bar in Fig. 9f. Beatrici previously found this result inthe context of cell segregation.When in PRW state, Fig. 9c, systems present the expected be- havior: for increasing N values, the CM moves less since particlesare uncorrelated. In Fig. 9d we multiply the effective velocitysquared by N to show that curves for different sizes collapse athigh- τ values. Since τ e is close to its single particle value (Table1), following Eq. 16, we expect the ring diffusion to scale withthe inverse of N . This is observed in Fig. 9f at large τ values.Fig. 9f summarizes the dependency of the diffusion constantwith the ring size in the different motion states. There is an evi-dent change in behavior with N : from a decrease in RW state toan increase in RUN state, crossing a region without dependence(marked with a vertical line) at the onset of collective motion. To characterize morphological changes in the shape of the activering, we use the gyration tensor, ℜ ( t ) , defined as ℜ ( t ) = N N ∑ i (cid:126) r i , CM ( t ) ⊗ (cid:126) r i , CM ( t ) , (18)where (cid:126) r i , CM = (cid:126) r i ( t ) − (cid:126) R CM ( t ) and ⊗ is the tensor product. In matrixform, ℜ ( t ) = (cid:34) R xx ( t ) R xy ( t ) R yx ( t ) R yy ( t ) (cid:35) . (19)To quantify the ring extension at time t , we measure the gyration’ssquared radius R g ( t ) ≡ Tr ( ℜ ( t )) = λ ( t ) + λ ( t ) , (20)where λ and λ are the gyration tensor eigenvalues. Anothershape measure is the asphericity , defined as A ( t ) = ( λ ( t ) − λ ( t )) ( λ ( t ) + λ ( t )) . (21)The limiting cases where A ( t ) = and A ( t ) = corresponds to acircle and to a rod, respectively. We use the stationary average ofthose quantities, (cid:10) R g (cid:11) , (cid:104) A (cid:105) and the asphericity fluctuation, δ A , tocharacterize the ring format.Fig. 10 shows the phase diagram for asphericity and its fluctua-tion. As in our previous diagram, Fig. 4, we fix the flexure number( F n = ) and vary parameters τ and Pe . On top of that, we sketchillustrative ring formats. We choose N = because the previ-ously used ring’s size, N = , shows small shape variations for Journal Name, [year], [vol.][vol.] , −8 −6 −4 −2 −3 −2 −1 M S D / σ t ∝ t ∝ t (a) RUN −8 −6 −4 −2 −3 −2 −1 M S D / σ t ∝ t ∝ t (b) Threshold to Collective Motion −8 −6 −4 −2 −3 −2 −1 M S D / σ t ∝ t ∝ t (c) PRW −1 ( v e / v ) N τ (d) −2 −1 −1 τ e / ( τ R N ) τ (e) −2 −1 −1 D e ff / v σ τ N = 2050100 (f)
Threshold to Collective MotionPRWRUN
Fig. 9
First panels show the change in behavior of
MSD of Center of Massas system goes from (a) RUN state, τ = . , to (c) PRW state, τ = ,crossing the (b) threshold region, τ = , for fixed set of parameters: Pe = , β = , B = and Fn = . We observe that in the RUN state,larger systems present higher values of effective diffusion constant andpersist longer in ballistic regime (inset). In the threshold region, diffusivebehavior is independent of system’s size. PRW region shows the expecteddecrease of diffusion constant with N . Panel (d) shows that v e approaches v with √ N in PRW region. Panel (e) shows that effective persistencetime increases increase with N in RUN region. This last quantity isresponsible for higher diffusion constant observed since v e is very closeto v for all N , as seen in panel (a). Panel (f) summarize the combinedeffect of both parameters, τ e and v e , in diffusion constant as function of τ and N . this flexure number.At high τ and low Pe values (Fig. 10), the ring maintains analmost circular format, that is, A ( t ) → and δ A → . In this limit,the active particles in the ring show short characteristic persis-tence time, and each particle quickly changes its direction, re-sulting in a mean circular shape with the boundary fluctuating atsmall scales.The circular shape is also stable in the collective motion region( τ / Pe (cid:28) ). Here, the alignment is responsible for moving parti-cles in the same orientation with the same self-propelled speed.The boundary fluctuates less than in the previous case.As both τ and Pe increase, the ring shape changes from an al- Fig. 10
State diagram ( Pe × τ ) for (a) asphericity, (cid:104) A (cid:105) , and (b) itsfluctuations, δ A , while parameters B = , Fn = , β = and N = are kept fixed. In RUN state, the shape changes are small, since allparticles move aligned with CM, and the ring preserves its initial circularformat. In RW state, the particles behave as uncorrelated ones and thering maintains its initial circular format but, now, with higher membranefluctuations than those of RUN state. Note that parameter axes are inlog scale but the color bar is in linear scale. most circular format to an elongated one. This change implies anincrease in both (cid:104) A (cid:105) and δ A , as shown along the main diagonalof Fig. 10. In this region, the active particles do not align globallywith each other (absence of collective motion). Still, similar towhat happens in the Run and Rotate state, the observed charac-teristic persistence time is high enough to ensure subgroups mov-ing persistently in different directions, causing ring deformation.The eigenvectors’ direction of ℜ fluctuate with time, and whilechanging direction, the system spends some time close to a circu-lar format. These low contributions of the circular shape to (cid:104) A (cid:105) are the reason for the low values observed in Fig. 10 despite theelongated aspect. These fluctuations are responsible for highervalues for δ A measured in our simulations, see Fig. 10b.To address the variation of shape with rigidity, we focus on theregion of collective motion. As the bending constant, responsi-ble for ring’s rigidity, decreases (flexure number increases), thering remains in an elongated format. Measuring the asphericity(Fig. 11a) we find that the flattening is more evident as the flex-ure number and the number of particles increase, thus, (cid:104) A (cid:105) → .In Fig. 11c we compare the gyration radio (cid:10) R g (cid:11) with the one froma circle, R , for different flexure number and number of parti-cles. For large values of Fn and N we observe an extremely de-formed active ring (black curve in the Fig. 11c), in that case wefind (cid:10) R g (cid:11) / R ∼ . . This value is close to the one found for arod, (cid:10) R g (cid:11) / R , which converges to π / √ = . as N increases(Supp. Mat.).When the system is in the ROT state, with an initial circularcondition and β = , particle velocities are tangential to the ring,causing no observable changes in its shape even when increasingthe flexure number (Figs. 11b and d).Finally, when the system is in RUN state and presents high de-formation, the ring becomes flattened. In that case, the collec-tive movement has a well-defined direction oriented parallel tothe larger ring dimension, which we may interpret as a sponta-neous emergence of a polarization direction. Whenever there is a Journal Name, [year], [vol.] , ig. 11 Observed changes in the shape of the ring in collective motion forRUN state, left panels, and ROT state, right panels, for set of parameters: Pe = , B = and τ = . . Left panels show simulation results for (a)average asphericity and (c) ratio between the average radius of gyrationand initial radius of gyration R as a function of flexure number fordifferent sizes of the ring, N , and initialization with β = . Both figuresindicate that larger systems present a more pronounced modification oftheir shape as systems become more flexible. It goes from a circularformat when N = to an elongated one, resembling an ellipse, for N = . Right panels (b) and (d) show same quantities for an initializationwith β = . When the system is in ROT state there is only a smalldeparture of the initial circular format independent of the ring’s flexibilityand size. Fig. 12
Center of Mass trajectory of the active ring for a simulationsample in RUN motion state with parameters: N = , Pe = , Fn = , β = , τ = . and B = . On top of the trajectory of CM, we showsnapshots of the active ring where it is possible to note that: (i) ringhas an elongated format; and (ii) it always moves in the most elongateddirection. These observations suggest the presence of spontaneous globalpolarization of the active ring. Colors indicate time (see color bar). SeeMovie5 in Supl. Material. change in the collective movement, there is a realignment of thatlargest dimension. This behavior is illustrated in Fig. 12, wherering snapshots are plotted along its trajectory. Colors indicate thetime. The causes for this emerging polarization remain an open question. In this work, we study a model for a ring composed of activeparticles. We establish the conditions for the emergence of collec-tive movement and study the ring’s deformation by tuning systemparameters such as rigidity constant, angular and translationalnoises, and angular relaxation time. We can identify different ringmotion states. In the limit τ / Pe (cid:29) , we observe a behavior com-patible with an ABP system, which results in an MSD for the cen-ter of mass equivalent to a system of N interconnected particlessubjected to random independent noise. When τ / Pe (cid:28) , the cor-relation between particles becomes pronounced, resulting in twoforms of collective movement: collective translational movement(RUN), where particles move with their speeds nearly parallel;and collective rotational movement (ROT), where particles rotatearound the CM. As far as we know, such rotational state has neverbeen observed in single cell experiments, and is probably due toan excess in degrees of freedom if compared, for example, withthe possible orientations found in actomyosin fibers . For smallrings, N < , we identify a dynamics where the system alter-nates between these states of collective movement (RRM). As thering’s size increases, the transitions between collective states ofmotion become unlikely.All simulations show MSD measurements compatible with theABP limit known solution. That is, ballistic for short times, fol-lowed by diffusion. The ballistic regime is preceded by a diffusiveone only when in the presence of translational noise . Evenat high flexure numbers, when membrane oscillations are large,angular noise by itself cannot produce a short time diffusive be-havior. Also, we found no super-diffusive regime intermediarybetween the ballistic and the diffusive ones.We fit effective self-propulsion speed, v e , and persistence time, τ e , based on the analytic results’ functional form. The ROT state’sfitting procedure resulted in the lowest values for the τ e , and v e since particles circulate the CM without generating a signif-icant displacement. On large time scales, we observe that theCM performs a diffusive process. The RUN state presented thehighest values for the effective parameters, with particles movingaligned with each other in a movement with considerable tempo-ral persistence and, consequently, large MSD . The RRM state’s fitpresents intermediary parameters since they depend on the frac-tion of time spent in the RUN state, with the ROT state poorlycontributing to CM displacement. Another interesting remark isthat even a small translational noise undermines the onset to col-lective movement deviating the transition to an order of magni-tude lower τ values.By varying the number of particles, N , we note a shift in the ef-fective parameters fitted for the MSD . As N increases in the RUNstate, the effective self-propulsion speed, v e , approaches v , andthe diffusion coefficient increases linearly with it. When in thethreshold between the collective movement and the PRW phase,fitted parameters are N independent. In the PRW state, both pa-rameters decrease with N as expected for particles dominatedby uncorrelated noise. A theoretical approach would help shedlight on N ’s effective parameters dependence at the transition to Journal Name, [year], [vol.][vol.]
Center of Mass trajectory of the active ring for a simulationsample in RUN motion state with parameters: N = , Pe = , Fn = , β = , τ = . and B = . On top of the trajectory of CM, we showsnapshots of the active ring where it is possible to note that: (i) ringhas an elongated format; and (ii) it always moves in the most elongateddirection. These observations suggest the presence of spontaneous globalpolarization of the active ring. Colors indicate time (see color bar). SeeMovie5 in Supl. Material. change in the collective movement, there is a realignment of thatlargest dimension. This behavior is illustrated in Fig. 12, wherering snapshots are plotted along its trajectory. Colors indicate thetime. The causes for this emerging polarization remain an open question. In this work, we study a model for a ring composed of activeparticles. We establish the conditions for the emergence of collec-tive movement and study the ring’s deformation by tuning systemparameters such as rigidity constant, angular and translationalnoises, and angular relaxation time. We can identify different ringmotion states. In the limit τ / Pe (cid:29) , we observe a behavior com-patible with an ABP system, which results in an MSD for the cen-ter of mass equivalent to a system of N interconnected particlessubjected to random independent noise. When τ / Pe (cid:28) , the cor-relation between particles becomes pronounced, resulting in twoforms of collective movement: collective translational movement(RUN), where particles move with their speeds nearly parallel;and collective rotational movement (ROT), where particles rotatearound the CM. As far as we know, such rotational state has neverbeen observed in single cell experiments, and is probably due toan excess in degrees of freedom if compared, for example, withthe possible orientations found in actomyosin fibers . For smallrings, N < , we identify a dynamics where the system alter-nates between these states of collective movement (RRM). As thering’s size increases, the transitions between collective states ofmotion become unlikely.All simulations show MSD measurements compatible with theABP limit known solution. That is, ballistic for short times, fol-lowed by diffusion. The ballistic regime is preceded by a diffusiveone only when in the presence of translational noise . Evenat high flexure numbers, when membrane oscillations are large,angular noise by itself cannot produce a short time diffusive be-havior. Also, we found no super-diffusive regime intermediarybetween the ballistic and the diffusive ones.We fit effective self-propulsion speed, v e , and persistence time, τ e , based on the analytic results’ functional form. The ROT state’sfitting procedure resulted in the lowest values for the τ e , and v e since particles circulate the CM without generating a signif-icant displacement. On large time scales, we observe that theCM performs a diffusive process. The RUN state presented thehighest values for the effective parameters, with particles movingaligned with each other in a movement with considerable tempo-ral persistence and, consequently, large MSD . The RRM state’s fitpresents intermediary parameters since they depend on the frac-tion of time spent in the RUN state, with the ROT state poorlycontributing to CM displacement. Another interesting remark isthat even a small translational noise undermines the onset to col-lective movement deviating the transition to an order of magni-tude lower τ values.By varying the number of particles, N , we note a shift in the ef-fective parameters fitted for the MSD . As N increases in the RUNstate, the effective self-propulsion speed, v e , approaches v , andthe diffusion coefficient increases linearly with it. When in thethreshold between the collective movement and the PRW phase,fitted parameters are N independent. In the PRW state, both pa-rameters decrease with N as expected for particles dominatedby uncorrelated noise. A theoretical approach would help shedlight on N ’s effective parameters dependence at the transition to Journal Name, [year], [vol.][vol.] , he collective movement and above it. A previous mean clusterstudy relating Vicsek’s collective movement parameter anddiffusion coefficient mass dependence correctly explained cellsegregation time scales. However, it offered no theoretical hintfor that relation.In the ring morphology study, we found that the ring maintainsits initial circular shape when τ or Pe is low enough. It presentslarge fluctuations in shape when τ and Pe increase. Larger ringspresent a substantial variation in the possible shapes they can as-sume when decreasing the curvature potential’s stiffness. Largesoft rings assume a slug-like form when in collective motion, witha well-defined movement polarization along the largest ring di-rection. Since we deal with correlated active particles, the emer-gence of polarization, in this case, can be interpreted as a non-linear instability of the center of mass, as proposed by Blanch-Mercader and coworkers , but the relation between the sluglarger direction and the global velocity remains an open question.The active ring system proposed aims to serve as a model forcells. Many models in Active Matter are single particle-based andunable to describe cells’ morphological properties. Furthermore,for being a bead-spring model, its use in phenomena such as duro-taxis, chemotaxis, cell segregation, cell crawling, or wound heal-ing is easy to implement with modest modifications by includinginteraction forces among cells or external chemical fields. Here,in this first work, we characterize the dynamic and morphologicalproperties of a single ring. In future works, we will study systemscomposed of many of these. Conflicts of interest
There are no conflicts to declare
Acknowledgements
This work is dedicated to the memory of Cássio Kirch. E.F.T.thanks the Brazilian funding agencies CNPq and Capes. H.C.M.F.acknowledges Universitat de Barcelona where part of this workwas developed. L.G.B. acknowledges the Max-Planck Instituteof Ploen, where part of this work was developed. All authorsacknowledge the suggestions and discussions with S. Lira. Thesimulations were performed on the IF-UFRGS computing clusterinfrastructure.
Notes and references
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