Active matter clusters at interfaces
AActive matter clusters at interfaces.
Katherine Copenhagen and Ajay Gopinathan University of California Merced, Merced CA
Collective and directed motility or swarming is an emergent phenomenon displayed by manyself-organized assemblies of active biological matter such as clusters of embryonic cells during tissuedevelopment, cancerous cells during tumor formation and metastasis, colonies of bacteria in a biofilm,or even flocks of birds and schools of fish at the macro-scale. Such clusters typically encounter veryheterogeneous environments. What happens when a cluster encounters an interface between twodifferent environments has implications for its function and fate. Here we study this problem byusing a mathematical model of a cluster that treats it as a single cohesive unit that moves in twodimensions by exerting a force/torque per unit area whose magnitude depends on the nature of thelocal environment. We find that low speed (overdamped) clusters encountering an interface witha moderate difference in properties can lead to refraction or even total internal reflection of thecluster. For large speeds (underdamped), where inertia dominates, the clusters show more complexbehaviors crossing the interface multiple times and deviating from the predictable refraction andreflection for the low velocity clusters. We then present an extreme limit of the model in the absenseof rotational damping where clusters can become stuck spiraling along the interface or move in largecircular trajectories after leaving the interface. Our results show a wide range of behaviors thatoccur when collectively moving active biological matter moves across interfaces and these insightscan be used to control motion by patterning environments. a r X i v : . [ phy s i c s . b i o - ph ] A p r INTRODUCTION.
Swarming is a widespread biological phenomenon characterized by long range order emerging in a system fromlocal interactions between agents[1], such as a swarm of flies[2, 3], or flock of birds[4]. Typically, a group of individualorganisms self-organize to form a cohesive cluster with directed motility in a spontaneously chosen consensus direction,for example a school of fish[5], cluster of cells during tumor growth, tissue development and repair[6, 7], or herd ofwildebeests[8]. These types of swarming systems will often encounter a change in the environment, such as a flock ofbirds flying into a cloud or area of lower air temperature, or a cluster of tumor cells invading different tissue types[9].Single cells have been shown to change their speed and direction when crossing sharp interfaces [10]. Swarms canalso use collective dynamics to find and localize themselves to preferred niches or microenvironments. For example ithas been observed that golden shiners,
Notemigonus crysoleucas , which prefer low lighting, will spend more time indark areas if it is part of a school due to cooperative sensing capabilities of the group[11]. Bacteria have also beenshown to take less time to reach a target in the presence of noisy chemical concentration gradients in the environmentwhen they are part of a cluster[12]. Also e. coli. clusters modify their own environment by secreting chemicalsto create an environmental change between the inside and outside of the cluster in order to trap the e. coli. andmaintain clustering behaviors[13]. It is therefore important to understand the effects of spatial environmental changeson a cohesive swarming group. In this paper we investigate finite swarming clusters moving through heterogeneousenvironments where agents change their speeds by exerting different forces within each environment. Such changescould arise from the agent’s sensing and response to a variety of environmental factors such as temperature, substratestiffness, or chemical composition. So, how is the path that a swarm takes affected by the presence of a boundarybetween different environments, and how does that depend on the properties of the cluster and the environmentalchange?Agent based models[14] and hydrodynamic continuum models[15] have been used with great success to modelnatural collective systems[1, 16], and reveal phases and transitions which emerge from the active, far from equilibriumnature of these systems[17, 18]. Agent based models are implemented by defining a set of interactions and update rulesfor individuals and then letting the system evolve in time, such systems show phase transitions driven by a wide varietyof quantities including noise, density[19], environmental disorder[20], behavioral heterogeneities[21, 22], and cohesiveinteraction details [23, 24]. They have also been used to study how swarms use cooperation to achieve specific goalswhich are useful to biological systems, such as cooperative decision making[25], agent segregation[26], and obstacleavoidance[27]. Hydrodynamic continuum models on the other hand do not treat each agent as an individual, butinstead study an average alignment and density profile within the system[28]. Hydrodynamic swarming models haveestablished active matter as a type of nonequilibrium complex fluid[29, 30] and provided a unified framework to studyphase transitions[28, 31], instabilities[32] and pattern formation[33] in active systems.To answer the question of swarms crossing from one environment into another, we utilize a simplified model fora swarm that assumes a polarized ordered state with velocities correlated across the system[34] and finite systemsize, which allows us to examine the overall behaviors of the swarm crossing an interface without necessitating detailsabout individual agent motion as in agent based models, or infinite system sizes as in hydrodynamic models. Weconsider a swarm as a single cohesive disk-shaped unit, or cluster, each individual agent within the swarm manifestsas a force per unit area applied to the disk in the direction of polarization. We then allow the cluster to cross aninterface between two differing environments where the portion of the swarm in each environment may apply strongeror weaker forces to the cluster depending on the substrate, resulting in a torque and therefore a curved trajectory.We can then map the trajectories of the cluster and measure the resulting relationships between the cluster motionbefore and after crossing the interface. We find four catagories for cluster behavior which depend on two importantcluster parameters. The two important parameters are cluster speed, i.e. low speed (overdamped) vs. high speed(underdamped) clusters, and the ratio of the rotational damping to translational friction. It is to be noted that thougha prescribed amount of rotational damping arises from the translational friction on the disk, the ratio is a parameterthat could be smaller (or bigger) depending on whether the cluster exerts torques to maintain (or resist) rotation.Our model has predictive capabilities for determining the curved path of a cell cluster at an interface betweensubstrates. The results of which could be extended to suggest possible methods of patterning substrates to direct cellcluster motion. Other regimes of our model provide insight into the behaviors of faster swarms, such as bird flocksand fish schools, moving between heterogeneous environments. Finally, for swarms that exert torques to maintainturning, we show that swarms can display circular paths and trapping at interfaces.
THE MODEL.
We use a mathematical model for a swarming cluster (Fig. 1) that treats it as a single cohesive unit that moves on atwo dimensional substrate by exerting a force per unit area in some cluster polarization direction (at an angle Φ with A A f f ˆ n y θ θ R R -y ˆ n p R y θ A f f ˆ n yR θ θ ϕ A f f ˆ n Y ϕ FIG. 1. View from above showing the two regions one above (light green) and below (dark green) the interface between twodifferent substrates shown in light and dark blue respectively. Each region of the cluster propels itself along the polarizationdirection (ˆ n at an angle of Φ with the positive x -axis) with a force per unit area of f for the top region and f for the bottomregion. The cluster center is at a height Y above the interface (in the diagram shown Y is negative because the center of thecluster is below the interface). respect to the positive x -axis). We then examine a single cluster moving across an interface between two differentsubstrates where the area of the cluster contained on substrate 1 (shown in light green above the solid horizontal linein Fig. 1) exerts a force per unit area of f and the portion on substrate 2 (shown by the dark green area in Fig. 1)exerts a force per unit area of f . The force applied on the cluster by every portion of the cluster is in the direction ofpolarization with a magnitude that depends on the areas within each substrate and the substrate dependent forces.The cluster also experiences a friction-like damping force resisting translational motion. Utilizing these details we cancalculate a force on the cluster at any height Y above a horizontal substrate interface (See appendix for details).If the cluster is polarized at an angle which is not normal to the interface there is an asymmetry of the forces oneither side of the interface, which results in a torque on the direction of polarization of the cluster which in turnrotates the direction of the force on the cluster and can result in a curved trajectory or a bend in the path of thecluster as it crosses the interface. This torque can be calculated from the force per unit area of the cluster on eachsubstrate along with the distance from the cluster center. From the derived expressions for force and torque (seeappendix) we can find the non-dimensionalized equations of motion for the cluster shown in eqs.1-3, where v and v are the equilibrium speeds of the cluster in the top and bottom substrates respectively, and C is the ratio of theangular damping to the translational friction on the disk. In the non-dimensionalized forms of the equations shownbelow the translational/frictional damping constant is incorporated into the equilibrium speeds v and v . d Φ dT = 43 π (cid:112) − Y (1 + Y )( v − v ) cos(Φ) − C d Φ dT (1) d XdT = (cid:16) / v + v ) + 1 /π ( v − v ) (cid:0) arcsin( Y ) + Y (cid:112) − Y (cid:1)(cid:17) cos(Φ) − dXdT (2) - -
20 20 40 - - - -
20 20 40 - - A n g u l a r l y d a m p e d ( C > ) - -
20 20 40 - - - -
20 20 40 - - Sp ec i a l ca s e ( C = ) High velocities (v =2)Low velocities (v =0.02) θ θ f FIG. 2. Representative trajectories of the system in each of the four limitting behaviors of the system. In all four cases v = 2 v . d YdT = (cid:16) / v + v ) + 1 /π ( v − v ) (cid:0) arcsin( Y ) + Y (cid:112) − Y (cid:1)(cid:17) sin(Φ) − dYdT (3)We can then use finite difference methods to solve these equations of motion and examine the system subjectedto different substrates and initial conditions. In the model the unit of length is set by the cluster radius, and theunit of time is set by the time taken for the cluster to accelerate from rest to the fraction (1 − / e) ∼ .
63 of itsequilibrium speed. The active nature of the system allows the angular damping and translational friction to not beequal, implying C (cid:54) = 1. Physically we can expect C >
C <
K. Mugil fish[35]. How, then, does the incident angle and ratio of equilibrium speeds in eachsubstrate affect the transmitted angle of the swarm? What is the effect of exerted torques which promote turning?
RESULTS.
Fig. 2 shows the four characteristic trajectories for limits of the two important parameters: low velocities (over-damped, friction dominated behaviors), high velocities (underdamped, inertia dominated behaviors), and clusterswith angular damping (
C > C = 0, relevant/applicableto agents that apply torques to promote turning which exactly cancel out any angular damping, resulting in uniquecluster behaviors. (a) xy (b) xy - -
10 10 20 - - - -
10 10 20 - - FIG. 3. Example trajectories of the cluster, each color is a different incident angle where the incoming cluster is shown bythe straight lines entering from the left of each plot ( x <
0) directed towards the origin, and the clusters are in contact withthe interface, which is at y = 0 (maroon dashed line), when the trajectory is within the orange band. The substrate on thebottom half ( y <
0) has twice the equilibrium speed as the top substrate ( v = 2 v ) in both cases. (a) For low speeds, wherefriction dominates cluster motion, the cluster curves while it is in contact with the interface (orange band), and once it leavesthe interface it has a well defined straight path along some rotated polarization direction. (b) High velocities, when inertiadominates, result in the cluster starting to curve when it comes into contact with the interface and then large sweeping curvesaway from the interface as the momentum of the cluster causes it to persist along the previous direction before graduallyadjusting to the new substrate. A. Cluster trajectories.
The overall behaviors of the cluster in the presence of angular damping fall into two different categories: refrac-tion/reflection (Fig 3(a)), and large sweeping curves (Fig 3(b)). In both cases the direction that the cluster activelypropels itself in (ˆ n at angle Φ) can only accelerate due to torques experienced while it is in contact with the interface.This means that while the cluster is on a single substrate the angular speed of ˆ n can only decrease due to rotationaldamping.In the low velocity cases ( v = 0 .
01 in Fig. 3(a)) friction dominates the cluster motion and the cluster moves in adirection parallel to the active propulsion of the cluster ( (cid:126)v || ˆ n ) at nearly all times. In this friction dominated limit,the system can refract or reflect off of the interface and only comes into contact with the interface once allowing usto measure the incident and refracted angle and make predictions about the behaviors and interactions of the clusterwith an interface.The high velocity case ( v = 5 in Fig. 3(b)) is characterized by the fact that the cluster’s translational inertiadominates the direction of cluster motion ( (cid:126)v ), for an extended period of time after crossing the interface. In this highvelocity case, as the cluster leaves the interface it will move in a direction which is not necessarily parallel to thedirection of ˆ n , resulting in the apparent angular acceleration away from the interface. However, after some time thesystem will reach equilibrium where (cid:126)v || ˆ n and the cluster will move in a straight line in some well defined direction.In this case the momentum of the cluster carries it away from the interface even if the direction of ˆ n is such thatthe cluster should be propelled towards the interface meaning that the cluster can rotate and return to the interfacea finite number of times before reaching a stable straight trajectory within a single substrate. This high velocity(inertia dominated) case could model the motion of a bird flock crossing an interface between hotter and colder airwhere their momentum will continue to carry the flock in one direction and the speed and trajectory will graduallystabilize as the flock adjusts to the new environment, potentially traveling in a different direction from that withwhich it entered. B. Refraction and reflection at an interface. v /v s i n ( θ f ) / s i n ( θ ) π π π π π π π π θ slow θ f a s t (a) (b) θ = π /16 θ = 2 π /16 θ = 3 π /16 θ = 4 π /16 θ = 5 π /16 θ = 6 π /16 θ = 7 π /16 FIG. 4. (a) The ratios of the sines of the incident ( θ ) and refracted ( θ f ) angles plotted against the ratios of the equilibriumspeeds in each material, with v = 0 . π/
2, resulting in a reflection, for each initial angle shown in the legend. When the cluster is reflected(right side of the vertical dashed lines) the horizontal line at sin θ f / sin θ = 1 shows that the reflected angle is equal to theincident angle. (b) The angle that the trajectory makes (with respect to the normal to the interface) on the faster ( θ fast ) andslower ( θ slow ) substrates for a cluster moving from the slower substrate to the faster one shown in blue, and the similar anglesfor a cluster moving from the faster to the slower substrate shown in red dashed line. In both these cases v = 0 . v = 0 . C = 1. In this section we examine the case where the velocity on each substrate is low enough that friction dominates,and the cluster experiences rotational damping (
C > θ f and θ respectively in Fig. 2) and compare them to the velocity ratio on the two substrates. To do this we plot theratio of the sines of the refracted and incident angles against the ratio of the equilibrium velocities on each substrate.Fig. 4 (a) shows the relationship between these ratios for many initial angles (see legend), along with a fit line tothe numerical data for the refraction angle which leads to the predicted reflections. Applying a similar fit to systemswith different values of C leads to the relationship shown in Eq. 4. This relation bears a remarkable resemblance toSnell’s law for optical paths except with an exponent of 0 . /C instead of unity. Additionally, when moving from aslower substrate to a faster one the cluster will be reflected off the interface if the transmitted angle should be π/ θ f > π/ θ sin θ = (cid:16) v v (cid:17) . /C (4)The trajectories of these clusters are also reversible in time as can be seen in Fig. 4(b) which shows the clustertrajectory angle with respect to normal on the faster substrate plotted against the angle of the trajectory on theslower substrate. The blue solid line shows the case where the cluster is moving from the slower substrate onto thefaster one, and drops down to the diagonal line when reflection begins to occur. The red dashed line shows the anglesfor the faster to slower case. The overlap of these two curves shows that whether moving from faster to slower orvice versa the angles depend only on the substrates and angles and not on the initial substrate of the cluster. Therefraction of the cluster into the slower medium while approaching at a large angle from the faster medium is similarto the reported behavior of the golden shiners that collectively turn into darker regions where they move slower[11]. C. Fast swarms and swarms with exerted torques to promote turning.
In the case where the velocity on each substrate is high enough for interia to dominate cluster motion, the clusterwill still adjust its direction as it passes through the interface but its own momentum will carry it straight across ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ (v /v ) s i n ( θ f ) / s i n ( θ i ) 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▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ θ = π /32 θ = 2 π /32 θ = 3 π /32 θ = 4 π /32 θ = 5 π /32 θ = 6 π /32 θ = 7 π /32 θ = 8 π /32 θ = 9 π /32 θ = 10 π /32 θ = 11 π /32 θ = 12 π /32 θ = 13 π /32 θ = 14 π /32 θ = 15 π /32 s i n ( θ f ) / s i n ( θ i ) (v /v ) v = 0.001 v = 0.01v = 0.1 (v /v ) s i n ( θ f ) / s i n ( θ i ) v = 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ 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The ratios of the sines of the initial and final trajectory angles measured with respect to the normal to the interface.The velocity on the top substrate ( v ) is labeled for each plot and the color legend for incident angle is consistent across allfour plots. Squares are for C = 2, triangles for C = 1 and circles for C = 0 .
5. The diagonal line is a fit for the low velocitycase and the vertical lines are predicted values for the reflections to begin for each incident angle shown by color according tothe legend. the interface before slowly adjusting it’s angle depending on the rotations caused by the interface which we can seequalitatively in Fig. 3(b). This results in the cluster often returning to the interface and interacting with it multipletimes making the refracted angle vary from the predictions made for the friction dominated case.Fig. 5 shows the ratio of the incident and refracted angles of clusters versus the ratio of velocities scaled by angulardamping similar to Fig. 4(a). In this plot the collapse at low velocities for different values of C is shown by the shapesof the markers, C = 0 . C = 1 (rotational damping arising from translational friction only) for thetriangles and C = 2 for the squares, the shapes and color legend are consistent across all four plots shown. We cansee that at high velocities the reflected angle varies greatly from the low velocity case where reflected angle was equalto incident angle. This is due to the fact that the cluster passes through the interface before turning around andthen passes through the interface again resulting in multiple interactions with the interface and final reflected anglesthat don’t relate in a well defined way to the incident angles. The refracted angle also differs from the low velocitycase due to the effects of momentum and inertia at high velocities, the trajectories become complicated and cease tofollow a well defined relationship. This behavior could be related to the complex fluid-like motions of starling flocksas they swoop and change directions in large arcing trajectories, possibly due to changing environments from flyingover trees or through different altitudes and air temperatures.Finally we consider a special case of a swarm that actively promotes rotations by exerting a torque that exactlycancels out any rotational resistance from friction-like damping (resulting in C = 0). In this case two very differentbehaviors emerge. The first is in the low velocity case where friction dominates the translational motion, thoughangular momentum is always important due to the fact that C = 0 independent of velocity. In this case the clusterwill again begin to rotate as it crosses the interface, and experience no angular acceleration once it loses contact withthe interface, however it will continue to rotate along a circular curve and return to the interface some time later.This causes the cluster to become trapped at the interface by always rotating around and returning to the interfacewithout being able to escape as seen in Fig. 6(a). The second case is for high velocities where the inertia of the clusterwill carry it past the interface and it may return to the interface or escape but the initial crossing of the interfacestarts the cluster rotating and the cluster will move in a circle somewhere away from the interface typically on theslower substrate as seen in Fig. 6(b). (a) xy (b) xy - -
10 10 20 - - - -
10 10 20 - - FIG. 6. (a) Trajectories for the C = 0 case with low velocities. v = 0 .
001 and v = 0 . v = 5 and v = 10. The cluster is incontact with the interface while the trajectory is shown in the orange band. DISCUSSION.
Collective directed motility is a phenomenon that is widespread in biological systems including cell clusters duringtissue development and tumor formation, as well as bacterial biofilms and flocks of birds. In these types of systemsit is reasonable to assume that clusters of collectively moving agents move through changing environments, be it achange in air temperature, stiffness in substrate or any other change which could result in speed change. We haveused a model which treats a swarming cluster as a single cohesive unit with a preferred direction, to examine theeffects of a swarm moving across an interface between two environments due to a change in speed that occurs in eachseparate environment.We found that clusters can display different broad behaviors. The most applicable of which is for slow movingswarms with some angular damping. In this case a swarming cluster approaching an interface at an angle willundergo some form of refraction or reflection resulting in a new direction which is predictable by a simple relationshipbetween incident and refracted angle and the ratio of the equilibrium speeds on each substrate. Clusters in this regimecan also display total internal reflection at the predictable angle where the refraction angle would exceed π/
2. Thisregime of our model could represent cell clusters on changing substrates and our predictions could be used to patterna substrate to direct cluster motion along a desired path.When the velocities of the cluster is increased on both substrates the trajectories gradually diverge from thepredicted refraction angles and reflections found for low velocities where friction dominates cluster motion. This isdue to the inertia of the clusters carrying it quickly across the interface with the inability of the cluster to changedirections at a comparable rate. As the cluster velocities continue to increase the trajectories become sensitive toinitial conditions and rotational damping, due to the cluster interacting multiple times with the interface or spendingless time in contact with the interface than necessary for the cluster to adjust its direction according to the torquespresent. These kinds of clusters display broad sweeping curved trajectories away from the interface and could provideinsight into the impressive collective motion seen in starling flocks and fast moving fish schools, such as sardines.In the special case where the cluster experiences no angular damping, possibly due to swarms exerting a torquewhich counter-acts the friction-like resistance that is present in the translational motion, the cluster will either becomestuck on the interface for low velocities or stabilize in circular trajectories on a single substrate for high velocities.These types of behaviors may be desirable for certain systems, and our predictions could be used to engineer cellclusters, or robot algorithms to follow desired paths. Trapping a cell cluster on an interface could be useful forseparating cells or subjecting them to specific conditions which could then be applied at the interface more easily tocell clusters as individual cells will not display the same kinds of collective modes at an interface. Cell cluster motionhas been shown to be controllable by hard boundaries[36], and our results suggest that a similar strategy could beused with softer substrate interfaces which can be crossed to manipulate cell cluster behaviors.Our results show possible predictive capabilities for slow moving clusters such as cells or bacteria moving acrosschanging substrates, as well as possible insight into ongoing questions such as the behaviors of starling flocks and fishschools as they spiral and curve while they span and cross interfaces between changing environments. Additionallyour results suggest possible mechanisms for directing collective systems by way of changing environmental conditions. [1] D J T Sumpter. The principles of collective animal behaviour.
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ACKNOWLEDGMENTS
This work was partially supported by National Science Foundation (NSF) grant EF-1038697 (to A.G.), a James S.McDonnell Foundation Award (to A.G.) and an NSF-IGERT graduate fellowship (to K.C.)]1 A A f f ˆ n y θ θ R R -y ˆ n p R y θ A f f ˆ n yR θ θ ϕ A f f ˆ n Y ϕ FIG. 7. Diagram of cluster setup, the parameters are described in the appendix text.
APPENDIX.
We use a mathematical model for the cluster that treats it as a single cohesive unit that moves on a two dimensionalsubstrate by exerting a force and torque per unit area whose magnitude depends on the nature of the substrate. Herewe examine the effects of a single cluster moving across an interface between two different substrates where eachportion of the cluster exerts a force per unit area depending on which substrate it is on. The force exerted on thecluster in direction of polarization (ˆ n ) is equal to the substrate dependent force per unit area ( f ) multiplied bythe area of the cluster on each respective substrate ( A ), with damping constant b . F = ( f A + f A )ˆ n − b(cid:126)v (5)The area of the partial circles within each region at any heigh y above the substrate interface can be calculated asshown below, where R is the radius of the cluster, and θ and θ are the angles to the intersections of the interfaceline and the rim of the cluster as shown in Fig. 7. Integrating and simplifying results in Eq. 6 as the net force on thecluster with radius R at height y above the interface. F = (cid:32) f (cid:90) θ θ (cid:90) R y sin θ rdrdθ + f (cid:90) π + θ θ (cid:90) R y sin θ rdrdθ (cid:33) ˆ n − b(cid:126)v F = (cid:32) f (cid:90) π − arcsin ( − yR )arcsin ( − yR ) (cid:16) R − y sin θ (cid:17) dθ + f (cid:90) π +arcsin ( − yR ) π − arcsin ( − yR ) (cid:16) R − y sin θ (cid:17) dθ (cid:33) ˆ n − b(cid:126)v F = 12 (cid:32) f (cid:16) R θ + y cot θ (cid:17) π − arcsin ( − yR )arcsin ( − yR ) + f (cid:16) R θ + y cot θ (cid:17) π +arcsin ( − yR ) π − arcsin ( − yR ) (cid:33) ˆ n − b(cid:126)v F = 12 (cid:32) f (cid:16) R ( π − arcsin( − yR )) + y cot( π − arcsin( − yR )) − ( R (arcsin( − yR )) + y cot(arcsin( − yR ))) (cid:17) + f (cid:16) R (2 π + arcsin( − yR )) + y cot(2 π + arcsin( − yR )) − ( R ( π − arcsin( − yR )) + y cot( π − arcsin( − yR ))) (cid:17)(cid:33) ˆ n − b(cid:126)v F = 12 (cid:32) f (cid:16) R ( π + 2 arcsin( yR )) + 2 y (cid:112) R − y (cid:17) + f (cid:16) R ( π − yR )) − y (cid:112) R − y ) (cid:17)(cid:33) ˆ n − b(cid:126)v F = 12 R (cid:32) f (cid:16) π + 2 arcsin( yR ) + 2 yR (cid:112) − ( y/R ) (cid:17) + f (cid:16) π − yR ) − yR (cid:112) − ( y/R ) (cid:17)(cid:33) ˆ n − b(cid:126)v The expression for force on the cluster due to the active force per unit area exerted by the cluster and damping isshown below. F = 12 R (cid:16) π ( f + f ) + 2( f − f ) (cid:0) arcsin( yR ) + yR (cid:112) − ( y/R ) (cid:1)(cid:17) ˆ n − b(cid:126)v (6)In addition to the substrate dependent force applied on the cluster, the polarization direction of the cluster willalso be subject to a torque due to the asymmetry of the variable forces on each area of the cluster. The net torque onthe cluster can be calculated by integrating the torque per unit area over the portions of the cluster on each substratesimilar to the calculation of force above. The torque per unit area is due to the force per unit area at each point onthe disk, i.e. f ˆ n × (cid:126)r , with angular damping − c ω . τ = (cid:90) A f ˆ n × (cid:126)rdA + (cid:90) A f ˆ n × (cid:126)rdA − c ω (7)The area of the portions of the cluster on each substrate is integrated as in the force case above, and simplified toget the expression for torque on the cluster shown in Eq. 8. τ = (cid:90) θ θ (cid:90) R y sin θ f ˆ n × (cid:126)rrdrdθ + (cid:90) π + θ θ (cid:90) R y sin θ f ˆ n × (cid:126)rrdrdθ − c ωτ = (cid:16) (cid:90) θ θ (cid:90) R y sin θ f r sin( θ − φ ) rdrdθ + (cid:90) π + θ θ (cid:90) R y sin θ f r sin( θ − φ ) rdrdθ (cid:17) ˆ z − c ωτ = (cid:16) (cid:90) θ θ f sin( θ − φ ) r (cid:12)(cid:12)(cid:12) R y sin θ dθ + (cid:90) π + θ θ f sin( θ − φ ) r (cid:12)(cid:12)(cid:12) R y sin θ dθ (cid:17) ˆ z − c ωτ = (cid:16) (cid:90) θ θ f (sin( φ ) cos( θ ) − cos( φ ) sin( θ ))( R − y θ ) dθ + (cid:90) π + θ θ f (sin( φ ) cos( θ ) − cos( φ ) sin( θ ))( R − y θ ) dθ (cid:17) ˆ z − c ωτ = (cid:16) (cid:90) θ θ f R φ ) cos( θ ) − cos( φ ) sin( θ )) − y sin( φ ) cos( θ )3 sin ( θ ) + y cos( φ )3 sin ( θ ) dθ + (cid:90) π + θ θ f R φ ) cos( θ ) − cos( φ ) sin( θ )) − y sin( φ ) cos( θ )3 sin ( θ ) + y cos( φ )3 sin ( θ ) dθ (cid:17) ˆ z − c ω τ = (cid:16) f (cid:32) R φ ) sin( θ ) + R φ ) cos( θ ) + y sin( φ )6 sin ( θ ) − y cos( φ ) cos( θ )3 sin( θ ) (cid:33) π − arcsin( − yR )arcsin( − yR ) + f (cid:32) R φ ) sin( θ ) + R φ ) cos( θ ) + y sin( φ )6 sin ( θ ) − y cos( φ ) cos( θ )3 sin( θ ) (cid:33) π +arcsin( − yR ) π − arcsin( − yR ) (cid:17) ˆ z − c ωτ = ˆ z (cid:16) f × (cid:32) R φ ) sin( π − arcsin( − yR ))+ R φ ) cos( π − arcsin( − yR ))+ y sin( φ )6 sin ( π − arcsin( − yR )) − y cos( φ ) cos( π − arcsin( − yR ))3 sin( π − arcsin( − yR )) (cid:33) − f (cid:32) R φ ) sin(arcsin( − yR ))+ R φ ) cos(arcsin( − yR ))+ y sin( φ )6 sin (arcsin( − yR )) − y cos( φ ) cos(arcsin( − yR ))3 sin(arcsin( − yR )) (cid:33) + f × (cid:32) R φ ) sin(2 π +arcsin( − yR ))+ R φ ) cos(2 π +arcsin( − yR ))+ y sin( φ )6 sin (2 π + arcsin( − yR )) − y cos( φ ) cos(2 π + arcsin( − yR ))3 sin(2 π + arcsin( − yR )) (cid:33) − f (cid:32) R φ ) sin( π − arcsin( − yR ))+ R φ ) cos( π − arcsin( − yR ))+ y sin( φ )6 sin ( π − arcsin( − yR )) − y cos( φ ) cos( π − arcsin( − yR ))3 sin( π − arcsin( − yR )) (cid:33)(cid:17) − c ωτ = ˆ z (cid:16) f (cid:32) R φ )( − y/R ) + R φ )( − (cid:112) R − y /R ) + y sin( φ )6 y /R − y cos( φ )( − (cid:112) R − y /R )3( − y/R ) (cid:33) − f (cid:32) R φ )( − y/R ) + R φ )( (cid:112) R − y /R ) + y sin( φ )6 y /R − y cos( φ ) (cid:112) R − y /R − y/R ) (cid:33) + f (cid:32) R φ )( − y/R ) + R φ )( (cid:112) R − y /R ) + y sin( φ )6 y /R − y cos( φ ) (cid:112) R − y /R − y/R ) (cid:33) − f (cid:32) R φ )( − y/R ) + R φ )( − (cid:112) R − y /R ) + y sin( φ )6 y /R − y cos( φ )( − (cid:112) R − y /R )3( − y/R ) (cid:33)(cid:17) − c ωτ = ˆ z (cid:16) f (cid:32) − R φ )( (cid:112) R − y ) − y cos( φ )( (cid:112) R − y )3 (cid:33) + f (cid:32) R φ )( (cid:112) R − y ) + 2 y cos( φ ) (cid:112) R − y (cid:33)(cid:17) − c ωτ = ˆ z (cid:112) R − y cos( φ ) (cid:16) f ( − R − y ) + f ( R + y ) (cid:17) − c ω The equation below shows the torque on the cluster due to the active forces on each substrate, and rotationaldamping. τ = ˆ z (cid:112) R − y cos( φ )( f − f )( R + y ) − c ω (8)From eq. 6 for force and eq. 8 for torque we can write down the equations of motion for the orientation as well asthe x and y coordinates of the cluster, as shown below.412 mR ¨ φ = 23 (cid:112) R − y cos( φ )( R + y )( f − f ) − c ˙ φm ¨ x = 12 R (cid:16) π ( f + f ) + 2( f − f ) (cid:0) arcsin( yR ) + yR (cid:112) − ( y/R ) (cid:1)(cid:17) cos( φ ) − b ˙ xm ¨ y = 12 R (cid:16) π ( f + f ) + 2( f − f ) (cid:0) arcsin( yR ) + yR (cid:112) − ( y/R ) (cid:1)(cid:17) sin( φ ) − b ˙ y We can then nondimensionalize the equations of motion through the following substitutions: φ = Φ x = RXy = RYt = mb TR b m d Φ dT = 23 R (cid:112) − Y (1 + Y )( f − f ) cos(Φ) − cbm d Φ dTRb m d XdT = 12 R (cid:16) π ( f + f ) + 2( f − f ) (cid:0) arcsin( Y ) + Y (cid:112) − Y (cid:1)(cid:17) cos(Φ) − Rb m dXdTRb m d YdT = 12 R (cid:16) π ( f + f ) + 2( f − f ) (cid:0) arcsin( Y ) + Y (cid:112) − Y (cid:1)(cid:17) sin(Φ) − Rb m dYdTd Φ dT = 4 mR b (cid:112) − Y (1 + Y )( f − f ) cos(Φ) − cR b d Φ dTd XdT = mR b (cid:16) π ( f + f ) + 2( f − f ) (cid:0) arcsin( Y ) + Y (cid:112) − Y (cid:1)(cid:17) cos(Φ) − dXdTd YdT = mR b (cid:16) π ( f + f ) + 2( f − f ) (cid:0) arcsin( Y ) + Y (cid:112) − Y (cid:1)(cid:17) sin(Φ) − dYdT In the limit where the cluster is entirely on one substrate, we can compute the equilibrium speed of the clusteron each substrate by letting the acceleration go to zero, and
Y >
Y < − mR b (cid:16) π ( f + f ) + 2( f − f )( ± π (cid:17) ˆ n − (cid:126)vv / = πmRb f / c due only to friction-like resistance is c = bR /
2. Substituting c = CbR /
2, where C is the ratio of the actual rotationalresistance to the frictional angular resistance, results in the following equations of motion. d Φ dT = 43 π (cid:112) − Y (1 + Y )( v − v ) cos(Φ) − C d Φ dT (9) d XdT = (cid:16) / v + v ) + 1 /π ( v − v ) (cid:0) arcsin( Y ) + Y (cid:112) − Y (cid:1)(cid:17) cos(Φ) − dXdT (10) d YdT = (cid:16) / v + v ) + 1 /π ( v − v ) (cid:0) arcsin( Y ) + Y (cid:112) − Y (cid:1)(cid:17) sin(Φ) − dYdTdYdT