Transient Pattern Formation in an Active Matter Contact Poisoning Model

One of the most notable features in repulsive particle based active matter systems is motility-induced-phase separation (MIPS) where a dense, often crystalline phase coexists with a low density fluid. In most active matter studies, the activity is kept constant as a function of time; however, there are many examples of active systems in which individual particles transition from living or moving to dead or nonmotile due to lack of fuel, infection, or poisoning. Here we consider an active matter particle system at densities where MIPS does not occur. When we add a small number of infected particles that can effectively poison other particles, rendering them nonmotile, we find a rich variety of time dependent pattern formation, including MIPS, a wetting phase, and a fragmented state formed when mobile particles plow through an nonmotile packing. We map out the patterns as a function of time scaled by the duration of the epidemic, and show that the pattern formation is robust for a wide range of poisoning rates and activity levels. We also show that pattern formation does not occur in a random death model, but requires the promotion of nucleation by contact poisoning. Our results should be relevant to biological and active matter systems where there is some form of poisoning, death, or transition to nonmotility.


INTRODUCTION
Active matter denotes systems of interacting particles that benefit from some form of self motility.It has been used to model biological systems, active colloids, artificial swimmers, social systems, and robotics [1][2][3][4][5][6].Due to their nonequilibrium nature, active systems exhibit a variety of irreversible behavior [7][8][9][10], boundary interactions [11][12][13][14][15], and novel transport effects [16].One of the most striking phenomena in particle based active matter systems is motility induced phase separation (MIPS) [17][18][19][20][21][22], where even for purely steric repulsive interactions that produce a uniform fluid in the Brownian limit, phase separation occurs at large activity into a high density solid surrounded by a lower density fluid or gas.The transition to the MIPS phase occurs as a function of increasing activity and/or particle density.
The propulsion in active systems can arise from chemical reactions on the surfaces of colloidal particles, from biological motors, or from mechanical motors in robots.It is generally possible for individual active particles to transition into a non-active or nonmotile state for several reasons, such as exhaustion of available fuel, finite motor lifetime, or death of a biological agent due to poisoning or excessive age.Depending on the scenario, the nonactive state may be permanent or temporary.Despite the wide variety of scenarios in which the lifetime of the activity is finite, relatively little is known about how this could impact pattern formation.For example, if the active system is in a regime where MIPS is absent, reducing the mobility should push the system further away from the MIPS regime.If the system is already in a MIPS phase, the addition of dead or nonmotile particles may be expected simply to break up the clustering.Some works have addressed mixtures of active and passive (nonmotile) particles in different limits.For example, a small number of active dopants added to a passive system can lead to the annealing of topological defects or can cause the passive particles to segregate into a crystalline-like arrangement [23][24][25][26][27].Other studies of active and passive binary mixtures have focused on how the active particles can cause the passive particles to form a dense cluster [28,29].Up until now, in studies of this type the ratio of active to passive particles has been held fixed, and it is not clear what would happen if there were some form of poisoning present in the system that could cause the number of passive particles to increase over time.Poisoning can be modeled by introducing nonmotile particles that can spread their nonmotile state to mobile particles through direct contact to create an epidemic-like transmission.A variant of this is to introduce a random death process where individual particles transition suddenly from mobile to nonmotile irrespective of any interactions with other particles.In both of these cases, mobile particles are only transiently present, and in the final state all of the particles are nonmotile.An open question is whether any kind of pattern formation occurs in such systems and whether it makes a difference if nonmotility is introduced through contact poisoning or through random death.
Motivated by recent models of epidemic behavior in active matter systems for susceptible-infected-recovered (SIR) dynamics [30][31][32][33], in this work we study active runand-tumble disks under non-MIPS conditions where we add a small number of nonmotile or dead particles that act as poisoning agents.With some finite probability, a mobile particle that is in contact with a nonmotile particle becomes nonmotile.We find that despite the apparent simplicity of this model, it exhibits a wide variety of transient pattern formation.At early times, the dead particles create small local clusters that act as nucleation sites for the growth of a large dense triangular cluster consisting of a dead central region surrounded by an accumulation of motile active particles.The accumulation process can be viewed as an example of active wetting or aggregation along walls, as observed in previous studies of active matter systems with barriers [34][35][36][37].The active particles in the accumulation region are progressively converted to nonmotile particles, driving the growth of the dense crystalline cluster.Once the nonmotile dense cluster has consumed a sufficiently large fraction of particles, the remaining active particles begin to create a fragmented zone along the edges of the cluster.The fragments consist of an intermediate density amorphous packing of passive particles crisscrossed by plow tracks from the active particles.The fragmented state penetrates the dense phase as a well defined front until the dense phase has become entirely fragmented.
Signatures of the pattern formation appear as a nonmonotonic time dependent behavior of the cluster size and the average number of contacts.We map out the transient cluster formation as a function of the activity level or run length and the poisoning probability β, and show that the pattern formation we observe is robust for a wide range of parameters.For low activity or low infectivity, pattern formation is absent.In a higher density system that can form MIPS states, the poisoning breaks up the MIPS.We show that the pattern formation we observe is due to the particle-particle information exchange, where the initial clustering of dead particles produces nucleation sites that cause active particles to become nonmotile or effectively stick to the cluster.In systems with random death, the pattern formation is lost.Our research reveals qualitative properties that can be hypothesized and tested in active biological systems such as bacteria that die after infection by bacteriophages and act as a poison to surrounding bacteria [38].Other experiments could be performed using active colloids with optical feedback effects [39][40][41].

METHODS
In previous work, we studied epidemic spreading in active matter systems with SIR models [42,43] or variations thereon [32,33].We considered a case where a susceptible (S) active particle comes into contact with an infected (I) particle, and during each simulation time step of contact, there is a probability β that the S particle will convert to I. If the only available states are suscepti-ble and infected, the model is known as the susceptibleinfected (SI) model, and, provided that every particle can come in contact with other particles, all particles will eventually transform to I. In the SIR model, I particles spontaneously transition to the recovered (R) state with a rate µ, opening the possibility that all the infected particle recover before the pool of susceptible particles is exhausted.As a result, over a range of β and µ values, a portion of the initial S particles never become infected.The epidemic size, defined as the fraction of recovered particles at the end of the epidemic, depends on the ratio of the transmission and recovery rates.In previous studies of SI and SIR epidemic dynamics on active matter, the infected particles remained mobile; however, in many physically relevant cases, infected particles would likely become nonmotile.In biological systems, a more realistic situation could be that the infected particles die, but remain capable of transmitting the infection or acting as a poison to other particles.
Here we consider a variation of the active matter SI model that we term the contact poisoning model.In this model, S particles are capable of becoming infected (poisoned), I denotes a nonmotile infectious or poisoning particle, and there is no spontaneous recovery.In this model, all the particles are eventually infected.We consider a two-dimensional system with periodic boundary conditions in the x and y directions of size L × L with L = 200.We typically focus on samples containing N = 4000 run-and-tumble active particles, which are modeled as harmonically repulsive disks of radius r a = 1.0.Unless otherwise noted, the disk density is φ = N πr 2 a /L 2 = 0.31415, low enough that the system does not enter the MIPS phase for the activity levels we consider.
The particles obey overdamped dynamics in which the time evolution of disk i is governed by the following equation of motion: Here the particle velocity is v i = dr i /dt, r i is the location of particle i, and the damping constant α d = 1.0.The disks interact sterically via a repulsive harmonic potential and the repulsive spring force constant is k = 20.The activity arises from a motor force F m i = F M mi of magnitude F M applied in a randomly chosen direction mi .A new random direction of motion is selected after each time period τ l ∈ [τ, 2τ ].This model has been used previously to identify the transition to a MIPS phase and to study active matter versions of SIR.[32,33,44,45].
Using the results from our previous work, we select a run length and motor force for which the system is in a uniform density phase rather than in the MIPS regime.On top of the discrete dynamics described above, each particle carries a variable indicating whether it is currently in the S or I state.Particles in the I state have F M set to F M = 0, rendering their motor inoperative; however, these particles can still move if pushed by direct contact with active particles.After each simulation time step update of the particle positions using Eq. ( 1), we update the status of the mobile active particles.For each pair of an I particle in direct contact with an S particle (i.e., |r ij | ≤ 2r a ), the S particle will change to I during that simulation time step with probability β.This adds an information exchange dynamic onto the active matter system.We consider values of β spanning the range 2 × 10 −6 ≤ β ≤ 3.2 × 10 −5 .To initialize the system, we place all particles at randomly chosen nonoverlapping positions and set all of the particles to S except for five randomly chosen poisoned I particles.

RESULTS
We summarize the dynamics of the epidemic using the temporal evolution of the fraction of infected individuals i(t).We want to relate this to the time evolution of the largest cluster size C and the average contact number Z per particle, both of which are determined by examining all particles in direct contact with each other.In Fig. 1(a,b), we show an example of how C and Z evolve for a system with φ = 0.31415, τ = 15000, F M = 1.5, and β = 8 × 10 −6 .The time t is normalized by the duration of the epidemic, defined as the time required for the last S particle to reach state I. Figure 1 nonmotile particles i.The data is obtained by averaging together ten different realizations in which time has been normalized by the duration of the epidemic, defined as the time required for the last S particle to reach state I so that all particles are nonmotile.Here, C and Z are nonmonotonic and reach a peak value near t = 0.45, indicative of transient cluster formation.In Fig. 2 we illustrate the positions of the S and I particles at different times for the system from Fig. 1.At t = 0.017 in Fig. 2(a), most of the particles are still in state S and form a weakly clustered fluid in which MIPS does not occur.In Fig. 2(b) at t = 0.214, larger clusters are beginning to nucleate around the growing number of I particles, but the clusters are amorphous and fluctuate with time.At t = 0.325 in Fig. 2(c) and t = 0.342 in Fig. 2(d) a dominant crystalline cluster containing a mixture of S and I particles is emerging, while at t = 0.410 in Fig. 2(e), there is now a dense triangular lattice composed entirely of I at the center of the cluster surrounded by a wetting layer of S on the outer edge of the cluster.In Fig. 2(f) at t = 0.504, the system consists of a dense triangular core of nonmotile particles with a halo-like re- gion of lower density amorphous phase containing tracks, which we call a fragmented phase.This fragmented phase moves into the dense region as a well-defined front driven by the plowing effect of the remaining mobile S particles.At t = 0.590 in Fig. 2(g), the dense core has been almost completely depleted and replaced by the fragmented phase.For t = 0.65 in Fig. 2(h), the system is completely fragmented and contains only a small remnant of moving S particles, while Fig. 2(i) shows the t = 1.0 final state in which all of the particles are dead and form low density amorphous clusters that are separated by tracks.
From the time-dependent images and behavior of C and Z, we can identify several different regimes.For 0 < t < 0.25 where Z, C and i are all low, only small clusters are present, as shown in Fig. 2(a,b), while for 0.25 ≤ t < 0.45, we see the emergence of a large dense cluster reflected by the increase of C and Z in Fig. 1(a,b).The cluster reaches its maximum size near the point at which i = 2/3 and s = 1/3.For 0.45 ≤ t < 0.75, the fragmentation region begins to form around the dense zone, resulting in a drop in C and Z in Fig. 1(a,b), while for t ≥ 0.75, the system is completely fragmented and contains only a small number of mobile S particles.
To further illustrate the behavior, we measure the time evolution of the local particle density φ loc in a slice of the sample taken along the y direction.We plot a height map of φ loc as a function of x position versus time t in Fig. 3(a) for the system from Fig. 2. The density remains uniform up to t = 0.2, but at later times a strong density peak appears in the center of the sample, corresponding to the formation of the dense crystalline phase.At larger times t, the fragmented state becomes visible with the appearance of strips of low density along the edges of the dense region.
By conducting a series of simulations at varied τ , we construct the phase diagram shown in Fig. 3 form of a height field of the average coordination number Z plotted as a function of τ versus t.For τ > 5000, the system can form a dense phase, indicated by the green region, while for τ ≤ 5000 the system remains in a uniform phase.This indicates that Brownian particles obeying the same S to I dynamics will only form a uniform nonmotile state, which we interpret to mean that the activity is essential for producing pattern formation.
We tested the impact of the poisoning probability on the pattern formation, as shown in Fig. 4(a) where we plot a height field of Z as a function of τ versus t for the same system as in Fig. 3(b) but for a lower β = 2 × 10 −6 .Here the clustering is strongly reduced.If we instead increase β to β = 3.2 × 10 −5 for the same system, the clustering is strongly enhanced, as shown in the heat map of Z in Fig. 4(b).In general, if β is very large, it is possible for a portion of the dense phase to persist to the end of the epidemic since the mobile particles become infected too quickly to be able to fragment the system.
Figure 4(c) shows the Z heat map phase diagram as a function of F M versus t for a system with φ = 0.31415 and τ = 15000 at β = 3.2 × 10 −5 , where a cluster always forms at intermediate time and the fragmented phase appears at late time.We have also considered a system with higher φ where a MIPS state appears at t = 0 for sufficiently large τ even without poisoning.We first allow the system to organize into a MIPS state prior to adding the poisoned particles.When the t = 0 MIPS state is present, the addition of poisoning dynamics only slightly increases the amount of clustering at intermediate times, but in general, the poisoning breaks up the MIPS at later times.In Fig. 4(d) we show the Z heat map phase diagram as a function of τ versus t for a higher φ = 0.393 at β = 8.0 × 10 −6 .Here, for τ > 9000 the system starts in a MIPS phase and shows a weak enhancement of the clustering prior to the onset of fragmentation, while for 4000 < τ < 9000 the system transitions from a fluid cluster to a fragmented state.For τ < 4000, the system is always in a fluid phase.
We have also considered the case of sudden random death, where there is no interaction between S and I particles, but instead S particles spontaneously transition to I with probability α.In Fig. 5(a,b) we plot C and Z versus time for a sudden death sample with φ = 0.31415 and τ = 15000, which corresponds to a regime where transient clustering occurs for the poisoning model.The probability for any single particle to die in a given time step is α = 8 × 10 −6 .The s and i vs t curves appear in Fig. 5(g).On average, the amount of simulation time required to transform the last S particle into I is much larger than for the contact poisoning model, leading to a more asymmetric shape of the curves as a function of rescaled time t.There are two effects contributing to this.First, in an SI model, particles making the greatest number of contacts are more likely to come into contact with infected particles and become infected more quickly, while the last surviving particles are typically the ones that have made the least contact with the infected particles.Such contact dynamics are irrelevant in the random death model.Second, there is no formation of a dense triangular phase in the random death model, and the largest cluster that forms is only of size C = 80.In Fig. 5(c), we plot the sudden death phase diagram in the form of a height field of Z as a function of τ versus t for the same system as in Fig. 5(a,b), indicating the lack of pattern formation over this range of τ .In Fig. 5(d,e) we plot C and Z versus t for a sudden death sample with the same τ = 15000 but at a higher density of φ = 0.393.The s and i versus t curves are unaffected by φ and appear in Fig. 5(g).In this case, the system starts in a MIPS phase at t = 0, and the sudden death breaks up the cluster without creating any new clustering.The height field plot of Z as a function of τ versus t in Fig. 5(f) for the φ = 0.393 sample indicates that the transient clustering found under interacting SI dynamics does not occur in the sudden death model.
Figure 6(a,b,c) illustrates the particle configurations in the sudden death model for the lower density φ = 0.31415 system in Fig. 5(a-c).At t = 0.1 in Fig. 6(a), small clusters appear throughout the sample.In Fig. 6(b) at t = 0.3, the clusters have decreased in size and the number of I particles has increased, while Fig. 6(c) shows the final state in which I particles are spread everywhere throughout the sample separated by voids marking the plow tracks of the final S particles.The φ = 0.393 sample from Fig. 5(d,e,f) is illustrated in Fig. 6(d,e,f).At t = 0.1 in Fig. 6(c), the system has not yet changed much from its initial MIPS state, but the MIPS cluster contains a moderate density of infected nonmotile I particles.In Fig. 6(e) at t = 0.3, the MIPS cluster is disintegrating, and in the final t = 1.0 configuration in Fig. 6(f), a lower density fragmented state appears.These results indicate that contact poisoning is essential for producing the transient clustering, and that the random death model does not give the same types of patterns.This is also consistent with the loss of patterning in the contact poisoning model for low β, shown in Fig. 4(a), where the odds of infection become low so that even if an S particle encounters a group of poisoned I particles, the S particle has a high chance of moving away before it becomes poisoned.For high values of β, shown in Fig. 4(b), the S particle has a higher chance to become infected and effectively stick to a cluster of dead I particles, which serve as cluster nucleation sites.In this way, the low infectivity limit is closer to the random death model.

DISCUSSION
In our model, even though the system is not in a MIPS regime, it can form a transient cluster similar to a MIPS phase that becomes fragmented by the remaining active particles.There are some differences between MIPS and the transient clustering.The active particles do not produce the transient clustering; instead, it is the result of the poisoning by dead I particles that act as nucleation sites and form dense regions in which the odds of a mobile S particle becoming poisoned are high.In contrast, for random death models where interactions between particles play no role in the transition to the I state, we do not observe any transient clustering.Our contact poisoning model should be relevant to physical active systems, since most such systems contain some constraint that can stop the mobility.This could include local depletion of resources needed for motion.A scenario in which individual particles carry their own source of fuel that can become exhausted would be more consistent with the random death model than with contact poisoning.
Another variation to consider is activation rather than poisoning, in which all of the particles are initially nonmotile but introduction of a small number of active particles can lead to the activation of additional particles through contact interactions.A competing scenario would be one in which nonmotile particles become active with some probability per time step, independent of interactions with other particles; this would be a sudden life model.In both cases, the final state would be that all particles would reach state S. It would be interesting to see whether the same sets of phases occur in reverse order or what would be the effect of adding spatial inhomogeneities [47][48][49].Finally, an active matter SIS model in which infected particles stop while infected and then restart moving when recovered presents a third scenario that combines both of the previous ones.We surmise that the latter may, once suitably rescaled, behave as a more standard quenched active matter system.

FIG. 1 .
FIG. 1.Time evolution of (a) cluster size C or number of particles in the largest cluster, (b) the average contact number Z per particle, and (c) the fraction s of susceptible particles (yellow) and infected nonmotile particles i (red) for a system with φ = 0.31415, τ = 15000, FM = 1.5, and β = 8 × 10 −6 .The curves are averaged over ten realizations.

Z
FIG. 3. (a) Heat map of the local density φ loc at a fixed value of y near the center of the system plotted as a function of x position versus time t for the system in Fig. 2 with φ = 0.31415, τ = 15000, FM = 1.5, and β = 8 × 10 −6 , showing the formation of the cluster.(b) Phase diagram in the form of a heat map of Z plotted as a function of τ versus t for the system in Fig. 1 with φ = 0.31415 and FM = 1.5.Clustering is present for sufficiently large τ only at intermediate times, as indicated by the green region.

ZFIG. 4 .
FIG. 4. Phase diagrams in the form of heat maps of Z. (a) Z as a function of τ vs t for the same system in Fig. 3(b) with φ = 0.31415 and FM = 1.5 at a lower infection rate of β = 2 × 10 −6 .where the amount of clustering is reduced.(b) The same for a system with a higher infection rate β = 3.2 × 10 −5 , where the clustering is enhanced.(c) Z as a function of motor force FM vs t for a system with φ = 0.31415 and τ = 15000 at β = 3.2 × 10 −5 .(d) Z as a function of τ vs t for a higher φ = 0.393 at FM = 1.5 and β = 8.0 × 10 −6 .At this density, MIPS clustering occurs at t = 0 for larger values of τ , but the poisoning process still destroys the cluster over time.

ZFIG. 5 .
FIG. 5.Results from the sudden death model, where S particles transition to I with probability α = 8×10 −6 per time step, independent of particle-particle interactions.(a) C and (b) Z vs t for a sample with φ = 0.31415 and τ = 15000.(c) The corresponding phase diagram in the form of a height map of Z as a function of τ vs t, showing that there is no pattern formation.(d) C and (e) Z vs t for a sample with φ = 0.393 and τ = 15000, where a MIPS state forms and the random deaths simply reduce the clustering.(f) The corresponding phase diagram in the form of a height map of Z indicates that the random death model breaks down existing clustering and produces no new clustering.(g) The s (yellow) and i (red) curves vs t are the same for both sudden death samples.