Abstract

We present a system of virtual particles that interact using simple kinetic rules. It is known that heterogeneous mixtures of particles can produce particularly interesting behaviors. Here we present a two-species three-dimensional swarm in which a behavior emerges that resembles cell division. We show that the dividing behavior exists across a narrow but finite band of parameters and for a wide range of population sizes. When executed in a two-dimensional environment the swarm's characteristics and dynamism manifest differently. In further experiments we show that repeated divisions can occur if the system is extended by a biased equilibrium process to control the split of populations. We propose that this repeated division behavior provides a simple model for cell-division mechanisms and is of interest for the formation of morphological structure and to swarm robotics.

1 Introduction

We investigate emergent behaviors found arising from the interactions within a heterogeneous swarm. The interactions are in the manner of that originally described by Craig Reynolds [26]. He introduced a simple algorithm showing that such a swarm could manifest flocking behaviors. Each particle is influenced only by other particles in its local neighborhood. Each update of the model represents a discrete time step. On each update every particle is drawn toward the center of mass of its neighbors, aligns its velocity with its neighbors, and is pushed away from any particles too close. Reynolds' swarms were homogeneous.

Sayama [28] extended this approach by allowing multiple swarms to interact. Each swarm may have different sets of parameters. A set of parameters may be thought of as defining a species. By mixing two or more species of swarms, unusual structures and dynamic behaviors have been seen [29, 30, 31]. Many swarms could be identified that have a distinct biological look to them: Cells, amoebas, and diatoms abound. It is tempting to see the dynamics of the so-called swarm chemistry as a simple model for the real-life counterparts of these forms.

We extend the heterogeneous swarm algorithm to include both growth and biased equilibrium mechanisms. Our explorations have found a set of species that show cell-division-like behavior. Density and entropy measures allow us to make broad categorizations of behaviors. Single homogeneous swarms show limited behaviors, but more complex emergent behaviors are apparent with just two interacting species. Our investigations explore the robustness of such behavior under parametric variation. Specifically we studied:

  • • 

    How cell division is affected by the total size of the swarm and the population of each subspecies.

  • • 

    The differences in the behavior exhibited in 2D and 3D environments.

  • • 

    How cell division is affected by variation of several of each swarm's defining parameters.

Structure and form abound in and between biological organisms. Much of this comes about via self-organization. One benefit of this is that its resultant emergent forms are, in some sense, available for free. Structure emerges from interactions without the need for it to be explicitly coded. An understanding of these rules and their application allow us the possibility of reusing this free structure in robotic systems. Self-organization of structures, self-repair, or growth without explicit command and control is beneficial. This approach may provide a model that allows us to look at the automatic creation of morphological artefacts and dynamic behaviors. The tendency of many swarms to mirror biological forms, albeit superficially, raises the question of whether they can also be a model of biological processes.

Theories on the origin of life often invoke mechanisms to assure that proto-replicators are held in close association: within rock fissures; through agglomeration at thermal vents; within the wind-blown organic foams in the sea. Self-organized structures offer options for such discussions. A similar argument has been made [14] with reference to artificial chemistry. However, this model is limited to the organizational dynamics arising from its kinetic interactions. Single-cell division and the dynamics of small multicellular groups contain the ebb and flow of chemical gradients, protein interactions, and gene expressions. While much is known, the precise chemomechanical details are still there for investigation. We propose that the dynamics of our cell-division swarms may offer a simple model that allows some of these investigations. In order to allow this we require that a robust repeating cell-division-like mechanism be implemented. Thus we also look at modifications made to enable the observed cell-division behavior to repeat.

2 Background

In swarm systems patterns and behaviors may arise that are not obvious from multiple simple interactions occurring between the individuals in the swarm. These are said to be emergent properties of the system. Biological swarms—for example, ants and honeybees—exhibit multiple behaviors that are necessary for the success of the colony. Each of these has had tens of behaviors described and listed [9, 39]. Observation and experiment allow us to unpack the relationships between low-level interactions and the coordination of these colony-wide behaviors. These understandings provide useful inspirations for algorithms or swarm robotic systems.

Similarly insight may be derived from the study of simple processes and consideration of how such mechanisms may be thought of as simple models of the forms and behaviors that life can take on. D'Arcy Thompson detailed many roles that physical processes might play in the morphological development of creatures and their artefacts [37]. For example, he observed the minimal surfaces formed by soap bubbles in a foam and noted that they bore resemblances to biological forms. He believed that this was not mere coincidence. It has been shown that this idea is indeed true—at least in part. Honeybees build sheets of honeycomb within their hives. In cross section the comb exhibits a hexagonal structure. One might be tempted to assume that this structure has arisen from a bubble-like formation mechanism, but that is not the case [1]. However, the packing of the four cones in the ommatidia of a fly's compound eye may be due to a mechanism of simple squeezing together as with bubbles. Ball [1] also documents work that notes that the spicule structures of sponges appear to form via a mechanism whereby a bubble array is created and then inorganic compounds are allowed to permeate the interstices of the bubble matrix. The creature is leveraging the free structure from what Ball refers to as a fossilized foam. The processes at all scales of life are complex compared to the simple mechanisms that our model uses. Finite subdivision rules have already been used to model cell division.

Reynolds' flocking algorithm [26] has been subjected to numerous variations, adding in assumed fear, or leadership roles, or desire to stay close to roost sites or the like. It has been shown [10] that in starlings it is the number of neighbors (not the radius) that is important, and that the influence of neighbors is spatially anisotropic. Nearest-neighbor interactions combined with an energy minimization argument have been used to generate line and V formation flocks [16]. The Reynolds model assumes that the elements of a flock (birds, say) are each in some way monitoring the distances and velocities of their neighbors and calculating their next step. Biologically it is not clear that this is plausible. These homogeneous swarm algorithms have been further extended by combining multiple species.

We should mention here again the studies of Sayama in particular on the relationship between 2D and 3D species [31]. An evolutionary approach was adopted to discover interesting heterogeneous swarms [29, 30]. The approach allowed colliding particles the possibility of exchanging one set of parameters for that of the other particle. The function used to control this exchange acted to favor one behavior over another. For example, if the colliding particle's recipe is favored over the collidee's, then predator-like behaviors tended to emerge. Sayama has observed many differing behaviors: blobs, amoebas, oscillators, rotating swarms, and jellyfish, to name a few. Even though the particle interactions are purely local and kinetic, one can't help using biological metaphors when observing the swarm chemistry in action. It is interesting to wonder how far these superficialities can be extended to become models. It is, however, not easy to determine the correct low-level interactions that will provide the desired emergent group behavior. This difficulty in designing the desired emergent outcome is a central problem in swarm interaction research.

The self-propelled particle (SPP) model predates the Reynolds algorithm and can be seen as a simpler version [2, 38]. In the pure kinetic SPP model the particles have a constant speed, and on each update the direction of travel is updated under the influence of each particle's neighbors. As the particle density rises, group motions emerge. This raises the question of whether there may be simpler formulations of swarm chemistry that exclude attractive and repulsive components. Self-driven particles are uncommon in physics, but they are typical in biological systems. The emergence of cooperative motion in this model has analogies with the appearance of spatial order in equilibrium systems. Dynamic variants of SPP are derived from consideration of attractive and repulsive Newtonian forces [18, 24]. SPP models may provide biological insights. Newman and Sayama explore the effect of a sensory blind spot on the emergent milling behavior of the SPP model they used. As the size of the blind zone increases the behavior disappears, perhaps suggesting a link between collective emergent behaviors and development of sensory apparatus in biology [24]. An SPP model has been shown to provide a model for cell sorting [5]. The model is parameterized and two cell types are modeled using different parameters. Mixtures of the two cell types are shown to separate into inner and outer areas. This is similar to the swarm-chemical approach described above.

Artificial chemistries have many similarities to swarm chemistry. Again they are concerned with aggregations of individual interacting elements. Here the swarms are less abstracted, tending to have mass, volume, and chemical interaction rules. While many formulations may be expressed as sets of differential equations, there are formulations that explicitly model the chemistries by mixing different swarms of particles. Typically the interactions may be expressed as reactions rather than kinetic interactions, but interesting structures can occur in such formulations, for example, the semipermeable membranes in [21]. The flip side of this approach may be described as chemical robotics. Here, real cell-like chemicals are designed to ensure that their interactions cause desirable behaviors. In [36] one species of particle is designed to flow against the gradient of a second species, perhaps for use as a drug delivery mechanism. Regular dynamic motions can be designed, such as the peristaltic polymer gel of [20].

The Los Alamos bug is a project to develop a minimal protocell [11, 13] that exhibits metabolism, heredity, and containment. These are the elements the authors propose are necessary for minimal life. The dynamics of the kinetic and structural interactions are noted as being an important problem to understand [11]. A dissipative particle dynamic (DPD) model is adopted to explore this. Interactions are derived from the interparticle forces, which contain three components + Fn. These represent cohesive, dissipative, and noise force components between particles i and j and thus bear comparison with swarm chemistry.

There are other biological inspirations that arise from simple local interactions. Quorum sensing is seen in a variety of biological sources [22]. In bacteria it refers to the fact that gene expression can be switched depending on local cell population densities. Vibrio fischeri is perhaps the most studied of these. Its quorum sensing system results in the presence or absence of bioluminescence. This bacterium exists symbiotically in a number of different creatures and serves various purposes. In the squid Euprymna scolopes it is used as an anti-predation device: hiding its shadow. In the fish Monocentris japonicus it is used to attract a mate. Other biological examples include slime molds and honeybees. Slime molds respond to local chemical concentrations to switch between amoeboid behavior and slime mold form. When honeybees swarm, the decision of which new hive site to select is made when the number of scouting bees promoting a particular site rises above a threshold level [33, 34].

Another well-known local interaction mechanism consists of the stigmergic microrules various insects employ, for example, nest building in some wasp species [8] or the well-known use of pheromones by certain eusocial ants. Honeybees have been well studied and show many behaviors that result from purely local interactions. Thermal regulation of their hive is vital for successful larval growth and honeybees are able to maintain constant hive temperatures against a wide range of ambient conditions. It has been shown [6, 15, 32] that bees can, through their local interactions, locate areas of a target temperature. Collectively they can either raise or lower the temperature. While these mechanisms differ from those in the swarm chemistry discussed here, they have in common the idea that nonobvious structures, behaviors, and decisions can arise globally from local processes. Similarly, a simple model of E. coli cell division (cells are tubes that grow and divide in two periodically) has been shown to demonstrate fractal structures that mimic those of real colony development [27].

We are therefore not short of examples that suggest that emergent behaviors offer the prospect of complex systems arising from the interactions of simple units. Such biological inspirations have informed swarm robotic work. Review documents [4, 23] highlight the extensive range of behaviors that may be implemented from swarm robotic interactions, including: pattern formation; aggregation; chain formation; self-assembly; coordinated movement; hole avoidance; foraging; self-deployment; grasping; pushing; caging. Swarm robotic morphogenesis is an interesting field. Historically different approaches can be placed in one of three categories: lattices of interconnecting units, agents organizing in chains or tree structures, or forms arising from mobile swarm interactions [41]. Algorithms such as those discussed above (Reynolds, swarm chemistry, SPP, etc.) belong to the mobile approach and could clearly be used to build complex dynamic structures or patterns.

We have seen that the interaction of simple units may result in high-level behaviors. It is common that the emergent forms are not easily apparent from the nature of the unit interactions. Thus it can be difficult to engineer low-level behaviors that produce the desired swarm-level outcomes. Swarm chemistry represents a system that demonstrates this problem. It is complex enough to show a range of intriguing behaviors but simple enough for us to believe that it may be possible to study and extract useful guidance on how to solve these problems. Biological inspirations offer the possibility of locating solutions analogous to those we seek. Analysis of the methods nature implements to solve these problems may illuminate the approaches we should take. Alternatively, searching though the swarm chemical system's parameter space may allow us a route to understanding the relationships between the parameters and the emergent behaviors. The insights from this project should be widely applicable.

3 Method

The basic heterogeneous swarm algorithm [31] gives each particle a set of parameters. Each particle's update of position and velocity is influenced only by the particles within a specific neighborhood radius. Each particle has a preferred normal speed, the speed being bounded above. Parameters c1, c2, and c3 scale the influence of the neighboring particles. The parameter c1 is a measure of cohesion, the strength of pull toward the mean neighbor position. The parameter c2 is a measure of alignment, the strength of pull toward mean neighbor velocity. The parameter c3 is a measure of avoidance, the strength of push from close neighbors. On each update of the swarm, each particle uses neighboring particles to update its position and velocity.

is the set of particles centered on particle i and being within the neighborhood radius of particle i. The average position of these is
formula
The average velocity of the particles within the neighborhood radius of particle i is
formula
The acceleration of particle i is given by
formula
The dynamics are further modified by the parameter c4, which is the probability of ignoring the neighbors' effects. The particle's velocity is updated using the acceleration ai:
formula
The magnitude of a particle's velocity has an upper bound. This is one of the swarm's parameters. Similarly, each swarm has a parameter that is the preferred magnitude of the particles' velocity. If a particle is not traveling at this preferred velocity vn, then the parameter c5 is used to nudge the velocity back toward its preferred velocity using
formula
Finally, each particle's position is updated using
formula

3.1 Quantification

The eight parameters (c1 through c5, neighborhood radius, speed, and maximum speed) define a large parameter space. It is likely that not all parameters will have the same scale of influence on the swarm's dynamics. However, what is clear is that there is potentially a huge parameter space that one could search for interesting behaviors. Watching the swarms as they interact and evolve allows different behaviors to be spotted. However, this is not a feasible means to detect interesting behaviors. Instead we require quantifiable measures to allow different behaviors to be located. Having located likely candidates, they can be checked visually.

Structure in an evolving swarm can most usually be seen in how particles tend to conglomerate. Often groups or clusters form; on other occasions the particles will disassociate. In our swarms we can calculate the average density of particles. This density measure differentiates single clusters from both dispersed swarms and multiple groups: Single clusters tend to show a higher density. We note that this may not always be true: A large hollow single cluster may be less dense than multiple groups that are close together. A second measure, a spatial entropy, allows differentiation between multiple groups and dispersed swarms. It has been suggested [7] that the spatial entropy can be defined as
formula
where P(k) is the fraction of particles found in patch k. Here H decreases as clusters form. We used patches that are always cubes of side 0.1 times the maximum extent of the swarm, so that the minimal cube containing the swarm is split into 1000 patches. Two similar treatments are made in [3] and [40].
We also use the Kullback-Leibler divergence from an evenly distributed population as a measure. This is defined by
formula
where P is the distribution of the particle positions and Q is the distribution of an evenly dispersed swarm. Note that since Q is evenly distributed, we have simply . For the cell-division-like behavior, DKL thus increases when the swarm has divided into separate clusters.

4 Results

4.1 Single-Species Characterization

A homogeneous swarm appears to exhibit behaviors drawn from a fairly limited palette of possible behaviors. We note four behaviors: full dispersal, cluster or sphere, multiple groups, and one we call a point swarm (all particles collapse toward a single point). In full dispersal the particles separate and move apart; there is little or no tendency to aggregate. In a cluster the particles form a sphere (or approximate sphere) or shell of a sphere. Multiple groups are simply a multiple version of the last form. Point swarms are seen for swarm parameters where the avoidance value is at or near zero. This results in all particles collapsing to a single point. This state tends not to show as a sphere or a point. Instead, the particles, which all exist in a tiny spatial volume, show as an irregular cluster of particles that jump about. Particles have discretized speeds, so at each update a particle tends towards the average position of the cluster, but the step size is larger than the size of the cluster, so that the particles are unable to actually occupy a single point.

We find that the four single-species states can be identified by the density and spatial entropy (or Kullback-Leibler divergence). By sweeping through the parameter space of the cohesion and avoidance parameters it was possible to find regions of each of the four swarm types. The spatial entropy and densities were measured for each. A visual check of the final state of the swarm was also made. The measures were plotted against the parameters to generate the surface plots shown in Figure 1. The types of behavior seen in these homogeneous swarms can be summarized by their location in density-entropy space as seen in Table 1.

Figure 1. 

Density and entropy measurements for a homogeneous swarm as a function of its cohesion (c1) and avoidance (c3) parameters. The logarithm of each measure is plotted in order to squash the vertical extent of the surface plots, as the range of values extends over several decades.

Figure 1. 

Density and entropy measurements for a homogeneous swarm as a function of its cohesion (c1) and avoidance (c3) parameters. The logarithm of each measure is plotted in order to squash the vertical extent of the surface plots, as the range of values extends over several decades.

Table 1. 

Summary of behaviors in homogeneous swarms as a function of density and entropy.

Entropy lowEntropy high
Density low Dispersed Multiple groups 
Density high Single cluster Point swarm 
Entropy lowEntropy high
Density low Dispersed Multiple groups 
Density high Single cluster Point swarm 

4.2 Cell-Division Behavior Species

We present two species of swarms that individually formed single clusters (multiple groups if their populations were large enough), but in combination resulted in cell-division-like behavior. Typical stages of this are shown in Figure 2. The values for the parameters used in these swarms are shown in Table 2.

Figure 2. 

Typical evolution of cell division in our swarm. Top left shows red particles as a toroid about the yellow ones. Top right shows the yellow swarm divided in two with a separate red swarm. Bottom left has the red particles rejoining the larger yellow cluster. Finally, in the bottom right, the process repeating and is shown at a larger scale.

Figure 2. 

Typical evolution of cell division in our swarm. Top left shows red particles as a toroid about the yellow ones. Top right shows the yellow swarm divided in two with a separate red swarm. Bottom left has the red particles rejoining the larger yellow cluster. Finally, in the bottom right, the process repeating and is shown at a larger scale.

Table 2. 

Parameter values for the cell-division-like behavior swarms.

spcradspdmspc1c2c3c4c5
20.5 1.94 20.7 18.6 0.05 
300 15.58 37.08 0.05 9.11 0.47 0.61 
spcradspdmspc1c2c3c4c5
20.5 1.94 20.7 18.6 0.05 
300 15.58 37.08 0.05 9.11 0.47 0.61 

Notes: Headings are: spc = swarm species, rad = neighborhood radius, spd = normal speed, msp = maximum speed, c1 = cohesion, c2 = alignment, c3 = avoidance, c4 = whim, c5 = speed control.

When displaying the particles, the parameters c1 through c3 are used to define the displayed color of the particles. Here, species 1 displays as yellow and species 2 as red (NB: if viewed in black-and-white copy, the red particles appear slightly darker). Clearly, the displayed colors change if we alter these parameter values. For descriptive convenience we choose to describe the two swarms as the yellow and the red swarm. Typically the swarm population is constructed with many more yellow particles than red particles (ratios from about 5 : 1 to 10 : 1 are variously used). When the swarms are created, the two species are initially spatially mixed. Quickly they separate, the yellow ones forming a central core, the red ones an outer shell. The yellow core tends to elongate, and the red ones form a toroid roughly positioned centrally along the yellow core. The toroid squeezes and separates the yellow core into two parts. These move apart, with the red particles forming a cluster in between. At some point the red ones are drawn into one of the yellow clusters and the process repeats. In the basic configuration the repetition only occurs within a single cluster of the yellow particles. The division of the yellow particles tends to be asymmetric (see Table 3), with the red particles always rejoining the larger group.

Table 3. 

Pre- and post-division populations.

Yellow populationRed populationLarger yellow cluster's populationSmaller yellow cluster's population
300 50 174 126 
70 177 123 
90 157 143 
350 50 211 139 
70 219 131 
90 214 136 
400 50 327 73 
70 277 123 
90 243 157 
Yellow populationRed populationLarger yellow cluster's populationSmaller yellow cluster's population
300 50 174 126 
70 177 123 
90 157 143 
350 50 211 139 
70 219 131 
90 214 136 
400 50 327 73 
70 277 123 
90 243 157 

4.3 Comparison with 2D Swarm Chemistry

We explored the differences in 2D and 3D behavior of our swarms. With no change to the swarm parameters, cell-division behavior still occurred. In 3D the red particles form a toroid about the yellow particle swarm and tend to squeeze this core until division occurs. The separated parts then travel apart. In the 2D case the red particles travel to the inside of a yellow circle of particles and cause division to occur from the inside out. It is notable that the separate parts do not travel apart, nor do the red particles get drawn back into one of the yellow clusters. See Figure 3 for example images.

Figure 3. 

2D cell division. Upper two rows show an inside-out division in eight frames. Initial mixture separates to an inner core of red particles and an outer ring of the more numerous yellow particles. Gradually the red particles eat their way out, while the yellow population develops a narrowing. Finally the two halves of the yellow population pull apart. In the lower two rows we show an outside-in division in eight frames. A pair of species with modified parameters recapture something of the original cell-division behavior. As the parameters have changed, the displayed colours have been altered accordingly. The yellows appear as cyan, and the reds as magenta. The reds circle the yellows. Gradually they constrict the larger subswarm, pinching it and eventually causing division.

Figure 3. 

2D cell division. Upper two rows show an inside-out division in eight frames. Initial mixture separates to an inner core of red particles and an outer ring of the more numerous yellow particles. Gradually the red particles eat their way out, while the yellow population develops a narrowing. Finally the two halves of the yellow population pull apart. In the lower two rows we show an outside-in division in eight frames. A pair of species with modified parameters recapture something of the original cell-division behavior. As the parameters have changed, the displayed colours have been altered accordingly. The yellows appear as cyan, and the reds as magenta. The reds circle the yellows. Gradually they constrict the larger subswarm, pinching it and eventually causing division.

An outside-in division, similar to what we have described in 3D, was achieved via modification of both swarms' parameters (also shown in Figure 3). As parameters have been changed, the particles no longer appear as red and yellow but as magenta and cyan respectively. Now the red particles form a ring around the yellow circle and squeeze it until division occurs. Again the separate parts do not travel apart.

It is possible that reintegration of the red particles with one of the yellow clusters would occur if the swarm were left to run. It is also possible that with further parameter modification a recipe might be found that would result in the split parts separating. Swarms in 3D that show dynamics akin to the cell divisions seen in 2D are also found as a result of changes to parameter values (see Section 4.5.2).

4.4 Robustness under Population Dynamics

4.4.1 Yellow versus Red Populations

We varied the two species' populations to determine the limits on the cell-division-like behavior. Each run lasted for 2000 time ticks. The density and entropy measures were captured at the end of each run. For confirmation the final state of the swarm was captured as an image. Yellow populations were varied over a range from 100 to 550 in steps of 50, and red population over the range 10 to 90 in steps of 10. Figure 4 shows the density and entropy measures as a surface plot for all combinations of these populations. Cell division is marked by low density (blue on left-hand plot) and high entropy (red on right-hand plot). We see that the cell-division behavior extends over a wide range of populations. Very low red or high yellow populations tend to never show cell division. The line between division and no division is noisy. We assume this is due to variability in the starting position of particles and/or the arbitrary duration of each run. We explore both of these possibilities.

Figure 4. 

Density and entropy measurements for a heterogeneous swarm as functions of its yellow and red populations (p1 and p2). The logarithm of each measure is plotted in order to squash the vertical extent of the surface plots, as the range of values extends over several decades.

Figure 4. 

Density and entropy measurements for a heterogeneous swarm as functions of its yellow and red populations (p1 and p2). The logarithm of each measure is plotted in order to squash the vertical extent of the surface plots, as the range of values extends over several decades.

We fixed the red population at 50, and executed five runs for yellow populations varying from 300 to 600 in steps of 25. When the yellow population was below 375, division always occurred. For populations above 450 it never occurred. In the range between, division may or may not occur. The difference between the runs was in the randomized initial positions of the particles in the swarms. The KL divergence and the density (averaged over the five runs) are summarized in Figure 5. The density increases and the KL measure drops above a population of 350, coinciding with the onset of swarms that fail to divide. When division never occurs, the values level off.

Figure 5. 

Density and Kullback-Leibler divergence measures as functions of yellow population after 2000 time ticks. For a fixed population of red particles (50), we vary the population of the yellow swarm (from 300 to 600).

Figure 5. 

Density and Kullback-Leibler divergence measures as functions of yellow population after 2000 time ticks. For a fixed population of red particles (50), we vary the population of the yellow swarm (from 300 to 600).

4.4.2 Effect of Lengthening Run Time

We repeated the previous investigation, allowing the model to run now for 10,000 steps. There is still no distinct population boundary between split and no-split behavior. Yellow populations less than 425 always result in division. Those greater than 475 never divide. Populations between these limits may divide. Figure 6 confirms this observation in that the step up in density occurs at higher yellow populations. Executing the swarm for still longer durations suggested that with relatively small red swarms the whole swarm may be unable to divide. However, when the red population was increased (to 180), the swarm, which appeared to be stable, would occasionally eject a small cluster of yellow particles. This suggests that such a swarm may slowly lose yellow particles until the remaining yellow cluster is small enough to show the normal division behavior.

Figure 6. 

Density and Kullback-Leibler divergence measures as functions of yellow population after 10,000 time ticks. For a fixed population of red particles (50), we vary the population of the yellow swarm (from 300 to 600).

Figure 6. 

Density and Kullback-Leibler divergence measures as functions of yellow population after 10,000 time ticks. For a fixed population of red particles (50), we vary the population of the yellow swarm (from 300 to 600).

4.4.3 Cell Division with Additional Species

The initial discovery of the cell-division behavior occurred with a three-species swarm recipe. It was subsequently found that only two of the species were required for the behavior. We briefly looked at the effect of the three species on the behavior dynamic under population variation and discussed above. The third species manifests as a shell of pink particles outside the other two swarms. Throughout the test the population was fixed at 92. The red and yellow populations were varied as before. The recipe for the third species is given in Table 4.

Table 4. 

Parameter values for the third species.

spcradspdmspc1c2c3c4c5
124.4 13.63 40 0.8 98.5 0.32 0.84 
spcradspdmspc1c2c3c4c5
124.4 13.63 40 0.8 98.5 0.32 0.84 

Notes: When the third species was added to our normal pair of swarms, cell division continued to occur. Headings are: spc = swarm species, rad = neighborhood radius, spd = normal speed, msp = maximum speed, c1 = cohesion, c2 = alignment, c3 = avoidance, c4 = whim, c5 = speed control.

Figure 7 shows the state at time tick 2000 of a three-species swarm with a yellow population of 550. When executing two-species swarms, no cell division was seen with a yellow population of 550. Previously, with just two species, division had not occurred within this time period with a yellow population of this size. It would appear that the inclusion of the third species results in a greater likelihood of cell division.

Figure 7. 

A swarm with three species, still showing division behavior.

Figure 7. 

A swarm with three species, still showing division behavior.

4.5 Robustness under Parameter Variation

A full search of the parameter space is currently too onerous. Therefore we choose a simpler approach. We look to vary single parameters while keeping all other parameters unchanged. We vary the parameter being studied until the cell-division behavior disappears.

4.5.1 Variation of Neighborhood Radius

Using a yellow : red population mix of 300 : 50, we varied, independently, the neighborhood radii of each swarm. We varied the red radius from 50 to 300 in steps of 25, and the yellow radius in the range 10 to 40 in steps of 10 for each red radius. Each swarm was run for 2000 time ticks. With yellow radius greater or equal to 30 the red particles end up inside the yellow swarm and do not cause it to split. When the yellow radius is 10, the yellow swarm disintegrates rather than showing the more ordered division process. It thus appears that the yellow radius of 20 is a sweet spot for the cell-division behavior. A red radius of 125 or above appears to be necessary for division to occur. Additional tests with a finer gradation of yellow radii suggest that at a distance of around 10 there is a quick and catastrophic division of the yellow particles into many small clusters. The red particles form a single cluster, apparently uninterested in joining the others. For yellow radii from about 13 to 25, cell division occurs with the behaviors already documented. The lower the yellow radius value, the easier, or faster, the divisions appear to occur. From 28 upwards the red particles and the yellow ones exist either as a single cloud or with the red particles held among the yellow ones. Figure 8 shows an example of swarm states at the end of each run for a range of the radius combinations studied.

Figure 8. 

Sample final states of each run (after 2000 time ticks). The population of each species is held constant (300 yellow : 50 red); all parameters are kept constant except the neighborhood radius of each species. A minimum radius of around 125 for the red species appears to be needed for cell-division behavior to manifest. The behavior shows for a small but finite range of yellow radius values, centered around 20. Larger yellow radii do not yield division behaviors; smaller values lead to disorderly swarm disintegration.

Figure 8. 

Sample final states of each run (after 2000 time ticks). The population of each species is held constant (300 yellow : 50 red); all parameters are kept constant except the neighborhood radius of each species. A minimum radius of around 125 for the red species appears to be needed for cell-division behavior to manifest. The behavior shows for a small but finite range of yellow radius values, centered around 20. Larger yellow radii do not yield division behaviors; smaller values lead to disorderly swarm disintegration.

4.5.2 Variation of Avoidance and Cohesion Parameters

We separately swept through combinations of avoidance and cohesion parameters. First we varied red avoidance between 5 and 40, yellow between 10 and 60. Then we varied the red and yellow cohesion values from 0.2 to 1.0. A number of different behaviors were noted. Several behaviors would not be distinguishable by the use of measurements alone, so each run was watched and categorized. A single run of each permutation was made. All runs lasted 2000 time ticks. Table 5 shows the results for avoidance variation, and Table 6 shows the results for cohesion variation.

Table 5. 

Division types as a function of avoidance parameter c3 for a selection of the parameter variations tried.

Avoidance valuesYellow = 10Yellow = 20Yellow = 30Yellow = 40Yellow = 50Yellow = 60
Red = 5 3D 2D+ 2D+ 2D+ 2D+ 
Red = 10 3D 2D+ 2D+ 2D+ 2D+ 
Red = 15 3D 3D 2D+ 
Red = 20 3D 2D 
Red = 25 
Red = 30 3D 
Red = 35 
Red = 40 3D 
Avoidance valuesYellow = 10Yellow = 20Yellow = 30Yellow = 40Yellow = 50Yellow = 60
Red = 5 3D 2D+ 2D+ 2D+ 2D+ 
Red = 10 3D 2D+ 2D+ 2D+ 2D+ 
Red = 15 3D 3D 2D+ 
Red = 20 3D 2D 
Red = 25 
Red = 30 3D 
Red = 35 
Red = 40 3D 

Notes: Categories are: 0—no division seen; reds may form a toroid around yellows. 3D—division seen; behavior was characteristic of the standard 3D cell division. 2D—considered the same as the 2D case: inside-out split, but clusters are largely static after split. 2D+—like the 2D case, but separate groups are more dynamic after split; reds may be drawn in. Y—yellows disintegrate into small groups; reds form their own cluster.

Table 6. 

Division types as a function of cohesion parameter c1 for a selection of the parameter variations tried.

Cohesion valuesYellow = 0.2Yellow = 0.4Yellow = 0.6Yellow = 0.8Yellow = 1.0
Red = 0.2 
Red = 0.4 3D 
Red = 0.6 
Red = 0.8 3D 
Red = 1.0 2D 2D 3D 3D 
Cohesion valuesYellow = 0.2Yellow = 0.4Yellow = 0.6Yellow = 0.8Yellow = 1.0
Red = 0.2 
Red = 0.4 3D 
Red = 0.6 
Red = 0.8 3D 
Red = 1.0 2D 2D 3D 3D 

Notes: Categories are: 0—no division seen; reds may form a toroid around yellows. 3D—division seen; behavior was characteristic of the standard 3D cell division. 2D—considered the same as the 2D case: inside-out split, but clusters are largely static after split. Y—yellows disintegrate into small groups; reds form their own cluster.

Cell-division behaviors exist over narrow ranges of both these parameters. Cell-division behavior of the sort we have been looking at is thus very sensitive to the values of both avoidance and cohesion parameters. As with the other parameter studies, whether this is true for other population and parameter mixes is unknown. It appears that for small yellow avoidance values (c3 ≤ 40) the red avoidance value needs to be around half the yellow one for any division to occur. Given the parameter set of the swarms, it appears that larger yellow cohesion values are needed to stop the yellow swarm from disintegrating. Perhaps above this level (around 0.6) the yellow swarm requires a greater “pull” from the red particles to begin to divide. As with the other parameter studies, whether this is true for other population and parameter mixes is unknown.

The 3D division behavior exists on an edge between no-division and division behaviors. This edge can be explored in an automated manner by sweeping through pairs of avoidance parameters with a finer granularity than was achieved with the manual checking. Here we are not interested in the subtleties of the type of division processes. Instead we simply detect any division, manifested by an increase in the KL divergence value for our swarm. We set a threshold of 2.5 for this, based on observations. We made increments in the avoidance parameter value for red particles from 5 to 40. For each red value the lower limit of the yellow parameter was estimated from the earlier observations (Table 5). The swarm was run, and if no division occurred in 2000 iterations, then the yellow avoidance parameter was incremented and the swarm rerun. This was repeated 10 times in order to gain some feel for the variability arising from particle initialization. We plot this data in Figure 9 as a boxplot, as this nicely visualizes the width of this division behavior edge. This plot shows the same pattern: At low values of red avoidance there is a linear relationship with the yellow avoidance values needed to manifest division. At higher red avoidance values the required yellow value appears broadly to be a constant. It is interesting to note that where these phases come together there appears to be a need to have a higher value for the yellow avoidance parameter than either of the two zones would suggest. This is perhaps suggestive that there is more than one mechanism at play here: responsible for the upper and lower zones, respectively. In the middle these mechanisms may be interfering and yielding the need for higher yellow avoidance values.

Figure 9. 

Detail of edge of division as a function of avoidance parameter c3. Table 5 suggests that the edge between division and no-division behaviors as one sweeps through combinations of avoidance parameters is where the 3D division behavior occurs. Here we explore this edge. For each value of the red avoidance we scanned through the yellow avoidance parameters using the KL divergence measure (threshold set to 2.5 by observation) to capture when a combination led to division behavior. This was repeated 10 times for each red avoidance value to allow us to capture a feel for the variation in onset of division behaviors. We visualize this in a boxplot. Swarms that show no division exist below the plotted points.

Figure 9. 

Detail of edge of division as a function of avoidance parameter c3. Table 5 suggests that the edge between division and no-division behaviors as one sweeps through combinations of avoidance parameters is where the 3D division behavior occurs. Here we explore this edge. For each value of the red avoidance we scanned through the yellow avoidance parameters using the KL divergence measure (threshold set to 2.5 by observation) to capture when a combination led to division behavior. This was repeated 10 times for each red avoidance value to allow us to capture a feel for the variation in onset of division behaviors. We visualize this in a boxplot. Swarms that show no division exist below the plotted points.

4.6 Repeated Division

The cell-division behavior in the previous sections splits a cluster of yellow particles in two. Only one of those groups will subsequently divide again. This occurs because the red particles tend to associate only with the larger cluster of yellow particles. In order for this division behavior to be seen as a possible model for real-world division, we needed a mechanism that would allow any yellow cluster to potentially divide. In [30] each particle is modeled as expressing one parameter set drawn from a group of parameter sets. This formulation allowed a natural extension to evolutionary techniques to be applied. We choose a similar approach. Each particle expresses itself either as a red or as a yellow particle. There is a small probability that any particle may change the behavior it expresses. This is modeled as a biased equilibrium process. Each yellow, on being chosen to pick a behavior, will select changing to red with probability 0.1. Each red will select changing to yellow with probability 0.9. This ensures a roughly 9 : 1 mix in the population, but allows any cluster of yellow particles to develop a red population. This mechanism only works because a divided cell tends to move apart. If the parts remain close, either by artificial confinement or as would be the case in the 2D version, then any new red particles in one cluster tend to be immediately sucked into the group with the larger red population.

This mechanism alone provides for each cluster to continue to divide over time. However, as groups do not tend to recombine, the ultimate future for this approach is a dispersed swarm. We added a growth mechanism to allow clusters to increase in size. New particles would be created close to randomly chosen existing particles. This can be viewed as new particles being recruited from the environment. Figure 10 shows some examples from a swarm that implements both the biased equilibrium and growth mechanisms. The swarm still tends to appear somewhat dispersed; however, there are still many groups that continue to divide.

Figure 10. 

Repeated division. Top left shows first division. Top right shows second division. Bottom left shows multiple groups with red particles from the bias equilibrium process. Bottom right shows multiple divisions occurring.

Figure 10. 

Repeated division. Top left shows first division. Top right shows second division. Bottom left shows multiple groups with red particles from the bias equilibrium process. Bottom right shows multiple divisions occurring.

5 Discussion

Prior to dividing, the red particles form a toroid but only because the yellow ones support it. There are configurations where this appears to be a long-lived phenomenon. Division occurs for a wide range of swarm sizes, but there appears to be a size above which the yellow swarm tends to stability. We found some evidence that such a swarm may gradually lose yellow particles, suggesting that cell division may reappear if the swarm runs for long enough. Balancing the growth and biased equilibrium can be hard, and the population will tend to fragment. It would be appealing to improve the linkage between these mechanisms so that division would become more regularly periodic.

We observed differences in the emergent behavior depending whether the swarm ran in a 2D or a 3D environment. If the parameter values used in a 3D environment were used, unchanged, in a 2D environment, then we observed an inside-out division. This resulted in a relatively static set of divided groups. By modifying the parameter values used, we were able to recapture the outside-in division seen in 3D. This still failed to show the full dynamics seen in 3D. However, the fact that there are parameter mixes that show behavior in 3D that matches that seen in 2D suggest the opposite may also be true.

The cell-division behavior was sensitive to the swarms' parameter recipes. The yellow neighborhood radius needs to be in a narrow band. The red neighborhood radius appears to have a lower limit; much larger values seem to result in division behavior. Cell-division behavior is seen only across a narrow band of both avoidance and cohesion parameters. On one side of the band no division is observed. On the other side, either an inside-out division similar to that seen in 2D, or a spontaneous yellow disintegration that requires no interaction with the red particles, is observed.

The inclusion of a biased population equilibrium and growth mechanisms enabled the swarm to show ongoing cell-division-like behaviors. Additionally, we showed that the division process is not limited to just two species: A third species, though it affected the swarm dynamics, was successfully included and retained the capability to show division. Is there scope to further complicate the set of recipes included in the swarm and yet retain the division mechanism? It would be interesting to explore modeling each swarm species with a probability distribution of parameter values, perhaps extending the swarm chemistry evolutionary approaches already explored by Sayama [29].

We presented a heterogeneous swarm that exhibits cell-division-like behavior. Lutkenhaus [19] outlines the sort of division process we draw comparisons with here. Prokaryotic cells divide via a mechanism whereby the FtsZ protein migrates to the center of the cell and polymerizes. As it does, it constricts the cell, initiating the division stage. Normally this Z ring provides a scaffold that many other proteins can connect to and contribute to the process. However, in simple bacteria (those lacking a cell wall), FtsZ is still present, suggesting it is sufficient to cause constriction in simple bacteria. In a Z-ring the filaments formed are too short, so the ring is incomplete. It is thought that the ring may be constructed via lateral interactions of the filaments. While cellular mechanisms are much more complex than our kinetic model, it is interesting to note that filamentary interactions are suggested as a possible part of the process. In our model the location of the formation of the split derives only from the kinetic interaction of the particles. In one suggested cellular division mechanism [12], the oscillations of one protein (MinE) essentially defines the midpoint of the cell. This drives a second protein (MinD) to vacate that location. The MinD protein couples with a third protein (MinC), which acts to prevent the FtsZ protein from polymerizing. Thus only at the midpoint does division occur. Again this mechanism is clearly hugely more complex than what we present; however, swarm chemistry has been shown to include a large repertoire of behaviors. Oscillatory motions have been shown [30]. Additionally we have shown that the cell division process is maintained even with the addition of another species of particle. It may be possible to combine these in the future.

Origin-of-life models attempt to explore mechanisms likely to have been available to a prebiotic Earth. Common to many are the need for containment and replication of assemblies of chemical mixtures. Various containers have been suggested: Oparin proposed gel coascervates as a suitable container [17]; lipid bilayers have appealing similarities to cell membranes [35]. Division of contained mixtures would then provide a means by which evolutionary selection could begin to take place. Such proto-replication would occur without the complex cellular mechanisms we currently see. The Los Alamos bug project [25] utilizes a DPD model to explore the necessary dynamics. Swarm chemistry, and its behaviors, may provide a simpler division mechanism.

In our model, we saw that division is usually asymmetric. This would be potentially harmful for cell division, but might have been of some use for early proto-replicators; for example, a simple mechanism that results in nearly even division of containers of chemical mixtures could be useful.

6 Conclusions

We have demonstrated a repeating cell-division-like behavior that emerges from the low-level kinetic interactions of a heterogeneous swarm. To achieve this it was necessary to add a biased equilibrium process to control the population split of the different particle types. The emergent behavior has some similarities to prokaryotic Z-ring-initiated cell division. The division behavior exists in a small but finite volume of the swarm chemistry parameter space. Locating other interesting behaviors that may exist in equally small volumes of this space will require improved diagnostic tools. Additionally we have seen that the behaviors differ depending on whether the swarm is moving in two- or three-dimensional space.

Acknowledgments

This work was supported by the Engineering and Physical Sciences Research Council (grant number EP/K503034/1).

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Author notes

Contact author.

∗∗

The University of Edinburgh, School of Informatics, IPAB, 10 Crichton Street, Edinburgh EH8 9AB, U.K. E-mail: a.erskine@sms.ed.ac.uk (A.E.); michael.herrmann@ed.ac.uk (J.M.H.)