Cell-Division Behavior in a Heterogeneous Swarm Environment

Adam Erskine*,** J. Michael Herrmann** University of Edinburgh

Keywords Heterogeneous swarm, swarm chemistry, cell division, emergent behavior

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 swarmʼs 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. Reynoldsʼ 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 * Contact author. ** The University of Edinburgh, School of Informatics, IPAB, 10 Crichton Street, Edinburgh EH8 9AB, U.K. E-mail: [email protected]. ac.uk (A.E.); [email protected] ( J.M.H.)

© 2015 Massachusetts Institute of Technology Artificial Life 21: 481–500 (2015) doi:10.1162/ARTL_a_00188

A. Erskine and J. M. Herrmann

Cell-Division Behavior in a Heterogeneous Swarm Environment

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 swarmʼs defining parameters. Structure and form abound in and between biological organisms. Much of this comes about via selforganization. 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 windblown 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 celldivision-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 lowlevel 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. DʼArcy 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 flyʼs compound eye may be due to a mechanism of simple squeezing together as with 482

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Cell-Division Behavior in a Heterogeneous Swarm Environment

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. Reynoldsʼ 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 particleʼs recipe is favored over the collideeʼs, 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 canʼt 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 particleʼs 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. Selfdriven 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 Artificial Life Volume 21, Number 4

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Cell-Division Behavior in a Heterogeneous Swarm Environment

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 Fij = Fijc + Fijd + F n. 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 systemʼs 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 particleʼs update of position and velocity is influenced only by the particles within a specific neighborhood 484

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Cell-Division Behavior in a Heterogeneous Swarm Environment

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. N is the set of particles centered on particle i and being within the neighborhood radius of particle i. The average position of these is hx i ¼

1 X xj : jN j j2N

(1)

The average velocity of the particles within the neighborhood radius of particle i is hv i ¼

1 X vj : jN j j2N

(2)

The acceleration of particle i is given by ai ¼ −c1 ðxi − hxiÞ − c2 ðvi − hviÞ þ c3

X

!   2   : xi − xj = xi − xj

(3)

j2N

The dynamics are further modified by the parameter c4, which is the probability of ignoring the neighborsʼ effects. The particleʼs velocity is updated using the acceleration ai : vi 0 ← vi þ ai :

(4)

The magnitude of a particleʼs velocity has an upper bound. This is one of the swarmʼs parameters. Similarly, each swarm has a parameter that is the preferred magnitude of the particlesʼ 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 vi ← c5 ðvn =jvi 0 j : vi 0 Þ þ ð1 − c5 Þ vi 0 :

(5)

Finally, each particleʼs position is updated using xi ← xi þ vi :

(6)

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 swarmʼs 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 Artificial Life Volume 21, Number 4

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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 H¼−

X

P ðkÞ log P ðkÞ;

(7)

k

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 DKL ¼

X

P ðkÞ log

k

P ðk Þ ; QðkÞ

(8)

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 DKL ¼ log Qð1x Þ −H. For the celldivision-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. 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 486

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Cell-Division Behavior in a Heterogeneous Swarm Environment

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.

stages of this are shown in Figure 2. The values for the parameters used in these swarms are shown in Table 2. 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-andwhite 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.

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.

Table 1. Summary of behaviors in homogeneous swarms as a function of density and entropy. Entropy low

Entropy high

Density low

Dispersed

Multiple groups

Density high

Single cluster

Point swarm

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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.

An outside-in division, similar to what we have described in 3D, was achieved via modification of both swarmsʼ 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. Table 2. Parameter values for the cell-division-like behavior swarms. spc 1 2

rad 20.5 300

msp

c1

1.94

20.7

1

1

15.58

37.08

1

0.05

spd

c2

c3

c4

18.6

0.05

1

0.47

0.61

9.11

c5

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.

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Table 3. Pre- and post-division populations.

Cell-Division Behavior in a Heterogeneous Swarm Environment.

We present a system of virtual particles that interact using simple kinetic rules. It is known that heterogeneous mixtures of particles can produce pa...
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