An Agent-Based Model for Collective Behaviors of Social Amoeba Dictyostelium discoideum Morphogenesis: Aggregation Phase

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An Agent-Based Model for Collective Behaviors of Social Amoeba Dictyostelium discoideum Morphogenesis: Aggregation Phase

PARHIZKAR, Mohammad, DI MARZO SERUGENDO, Giovanna

Abstract

Abstract: Dictyostelium discoideum is a social amoeba exhibiting distinct self-organizing behavior at different phases of its life - moving in a coordinated way towards areas with food, signaling lack of food and recruiting partners to create a single super-organism (fruiting body).

These phases and their underlying mechanisms are excellent models useful for understanding other natural cells' behavior (e.g. cancer cells), as well as to engineer artificial systems such as swarms of robots. This paper focuses specifically on the aggregation phase of Dictyostelium discoideum. We present a detailed agent-based - “bottom-up” - model, which exhibits a series of individual, collective behaviors and emergent properties of social amoeba Dictyostelium discoideum. We extended previous models of the aggregation phase with: a pre- aggregation phase; and three different levels of quorum sensing allowing collective decisions to be taken in a decentralized manner for (1) identifying the time for aggregation phase; (2) providing aggregation territories of homogeneous size; (3) allowing the appearance of late centres. The key and [...]

PARHIZKAR, Mohammad, DI MARZO SERUGENDO, Giovanna. An Agent-Based Model for Collective Behaviors of Social Amoeba Dictyostelium discoideum Morphogenesis:

Aggregation Phase . SWARM 2017: The 2nd International Symposium on Swarm Behavior and Bio-Inspired Robotics : 2017

Available at:

http://archive-ouverte.unige.ch/unige:116286

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An Agent-Based Model for Collective Behaviors of Social Amoeba Dictyostelium discoideum Morphogenesis: Aggregation Phase

Mohammad Parhizkar

and Giovanna Di Marzo Serugendo

Institute of Information Service Science (ISS), Geneva School of Social Sciences, University of Geneva, Switzerland

(¹Email: Mohammad.Parhizkar@etu.unige.ch) (²Email: Giovanna.DiMarzo@unige.ch)

Abstract: Dictyostelium discoideumis a social amoeba exhibiting distinct self-organizing behavior at different phases of its life - moving in a coordinated way towards areas with food, signaling lack of food and recruiting partners to create a single super-organism (fruiting body). These phases and their underlying mechanisms are excellent models useful for understanding other natural cells’ behavior (e.g. cancer cells), as well as to engineer artificial systems such as swarms of robots. This paper focuses specifically on the aggregation phase ofDictyostelium discoideum. We present a detailed agent-based - “bottom-up” - model, which exhibits a series of individual, collective behaviors and emergent properties of social amoebaDictyostelium discoideum. We extended previous models of the aggregation phase with: a pre- aggregation phase; and three different levels of quorum sensing allowing collective decisions to be taken in a decentralised manner for: (1) identifying the time for aggregation phase; (2) providing aggregation territories of homogeneous size; (3) allowing the appearance of late centres. The key and unique character of our model is the cells’ self-assessment and self-generated gradients arising from six chemical factors: PSF, CMF, Adenosine, cAMP, PDE and CF released by each individual amoeba. We programmed our model in Matlab. Our results show a series of behavior close to individual and collective behavior of livingDictyostelium discoideum. In particular we observed: inhibition of centers too close to each other; appearance of late centers when aggregation territories are too large; and measured homogeneous size aggregation territories emerging from the cells’ behavior. Future work will address the remaining phases ofDictyostelium discoideum life cycle (migration and culmination).

Keywords: Bio-inspired Swarm Modeling, Multi-agent Systems, Slime Mold, Dictyostelium discoideum, Self- Organization, Quorum Sensing, Unicellular Communication

1. INTRODUCTION

Understanding collective behaviors in nature and its potential links to engineering of collective artificial be- haviors attracts many researchers from biology, computer science and swarm robotics. It impacts different scientific and industrial topics such as cell-biology, cancer study, environment cleaning, swarm of drones and unmanned robots. For instance, cancer cells exhibit collective be- haviors, biomedicine researchers look for different exam- ples from nature to design anti-cancer drugs to shrink tu- mors in human bodies. On the engineering side, swarm roboticists take inspiration from natural cells collective behavior to build autonomous and robust systems.

A classic and interesting example of a collective sys- tem is demonstrated by the multicellular development process ofDictyostelium discoideum. The social amoe- bae of the cellular slime moldDictyostelium discoideum is one of the eight species selected as unicellular model organisms for biomedical researches [1]. Despite its unique and relatively simple life cycle, it has been ex- ploited for the study and modeling of various behav- ior, such as cell motility, chemotaxis, pattern forma- tion, phagocytosis, cell-cell contact, gene-expression, cell death. Studies of the developmental cycle inD. dis- coideumrepresent the best examples of the use of math- ematical modeling in developmental biology [2]. These

†Mohammad Parhizkar is the presenter of this paper.

unicellular motile amoebae with typical lengths about 10 µmin diameter, cooperate and show notable social be- havior when they are deprived of food. Figure 1 shows the different phases of D. discoideum behavior. Star- vation, prompts the solitary cells to interact by means of a self-generated chemical signal to synchronize their, otherwise random and unorganized, motion (Aggregation phase). Upon exhaustion of nutrients, unicellular starv- ing amoebae initiate early cellular differentiation and un- dergo a multicellular phase to form a mound which de- velops into a moving slug looking for favorable condi- tions (Migration phase), and transform later into a fruit- ing body consisting of a head of spores and a stalk of life- less cells (Culmination phase) [3]. This ability to alter- nate between unicellular and multicellular forms makes D. discoideuman ideal organism to study social and self- organized behaviors, as well different levels of emergent properties.

As computer scientists, we are particularly interested in understanding the different phases of D. discoideum behavior, and providing new agent-based models we can later integrate in artificial engineered systems such as swarms of robots. This paper presents a detailed descrip- tion of our agent-based model for a part of the aggrega- tion phase ofD. discoideumlife cycle, which goes from starvation, to streaming and aggregation territories.

The paper is structured as follows. In Section 2, we briefly review previous related models. Section 3 in-

Fig. 1: D. discoideum life cycle:the development process is highly regulative and depends on the local cell density. From 100 to 100,000 cells aggregate together (after 6hr of starvation) in response to cAMP signals emanating from the center of an aggregation territory. During each phase every cell acquires some competences that it did not pos- sess in the previous state. Additionally,D. discoideumhas one of the well-known self-regulation proportion in nature is the proportion of two different cell types, prestalk (20%) and prespore (80%) cells, in the slug phase. In this paper, we investigate and model the red arc of the image.

troduces the biological mechanism for growth and ag- gregation phases of Dictyostelium cells up to stream- ing. In Section 4, we introduce a decentralized agent- based model that incorporates this biological mechanism.

We present our results highlighting specific behavior of D. discoideum(inhibition of centers, late centers appear- ances, and homogeneous size territories) in Section 5.

Conclusions and future work are discussed in Section 6.

2. STATE OF THE ART

Aggregation is modeled in different previous works such as Mackay [14] and Vasieva et al. [17], or Van Oss et al. [29]. In these discrete models, each cell has three status: sensitive, refractory and rest. ‘Local excita- tion and global inhibition’ is trending to model the gradi- ent sensing ofD. discoideum, during last years [32–34].

Compared to these models we provide here an integrated, novel model that: starts from pre-aggregation and goes up to streaming; makes use of the most significant signals re- leased by the cells, namely PSF, CMF, Adenosine, PDE, CF and 3’,5’-cyclic adenosine monophosphate (cAMP) (see below for more details); and brings in a series of quo- rum sensing algorithms for decentralized collective deci- sions based on various gradients provided by the above signals.

3. DICTYOSTELIUM DISCOIDEUM

In this paper, we focus on how starvation, self-selected centers, inhibition of centers, collective decisions and the movement behavior of individual amoebae leads to the collective behavior where cells aggregate into emergent

territories of homogeneous size of approximatively5× 10³cells.

This whole transformation process is regulated by 349 different proteins secreted by developing cells [4]. The most significant chemical agents are PSF, CMF, Adeno- sine, PDE, CF and 3’,5’-cyclic adenosine monophos- phate (cAMP). A key role in aggregation is caused by periodic cAMP secretion, amplified by surrounding cells, resulting in cell polarization [13].

3.1. Pre-aggregation

Nutrient sensing and vegetative state: as long as food is present, the vegetative cells monitor food bacteria within a limited area and respond to folic acid (FA) as a derived metabolite [9]. Each cell reacts autonomously to FA as a chemoattractant generating a spatial cue that cells use to hunt bacteria [10].

Prestarvation Factor (PSF): Fig. 2 demonstrates the releasing time of various signals, among them PSF. Star- vation makes the cells less responsive to FA and cause also the emergence of different new genes which are es- sential for chemotaxis towards cAMP [12]. PSF is an autocrine factor that is secreted by growing cells until early multicellularity development. It has two significant roles: measuring cells’ density and the ratio of bacteria to D. discoideumcells [10].

starvation t = 0

PSF CMF

t = 5 hr

adenosine, (cAMP + PDE) , CF t = 12 hr

aggregation

Fig. 2: Times of continuously synthesizing and secreting autocrine factors before and after starvation.

Conditioned Media Factor (CMF): another crucial factor for D. discoideum development is CMF (a 80- kDa glycoprotein), which is secreted by starved cells slowly and simultaneously as a preparation for aggrega- tion. CMF secretion serves to synchronize the beginning of the aggregation, signaling when the number of starved cells passed a critical threshold.

3.2. Single-cell Self-analysis

We discuss here the behavior of single cells, how they analyze themselves and undertake differentiation towards becoming regular cells or centers (i.e. recruiting regular cells to form an aggregation territory, which will later be- come a slug).

Cell growth cycle and cell differentiation: aggregation begins for starved cells, which are in S phase of their growth cycle; this occurs between 2 to 3 hr and also be- tween 6 to 7 hr after release [7]. When cells synchronized in S phase, approximately 50% of the population initiates centers. When cells are synchronized in late G2 phase (T7 cells) only 7.5% of the population initiates centers [7]. Thus, in a population containing homogeneous cells in different phases of their cell cycles, it may be the S-

phase cells which differentiate earliest and possibly initi- ate centers.

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Fig. 3: D. discoideumcell growth cycle: the whole doubling time is approximately 7.2 hr and most of it, isG2phase. Inspired from Maeda’s work [9].

Center selection: as we explained in the introduction section, all starved amoebae have the potential to initi- ate the aggregation centers but there are three facts we should keep in mind. First, it is now commonly stated that cells capable of initiation are not genetically differ- ent but arise at random in a starving population [8]. But theses randomly selected cells need first to reach a crucial threshold of neighbors as a physiological state by sensing the concentration of CMF. Second, they are sensitive to the inhibition action of adeno- sine, released by already initiated centers. Concentration of adenosine at5mM impedes aggregation center initiation without any compe- tition with respect to cAMP and to the signal relay [31].

Third, amoeba’s position in its growing cell cycle at the onset of development influences the differentiation on the population. Approximately 400 per eachmm²is critical density to reach cAMP concentration threshold, less than it, the aggregative signal cannot be stimulated in the population [12]. InD. discoideumaggregation, the movement of cells consists of cells adhesion and making streams, then the streams converge to the big streams and finally they gather in the centers.

Sussman’s work [27] indicates that the number of ag- gregative centers is both a function of the number of the cells and the population density. Then, in the optimal population densities, the number of centers is constant at the different developmental stages.

3.3. Aggregation Territory Size and Later Centers As the D. discoideum cells grow and population be- gins to rise, largely unaware of each other, a set of factors is used to measure the ideal amoeba numbers required to form a complete multicellular organism. Basically, cells use three substances to monitor the population size, ex- tracellular environment and control the choice between growth and differentiation:

Counting Factor (CF): in addition to CMF, CF has great affects on the development process in order to reg-

ulate the size of a group of cells. It is involved in sensing the number of cells in an aggregation stream. In the high concentration of CF, large aggregation streams will be di- vided into small groups.

To achieve the optimal spore dispersal, a fruiting body has to hold the spore mass as high as possible from the ground. In that way, there might be a relation between the number of dead cells in the stalk and the number of spores in the sporangium. Thus, in one hand the process tries to survive the maximum number of cells and on the other hand, increasing the number of cells in the sporangium, there should be a proper strong stalk to keep them safe.

Thus, in a field of starving D. discoideum, there are a upper threshold and lower threshold on number of cells in each aggregation center. In the other words, depending on the planting density and the type of species the average size territories usually are the same [28].

3.4. Chemotaxis

Chemotaxis in D. discoideum morphogenesis is the movement of the amoebae as a reaction to starvation, which is determined by spatial and temporal leads of the dynamic cAMP gradients as a chemical stimuli.

Relaying threshold Chemotaxis threshold cAMP concentration level

for a regular cell C

B

A New center threshold

Fig. 4: Different levels of cAMP thresholds Cyclic Adenosine Monophosphate (cAMP): in D. dis- coideum life cycle, collective behaviors begin by starva- tion of population, which is signaled by small molecules of cyclic-3’, 5’-AMP (cAMP). In each aggregation terri- tory, up to 100,000 starved amoeba gather by periodically synthesizing and secreting cAMP into the extracellular medium and also responding to it by small step of move- ment. Individual amoebae acquire new and unique abil- ities such as synthesize, detect, and degrade cAMP after starvation. When the number of starved cells reaches a critical threshold, only a few cells start to secret cAMP, which are called centers, or autonomous initiators. Dur- ing aggregation and even after it these centers release short, periodic pulses of cAMP autonomously. The other cells make a positive feedback loop, by replaying the re- ceived signal and increase the concentration of cAMP.

In the meantime, the cells measure the concentration of cAMP as a chemoattractant to find the spatial gradient and begin the movement. Even during later phases of life cycle, inside the slugs, cAMP and phosphodiesterase are secreted at the tips like the centers in the aggregations.

The waves of cell movement due to cAMP relay is the

same used in the aggregation stage.

Phosphodiesterase (PDE): secreted cAMP is the prin- ciple signal in the development of D. discoideum and is regulated by a cyclic nucleotide phosphodiesterase (PDE). It is specifically responsible to convert cyclic-3’, 5’-AMP to 5’-AMP [11].

Refractory period: as a highly organized system, dur- ing the cAMP signal propagation, there is a delay be- tween receiving and releasing the cAMP pulse, which is called “Refractory period”. As the cAMP waves move through the population directionally from the center, the responding amoebae must become insensitive for a small time after relying, for otherwise they would react to re- flection of their own relayed signals.

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Fig. 5: The left image demonstrates theD. discoideumcell movement towards cAMP emitted from a source. The right image demonstrates the chemical diffusion from cell to the extracellular medium. In our calculation, we consider a cell as a cylinder, not as a simple point like Mackay’s model:l²= 10µm×2µm×π, wheredis the side length of the square andl≈8µm.Then we can assume a two dimensional dis- tribution model in which amoebae are distributed on a grid that consists of squares of side8µm.

4. AGENT-BASED MODEL

Fig. 6 demonstrates the whole model for individual cells, and follows the description of the natural behav- ior discussed in Section 3. The model is composed of two parts: the cells’ behavior during the pre-aggregation phase (starvation and synchronization for starting the ag- gregation); and the cells’ behavior during the aggrega- tion phase (regular cell vs center). The model describes amoeba cells as discrete self-organized agents which can measure and sense different chemical factors in their neighborhood. Based on different concentrations of these chemicals, individual cells decide to boost some particu- lar factors or move toward spatial gradient of cAMP.

4.1. Novel Features of the Model:

We started from Mackay’s work [14] (in white in Fig- ure 6), to which we added several interesting points (in gray in Figure 6). The unique and novelty part of our model can be categorized into these points:

• Pre-aggregation phase: coordinated by self-generated diffusive extracellular PSF and CMF; synchronizes cells to start the aggregation phase.

• Centers’ selection, centers’ competition and inhibition and new centers appearance at later stages of the aggre-

gation (for regulating aggregation territories sizes);

• Single cell self-analysis: cell’s age and different prob- abilities to become a center depending on their individual growth cycle;

• Three different levels of quorum sensing (red circles in Fig. 6) based on the measurement of each amoeba of several chemical substrates (gradients), which originate outside their borders. Each of these quorum sensing pro- cesses has different responsibility in the aggregation of amoebae, and trigger decentralized collective decisions:

1. Identify starvation (1a) and start the aggregation based on the high population density (1b); 2. Regulate aggrega- tion territories size according to the density of their local population, 3. Coordinate the formation of the new (later) centers.

• To build a more realistic model of in vivo cell behav- iors, we consider cells as cylinders (Fig. 5 on the right) and not just as simple points. Fig. 5 also demonstrates, on the left, the cell’s size and shape when it undergoes a conformational change, and navigates towards cAMP spatial gradient.

Algorithm1illustrates the global model and how the simulation engine executes each cell (agent) behavior.

For each cell we keep its current status, and separately with keep a cAMP firing queue, storing for each cell the amount of cAMP it has to release and the simulation time when this should occur. We start the simulation with the pre-aggregation part. Then, at each simulation step, we execute each cell behavior (aggregation part of the model in Fig. 6). At the end of each simulation time, we sepa- rately look at the cAMP firing queue (progressively built while the cells are executing), and update the cAMP val- ues in the simulation environment. It is also important to notice that while a given cell is executing, it also looks at the cAMP firing queue to update its own status (or knowl- edge about gradient values).

4.2. Pre-aggregation

Starved population size estimation: as described in Section 3.1 and demonstrated in simulation results ob- tained through Matlab (Fig. 7-a1), cells use PSF to sense the density of their local population of bacteria. After a delay between the production of PSF and processing the environment (Fig. 7-b1), whenever the concentration of PSF reaches a special threshold (the first quorum sensing) around the cells (Fig. 7-c1), they start to release CMF as an acknowledgment (Fig. 7-c₂). Fig. 7-a₂ and Fig. 7-b₂ demonstrate the concentration of CMF is still zero at that time.

When the other cells realize the high concentration of CMF around them, they stop the production of PSF (Fig. 7-d1) and begin to relay CMF to inform the whole population to start the morphogenesis development to- gether as a consensus decision (quorum sensing 1b) (Fig. 7-d₂). Eventually after a while, whenever the con- centration of CMF is very high at some points, the cells enter the ‘single-cell self-analysis’ phase. The location of the highest CMF concentration is the potential position of the first autonomous centers, which will be described in

Fig. 6: Proposed model for an individual cell: our model for an in- dividual cell consists of two big phase: “pre-aggregation” and “aggre- gation”. In pre-aggregation, cell uses quorum sensing (1a and 1b) to investigate the environment. At 1a if the concentration of PSF is above the threshold that means the starvation has happened and the cells need to inform the others by releasing CMF. Whenever the concentration of CMF is above the threshold, cells go to the aggregation part. In the aggregation phase the first thing, is the selection of the centers. Becom- ing an autonomous center depends on cell’s age, also the cell can be an autonomous center when the concentration of CMF is high and concen- tration of adenosine is low. In the “Territory size + New centers” part, cell uses two levels of quorum sensing, one to realize the proper aggre- gation cluster (red circle-2) and other for formation of the new centers (red circle-3). When the concentration of cAMP and CF is both high enough the cell discovers that the population needs a new center. In the last part of the model “Chemotaxis” cells decide to move and join to the territories by following to the high gradient of cAMP.

the next sections.

4.3. Single-cell Self-analysis

Individual cell cycle of aDictyostelium discoideum:

the second part of Fig. 6 shows the behavior of each indi- vidual amoeba after the onset of the aggregation. In the first place, after starvation with probabilityP₁amoeba is in itsT7phase and with probability ofP2it is in theS phase of its growth cycle. As explained in Section 4.2, cells in theirT7phase (see Figure 3,G2phase between

Fig. 7: Relationship between concentrations of PSF and CMF: the concentration level and diffusion of PSF, which are illustrated ina1, b1,c1andd1. The concentration level of CMF, which is exhibited in a2,b2,c2,d2in different time steps. The concentration of CMF de- pends on the concentration of PSF. Ina2andb2the concentration level of PSF is not high enough, though the cells have not yet release CMF (CM F concentration= 0). At the end of the simulation, the poten- tial locations of the future centers which are popped up at the highest concentration of CMF.

7h and 8h) are going to be autonomous centers with prob- abilityP3and with probabilityP4they will become a reg- ular cell. Same forSphase cells, with probability ofP5

they will become an autonomous center and with proba- bility ofP₆they will become a regular cell.

Algorithm 1: Aggregation

1: procedureENGINE OF MULTI-AGENT SYSTEM

2: Createncells in random positions . n: population size 3: forcell_id= 1tondo

4: Execute the pre-aggregation model for an individual cell 5: end for

6: Save cells locations and the concentration of CMF in a file 7: cAMP releasing queue=empty .a dynamic array 8: fort= 0totmaxdo . t: engine time 9: forcellid= 1tondo

10: Execute the aggregation model for an individual cell 11: end for

12: U pdatecAMP releasing queue 13: end for

14: end procedure

Centers competition: on the high concentration of cAMP and the high concentration of CF, a random cell would be an autonomous center if its age is fine and it is not inhibited by the other early centers. According to Section 3.2, early centers use adenosine to prevent the late centers formation around them (through morphogen- esis). Fig 8 illustrates a population of 5,000 cells with two early centers. Among this population, there are 500 I-cells, which 498 of them become activated, but will be shuted down by the early two centers. Interestingly the model will stop when all the I-cells have tried their chance to become initiator, and most of them failed.

Fig. 8: Effect of adenosine: the number of aggregation centers during a successful aggregation process with 5,000 amoebae (green points). a) formation of the second center (red point), which succeeds and becomes an initiator, b) appearance of the third center (blue point), which will be shuted down eventually, c) the third center is imposed to become a regular cell by the two early centers.

4.4. Regulation of Aggregation Territory Size and New (late) Centers Formation

As explained in the previous section, ‘center selection’

is a cell-cycle-dependent process beyond its dependency to the chemical factors. As demonstrated in Fig. 6, if the investigated cell was an autonomous center and it was ready to release cAMP, its location and the exact releas- ing time would be put in a cAMP releasing queue. In contrast, if it is not ready to fire, the model goes to the next cell, as shown inAlgorithm1.

Besides, if the cell was a regular cell, and it is not in a refractory period, it will inspect its status. If the last thing was not firing, and it is close enough to a sensitive cell, this cell’s position and its firing time will be added to the cAMP releasing queue.

Aggregation territory size regulation: eventually, once all the queue has been inspected, the cell will check the cAMP concentration. If the concentration is above the movement threshold it starts to move. But, in the mean- time if the concentration of CF is so high, the cell knows there are so much cell around the center and it should change its spatial gradient (by releasing PDE). Then, it takes a step and move. If the concentration was above the relay threshold, it goes to 1 in the figure and will behave like an autonomous cell and release a pulse of cAMP. Depending on the cAMP and CF levels, the cell will then decide to become a center; to move towards a center, release PDE to decrease the cAMP (regulatory ef- fect) or simply relay cAMP. This is a fully decentralized approach regardless of the number of population and cell density.

New (late) centers: at the beginning of the model, we assume that only 10% of population has the chance to be-

come an aggregation center based on the cells’ age. As explained in Section 4.2, before the start of movement, depending on the levels of PSF, CMF and adenosine, early centers were selected. Then after the movement started, when the concentration of cAMP is too high and the concentration of CF is sufficiently high enough also, a cell may decide to become an aggregation (late) center.

But, becoming a center depends also on the concentra- tion of adenosine, as well as on the CMF concentration too. Thus, each cell has to check the adenosine and CMF concentration level, before becoming a center. Figure 9 shows the case of new late centers appearance.

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Fig. 9: Self-appearance of new autonomous centers: the simulation starts with 5,000 regular cells (blue points) and two autonomous centers (red point 1, red point 2).Fig. a)There are still two autonomous centers, b)Third center (red point 3) has appeared and starts to release cAMP pulses,c)There are still three active centers andd)End of the simulation with three active centers.

4.5. Chemotaxis: cAMP Signaling and PDE action As explained in Section 3.4 and demonstrated in Fig.10, each cell measures the local concentration of cAMP and responds by moving 20 µm in 100 s to the spatial gradient, when the concentration of cAMP is above the threshold of10⁻⁹M. We use a diffusion-based model to simulate cAMP and other chemical factors. The concentration of cAMP decreases from the source, is de- graded by PDE and the diffusion away. Regarding hemi- spherical diffusion process, concentration level follows a mathematical description of the change in concentration of one or more substances. In this work, we adapt the standard first-order Michaelis-Menten enzyme kinetics to our model:

δq(r, t) = η

(4πDt)³²exp(−r²/4Dt)exp(−t/τ0) where q(r, t) is the distribution of the molecules as a function of time and distance r from the source, D is diffusion coefficients, andη is the number of molecules released by one individual cell andτ0=Km/Vmaxgives the measure of PDE activity. In order to calculate the spa-

tial gradient, equation below is used in the simulation:

5c~ = −2C 4Dt~r

Fig. 10: Diffusion of cAMP: 2 centers through 1,000 cells (red points): Each center releases cAMP with random frequency, with a ran- dom starting time. The green points exhibits the concentration of cAMP in two random position of the field.

5. RESULTS

5.1. Initial Conditions

Table 2 illustrates the initial values of parameters which were used in our model. Some of them are sug- gested in the Mackay’s paper [14].

Table 1: Simulation initial parameters [14]

Parameter Initial value

No. of molecules release by each cell 10⁷

Diffusion constant of cAMP 9.7×10⁻⁶cm²/s

τ0 1-10 s

Relay threshold 8×10⁻⁹M

Chemotaxis threshold 10⁻⁹M

Delay for relay 15s

Refractory period 10−3min

Refractory period for chemotaxis 100s

Chemotaxis step 20µmin100s

Random motion speed 5µm/min

No. of cells 1000,3000,4000,5000

Area 4e⁻³×4e⁻³

New center formation cAMP threshold 2×10⁻⁶

Adenosine threshold 10⁻⁵M

Released adenosine molecule 10³

CF threshold 10⁻⁵M

Released adenosine molecule 3×10³

CMF threshold 10⁻⁵M

Released adenosine molecule 3×10³

Max time simulation 3×60×60

Each time step equals to1min

P1 1/7×100

P2 1/14×100

P3 7.5

P4 50

5.2. Aggregation territory size

As explained through previous sections, in realD. dis- coideum development process, centers are autonomous sources of cAMP, which release pulses every420sfor a period of60s. Centers begin the aggregation by trigger- ing their neighbors to propagate the signal. In each aggre- gation territory, a center-inhibiting substance (adenosine) is diffusing outward from the first formed centers [15].

The centers do not appear at the same time but slowly over time with different pulse frequencies. Center for- mation was stated to be maximal at a cell density of200

myxamebas per square millimeter and that the final ra- tio of centers to cells under their test conditions was ap- proximately1 : 2100. Large cells (I-cells) are in fact the initiator cells postulated earlier upon the basis of popu- lation studies, whereas the small less active cells are the responder cells (R-cells).

In our model, we assume only10%of population has the chance to become an autonomous center. Each fired center starts to release a chemical substance to prevent the others to become a initiator. Cells can become a center if and only if, the concentration of cAMP and CF around it are above some critical thresholds.

As shown in Figure 11 and Table 2 despite varying density and population size, aggregation territory exhibit emergent homogeneous sizes.

Table 2: Result: the number of aggregation fields at the end of aggre- gation phase. With different cell population and different cell densities.

Population Initial I-cells number Inter-amoeba aver- age distance

Number of firing cen- ters in 5 experiments

2000 200 1.044 2 1 2 1 1

2200 220 1.047 1 2 1 2 1

2400 240 1.044 1 2 1 1 2

2600 260 1.038 1 2 1 1 1

3000 300 1.039 2 2 1 2 1

3500 350 1.039 2 2 1 1 2

3800 380 1.038 1 2 3 1 2

4000 400 1.04 3 2 3 3 2

4200 420 1.036 2 3 2 3 1

4500 450 1.035 2 3 4 3 2

4700 470 1.035 2 1 4 2 3

5000 500 1.034 3 2 2 3 2

5500 550 1.032 2 3 2 1 2

6300 630 1.03 4 2 3 2 2

6600 660 1.027 2 3 4 2 2

6. CONCLUSION

The model and simulation outlined in this paper pro- vide a practical approach to investigate the aggregation phase of D. discoideum development. One of the in- tended uses of this work is to help biologists and medical researchers to perceive slim mold’s more complex social behaviors. Nowadays, the researchers prefer to investi- gate relatively simple organisms such as,Dictyostelium discoideumto understand indirectly a wide range of cel- lular behaviors arising in more complex species including humans disease genes and the crosstalk between host and pathogen [18–21, 23, 25, 26].

Simulations showed interesting multicellular behav- iors, such as stream formation, homogeneous size terri- tories, late centers or centers inhibition. The obtained results show a simple and effective model of individual cells, exhibiting emergent property resulting from the in- teraction among individual cells. Future work will con- sider models for the remaining phases of D. discoideum behavior, and translation of this model into artificial sys- tems, such as swarms or micro-robots.

a.1 a.2 a.3 a.4 a.5

b.1 b.2 b.3 b.4 b.5

c.1 c.2 c.3 c.4 c.5

Fig. 11:Same aggregation territory size: three different simulations in different cells numbers: Part-a) Population size: 3,000 and final centers number:

1. Part-b) Population size: 4,000 and final centers number: 2. Part-c) Population size: 7,500 and final centers number: 3.

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