Poster
Reference
Agent-based models for first- and second-order emergent collective behaviors of social amoeba Dictyostelium discoideum aggregation
and migration phases
PARHIZKAR, Mohammad
PARHIZKAR, Mohammad. Agent-based models for first- and second-order emergent collective behaviors of social amoeba Dictyostelium discoideum aggregation and migration phases. In:
CUSO Winter School Lenk 2019 , Lenk, Switzerland, 2019
Available at:
http://archive-ouverte.unige.ch/unige:114230
Disclaimer: layout of this document may differ from the published version.
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time: 25 time: 29
time: 33 time: 35
stream breakup
c.
a. b.
d.
one stream
reattainment aggregation
a. b. c. d.
e. f. g. h.
i. j. k. l.
m. n. o. p.
• Focus on the aggregation and migration phases of D. discoideum.
• Two agent-based models, which exhibit series of individual, collective behaviors and emergent properties of social amoeba D. discoideum.
• The key character of our models 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.
• Understand, simulate and visualize the individual and intercellular activities of cells (growth, division, movement, secretion, differentiation, etc.)
• Modeling and simulating the emergence of slug behaviors
• Phototaxis
• Ammonia effect
• Merging of two small slugs
• The relationship between slug’s speed, age, length
• Appealing to inspire the engineering of swarm robotics
D. discoideum life cycle Red arc: first part, first-order collective behaviors Blue arc: second part, second-order collective behaviors Ref: https://ww.wikipedia.org
Mohammad Parhizkar
Centre Universitaire d’Informatique (CUI) University of Geneva, Switzerland Email: mohammad.parhizkar@unige.ch
Giovanna Di Marzo Serugendo Centre Universitaire d’Informatique (CUI) University of Geneva, Switzerland Email: giovanna.dimarzo@unige.ch
Aggregation Phase: Implementation
Cells are seeded in a 2D surface, initially populated uniformly at random within the domain.
The simulation is updated at discrete time intervals, using the 6 different chemical factors concentration rules.
Our results show a series of behavior close to individual and collective behavior of living D.
discoideum:
1. Dynamic center selection process and centers inhibition: at the beginning of the simulation, 10% of the population has the ability to become an autonomous center. According to the role of PSF, CMF and Adenosine only a few of them actually release autonomous pulses. At the end of the pre-aggregation phase we already observed approximate locations for potential centers. At later stages, based on CF and cAMP concentration, we observe regular cells turning into centers, thus splitting large aggregation territories into smaller ones.
2. Streams formation: without any consideration of other mechanisms, such as incorporating cell adhesion, cell death; we observed the movement of the cells in the stream manner towards the source of cAMP.
3. Homogeneous aggregation territories size: CF and cAMP combination helps the cells to locate the best aggregation field around them. Center formation was stated to be maximal at a cell density of 200 myxamebas per square millimeter and we observed at the end of simulations an homogeneous ratio of centers to cells of approximately 1 : 2100. In over-crowded populations the aggregation process runs fasters with more interesting results.
Publications
• M. Parhizkar and G. D. M. Serugendo, Social Amoeba Dictyostelium discoideum as an Inspiration for Swarm Robotics, IEEE 9th International Conference on Self- Adaptive and Self-Organizing Systems, 2015 , USA .
• M. Parhizkar and G. D. M. Serugendo, An Agent-Based Model for Collective Behaviors of Social Amoeba Dictyostelium discoideum Morphogenesis: Aggregation Phase, SWARM- The 2nd International Symposium on Swarm Behavior and Bio- Inspired Robotics, 2017, Japan.
• Mohammad Parhizkar and Giovanna Di Marzo Serugendo, Agent-based models for first- and second-order emergent collective behaviors of social amoeba Dictyostelium discoideum aggregation and migration phases, Artif Life Robotics, 2018, Volume 23, Issue 4, pp 498–507.
Diffusion of cAMP – two centers and 1,000 regular cells in two different time steps http://www.formedium.com/
Future Works
• Validation of all models and simulations with biologists
• Improving the first-order and second-order models
• Continue the work on stream breaking mechanism
• Continue the work on slugs speed
• Cell-cell adhesion: considering the cells more than one discrete point
• Translating the models and mechanisms to swarms of Kilobots
First-order Collective Behaviors-Model
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
Comparison of same aggregation territory size:
Part-a) Population size: 3,000, final centers number: 1, a4 shows the commencement of stream formations at time step 3,620.
Part-b) Population size: 4,000, final centers number: 2, b2 shows the activation of second center at time step 1,260.
Part-c) Population size: 7,500, final centers number: 3, c2-5 show the same aggregation size at the end f simulation at time step 10,800.
First-order Collective Behviors Simulation Results Times of continuously synthesizing
factors before/after starvation Effect of adenosine – The number the centers in 5,000 cells
Research Domain
• Implemented an agent-based model to simulate the formation of cell aggregation using 6 different chemical factors.
• Our model follows a completely decentralized and self-organizing process.
Simulations showed interesting multicellular behaviors, such as stream formation, homogeneous size territories, late centers or centers inhibition.
• This model considers D. discoideum amoeba populations composed of a single cell type without differentiation between pre-stalk and pre-spore. The formation of aggregates consisting of two or more cell types and sorting process during the aggregation is also a point of interest for future works.
Future work will consider models for the remaining phases of D. discoideum behaviour, and translation of this model into artificial systems, such as swarms or micro-robots.
• Understand, simulate aggregation and slug-formation phases of D. discoideum life cycle.
• Second and higher-order emergence of slug behaviors
• Design patterns for second-order emergence and engineering methods
• Implementation of the first and second order collective behaviors models with a swarm of small Kilobots.
First-order Collective Behaviors-Method
We extended Mackay’s model with an agent-based model with releasing and aggregating 6 chemical signals:
A. Dynamic center self-selection: first centers appear based on cells’ age, high concentration of CMF and low Adenosine. During streams formation, late centers appear when cells’ density is too high (CF high, cAMP high, and Adenosine low).
B. Three levels of quorum sensing: the 6 signals provide gradients for collective decentralized decisions:
1. Pre-aggregation: PSF and CMF concentrations trigger identification of starvation and aggregation;
2. Aggregation: CF, cAMP concentrations lead to late centers formation;
3. Density and territory size: cells measure density levels through CF; trigger more PDE regulating territory size.
C. Chemotaxis: cAMP concentration causes cells to move towards centers and form streams, following:
1. Cells follow Michaelis-Menten equation enzyme kinetics (for cAMP and PDE);
2. Three cAMP levels: late center formation, relay response, chemotaxis response;
3. cAMP and PDE secretion depends on cells density.
Agent-based models for first- and second-order emergent collective behaviors of social amoeba Dictyostelium discoideum aggregation and migration phases
Long-term objectives Short-term objectives
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 and d) End of the simulation with three active centers.
Local activator and global inhibitor Relationship between concentrations of PSF and CMF: the concentration level and diffusion of PSF, which are illustrated in a1, b1, c1 and d1. The concentration level of CMF, which is exhibited in a2, b2, c2, d2 in different time steps. The concentration of CMF de- pends on the concentration of PSF. In a2 and b2 the concentration level of PSF is not high enough, though the cells have not yet release CMF (CMF concentration = 0). At the end of the simulation, the potential locations of the future centers which are popped up at the highest concentration of CMF.
Bio-inspired engineering of artificial systems
• From biological behaviour to design patterns
• Swarm behavior: first-order emergence
• Organism behavior: second- higher-order behavior
• Investigate D. discoideum to define design patterns for higher-order emergence
• Application to artificial systems, e.g. swarm robotics
D. discoideum life cycle properties:
1. The simplicity of its life cycle always fascinates researchers to simulate the whole development process or some of its important stages, cyclic adenosine monophosphate (cAMP).
2. The life cycle duration is relatively short, lasting only one or two days.
3. Cells movement, intercellular chemical signaling and the developmental process, which are applicable to the other search of studies.
4. The exhibit collective system has the plasticity ability in its development process. In its life cycle, we can see horizontally and vertically self-organizing different developmental processes.
Different levels of cAMP thresholds:
• C > B > A
• New center: when the number of independent centers are not enough
Simulation initial parameters
D. discoideum cell 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, where d is the side length of the square and l ≈ 8µm
D. discoideum cell growth cycle Center- selection criteria
The whole doubling time is approximately 7.2h and most of it, is G2 phase. The aggregation begins for starved cells, which are in S phase of their growth cycle;
this occurs between 2 to 3h and also be- tween 6 to 7h after release. 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.
The aggregation pattern in the simulation results were obtained:
• Average amoebae concentration d=0.5
• Corresponds to a cell density of 5 x 105 cells/cm2
• Amoeba size: 10 x 20 µm2
• Stream formation: 0.4 < d < 1 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.
In the “Territory size + New centers” part, cell uses two levels of quorum sensing, one to realize the proper aggregation cluster (red circle-2) and other for formation of the new centers (red circle-3).
a. b.
1 2 3
4{ { { {0{ 43210
c.
Individual agent behaviour model - Second-order emergence - Migration phase
Blocks movement and major-voting in each block First- and Second-order emergence
- First-order Michaelis-Menten enzyme kinetics to our model: 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, and η is the number of molecules released by one individual cell and τ0 = Km/Vmax gives the measure of PDE activity.
Second-order Collective Behaviors: Model
pst cells psp cells
(a) (b) (c) (d) (e) (f)
Second-order Collective Behaviors Simulation Results
(a) (b)
(c) (d)
(a) (b)
(c) (d)
(a) (b) (c) (d)
(e) (f) (g) (h)
experiment 1 experiment 2
experiment 3 experiment 4
Slug’s Length, Speed
Aggregation Phase: Stream Breaking
a. b. c.
d. e. f.
early center stream breaks into groups
new aggregation center
a. b. c.
d. e. f.
early center stream breaks into groups
new aggregation
"centers competition" and “one- aggregation with several fruiting bodies.”
Stream breaking: The aggregation mass can move toward a stable center.
4 Mohammad Parhizkar and Giovanna Di Marzo Serugendo 3. All cells move in the same direction with same speed and have a relatively
same position.
4. Each cell moves actively with a constant motive force.
5. Each cell meets with an intrinsic resistance related to its speed 6. The cells in the tip of the slug and the sheath helps the cell to move forward.
7. There is no resistance from the sheath on the sides nor on the back of the slug
Also, in Innouye’s work [5] we can find an equation between the slug’s length, width and its velocity. They have extracted this formula by making a multiple regression analysis on the data of 27 di↵erent slugs.
1 v= 0.271
L+ 0.00751
!+ 0.26 (1)
Where,vdenotes the migrating velocity andL,!for length and width of the slug respectively. In this equation, the coefficient for the width of the slug is so small so that it can be regarded as zero. Thus, we can conclude the formula based on the length of the slug:
1 v= 0.271
L+ 0.26 (2)
Fig. 3 illustrates the flowchart of decision making for each self-organised cell.
In the beginning, each cell can be one of the three cell types: PST, PSP or Pacemaker. If it is a PSP cell, it will just follow the tip cAMP signal. However, if it is a PST cell, it will follow the whole flowchart. The PST cells notice when they become a slug (identification of emergent property), i.e. when they realise that they are surrounded by the same type of cells (they are surrounded by other PST cells). In this model, we have two quorum sensing phenomena which are indicated by red circles (one and two). They also become sensitive to the light.
Additionally, we use a di↵usion-based approach based on a chemical signal (DIF) that helps PST cells inside the anterior part to measure the population size The model consists of two second-order collective behaviours: (1) slugs merging (the blue part) and (2) phototaxis orientation and movement towards the light (the red part). Both abilities are made possible by measuring the concentration of chemical signals and light using two-dimensional di↵usion functions.
3.2 Model: Considering the Slug Age
In this model, velocity and length of the slug is a team work of both kind of the cells (PST and PSP). Nevertheless, the switch from crawling slug to a still fruiting body formation is not age-related.
4 Simulation In our simulation,
4 Mohammad Parhizkar and Giovanna Di Marzo Serugendo
3. All cells move in the same direction with same speed and have a relatively same position.
4. Each cell moves actively with a constant motive force.
5. Each cell meets with an intrinsic resistance related to its speed 6. The cells in the tip of the slug and the sheath helps the cell to move forward.
7. There is no resistance from the sheath on the sides nor on the back of the slug
Also, in Innouye’s work [5] we can find an equation between the slug’s length, width and its velocity. They have extracted this formula by making a multiple regression analysis on the data of 27 di↵erent slugs.
1
v= 0.27 1
L
+ 0.0075 1
!
+ 0.26 (1)
Where,
vdenotes the migrating velocity and
L,!for length and width of the slug respectively. In this equation, the coefficient for the width of the slug is so small so that it can be regarded as zero. Thus, we can conclude the formula based on the length of the slug:
1
v= 0.27 1
L
+ 0.26 (2)
Fig. 3 illustrates the flowchart of decision making for each self-organised cell.
In the beginning, each cell can be one of the three cell types: PST, PSP or Pacemaker. If it is a PSP cell, it will just follow the tip cAMP signal. However, if it is a PST cell, it will follow the whole flowchart. The PST cells notice when they become a slug (identification of emergent property), i.e. when they realise that they are surrounded by the same type of cells (they are surrounded by other PST cells). In this model, we have two quorum sensing phenomena which are indicated by red circles (one and two). They also become sensitive to the light.
Additionally, we use a di↵usion-based approach based on a chemical signal (DIF) that helps PST cells inside the anterior part to measure the population size The model consists of two second-order collective behaviours: (1) slugs merging (the blue part) and (2) phototaxis orientation and movement towards the light (the red part). Both abilities are made possible by measuring the concentration of chemical signals and light using two-dimensional di↵usion functions.
3.2 Model: Considering the Slug Age
In this model, velocity and length of the slug is a team work of both kind of the cells (PST and PSP). Nevertheless, the switch from crawling slug to a still fruiting body formation is not age-related.
4 Simulation In our simulation,
Linear Relationship between migrating slug velocity and its length:
Inouye (1979)
Simulation of 8 different slugs : phototaxis, ammonia affect, merging
Slug’s behaviors: phototaxis Slug’s behaviour: merging Stream breaking: The formation of a new stable center.
Project website:
https://www.unige.ch/cui/cas/research/dicty Dictyostelium discoideum Life cycle
Slug’s block movement: leader- follower-16 possible movements
Acknowledgement
This project is supported by the Swiss National ScienceFoundation (SNSF) Grant number 205321 179023.
Migration Phase: Implementation
Slug’s structure: If the anterior part (pre-stalk zone) of the slug is cut off, the remaining slug reorganizes and rebuilds the anterior part again.
Moving towards light sources: Phototactic behaviour
• In our desired model, individual amoebae are not able to measure the direction from which the light comes, and differences in light intensity do not lead to differentiation in motion velocity.
• Nevertheless, the whole slug orientates itself towards the light. The tip of the slug, formed by pre-stalk (pst) cells leads the slug towards the light.
- Here, we identify and simulate the characteristics of the migratory stage of D.
discoideum multicellular development such as slug’s length, speed, etc.
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⇥106cm2/s
⌧0 1-10 s
Relay threshold 8⇥109M
Chemotaxis threshold 109M
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 4e3⇥4e3
New center formation cAMP threshold 2⇥106
Adenosine threshold 105M
Released adenosine molecule 103
CF threshold 105M
Released adenosine molecule 3⇥103
CMF threshold 105M
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- coideumdevelopment 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 ofD. discoideumdevelopment. 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.
Emergent properties of the second-order collective behaviors:
1. Slugs collective movement 2. Phototaxic 3. Merging 4. Uniformly distributed
Emergent properties of the first-order collective behaviors:
1. Signaling and Chemotaxis 2. Inhibition of centers 3. Appearance of late centers 4. Homogeneous aggregation territories size
