# DictyosteliumColonyFormation

### Contributors

Brandenburg University of Technology Cottbus, Germany:
Fei Liu

Otto-von-Guericke Universität Magdeburg, Germany: Mary Ann Blätke, Wolfgang Marwan

Surrey University, UK: Andrzej Kierzek

Université Laval, Canada:
Simon Hardy

### Background

Dictyostelium discoideum is a species of soil-living amoeba belonging to the phylum Mycetozoa. Commonly referred to as slime mold, it has uni-cellular and multicellular stages in its life cycle. It is a eukaryote that transitions from a collection of unicellular amoebae into a multicellular slug and then into a fruiting body.

D. discoideum has a unique asexual lifecycle that consists of four stages:

- vegetative,
- aggregation,
- migration, and
- culmination.

The cells involved in the life cycle undergo movement, chemical signaling, and development, which are applicable to human cancer research. The simplicity of its life cycle makes D. discoideum a valuable model organism to study genetic, cellular, and biochemical processes in other organisms.

Our literature search found several approaches to modelling aspects of the life-cycle behaviour in D. discoideum. We decided to focus on pattern formation, and have taken as our motivational paper (Kessler 1993) which was refined in(Levine 1996).

The overall scenario is described in (Kessler 1993):*The cellular slime mold Dictyostelium discoideum exhibits a variety of spatial patterns as it aggregates to form a multicellular slug. These patterns arise via the interaction of the aggregating amoebae, either via contact or as mediated by chemical signals involving cyclic adenosine monophosphate (cAMP).*

The cells aggregate together in a centralising manner, and the concentration of the cAMP forms spiral patterns:

Problem definition summary:

- A population of cells of a single cell protozoan eukaryote.
- The cells can be distributed in space. The accepted scenario is in 2 dimensions (i.e. in a Petri dish). However we can also consider 1 dimension (linear colony) or 3 dimensions.
- The presence of a chemoattractant, cAMP, which can be secreted by the cells.
- cAMP can diffuse in space and create a gradient.
- Cells can move in the direction of the cAMP gradient with no overlap of cells permitted.
- The detection of cAMP by cells initiates an intracellular signalling cascade to produce more cAMP.

The BioSystem behaviour is as follows:

**Initial state:**

- Starved cells, quiescent.
- Randomly sparsely distributed.
- At t0, there is some distribution of the concentration of cAMP.

**Actions**

- Consider both cells & cAMP
- 1 (or more) pacemaker cell starts to secrete cAMP
- A quiescent cell will start to emit cAMP when it senses [cAMP]> threshold (Gaussian distributed threshold)
- Any cell will try to move towards a higher concentration of cAMP

**Output:**

- Pattern of cAMP concentration over the surface (Petri dish)
- Distribution of cells over the surface

### Modelling challenge

We needed to define a neighbourhood function for 1-dimensional, 2-dimensional and 3-dimensional situations. This is essentially a constraint over the grid positions which can be accessed from any one cell. We used the constraint logic programming system clpr to help to develop and validate our definitions:

% clpr code: % ';' is or '|' % ',' is and '&' neighbour1D(X, A, D1):- (A=X; A = X+1; A = X-1), not(A=X), A <= D1, A >= 1. neighbour2D((X,Y), (A,B), (D1,D2)):- (A=X; A = X+1; A = X-1), (B=Y; B = Y+1; B = Y-1), not((A=X,B=Y)), A <= D1, B <= D2, A >= 1, B >= 1.

### Modelling

Step-wise modelling:

We first of all created a plain (uncoloured) 1-dimensional model with 3 layers.

- The layer given in green models cAMP diffusion, synthesis and degradation.

- The layer given in blue models the cell behaviour, which includes the movement to neighbouring grid positions.

- The layer given in red models the mutual exclusion (mutex) on grid positions to prevent multiple occupancy by moving cells:

We then made coloured models: note that the 1-dimensional coloured model (left) is essentially the same as the 2-dimensional coloured model (right), except for the neighbourhood function.

The coloured models represent generic patterns.

- They can easily be scaled to different grid sizes by only changing the constants.
- The extension to 3D grids is straightforward.

Finally, we constructed a more sophisticated description of the behaviour of a cell based on the 3-state model described in (Kessler 1993).

**Remark:**

- neighbourhood function, scaling by sqrt(2).
- future
- increase grid resolution
- 2D -> 3D

### Analysis

**Simulation results**

Gradient of cAMP diffusion over a 1-dimensional grid (Timeline plot).

Cell movement over 1-dimensional grid with cAMP concentration: 2 cells (Timeline plot)

Clumping behaviour of cells in a linear 1-dimensional grid (no mutex condition)

Waves of total cAMP concentration with population of static cells (3-state internal model), oscillating over time, 20x20 grid, indicating time-dependent patterning of cAMP:

*Media:3state_cells_oscilation_1.mpg (movie)*

cAMP diffusion over a 20x20 2-dimensional grid*Media:2D_Gradient_1.mpg (movie)*

Cells and cAMP concentration on a 20x20 2-dimensional grid, 80 cells (20% density)*Media:80cells_10ks.mpg (movie)*

### Conclusions

Outlook

- We will model more defined intra-cellular machinery, based on the cAMP cascade.
- We are actively looking at modelling biological systems with explicit Cell-cell communication
- We need to design methods to analyse and logically reason about the physiological readout behaviour (pattern formation etc.).
- We would like to carry out biological validation - computational prediction of mutant behaviour.

### References

- D Kessler, H Levine:

Pattern formation in Dictyostelium via the dynamics of cooperative biological entities;

Phys. Rev. E 48, 4801–4804 1993. [ link ] - H Levine, I Aranson, L Tsimring, TV Truong:

Positive genetic feedback governs cAMP spiral wave formation in Dictyostelium;

Proc. Natl. Acad. Sci. USA, Vol. 93, pp. 6382-6386, June 1996. [ link ]

*latest update: December 07, 2011, at 09:33 PM*