Myxobacteria are ubiquitous gram-negative social bacteria

which move by gliding.

They form fruiting bodies in response to starvation.

Introducing Myxobacteria

Stochastic LGCA Models for Myxobacteria Fruiting Body Formation

Maria KiskowskiDepartment of MathematicsUniversity of Notre Dame

Research Support:The Center of Applied Mathematics, University of Notre Dame

The Interdisciplinary Center for the Study of Biocomplexity, University of Notre Dame

DOE under contract W-7405-ENG-36

Presentation Outline

Stages of Myxobacteria Fruiting Body

Formation C-Signaling LGCA Model for Rippling LGCA Model for Aggregation

Fruiting Body Formation in Myxobacteria

Under starvation conditions, Myxobacteria aggregate into huge fruiting bodies.

Within the fruiting bodies, the cells differentiate into myxospores that can survive for years in tough conditions.

Stages of Myxobacteria Fruiting Body Formation

I. AlignmentII. StreamingIII. RipplingIV. AggregationV. Fruiting Body Formation

I. Alignment

Myxobacteria cells are very elongated,with a 1x10 aspect ratio. When settling from suspension, cells immediately align within patched arrays.

Behmlander and Dworkin Hans Reichenbach

II. Streaming

Myxobacteria cells move cooperatively by streaming. They move by extending and retracting

pili (social motility) and by secreting slime (adventurous motility).

Dale Kaiser Hans Reichenbach

III. Rippling

During rippling, myxobacteria form evenly spaced, high density ridges which appear to advance as traveling waves.

Rippling has been modeled by our group and others (Mogilner et al., Deutsch et al., Leutscher et al.)

Shimkets and Kaiser

Alber et al.

IV. Aggregation

During rippling, myxobacteria form evenly spaced, high density ridges which appear to advance as traveling waves.

Rippling has been modeled by our group and others (Mogilner et al., Deutsch et al., Leutscher et al.)

Kuner and Kaiser

V. Fruiting Body Formation

Aggregation centers form mature fruiting bodies which are complex, beautiful and species-specific.

Within mature fruiting bodies, spores develop which are heat resistant and can survive

for years without food or water.

Hans Reichenbach

C-SignalingC-SignalingC-signaling is required for all stages of fruiting body formationand higher levels of C-signal trigger more advanced stages.

Myxobacteria C-signal only when they transfer C-factor via their cell poles. It appears that rippling is controlled by interactionsbetween head poles (collisions) while aggregation is controlled by interactions between a head and a tail (trail-following).

C-signal levels increase throughout development.

The C-signal transduction pathway contains two signal amplification loops.

More Detail:Rippling in Myxobacteria

The exact mechanism for rippling is unknown and enigmatic. Ripples waves are unusual because

1. they propagate with no net transport of cells, and,

2. there is no interference, either constructive or destructive, where waves overlap.

Facts about Rippling

Isolated cells oscillate with a mean period of about 5-8 minutes.

Cells in a rippling swarm oscillate more quickly and travel about one wavelength between reversals.

Cell motion is approximately 1-D. The majority of cells move in lines parallel to one another, with or against the axis of wave propagation.

Hypothesis of Precise Reflection

Sager, B. and Kaiser, D. [1994] Intercellular C-signaling and the traveling waves of Myxococcus Xanthus

Head-on collisions between cells cause cell direction reversals. Thus, when two wave fronts collide, the cells reflect one another pair by pair in a precise way that preserves the wave structure in mirror image.

LGCA Model For Rippling

Modeling cells:Cells are modeled as 3xl rectangles on a square lattice. Each cell occupies a single velocity channel with its center of mass. Two C-signaling pole regions are defined.

Modeling cell movement:All cells are parallel, and are right-directed or left directed. Cells move one unit per timestep.

Collisions occur when the head pole regions of two cells collide.

C-signal collisions speed up an internal biochemical clock which regulates reversals. More collisions shorten the period of time between reversals. After reversing, cells are insensitive to further collisions for a period of time called the refractory period.

Mogilner et al.

Possible Components of a Cell Clock:

Refractory period: period for which cells are insensitive to C-signal immediately after turning

Minimum response time: minimum period of time required for a sensitive cell to be stimulated to turn

Maximum oscillation period: maximum period of time before a cell will turn again

(Minimum oscillation period=refractory period+minimum response time.)

Modeling question: Which of these components are necessary to quantitatively reproduce experimental rippling?

Ripple Model Results

1. Our model based on the hypothesis of precise reflections reproduces rippling for a wide range of parameters. For example, ripples result with and without an assumed maximum oscillation period.

2. The refractory period is an essential feature of the model.

3. The minimum reponse time must be small compared to the refractory period.

4. We deduce from (2) and (3) and experimental data that the cell refractory period is 2-3 minutes.

5. Our model predicts there is no maximum oscillation period.

Solid line: no max osc period.

Dotted line: with max osc period.

6. Wavelength decreases with density. (dotted line)

7. This density-dependant decrease in wavelength completely accounts for the observes increase in wavelength when C-signaling cells are diluted with non-C-signaling cells.

Shown is ripple wavelength verses wild-type density with no csgA-minus cells (dotted line) and when the density of csgA-minus cells is increased so that the total cell density remains 1.6 (solid line).

When two ripples meet, it appears that they inter-penetrate.

In fact, each ripple reverses by precise reflection. Depending on when the cells reverse and their cell length, there may be a small phase shift between the two waves.

8. "Soliton" Interaction

9. Rippling Evenly Disperses Cells Much Faster Than Random Diffusion

Initial conditions: A central stripe of cell density 15 is added to a 5x500 lattice of cell density 3. Randomcell positions.

Rippling Random Walk

Aggregation in Myxobacteria

1. Aggregates form from a large area of swarming cells. These swarming cells align from a random distribution and form long chains that stream into aggregation centers.

2. The aggregates are at first flat, then round up into a mound.

3. In both flat and rounded aggregates, cells tend to be arranged tangent to a hollow inner region and move in a circular or vortex-like fashion around the hollow center.

4. Little is known about how cells organize during aggregation, but C-signaling plays a pivotal role.

LGCA Model Of Aggregation

Model assumption: Cells align by C-signaling.

In particular, we use a Monte Carlo process so cells turn preferentially in directions which maximize overlap of C-signaling nodes (head pole nodes with tail pole nodes).

Initial conditions:Cells are randomly distributedon an nxn hexagonal lattice sothat each node is covered byapproximately 10 layers of cells.

Local rules:At each time-step, there is a 1. C-signal alignment step, and2. a transport step.

Cell Movement

We model cell movement on a triangular lattice. Each cell occupies only one velocity channel with its center of mass.

Cells move exactly one node per timestep, so there are six allowed velocity channels.

By the exclusion principle, only one cell center may occupy each velocity channel.

Modeling Cells

Each cell is specified by:

1. A cell center located at the node (i,j),2. An assigned direction, and3. Two local interaction neighborhoods: a head pole area and a tail pole area.

TAIL HEAD

For a realistic cell shape, we model 3x21 cells on a triangular lattice. Each cell occupies only one velocity channel with its center of mass.

1. Alignment 2. Small Aggregates 3. Streams 4. Stable Aggregates

25 TS 200 TS 900 TS 25,000 TS

Cell density shown (cells per unit area).

Aggregation results: Stages of pattern formation on a 500x500 lattice.

"Round Aggregate" Formation

"Stream Aggregate" Formation

200-400 Timesteps

Once formed, roundaggregates are stableunless they are perturbed by interactionswith other aggregates.

Streams are transient.Interactions with otheraggregates cause them to extend and shrinkdynamically. Withoutinteractions, streams eventually condense into a round aggregate.

Novel mechanism: Two-stage aggregate formation via streams

1. Round aggregates appear in random positions and cells join aggregates by random walk.

2. Transient streams form from round aggregates which are within interaction distance.

3. Streams span large distances and cells reorganize by moving within streams.

4. Streams condense into round aggregates and there are fewer, larger round aggregates remaining.

Round Aggregate Shapeas Cell Number Increases

Cell center geometries of stationary aggregates listed in order of increasing number of aggregate cells .

Area density phase diagram for 186 aggregates found in 2 simulations over 25,000 time-steps defines an "attractor" region.

Effect of Noise: Aggregate Response to Adiabatic perturbation

Cells are added slowly to an initially small aggregate over time.

Area density phase diagram as cells are added over 1000 timesteps.

Aggregate shape as cells are added.

Effect of Noise: Aggregate Response to Non-Adiabatic perturbation

An aggregate (star) is constructed by the fusion of two adjacent aggregates. Upon the initial interaction; aggregate area doubles.

Area density phase diagram over 600 timesteps.

Comparisons With Biology

Myxobacteria fruiting bodies are hollow, with cells

tightly packed alongthe periphery.

Myxobacteria cells move inconcentric

circlesor spirals.

Cell Density of aTypical Simulation

Aggregate(Darker gray corresponds

to higher cell density.)

The path of a tracked cell within

a simulation aggregate.

SeveralMyxobacteria fruiting bodies may fuse to formsporangioles.

Aggregates at a distance have been observed to grow while

anothershrinks.

Cell centers of another typical

aggregate.

Two InteractingSimulation Aggregates

Lattice Boltzman Model

The LB simulation evolves from the LGCA. Instead of considering individual cells, the state variable is the probability of a

particle at a node and represents the average LGCA behavior. Instead of using a Monte

Carlo process for alignment, we use a deterministic collision function.

Lattice Boltzmann Equation

For the evolution of f, the distribution function,given each velocity channel i and a collision function Omega.

f i rri ,t1 f i r,t i n

Cell density for lattice boltzmann simulation after100, 500, 2200 and 3400 timesteps.

Without the internal noise in cell decision making of the LGCA,the LB model results in streams which are smaller. Final aggregate sizes are smaller and more homogeneous.