# PhaseVariationInBacterialColonyGrowth

### Contributors

Brunel University, UK: Nigel Saunders

Otto-von-Guericke Universität Magdeburg, Germany: Jan Wegener

### Background

Gene expression is controlled by three types of processes:

- programmed regulation (involving transcriptional regulators),
- epigenetics (usually reinforcing phenotypes), and
- stochastic processes (generating alternate behaviours).

A common (microbial) stochastic mechanism is **phase variation**, in
which gene expression is controlled by a reversible genetic mutation,
re-arrangement, or modification.

**Phase variation** has traditionally been considered in the context of
'contingency genes' in which a sub-population is continuously generated
which is pre-adapted to repeated environmental transitions, often to immune
selective changes. However recent re-consideration, in the light of
stochastic processes in genes under other forms of regulation, suggests an
important potential role in

- bacterial specialization and differentiation, and
- the generation of structured bacterial populations.

The diversification of mutation-mediated phase variation is context
independent, thus the process can be observed and studied in in-vitro
culture conditions and occurs within bacterial colonies. This is the
simplest culture condition and can be used to **assess basic properties** of the
process, such as

- rate of variation (mutation rate),
- contributions of differing fitness,

It also helps to establish the underlying framework for future more complex modelling and determination of local population sub-structures.

### Modelling challenge

We are looking for

- a generic spacial model for bacterial colonial phase variation, which
- allows to substantially extend the method developed in (Saunders 2003) to computationally predict rates of phase variation.

Previously, phase variation has been described by deterministic models, which describe synchronous growth in cell colony without reflecting how a colony develops in space. Our Petri nets adopt an asynchronous modelling approach so that cells divide individually, and explicitly consider spreading in space.

### Modelling

Step-wise modelling:

- Encode the basic model of (Saunders 2003) as stochastic Petri net,

or as extended stochastic Petri net. The read arcs (given in read) reflect the interpretation that one parent gets one offspring.

*Tip: Click on image opens animation window. Animation does not consider stochastic rates.* - Consider the displacement of cells in space, with no cell division, where the surface of the growth medium is modelled by a rectangular grid. Coloured (stochastic, continuous) Petri nets permit scalable models.
- Combine the basic and displacement models, where the parent remains in-situ, and the offspring may displace by one grid position or stay in its current location according to local cell density.
- Control thickness and speed of spread of the colony.

Model parameters:

- grid size
- mutation rates
*alpha*(forward mutation rate),*beta*(reversion mutation rate) - fitness
*da*(probability of A to survive until next reproduction),*db*(likewise for B) - preference of an offspring to stay with parent
- total number of cells (colony size), e.g.
- 25 generations: 33.5 E+06
- 26 generations: 67 E+06

Model assumptions:

- 3D colony is represented by a 2D grid with a finite capacity on each grid position.
- Equal height of the cell colony (all grid positions have the same capacity).
- If phase variation occurs, the progeny consists of one A and one B (Saunders 2003).
- It is always the mutant, who goes to a neighouring position, if any.
- The simulation starts with one bacterium of phenotype A.

### Analysis

All computational experiments are done on the automatically unfolded Petri nets. With other words, the coloured stochastic Petri nets serve as high-level description of the stochastic systems under study.

For example, unfolding our coloured Petri net for a 101x101 grid yields a plain Petri net with

- 30,604 places,
- 362,404 transitions,
- 1,087,212 standard arcs, and
- 362,404 read arcs

The size of the model suggests the use of the Gibson & Bruck algorithm for stochastic simulation, which turns out to be substantially faster than the Gillespie algorithm in the given setting.

We define for each grid position auxiliary variables:

- total number of cells, i.e., A+B
- proportion of A = A/(A+B)
- proportion of B = B/(A+B)

Converting the stochastic simulation traces into 2D representations (heat map style) allows us to observe the development of a cell colony in time and space.

The model permits to predict mutation rates and fitness by counting and measuring mutation sectors.

### Conclusions

Our model is generic and scalable; it contains potential for a couple of model extensions, including:

- fine tuning biofilm thickness
- multiple gene on/off and their dependencies
- log pedigree and/or mobility

Publication in preparation.

### References

- NJ Saunders, ER Moxon, MB Gravenor:

Mutation rates: estimating phase variation rates when fitness differences are present and their impact on population structure;

Microbiology 149(2003)2, pp. 485–495. [ link ]

*latest update: July 03, 2012, at 08:53 PM*