Day 3: Multi-level modelling in morphogenesis

The third day of the multi-level modelling in morphogenesis course started off with a lecture by Dr Veronica Grieneisen, giving an introduction to the cellular Potts model.

The talk started off with a statement that biophysics, or simply physics, constrains and drives tissue development. And that (embryonic) tissues share properties with fluids.

So why do clusters of cells form similar structures to froths and bubbles? Basically some of the principles are the same: the tendency to minimise area and conserve (topological) constraints give rise to a frustration which generates characteristics configurations.

In 1964 Steinberg formulated the differential adhesion hypothesis in which he made the comparison between cells and immiscible fluids. He used the idea that cell types present different adhesive and cohesive interactions to postulate that the final configurations are established by obtaining a minimal interface free energy through successive changes in cell contacts.

This means that differential inter-cellular adhesion is one of the most important factors in cell sorting.

It did however take some time before this could be modelled. People tried doing it using cellular automata. However, this turned out to be (close to) impossible.

Some time later people started experimenting with the cellular Potts model. A significant difference between the cellular Potts model and cellular automata is the representation of a biological cell. In cellular automata a cell is represented by a single pixel, whereas in the cellular Potts model a biological cell is represented by lots and lots of pixels. The latter is therefore able to represent cell shape.

The cellular Potts model is simply a Hamiltonian consisting of terms for adhesion, volume conservation and cortical tension. The latter being a more recent addition.

The cellular Potts model is driven by Monte Carlo sampling where the edges of the cells are allowed to change state into neighboring cells. The energy of each pixel is then evaluated and if the energy is lowered the change is accepted. If a change increases the energy the change can still be accepted. The likelihood of accepting an energetically unfavourable change is evaluated using a probability function that makes it more unlikely as the energy difference increases. The reason for accepting some energetically unfavoured changes is so that the system can be driven towards the global minimum over time.

As it turns out the cellular Potts model can capture both the stochastic nature of cell dynamics and cell sorting behaviour.

The formalisms of the cellular Potts model were then described in some detail before the talk was concluded by stating that the cellular Potts model is an energy-based model that describes surface mechanics (adhesion, membrane fluctuations, internal pressures, cortical tension). As a result one can use it to talk about macroscopic phenomena such as cell shape, cell sorting, tissue surface tension, cell movement, stresses and shape changes, as well as stresses and strains through a tissue.

It is also possible to extend the cellular Potts to model specific biological problems such as chemotaxis and cell differentiation. Furthermore it can be used in conjunction with other models to look at sub-celluar details such as gene regulatory networks. It therefore serves as a great tool for cell-based modelling of morphogenesis.

The lecture was followed by a practical session where the participants got hands on experience of using the cellular Potts model for studying cell sorting.

After lunch Dr Stan Maree took over where the morning’s lecture had ended by illustrating how the cellular Potts model can be used to model cellular movement and morphogenesis.

The first example illustrated how the cellular Potts model could be extended to model movements of cells in lymph nodes. The movement of T-cells in the lymph node can be described as random persistent motion. The reason for this type of movement is that T-cells want to move past as many dendritic cells as possible (and vice versa) in a short a time as possible.

The model was created from:

• T-cells set to be persistently moving
• Dendritic cells (including extensions)
• Reticular network (undeformable)
• Correct sizes, densities, and shapes of cells
• Fitting speed and motility

Where the persistent T-cell movement was created from:

• Continuously adjusting the target direction
• Continuously adjusting the directional persistence
• Adjustment according to the reticular network

The model created managed to reproduce: short term persistent motion, long term random motion, and the experimentally observed “stop-and-go” behaviour. The latter had not been incorporated into the model and was previously thought to occur from a syncronised clock in the T-cells. These and other simulations then promted further and longer time-lapse experiments from the experimental biologists that disproved the internal clock hypothesis.

The second example illustrated how the cellular Potts model could be combined with chemotaxis, cell differentiation and gene regulatory networks to model complex developmental changes during gastrulation.

Dr Maree’s lecture was followed by a keynote talk by Professor Shigeru Kondo.

Professor Kondo gave a fascinating talk describing his quest to find evidence of reaction-diffusion system animals.

He did this by turning the traditional work flow of a molecular biologist on its head. Usually a molecular biologist would:

1. Find mutants
2. Identify all the genes involved in the phenomenon
3. Identify the functions of all those genes
4. Clarify the whole interaction network
5. Do calculation to make sure the identified system can reproduce the interested phenomenon

However professor Kondo decided that if he was to find evidence for the reaction-diffusion system he would need to use the theory before doing the experiments. As such he set out to:

1. Do many computer simulations
2. Extract important characteristics
3. Predict something unexpected
4. Show that it can happen!

He then illustrated how his group had applied this methodology of modelling first and experimenting second to find extraordinary evidence of the reaction-diffusion system in fish.

For example, one prediction made from the Turing reaction-diffusion system was that fish stripes should be able to migrate, specifically stripes that bifurcate. At the time Professor Kondo asked leading experts of fish developmental biology if this had every been observed. However, they replied that it had not. Undeterred Dr Kondo started looking for evidence of this behaviour and was able to find it in the skin of maring anglefish, see Kondo and Asai; Nature (1995).

After several other striking examples of prediction followed by experimental evidence Professor Kondo concluded by stating that Turing systems had proved an effective tool for understanding patterning. However, he made the point clear that the underlying mechanism is probably encoded in cell motility rules rather than necessarily in an activator and inhibitor.