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Day 1: Multi-level modelling in morphogenesis

Today was the first day of a two week course on multi-level modelling in morphogenesis.

During the introductory lecture, given by Dr Veronica Grieneisen, the goals of the course were outlined.

A goal of the course is to spread a broader understanding of developmental biology and biological modelling. More specifically to:

  • Define and understand generic principles guiding developmental biology
  • Learn how to identify and unravel processes
  • Understand and discuss at what level one should “model” a phenomena
  • Get exposed to different biological paradigm systems, as well as modelling formalisms
  • How to obtain models with predictive value and explanatory power that create isomorphisms

In this context isomorphisms are corresponding abstractions and conceptual models that can be applied to different phenomenon.

At another level a goal of the course is to discuss practical aspects of modelling. As scientists we need techniques to be able to express ourselves. As such the course aims to open the black boxes that biologists often make use of.

At yet another level the course is about communication. In particular enabling communication with a common language. The participants of the course are intentionally a mix of experimental and computational biologists from many different fields of biology. Bridging the gap between experimental and computational biologists is a central theme of the course. As is learning how to express oneself to people from different fields.

The section describing the goals was followed by a brief introduction to partial differential equations starting from the conservation equation (the differentiation form of the continuity equation), leading into a discussion about flux, Fick’s first law and it’s relation to diffusion.

The overview of partial differential equations was followed by a discussion on cellular automata, “to have an object or not to have an object”. Three examples were described:

  1. Majority voting rule
  2. Conway’s game of life
  3. Margolous diffusion/alternation

The participants were then immersed in a hands on workshop exploring majority voting and Conway’s game of life. Followed by an exploration of diffusion simulated by partial differential equations and Margolous alternation. The latter was accomplished by an exercise exploring diffusion-limited aggregation. The purpose of these exercises was to make biologist more familiar with thinking algorithmically and for everyone to think critically about the impact of the choice of modelling technique. What can be seen as a feature in one instance can be an artifact in another. It all depends on the phenomena that one is trying to model.

Then it was time for lunch and socialising.

After lunch Dr Stan Maree introduced three seemingly different phenomenon: cellular slime mold chemotaxis, Belousov-Zhabotinsky reaction (chemistry) and action potentials in neurophysiology.

The Hodgkin-Huxley model was described in all its complexity. Followed by a statement that it can be described as “unpleasantly complex” and a quote from FitzHugh that “the usefulness of an equation to an experimental physiologist (…) depends on his understanding of how it works”. The FitzHugh-Nagumo model was then briefly introduced. However, the details of it and the implications of the model were not described as it was to be explored during the afternoons practical session.

Instead the focus shifted to how one can gain an understanding of systems of linear ordinary differential equations. Time plots were contrasted with phase plane plots. And the importance of visualising nullclines as lines of zero change for a particular parameter was highlighted. In particular the fact that one can identify all equilibria from the intersections of nullclines in a phase plane plot.

Stability of equilibria was then discussed and simple rules for quickly analysing the stability of equilibria were derived from the fact that:

  1. For an equilibria to be stable its eigenvalues need to be negative
  2. Summing two eigenvalues results in the trace
  3. Multiplying two eigenvalues results in the determinant

So by plotting the trace vs the determinant we can get a plot illustrating different types of equilibria, see also.

trace-determinant plot

The Jacobian matrix was then introduced and the idea that the Jacobian can be approximated by plotting the nullclines on the phase plane plot and making small perturbations around the equilibria.

The participant where then invited to explore the temporal dynamics of the FitzHugh-Nagumo model using the software grind. This was followed by exercises looking at the spatio-temporal dynamics of the same model using partial differential equations. This led on to looking at spirals formed when introducing introducing a temporary barrier and it was highlighted that these spirals could never have been identified if one did not take the spatial regime into account. Finally the link to the other phenomenon outlined at the beginning of the afternoon session, slime mold chemotaxis and Belousov-Zhabotinsky reaction was pointed out and the isomorphic nature of these phenomenon was highlighted.