Introduction to Systems Biology - Tutorial 2

Program of the lecture "Introduction to Systems Biology" (WS0506)

This tutorial for the module "Space" uses the tools

• CA: Cellular Automaton simulations with local microscopic rules for moving interacting particles, e.g. molecules or cells, that may lead to macroscopic patterns with extensions independent from the size of the particles.
• PDEsim: Simulation of coupled nonlinear partial differential equations with the online platform PDEsim which should be used in the explore mode and the public domain software XPP-AUT known from tutorial 1.

Explore the online tools by testing different models, options and parameter values. In particular focus on the initial steps of the simulations which mostly start from a homogeneous state plus some noise. For the initial dynamics close to the homogeneous state, often an analytical understanding (see the next lecture on Turing patterns) is available and predicts typical wavelength of the intermediate patterns. Does this wavelength change compared to later stages? After having some feeling for the simulations consider the following more specific exercises. The cellular automaton and XPP-AUT run locally but PDEsim runs remotely on a single computer, hence you should continue with the local ones if PDEsim runs slowly because others are using it at the same time.

• Brownian motion with CA
  Start with a small number of particles (radius 3 under parameters) and observe the spreading in the channel mode (under visualization). This process underlies the diffusion term that is used in partial differential equations.
• Turing patterns with CA
  Run the Turing model with lattice size 100x100, density plot. How would you characterize the pattern, which properties would you measure and record?
  Under parameters change the diffusion property (while the simulation keeps running) of the inhibitor (initially mI=11 is much faster than the activator mA=1) with some steps from 1 to 20. How do your selected measures change?
  Similar models are used to reproduce periodic pigment patterns in tissues.
• Excitable medium with CA
  While Turing patterns are stationary, excitable media allow for moving patterns, here a front between the unstable empty state and a state of spatio-temporal chaos. The muscle tissue of the heart behaves like an excitable medium for electrical excitation. The wave front triggers muscle contraction and then the muscle returns to its rest state. However, also spatio-temporal chaos may occur and lead to heart failure.

The simulations of PDEs with 1 space dimension are displayed as space-time plots and are controlled via two sets of parameters:

Simulator parameters

• amin for plot of variable 1 = smallest value to resolve by color scale (values below are set to red)
• amax for plot of variable 1 = largest value (above set to yellow)
• pmin for plot of variable 2 = smallest value
• pmax for plot of variable 2 = largest value
• dt = step size in time
• dx = step size in space (choose it larger than dt for correct integrations, else you see only red)
• per = selects initial condition
  0: homogeneous
  1: periodic
• bc = selects boundary conditions
  0: zero flux (no exchange across boundary)
  1: periodic (left and right boundary glued together)
  2: Dirichlet (fixed value of the variables on left boundary, zero flux right)
  r = fixed value to be used if bc=2
• l = system length in space (number of steps)
  total system size is dx * l
• step_per_line = number of internal time steps before plotting next line
• lines = number of plotted time steps
  total time is product dt * step_per_line * lines
• noise = amount of noise to be added to initial condition
  seedval = start of random number generator, to reproduce the same noise again

Custom parameters
which are defined in the model equations. There are diffusion constants df, ds and kinetic parameters. The cross-diffusion b is internally set to zero if not mentioned.

• Gradient formation
  Monique's model was used to describe Fgf8 morphogen gradients in zebrafish that emerge from a fixed source of Fgf8 production at the left boundary (so-called organizer).
  Change the strength of the source and the diffusion constants to get shorter or longer gradients.
• Turing patterns with PDEsim
  The Gierer-Meinhardt model was suggested in 1972 to produce gradients in a homogeneous system, e.g. to establish an organiser in a short system compared to a characteristic length scale.
  Start in explore mode from the preset -gradientformation- to see the original idea working. A gradient (yellow=high to red=low in plot) of activator (left) and inhibitor (right) concentration is seen in the plot.
  Now double the system length using parameter l (or the preset -doublegradient-). Does the gradient idea still work?
  Try periodic bc. instead of zero flux. Where do the maxima form?
  Go to a larger system with preset -randominitcond- and periodic bc. What selects the spacing between the stripes?
  Run the same with zero noise.
  Run the same with fixed bc without noise (or preset -standard-).
  Why and how do the stripes form during the initial transient?
  Interestingly the same model makes usual Turing patterns if the system size is long and only a single gradient if the size is small. Hence it possesses a preferred length scale which can be calculated analytically from the parameter values which we kept constant in all presets.
• Wave patterns with PDEsim
  Similar to the excitable medium in CA, here the Ginzburg-Landau model is implemented which shows travelling waves and can also be driven into spatio-temporal chaos. The important parameters are b and c.
  

Run bifurcation diagram for MAPK
If you did not get the bifurcation diagram of the MAPK model in tutorial 1 but have meanwhile seen the examples in Diana's lecture on bifurcation analysis, then it may be worth looking at the task from tutorial 1 again.

Simulation of PDEs with XPP-AUT
XPP-AUT is not only good for bifurcation analysis but can also simulate 1d PDEs like PDEsim. Browse the XPP-AUT website and tutorial to see how to get the space-time plot for example models that are provided on the XPP-AUT website.