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G. Fath Institute of Theoretical Physics University of Lausanne Switzerland

Propagation Failure of Travelling Waves. G. Fath Institute of Theoretical Physics University of Lausanne Switzerland. 1. Motivation - a biological example:. Retina differentiation in Drosophila. Orchestrated series of events including. }. Cell divisions

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G. Fath Institute of Theoretical Physics University of Lausanne Switzerland

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  1. Propagation Failure of Travelling Waves G. Fath Institute of Theoretical Physics University of Lausanne Switzerland

  2. 1 Motivation - a biological example: Retina differentiation in Drosophila Orchestrated series of events including } • Cell divisions • Cell differentiation • Pattern formation • Cell death in 2 days Eye imaginal disc regular pattern of ~800 ommatidia ommatidium = 8 photoreceptors + 12 other cells different genes are expressed around and behind the wave front: e.g. hh = hedgehog morphogenetic furrow /a propagating wave/ initiates differentiation temperature shift propagation halts back to normal temp. propagation continues Experiment:

  3. 1b Drosophila melanogaster head - anterior

  4. u a 0 1 2 Travelling waves We consider the following scalar reaction-diffusionequation in 1D: /continuous model/ diffusion nonlinear reaction /self-reaction/ The reaction function is bistable: or Nagumo-function Uniform steady states: Are there anykink-typesteady states? Only exists if kink-M Else: All reasonable initial conditions develop into a travelling wave. a x M

  5. 3 Def:travelling wave(TW)a self similar solution wave speed 1 wavefront profile (shape) a z 0 u c x After a short transient period the travelling wave keeps its shape as it propagates. • Interesting questions: • Existence of steady states and TWs • U(z) as a function of the speed c • Stability of U(z) • c as a function of D and f = f(u) • Speed selection • Effect of discreteness

  6. u a 0 1 3b It is easy to solve the Continuous model when f(u) is piece-wise linear: with equal slope wavefront profile inverse diffusion length wave speed

  7. u a 0 1 4 Continuous model summary Wavefront profile: with Speed vs a : TW solutions always exist 5 c d D=1 e e p a s 0 0 . 0 0 . 5 1 . 0 Propagation failure (pinning) only occurs if a = 1/2. - 5 Phase diagram:

  8. u a 0 1 5 Discrete model Note that biological cells are discrete objects. f(u)is piece-wise linear & bistable as in the case of the continuous model, but space is discretized: This introduces a new lengthscale D cannot be scaled to 1 as in the continuous case They exclude each other, thus there are domains in the (a,D) parameter space where only KSSs exist and other regions where TWs propagate.

  9. 5 D=fixed c speed pinning a 0 0 0 . 5 1 . 0 - 5 6 Propagation failure /pinning/ Due to discreteness, not to impurities! Speed vs a: There is an extended region where TW propagation is impeded. This is called the pinning region. Phase diagram: The pinning region lies between two curves: pinning limits for a given D

  10. u a 0 1 7 • Goals: • Understand the pinning • Find the pinning limits • Find c=c(D,a) • Find • Find the universlity class Why is the kink pinned? Easy to understand in the small D limit: Concentration values are close to either one or the other stable fixed points 0 or 1. D=0 D>0 but small 1 no points in this region when a is here KSS exists 0 M M+1 position n position of the kink When D is small the value of a can be varied freely in a large interval without effecting the solution.

  11. 8 Wavefront profiles - numerical simulation Wave front profiles near the pinning transition for D = 1. The corresponding exact critical point is a_=.27639320…. The curves, shifted for better visibility, belong to different a parameters and TW velocities: (a) a = a_, c = 0, (b) a = 0.276393, c = 0.0780, (c) a = 0.276, c = 0.1782, (d) a = 0.27, c = 0.3246. • Note: • Exactly on the transition line has jump discontinuities • Close to the transition point the front propagates in a 2-state manner: “step ahead” stage (narrow in z fast) “wait” stage (wide in z slow)

  12. u a 0 1 1 a 0 position n M M+1 9 ‘Real’ and ‘virtual’ steady states How to calculate a kink-M steady state ? M is arbitrary /translational invariance/ It is useful to introduce a slightly modified problem (the “virtual” problem) whose solution is easier. ‘real’ problem: ‘virtual’ problem: 1 0 position n M M+1 virtual steady state real steady state We solve the virtual problem! Consistency condition: Consistency must be checked to see whether we obtained a valid solution of the original (“real”) problem.

  13. The consistency condition holds if 10 pinning travelling waves The virtual steady state always exists. It is given by 1 If a is too small the consistency condition fails ! a 0 position n M M+1 We find The pinning limits /within which steady state solutions exist/ are

  14. modified Bessel function of integer order 10b Stability of the /virtual/ steady state stable Fundamental solution:

  15. 1 1 a a 0 0 n n M M+1 M M+1 1 1 a a 0 0 n n M M+1 M+2 M M+1 1 1 a a 0 0 n n M M+1 M+2 M+3 M M+1 M+2 Dynamics in the travelling wave region 11 consistency fails VSS jumps ahead consistency fails VSS jumps ahead A schematic illustration how the TW proceeds in the model when the actual value of a is outside the pinning limits, ie. a < a_. Whenever the concentration value on the right hand side of the kink reaches the value of a the consistemcy condition breaks down and the virtual steady state (the virtual attractor) jumps a lattice site ahead. The wave speed is c = 1/T .

  16. Schematic illustration of the role of virtual steady states in the dynamics. The continuous curve represents the time evolution of the system in the infinite dimensional vector space . When a virtual attractor (black dot) is approached as close as the red circle, the consistency condition breaks down and a new stable-looking attractor emerges one lattice site ahead. 12 Dynamics = Evolution towards actual virtual steady state 0 ~T 2T 3T t kink position: M = 1 M = 2 M = 3 4 Whenever the consistency condition breaks down the virtual steady state jumps ahead virtual attractors: Inside the red circles the consistency condition does not hold. stable (virtual) attractors

  17. rapid slow 13 Wavefront profile for c << 1 modified Bessel function of integer order We find that the n = 0 correction starts with a finite derivative

  18. rapid slow 1 a 0 n M M+1 14 Speed scaling as In the large t limit we find The jump of the virtual attractor (virtual steady state) occurs at time T when distance from the critical point The wave speed scales logarithmically around the critical point

  19. u u a a 0 1 1 0 u a 0 1 15 Summary • Lattice discreteness may cause propagation failure • Near the pinning transition the front propagates in two steps: • “waiting” and “jump-ahead” periods • When c > 0, singularities in the wavefront profile round • off except at one point where a cusp remains • The wave speed scales logarithmically close to the pinning • transition • A simple model is enough to understand the failure of • propagation during the retina formation in the Drosophila Is the scaling near the transition universal? Same universality class for the cases: logarithmic e.g. Nagumo power law

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