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← 08x-06 Activator Inhibitor

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Showing Revision 1 created 10/24/2012 by Amara Bot.

  1. And now we have our final exam question or rather our final, final exam question.
  2. This one is about something called morphogenesis.
  3. This relates to how an organism knows where to grow its legs
  4. or where to form colored stripes on its coat and things like that.
  5. This really concerns pattern formation then.
  6. And Alan Turing of all people--what a brilliant man--had an idea that an initially only
  7. slightly random distribution of chemicals can evolve into a stable pattern over time.
  8. So we're going to look at one model of how that could happen.
  9. Let's say that we have two different chemicals.
  10. We have one that we're going to call the activator and one that we're going to call the inhibitor.
  11. Now the activator causes production of both itself and of the inhibitor
  12. and also diffuses slowly through the space that we're considering.
  13. The inhibitor on the other hand causes a reduction of both the activator and itself,
  14. and it diffuses very quickly through the space.
  15. So we can look at our graph over here. The x axis is just the space in general.
  16. And the y axis is concentration of either chemical.
  17. Now, if we consider what's happening in the curves inside this blue circle,
  18. we can see that where there's a little bit more activator than inhibitor
  19. the activator level rises due to diffusion.
  20. The inhibitor that is also produced then flows quickly away because this is fast diffusion,
  21. and it stops other peaks from developing nearby.
  22. We have equations for how the concentration of the activator and inhibitor change with time.
  23. We're going to call the concentration of the activator a and the concentration of the inhibitor b.
  24. This constant right here shows us that yes in fact the activator is very slow moving
  25. as opposed to this constant over here which shows that the inhibitor acts on a much shorter time scale.
  26. Back over in the equation for da/dt this factor of 1-a²
  27. limits the growth of a when a turns to 1 or turn -1.
  28. This is basically similar to logistic growth.
  29. Back in the equation for db/dt, you see that when a is greater than 0 since a term of just +a
  30. is added in, this forces both a and b upward.
  31. Since we have -0.7b, however, whenever b is greater than 0
  32. this forces a and b both downward.
  33. Looking at the code, the final task of the problem is actually going to be to use
  34. the explicit finite difference scheme for this activator-inhibitor model that I just showed you--
  35. the equations for da/dt and db/dt.
  36. However, before you get to that step, there are two other tasks that you must complete.
  37. Both task one and task two concern using periodic boundary conditions.
  38. Just to help you visualize this, I'm going to draw a little diagram.
  39. We're considering a grid of some length by some length,
  40. and it's made up of all these little squares just like we've seen so many times before.
  41. We know how the conditions in this square relate to the conditions
  42. in the adjacent square and so on and so forth.
  43. We can progress horizontally across the square until we reach the boundary.
  44. Now, however, we have a problem.
  45. What am I going to do with all the information that I have about this square?
  46. In the same way, where does the information about this square come from?
  47. Periodic boundary conditions say if I pretend to create one more square
  48. outside of the boundaries of the grid then this square is going to be identical
  49. to the first square on the opposite side where we started.
  50. We can do the same thing moving in the opposite direction horizontally
  51. and also in both directions vertically.
  52. So for task one and task two of this final problem, you're going to figure out
  53. what expressions are needed to ensure that these boundary conditions are put in place.
  54. In the end, you should come up with some pretty interesting looking patterns.
  55. I'm just going to give you a sneak peak of what you should end up with.
  56. So here's what you'll get with your final plot.
  57. You have one graph showing the concentration of the activator throughout the entire space
  58. we're considering and the other graph showing the concentration of the inhibitor.
  59. Congratulations on some awesome work in this course.
  60. I hope you've had fun seeing how many different kinds of problems could be solved
  61. using numerical approximation methods.
  62. I know that this course has taught me to look at the world in a totally different way.
  63. Great job on the final exam.