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The Structure, Function, and Evolution of Biological Systems

The Structure, Function, and Evolution of Biological Systems. Instructor: Van Savage Spring 2011 Quarter 5/10/2011. Global signature of diffusion. Random walk x(t+1)=x(t)±1 ->x 2 (t+1)=x 2 (t)+2x(t)+1(1/2 of time)

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The Structure, Function, and Evolution of Biological Systems

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  1. The Structure, Function, and Evolution of Biological Systems Instructor: Van Savage Spring 2011 Quarter 5/10/2011

  2. Global signature of diffusion Random walk x(t+1)=x(t)±1 ->x2(t+1)=x2(t)+2x(t)+1(1/2 of time) =x2(t)-2x(t)+1(1/2 of time) <x2(t+1)>=(1/2)* <(x2(t)+2x(t)+1)> +(1/2)*<(x2(t)-2x(t)+1)> =<x2(t)>+1=<x2(t-1)>+2 Iterating this gives: <x2(t+1)>=Number of time steps~t

  3. Combined Effects Person trying to walk north (directional) through a busy intersection (nondirectional) Net Flow=Directional Flow+Nondirectional Flow Diffusion Equation (Also known as Kolmogorov forward equation)

  4. Komarova et al.

  5. Two-cell system Crypts contain 1000-4000 cells and 4-6 stem cells

  6. Two-cell system Birth probabilities Death probabilities

  7. Transition probabilities Transition from i to j B cells have population size i.

  8. Master equation Working from last slide, this is wrong. r’s and u’s are reversed. N’s are missing. Correct version on board.

  9. Kolmogorov forward/Fokker-Planck Paper has this derivation and these identifications wrong.

  10. Under weak selection the Kolmogorov forward equation becomes In this limit, their result is correct

  11. Absorption times and probabilities Solve with BC and recursion to find

  12. Condition for fixation and then for tunneling

  13. Three-cell and three-step process Transition probabilities for B cells

  14. Stochastic tunneling is possible

  15. Conditions for tunneling under different selection regimes

  16. Conditions for tunneling under different selection regimes

  17. Three-cell system combined with chromosomal instability (CIN)

  18. Number of genes necessary for CIN to occur before mutations do (N=6 stem cells) But, are stem cells and other cells equally likely to mutate?

  19. Number of genes necessary for CIN for that to occur before mutations (N=100 crypt cells)

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