The Structure, Function, and Evolution of Biological Systems

Instructor: Van SavageSpring 2011 Quarter

5/10/2011

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

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⇒ | x |= x 2 = t

Combined EffectsPerson trying to walk north (directional) through

a busy intersection (nondirectional)

Net Flow=Directional Flow+Nondirectional Flow

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∂f∂t= −∂(vf )

∂x+ ∂

2(Df )∂x 2

Diffusion Equation(Also known as Kolmogorov forward equation)

Komarova et al.

Two-cell system

Crypts contain 1000-4000 cells and 4-6 stem cells

Two-cell system

Birth probabilities

Death probabilities

Transition probabilities

Transition from i to jB cells have population size i.

Master equation

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

Kolmogorov forward/Fokker-Planck

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M = (1− r)η (1−η ) − u(1−η )1+η (r −1)

V = (1− u)η (1−η )1+η (r −1)

Paper has this derivation and these identifications wrong.

Under weak selection the Kolmogorov forward equation becomes

In this limit, their result is correct

Absorption times and probabilities

Solve with BC and recursion to find

Condition for fixation and then for tunneling

Three-cell and three-step process

Transition probabilities for B cells

Stochastic tunneling is possible

Conditions for tunneling under different selection regimes

Conditions for tunneling under different selection regimes

Three-cell system combined with chromosomal instability (CIN)

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?

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