sb3-hw14-2016 - JHU 580.429 SB3 HW14 Diffusion and spatial...

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JHU 580.429 SB3 HW14: Diffusion and spatial patterning. 1. Diffusion. (a) The diffusion constant of water in water is D = 10 - 5 cm 2 / sec. We have interpreted D as ( 1 / 2 ) Δ x 2 / Δ t , where Δ x is diffusive step and Δ t is the typical time between steps. For water, a typical step will be the diameter of a water molecule, about 0.3 nm. What is the corresponding Δ t in picoseconds? (b) Suppose a hormone has to diffuse through a cell to activate a gene in the nucleus. A typical diffusion constant for small peptides or drugs is D = 10 - 6 cm 2 / sec. If the diffusion distance is 5 μ m, approximately how much time does the hormone require to diffuse to the nucleus? (c) Suppose a morphogen has a diffusion constant of D = 10 - 6 cm 2 / sec and has a lifetime of 1 hr. The distance over which patterning can occur is about equal to its root-mean- square diffusion length over its lifetime. What is this distance? At what developmental stage does a human embryo have this diameter, and how many cells does it have at this point? 2. Morphogen gradient patterning, transient solution. Suppose a morphogen diffuses in one dimension according to the partial differential equation ˙ ρ ( x , t ) = D ( d / dx ) 2 ρ ( x , t ) - αρ ( x , t ) . At time 0, the density is concentrated at the origin, ρ ( x , t = 0 ) = n 0 δ ( x ) . (a) Convert the PDE for ρ ( x , t ) into an ODE for ˆ ρ ( k , t ) . (b) Evaluate ˆ ρ ( k , t = 0 ) in terms of model parameters. (c) Solve the ODE to obtain ˆ ρ ( k , t ) in terms of model parameters. (d) Use an inverse Fourier transform to obtain ρ ( x , t ) . (e) For no decay, α = 0, find the time t ? ( x ) when ρ ( x , t ) has its maximum value. (f) Whenever you have a diffusion problem, R 2 = 2 Dt is a good guess for relating time, distance, and diffusion constant. For the previous problem, a reasonable guess would therefore be t ? ( x ) = x 2 / 2 D . How does this guess compare to to the analytical answer? (g) Evaluate the maximum morphogen density at position x , equal to ρ ( x , t ? ( x )) . (h) For threshold K , find the patterning length x ? defined as ρ ( x ? , t ? ( x ? )) = K . 3. Morphogen gradient patterning, method of images for an absorbing barrier. Suppose a mor- phogen diffuses in one dimension according to the partial differential equation ˙ ρ ( x , t ) = D ( d / ) 2 ρ ( x , t ) . At time 0, the density is concentrated at the origin, ρ ( x , t = 0 ) = n 0 δ ( x ) . The cell membrane is at x = L . Suppose that morphogens are absorbed and degraded at the cell membrane so that Version: 2014/12/02 09:51:56 1 of 4
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JHU 580.429 SB3 HW14: Diffusion and spatial patterning.
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  • Fall '15
  • SBE3, Diffusion And Spatial, Patterning, Diffusion Patterning, Spatial Patterning, SB3, Partial differential equation, x,t, JHU

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