HW_4_2011_v2

# P bound lk d 1 lk d first use the steady state

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p bound = ( L/K d ) / (1+( L/K d ))). First, use the steady-state diffusion equation for a spherically symmetric source to estimate the concentration at 0.6 mm from the pipette. The idea is to solve the 3D diffusion equation in spherical coordinates, given that the concentration at the source (i.e. the pipette) is 1 mM and that the concentration in the far field is zero. (NOTE: we work this out in chap. 13 of PBoC in a different context, but the ideas are all the same.) Do you agree with them about the concentration being 8 μ M at a distance of 0.6 mm? Next, examine the statement about the consequences of a 10- μ m 3

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run and also about the fractional change in occupancy. Do you agree with their numbers? Do you agree with the qualitative thrust of their statements? In his commentary on the paper of Sourjik and Berg, Dennis Bray says: “The mystery can be expressed in a different way. Estimates of the binding affinity of aspartate to the membrane receptor of wild-type E. coli typically give a dissociation constant in the range 15 μ M. A bacterium responding to a change in occupancy of 0.1% is therefore sensing concentrations of aspartate of a few nanomolar. And yet we know from decades of observations that the same bacterium is also capable of responding to gradients of aspartate that extend up to 1 mM. Somehow, E. coli is able to sense aspartate over a range of at least 5 orders of magnitude in concentration by using just one molecular species of receptor!” Your job is to actually do the estimate/calculation that supports the claim made by Bray. In particular, examine a 0.1% change in occupancy and see what that means about the change in concentration given that the K d has the value claimed. Also, if there is a change of concentration of order a few nanomolar, how many fewer molecules are there in a box of size 1 μ m 3 due to such a concentration difference at the front and back of a cell? 4
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