Work out the concentration of fluorescent proteins in the cell as a function of
ID: 168061 • Letter: W
Question
Work out the concentration of fluorescent proteins in the cell as a function of position and time in analogy with the one-dimensional treatment of the problem done in the chapter. Compute the number of molecules in the hole after photobleaching as a function of time. (a) Consider a sphere of radius R in water. Due to random and collisions with the water molecules, the sphere will rotationally diffuse. The diffusion law in this case is analogous to the one obtained for translational motion, (delta theta^2) = 2 D_ t. What are the units of the rotational diffusion coefficient D_ ? Write down the formula for D_ using the Einstein relation and the rotational friction coefficient obtained in Problem 12.5, and convince yourself that the units are correct. dimensions, this leads to a simple and beautiful "flux distribution" function. In this problem, consider two adjacent planes, one of which has N_1 molecules and the other of which has N_2 molecules, and derive and flux distribution function. (b) Estimate how long it takes for an E. coli to diffuse over an angle equal to 1 radian. What is the distance traveled by the bacterium during that time?Explanation / Answer
13.6
(a) The units of the rotational diffusion coefficient Dr is radians2/s.
The formula for Dr using Einstein relation and the rotational friction coefficient obtained in problem is Dr kBT/fkBT/(8)(r)3
Where Dr = rotational diffusion coefficient, r = radius of the sphere, kB= Boltzmann's constant and T = temperature (Kelvin), friction=f(8)(r)3 , = viscosity of the medium.
(b) E. coli to diffuse over an angle equal to 1 radian
As per the equation of rotational diffusion:
t 2/ 2Dr
= 1 /2 (KbT/ 8R3)
= 4R3/ kBT
Assuming that the E. coli as a sphere
And radius R 1µm
The time = t:
t = 4R 3 /KBT 4 × 103Pas1018m3 /4 × 1021Nm 3s.
Assuming E. coli travels 20µm/s it would have travelled its body length by that time i.e. 60µm.
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