# membrane permeability

BIOE2397 Homework #6

(Homework #6 is due on Wednesday, 12/5, 11:59pm)

Problem 1 [5 points]: If resting membrane potential of a cardiac cell is -90mV, the concentration of 𝐾𝐾+ outside the cell is 5mM and membrane permeability for all cations (except for 𝐾𝐾+) are zero. What is the concentration of 𝐾𝐾+ inside of the cell?

Problem 2 [10 points]: A cell’s resting membrane potential is -50mV, and the cell’s diameter is about 10μm. The capacitance of the cell membrane is about 0.5 μF/cm2, and the internal concentration of 𝐾𝐾+ is 150mM. What is the maximum surface density of 𝐾𝐾+ in the cell? What is the surface density of 𝐾𝐾+ needed to produce the resting membrane potential?

Problem 3 [25 points]: The relative importance of advection versus diffusion is described by a non-dimensional parameter called the Peclet number, 𝑃𝑃𝑃𝑃 = 𝑓𝑓(𝑢𝑢,𝐷𝐷, 𝑥𝑥), where 𝑢𝑢 is the velocity, 𝐷𝐷 is the diffusion coefficient, and 𝑥𝑥 is the distance to the point of interest. Using dimensional analysis, find the form of 𝑃𝑃𝑃𝑃 such that 𝑃𝑃𝑃𝑃 ≪ 1 is advection dominated, and 𝑃𝑃𝑃𝑃 ≫ 1 is diffusion dominated. For stream with 𝑢𝑢 = 0.3 m/s and 𝐷𝐷 = 0.05 m2/s, where are diffusion and advection equally important?

Problem 4 [40 points]:

A student injects 5 mL of 20% Rhodamine solution in water (with specific gravity 1.15 g/mL) instantaneously and uniformly over the tube cross-section (𝐴𝐴 = 0.8 cm3) at the point 𝑥𝑥 = 0 and the time 𝑡𝑡 = 0. The tube is filled with stagnant water. Assume the molecular diffusion coefficient if 𝐷𝐷 = 0.13·10-4 cm2/s.

• What is the concentration at 𝑥𝑥 = 0 at the time 𝑡𝑡 = 0?

• What is the standard deviation of the concentration distribution 𝑡𝑡 = 1 second after injection?

• Plot the maximum concentration in the tube, 𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚(𝑡𝑡), as a function of time over the interval 𝑡𝑡 = [0, 24 h].

• How long does it take until the concentration over the region 𝑥𝑥 = [-1, +1] meters can be treated as uniform? (A uniform distribution is one where the minimum concentration within a region is no less than 95% of the maximum concentration within that same region.)

BIOE2397 Homework #6

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