Question

A red blood cell has a disk diameter of 0.6 micrometers and an intracellular isotonic osmolarity of 290mOsmol/l at a temperature of 293 kelvin. The hydraulic permeability of the red blood cell is 2.4 micrometer atm^-1min^-1 with a b inactive fraction intercept of 0.2. Calculate the normalized cell volume and plot it against osmolarity variations between 50mOsm to 850mOsm. Plot another graph that represents cell volume variations against time when the extracellular isotonic osmolarity varies from 290mOsmol/l to 990mOsm/l.

Answer 1

For an ideal solution, where inter-particle interaction doesn't exist, the solution can be modeled similar to an ideal gas. Hence we have a van't Hoff equation which states

, where P is the osmotic pressure, V is the volume of the solution, n is the moles of solute molecules present, R is the gas constant and T is the temperature of the solution in Kelvin.

The equation can be rearranged to obtain

now

where, is the osmolarity of the solution.

Hence, we have

Now, since

Dividing both sides by density of water

Assuming that the volume of solute is negligible, V can be considered as the volume of the solution. Thus, the equation on right is the molality of the solute, m. Hence, we have

, where is the Osmolality of the solution.

Now, coming back to erythrocytes, for the purpose of our question, we can consider the erythrocyte membrane to be strictly permeable to water. Then the number of solutes after dipped in new solution () is the same as the number of solutes in isotonic solution ().

Therefore, we have

Hence,

where, m is the molality of the solute, rho is the density and V is the volume of solution which is water.

Now, the Volume of the solution is composed of osmotically active and osmotically inactive volumes.

Thus, , b is the osmotically inactive fraction volume.

Cancelling rho's and subsituting this back in the previous equation gives:

Rearranging gives

But we know that molality is equal to Osmololality. Therefore, we have the van't Hoff Boyle equation which can be stated as

In the given problem, the RBC can be modeled as a sphere, with a diameter of 0.6 micrometers. Now, the volume of the sphere can be given by

And the isotonic osmolality is given as 290mOsmol/L, and the osmotically inactive fraction b = 0.2.

Substituting these values in the van't Hoff Boyle equation gives

Substituting the values of Osmolality in the range 50-850mOsmol/L, we can plot a graph as follows:

Now, coming to the second part of the problem

Hence, , where P is in pascal. Now, 1Pa = 9.869*10^-6 atm. Substituting the values at isotonic condition, we have

Now, for 1atm, the radius changes by 2.4um in 1min. Thus, for P atm, the radius change in 1 min will be

This can now be plotted. The plot is left for you as an exercise.