A spherical cell has volume and contains only impermeant solutes

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Biophysics & Medical Physics · Molecular machines, transport and bioenergetics

Life at Thermal Energy, Ch The membrane: barrier, capacitor and pump

Problem

A spherical cell has volume

V_0 = 400\,\mathrm{fL}

and contains only impermeant solutes, at an internal osmolarity of

320\,\mathrm{mOsm}

. It is isotonic with its original bath. The cell is now transferred into a very large bath (so its concentration does not change) of impermeant solute at

200\,\mathrm{mOsm}

, at

310\,\mathrm{K}

. Assume the cell behaves as a perfect osmometer: water crosses until the osmotic pressure inside equals that outside, and the amount of impermeant solute inside stays fixed. ((a)) Using

\Pi = ck_BT

for the solute on each side, predict the cell's new equilibrium volume

V_1

. ((b)) At its new (swollen) equilibrium radius, the cell's membrane has a permeability to glucose of

P_{\rm s} = 2.0\times10^{-6}\,\mathrm{m\,s^{-1}}

, and the cell's metabolism holds a steady glucose concentration difference of

\Delta c = 5.0\,\mathrm{mM}

across the membrane (higher outside). Using

J = P_{\rm s}\Delta c

, find the total rate at which glucose molecules enter the cell (in molecules per second). ((c)) Glucose diffuses inside the cytoplasm with

D = 6.0\times10^{-10}\,\mathrm{m^2\,s^{-1}}

. Using the one-dimensional diffusion-time relation from Chapter 2, estimate the time for glucose entering at the membrane to spread a distance equal to the new cell radius, and say whether this is fast or slow compared with the seconds-to-minutes timescale of cell metabolism.

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