Question

How long would it take 100 molecules of valinomycin to transport enough K⁺ to change the concentration inside an erythrocyte of volume 100 µm³ by 10 mM? (Assume that the valinomycin does not also transport any K⁺ out of the cell, which it really does, and that the valinomycin molecules outside the cell are always saturated withK⁺.)

Short answer

The time taken by 100 molecules of Valinomycin to transport enough K⁺ ions to change the concentration inside an erythrocyte is 10 mins.

Step-by-step solution

Valinomycin

Valinomycin is a well-studied ionophore that binds specifically to a K⁺ion. It's a cyclic ionophore with ester linkage, peptide bonds, and D and L-amino acids. The channel-forming ionophore valinomycin generates transmembrane channels or pores through which chosen ions can flow.

Calculating the number of ions transported

The volume of erythrocyte is$100{\mathrm{\mu m}}^3$

The change in concentration inside the cell is$10\mathrm{mM}$

The number of ions is$\left(6.023\times {10}^{23}{\mathrm{ionsmol}}^{-1}\right)\left(100{\mathrm{\mu m}}^3\right)\left(10\mathrm{mM}\right)$

On calculating,

$$\begin{gathered} =\left(0.010{\mathrm{molL}}^{-1}\right)\left({10}^{-13}\mathrm{L}\right)\left(6.02\times {10}^{23}{\mathrm{ionsmol}}^{-1}\right) \\ =6.02\times {10}^8\mathrm{ions} \end{gathered}$$

Calculating the time required

There are 100 ionophores and therefore each transport 6.02x10⁶ions.

The time needed for the transport is,

$$\begin{gathered} \mathrm{Time}=\left(6.02\times {10}^6\mathrm{ions}\right)\left(\frac{1\mathrm{s}}{{10}^4\mathrm{ions}}\right) \\ \mathrm{Time}=602\sec \\ \mathrm{Time}=10\min \end{gathered}$$

Hence, 100 molecules of Valinomycin take 10 minutes to transfer enough ions to alter the concentration inside one erythrocyte.