Question

Cells in a multicellular organism often signal to eachother by having specialized cells that emit signaling chemicals. Those signalling chemicals diffuse through the fluid medium in which the cells are embedded. Consider two cells that communicate by diffusing a chemical that has a diffusion constant D = 250 micro meters squared per second. The cells are a few micrometers in diameter. For simplicity, we will consider this a one-dimensional diffusion problem.

a. Suppose that two of the signalling cells are a distance of 1000 micrometers apart (left to right). About how long would it take for a signal to diffuse from the first cell on the right to the second cell on the left?

b. One way that organisms improve this signalling time is by having additional cells that relay the signal. That is, instead of having just two signalling cells 1 mm apart, there are an additional nine signalling cells (in between the two cells at the end) located 100 micrometers apart. As soon as one of the signalling cells detects the diffusing chemical from the first cell on the right, it immediately releases its own signalling chemical that begins to diffuse. The chemical released by the second cell is then detected by the third cell, and so on. Calculate how long it takes the chemical to diffuse from the first cell to the second cell.

c. Based on the calculations in part b, how long will it take the chemical signal to reach the last cell, 1 mm away from the first cell? How does this compare to the time it takes with just the two cells separated by a distance of 1 mm?

Answer 1

To calculate the time,t, taken for diffusion of a chemical molecule through a distance of R with a diffusion constant of D, assuming one-dimensional diffusion we use the formula:

t = R²/2*D

a. Here, R = 1000µm, D = 250µm² per sec, thus t = (1000)²/2*250 = 2000 seconds

b. To diffuse through each cell, the distance of diffusion is 100µm. So, R' = 100µm, D = 250µm² per sec,

thus t' = (100)²/2*250 = 20 seconds

c. Now to reach the tenth cell the diffusing molecule has to cross 9 cells. And to reach to from the left to right of the tenth cell another 100µm of diffusion distance has to be traversed.

Thus, the time taken by the chemical molecule to reach the left end of the tenth cell = 9*20 =180 seconds.

And the time taken by the chemical molecule to reach the right end of the tenth cell = 10*20 = 200 seconds.

If we compare the relaying between 10 cells and the direct diffusion between 2 cells, we can observe that the relaying event makes diffusion 10 times faster (2000/200) for the same distance traversed.