Question

Most eukaryotic cells are about 10µm in diameter, but a few cells in your body are about meter long. These are the neurons running from you spinal cord to your feet. They have a normal-sized cell body, with various bits sticking out, notably the “axon.” Neurotransmitters are small molecules synthesized in the cell body, but needed at the tip of the axon. One way to get them to their destination is just to let them di?use there. Model the axon as a tube 1m long and 1µm in diameter. At one end of the axon, the concentration of a small molecule is maintained at one millimolar (that is, (10^-3 mole)/(10?3 m3)). Some process removes all the molecules arriving at the other end.

a. Estimate how many molecules per second arrive at the end.

b. Real neurons package neurotransmitter molecules in packets containing about 10000 molecules. To send a signal to the muscle, a motor neuron must release about 300 of these packets. Using the model outlined above, estimate how often the neuron could send a signal if difusion were the only means of transport.

Answer 1

Here It is given that model axon as tube of 1 m long and 1µm Diameter and we have to calculate diffusion rate or rate of molecules per second at the end.

1)To solve this we have to use two equations

1. Fick's law- Diffusion rate (Let say it R)= -Dc/L

Where, D= Diffusion constant

c= Concentration

L= Length

Now, we don't know the value of D for this we have to use next equation

Stokes-Einstein equation- D= (k*T)/(6*pi*n*r)

Where, D= Diffusion rate

k= Boltzmann Constant

T= Tempreture (Assume= 310K)

n= dynamic viscosity

r= radius of tube

Inserting the value of D in ficks equation

R=(k*T)/(6*pi*n*r)*c/L

R= (1.38^-23*310)/(6*3.14*10^-3*1*51*10^-9)

R= 45.5*10^-7 molecules/second

2).Now it releases 300 molecules and we have to calculate how often it sends we know the rate of diffusion

Time= R*300= 0.001365 sec