138 Chapter 4. Random walks, friction, and diffusion[[Student version, December 8, 2002]]
Table 4.2:Sizes and diffusion constants of some molecules in water at 20◦C.
Molecule Molar mass,g/mole Radius,nm D× 109 ,m^2 s−^1
Water 18 0.15 2.0
Oxygen 32 0.2 1.0
Urea 60 0.4 1.1
Glucose 180 0.5 0.7
Ribonuclease 13 683 1.8 0.1
β-lactoglobulin 35 000 2.7 0.08
Hemoglobin 68 000 3.1 0.07
Collagen 345 000 31 0.0074.3Limitations of passive transport
Most eukaryotic cells are about 10μmin 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 diffuse there. Model the axon
as a tube 1mlong and 1μmin 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 m^3 )). 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 10 000 molecules.
Tosend 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 diffusion were the only
means of transport.
4.4Diffusion versus size
Table 4.2 lists the diffusion constantsDand radiiaof some biologically interesting molecules in
water. Consider the last four entries. Interpret these data in light of the diffusion law. [Hint: Plot
Dversus 1/a,and remember Equation 4.14.]
4.5Perrin’s experiment
Here are some experimental data on Brownian motion taken by Jean Perrin (Figure 4.17). Perrin
took colloidal particles of gutta-percha (natural rubber), with radius 0.37μm.Hewatched their
projections into thexyplane, so the two-dimensional random walk should describe their motions.
Following a suggestion of his colleague P. Langevin, he observed the location of a particle, waited
30 s,then observed again and plotted the net displacement in that time interval. He collected 500
data points in this way and calculated the root-mean-square displacement to bed=7. 84 μm.The
circles drawn on the figure have radiid/ 4 , 2 d/ 4 , 3 d/ 4 ,....
a. Find the expected coefficient of friction for a sphere of radius 0. 37 μm,using the Stokes formula
(Equation 4.14). Then work out the predicted value ofdusing the Einstein relation (Equation 4.15)
and compare to the measured value.
b. T 2 How many dots do you expect to find in each of the rings? How do your expectations
compare with the actual numbers?