4.6. Biological applications of diffusion[[Student version, December 8, 2002]] 121
Toget the sort of graphs shown in Figure 4.11, weslicethe surface-graph along a line of constant
time; to get the graph made by our stationary observer we instead slice along a line of constantx
(heavy lines in the figures).
Figure 4.12a shows the sort of behavior we’ll find for the solution to the diffusion equation.
Your Turn 4d
Examine Figure 4.12a and convince yourself visually that indeed a stationary observer, for ex-
ample located atx=− 0 .7, sees a transient increase in concentration.
In contrast, Figure 4.12b depicts a behavior very different from what you just found in Your Turn 4d.
This snake-under-the-rug surface shows a localized bump in concentration, initially centered on
x=0,which moves steadily to the left (largerx)astime proceeds, without changing its shape.
This function describes atraveling wave.
The ability to look at a graph and see at a glance what sort of physical behavior it describes is
akey skill, so please don’t proceed until you’re comfortable with these ideas.
4.6 Biological applications of diffusion
Up to now, we have admired the diffusion equation but not solved it. This book is not about
the elaborate mathematical techniques used to solve differential equations. But it’s well worth our
while to examine some of the simplest solutions and extract their intuitive content.
4.6.1 The permeability of artificial membranes is diffusive
Imagine a long, thin glass tube (or “capillary tube”) of lengthL,full of water. One end sits in
abath of pure water, the other in a solution of ink in water with concentrationc 0 .Eventually
the containers at both ends will come to equilibrium with the same ink concentration, somewhere
between0andc 0 .But equilibrium will take a long time to achieve if the two containers are both
large. Prior to equilibrium, the system will instead come to a nearly steady, orquasi-steadystate.
That is, all variables describing the system will be nearly unchanging in time: The concentration
stays fixed atc(0) =c 0 at one end of the tube andc(L)=0atthe other, and will take various
intermediate valuesc(x)inbetween.
Tofind the quasi-steady state, we look for a solution to the diffusion equation with dc/dt=0.
According to Equation 4.19 on page 118, this means d^2 c/dx^2 =0.Thusthe graph ofc(x)isa
straight line (see Figure 4.11b), orc(x)=c 0
(
1 −x/L)
.Aconstant fluxjs=Dc 0 /Lof ink molecules
then diffuses through the tube. (The subscript “s” reminds us that this is a flux ofsolute, not of
water.) If the concentrations on each side are both nonzero, the same argument gives the flux in
the +ˆxdirection asjs=−D(∆c)/L,where ∆c=cL−c 0 is the concentration difference.
The sketch in Figure 2.30 on page 57 shows cell membranes as having channels even narrower
than the membrane thickness. Accordingly, let’s try to apply the preceding picture of diffusion
through a long, thin channel to membrane transport. Thus we expect that the flux through the
membrane will be of the form
js=−Ps∆c. (4.20)
Here thepermeation constantof solute,Ps,isanumber depending on both the membrane and
the molecule whose permeation we’re studying. In simple cases, the value ofPsroughly reflects the
width of the pore, the thickness of the membrane (length of the pore), and the diffusion constant
for the solute molecules.