Biological Physics: Energy, Information, Life

120 Chapter 4. Random walks, friction, and diffusion[[Student version, December 8, 2002]]

 2

 1

0

1

2

x

0

1

2

3

t

1

2

c

 2

 1

0

1

 2

0

2

4

x

0

1

2

3

t

0

0.5

1

1.5

c

 2

0

2

4

x

ab

Figure 4.12:(Mathematical functions.)(a)Diffusion in space and time. The vertical axis gives the concentration
(arbitrary units) after a localized lump of solute begins to spread. Notice that time is drawn as increasing as we
move diagonally downward in the page (arrow). Section 4.6.5 derives the function graphed here. The heavy line is
the concentration profile at one particular time,t=1.6. (b)Ahypothetical travelingwave.Chapter 12 will find this
sort of behavior in the context of nerve impulses. The heavy line is the concentration as seen by an observer fixed
atx=0.7.

of neurotransmitter at one point, creating such a bump distribution in three dimensions. Looking
at the slope of the curve, we see that the flux will be everywhereawayfrom 0, indeed tending to
erase the bump. More precisely, the curvature of this graph is concave-down in between the two
starred points. Here the diffusion equation says that dc/dtwill be negative: The height of the bump
goes down. But outside the two starred points, the curvature is concave-up:dc/dtwill be positive,
and the concentration grows. This conclusion also makes sense: Particles leaving the bump must
go somewhere, enhancing the concentration away from the bump. The starred points, where the
curvature changes sign, are calledinflection pointsof the graph of concentration. We’ll soon see
that they move apart in time, thereby leading to a wider, lower bump.
Suppose you stand at the pointx=Aand watch. Initially the concentration is low, because
you’re outside the bump. Then it starts to increase, because you’re outside the inflection point.
Later, as the inflection point moves past you, the concentration again decreases: You’ve seen awave
of diffusing particles pass by. Ultimately the bump is so small that the concentration is uniform:
Diffusion erases the bump and the order it represents.

4.5.2 A function of two variables can be visualized as a landscape

Implicit in all the discussion so far has been the idea thatcis a function oftwo variables,spacex
and timet.All the pictures in Figure 4.11 have been snapshots, graphs ofc(x, t 1 )atsome fixed
timet=t 1 .But the stationary observer just mentioned has a different point of view: She would
graph thetimedevelopment byc(A, t)holdingx=Afixed. We can visualize both points of view
at the same time by drawing a picture of the whole function as asurfacein space (Figure 4.12). In
these figures, points in the horizontal plane correspond to all points in space and time; the height
of the surface above this plane represents the concentration at that point and that time. The two
derivativesddxcandddctare thenbothinterpreted as slopes, corresponding to the two directions you
could walk away from any point. Sometimes it’s useful to be ultra-explicit and indicate both what’s
being variedandwhat’s held fixed. Thus for example ddxc

∣∣

tdenotes the derivative holdingtfixed.