112 Chapter 4. Random walks, friction, and diffusion[[Student version, December 8, 2002]]
include globular protein!seeprotein such as serum albumin. We can crudely classify polymers as
“compact” or “extended” by comparing the volume occupied by the polymer to the minimal volume
if all its monomers were tightly packed together. Most large proteins and nonbiological polymers
then fall unambiguously into one or the other category; see Table 4.1.
Table 4.1:Properties of various polymers. The table shows the measured radius of gyration for a few natural and
artificial polymers, along with the radius of the ball the polymer would occupy if it were tightly packed, estimated
from the molar mass and approximate density. [From (Tanford, 1961)]
Polymer Molar mass,g/mole RG,nm,Packed–ball radius,nm Type
Serum albumin 6. 6 · 104 32 compact
Catalase 2. 25 · 105 43 compact
Bushy stunt virus 1. 1 · 107 12 11 compact
Myosin 4. 93 · 105 47 4 extended
Polystyrene 3. 2 · 106 49 8 extended
DNA, in vitro 4 · 106 117 7 extended
Even if a polymer does not collapse into a packed coil, its monomers are not really free to
sit anywhere: Two monomers cannot occupy the same point of space! Our treatment ignored
this “self-avoidance” phenomenon. Remarkably, introducing the physics of self-avoidance simply
ends up changing the scaling exponent in Idea 4.16 from 1/2 to another, calculable, value. The
actual value of this exponent depends on temperature and solvent conditions. For a walk in three
dimensions, in “good solvent” the corrected value is about 0.58. The experiment shown in Figure 4.7
is an example of this situation, and indeed its exponent is seen to be slightly larger than the simple
model’s prediction of 1/2. Whatever the precise value of this exponent, the main point is that
simple scaling relations emerge from the complexity of polymer motions.
Figure 4.8 shows a particularly direct test of a scaling law for a polymer conformation. B. Maier
and J. R ̈adler formed a positively charged surface, then let it attract single strands of DNA, which
is negatively charged. They then took successive snapshots of the DNA’s changing conformation
(the DNA contained a fluorescent dye to make it visible). The DNA may cross over itself, but
each time it does so there is a cost in binding energy, as the negatively charged upper strand does
not contact the positive surface at the point of crossing, and instead is forced to contact another
negative strand. Thus we may expect the coil size to follow a scaling relation appropriate to a
two-dimensional, self-avoiding, two-dimensional, random walk. Problem 7.9 will show that the
predicted scaling exponent for such a walk is 3/4.
Once bound to the plate, the strands began to wander (Figure 4.8b). Measuring the fluorescence
intensity as a function of position and averaging over many video frames allowed Maier and R ̈adler
to compute the polymer chain’s “radius of gyration”RG,which is related to the chain’s mean-
square end-to-end distance. The data in Figure 4.8d show thatRG∝M^0.^79 ,close to the 3/4 power
law predicted by theory.
T 2 Section 4.3.1′on page 133 mentions some finer points about the conformation of random-coil
polymers.
4.3.2 Vista: Random walks on Wall Street
Stock markets are interacting systems of innumerable, independent biological subunits—the in-
vestors. Each investor is governed by a personal mixture of prior experience, emotion, and incom-