110 Chapter 4. Random walks, friction, and diffusion[[Student version, December 8, 2002]]
2 Ln+1
√
3 LnFigure 4.6:(Schematic.) A fragment of a three-dimensional random walk, simplified so that every joint can make
any of eight possible bends. In the configuration shown, the step from jointnto jointn+1is the vector sum of one
step to the right, one step down, and one step into the page.
4.3 Other random walks
4.3.1 The conformation of polymers
Up to this point, we have been thinking of Figure 4.2 as a time-lapse photo of themotionof apoint
particle. Here is another application of exactly the same mathematics to a totally different physical
problem, with biological relevance: the conformation of a polymer, for example DNA.
Section 3.3.3 on page 89 described Max Delbr ̈uck’s idea that the physical carrier of genetic
information is a single long-chain molecule. To describe the exact state of a polymer, we’d need an
enormous number of geometrical parameters, for example, the angles of every chemical bond. It’s
hopeless topredictthis state, since the polymer is constantly being knocked about by the thermal
motion of the surrounding fluid. But here again, we may turn frustration into opportunity. Are
there someoverall, averageproperties of the whole polymer’s shape that we could try to predict?
Let us imagine that the polymer can be regarded as a string ofNunits. Each unit is joined
to the next by a perfectly flexible joint, like a string of paperclips.^6 In thermal equilibrium, the
joints will all be at random angles. An instantaneous snapshot of the polymer will be different at
each instant of time, but there will be a family resemblance in a series of such snapshots: Each
one will be a random walk. Following the approach of Section 4.1.2, let us simplify the problem
bysupposing that each joint of the chain sits at one of the eight corners of a cube centered on
the previous joint (Figure 4.6). Taking the length of the cube’s sides to be 2L,then the length of
one link is
√
3 L.Wecan now apply our results from Section 4.1.2. For instance, the polymer is
extremely unlikely to be stretched out straight, just as in our imaginary checker game we’re unlikely
to take every step to the right. Instead the polymer is likely to be a blob, orrandom coil.
From Equation 4.4 on page 104 we find that the root-mean-square distance between the ends
of the random coil is
√
〈rN^2 〉=√
〈xN^2 〉+〈yN^2 〉+〈zN^2 〉=√
3 L^2 N=L
√
3 N.This is an experi-
mentally testable prediction. The molar mass of the polymer equals the number of units,N,times
(^6) In a real polymer the joints will not be perfectly flexible. Chapter 9 will show that even in this case the freely
jointed chain model has some validity, as long as we understand that each of the “units” just mentioned may actually
consist of many monomers.