Biological Physics: Energy, Information, Life

162 Chapter 5. Life in the slow lane: the low Reynolds-number world[[Student version, December 8, 2002]]

rotation of

the DNA duplex

needed here

2 R = 2nm

lagging strand

leading strand

DNA polymerase

new DNA chain

Figure 5.12: (Schematic.) Replication of DNA requires that the original double helix (top) be unwound into its
twostrands. Molecular machines called DNA polymerase sit on the single strands synthesizing new, complementary
strands. The process requires the original strand to spin about its axis, as shown. Another molecular machine called
DNA helicase (not shown) sits at the opening point and walks along the DNA, unwinding the helix as it goes along.
[After (Albertset al.,2002).]

Consider cranking a long, thin, straight rod in water (Figure 5.11b). This model is not as
drastic an oversimplification as it may at first seem. DNA in solution is not really straight, but
when cranked it can rotate in place, like a tool for unclogging drains; our estimate will be roughly
applicable for such motions. Also, the cell’s cytoplasm is not just water, but for small objects
(like the 2nmthick DNA double helix) it’s not a bad approximation to use water’s viscosity (see
Appendix B).
The resistance to rotary motion should be expressed as atorque.The torqueτwill be propor-
tional to the viscosity and to the cranking rate, just as in Equation 5.4 on page 146. It will also
beproportional to the rod’s lengthL,since there will be a uniform drag on each segment. The
cranking rate is expressed as an angular velocityω,with dimensionsT−^1 .(Weknowωonce we’ve
measured the rate of replication, since every helical turn contains about 10.5 basepairs.) In short,
we must haveτ∝ωηL.Before we can evaluate this expression, however, we need an estimate for
the constant of proportionality.
Certainly the drag will also depend on the rod’s diameter, 2R.From the first-year physics
formulaτ=r×fwefind that torque has the same dimensions as energy. Dimensional analysis
then shows that the constant of proportionality we need has dimensionsL^2 .The only parameter in
the problem with the dimensions of length isR(recall that water itself has no intrinsic length scale,
Section 5.2.1). Thus the constant of proportionality we seek must beR^2 times some dimensionless