5.4. Excursion: The character of physical Laws[[Student version, December 8, 2002]] 163
numberC,or:
τ=−C×ωηR^2 L. (5.18)
Problem 5.9 shows that this result is indeed correct, and thatC=4π,but we don’t need the precise
value for what follows.
The rate at which we must do work to crank the rod is the product of the resisting torque times
the rotation rate:−τω=Cω^2 ηR^2 L.Since the rod rotates through 2πradians for each helical turn,
wecan instead quote the mechanical work needed per helical turn, as
Wfrict=− 2 πτ=2πC·ωηR^2 L. (5.19)An enzyme called DNA polymerase syntheszes new DNA inE. coliat a rate of about 1000 basepairs
persecond, orω=2πrevolutonradian × 1000 b.p.s
− 1
10 .5b.p./revolution=^600 s− (^1). Equation 5.19 then givesWfrict ≈
(2π)(4π)(600s−^1 )(10−^3 Pa s)(1nm)^2 L≈(4. 7 · 10 −^17 Jm−^1 )L.
Asecond enzyme, calledDNA helicase,doesthe actual cranking. Helicase walks along the DNA
in front of the polymerase, unzipping the double helix as it goes along. The energy required to do
this comes from the universal energy-supply molecule ATP. Appendix B lists the useful energy in a
single molecule of ATP as≈ 20 kBTr=8. 2 · 10 −^20 J.Let us suppose that one ATP suffices to crank
the DNA by one full turn. Then the energy lost to viscous friction will be negligible as long asLis
muchsmaller than (8. 2 · 10 −^20 J)/(4. 7 · 10 −^17 Jm−^1 ), or about two millimeters, a very long distance
in the nanoworld. Levinthal and Crane correctly concluded that rotational drag is not an obstacle
to replication.
Today we know that another class of enzymes, the “topoisomerases,” remove the excess twisting
generated by the helicase in the course of replication. The above estimate should thus be applied
only to the region from the replication fork to the first topoisomerase, and hence viscous rotary
drag is even less significant than the previous paragraphs makes it seem. In any case, a physical
argument let Levinthal and Crane dismiss an objection to Watson and Crick’s model for DNA, long
before any of the details of the cellular machinery responsible for replication were known.
5.4 Excursion: The character of physical Laws
Weare starting to amass a large collection of statements called “laws.” (This chapter alone has
mentioned Newton’s Law of motion, the Second Law of thermodynamics, and Ohm’s and Fick’s
laws.) Generally these terms were born like any other new word in the language—someone noticed
acertain degree of generality to the statement, coined the name, a few others followed, and the
term stuck. Physicists, however, tend to be a bit less promiscuous in attaching the term “physical
Law” to an assertion. While we cannot just rename terms hallowed by tradition, this book attempts
to make the distinction by capitalizing the word Law on those statements that seem to meet the
physicist’s criteria, elegantly summarized by Richard Feynman in 1964.
Summarizing Feynman’s summary, physical Laws seem to share some common characteristics.
Certainly there is an element of subjectivity in the canonization of a Law, but in the end there is
generally more consensus than dispute on any given case.
- Certainly we must insist on avery great degree of generality,an applicability to an
extremely broad class of phenomena. Thus, many electrical conductors do not obey
“Ohm’s law,” even approximately, whereas any two objects in the Universe really