- Track 2[[Student version, January 17, 2003]] 337
- Much of the power of Idea 9.1 comes from the word “local.” In a system with local interactions
wecan arrange the actors in such a way that each one interacts with only a few “near neighbors,”
and the arrangement resembles a meshwork with only a few dimensions, typically two or three. For
example, each point on a cubical lattice has just six nearest neighbors (see Problem 1.6).
Idea 9.1 is not strictly applicable to problems involving nonlocal interactions. In fact, one
definition of a “complex system” is “Many non-identical elements connected by diverse, nonlocal
interactions.” Many problems of biological and ecological organization do have this character, and
indeed general results have been harder to get in this domain than in the traditional ones. - Section 9.1.1 stated that a fluid membrane has one elastic constant. This statement is a slight
simplification: There are actuallytwoelastic bending constants. The one discussed in Section 8.6.1
discourages “mean curvature,” while the other involves “Gaussian curvature.” For more informa-
tion, and to see why the Gaussian stiffness doesn’t enter many calculations, see for example Seifert,
9.1.2′
- Technicallyωis called apseudoscalar. The derogatory prefix ‘pseudo’ reminds us that upon
reflection through a mirrorωchanges sign (try viewing Figure 2.19 on page 47 in a mirror), whereas
atrue scalar likeudoes not. Similarly, the last term of Equation 9.2 is also pseudoscalar, being
the product of a true scalar times a pseudoscalar. We should expect to find such a term in the
elastic energy of a molecule like DNA, whose structure is not mirror-symmetric (see Section 9.5.1).
The twist-stretch coupling in DNA has in fact been observed experimentally, in experiments that
control the twist variable. - We implicitly used dimensional-analysis reasoning (see Section 9.1.1′on page 336) to get the
continuum rod-elasticity equation, Equation 9.2. Thus the only terms we retained were those with
the fewest possible derivatives of the deformation fields (that is, none). In fact,single-stranded
DNA is not very well described by the elastic rod model, because its persistence length isnotmuch
bigger than the size of the individual monomers, so that Idea 9.1 does not apply. - We also simplified our rod model by requiring that the terms have the symmetries appropriate
to a uniform, cylindrical rod. Clearly DNA is not such an object. For example, at any points
along the molecule, it will be easier to bend in one direction than in the other: Bending in the
easy direction squeezes the helical groove in Figure 2.17. Thus strictly speaking, Equation 9.2 is
appropriate only for bends on length scales longer than the helical repeat of 10. 5 × 0. 34 nm,since
on such scales these anisotropies average out. For more details see Marko & Siggia, 1994. - It is possible to consider terms inEof higher than quadratic order in the deformation. (Again
see Marko & Siggia, 1994.) Under normal conditions these terms have small effects, since the large
elastic stiffness of DNA keeps the local deformations small. - We need not, and should not, take our elastic-rod imagery too literally as a representation of
amacromolecule. When we bend the structure shown in Figure 2.17, some of the free energy
cost indeed comes from deforming various chemical bonds between the atoms, roughly like bending
asteel bar. But there are other contributions as well. For example, recall that DNA is highly
charged—it’s an acid, with two negative charges per basepair. This charge makes DNA a self-
repelling object, adding a substantial contribution to the free-energy cost of bending it. Moreover,
this contribution depends on external conditions, such as the surrounding salt concentration (see