- Track 2[[Student version, January 17, 2003]] 341
9.3.2′ The ideas in Section 9.3.2 can be generalized to higher-dimensional spaces. A linear vector
function of ak-dimensional vector can be expressed in terms of ak×kmatrixM.The eigenvalues
of such a matrix can be found by subtracting an unknown constantλfrom each diagonal element of
M,then requiring that the resulting matrix have determinant equal to zero. The resulting condition
is that a certain polynomial inλshould vanish; the roots of this polynomial are the eigenvalues
ofM.Animportant special case is whenMis real and symmetric; in this case we are guaranteed
that there will beklinearly independent real eigenvectors, so that any other vector can be written
as some linear combination of them. Indeed, in this case all of the eigenvectors can be chosen to
bemutually perpendicular. Finally, if all the entries of a matrix are positive numbers, then one of
the eigenvalues has greater absolute value than all the others; this eigenvalue is real, positive, and
nondegenerate (the “Frobenius–Perron theorem”).
9.4.1′Even though we found that the 1d cooperative chain fit the experimental data slightly better
than the one-dimensional FJC, still it’s clear that this is physically a very unrealistic model: We
assumed a chain of straight links, each one joined to the next at an angle of either zero or 180◦!
Really, each basepair in the DNA molecule is pointing in nearly thesamedirection as its neighbor.
Wedid, however, discover one key fact, that the effective segment lengthLsegis tens of nanometers
long, much longer than the thickness of a single basepair (0. 34 nm). This observation means that we
can use our phenomenological elastic energy formula (Equation 9.3) as a more accurate substitute
for Equation 9.17.
Thus, “all” we need to do is to evaluate the partition function starting from Equation 9.3,
then imitate the steps leading to Equation 9.10 to get the force–extension relation of the three-
dimensional elastic rod model. The required analysis was begun in the 1960’s by N. Saito and
coauthors, then completed in 1994 by J. Marko and E. Siggia, and by A. Vologodskii. (For many
more details see Marko & Siggia, 1995.)
Unfortunately, the mathematics needed to carry out the program just sketched is somewhat more
involved than in the rest of this book. But when faced with such beautifully clean experimental
data as those in the figure, and with such an elegant model as Equation 9.3, we really have no
choice but to go the distance and compare them carefully.
Wewill treat the elastic rod as consisting ofNdiscrete links, each of length. Our problem
is more difficult than the one-dimensional chain because the configuration variable is no longer
the discrete, two-valuedσi=±1, but instead the continuous variableˆtidescribing the orientation
of link numberi.Thusthe transfer matrixThascontinuous indices.^11 TofindT,wewrite the
partition function at a fixed external forcef,analogous to Equation 9.18:
Z(f)=∫
d^2 ˆt 1 ···d^2 ˆtNexp[ N∑− 1
i=1(
f
2 kBT
(cosθi+cosθi+1)−A
2
(Θi,i+1)^2)
+ f
2 kBT
(cosθ 0 +cosθN)]
. (9.36)
In this formula theNintegrals are over directions—eachˆtiruns over the unit sphere. θiis the
angle between linki’s tangent and the directionˆzof the applied force; in other words, cosθi=ˆti·zˆ.
Similarly, Θi,i+1is the angle betweenˆtiandˆti+i;Section 9.1.3′on page 338 showed that the elastic
energy cost of a bend is (AkBT/ 2 )Θ^2 .Since each individual bending angle will be small, we can
(^11) This concept may be familiar from quantum mechanics. Such infinite-dimensional matrices are sometimes called
“kernels.”