In this section we focus on the spontaneous thermal bending fluctuation of biofilaments.

For short filaments, thermal bending fluctuations only introduce a slight angle of bending. For long filaments, many random bends occur, and a really long biofilament therefore has many of the properties of a flexible polymer.

The characteristic length of a filament over which thermal bends of about a radian occurs is called the persistence length.

For a short enough filament, thermal bends will only introduce a slight angle. Over a short filament, we may imagine that a bend occurs by some (small) angle q, as a smooth circular bend. If we measure angles in radians, then small angle means q < 0.1 (a few degrees or less).

The above figure defines the geometry for most of this lecture.

As usual we talk about a rod of length L, with a bending modulus B. The energy of a smooth circular bend by angle q (in radians) is just

So, in thermal equilibrium, we can expect this bending energy to have the average value k_B T/2, which tells us

Furthermore - this formula indicates that over distances of B/(k_B T), thermal bending fluctuations occur which deflect the tangent direction of the rod by about one radian (57 degrees).

We call the distance of B/(k_B T) = A the persistence length of the rod. If the rod is much shorter than this distance, it is essentially straight.

If it is much longer than this, many bends in random directions occur along the total contour length of the rod, and it will look like a random walk.

We can rewrite

We already learned (previous section) that dsDNA has a bending constant of B » 2 ×10^-28 J m, and its persistence length is just this divided by k_B T, or A = [2×10^-28] / [4×10^-21 m] = 5 ×10^-8 m or 50 nanometers (nm). This is about the value of A that has been measured in a wide variety of experiments.

Thus dsDNA is essentially straight over 50 nm = 150 bp stretches; then it is thermally fluctuating over longer contour lengths.

We saw actin had B » 6 ×10^-26 J m, which gives A = 1.5 ×10^-5 m or about 15 microns. You can read the reference Gittes et al JCB 1993 from the previous section to find the experimentally accepted value of about 17 microns.

Finally, microtubules have B » 1.5 ×10^-23 J m which indicates a persistence length A » 4 ×10^-3 m, about 4 mm. The paper by Gittes et al experimentally finds a value right close to this value.

We can just use our previously derived formula for the mean-squared-angle, and apply it to the distance between adjacent base pairs, L = 0.3 nm:

This is exactly consistent with thermal stretching of the distance between adjacent base pairs by about 0.03 nm, in the sense that such a stretch on only one backbone would generate a bend angle of around 0.03/0.3 » 0.1 radian.

This use of elasticity down to the nearly atomic scale is a bit suspect, but it works!

Bending a rod always reduces its end to end distance (the chord D in the figure above).

All we have to do is estimate by how much. A rod of length L bent through a small angle q has an end-to-end distance which is the base of an isoceles triangle with two sides of length R = L/q and an apex angle q. The end-to-end distance is therefore

So the rod has its end-to-end distance shortened by an amount » L²/A, a formula obviously only valid for L < A. Much fancier calculations for this shortening give the same form, with a change in the numerical prefactor.

The above figure shows the geometry, although with a greatly exaggerated bend.

We know that the square of the angle of bending is on average < q² > = L/A for L << A. This means that the typical bending angle is roughly Ö[(L/A)]. The perpendicular displacement is therefore (using the reduced end-to-end distance from above)

All of the above has treated bending fluctuations from the point of view of a single angular bend degree of freedom. This single degree of freedom must involve about k_B T of free energy, so to force this degree of freedom to stop fluctuating should require work done of about k_B T.

This work done will extend the rod to nearly its relaxed length, which will be an extension of about L - D » L²/A. Writing the work done by this force:

We now move on to long filaments, i.e. L >> A. We will see that at low forces, these long filaments behave as flexible polymers. Under large forces, there is a new type of nonlinear elastic response distinct from that of the completely flexible random-walk polymer analyzed previously. This problem gives a good idea of a lot of the ideas used in modern condensed matter physics, as well as giving us a detailed picture of the stress-strain response of biofilaments.

Random bends occur every A or so, resulting in the biofilament being rather like a random walk of » L/A steps. We should guess from our previous calculations for random walks then that its end-to-end distance should be about < D² > » (L/A) A² = A L.

In fact the exact result is

Well, if a long filament is just a flexible polymer of segment length b = 2A and N = L/(2A) segments, for low enough forces, the force response should just follow our flexible polymer formula:

The regime where this formula will be accurate will be when the polymer is not too strongly stretched, or x << L, equivalently f << k_B T/A.

This result can be confirmed by a detailed (and complicated) calculation for a long filament.

What happens at large forces? Well, the filament obviously gets straightened out. Above we saw that a force f has associated a length l of filament satisfying

So now we see that this formula is telling us about the organization of a highly stretched filament. Instantaneously the filament has the structure of a sequence of short filaments of length l, joined end-to-end.
 

For large forces f >> k_B T/A, the bending fluctuations along the filament have a characteristic length l = Ö[(k_B T A / f)]. Each one of these pieces of the chain of length l has inside it a little bit of `stored length' l²/A = k_B T A/f. Therefore the difference between the total length L and the end-to-end extension x is

So, the force diverges as x/L ® 1, with a pretty strong divergence,

The fully flexible polymer has no intrinsic bending rigidity.

The formula

More precise interpolation formulae are available in the research literature, and are useful in interpretation of force-extension data on biofilaments.   This kind of formula has been used to fit dsDNA force-extension data, as well as other biofilaments.

There have been marvelous experiments done since 1993 studying the mechanical properties of single molecules.  One of the most heavily studied examples is dsDNA, which can be relatively easily obtained in long (10's of mcirons) lengths, and attached to e.g. glass surfaces and colloidal particles.

The first high-precision experiments on DNA elasticity were done in the lab of Carlos Bustamante at the University of Oregon.  Steve Smith in that lab played a large role in developing the techniques used.   Data from experiments published in 1992 looked like this (Smith et al, Science 258, 1122 (1992)):

The experimental data are shown by boxes; the units of force are in kT/nm = 4.1 pN.   The solid curve is the exact result of the flexible filament theory (the `real' version of the interpolation formula of the previous section).  As you can see, over the force range shown (about 0.04 to 4 pN) the simple filament theory fits the data pretty well (although always beware of logarithmic axes!).  Really the only fit parameter is the persistence length A=53 nm.  And this value is roughly known from lots of other experiments.

The dashed line shows the result of the simple flexible polymer (random-flight) model we previously discussed, fit so that it agrees with the low-force behavior.  As you can see it does not work at all at high forces.  This is because the flexible-polymer model does not have any bending stiffness.  For more on the details of the theory that can be developed for the thermally fluctuating filament, see Marko and Siggia, Macromolecules 28, 8759 (1995)).

The inset shows a plot of the inverse of the square root of force, versus extension.  It shows a linear dependence for large force, in accord with theory.

In case you are wondering about the reproducibility of these sorts of experiments, here is data from the group of Vincent Croquette and David Bensimon in Paris (ENS):

It turns out that these data indicate almost the same value of A=50 nm.  You will do the fitting on a problem set.

Finally it should be noted that at slightly larger forces (10 pN) this simple filament-bending elasticity breaks down, and you start to really stretch the secondary structure of the double helix.  At first you see lengthening of the double helix, from which one can directly measure the stretching modulus.  Finally there is a sharp `phase transition' to a stretched form of the double helix, at about 60 pN.    You might want to look at Cluzel et al Science 271, 792-794 (1996) or Smith et al Science 271 795-799 (1996) where this behavior was first observed.

The following experimental data (Leger et al, PNAS USA 95 12295-12299 (1998)) shows the gradual stretching of the double helix for 10-50 pN, then a transition in  lambda-DNA (EMBL3) at about 55 pN.  The dashed curve shows a similar transition in a molecule which is very AT-rich, and in this case it is thought that the two strands separate at the transition.