7.2 Beyond kinking
7.2.1 AFM measurements of the bending energy of DNA
Individual DNA molecules are directly observable on the nm length scale in 2D via AFM [10,11] and EM experiments[12], and in 3D via Cryo EM Tomography [13]. If these molecules are in thermal equilibrium, the chain statistics is also directly observable since it can be computed from the observed conformations. For example, the tangent distribution functionG(~tf;~ti;L) for contour lengthLis the conditional probability density of a final tangent~tf at contour lengthLgiven an initial tangent~ti at contour length 0. This distribution function can be computed directly from observed conformations by histograming the deflection angles (θ≡arccos~tf·~ti) for contour lengthLsegments of the chain.
The bending energy for the contour lengthL segment is then defined by the Boltzmann distri- bution
EL(θ) =−kTlogG(~er(θ);~ez;L), (7.1) where~er and~ez are the unit vectors in ther andz directions respectively. Of course, the bending energy as we have defined it here is really a bending free energy. For chain statistics calculations, the distinction between energy and free energy is not meaningful since we describe a coarse grained model. Note also that I have explicitly restored the thermal units in this equation in the interest of clarity.
A number of authors have studied the chain statistics of DNA adsorbed to mica [10,11]. Busta- mante and coworker showed that when DNA is adsorbed in the presence of Mg2+ at concentrations smaller than 10 mM, the persistence length of DNA is nearly identical to that measured in solution [10]. The authors interpreted these results to imply that at low magnesium concentration, (i) the DNA was weakly bound to the mica substrate and was free to equilibrate on the surface and (ii) the bending energy of the chain was not significantly altered by interaction with the surface.
Dekker, van Noort, and coworkers recently employed the mica deposition technique to study the effect of the DNA repair protein Rad50 on DNA curvature [11]. As a control, they computed the DNA bending energy as a function of the deflection angle for 5 nm segments of DNA. These measurements posit a bending energy of DNA, on the 5 nm length scale, that is poorly modeled by
0 0.1p 0.2p 0.3p 0.4p 0.5p µ
0 2 4 6 8
E(µ)kBT)(
Expt SEC WLC t1
~
t2
~
t3
~
1 2
Not predicted by AFM data
Figure 7.3: The bending energy of DNA on short length scales? The tangent distribution function is measured as a function of deflection angle for 5 nm sections of long sequences of DNA absorbed to mica via AFM [11]. The bending energy deduced from the tangent distribution function (Eq. 7.1) (black dots) is significantly non-harmonic and can be approximately fit to the functional formA|θ|/`
(red curve). For comparison, the WLC bending energy is also shown (blue curve) for persistence length 53 nm which accurately describes the long-length-scale DNA statistical mechanics but fits the short-length-scale experimental data very poorly. We shall call the model based on the fit to the experimental data the SEC model.
an elastic rod model. In fact the data is better fit by the bending-energy
E(θ) =A|θ|= 5.3|θ|. (7.2)
It is clear from Fig. 7.2.1 that the model is softer at high curvature than the WLC model with persistence length 53 nm (we shall show that the persistence length of the model defined by Eq.7.2 is 53 nm) and the energy is not non-convex. We will refer to this model as the Sub-Elastic Chain model (SEC) since the constitutive relation (the bending energy) has a weaker dependence on the magnitude of the curvature than the elastic rod model.
It seemed immediately clear to all of us that this bending energy must be incorrect since the energy did not have a wide quadratic region at small deflection. The WLC model work well for force-extension experiments and long-contour-length cyclization measurements, both of which were sensitive to bending in this regime. To our surprise, when we computed the persistence length from the SEC energy, it was 53 nm, correct for DNA. I then proceeded to compute the long-contour- length tangent distribution functions. They were indistinguishable from the WLC model! How could a theory that was so different from the WLC model on short-length-scales still give the correct answer at long length scales? The answer was thermal fluctuations and the renormalization group.
The renormalization group was well known to me in other contexts (high-energy particle physics) but I had forgotten it should also apply for systems as simple as DNA bending. Many physical properties of complicated condensed matter systems have been described by a small set of theories described in terms of renormalizable operators [14]. Regardless of the complicated structure of
the theory at short length scales, the Renormalization Group guarantees that the long-length-scale chain statistics will be described by a theory of renormalizable operators only. For stiff polymers, only one such renormalizable operator exists with the right symmetries. As a consequence, all stiff polymers share generic long-length-scale behavior: that described by the WLC model. Physically, this loss of information is due to the averaging effect of thermal fluctuations. (Many microstates contribute to any given macrostate.) But, on short enough length scales, the underlying structure of the theory becomes important. Violations of the linear elastic theory, analogous to those observed in macroscopic bending, are therefore predicted in experiments that probe the short-length-scale bending of DNA: experiments like the cyclization experiments of Cloutier and Widom.
The renormalization group and the inextensibility of DNA essentially guaranteed that the force- extension of the SEC model would be indentical to the WLC model. To check this, it was necessary to expand the machinery developed by Spakowitz and Wang [15,16, 17] for computing the spatial distribution of the WLC model to analyze general theories. This turned out to be surprisingly easy!
As expected, the force-extension of the SEC model was essentially indistinguishable from the WLC model. The last calculation was the most surprising. When we computed the J factor, it fit the Cloutier and Widom data [1] without a fitting parameter.
We had a cute story and no kinking was required. Was it really so surprising that DNA was not described by an elastic rod on short length scales? DNA was a complicated biomolecular polymer.
How could the elastic model do it justice? The only reason the elastic rod model worked was that the renormalization group hid all the details.
7.3 On the experimental front...
Phil told our story at the Aspen Single Molecule Biophysics (Steve M. Block) conference. Cees Dekker, whose data was the basis for our model, was in the audience. Cees’ paper had not been about DNA mechanics. Had it been, he would have had much better statistics. When Cees returned to Delft, he asked Fernando Moreno and Thijn van der Heijden to repeat the measurement of the bending energy more carefully. Two weeks after Aspen, we had a new set of data from the experimentalists (Fig.7.3) which was as puzzling as it was exciting.
We had assumed that the bending energy was approximately linear in the deflection angle in part due to poor statistics. But the new data was amazingly linear! Back of the envelope estimates suggested that Dekker and coworkers had the resolution to measure the bending energy for 5 nm segments of DNA. Most disturbing was the fact that the linear elastic regime which we assumed must be there intuitively was nowhere to be seen. Cees seemed reasonably confident about the data, but it seemed to us too good (too linear) to be true3. The other problem was what kind of microscopic
3That being said, hind sight is always 20-20.
Figure 7.4: The new bending energy measurements of Dekker and coworkers. These new measure- ments of the bending energy were perplexingly linear in the deflection angle. Even at small deflection, where we intuitively expect the bending energy to be quadratic, the dependence resembled the SEC model (E=|θ|).
mechanism could be responsible for this linear dependence. Why was it so linear? Like us, everybody who had seen the bending energy was also bothered by this linearity, especially at small deflection.
Carlos Bustamante had basically told us he didn’t believe any of the data. Eventually, Rob and I decided that I should go to Delft so I could learn more about the details of the experiment.