by PhilipJ on 4 July 2006
In first year physics we all learned about Hooke’s law, where certain materials possess the property of exerting forces based on how much strain they are under. In the case of simple springs you can think of it as a force arising from the compression or elongation of the spring from its equilibrium state. Mathematically we’d write
where the force f is related to how much the spring is stretched or compressed from equilibrium, x, by a “spring constant”, k, with a minus sign signifying it is a restoring force. The spring “wants” to go back to zero elongation.
Despite having been given law status, as with most laws (Ohm, I’m looking at you), it isn’t much of a law at all1. Indeed, it only applies to “Hookean” or linear materials, those, circularly defined, which obey Hooke’s law. As one can imagine, this model fails spectacularly when it comes to more interesting materials than metallic springs (though even those display non-linearities if you measure things carefully enough). I’m thinking in particular about the force-extension curves I measured for double-stranded DNA (dsDNA). For anyone who’s forgotten, it isn’t a straight line as predicted by Hooke’s law. DNA isn’t Hookean.
So what is it? DNA is much longer than it is wide – λ DNA (which I’ve mentioned before) is about 16.5 µm upon full extension, but the molecule’s diameter is only about 2 nanometres, or some four orders of magnitude smaller. An individual base-pair is only some 0.338 nm in length, five orders of magnitude smaller than the full extension, so it isn’t unreasonable to imagine that the mechanical properties won’t depend on the microscopic structure of the molecule. Phil Nelson, in his book Biological Physics, puts it best:
When we study a system with a large number of locally interacting identical constituents on a far bigger scale than the size of the constituents, then we reap a huge simplification: Just a few effective degrees of freedom describe the system’s behavior, with just a few phenomenological parameters.
An example Nelson gives of phenomenological parameters that we’re familiar with are those of liquids, namely mass density (ρ) and viscosity (η). These two parameters tell us basically nothing of the underlying network of molecules that comprise the fluid, but at the scale with which macroscopic objects interact with fluids, they are the only ones which matter.
Freely jointed chains
Getting back to DNA, given that it is basically an extremely long cylinder with a very small diameter relative to its length, we can expect a simple model of a cylinder with some phenomenological parameters to give us a reasonable starting point to describe what is, at the microscopic level, a fairly complex system. The simplest possible model is called the freely jointed chain (FJC), and it treats the molecule as a chain of perfectly rigid subunits of length b joined by perfectly flexible hinges, as shown here:
You can think of b as the length of the repeating subunits, and is called the Kuhn length. If the molecule is under an applied force f as above, the effective energy for the chain is given by
If there were no applied force, all configurations have equal energy (and therefore the system has large configurational entropy), and the chain orients itself in any which way—analogous to a random walk. When a force is applied, the molecule is elongated and the work being done is that to remove entropy in the system.
An analytic force-extension relation has long been derived for the freely jointed chain, so I’ll simply quote it here:
In the limit of low streching (expand hyperbolic cotangent about f=0), we get back Hooke’s law (indeed, all polymer models reduce to a linear relation in the limit of low force, and DNA acts Hookean at very low forces), but in the high force limit the model predicts z approaches the contour (total) length Ltot as 1/f, which does not follow the force extension curve data for dsDNA at all. It isn’t a bad fit for ssDNA, where each individual non-paired base acts as an individual segment, but it fails miserably at high forces for dsDNA.
A more realistic model for dsDNA is actually quite similar to the freely jointed chain, but changes from discrete segments to a continuous elastic medium. This can be done because DNA is actually a rather stiff molecule, with successive segments displaying a sort of cooperativity—all pointing in roughly the same direction. The figure below is a cartoon of the wormlike chain (WLC) model, where now we define r(s) as the position as a fuction of the relaxed-state contour length, s. Also shown is the tangent vector t(s), which is the first derivative of r(s) with respect to a line segment ds.
There is now another added term to the effective energy of the chain which is related to the curvature (itself proportional to the square of the tangent vector), and the summation is now replaced by integration along the entire contour length:
A new phenomenological parameter has entered the energy term, A, which is a measure of the persistence length of the chain, or how long a segment of the chain will have tangent vectors all pointing in nearly the same direction. Indeed, the tangent-tangent correlation function for the wormlike chain at zero stretching force is given by
or, the similarity in directionality for the chain decays as an exponential in the persistence length.
An analytic solution for the force-extension relation for wormlike chains is not currently known, but the above equation has been solved numerically, and an interpolation formula to the numerical solution is most commonly used in DNA force-extension work today. At low force it again displays a Hookean linear relation, but as the extension nears the contour length of the molecule, it scales not as 1/f as predicted by freely jointed chains, but as 1/f1/2, in significantly better agreement with the data. The full interpolation formula is most commonly written today as
where A has become Lp and the contour length is written as L.
An example of the quality of fit one can obtain from the WLC model, here’s some recent data I took of a 12kb piece of dsDNA and the associated WLC fit:
What’s happening at forces > 20 pN?
The above fit is good to about 10-15 pN, above which the model keeps increasing quicker than the experimental data. This is because the WLC model is an inextensible model, where the chain contour length is constant2. Further modifications to the wormlike chain have introduced an enthalpic correction for higher forces, where the applied force is no longer simply extending the molecule, but has started doing work on the structure itself, deforming it from its regular B-DNA form. This addition makes the model valid up to approximately 60 pN, at which point DNA undergoes a structural phase transition and the force-extension relation changes quite rapidly back to a more Hookean form, extending up to 1.6x its normal contour length before displaying further non-linearities. No one is sure what the structure of DNA is after undergoing this structural phase transition, or how exactly it is able to stretch so significantly, though it has been proposed that the bases flip out away from the phosphate backbone (not unlike some of the early models proposed by Pauling). Other work on DNA elasticity attempts to explain the shortening or lengthening of the persistence length with salt concentrations, or AT/GC asymmetries in the sequences allowing more or less stretching and bending. So while my lab uses dsDNA force-extension relations as a calibration tool, there is still quite a bit of work on the elastic properties of DNA.
1. Though that isn’t to say it doesn’t crop up in the most curious of places, like optical trap calibration.
2. Phil Nelson remarks in his book that “wormlike chain” isn’t a particularly awesome name, as real worms are quite extensible!
P.S.—The FJC and WLC figures are inspired by C. Storm and P. Nelson, Physical Review E 67, 051906 (2003).