From top to bottom the curves are for polypeptide, ssNA, and dsDNA. Note that on the logarithmic force scale the shapes of the curves are all the same; the different values of b just shift them in force by different amounts. The use of the logarithmic force scale also allows us to see all the curves at once.

(d) Reading off the graph we see that 60% extension is achieved for forces of about 35 pN, 15 pN, and 0.1 pN for polypeptide, ssNA and dsDNA, respectively. The important lesson is that as b gets larger (i.e. polymer stiffer), there are fewer thermally fluctuating degrees of freedom per length, reducing the strength of entropic elasticity less.

We'll revisit this concept when we discuss how to think about polymers with bending elasticity rather than freely-jointed segments. Bending elasticity is essential to really quantitatively describing many biopolymers including dsDNA.

Minimizing this with respect to R gives the condition for the equilibrium polymer size R:

(c) In the extreme case of d = 1 we have R µ N from the Flory theory. This must be, since in d = 1 if there are repulsive segment-segment interactions, the polymer must just have its segments arranged head-to-tail going in either the left or right direction.

(d) A flexible polymer somehow loosely plastered down to a surface by transient `adsorption' interactions should show 2d self-avoiding statistics. A few groups are trying to show the 3/4 exponent, some using lipid membranes as substrates, and others using solid surfaces. The trick to having the polymer behave as truly a two-dimensional polymer is to have a smooth substrate (so that the polymer will be sensitive to itself, rather than to lumps and holes on the surface), and to have the adsorption interactions tuned so that they are not so strong that the polymer never moves, nor too weak so that the polymer doesn't float away into the solution.

It is worth noting that cell membranes have tons of things from folded proteins to flexible molecules attached to them. So in general, thinking about macromolecules moving around in two dimensions is biologically relevant.

(e) For d > 4, you can see that n = 3/(d+2) < 1/2. This is impossible, since we are considering purely repulsive interactions, yet this result implies that for d > 4, polymers are shrunk relative to random walks by the repulsive interaction.

What must happen for d > 4 is that collisions between segments are so rare (space is so `big') that the repulsive interaction does not change the polymer size from its random-walk value R » N^1/2 b.

Another way to see this is to just directly estimate the free energy contributed by the interaction term for the case of a random walk (R » N^1/2 b)

In order to confuse the uninitiated, theoretical physicists like to say

the local repulsive interaction of a polymer with itself is irrelevant above four dimensions

or even more cryptically

d=4 is the upper critical dimension for the self-avoiding walk

These statements both mean that in d > 4, the repulsive interactions don't change the noninteracting problem (i.e. life is easy for theorists), while in d < 4 even weak interactions result in a theory which is very hard to get accurate results from. In condensed matter the first question asked of any new type of interaction in any many-body problem is what its upper critical dimension is.

Something else which is rather amusing is that the self-avoiding polymer problem can be mapped to a special case of a quantum field theory of particles which interact by 4-particle scattering interactions (e.g. X interaction vertices, rather like Fermi's old theory of the weak interactions). This is discussed in a surprisingly straightforward way in de Gennes' book on polymers.

You may recall that we live in d = 4 space-time dimensions - i.e. at the critical dimension for many quantum field theories. This funny coincidence results in weak-interaction field theories (e.g. QED and the standard electroweak model) being just easy enough to solve that it is possible to do accurate calculations with pencil and paper, but just hard enough for those calculations to be really impressive.

If we had an extra space dimension, you would not have to worry about the `running coupling constant' or `renormalization' of QED because perturbation theory would be highly accurate. If we had one fewer space dimension, QED would be a theory with phase transitions in it (i.e. more like QCD), and would have to be solved numerically and (and on flat computers).

Finally, somewhat similar things happen in the theory of quantum strings: in some currently popular theories d = 11 is analogous to d = 4 for usual field theory, which has motivated string theorists to assert that space-time actually has 11 dimensions. Experimenters are then left with the question of where the other 7 dimensions are.

Minimizing this free energy with respect to R gives us the equation

This problem provides a way to start thinking about charged flexible polymers or polyelectrolytes, which leads quickly to current, open research problems.

In reality there are always ions around which screen electrostatic interactions. So one prevailing ideology is that electrostatic interactions stretch polymers at short scales (i.e. smaller than the Debye screening length l_D, but that long polymers should be flexible at longer length scales.

There are even more complications. Usually electrically charged polymers have a mixture of hydrophilic (the charges) and hydrophobic patches along them. ssDNA is a good example - the bases are to some degree hydrophobic, preferring to stack and to form hydrogen bonds among them rather than disrupting the water hydrogen-bonding, while the sugar-phosphate backbone is really well soluble in water. This, mixed with the complications of screening of electrostatic interactions, makes ssDNA (a highly charged flexible polymer) have a complicated, fluctuating and poorly-understood secondary-structure.

5'-aggtcgccgccc-3'
3'-tccagcggcggg-5'

Running down the top strand from 5' to 3' we have the 11 nucleotide pairs, and DH, DS and DGs:

ag  7.8 20.8 1.6
gg 11.0 26.6 3.1
gt  6.5 17.3 1.3
tc  5.6 13.5 1.6
cg 11.9 27.8 3.6
gc 11.1 26.7 3.1
cc 11.0 26.6 3.1
cg 11.9 27.8 3.6
gc 11.1 26.7 3.1
cc 11.0 26.6 3.1
cc 11.0 26.6 3.1

Note that the energies come from Table 2 of Breslauer. Now all we do is add up the DH and DS values:

DH = 7.8 + 4 ×11.0 + 6.5 + 5.6 + 2 ×11.1 + 2×11.9 = 109.9 kcal/(mol bp)

DS = 20.8 + 4×26.6 + 17.3 + 13.5 + 2 ×27.8 + 2×26.7 = 267 cal/(mol K) = 0.267 kcal/(mol bp K)

Breslauer's paper also implies that we should add a `helix initiation entropy' of (5/297) kcal/(mol bp K) for GC-rich DNA, giving a net DS = 0.284 kcal/(mol bp K).

Therefore the total free energy is

DG = DH - T DS = 109.9 - 0.284 T    kcal/(mol bp)

where T is in Kelvin.

The melting temperature, when DG = 0, is therefore T_m = 109.9/0.284 = 386 K, or around 113 C.

This is a pretty high melting temperature because the data of Breslauer et al refer to 1 M NaCl concentration, where the double helix is very stable.

As you can see, the probability for unzipping rapidly goes to zero.

(b) A simple estimate for the time waited for spontaneous dissociation would be the inverse of the relative probability calculated above (i.e. the number of random states you'd have to look at to find one dissociated state), times t₀, or t = t₀ e^3N. This is plotted below:

As you can see, the lifetime goes up rapidly with base-pairs, passing one year (note 1 year = 3.12x10⁷ sec) at around 12 bp.

The models presented in this section are VERY rough, but they serve to illustrate that a bunch of weak interactions put in parallel can make a self-assembled structure very stable.

(b) If we suppose that n base pairs are unzipped, and the ends of the ssDNA pulled to be a distance x apart, the free energy will be the sum of the stretching free energy of a polymer 2n segments long stretched to an extension x, and the energy cost of unzipping the n base-pairs:

Something we didn't ask for but should have is the Flory theory's critical force for unzipping. This follows from the stretching of the polymer as it unzips, which is

Experiments (e.g. Essevaz-Roulez et al, PNAS 94, 11935, 1997) indicate a value between 10 and 15 pN dependent on sequence, with x µ n. Some people dream of sequencing DNA by unzipping it, and observing the variation of force.

One caveat is that the simple flexible polymer model used here for ssDNA is an oversimplification. But the simple model described here is still very useful in thinking about what is going on.