Lecture notes. Written from the material of MIT 8.592J, licensed CC BY-NC-SA 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.

8. The Poland-Scheraga Melting Model¶

What this covers¶

Why does a DNA molecule, packed into a cell many orders of magnitude smaller than its own contour length, come apart smoothly when heated in some cases and snap apart in others? This chapter treats DNA as a polymer subject to two competing statistical-mechanical effects — the pairing energy of the Watson–Crick bonds and the entropy gained by letting a strand flop around once it is freed of its partner — and asks how those two effects, together with the geometry of closing a loop, decide whether "melting" the double helix is a gentle crossover or a sharp phase transition. It assumes the polymer-physics vocabulary built up earlier in the course: a chain as a random walk with a persistence length, radius of gyration scaling, and the excluded-volume (self-avoiding walk) exponent $\nu$.

DNA in the cell: scale and packaging¶

DNA spans an enormous range of length scales: a $\lambda$-phage genome is about 50,000 monomers, the human genome is $6\times10^9$, and the lily genome runs to $9\times10^{10}$ nucleotides — which, stretched out, would be around thirty metres long. Treated as a random, non-self-avoiding chain with persistence length $\xi_p \approx 50\ \text{nm}$, a chain of contour length $L$ has a typical size $R_g \sim \sqrt{L\,\xi_p}$; for the human genome this comes to roughly $0.2\ \text{mm}$, and excluded-volume effects only push that further out. All of these sizes dwarf a cell, which is why DNA cannot simply sit inside one as an unconstrained coil: eukaryotes compactify it by wrapping the chain around histone proteins to form nucleosomes, which are then packed together in turn.

Melting the double helix¶

At the microscopic level the two strands are held together by Watson–Crick base pairs: G–C pairs, with a binding energy of around $4k_BT$, bind roughly twice as strongly as A–T pairs. That binding energy competes with an entropy cost — separating the two strands frees them to explore many more configurations than they have while braided together — and the competition is temperature-dependent in the usual way: at around $80^\circ\text{C}$ the entropy term wins enough of the free energy that the double strand starts to unravel, or denature ("melt"), opening into bubbles where the two strands have come apart locally. Because A–T pairs are weaker, A–T-rich regions open up at lower temperature than G–C-rich regions. This shows up experimentally in ultraviolet absorbance versus temperature: a short DNA molecule, with only a few possible melting domains, shows distinct blips as each domain opens in turn; a very long, compositionally heterogeneous molecule has so many overlapping melting events that the same measurement looks like one continuous curve. Software exists that predicts, for a given sequence, how it will unravel as temperature is raised, by computing a free energy from a model of the pairing energies (for instance summing stacking energies between successive base pairs) together with the entropy cost of opening a bubble of a given size.

Loop-closure entropy¶

That entropy cost is observed to depend on the length $l$ of a denatured segment as

\[S(l) \approx bl + c\log l + d, \qquad c \approx 1.8\,k_B. \tag{2.71}\]

The extensive term $bl$ is just the number of configurations a flexible single strand gains per added monomer; the logarithmic correction is the interesting piece, and it comes from loop closure: the two single strands of a bubble of length $l$ each must still meet the double helix at the same point in space at both ends, so the pair of them behaves like a closed loop of total length $2l$, and closing a loop costs more than just being a free chain.

This can be made quantitative by first working out $\Omega(l)$, the number of configurations of a bubble, for a non-self-avoiding (Gaussian) chain in $d$ spatial dimensions. Split the loop into two independent strands of length $l$ that must arrive at the same displacement $\vec r$ from where they started. If $W(\vec r,l)$ is the number of $l$-step configurations ending at $\vec r$, the loop of total length $2l$, closed at a definite point $\vec r$, is counted by the product of the two halves,

\[W(\vec r, 2l) = W(\vec r, l)^2 = g_1^{2l}\exp\left[-\frac{dr^2}{2l\xi_p}\right] \frac{1}{(4\pi l\xi_p/d)^{d}}, \tag{2.72}\]

where $g_1$ counts the configurations available per monomer to one flexible strand. Since the bubble can close at any point in space, integrating over all positions of that meeting point gives

\[\Omega(l) = \int d^dr\, W(\vec r, 2l) = \left(\frac{d}{8\pi\xi_p}\right)^{d/2}\frac{g^l}{l^{c}}, \qquad g = g_1^2,\quad c = \frac{d}{2}. \tag{2.73}\]

So for an ideal chain the loop-closure exponent is simply $c=d/2$ — in three dimensions, $c=1.5$.

Real single strands are not ideal chains: they cannot cross themselves, so the relevant walk is self-avoiding. Scaling arguments replace the Gaussian form with

\[W(\vec r, 2l) = \frac{g^l}{R^d}\,\Phi\!\left(\frac{\vec r}{R}\right), \qquad R\sim l^{\nu}, \qquad \Omega(l) = \frac{g^l}{l^{d\nu}}, \tag{2.74}\]

so that now $c = d\nu$. The exponent has an intuitive reading: with no closure constraint the free end could sit anywhere in a volume $\sim R^d \propto l^{d\nu}$, and demanding that it come back to meet the other end removes exactly that many choices, which is why $\Omega(l)$ picks up a factor of $1/l^{d\nu}$ on top of the per-monomer weight $g^l$. In three dimensions, with the self-avoiding exponent $\nu\approx0.588$, this gives $c=d\nu\approx1.76$ — close to the $c\approx1.8$ quoted from melting-curve measurements above, which is a sign that a melted DNA strand behaves like a self-avoiding chain rather than an ideal one, consistent with the excluded-volume effects already mentioned. The ideal-chain result is the special case $\nu=1/2$.

The Poland–Scheraga picture: a chain as an alternating sequence of paired double-helix rods, each contributing a weight $w^{l}$ per unit length, and open single-stranded bubbles, each contributing $\Omega(l)\propto g^{l}/l^{c}$.

The Poland–Scheraga model¶

The Poland–Scheraga model treats a partly-melted DNA molecule exactly as the picture above suggests: a sequence of alternating rods (still-paired double helix) and bubbles (open single strands), each segment contributing its own statistical weight. A rod of length $l$ is energetically favoured and has little entropy of its own, so it contributes a Boltzmann weight

\[w^l, \qquad w = e^{-\beta\epsilon} > 1 \quad (\epsilon < 0),\]

where $\epsilon$ is the (negative) pairing energy per base pair. A bubble of length $l$ is entropically favoured instead, contributing the weight worked out above,

\[\Omega(l) \propto \frac{g^l}{l^{c}}, \qquad g>1,\]

with $g$ counting the configurations available to the flexible open strands and $c$ the loop-closure exponent. The full partition function of an $N$-monomer chain is built by threading these two kinds of segment along the contour in every possible way their lengths can sum to $N$ — a renewal-type sum that is naturally handled with a generating function in a fugacity $z$ conjugate to segment length. That algebra is not part of the material available for this chapter (see Sources); what follows is the result it leads to.

The order of the melting transition¶

The rod weight $w^l$, summed over $l$, has its own generating-function singularity at $z=1/w$; the bubble weight, summed as $\Sigma(z) = \sum_l (gz)^l/l^{c}$, is controlled by the fixed point $z=1/g$. As temperature rises, $\beta$ falls and so $w=e^{-\beta\epsilon}$ falls toward $1$, meaning $1/w$ rises toward $1/g$. Whichever singularity sits closer to the origin dominates the chain's free energy: below the temperature where $w=g$ the rod dominates and the chain is mostly paired; above it the bubble sector takes over and the chain unravels. Whether the handover at $w=g$ is a smooth crossover, a continuous transition, or an abrupt jump depends entirely on how $\Sigma(z)$ behaves as $z\to(1/g)^-$ — which is exactly the question of whether $\sum_l l^{-c}$ converges, i.e. on the value of the loop exponent $c$:

$c\le1$. $\Sigma(z)\to\infty$ as $z\to1/g$ (the harmonic-type sum diverges), so the bubble sector never "runs out" of weight to hand over gradually. There is no genuine singularity in the free energy at all — melting is a smooth crossover with no sharp temperature.
$1<c<2$. $\Sigma(1/g)=\zeta(c)$ is now finite (the sum converges), but its derivative in $z$ still diverges. The notes mark $\zeta(3/2)\approx2.612$ — the same constant that sets the condensation threshold of an ideal Bose gas, the same mathematics appearing here for the same reason: a sum over a power law that just barely converges. This is enough for a genuine, continuous phase transition at a melting temperature $T_m$ (given by $w(T_m)=g$), with the fraction of still-paired bases $\theta$ vanishing as $$\theta \propto (T_m - T)^{\beta}, \qquad \beta = \frac{2-c}{c-1}.$$ The exponent is non-universal: it depends continuously on $c$ rather than being fixed once and for all.
$c>2$. Now even $\sum_l l^{-(c-1)}$ converges, so $\Sigma(z)$ and its derivative are both finite at $z=1/g$ — nothing in the bubble sector is singular anywhere near the transition. With no singular behaviour available to soften the handover, the transition is forced to be first order: $\theta$ drops discontinuously at $T_m$.

How the paired-base fraction θ falls with temperature depends on the loop exponent c: a smooth crossover for c ≤ 1, a continuous transition θ ∝ (T_m − T)^β for 1 < c < 2, and a discontinuous jump for c > 2.

Putting a number on this with the values already in hand: the marked example $c=3/2$ (the ideal, non-self-avoiding value in three dimensions) gives $\beta=(2-1.5)/(1.5-1)=1$, a transition that vanishes linearly. The measured, self-avoiding value $c\approx1.8$ instead gives $\beta=(2-1.8)/(1.8-1)=0.25$ — still, strictly, a continuous transition, but a much steeper one, close to the $c=2$ boundary beyond which it would become a discontinuous jump. Real DNA melting therefore sits in the continuous regime, but only just, which is part of why melting curves for long DNA can look almost as sharp as a true first-order transition even though they are not one.

Sources¶

Body text, Eqs. (2.71)–(2.74), and the section heading "2.4.1 The Poland–Scheraga model for DNA Denaturation" are from lectures/16-slides.md (course computational-biology/mit-ocw/8592j, lecture 16 slide file, "2.4 DNA structure" — MIT OCW 8.592J, Statistical Physics in Biology, Spring 2011, CC BY-NC-SA 4.0). That file is itself a model's reconstruction of a scanned PDF with no text layer and carries its own warning that the prose is paraphrase and every equation is unverified; the extracted text breaks off mid-sentence ("In practice, the crossover in behavior occurs") right after the Poland–Scheraga heading.
The physical picture of alternating rods and bubbles, the statistical weights $w^l$ and $\Omega(l)\propto g^l/l^c$, and the definitions $g=g_1^2$, and the re-derivation of $c=d/2$ (ideal) and $c=d\nu$ (self-avoiding), are read from the hand-written page 16-slides/figures/p002-1.png — a scan headed "8.592, 3/21/07, Lec. 13 (1)" — cross-checked against Eqs. (2.72)–(2.74) in the slide text, which they match.
The three-regime classification of the melting transition (smooth crossover for $c\le1$, continuous transition with $\beta=(2-c)/(c-1)$ for $1<c<2$, discontinuous jump for $c>2$), the marked constant $\zeta(3/2)\approx2.612$, and the comparison of $T_m$ at $c=1.8$ against $c=3/2$, are read from the hand-written page headed "Lec. 13 (4)", extracted twice as identical images 16-slides/figures/p005-1.png and 16-slides/figures/p006-1.png.
The intermediate pages of that same hand-written derivation — the generating-function sum over rod and bubble segments that connects the weights on page (1) to the graphical results on page (4), which the lecture presumably worked through on pages (2)–(3) — were not extracted into the supplied file and so are not reconstructed here.
No transcript, written notes, or exercises were supplied for this lecture.

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