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.

7. Coil-Globule Transition and Folding¶

What this covers¶

This chapter finishes what the previous one set up: a mean-field trial free energy for a polymer interacting with itself and its solvent (Eq. 2.49). It asks what shape that free energy predicts in a good solvent versus a bad one, and then pushes further — once the polymer is a heteropolymer with many chemically distinct monomers, like a protein, can it freeze into one preferred compact shape? Answering that needs a new tool, the random energy model, and a "designed" version of it that resolves a real puzzle: the plain model predicts folding as slow as glass formation, while real proteins fold fast. Assumes the entropic-elasticity result for an ideal chain, the excluded-volume expansion, and the Flory–Huggins parameter $\chi$, all from the previous chapter's construction of $\ln Z(N,R)$.

Where the last chapter left off¶

The trial free energy for $N$ monomers of size $a$ and persistence length $\xi_p$ confined to a sphere of radius $R$ was (Eq. 2.49)

\[\ln Z(N,R) = N\ln g - \frac32\ln\!\Big[\frac{4\pi Na\xi_p}{3}\Big] - \frac{3R^2}{4Na\xi_p} - \frac{N^2}{2}\Big(\frac aR\Big)^3 - \frac{N^3}{6}\Big(\frac aR\Big)^6 - \cdots + \chi N^2\Big(\frac aR\Big)^3,\]

where $g$ sets the growth rate of the number of configurations of the free chain and $\chi$ is the Flory–Huggins parameter: positive $\chi$ favours monomer–monomer contact over monomer–solvent contact. Every term here has a definite sign except one combination: the excluded-volume repulsion and the $\chi$-driven attraction sit inside the same power of $(a/R)$ and combine as $-\tfrac12(1-2\chi)N^2(a/R)^3$. Repulsion and attraction compete inside this single term, and its sign — set by whether $\chi$ is above or below $1/2$ — decides between two qualitatively different equilibrium shapes. That competition is the subject of this chapter.

Good solvent: the swollen coil¶

For $\chi<1/2$ the repulsive part dominates, and the only thing opposing indefinite swelling is the entropic cost of stretching, the spring term $-3R^2/(4Na\xi_p)$. Keeping just these two competing terms — one can check afterwards that the terms dropped really are smaller in this regime — gives

\[\ln Z(N,R) = \text{const} - \frac{3R^2}{4Na\xi_p} - \frac{1-2\chi}{2}N^2\Big(\frac aR\Big)^3 + \cdots \tag{2.50}\]

Extremising over $R$,

\[\frac{\partial\ln Z}{\partial R} = -\frac{3R}{2Na\xi_p} + \frac{3(1-2\chi)}{2}N^2\frac{a^3}{R^4} = 0 \;\Longrightarrow\; \overline R^{\,5} = (1-2\chi)a^4\xi_pN^3,\]

\[\overline R = (1-2\chi)^{1/5}(a^4\xi_p)^{1/5}N^{3/5}. \tag{2.51}\]

Compare this with the non-interacting chain, where $\overline{R^2}\approx 2Na\xi_p$, i.e. $\overline R\propto N^{1/2}$ (previous chapter, Eq. 2.39). Excluded volume does not just rescale the prefactor — it changes the exponent. Writing $R\propto N^\nu$, the ideal chain has $\nu=1/2$; the swollen, self-avoiding coil has $\nu=3/5$ in this variational treatment, the Flory exponent. A larger exponent means each added monomer buys more volume: a self-avoiding chain fills space more generously than a random walk that is free to cross itself.

Going beyond this mean-field argument is genuinely hard, but one of the successes of renormalization group theory is an essentially exact value, $\nu = 0.591\ldots$, remarkably close to the crude variational estimate of $3/5$.

The exponent in other dimensions¶

The same extremisation goes through for a self-avoiding walk confined to $d$ spatial dimensions — e.g. a polymer stuck to a $d=2$ surface — dropping the attractive part of the interaction and keeping only the repulsive core. The free energy of Eq. (2.50) generalises to

\[\ln Z(N,R) = \text{const} - \frac{dR^2}{4Na\xi_p} - \frac{N^2}{2}\Big(\frac aR\Big)^d, \tag{2.52}\]

and extremising exactly as before gives

\[\overline R = \big(a^{d+1}\xi_p\big)^{\frac1{d+2}}N^{\frac3{d+2}}, \qquad \nu_F(d) = \frac{3}{d+2}. \tag{2.53}\]

This generalised Flory exponent is exact at $d=1,2,4$, giving $\nu=1,\,3/4,\,1/2$ respectively — an unusual case of a mean-field estimate landing exactly right outside its naive regime of validity. Above four dimensions the excluded-volume constraint stops mattering altogether and $\nu$ stays pinned at the ideal-chain value $1/2$: in high enough dimension two strands of the same walk essentially never meet anyway.

Bad solvent: collapse at the theta point¶

Lowering the temperature typically increases $\chi(T)$ — the solvent loses less by staying pure than the monomers gain by aggregating. The sign of $(1-2\chi)$ flips at the theta point, $\chi(\theta)=1/2$, and for $T<\theta$ attraction wins: the polymer collapses into a compact globule with a finite monomer density $\rho = N(a/R)^3$, rather than the vanishing density of the swollen coil. Keeping terms in $\rho$ to the order needed, the free energy per monomer is

\[-\frac{\ln Z(\rho)}{N} = -\ln g + \frac{1-2\chi}{2}\rho + \frac{\rho^2}{6} + \cdots \tag{2.54}\]

Minimising over $\rho$,

\[-\frac1N\frac{d\ln Z}{d\rho} = \Big(\frac12-\chi\Big) + \frac{\overline\rho}{3} + \cdots = 0 \;\Longrightarrow\; \overline\rho = 3\Big(\chi-\frac12\Big) + \cdots, \tag{2.55}\]

so the equilibrium density grows linearly from zero just below the theta temperature (the higher-order terms in $\rho$ are what eventually saturate this growth, since $\rho$ cannot exceed 1 — a fully packed globule).

The attractive interactions contribute an energy per monomer

\[\frac{E_{\text{att.}}}{N} = -\overline\rho\,\chi\,k_BT, \tag{2.56}\]

using the previous chapter's $-\beta E_{\text{att.}} = \chi N^2(a/R)^3$, and the entropy per monomer follows from $S = (E-F)/T$:

\[\frac{S}{Nk_B} = \frac{\ln Z}{N} - \overline\rho\,\chi = \ln g + \frac32\Big(\chi-\frac12\Big)^2 + \cdots - \overline\rho\,\chi \;\approx\; \ln g - \frac{\overline\rho}{2} + \mathcal O(\overline\rho^2), \tag{2.57}\]

keeping only the leading term as $\chi\to1/2$. Entropy is reduced, at first linearly, as the globule condenses — exactly the mean-field picture of a gas condensing into a liquid, where the liquid retains many configurations, just fewer than the gas did. Cooling an ordinary liquid further eventually freezes it into a low-entropy solid. That analogy is what motivates the rest of the chapter: is there a freezing transition for a compact polymer too, and — since a protein is a heteropolymer — can it freeze into one particular shape rather than a disordered solid?

Compact heteropolymers and the random energy model¶

Deep in the globular phase, the accessible configurations are the maximally compact ones. On a lattice these are Hamiltonian walks — self-avoiding walks that visit every site, leaving none empty. Their number also grows exponentially, as $g'^N$, but $g'\ll g$: full compactness is a much more severe constraint than mere self-avoidance.

For a homopolymer every Hamiltonian walk has the same energy, so all $g'^N$ compact configurations are equally likely. A heteropolymer — built from chemically distinct monomers, as a protein is from its amino acids — breaks that symmetry: different folds bring different pairs of monomers into contact, so different compact configurations $\alpha$ carry different energies,

\[E_\alpha = \sum_{\langle ab\rangle} V_{ab}, \tag{2.58}\]

summed over non-covalent nearest-neighbour contacts, giving a partition function

\[Z = \sum_\alpha e^{-\beta E_\alpha} \tag{2.59}\]

over the $g'^N$ compact states. Real biomolecules combine non-specific attractions that pull all monomers together (driving the collapse of the previous section) with specific interactions that could in principle select one particular compact shape — the question is whether, and at what temperature, they actually do.

Summing over $g'^N$ configurations of a fixed, specific sequence is intractable directly. The random energy model (REM) makes it tractable with a drastic simplification: treat the bond energies $V_{ab}$ as independent random variables. Each $E_\alpha$ is then a sum of $N_B = (z-1)N$ such terms ($z$ the lattice coordination number, minus the one bond that is polymeric), so for large $N$ it is approximately Gaussian, with

\[\langle E_\alpha\rangle = N_B\langle V_{ab}\rangle \equiv N\varepsilon_0, \qquad \langle E_\alpha^2\rangle_c = N_B\langle V_{ab}^2\rangle_c \equiv N\sigma^2, \tag{2.60}\]

\[p(E) = \frac{1}{\sqrt{2\pi N\sigma^2}}\exp\Big[-\frac{(E-N\varepsilon_0)^2}{2N\sigma^2}\Big]. \tag{2.61}\]

The REM's key idealisation is treating the energies of different configurations as statistically independent of one another — it throws away any correlation between similar folds — which turns the counting problem into an extreme-value problem in disguise.

The REM's entropy, and its glass transition¶

The density of states is the number of configurations, $g'^N$, times the probability that a given one lands at energy $E$, so $\Omega(E) = g'^Np(E)$, and the entropy follows:

\[S(E) = k_B\ln\Omega(E) = k_B\Big[N\ln g' - \frac{(E-N\varepsilon_0)^2}{2N\sigma^2}\Big] - \frac{k_B}{2}\ln(2\pi N\sigma^2). \tag{2.62}\]

The last term is sub-extensive and can be dropped. What remains is an inverted parabola in $E$, but two thermodynamic constraints cut down how much of it is physical:

Positive temperature. $T^{-1} = dS/dE$ must be positive, restricting attention to $E < N\varepsilon_0$, the branch below the mean energy.
Non-negative entropy. $S(E)$ cannot go below zero. The parabola would formally go negative below some energy; instead the true entropy sticks at $S=0$ for $E<E_c$, where

\[S(E_c) = 0 \;\Longrightarrow\; \frac{E_c}{N} = \varepsilon_0 - \sigma\sqrt{2\ln g'}. \tag{2.63}\]

($E_c$ is exactly the extreme-value answer for the mean of the lowest of $g'^N$ independent draws from $p(E)$ — the connection to extreme-value statistics referred to, but not re-derived, here.)

Below $E_c$ there are essentially no compact states left in a typical sequence — not a continuum with small but positive entropy, but a genuine cliff, and a kink in the entropy signals a phase transition. Its temperature is read off from the slope of the parabola exactly at $E_c$:

\[\frac1{T_c} = \frac{dS}{dE}\Big|_{E_c} = k_B\frac{\sqrt{2\ln g'}}{\sigma} \;\Longrightarrow\; k_BT_c = \frac{\sigma}{\sqrt{2\ln g'}}. \tag{2.64}\]

For $T\le T_c$ the system is confined to the handful of states near $E_c$: a transition into a glass — not a unique ground state, but a small, disordered pool of nearly-degenerate low-energy configurations that the chain freezes into.

Why real proteins are not typical random heteropolymers¶

It is tempting to identify this glass transition with protein folding, but the REM's transition has the wrong character for that. As $T\to T_c^+$ the number of accessible states collapses drastically, and the handful of states surviving near $E_c$ have no particular reason to resemble one another — reaching a different one typically means rearranging many monomers at once, climbing a large energy barrier in the process. So the REM polymer's kinetics should slow down sharply on approaching $T_c$, the signature of an ordinary structural glass. Real proteins do the opposite: most fold quickly and reliably, on laboratory rather than glassy timescales.

The resolution is that proteins are not typical random heteropolymers: evolution has "designed" real sequences both for function and, apparently, for ease of folding. The REM can mimic this design with one small addition — alongside the Gaussian continuum of $g'^N$ random compact states, add a single distinguished state at energy $E_n$, with $E_n < E_c$, representing the native fold.

Designed REM: the tangent construction¶

With the native state added, the system has somewhere better than the glass to go: a folding transition to this state at a temperature $T_f$ that sits above $T_c$ — high enough that there are still many competing states left to explore on the way there, avoiding the glassy kinetics of the previous section.

$T_f$, and the energy $E_f$ of the competing continuum states right at the transition, are pinned down by equating the free energy of the native state to that of the continuum. Geometrically this is a common-tangent construction on the entropy curve: a straight line from the point $(E_n,0)$ (a single state has zero entropy, $\ln 1 = 0$) drawn tangent to the curve $S(E)$ at $(E_f, S(E_f))$:

\[\beta_f = \frac{S(E_f)/k_B}{E_f - E_n} = \frac{N\ln g' - (E_f - N\varepsilon_0)^2/(2N\sigma^2)}{E_f - E_n}. \tag{2.65}\]

Eₙ Ec Ef Nε₀ S(E)

The REM entropy is zero for $E

To see why this tangent condition is the right one, work in the canonical ensemble. The probability of finding the system in the native state is $$p_n = \frac{e^{-\beta E_n}}{Z(\beta)}, \qquad Z(\beta) = e^{-\beta E_n} + \int dE\,\Omega(E)e^{-\beta E}. \tag{2.66}$$ In the thermodynamic limit ($N\to\infty$, with $E_n\propto N$ so the model has a well-defined extensive limit), the integral is dominated by a single saddle-point energy. That saddle sits above $E_f$ for $\beta\le\beta_f$ — the continuum wins — and switches to $E=E_n$ for $\beta>\beta_f$ — the native state wins. The switch is a genuine discontinuity, $p_n$ jumping from 0 to 1, occurring exactly where the two contributions to $Z$ are equal: $$e^{-\beta_fE_n} = \Omega(E_f)e^{-\beta_fE_f}, \tag{2.67}$$ which after taking logarithms is precisely the tangent condition, Eq. (2.65). Solving for $\beta_f$ needs two more relations: the saddle-point energy at inverse temperature $\beta$, from $dS/dE = k_B\beta$ applied to the parabola, is $E(\beta) = N\varepsilon_0 - N\sigma^2\beta$, and Eq. (2.64) can be rewritten $\ln g' = (\beta_c\sigma)^2/2$. Writing $\beta_n \equiv (N\varepsilon_0 - E_n)/(N\sigma^2)$ for the same construction applied formally to the native energy — the reconstructed slide prints this definition with the opposite sign, $(E_n - N\varepsilon_0)/(N\sigma^2)$, but only the sign used here is consistent with Eq. (2.64)'s convention and reproduces Eqs. (2.69)–(2.70) below, which is the check available without a transcript or the original PDF to compare against — Eq. (2.65) reduces to a quadratic in $\beta_f$ (a rearrangement of Eq. 2.68), $$\beta_f^2 - 2\beta_n\beta_f + \beta_c^2 = 0 \;\Longrightarrow\; \beta_f = \beta_n - \sqrt{\beta_n^2 - \beta_c^2}, \tag{2.69}$$ and hence $$\frac{T_f}{T_c} = \frac{\beta_c}{\beta_f} = \frac{\beta_n}{\beta_c} + \sqrt{\Big(\frac{\beta_n}{\beta_c}\Big)^2 - 1}. \tag{2.70}$$ Since $E_n < E_c$ makes $\beta_n>\beta_c$, this ratio is always $\ge 1$: the folding temperature sits at or above the glass temperature, and the gap between them grows with $\beta_n$ — that is, with how far below $E_c$ the designed native energy sits. Push the native energy far enough below the glassy states and folding happens at a comfortably high temperature, where the continuum of unfolded states is still large and easy to explore, rather than at the sluggish glass transition itself. ## Sources - Slides: `lectures/11-slides/01-2-3-2-swollen-coil-polymers-in-good-solvents.md` through `04-2-3-5-designed-rem-for-protein-folding.md` (course `computational-biology/mit-ocw/8592j`, MIT 8.592J/HST.452J *Statistical Physics in Biology*, Spring 2011, MIT OpenCourseWare), covering textbook sections 2.3.2–2.3.5. These files are flagged in the library as reconstructed by a model from a PDF with no text layer, with "every equation... unverified" — treat the displayed equations here as a pointer back to the original PDF rather than a checked transcription. One internal inconsistency was found and resolved by hand: the printed sign of $\beta_n$ in slide 04's Eq. (2.68) does not reproduce the printed Eqs. (2.69)–(2.70); the opposite sign, used above, does. - Continues directly from the previous chapter's slides (`lectures/10-slides/01-2-2-fluctuating-polymers.md`, `02-2-3-interacting-polymers.md`, same course): the entropic-elasticity result $\overline{R^2}\approx 2Na\xi_p$ (Eq. 2.39), the end-to-end distribution (Eq. 2.40), and the trial free energy Eq. (2.49) are used here without re-derivation. - No transcript, written notes, or exercises were supplied for this lecture; nothing in this chapter is drawn from spoken commentary, and no exercises are given. - The slides refer to, but do not contain, "the extreme value problem studied earlier" (in connection with Eq. 2.63) and the renormalization-group calculation behind the exact value $\nu=0.591\ldots$ of the Flory exponent; neither is part of the supplied material. --- [← 6. Fluctuating and Interacting Polymers](06-fluctuating-and-interacting-polymers.md) · [Contents](index.md) · [8. The Poland-Scheraga Melting Model →](08-the-poland-scheraga-melting-model.md)