Past exam of the quantum condensed matter course of the University of Cambridge / 2006 / Section A / A1

A protein is synthesized as a polypeptide whose sequence constrains its possible interactions. Protein folding is the acquisition of a functional three-dimensional structure: local hydrogen bonds organize secondary structure, while packing, electrostatic interactions and burial of nonpolar groups through the hydrophobic effect stabilize the overall fold. Some proteins also form protein disulfide bonds. Folding reduces the chain's conformational entropy, but the relevant balance includes the entropy of the surrounding water; it is the total Gibbs free energy that is minimized at fixed pressure and temperature.

A folding funnel describes a biased but rugged energy landscape: increasingly native-like conformations become favorable, while barriers can trap intermediates. The chain therefore does not have to test every possible conformation in a uniform random search. Molecular chaperones can prevent protein aggregation and help escape kinetic traps; some proteins remain disordered or require partners, so a unique isolated compact fold is not universal.

The ribonuclease A refolding experiment connects this thermodynamic account to a measurement. Unfolding the enzyme with urea while reducing its disulfide bonds removes activity; suitable removal of the denaturant and restoration of disulfide bonds recover activity. Oxidizing while the chain remains unfolded instead produces incorrectly paired disulfides and little activity. Thus sequence, solvent conditions and the route past kinetic traps all matter; restoration of enzyme activity provides a functional test of folding.

An amphiphile contains a solvent-compatible head and a solvent-avoiding tail. In water, aggregation can lower the Gibbs free energy by reducing exposed hydrocarbon area through the hydrophobic effect. This competes with loss of translational entropy, head-group repulsion and the cost of deforming the tails. Above the critical micelle concentration, added amphiphiles mainly enter aggregates rather than continuing to increase the isolated-molecule concentration proportionally.

The amphiphile packing parameter $P=v/(a_0{\ell }_c)$ compares tail volume $v$, preferred head area $a_0$ and tail length ${\ell }_c$. Rough geometric ranges favor spherical micelles for $P<1/3$, cylindrical aggregates for $1/3<P<1/2$, and lipid bilayers for $1/2<P<1$. These are organizing estimates rather than universal sharp phase boundaries. A bilayer can close into a vesicle to eliminate exposed edges; concentration, salt and temperature can change the preferred aggregate.

A simple experimental test varies concentration and measures surface tension: its slope against log concentration changes near the critical micelle concentration, while light scattering detects the appearance of larger aggregates. These observations distinguish the concentration threshold from an assumption that every molecule forms an aggregate at arbitrarily low concentration.

The fluctuation-dissipation theorem relates spontaneous fluctuations in thermal equilibrium to the dissipative part of the linear response to a weak conjugate force. For an overdamped trapped coordinate, the Langevin equation isThe fluctuation-dissipation relation for a Langevin particle fixes the noise strength from the same drag coefficient $\gamma$ that dissipates motion. With a drive proportional to $e^{-i\omega t}$, the linear response function is $\chi (\omega )=1/(K-i\gamma \omega )$. The two-sided angular-frequency power spectral density then obeysIts integral $\int S_{xx}d\omega /(2\pi )=k_{B}T/K$ agrees with the equipartition theorem. Without a restoring force, the same noise-drag relation gives the diffusion-mobility relation $D=k_{B}T/\gamma$.

In thermal calibration of an optical trap, record a bead's spontaneous displacement fluctuations and compare their spectrum with its response to a weak calibrated oscillatory force. The overdamped harmonic-trap power spectrum and the dissipative response must be consistent with the same temperature and drag. The equilibrium and weak-drive conditions are essential: an actively driven bath need not obey this equilibrium relation.

A liquid crystal has some molecular order while retaining liquid-like motion. The distinction between orientational and positional order separates its main phases.

A nematic liquid crystal has long-range orientational order about a nematic director $\mathbf{n}$, with $\mathbf{n}$ and $-\mathbf{n}$ equivalent for apolar molecules, but no periodic density lattice. Its nematic order parameter is $S=⟨(3{\cos }^2\theta -1)/2⟩$, where $\theta$ measures molecular alignment with the director. A cholesteric liquid crystal is a chiral nematic whose director rotates in space, producing a helical pitch while remaining locally nematic.

A smectic phase adds a periodic density modulation in one direction, forming layers with liquid-like motion within each layer. In a smectic A phase, the director is perpendicular to the layers; in a smectic C phase, it is tilted relative to the layer normal. A columnar liquid crystal instead has columns with a two-dimensional positional lattice and comparatively fluid motion along them.

These orders have different experimental signatures. Birefringence under crossed polarizers reveals orientational anisotropy, while X-ray diffraction can detect a smectic layer spacing $d$ through a peak near $q=2\pi /d$. Optical anisotropy alone does not establish a layer lattice.