The Nucleolus Is a Droplet: Compartments Made by Phase Separation — Epoche C2
The textbook answer to how a cell keeps incompatible reactions apart is a membrane. Yet the nucleolus, where ribosomes are assembled, has no membrane, and neither do stress granules, Cajal bodies or germ granules. The question they pose is mechanical rather than descriptive: if nothing encloses them, what holds them together, and what sets their size? The answer that has emerged is that they are not enclosed structures at all but droplets — a dense liquid phase coexisting with a dilute one, as oil coexists with water. This essay traces the chain from the interactions that produce demixing, through the measurements establishing liquidity, to two consequences: a concentration jump with no barrier, and the possibility of ageing into a solid. The cause: multivalency lowers the threshold for demixing Consider a chain of $N$ segments dissolved in a solvent at volume fraction $\phi$. The Flory–Huggins free energy per lattice site counts the entropy of placing chains and solvent on a lattice and adds one mean-field contact term: $$\frac{f}{k_B T} = \frac{\phi}{N}\ln\phi + (1-\phi)\ln(1-\phi) + \chi\,\phi(1-\phi),$$ where $k_B T$ is the thermal energy and $\chi$ is the cost of a chain–solvent contact relative to the average of chain–chain and solvent–solvent contacts. The mixture separates when the curvature of $f$ turns negative; setting the second and third derivatives with respect to $\phi$ to zero locates the critical point at $\phi_c = 1/(1+\sqrt{N})$ and $\chi_c = \tfrac{1}{2}(1 + N^{-1/2})^2$. The dependence on $N$ is the whole point. The entropy resisting demixing is that of the chain's centre of mass, and one chain of $N$ segments carries $N$ times less of it than $N$ free monomers. For $N = 100$ the critical volume fraction is $1/11 = 0.091$ and $\chi_c = (11)^2/200 = 0.605$; for $N = 10^{4}$ they become $1/101 = 0.0099$ and $\chi_c = 0.510$. A highly multivalent molecule thus demixes at a volume fraction of one per cent and at an interaction strength barely above the value at which the solvent is indifferent. This is why condensate scaffolds are always the same kind of molecule: long disordered regions, arrays of repeated binding modules, or RNA, each presenting many weak sites rather than one strong one. The mean-field number should not be trusted too far, and the discrepancy is informative. For a scaffold of molar mass $5\times10^{4}\ \mathrm{g\ mol^{-1}}$ and partial specific volume $0.73\ \mathrm{cm^{3}\,g^{-1}}$, a molar volume of $3.65\times10^{4}\ \mathrm{cm^{3}\,mol^{-1}}$ is about $2\times10^{3}$ water-sized sites, so $N = 2\times10^{3}$ and $\phi_c = 1/(1+\sqrt{N}) = 0.022$, which is $30\ \mathrm{g\,L^{-1}}$, about $590\ \mu\mathrm{M}$. Measured saturation concentrations are frequently near $10\ \mu\mathrm{M}$, some sixty times lower — the gap a mean-field $\chi$ has to absorb, because it smears out an interaction that in reality sits on specific sticker motifs joined by flexible spacers. The gap signals that the attraction is not uniform: specific "sticker" motifs joined by flexible spacers condense far more readily than a smeared-out $\chi$ allows. The evidence: the dense phase behaves mechanically as a liquid Demixing alone would not distinguish a droplet from a precipitate. Three mechanical observations do, and all three were made on germ granules in the nematode embryo. Shape. The bodies are spherical, which means an interfacial tension $\gamma$ is minimising area against thermal agitation. Fusion. Two touching droplets merge and relax back to a sphere. In the inertialess regime the only timescale available comes from balancing the viscous stress $\eta/\tau$ inside the droplet against the capillary stress $\gamma/R$, which gives $\tau \sim (\eta/\gamma)R$ — linear in radius, with the slope $\eta/\gamma$ an inverse capillary velocity. Germ granules give $\eta \approx 1\ \mathrm{Pa\,s}$ and $\gamma \approx 1\ \mu\mathrm{N\,m^{-1}}$, so $\eta/\gamma = 10^{6}\ \mathrm{s\,m^{-1}}$ and a droplet of radius $1\ \mu\mathrm{m}$ rounds in about $1\ \mathrm{s}$. For water, $\eta/\gamma = 10^{-3}/7.2\times10^{-2} = 1.4\times10^{-2}\ \mathrm{s\,m^{-1}}$, and the same droplet rounds in $1.4\times10^{-8}\ \mathrm{s}$: comparing $1\ \mathrm{s}$ with $1.4\times10^{-8}\ \mathrm{s}$ is a gap of nearly eight orders of magnitude. Condensates are liquids, but extraordinarily sluggish ones, which is why fusion is visible at all. Exchange. Bleaching a spot of radius $w$ and watching fluorescence return probes internal mobility, with a recovery time scaling as $w^{2}/D$. For $w = 1\ \mu\mathrm{m}$ and an in-condensate diffusion coefficient of $0.1\ \mu\mathrm{m^{2}\,s^{-1}}$ that is about $10\ \mathrm{s}$: the contents turn over on the timescale of a breath, not of a cell cycle. One caveat. Sphericity and fusion establish that a surface tension exists and that the interior flows during the observation; they do not exclude a soft gel watched for too short a time. The honest criterion is the ratio of the material's own relaxation time $\eta/G$, with $G$ the elastic modulus, to the duration of the experiment. Two consequences of the same physics The functional consequence is a concentration jump without a wall. At coexistence the chemical potential of each component is equal in the two phases, so a client molecule partitions with $K = c_{\mathrm{dense}}/c_{\mathrm{dilute}} = \exp(\Delta\mu/k_B T)$, where $\Delta\mu$ is its transfer free energy. A hundred-fold enrichment therefore requires only $\ln 100 = 4.6\,k_B T$, which at 310 K is $8.314 \times 310 \times 4.605 = 1.19\times10^{4}\ \mathrm{J\,mol^{-1}}$, or about $12\ \mathrm{kJ\,mol^{-1}}$ — roughly one hydrogen bond in water. A cell can therefore concentrate a reactant a hundredfold using interactions weak enough to be reversed by a single phosphorylation, and without paying for a transporter. Coexistence also predicts buffering: by the lever rule, adding scaffold should enlarge the dense phase while leaving the dilute concentration pinned at s