Past exam of the genetics course of the University of Cambridge / 2024 / Paper 1 / 7 / c

At fixed volume and temperature, equilibrium minimizes the Helmholtz free energy $F=U-TS$. Mixing increases translational entropy, whereas favourable contacts between chromatin segments can lower the interaction contribution to internal energy. At low enough temperature, favourable contacts can compensate the entropy lost on concentrating the arrays; at higher temperature, the mixing contribution can dominate. This gives the upper-critical dome assumed in (a).

A simple illustrative Flory–Huggins solution theory writes the dimensionless mixing free energy per site asHere $N$ is an effective polymer size, not automatically the number 12, and $A>0$ represents an effective energetic incompatibility of polymer and solvent. The common-tangent construction determines coexisting compositions. Solving $g^{\prime \prime }=g^{(3)}=0$ givesSince $\chi$ decreases on heating, the coexistence interval narrows and closes at $T_c=A/{\chi }_c$. For fixed pressure the corresponding potential is Gibbs free energy, with the enthalpic term in $G=H-TS$; enthalpy and internal energy should not be interchanged without stating the mechanical constraint. Hydration, ion release or temperature-dependent effective interactions can change this simple temperature trend.