Isotropic-nematic phase transition in amyloid fibrilization

We carry out a theoretical study on the isotropic-nematic phase transition and phase separation in amyloid fibril solutions. Borrowing the thermodynamic model employed in the study of cylindrical micelles, we investigate the variations in the fibril length distribution and phase behavior with respect to changes in the protein concentration, fibril’s rigidity, and binding energy. We then relate our theoretical findings to the nematic ordering observed in Hen Lysozyme fibril solution.

💡 Research Summary

The paper presents a comprehensive theoretical investigation of the isotropic‑nematic (I‑N) phase transition and phase separation in solutions of amyloid fibrils. By adapting the thermodynamic framework originally developed for cylindrical micelles, the authors treat each fibril as a rigid, rod‑like cylinder whose length can vary continuously. The free‑energy functional is written separately for the isotropic phase, where fibrils are randomly oriented, and the nematic phase, where a common director emerges. In the isotropic case the free energy contains an entropic term associated with the length distribution and a binding‑energy term (ε) that accounts for monomer‑monomer association along the fibril axis. In the nematic case an additional orientational entropy penalty and a bending‑energy contribution proportional to the fibril rigidity (κ) are introduced.

Using a variational approach, the equilibrium length distribution ρ(L) that minimizes the total free energy is derived. The solution is essentially exponential but with a markedly larger mean length ⟨L⟩ and a narrower spread in the nematic phase, reflecting the fact that alignment promotes elongation and suppresses fluctuations. The authors then explore how three key parameters—protein concentration (c), binding energy (ε), and rigidity (κ)—affect the phase behavior. A critical concentration c* is identified; above this value the system enters a coexistence region where isotropic and nematic domains share the same chemical potential and pressure. Increasing ε (stronger monomer binding) lowers c* and expands the coexistence window, allowing nematic ordering at relatively low concentrations. Conversely, raising κ (making fibrils stiffer) raises the energetic cost of bending, suppresses nematic ordering, and shifts the coexistence region toward higher concentrations.

The coexistence curve is constructed by solving the common‑tangent conditions for the two free‑energy branches, yielding the volume fractions of the nematic (φ_N) and isotropic (φ_I) phases as functions of c, ε, and κ. The resulting I‑N‑I “sandwich” topology—an isotropic‑rich phase at low concentration, a nematic‑rich middle band, and a return to isotropic at very high concentration—matches the experimentally observed double‑phase separation in lysozyme fibril solutions.

To validate the model, the authors compare its predictions with experimental data on hen lysozyme fibrils. Measured nematic ordering angles, average fibril lengths, and the concentration at which nematic domains first appear are reproduced quantitatively when the binding energy is set to roughly –8 k_BT and the bending rigidity to about 10 pN·nm². This agreement demonstrates that the cylindrical‑micelle‑based description captures the essential physics of amyloid fibril self‑assembly and ordering.

Beyond reproducing known observations, the study offers practical insights. By tuning ε (through pH, ionic strength, or chemical modifiers) or κ (via cross‑linking or mutation), one can deliberately shift the I‑N transition, providing a route to engineer fibril‑based materials with desired mechanical anisotropy. Moreover, understanding how fibril length distribution couples to nematic ordering may illuminate pathological aggregation processes in neurodegenerative diseases, where fibril alignment could influence tissue mechanics and toxicity.

In summary, the paper establishes a robust, analytically tractable framework for predicting isotropic‑nematic transitions in amyloid fibril solutions, links the theory to real‑world lysozyme data, and highlights avenues for both material design and biomedical research.