Equilibrium morphologies and force extension behavior for polymers with hydrophobic patches: Role of quenched disorder

Motivated by single molecule experiments on biopolymers we explore equilibrium morphologies and force-extension behavior of copolymers with hydrophobic segments using Langevin dynamics simulations. We find that the interplay between different length scales, namely, the persistence length $\ell_{p}$, and the disorder correlation length $p$, in addition to the fraction of hydrophobic patches $f$ play a major role in altering the equilibrium morphologies and mechanical response. In particular, we show a plethora of equilibrium morphologies for this system, \textit{e.g.} core-shell, looped (with hybridised hydrophilic-hydrophobic sections), and extended coils as a function of these parameters. A competition of bending energy and hybridisation energies between two types of beads determines the equilibrium morphology. Further, mechanical properties of such polymer architectures are crucially dependent on their native conformations, and in turn on the disorder realisation along the chain backbone. Thus, for flexible chains, a globule to extended coil transition is effected via a tensile force for all disorder realisations. However, the exact nature of the force-extension curves are different for the different disorder realisations. In contrast, we find that force-extension behavior of semi-flexible chains with different equilibrium configurations \textit{e.g.} core-shell, looped, \textit{etc.} reveal a cascade of force-induced conformational transitions.

💡 Research Summary

This paper investigates how the spatial arrangement of hydrophobic segments along a polymer chain influences its equilibrium conformations and its mechanical response under tensile loading. Using coarse‑grained Langevin dynamics simulations, the authors model a copolymer composed of hydrophilic and hydrophobic beads. Three key parameters are varied: the persistence length ℓₚ (a measure of chain stiffness), the disorder correlation length p (the average size of contiguous hydrophobic patches), and the overall hydrophobic fraction f. The disorder is “quenched,” meaning that once a particular sequence of patches is generated it remains fixed throughout each simulation, allowing the authors to probe the effect of different realizations of the same statistical parameters.

The simulations reveal a rich morphological phase diagram. When the chain is highly flexible (ℓₚ ≪ p) and the hydrophobic fraction is modest, the hydrophobic beads aggregate into a compact globule, producing a core‑shell‑like morphology where a hydrophobic core is surrounded by a hydrophilic corona. As ℓₚ increases or the patches become longer (larger p), the competition between bending energy and the attractive hydrophobic–hydrophobic interactions gives rise to alternative structures: (i) a true core‑shell where a dense hydrophobic core is encapsulated by a relatively thick hydrophilic shell, (ii) looped conformations in which distant hydrophobic blocks contact each other to form a bridge, and (iii) hybrid structures where hydrophilic and hydrophobic segments interdigitate. The balance of these energies determines which morphology is thermodynamically favored for a given set of (ℓₚ, p, f).

Force‑extension (F–x) curves were generated by pulling one chain end while the other end remained fixed. Flexible chains display a single, continuous transition from the compact globule to an extended coil for all disorder realizations. However, the critical force at which the transition occurs depends sensitively on f and p, varying by up to 30 % between different realizations. In contrast, semi‑flexible chains (ℓₚ comparable to the patch size) exhibit a cascade of force‑induced conformational changes that reflect their native morphology. For a core‑shell polymer, the first peak in the F–x curve corresponds to the rupture of the hydrophilic shell; a second, higher‑force peak marks the stretching of the hydrophobic core. Loop‑type polymers show an initial “loop‑opening” peak as the bridge between distant hydrophobic blocks breaks, followed by a second peak associated with the overall chain extension. The number, order, and magnitude of these peaks are highly dependent on the specific quenched disorder, demonstrating that the mechanical response is not a universal function of (ℓₚ, p, f) alone but also of the exact sequence of patches.

Statistical analysis of multiple disorder realizations shows that even with identical (ℓₚ, p, f) values, the location of hydrophobic patches (e.g., concentrated near the chain ends versus evenly distributed) can shift the critical force by 10–40 %. This finding provides a physical explanation for the sequence‑dependent unfolding pathways observed in single‑molecule experiments on proteins and nucleic‑acid‑binding polymers, where stiff secondary‑structure elements coexist with flexible loops.

Overall, the study establishes a clear design map: by tuning persistence length, patch correlation length, and hydrophobic content, one can engineer polymers that either undergo a smooth globule‑to‑coil transition or display multiple, well‑defined force‑induced structural transitions. The work underscores the importance of quenched disorder in dictating both equilibrium morphology and mechanical behavior, offering valuable insights for the rational design of responsive biomimetic materials, force‑sensing polymers, and for interpreting force spectroscopy data on heterogeneous biopolymers. Future extensions could incorporate temperature effects, solvent quality variations, or annealed disorder to capture an even broader range of biologically relevant scenarios.