Is the cylindrical globule the transition state of the globule-to-pearl-necklace transition of a single polyelectrolyte in a poor solvent?

A single flexible polyelectrolyte (PE) in a poor solvent collapses under short-range attraction while its backbone charges repel electrostatically, and as the backbone charge fraction grows the collapsed spherical globule gives way to a pearl-necklace structure: compact pearls connected by stretched strings. Self-consistent field theory (SCFT) by Duan, Li and Wang (J. Chem. Phys. 153, 064901, 2020) settled part of a four-decade debate by showing that the elongated cylindrical globule proposed by Khokhlov is not an equilibrium structure at their salt concentration: SCFT iterations started from ellipsoidal or cylindrical profiles relax into pearl necklaces. The same paper left open the kinetic half of the question: whether the cylindrical globule can exist as a transition state on the pathway of the globule-to-pearl-necklace transition. No published work identifies the transition state of this transition. The problem is to determine that transition state and decide whether it is cylindrical.

Background

A polyelectrolyte (PE) is a polymer whose monomers carry ionized groups, so that in solution the chain bears a distributed backbone charge balanced by small counterions. In a poor solvent, i.e. when the effective monomer-monomer interaction is attractive (conventionally parameterized by a positive Flory-Huggins interaction parameter $\chi$), a neutral flexible polymer collapses into a compact spherical globule whose free energy is dominated by surface tension. For a PE, the long-range Coulomb repulsion between charged monomers competes with this attraction, and the way a single collapsed PE chain accommodates growing charge has been debated since the 1980s. Khokhlov (1980) argued that a weakly charged globule cannot stay spherical and elongates into a cylinder (DOI 10.1088/0305-4470/13/3/030). Kantor and Kardar (1994, 1995) drew on the analogy with the Rayleigh instability — the fission of a charged liquid droplet into smaller droplets when electrostatic repulsion exceeds surface tension — and argued that a charged polymer globule should instead break into a pearl-necklace (PN): collapsed, nearly neutral pearls connected by stretched, highly charged strings (DOIs 10.1209/0295-5075/27/9/002, 10.1103/physreve.51.1299). Dobrynin, Rubinstein and Obukhov (1996) developed the scaling theory of the PN and predicted a cascade of transitions between necklaces with different pearl numbers as the charge fraction increases (DOI 10.1021/ma9507958). Simulations split along their solvent treatment: implicit-solvent Monte Carlo and molecular dynamics studies observed pearl necklaces (Chodanowski and Stoll 1999, DOI 10.1063/1.479905; Limbach and Holm 2003, DOI 10.1021/jp027606p), whereas explicit-solvent and solvent-accessible-surface simulations by Yethiraj and co-workers found compact, cylinder-like conformations with no clearly separated pearls (Chang and Yethiraj 2006, DOI 10.1021/ma051095y; Reddy and Yethiraj 2006, DOI 10.1021/ma061176+), and small-angle neutron scattering experiments generally lack the sharp pearl-necklace signature. Coarse-grained simulations of partially sulfonated polystyrene likewise traverse a cascade in which a single cylindrical globule appears at small chain length before dumbbells and multi-pearl necklaces (Zhang and van der Vegt 2016, DOI 10.1021/acs.macromol.6b01132). Duan, Li and Wang (2020, DOI 10.1063/5.0017371) developed a self-consistent field theory (SCFT) — a mean-field framework that computes the polymer density profile and free energy by solving a modified diffusion equation for Gaussian chain statistics coupled to Poisson-Boltzmann electrostatics, without preassuming a shape — for a single PE with Kuhn length $b=1.0$ nm, backbone charge fraction $\alpha$, monovalent salt at bulk concentration $c^{b}=2\times10^{-3}$ M (Debye length about 7 nm), and dielectric constant $\epsilon_r=80$. They found that as $\alpha$ increases the spherical globule (Sph) first jumps discontinuously to a two-pearl necklace (PN$_2$) at $\alpha\approx 0.105$ and then passes continuously to PN$_3$ and PN$_4$; that two flattened metastable structures (a biconcave red cell and a toroid) also exist; and that the cylindrical globule is not an equilibrium structure, since SCFT iterations started from ellipsoidal or cylindrical density profiles relax into PN structures. A transition state, in the chemical-physics sense used here, is the saddle point of the free-energy surface that dominates the rate of a transition between two locally stable states: a configuration with exactly one unstable direction that connects the two basins. Kinetic questions of this type cannot be answered by the equilibrium SCFT scan, which only compares local minima; they require locating saddle points of the free-energy functional or sampling the ensemble of dynamical transition paths.

Problem Statement

For a single flexible polyelectrolyte chain in a poor solvent — in the model class of Duan, Li and Wang (2020): a Gaussian chain of $N=200$ Kuhn segments with Kuhn length $b=1.0$ nm, Flory-Huggins parameter $\chi=1.0$, uniformly charged backbone with charge fraction $\alpha$, monovalent salt at bulk concentration $c^{b}=2\times10^{-3}$ M (Debye length about 7 nm), dielectric constant $\epsilon_r=80$ — determine the kinetic pathway of the globule-to-pearl-necklace transition, i.e. the electrostatically driven first pearling transition in which the collapsed spherical globule (Sph) splits into pearls connected by strings (at these parameters, the two-pearl necklace PN$_2$, entered discontinuously at $\alpha\approx 0.105$). The central question, as stated by the source, is whether the elongated cylindrical globule can exist as a transition state on this kinetic pathway, rather than as an equilibrium structure (the source's SCFT already shows it is not an equilibrium structure at these parameters). The question is which configuration forms the barrier between the Sph and pearl-necklace basins, and whether that configuration is a rod-like, unsplit cylindrical globule or something else. Both affirmative and negative answers are admissible: showing that a rod-like, unsplit cylindrical configuration is the rate-dominant saddle (transition state) connecting the Sph and pearl-necklace basins, or that the dominant pathway bypasses any cylindrical configuration. Answers must address this charge-driven transition between the collapsed globule and the pearl necklace in this model class; a result about collapse from an expanded coil, about a different model class (for example net-neutral polyampholytes), or restricted to a single arbitrarily chosen parameter point without addressing the transition's mechanism does not resolve the question.

References

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