Reentrant condensation of sparsely charged polyelectrolytes induced by multivalent salts: Interplay between electrostatic and non-electrostatic interactions

We study the reentrant condensation of sparsely charged polyelectrolytes in dilute solutions of multivalent ions, whose phase-transition mechanism remains under debate. We propose a mean-field model by recasting the concept of a "physical crosslinking" effect, which can predict the essential features of the reentrant condensation including the phase diagram of sparsely charged polyelectrolyte. The model unveils that the strong adsorption between the ionic monomers and multivalent ions can be at the origin of the peculiar phenomenon that rather low concentrations of multivalent salts trigger both collapse and re-entry transitions. The analytical solution of the model indicates that a minimum of coupling energy due to sharing multivalent salt ions between ionic monomers is essential for a phase transition to occur, which can explain the enigmatic observation that the sparsely charged polyelectrolytes can only show phase transition in a dilute solution of salts with selective multivalency. Our analytical calculations also show that the incompatibility of the uncharged moieties of the polyelectrolytes with solvent (water) is critical to regulate phase behaviors of sparsely charged polyelectrolytes in aqueous solutions. This is in agreement with recent experimental investigations on solution properties of amphiphilic proteins. We envisage that this work will shed light on the understanding of reentrant condensation in biological processes where the multivalent ions bound to biopolymers (such as RNAs and proteins) plays an essential role, which remains understood poorly.