Transport and phase separation of charged polymer

A vast majority of intrinsically disordered proteins are either strong polyampholytes or strong polyelectrolytes. These systems can form mesoscale biomolecular condensates via simple or complex coacervation. Recent measurements have shown evidence for differential partitioning of solution ions and protons across phase boundaries. These ion gradients generate equilibrium Donnan and Nernst potentials that directly affect the electrochemical properties of two-phase systems and at the interfaces of condensates. Existing mean-field models partially explain how charged polymers can aggregate into specific sizes at equilibrium, revealing the dependence of condensate size on polymer lengths and charge asymmetries. These models do not incorporate dynamics, nor do they explain the transport of charge polymers across interfaces in two-phase systems. We present a theoretical model to construct free energies that describe the interactions between different charged polymeric species and solution ions. We formulate and numerically solve transport equations to capture the spatiotemporal evolution of the polymers and ionic species towards their respective equilibrium states. Results, including stability analysis, phase behaviors deviating from classical liquid-liquid phase separation, and distinct pattern formation, will be discussed. By studying a wide range of parameters representing various macromolecular systems, we demonstrate the general applicability of our theoretical and computational approach. These findings provide insights into the fundamental physics of the dynamical assembly and phase behaviors of charged polymers and the interplay between electrostatic interactions and polymer structure.