Dealing with long‐range interactions in the determination of polyelectrolyte ionization properties. Extension of the transfer matrix formalism to the full range of ionic strengths

Abstract

The ionization state of charged macromolecules in solution is usually determined by the extent of the binding processes. These processes are very sensitive to the ionic strength of the medium, which are of long-range nature. The ionization properties of weak polyelectrolytes can be described by means of Ising-type models, which is only feasible when long-range interactions are neglected. Here, this formalism is extended to include long-range interactions by introducing a modified free energy involving only effective short-range interaction parameters. These parameters can be systematically calculated by using the GibbsBogoliubov variational principle. The technique is illustrated with the calculation of titration curves of homogeneous and heterogeneous polyelectrolytes in 1 a wide range of ionic strengths. The correction of the site protonation free energy (first order correction) is enough to obtain an excellent agreement between theory and Monte Carlo simulations. Corrections to other cluster parameters (higher order corrections) are also implemented. In general, the correction to a particular parameter represents the average change in the long-range energy when a new interaction is created in the polyelectrolyte. The method presented here represents an improvement in the description of the ionization state of polyelectrolytes that can be relevant in a wide range of areas.

Coupling of Charge Regulation and Conformational Equilibria in Linear Weak Polyelectrolytes: Treatment of Long Range Interactions via Effective Short-Ranged and pH-Dependent Interaction Parameters

“…For instance one could decide to correct, not only the pK-value (pK → pK − x), but also the nearest neighbour interaction energy ( → − x ). Proceeding in the same way, it can be shown that x and x fulfill the non-linear system of equations [56]…”

“…This is an important limitation of the method when dealing with polyelectrolytes, since it can be only used at high enough ionic strengths, for which the screening is enough to avoid the LR interactions. In a recent paper [56] we introduced a method which allows to include the LR interactions in a very accurate way. In this approach, the full free energy (1) is replaced by a new one involving only short range (SR) interaction parameters, accounting for the LR interactions in an effective way.…”

“…The details of the derivations are given in Ref. [56]. Although the following arguments can be readily generalized to the general form of the free energy (1), let us consider the simplest case of a rigid linear chain with identical sites, such as the one sketched in Fig.…”

“…to determine the optimal value of x, where Ω 0 (x) = − ln Ξ 0 and · · · 0 represent the free energy and the thermal average corresponding to H 0 , respectively. Minimizing Ω with respect to x, it is found that x fulfills the equation [56]…”

On the origin of non-membrane-bound organelles, and their physiological function

“…While mean-field theories may be applicable to ionic interactions with planar charged surfaces, these theories break down for polyelectrolytes due to the short-range nature of the interactions between neighboring charged residues (Borkovec et al, 1997). Hence, the ionization state of a polyelectrolyte, which is important for the formation of dense polyelectrolyte-protein phases, depends on both the long-range effects due to the ionic strength of the medium, and the short-range interactions between neighboring charges on the polyelectrolyte (Garcés et al, 2017). …”