Simulation of polyelectrolyte solutions. The density of bound ions J.M.G. Sarraguc ¸a * , A.A.C.C. Pais Departamento de Quı ´mica, Universidade de Coimbra, Rua Larga, 3004-535 Coimbra, Portugal Received 7 May 2004; in final form 9 September 2004 Available online 5 October 2004 Abstract We discuss a simple approach to describe the ion density around a polyelectrolyte chain, quantifying bound and bulk counteri- ons, and allowing for the renormalization of the charge in the polyion. This approach is both physically motivated and readily exten- sible to systems containing other types of highly charged ions. The method addresses the problem in simulation experiments and allows to correlate ion condensation and compaction. Ó 2004 Elsevier B.V. All rights reserved. 1. Introduction Coulomb interaction and thus the counterion atmos- phere and the induced degree of neutralization are to a large extent responsible for the behavior of highly charged polymers such as RNA and DNA. The difficulty of quantitatively assessing the ion density about the chain is both experimental [1,2] and theoretical [3,4]. The simplicity of the Debye–Hu ¨ ckel and Poisson–Boltz- mann (PB) approximations, which may fail in some cases (especially in highly correlated systems with multivalent ions [5,6]), make them prevalent and, in biophysics, the Manning–Oosawa [7] counterion condensation (CC) and PB theories are still of major importance. For infi- nitely long rigid rods, ManningÕs two-state picture finds an exact counterpart in the exact solution of the cylindri- cal PB equation [8]. Extension of the PB equation for di- lute solutions of finite length rod-like polyelectrolytes implies the use of a modified CC theory [9]. Studies on ion condensation are not only an attempt to provide a reference point [10] in polyelectrolyte solu- tions corresponding to the Debye–Hu ¨ ckel limiting law, but have a counterpart in the chain conformational behavior. The coil-globule coexistence phenomenon and compaction behavior of DNA under various charged compaction agents are clearly linked to changes in electrostatic interactions (due to salt valence and con- centration) and to variations in the ionic atmosphere in the close vicinity of the polyion. The number of ions around the chain may be directly counted [11] in simulations, but the distinction between bound ions and those in the bulk is not clear, being the ion density established as a function of the distance from the chain. This distance concept has motivated several approaches, in terms of what is usually designated as Manning radius (see e.g. [12]). The Manning radius (R M ) is associated, for cylindrical symmetry around a linear polyion, with the axial distance that encloses the fraction of bound ions, but can be extended to other central colloidal particles [13]. Two major approaches have been used to establish R M : (i) the search for a char- acteristic point [12,13], such as an inflection point in the running coordination number (see below) curve, RCN(r), or (ii) a physically motivated criterion for a pri- ori assesment of R M [14–17]. The first approach implies that the separation surface between bound and bulk ions should impart some alteration in the RCN behavior. The alteration would thus be visible in representations of RCN vs 1/r [13] or ln r [12] based on the PB frame- work but considered to apply beyond this framework. In the a priori vision R M may correspond to the value 0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.09.042 * Corresponding author. Fax: +351 239827703. E-mail address: jsarraguca@qui.uc.pt (J.M.G. Sarraguc ¸a). www.elsevier.com/locate/cplett Chemical Physics Letters 398 (2004) 140–145