PHYSICAL REVIEW E 83, 041803 (2011) Manning free counterion fraction for a rodlike polyion: Aqueous solutions of short DNA fragments in presence of very low added salt T. Vuleti´ c, 1,* S. Dolanski Babi´ c, 1 D. Grgiˇ cin, 1 D. Aumiler, 1 J. R¨ adler, 2 F. Livolant, 3 and S. Tomi´ c 1 1 Institut za fiziku, 10000 Zagreb, Croatia 2 Ludwig-Maximilians-Universit¨ at, Sektion Physik, Geschwister-Scholl-Platz 1, D-80539 Munich, Germany 3 Laboratoire de Physique des Solides, Universit´ e Paris Sud, F-91405 Orsay, France (Received 27 December 2010; published 18 April 2011) We quantified the Manning free (uncondensed) counterions fraction θ for dilute aqueous solutions of rodlike polyions: 150 bp DNA fragments, in the presence of a very low concentration of monovalent salt c salt < 0.05 mM. Conductivity measurements of these solutions for DNA base pair concentration range 0.015  c  8 mM were complemented by fluorescence correlation spectroscopy (FCS) measurements of the DNA polyion diffusion coefficient D p (c). We observed a crossover in the normalized conductivity σ (c)/c that nearly halved across the c = 0.05–1 mM range, while D p (c) remained rather constant, as we established by FCS. Analyzing these data we extracted θ (c) = 0.30–0.45, and taking the Manning asymmetry field effect on polyelectrolyte conductivity into account we got θ (c) = 0.40–0.60. We relate the θ (c) variation to gradual DNA denaturation occurring, in the very low salt environment, with the decrease in DNA concentration itself. The extremes of the experimental θ (c) range occur toward the highest, above 1 mM, and the lowest, below 0.05 mM, DNA concentrations and correspond to the theoretical θ values for dsDNA and ssDNA, respectively. Therefore, we confirmed Manning condensation and conductivity models to be valuable in description of dilute solutions of rodlike polyions. DOI: 10.1103/PhysRevE.83.041803 PACS number(s): 82.35.Rs, 87.15.hj, 66.30.hk I. INTRODUCTION Most biologically relevant macromolecules (DNA, pro- teins, polysaccharides) are polyelectrolytes with a very distinct behavior compared to neutral polymers or simple electrolytes [1,2]. When dissolved in polar solvents polyelectrolytes dissociate into a highly charged polyion (a macromolecule of extended shape) and many small counterions of low valency. The long-range nature of the electrostatic interactions and the entropy effects due to inhomogeneities in the counterion distributions and to a myriad of polyion configurations control their phenomenology. The strong linear charge of the polyion tends to attract the counterions to its immediate vicinity. The condensation occurs for polyions with the Manning parameter u = l B /b > 1, where l B is the Bjerrum length, the length at which two elementary charges interact in a given solvent with energy equal to the thermal energy kT , and b is the average distance between the charges on the polyion backbone. If there is more than one charge per Bjerrum length, the condensation will tend to effec- tively reduce the linear charge density down to the 1/l B level. The condensed ions fraction is then equal to 1 − 1/u, and the free, uncondensed counterions fraction is θ = 1/u. The con- densation was modeled for an infinitely long and thin polyion in pure water, with no added salt, which might appear as a rather unrealistic proposition, with no biological relevance [3]. Counterion condensation is therefore more easily experi- mentally studied and the results theoretically interpreted for a dilute solution of rigid, monodisperse polyions, which do not change conformation with concentration. In a dilute solution, effectively, the condensed fraction of counterions may be considered to be found in a cylindrical cell around the polyion, while the rest may be taken to be free inside a larger volume that * http://tvuletic.ifs.hr/; tvuletic@ifs.hr belongs to a given polyion [4]. According to the theory, since the condensed counterions are not chemically bound to the polyion, the free and condensed counterions exchange between the two concentric regions, and only a continuous radial coun- terion distribution [5] can exist around the polyion. In other words, there should be no step in the radial counterion distri- bution that would define the limit of the cylindrical zone, as shown experimentally by electron paramagnetic resonance [6]. Besides theoretical works considering two types of ions, experiments also attempt to quantify the condensed and free counterion fractions. Since only uncondensed, free counteri- ons contribute to the osmotic pressure of a polyelectrolyte [7,8], the measured osmotic pressure of a polyelectrolyte solution evaluates the free counterions fraction [9–11]. The condensed counterions are those that move together with the polyion when an electric field is applied, while the free counterions would move in the opposite way due to their opposite charge [12–14]. Thus, the concept of two types of counterions gets a physical meaning. Thus, the transport techniques may contribute to our knowl- edge of condensation in polyelectrolytes. The techniques range from electrical transport measurements like conductometry [12,15–17] and capillary electrophoresis [18,19] to diffusion measurements by dynamic light scattering [18–21] or fluores- cence correlation spectroscopy [22,23]. Manning [24,25] has proposed a rather comprehensive and convincing conductivity model for polyelectrolytes, and Bordi et al. [14] worked on including the scaling theories by Rubinstein et al. [26], in order to separate the influences from the polyion (conformation and charges), the counterions, and the added salt. For a successful quantitative study of Manning conden- sation by the transport experiments one has to use the simplest possible system: a dilute solution of monodisperse polyelectrolytes with no added salt. Also, an experimental method is needed to separate the influence of the charge 041803-1 1539-3755/2011/83(4)/041803(8) ©2011 American Physical Society