pubs.acs.org/Macromolecules Published on Web 06/22/2009 r 2009 American Chemical Society Macromolecules 2009, 42, 5851–5860 5851 DOI: 10.1021/ma9008143 Scale-Dependent Electrostatic Stiffening in Biopolymers Alexander Gubarev, †,‡ Jan-Michael Y. Carrillo, † and Andrey V. Dobrynin* ,† † Polymer Program, Institute of Materials Science, Department of Physics, University of Connecticut, Storrs, Connecticut 06269-3136, and ‡ Department of Physics, Saint-Petersburg State University, Saint-Petersburg, Russia Received April 14, 2009; Revised Manuscript Received May 29, 2009 ABSTRACT: Using a combination of the molecular dynamics simulations and theoretical calculations, we have demonstrated that bending rigidity of biological polyelectrolytes (semiflexible charged polymers) is scale-dependent. A bond-bond correlation function describing a chain’s orientational memory can be approximated by a sum of two exponential functions manifesting the existence of the two characteristic length scales. One describes the chain’s bending rigidity at the distances along the polymer backbone shorter than the Debye screening length, whereas another controls the long-scale chain’s orientational correlations. The short- length scale bending rigidity is proportional to the Debye screening length at high salt concentrations and shows a weak logarithmic dependence on salt concentration when the Debye screening length exceeds a crossover value of κ cr -1 µ (l B R 2 /l p ) -1/2 (where l B is the Bjerrum length, R is the fraction of ionized groups, and l p is a bare persistence length). The long-scale chain’s bending rigidity has a well-known Odijk-Skolnick- Fixman form with a quadratic dependence on the Debye radius. Simulation results and a theoretical model demonstrate good qualitative agreement. 1. Introduction Electrostatic interactions play an important role in controlling properties of biological objects, such as DNA, F-actin, micro- tubules and filamentous viruses. 1-3 The change in the ionic environment is known to influence the conformational properties of DNA significantly. Condensation of multivalent ions triggers an abrupt collapse of T4 phage DNA from coil-like conforma- tions to compact structure inside the viral capsid. 1 The decrease in the volume of the DNA molecule could be significant and reaches the volume compression ratio of 6900 times. 1 The force-elongation measurements, 4 diffusion coefficient mea- surements, 5 and studies of the effect of confinement of the λ-DNA 6 show that with increasing solution ionic strength, the chain stiffness decreases. The ability to manipulate mole- cular properties, such as the chain’s bending energy, by con- trolling its environment was utilized in DNA separation 7 and gene mapping 8 and is at the heart of the future of nanotech- nology. The concept of the electrostatic persistence length of semiflex- ible polyelectrolytes was introduced by Odijk 9 and by Skolnick and Fixman 10 (OSF). They studied bending rigidity of a semi- flexible polyelectrolyte chain with ionic groups interacting via the screened Debye-Huckel potential. This is a crude model of DNA in salt solutions where electrostatic interactions are exponentially screened by salt ions on the length scales larger than the Debye screening length, κ -1 . However, despite the exponential decay of the strength of the electrostatic interactions, the covalent bonding of ionic groups into the polymer chains extends their range beyond the Debye screening length, leading to strong orienta- tional correlations between chain segments. A bond orientational memory propagates far beyond the Debye screening length, κ -1 , inducing extra chain stiffening, which is proportional to the square of the Debye radius. The OSF theory has been challenged. Different authors have shown that the electrostatic part of the chain’s bending energy scales linearly with the Debye radius. (For a recent overview of the subject, see refs 11-14). In this article, we present a new model of the electrostatic induced rigidity of biological (semiflexible) polyelectrolytes by extending the idea of a scale dependence of the chain’s persistence length developed in refs 11, 15, and 16. We will show that the electrostatic-induced stiffening of a semiflexible polyelectrolyte is a multiscale process that can be approximated by two character- istic length scales. One describes the decay of the bond-bond orientational memory at the distances along the polymer back- bone shorter than the Debye screening length, whereas another controls long-scale orientational correlations. At high salt con- centrations, when intrachain electrostatic interactions are signifi- cantly screened, the short-length scale correlation length (bending rigidity) is proportional to the Debye screening length, whereas the long-scale correlation length is proportional to the square of the Debye radius. 9,10 However, when the Debye screening length exceeds a crossover value, the short-length scale correlation length has a weak logarithmic dependence on the Debye screen- ing length. Note that a logarithmic dependence of the chain’s orientational correlations is a characteristic feature of a semi- flexible chain under tension generated by local electrostatic interactions. A crossover shifts toward larger values of the Debye screening lengths with increasing bare chain persistence length. The rest of the article is organized as follows. In Section 2, we present the results of the molecular dynamics simulations of semiflexible polyelectrolyte chains. Section 3 discusses the results of the high-temperature expansion and variational calculations. At the end of the section, we compare derived scaling relations with simulation results. Finally, Section 4 summarizes our results and discusses the possibility of an experimental verification of our findings. 2. Model and Simulation Results To elucidate the factors controlling the electrostatic-induced chain’s stiffening of biopolymers, we have performed molecular *Corresponding author. E-mail: avd@ims.uconn.edu. Downloaded by UNIV OF CONNECTICUT on August 25, 2009 | http://pubs.acs.org Publication Date (Web): June 22, 2009 | doi: 10.1021/ma9008143