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CU-Boulder PHYS 7450 - Complexation of semiflexible chains with oppositely charged cylinder

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Complexation of semiflexible chains with oppositely charged cylinderAndrey G. Cherstvya)and Roland G. WinklerInstitut fu¨r Festko¨rperforschung, Theorie-II, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany共Received 14 November 2003; accepted 24 February 2004兲We study the complexation of long thin semiflexible polymer chains with an oppositely chargedcylinder. Starting from the linear Poisson–Boltzmann equation, we calculate the electrostaticpotential and the energy of such a charge distribution. We find that sufficiently flexible chains preferto wrap around the cylinder in a helical manner, when their charge density is smaller than that of thecylinder. The optimal value of the helical pitch is found by minimization of the sum of electrostaticand bending energies. The dependence of the pitch on the number of chains, their rigidity, and saltconcentration in solution is analyzed. We discuss our results in the light of recent experiments onDNA complexation with cylindrical dendronized polymers. © 2004 American Institute of Physics.关DOI: 10.1063/1.1707015兴I. INTRODUCTIONCompactification of DNA into ⬃100-nm large com-plexes with oppositely charged proteins and polycations is animportant tool in gene therapy1to deliver DNA into an in-fected cell.2In particular, DNA condensates withpolylysine3,4and polyamines5,6are used for these purposes.Nucleosome core particles and supercoiled DNA’s are alsofundamentally important biological objects which involve aDNA arrangement in a helical manner. Although the com-plexation and aggregate formation of chains of various flex-ibilities and of different lengths with oppositely chargedspheres and cylinders has been studied theoretically,7–16experimentally,4,17–22and by computer simulations,23–30sev-eral features of this phenomenon are still not completely un-derstood.One of these features is overcharging of a cylinderby adsorbed oppositely charged chains predictedtheoretically15,16and recently reported experimentally for cy-lindrical DNA/dendrimer complexes.19In contrast, DNAwrapped around spherical dendrimers was shown to formpreferentially neutral complexes.18,21,22Nature also usessimilar techniques to compact the genetic material in cells,where the nucleosome can be substantially overcharged by aDNA molecule wrapped around it.31,32Another aspect is thetheoretically predicted release of a chain from a sphereat high salt concentrations8that reminds salt-induced DNArelease from the nucleosome observed experimentally.33,34Inexperiments in vitro at higher salt concentrations, it has beenreported that DNA attaches stronger to cylindricaldendrimers20but weaker to spherical ones.21The physicalorigin of this effect is not clear yet.In many cases, electrostatic interactions play a dominantrole in the formation of complexes between highly chargedmacromolecules. Wrapping of DNA duplexes around acylindrical/spherical object is, however, a formidable electro-static problem, which can involve chiral interactions betweenDNA helices.35The details and discreteness of charge distri-bution on DNA might also be important 共DNA overwindingfrom 10.5 bp/turn in solution36to about 10.2 bp/turn innucleosomes31,32,37occurs兲. Large charge densities of bothobjects introduce additional complications, since the linearPoisson–Boltzmann theory may not apply. However, to un-derstand the basic physical properties of complexes, it mightbe sufficient to start from the simplest model.In this article we consider the wrapping of thin semiflex-ible charged chains around an oppositely charged cylinder.We calculate the electrostatic potential and energy of thehelical complex using linear Poisson–Boltzmann theory.Hence, our calculations extend those of Ref. 12, because thefull solution of the linear Poisson–Boltzmann equation isconsidered for the particular charge distribution and not sim-ply a superposition of Debye–Hu¨ckel potentials. This hasconsequences for the amount of DNA adsorbed on a cylin-der. Within our model neutral and undercharged complexesare preferred, whereas the calculations of Ref. 12 predict asignificant overcharging. We compare the predictions of themodel with available theoretical and experimental data. Inthe end we discuss some possible extensions of the model.The paper is organized as follows. In Sec. II the model isoutlined and the electrostatic energy for a helical charge dis-tribution on a cylinder is calculated. In Sec. III the electro-static energy for a helical charge distributions is determined.Results for the helical pitch are presented and discussed inSec. IV. Finally, Sec. V summarizes our findings.II. ENERGY CALCULATIONA. Model and approximationsWe solve the linear Poisson–Boltzmann equation for aninfinitely long, positively charged cylinder with negativelycharged semiflexible strings adsorbed on its surface in a he-lical conformation. The strings are considered to be infinitelythin charged lines. No fluctuations of the strings on the cyl-inder surface are considered 共zero-temperature solution, thehelices are ideal兲. The cylinder has the radius a and the sur-face charge density␴c共the linear charge density is e0␶c, e0is the elementary charge兲. For simplicity, no distribution ofa兲Corresponding author. Electronic mail: [email protected] OF CHEMICAL PHYSICS VOLUME 120, NUMBER 19 15 MAY 200493940021-9606/2004/120(19)/9394/7/$22.00 © 2004 American Institute of PhysicsDownloaded 27 Sep 2004 to 129.105.213.90. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jspthe electrostatic potential inside the cylinder is considered.Every string is characterized by a linear charge density e0␶pand the mechanical persistence length lp. The water is con-sidered as a dielectric continuum 共no discreteness effects兲with the dielectric constant ␧⬇80. The dielectric constant ofthe cylinder interior is the same as in the bulk solution 共noimage forces兲. No effects of charge fluctuations and correla-tions are considered.B. Electrostatic energyThe electrostatic energy of such a charge distributionfollows from the electrostatic potential␾(rជ) with the chargedensity␳(rជ) according toEel⫽12冕d3rជ␾共rជ兲␳共rជ兲⫽a2冕02␲d␸冕⫺ ⬁⬁dz␾共z,␸,a兲␴共z,␸兲. 共1兲The volume charge density␳is related to the surface chargedensity␴of the complex via␳(z,␸,r)⫽␦(r⫺a)␴(z,␸).Cylindrical coordinates are used, where the


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CU-Boulder PHYS 7450 - Complexation of semiflexible chains with oppositely charged cylinder

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