Electrostatic complexation of spheres and chains under elastic stressH. Schiessela)Max-Planck-Institute for Polymer Research, Theory Group, P.O. Box 3148, 55021 Mainz, GermanyR. F. BruinsmaInstituut-Lorentz for Theoretical Physics, Universiteit Leiden, Postbus 9506, 2300 Leiden, The NetherlandsW. M. GelbartDepartment of Chemistry and Biochemistry, University of California, Los Angeles, California 90095-1569共Received 22 May 2001; accepted 25 July 2001兲We consider the complexation of highly charged semiflexible polyelectrolytes with oppositelycharged macroions. On the basis of scaling arguments we discuss how the resulting complexesdepend on the persistence length of the polyelectrolyte, the salt concentration, and the sizes andcharges of the chain and the macroions. We study first the case of complexation with a single sphereand calculate the wrapping length of the chain. We then extend our consideration to complexesinvolving many wrapped spheres and study cooperative effects. The mechanical properties of sucha complex under an external deformation are evaluated. © 2001 American Institute of Physics.关DOI: 10.1063/1.1403688兴I. INTRODUCTIONThe complexation of polyelectrolytes and oppositelycharged macroions is a primary ingredient in biological pro-cesses. The nonspecific part of the interaction between pro-teins and DNA is governed by electrostatics. A well-knownexample of this form of complexation is the association ofDNA with oppositely charged octamers of histone proteins,an essential step in chromosomal DNA compaction.1Com-plexation of macroions is also encountered in several tech-nological applications. For instance, the complexation ofsynthetic polymers with colloidal particles2,3and chargedmicelles4is of practical importance for modifying macroionsolution behavior.A number of experimental5–8and theoretical studies9–13have demonstrated that complexation of highly charged mac-roions is governed by an unususal electrostatics mechanism:counterion release. The electrostatic free energy of associa-tion of oppositely charged macroions is dominated by theentropy increase arising from the release of counterions thathad been condensed onto the macroions before association.This electrostatic free energy gain must compete with a freeenergy cost induced by deforming either or both of the mac-roions so as to bring the fixed macroion charges of oppositesign in close contact.A simple example of this competition, first discussed byMarky and Manning,14is the association of a charged spherewith an oppositely charged semiflexible chain. If R is theradius of the sphere, lPthe persistence length of the chain共i.e., the bending modulus of the chain equals kBTlPwithkBT the thermal energy兲, and the electrostatic free energygain per unit length of adhesion, then the free energy cost ofassociation isF共l兲⬵冉kBTlPR2⫺ 冊l⫹ O共l2兲共1兲with l the length of chain wrapped around the sphere. Thefirst term in the brackets represents the bending energy perlength of the adsorbed chain, taking into account the fact thatits curvature is of the order of 1/R.15Under conditions wherecounterion release dominates 共i.e., high bare charges兲, is ofthe order kBT/b with b the spacing between charges alongthe chain 共cf. the discussion in Sec. II兲. According to Eq. 共1兲,complexation starts when exceeds kBTlP/R2. It would ap-pear reasonable that wrapping continues until the charge ofthe wrapped part of the chain has compensated the charge ofthe central sphere. The counterion release mechanism pro-duces a surprise: An analysis10for this case found that, forsmall enough persistence lengths, the chain/sphere complexis overcharged: more chain is wrapped on the sphere thenrequired for charge compensation. ‘‘Charge-reversal’’ nor-mally is associated with short-range correlations between thecharges,16but here it is again due to entropy increase of thecounterions.In the present paper we extend the above-mentionedanalysis to examine complexation of a flexible charged chainplaced in a solution of oppositely charged macroions, assum-ing that the complex adopts a ‘‘beads-on-a-string’’ geometry.共This particular geometry is, for example, encountered forDNA/histone complexes at low salt concentrations兲. A simi-lar system of charged spheres and chains was also recentlyinvestigated by Nguyen and Shklovskii.17Their study fo-cuses on weakly charged systems where counterions are notimportant 共a recent preprint of their work also considers therole of condensed counterions in highly charged systems18兲.As we discuss in the conclusion of this paper their systemshows nevertheless many features that are characteristic ofour system. It should be noted though that such simple mod-els are only useful for a discussion of the generic aspects ofa兲Electronic mail: [email protected] OF CHEMICAL PHYSICS VOLUME 115, NUMBER 15 15 OCTOBER 200172450021-9606/2001/115(15)/7245/8/$18.00 © 2001 American Institute of PhysicsDownloaded 27 Sep 2004 to 128.210.142.204. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jspcomplexation of charged linear macromolecules with spheri-cal macroions. For any particular case, the specific aspects ofthe molecular interactions for that situation must be ac-counted for.The central claim of this paper is that whereas complex-ation of a chain with a single sphere leads to spontaneousovercharging, complexation in a solution of spherical mac-roions leads to spontaneous undercharging, even though bothare due to the same counterion release mechanism. The sur-prising role reversal is reflected in force-extension curves ofthe kind now routinely measured for long biopolymers. Forthe case of individual sphere/chain complexes the effect ofan external tension f can be accounted for by adding, in Eq.共1兲, a term fl. At a critical tension equal to ⫺ kBTlP/R2, thechain–sphere complex dissociates. The measurement of theforce-extension curve thus gives information on the adhesionenergy per unit length. For the case of a chain under tensionin chemical equilibrium with a solution of spherical macro-ions, however, we find that with increasing tension more andmore spheres condense on the chain and that the critical ten-sion to add one additional sphere vanishes in the thermody-namic limit of an infinitely long chain. We also consider achain complexed with a fixed number N of spherical macro-ions under an
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