Manning free counterions fraction for a rod-like polyion - short DNA fragments in very low salt T. Vuleti´c,1,∗ S. Dolanski Babi´c,1 D. Grgiˇcin,1 D. Aumiler,1 J.Ra¨dler,2 F. Livolant,3 and S. Tomi´c1 1Institut za fiziku, 10000 Zagreb, Croatia 2Ludwig-Maximilians-Universit¨at, Sektion Physik, Geschwister-Scholl-Platz 1, D-80539 Munich, Germany 3Laboratoire de Physique des Solides, Universit´e Paris Sud - F-91405 Orsay, France (Dated: January 6, 2011) We quantifiedthe Manning free (uncondensed) counterions fraction θ for dilute solutions of rod- like polyions - 150bp DNA fragments, in very low salt < 0.05mM. Conductivity measurements of 1 aqueous DNA solutions in the concentration range 0.015 ≤c≤8 mM (bp) were complemented by 1 fluorescencecorrelation spectroscopy (FCS) measurements of theDNApolyion diffusion coefficient 0 2 Dp(c). We observed a crossover in the normalized conductivity σ(c)/c which nearly halved across c=0.05−1mMrange,whileDp(c)remainedratherconstant,asweestablishedbyFCS.Analyzing n these data we extracted θ(c) = 0.30−0.45, and taking the Manning asymmetry field effect on a polyelectrolyte conductivityinto account we got θ(c)=0.40−0.60. Werelate theθ(c) variation to J gradual DNA denaturation occuring, in the very low salt environment, with the decrease in DNA 5 concentration itself. The extremesof theexperimental θ(c) range occur towards thehighest, above 1 mM and the lowest, below 0.05 mM, DNA concentrations, and correspond to the theoretical θ ] t values for dsDNA and ssDNA, respectively. Therefore, we confirmed Manning condensation and f conductivitymodels to bevaluable in description of dilute solutions of rod-like polyions. o s . PACSnumbers: 82.35.Rs87.15.hj66.30.hk t a m I. INTRODUCTION logical relevance [3]. - d Counterion condensation is therefore more easily ex- n perimentally studied and the results theoretically in- Mostbiologicallyrelevantmacromolecules(DNA, pro- o terpreted for a dilute solution of rigid, monodisperse c teins, polysaccharides) are polyelectrolytes with a very polyions, which do not change conformation with con- [ distinct behavior compared to neutral polymers or sim- centration. Inadilutesolution,effectively,thecondensed ple electrolytes [1, 2]. When dissolved in polar solvents 3 fractionof counterionsmay be consideredto be found in polyelectrolytes dissociate into a highly charged polyion v a cylindrical cell around the polyion, while the rest may 0 (a macromolecule of extended shape) and many small be taken to be free inside a larger volume that belongs 3 counterions of low valency. The long range nature of to a given polyion [4]. According to the theory, since 7 the electrostatic interactions and the entropy effects due 0 to inhomogeneities in the counterion distributions and the condensed counterions are not chemically bound to . thepolyion,thefreeandcondensedcounterionsexchange 0 to a myriad of polyion configurations control their phe- between the two concentric regions and only a continu- 1 nomenology. 0 ous radial counterion distribution [5] can exist around Thestronglinearchargeofthepolyiontendstoattract 1 the polyion. In other words, there should be no step in the counterions to its immediate vicinity. The conden- : theradialcounteriondistributionwhichwoulddefinethe v sation occurs for polyions with the Manning parameter limit of the cylindricalzone, as shownexperimentally by i X u=lB/b>1,wherelB istheBjerrumlength,thelength EPR (electron paramagnetic resonance) [6]. at which two elementary charges interact in a given sol- r Besidestheoreticalworksconsideringtwotypesofions, a vent with energy equal to the thermal energy kT, while experiments also attempt to quantify the condensedand b is the average distance between the charges on the free counterion fractions. Since only uncondensed, free polyion backbone. If there is more than one charge per counterionscontribute to the osmoticpressureofa poly- Bjerrumlength, the condensationwill tend to effectively electrolyte [7, 8], the measured osmotic pressure of a reduce the linear chargedensity downto 1/l level. The B polyelectrolyte solution evaluates the free counterions condensed ions fraction is then equal to 1−1/u and the fraction[9–11]. Thecondensedcounterionsarethosethat free, uncondensed counterions fraction is θ = 1/u. The move together with the polyion when an electric field is condensationwasmodeledforaninfinitely longandthin applied,whilethefreecounterionswouldmoveintheop- polyion in pure water, with no added salt, which might posite way due to their opposite charge [12–14]. Thus, appear as a rather unrealistic proposition, with no bio- the concept of two types of counterions gets a physical meaning. Thus, the transport techniques may contribute to our ∗URL: http://tvuletic.ifs.hr/; Electronic address: knowledgeofcondensationinpolyelectrolytes. Thetech- [email protected] niquesrangefromelectricaltransportmeasurementslike 2 conductometry [12, 15–17] and capillary electrophoresis [18,19],todiffusionmeasurementsbydynamiclightscat- tering [18–21] or fluorescence correlation spectroscopy c=Ncp (4) [22, 23]. Manning [24, 25] has proposed a rather com- where N is the polyion degree of the polymerization. prehensive and convincing conductivity model for poly- Also,thepolyionchargeisrelatedtothemonomercharge electrolytes and Bordi et al.[14] worked on including the z : scaling theories by Rubinstein et al.[26], in order to sep- p arate the influences from the polyion (conformation and charges), the counterions and the added salt. Z =Nz (5) p p ForasuccessfulquantitativestudyofManningconden- sation by the transport experiments one has to use the Due to the electroneutrality of the solution simplest possible system: a dilute solution of monodis- perse polyelectrolytes with no added salt. Also, an ex- perimentalmethodis neededto separatethe influence of Zpcp =zici =zpc (6) the chargeand conformationof the polyion on the (elec- Thus the conductivity of a polyelectrolyte solution prin- trical) transport. Few experimental works met those re- cipally depends on the concentration of the monomers quirements [9]. In other cases, there was a necessity to c: introducethe modelfortheconformationofthepolyions into the interpretation of the conductivity data [12, 15– 17] which hinders the quantification of the Manning free σ =z c(λ +λ ) (7) counterions fraction. p p i Electrical conductivity in the system under study is a Here we remind that the polyion charge is effectively re- product of three separate factors characterizing the mo- duced, Z = θNz due to the counterion condensation, p p bile charge carriers, summed over all charge species i in and also that only the free fraction θc of counterions is i the system: their charge z e, their concentration n and i i consideredtotakepartinelectricaltransport. Therefore: theirmobilityµ (ratioofcarriervelocityandtheapplied i electric field). σ =θz c(λ +λ ) (8) p p i σ =X(|zi|e)niµi (1) The polyion conductivity λp, being defined by polyion i mobility,actuallystemsfromtheself-diffusioncoefficient of the polyion D and its charge Z , according to Ein- For simple electrolytes (cf. [12, 14]), it is convenient to p p stein’s relation for a charged particle: work with molar concentrations c =n /N and equiva- i i A lentconductivitiesλ =Fµ (FaradayconstantF =eN i i A andNA isAvogadronumber). Theconductivityisasum D = kTµp (9) ofequivalentconductivitiesoftheionicspeciespresentin p eZ p solution, multiplied by the charge (valence) z and con- i centration c of the respective ion: and thus i D σ =Xziciλi (2) λp =FZpekTp (10) i The diffusion coefficient depends on the size and shape ForpolyelectrolytestheexpressionofEq.2isstillvalid. of the particle, as well as on the viscosity of the solution Amonodispersedilutepolyelectrolytewithnoaddedsalt inwhichtheparticleismoving. InsertingEq.10intoeq.8 will contain only two ionic species. For one species, the we get polyion, a large molecule with a relatively small con- centration c and a proportionally large charge Z , the p p equivalent conductivity λ is dependent on its size and D p σ =θz c(FθNz e p +λ ) (11) conformation, thus p p kT i Consequently, the conductivity of a monodisperse σ =Z c λ +z c λ (3) polyelectrolyte without added salt is primarily governed p p p i i i by the self-diffusion coefficient D of its polyion and the p Here we note that c is the concentration of counterions free counterion fraction θ. i released from the polyelectrolyte upon solvation, and is In order to quantify the effects of the diffusion and proportional to the concentration of monomers c con- electrostatics in a polyelectrolyte we used nucleosomal stituting the polyion. The monomer concentration c is DNA fragments 150 bp (50 nm) long. These are ex- related to the polyion concentration c via: pected to be rather rigid and rod-like since the DNA p 3 persistence length is 50 nm [27]. The dilute-semidilute with a large volume of 10mM Tris-Cl- buffer and then crossover concentration for these fragments is ≈2 mM spin-concentratedtothe originalvolume. Theprocedure [28]. The details ofmaterialpreparationandexperimen- wasrepeated3times. Inthe resultingsolution,(denoted talmethodsaregiveninSec. II.AspresentedinSec. III, DNA110*with*toindicatethefluorescentlabeling),the conductivity measurements were complemented by fluo- DNA and Cy5 concentrations were respectively 0.5mM rescence correlation spectroscopy (FCS) measurements and 5µM. of the DNA polyion self-diffusion coefficient Dp. Our For FCS measurements, 2 µL of DNA110* stock were proposition, discussed in Sec. IV is that the conductiv- added into 500 µL DNA146 of varying concentrations ity crossover observed in c = 0.05−1 mM (in basepair) (0.0015-8 mM concentration range) to achieve a 20 nM DNA concentrationrangeresults fromthe DNA denatu- Cy5 concentration. The 10mM Tris of the DNA110* rationthatinducesaconcomitantchangeintheextentof stock was diluted 250 times. Therefore, all experiments Manning condensation. Eventually, we estimate the free were performed at very low salt (c < 0.05mM). For salt counterionsfractionθ andcomparethemwiththevalues FCS calibration, Cy5 fluorophore alone was diluted in predicted by Manning for both ssDNA and dsDNA. pure water to 20 nM. Since the fluorophore concentra- tion can deviate only less than one order of magnitude from this concentration, the amount of fluorescently la- II. MATERIALS & METHODS beled DNA110* was fixed whereas the concentration of DNA146 spans over severalorders of magnitude. A. Monodisperse DNA We will express DNA concentrations as molar concen- B. Fluorescence correlation spectroscopy trations of basepairs (bp) (1g/L equals 1.5 mM bp). Large quantities of practically monodisperse nucleoso- mal DNA fragments were prepared as described in Siko- Fluorescence correlation spectroscopy inherently rav et al.[30] by enzymatic digestion of H1 depleted calf probes the system under study both at single molecule thymus chromatin [31]. This DNA, denoted DNA146, andensemble levels. FCS observesfluorescenceintensity containsfragments150±10bplong(50nm)togetherwith fluctuations emitted by fluorescently labeled objects traces of 300-350 bp fragments that correspond to two diffusing through a small open volume (< 1 fL) defined nucleosomal DNA fragments connected by undigested by the profile of the laser beamand the optics, objective linker DNA. DNA fragmentswere precipitatedwith cold of the microsope. That is, number fluctuations of ethanol, dried and stored at 4oC. The stock solution the molecules entering and leaving the focal volume was prepared by dissolving 10 mg of the Na-DNA pel- are registered as fluorescence variation, which is then let in 0.55 mL pure water. A low protein content was recorded and autocorrelated. Thus following practically verified by UV absorption. DNA146 solutions (0.015 - singlemoleculesweobtainthepropertiesoftheensemble 8mM bp) were prepared by dilution with pure water of [33, 34]. We have used a commercially availabe Zeiss aliquots fromthis 27mMmother solution. To check that ConfoCor II FCS instrument, where the measure- no salt was released from the pellet in addition to the ment volume was defined by a Zeiss Plan-NeoFluar Na+counterions (2 Na+ per bp), an aliquot of the pellet 100x/NA1.3water immersion objective, epi-illumination was dissolvedin 10 mM NaCl, diluted 5 times with pure wasbyHe-Ne632.8nm5mWlaser,forexcitationofCy5 water and spin-filtered to the original volume. This pro- fluorophore. Measurements were performed at 25oC, cedurewasrepeated3times. Anothersamplewassimply the ambient temperature of the temperature stabilized dissolved in pure water. The two samples had similar clean-room. Zeiss proprietary software was used for conductivities (normalized for concentration). We con- autocorrelation function calculation and extraction of cluded that any salt that may have been present in the diffusion times by non-linear least squares fitting [36]. pelletdidnotraisetheconductivitymorethantheequiv- The physical principles of such an experimental set-up alent of 0.2 Na+ ions per basepair. and theoretical background of FCS have been described 110bp dsDNA was prepared as follows. Two separate elswhere [34, 35]. The manner used to obtain the oligonucleotides (ssDNA, 110nt) were purchased (My- self-diffusion coefficient of the molecule under study, in crosynth A.G., CH) [32]. The two sequences were com- our case 110 bp Cy5 labelled dsDNA, is presented in plementaryandoneofthem waslabeledatone endwith brief in the following. The instrument directly measures a covalently bound Cy5 fluorophore. The dry comple- fluorescence intensity for e.g. 30 seconds. The autocor- ments were dissolved in 10 mM Tris-EDTA (TE) buffer relation function G(τ ) is calculated for the intensity c withupto60mMNaCl,mixedandheatedto97oCfor15 trace, with the correlation time τ as the variable. The c minutes to remove any hairpin loops previously formed fluorescence intensity autocorrelation function, G(τ ), is c and then left to cool down for several hours to slowly fitted with a diffusion time, τ. This FCS diffusion time hybridize and form 110 bp long dsDNA. Hybridization relates to the characteristic time for fluorescent particle was checked to be complete on an agarose gel. To re- to diffuse through the focal volume. Autocorrelation move any NaCl excess, the solution was diluted 4 times function decays exponentially and is fitted to 4 104 DNA146 a) 1 1 1 T τ salt<0.01mM c G(τc)= Nf·1+ ττc q(1+(wz00)2ττc)(1+1−Texp(−τT)) -1m) 103 c (12) S 102 Here N is average number of fluorescent molecules f m( in the confocal detection volume. The transition of the s 101 Cy5 fluorophore to the first excited triplet state and a denatured relatively slow relaxation to ground state influence the untreated 100 observed autocorrelation curve. Thus, T, average frac- tion of fluorophores in the triplet state, and τ , lifetime T of the triplet state of the fluorophore are taken into ac- count when fitting. Another fit parameter is z0/w0, the M) structure parameter, i.e.the ratio of the axial and radial m extension of the focal volume. The structure parameter / 200 1 z /w ≈10isobtainedfromfitstoautocorrelationcurves -m 0 0 c measuredforCy5moleculesinpurewatersolution. Then S it is kept as a fixed parameter when τ is later being ex- 100 m( tractedforDNA110*. Theself-diffusioncoefficientD of c p / anyparticleiseasilyobtainedfromitsFCSdiffusiontime s b) 25 oC τ as these are inversely proportional. Since the diffusion 0 coefficientofCy5isknown,D =3.16·10−10m2/s[36] Cy5 10-3 10-2 10-1 100 101 and the diffusion time τ we found to be about 50 µs, Cy5 this provides means for conversion of the diffusion times c (mM) τ into Dp: FIG. 1: Conductometry data for DNA146 solutions before τ D =D Cy5 (13) (black circles) and after denaturation at 97oC (open circles). p Cy5 τ (a) DNA146solution conductivityversusDNAbasepair con- centration. (b)conductivitynormalizedbyconcentrationver- sus DNA basepair concentration. The dotted line shows the C. Conductometry average value for denatured samples, 180 µScm−1/mM. The valuesforuntreatedsamplesatthelowestconcentrationsalso approach this value. Shaded rectangle denotes the crossover Dielectric spectroscopy in the range 100Hz-110MHz concentrationregion. Measurementswereperformedat25oC. was performed with Agilent 4294A impedance analyzer. All the measurements were performed at 25oC. Conduc- tometrydatawasextractedfromthesespectra. Conduc- ter,andnotfromthesolutes. Thusallthesamplesshould tivity was calculated from conductance at 100 kHz and havethe samecapacitanceiftheyhavethe samevolume. capacitance was read at 10 MHz. Conductivity at 100 kHz shows a minimal influence from the electrode polar- ization effects, as well as from the conductivity chamber III. RESULTS resonance at 100 MHz. Basically, one has to measure a spectrum [37], to be able to confidently extract conduc- A. Electrical transport tivity values. Only in this manner, the obtainedconduc- tivitymayberegardedasdcconductivity,theconductiv- ity related to currents of freely mobile charges (polyions We present the dc conductivity data of 0.015−8 mM and free counterions) and not due to polarization cur- DNA146solutions. Experimentswereperformedat25oC rents. We emphasize that all the conductivities of poly- in the absence of added salt (concentration of Na+ or electrolytes have been deducted for 1.5 µS/cm, the con- Trisions<0.05mM)ontheuntreatedDNAsolutionand ductivityofthesolvent[38],i.e.pure water(Milli-Q,Mil- after DNA denaturation. Fig.1(a) emphasizes a general lipore). This residual conductivity is due to the ambient power-lawdependence ofpolyelectrolyteconductivityon CO2 dissolved in pure water. In this manner, pure wa- monomer concentration (see Eq.8). However, a slight S- ter solutions may be regarded as very low salt solutions, shaped bending may be noted in σ(c) for the untreated c <0.01mM,andwelabeledthemappropriately. The sample (black circles). After 20 min at a temperature of salt pH of pure water exposed to air is about 5.5, however 97oC, followed by a quenching to 4oC for a minute, the this is unbuffered. The capacitance at 10MHz serves as conductometry was performed at 25oC. For these, dena- a check of the sample volume for our experimentalsetup tured samples (open circles)the power-lawis apparently [39]. At this high frequency the contribution to the ca- better defined. pacitance comes from the dielectric constant of pure wa- If we normalize the conductivity with concentration, 5 thendatamaybepresentedinaphysicallymorerelevant s) / form,Fig.1(b). Thatis,normalizedconductivityconcen- 2m 5 DNA146 tration dependence is directly related to the behavior of 1 1 salt<0.05mM the molar conductivities of Na+ counterions λ = λ -0 i Na+ 1 and DNA polyions λ : ( 4 p t σ n c =2θ(λNa+ +λp) (14) cie 3 i f f Here, the factor 2 stands for DNA monomer charge e o (izveadlecnocned)zupct=ivi2ty((soeepeEnqc.i8r)c.leWs)eofdteheemdetnhaattutrheedsnaomrmplaels- n c 2 25oC o is, within the data scatter, constant, with a value of 180 si u µS/cm. The data for denatured samples show a rather f 10-3 10-2 10-1 100 101 f highscatter,whichweascribetothedenaturationproce- di c (mM) dure. Eitherdenaturationdidnotproceedtothe fullex- tentforallthe samplesorsomeofthe DNArenaturedin hairpinsduringquenching[40]andthisintroducedacon- FIG. 2: Diffusion coefficient D1e4x6p(c) for DNA146 polyion, ductivity variation. Gradual renaturation, after quench- obtained by fluorescence correlation spectroscopy (FCS) is ing, and during the measurement at 25oC was not an shown versus DNA basepair concentration. Shaded rectan- gledenotesthecrossoverconcentrationregion identifiedfrom issue, as the samples held in our conductivity chamber conductivity measurements. Black triangle denotes diffusion showed a stable conductivity for at least an hour, and coefficient Dss derived for 146 bp ssDNA. the measurement itself lasted for only 2 minutes. Contrary to the denatured DNA, the normalized conductivity of the untreated DNA146 samples shows Here L = Nb is contour length, d is polyion diameter, c a crossover in the 0.05-1 mM concentration range η is viscosity of water (T =298 K). Stellwagen et al.[19] (the crossover region is denoted by a shaded rectan- have reviewed the literature and shown that the expres- gle). Above 1 mM it attains a constant value of 100 sion by Tirado et al. is well applicable to experimental µScm−1/mM, while at the lowest concentration it ap- dataobtainedforDNA moleculesinsizefrom10to1000 proachesthe180µScm−1/mMvaluefortheheattreated, basepairs. Then, the relationship which holds between denaturedDNA146samples. Thisconductivitycrossover thetheoreticalvaluesshouldalsoholdfortheexperimen- has not, to our knowledge, been reported previously, for talvaluesobtainedatvaryingDNA146 concentrationsc. any DNA sample. Thus, Dth B. Polyion diffusion Dexp(c)= 146 Dexp (c) (16) 146 Dth 110∗ 110∗ We had to check whether the observed conductiv- Using Eq.15 to get Dth and Dth and Eq.13 to get ity crossover relates to a change in DNA146 conforma- Dexp fromthediffusion14t6imesτ m1e1a0∗suredforDNA110*, tion due to DNA denaturation expected in the very low we11d0i∗rectly convert τ into Dexp. In this manner, fluores- salt environment [29]. Therefore, we had to obtain the 146 cence correlationspectroscopy providesthe self-diffusion concentration dependence of the self-diffusion coefficient coefficient of DNA146 polyion, Dexp(c) at varying con- D for the DNA146 polyion for the concentration range 146 p centrations (c = 0.0015−8 mM, basepair). The results studied by conductometry. However, the FCS diffusion are shown in Fig.2. timesτ weremeasuredforfluorescentlylabeledDNA110* First, we note that Dexp(c) is practically constant in polyion diffusing freely along the DNA146, but not for 146 the crossover concentration region c = 0.05 − 1 mM DNA146 itself.The labeled DNA is somewhat shorter identified from conductivity measurements (denoted by than the bulk of DNA in the sample solution. Thus dif- fusion coefficients Dexp (c) for DNA110* that may be a shaded rectangle). D1ex46p(c) only starts to vary above 110∗ 1mM.Thiscoincideswiththedilute-semidilutecrossover derivedaccordingtoEq.13hadtobeextrapolatedtoob- tain Dexp(c) values for DNA146. That is, DNA110* and concentrationfor50nmlongDNA146molecules[28]. At 146 higher concentrations the polyions start to overlap and DNA 146, 38 and 50 nm long, respectively, have lengths the apparentviscosity of the solutions changes,inducing comparable to the dsDNA persistence length L = 50 p thedecreaseofthediffusioncoefficient[41]. Thefactthat nm [27]. Thus, an extended rod-like configurationmight ourprobeDNA110*”feels”thephenomenon(the dilute- be expected, especiallyatlowsaltconditions. According semidilute crossover)due to DNA146 demonstrates that to Tirado et al.[18] the translational diffusion coefficient DNA110*diffusionpropertiesindeedreflecttheDNA146 calculated for a rod-like macromolecule is given by diffusion. kT ln(L /d)+0.312 Second, below the crossover range Dexp(c) starts to Dth = c (15) 146 3πη L increase towards the value Dss (black triangle in Fig.2) c 6 calculated, according to Eq.13 and Eq.16 from τss ob- DNA polyion charge, i.e. counterion condensation and 110∗ tainedforthe110baseslongssDNAinpurewater,with- the DNA polyion conformations on our conductometry outDNA146. ThisssDNAisasampleoftheCy5labeled data which is a function of both. We start with DNA synthetic oligonuclotide, dissolved in pure water, before polyion molar conductivity, defined by any treatment (before mixing and hybridization with its complement). It is conceivable that Dss is the limiting D p value for a series of decreasing DNA146 concentrations. λp =F2θNe (17) kT That is, due to the very low salt (practically without First we note that this applies both for ssDNA and ds- it: c < 0.05mM) and rather low DNA concentration salt DNA. Comparing this expression with Eq.10 and 11, we and correspondingly low counterion concentration [29], find that for the valence we inserted z =2. This is due we presume that DNA denatures below 0.05 mM and p to two negative charges (phosphate) being found on a becomes ssDNA. single basepair in native dsDNA, which are still present on two separate nucleotides on two separated strands of ssDNA. Certainly, for an ssDNA of similar N as an ds- IV. DISCUSSION DNA z equals one. However,since two ssDNA polyions p appear in solution as a result of melting of one dsDNA OsmometryfordsDNA[9–11]hasinsofarbeenthepri- molecule,the ssDNA concentrationis doubledcompared mary experimental source of θ data of sufficient quality todsDNA.Thiscancelsthehalvedz ,sothereisnoeffect p to validate extensions to Manning theory [7]. Conduc- of melting on the polyelectrolyte conductivity σ, beyond tometry has been performed on different synthetic poly- the variation in θ or in D . Thus, for the sake of clar- p mers,andhasinsofargivenresultsforθ whichonlyagree ity, we can proceed by keeping the factor 2 within λ , p with Manning within a prefactor of the order of unity, nevermind the DNA state. andmaydependstronglyonthe monomerconcentration Inserting Eq.17 into the expression for DNA conduc- even in dilute solution [12, 15–17, 42]. We note that tivity Eq.14 we get (see also Eq.11) these experiments were either performed in semi-dilute solutions or with polydisperse samples and, most impor- σ(c) Fe tantly, the synthetic polymers used were usually rather =2θλ +4θ2ND (c) (18) flexible. We remindthat inthese casesthe conformation c Na+ p kT of the polyion is not well defined and renders analysis This is a quadratic equation for θ(c) as a variable and difficultduetothenecessitytointroduceamodelforthe D (c) and σ(c) as the parameters: p conformation, besides the model for condensation and conductivity. However, modeling conformation of a flex- λ σ(c) 1 ible polyion in varying salt and monomer concentration θ(c)2+ Na+ θ(c)− =0 (19) D (c)·cte. c 2D (c)·cte. is an elaborate problem in itself [14, 26]. p p On the contrary, our DNA146 has well defined and Here cte. stands for a product of several constants (de- simple rod-like conformation, it is highly monodisperse fined previously): 2NFe/kT. Our measurements of the and forms a dilute solution. DNA in the very low salt DNA146 polyelectrolyte conductivity σ(c) and our inde- conditions is also distinct as it is expected to go through pendent probe of DNA146 diffusion coefficient D (c) = p melting transition with decreasing concentration, so we Dexp (cf. Fig.1 and Fig.2 allow for the equation to be 146 could have ssDNA or dsDNA in solution, depending on solvedforthefreecounterionfractionθ,withoutaneces- concentration [43]. That is, this may allow us to com- sitytomodelthe DNAconformation. Theequationisto pare θ(c) results to Manning values for both ssDNA and be solved repeatedly for each concentration c, resulting dsDNA in one experiment. Actually, counterionconden- in a concentration dependence θ(c). We take only the sation is related to the DNA stability: entropic cost to positive solutions as the physically meaningful. condense or confine the counterions compares with the Theconcentrationdependenceθ(c),accordingtoEq.19 gainin electrostaticfree energyuponDNA denaturation forDNA146inpurewaterisshown(squares)inFig.3. At [3, 44]. This gain is due to the single stranded DNA lower concentrations it reaches a value θ = 0.45. Above (ssDNA) having a lower linear charge density parameter 0.05 mM, in the conductivity crossover regime it starts than dsDNA, u = 1.7 and u = 4.2, respectively. Ac- to decrease,and above about 1 mM, outside crossoverit cordingly, the free counterions fraction should be higher becomes constant at θ = 0.30. It is apparent that the for ssDNA, θ = 0.59 than for dsDNA, θ = 0.24. We experimentally derived range of values for θ falls within have shown in the Introduction how such an increase in the theoretical Manning values for ssDNA and dsDNA, θ would lead to an increase in the polyelectrolyte con- asdenotedbydashedlinesinFig.3. Also,itmaybenoted ductivity, see Eq.11. thatthe50%variationinθ coincideswiththeconductiv- Most importantly, we have measuredthe self-diffusion ity crossover regime (denoted by the shaded rectangle). coefficient D (c) = Dexp(c) of DNA146 as a function This is not surprising, as σ(c)/c is the only variable pa- p 146 of DNA concentration. As we will show in the fol- rameter in Eq.19, while the polyion diffusion coefficient lowing, this allowed us to deconvolute the influence of D is rather constant in this regime. p 7 The preceding calculation did not take into account 1.0 the asymmetry field effect, due to the distortion of the DNA146 counterion atmosphere surrounding the polyion, occur- salt<0.05mM ringwhenthepolyionissubjectedtoanexternalelectric 0.8 field [3, 16, 24]. The original work by Manning was re- viewedandpresentedbyBordiet al.[12]inthe formpre- sentedhere. Asymmetryfieldeffectcanbetaken[16]asif ssDNA 0.6 it corrects θ which appears in expressions for conductiv- ity, Eq.8 and Eq.14, by a factor B =0.866. The factor, calculatedbyManning,originatesinthedifferenceinthe q diffusioncoefficientsofcounterionsinthelimitofinfinite 0.4 dilutionandinthepresenceofpolyions. Asymmetryfield also influences the effective observable molar conductiv- ity λp ofthe polyionthatfigures inthe abovementioned 0.2 dsDNA expressions. First,itcorrectstheeffectivepolyioncharge through the factor B: 25oC 0.0 ′ ′ Fe 10-3 10-2 10-1 100 101 λ =2θBND (20) p pkT c (mM) Second,thediffusioncoefficientofthepolyionisalsocor- rected due to the asymmetry field: FIG. 3: Free counterion fraction θ for 146bp nucleosomal DNA (DNA146) in pure water versus the DNA concentra- 1 D′ =D (21) tion c (in mM basepairs). Squares denote θ(c) calculated p p1+ NDp ·2θB· 1−B according to Eq.19. Diamonds denote θ′(c), the results of a DNa+ B calculation where the assymetry field effect was taken into account, Eq.23. Dashed lines denote the theoretical value of Importantly,thediffusioncoefficientofthepolyionD = p Dexp(c) we have experimentally obtained by FCS, with- θ = 0.24 for dsDNA and θ = 0.59 for ssDNA, as labeled. 146 Shaded rectangle denotes the crossover concentration region out electric field and thus without the asymmetry effect. identified from conductivity measurements. Also, D is the diffusioncoefficientoffree Na+ ionsin Na+ a simple, dilute electrolyte, 1.33·10−9m2/s. Combining the above, the Eq. 14 becomes region (denoted by the shaded rectangle), and the rela- σ ′ ′ tive change of θ in the crossover region remains about =2θB(λ +λ ) (22) c Na+ p 50%. However, the absolute values are different. At low ′ concentrations θ (c) = 0.60 reaches the theoretical value Inserting the Eqs. 20 and 21 into Eq. 22 we get another for ssDNA θ = 0.59, while at high concentrations it de- equationwhichmaybeusedtoobtainthefreecounterion creases only down to θ =0.4. fraction: The crossover in conductivity that we have observed for DNA in very low salt reflects as the crossoverin θ(c) σ NDp 1 (or θ′(c)), the Manning free counterion fraction. The =2θB·(1+2θB ) c·λNa+ DNa+ 1+ NDp · 1−B ·2θB exact θ values may depend whether basic corrections DNa+ B to polyelectrolyte conductivity are taken into account. (23) Notwithstanding the details of the conductivity model InanalogywithEq.19,werewritethisintoaquadratic [17], we emphasize that the obtained extremalvalues for equation: θ correspond to Manning model predictions both for ss- 1−B DNAanddsDNA(denotedintheFig.3bydashedlines). M/Bx(c)2+(1−AM )x(c)−A=0 (24) B This also corroboratesthe expected DNA melting across the studied DNA concentration range. M stands for NDp and A stands for σ . We remind Notably, our result complements the unique result by DNa+ cλNa+ thatbothM andAareobtainedexperimentallyasfunc- Auer and Aleksandrowitz [9] obtained by osmometry tions of c, and that the equation was solved separately for DNA solutions without added salt. These authors at each c value, to get x(c) (we only take the positive, studied somewhat higher DNA concentrationrange 2-10 physical solution). mM. For dsDNA they obtained an osmotic coefficient ′ InFig.3weshowtheasymmetryfieldcorrectedθ (c)= φ = 0.16 that would correspond to θ = 0.32 and for 0 ′ x(c)/2B (diamonds). Overallbehaviourofθ (c)isanalo- ssDNA they got φ = 0.24 corresponding to θ = 0.48 0 goustoθ(c)behaviourcalculatedwithEq.19. Thevaria- for ssDNA. The relationship between θ and φ is given 0 ′ tioninθ(c) alsooccurswithintheconductivitycrossover by Manning [3, 17]. We find that it is very significant 8 that both osmometry, and our technique find θ for ss- tion molar conductivity normalized by DNA concentra- DNA only 50% larger than for dsDNA, while Manning tion, attains almost 100% higher value below 0.05 mM condensationtheorypredictsmorethan100%! Whilethe than above 1 mM (basepair). The results for solutions details of DNA conformations (e.g. coiling or formation of ssDNA (actually, samples of thermally denatured ds- of hairpins in ssDNA) might be in the origin of this dis- DNA) lacked this conductivity crossover. Then, we ap- crepancy, the limitations of Manning model should also plied fluorescencecorrelationspectroscopy(FCS) to find be acknowledged - DNA is not a simple line charge. that the diffusion coefficient of DNA polyion D is prac- p The correspondence between the osmotic and trans- tically constant in the crossover region. Thus, we have portmeasurementsdrawsourfinalremark. Thatis,both shown that the origin for the conductivity crossover lies techniques independently validate Mannings notion that inthe increaseoffreechargefractionanddecreaseofthe counterions differentiate into two functionally separate effective polyioncharge,due to changesinManning con- populations. However, there is no a priori reason for densation, which we were able to quantify. Depending if these experiments to findsimilar fractionsfor these pop- the Manning asymmetry field effect was taken into con- ulations. Thetransporttechniquesmeasurethecontribu- ductivitymodelornot,weobtainedthevalueswithinthe tionto polyelectrolyteconductivity ofthe polyionwhose ranges θ =0.40−0.60 or θ =0.30−0.45, respectively. charge is reduced due to the condensed counterions that The conductivity crossover and θ variation are easily move along, as well as the contribution of free counte- related to be due to DNA denaturation. However, the rions that move opposite to the polyion in the external 50% variation in θ that we observe is smaller than what electricfield. Osmometryidentifiesasfreethecounterion Manningcondensationtheorypredictsasadifferencebe- fraction that contributes to the osmotic pressure of the tweendsDNAandssDNA(morethan100%). Neverthe- solution. That is, those counterions that diffuse freely less, a 50% difference in θ between ssDNA and dsDNA at distance from the polyion. However, as mentioned in wasalsoobtainedinosmoticpressurestudiesbyotherau- theIntroduction,theradialdistributionofcounterionsis thors. We also found surprising that variations in DNA continuous and, beyond Manning model, in calculations conformation due to denaturation appear to be of lesser basedon Poisson-Boltzmann(PB) theory it is rather ar- influence on the polyion conductivity. The above two bitrary to define any given distance from the polyion as issuesleadtothe questionhowDNAconformationspop- theextentofcondensedcounterionszone[5]. Specifically, ulationchangeswithadecreaseinDNAconcentrationin a nonlinear PB model has been worked out for a system the verylow saltenvironment. This is the subject ofour very similar to our experimental one - rod-like polyelec- following paper [45]. trolyte dilute solutions in very low salt, and is based on Further application of FCS, with samples subjected defining the two (condensed and free) zones around the to an external electric field (similar as used for conduc- polyion[4]. Now,accordingtoour experiments,the con- tometry) could quantitate asymmetry field effect on the densed counterions zone radius should be less arbitrary. diffusioncoefficientof DNA146polyionandrevealinde- That is, as our conductometry study detects a reduced tailto whatextentthecondensedcounterionsmovewith DNA polyion chargedue to condensationandas the dif- the polyion. Finally, combined conductometry and FCS fusionresultsindicateaDNApolyiondiameterof2.6nm, studies of dilute monodisperse DNA in added salt solu- then this is also the condensed counterions zone diame- tions could extend the studies of θ and further comple- ter. The condensed counterions are to be found in the ment the data obtained by osmometry. immediate vicinity of the polyion, as initially suggested by Manning, and the cylindrical condensed counterions zone depicted in [4] should be very thin. Acknowledgement We gratefully acknowledge A.S. Smith and R. Pod- V. SUMMARY AND CONCLUSION gornik for illuminating discussions. T.V. is thankful to S. Kempter for all her assistance in the lab. This work In this work we have quantified Manning free (uncon- is based on the support from the Unity through Knowl- densed)counterionsfractionθ fordilutesolutionsofrod- edge Fund, Croatia under Grant 22/08. The work was like polyions - 150bp DNA fragments, in very low salt in part funded by IntElBioMat ESF activity. 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