DNA AS A POLYELECTROLYTEEric Raspaud

Laboratoire de Physique des SolidesBt. 510,Universit Paris Sud91405 Orsay, FranceEmail : raspaud@lps.u-psud.frCargse, 2002 - *

Cargse, 2002 - *Unlike proteins, each of the DNA basic units (the base pair) is composed of two identical ionisable groups, the phosphate groups.Projection of the two phosphates on the double-helix axis :(B-DNA)

Linear charge density :2 e / 3.4 ~ 6 e / nm

equivalently,1 e / 1.7

One of the most highly charged systems

Cargse, 2002 - *Amino-acid size ~ 4

Typical Linear Charge density ~ 1 /( 4 4 ) ~ 0.6 e / nmFor proteins,5 / 20 basic units may be ionized :

basic : lysine, arginine, histidine (-NH2+)

acidic : aspartic, glutamic (-COO-)For the Cell membrane,

Typical Surface Charge Density ~ 0.1- 1 e / nm2

Proteins + (RNA or DNA) complexes :Cargse, 2002 - *Nucleosome Core ParticleSurface Charge Density ~ 6 e / nm2Linear Charge densityTMV ~ 2 e / nmfd-virus ~ 5 e / nmTobacco Mosaic Virus

Cargse, 2002 - *(Bloomfield, 1999)

Cargse, 2002 - * 3

Cargse, 2002 - *1_ Brief Introduction on the Ionic dissociation and hydration1_a) simple salt1_b) DNA

2_ Simple salts : the Debye-Hckel model2_a) spatial distribution2_b) thermodynamic consequences

3_ Polyelectrolyte and the counterions distribution3_a) the Poisson-Boltzmann equation3_b) the Manning model

4_ Thermodynamic coefficients and the free counterions4_a) the radius effect4_b) the length effect

5_ Conformation and Interaction5_a) Thermal denaturation5_b) Interaction and chain conformation

6_ Multivalent Counterions6_a) collapse and phase diagram6_b) ionic correlation Menu

Cargse, 2002 - *Coulombs law (1780)

Attraction force between two elemantary charges e separated by a distance dand its associated energy :For a typical ionic crystal,Thermal Energy,kT 10-21 Joules (0.6 kcal/mol)In water,The salt dissociatesin water1_ Brief Introduction on the Ionic dissociation and hydration1_a) simple saltRoughly

In general, one defines a characteristic lengthof the ions dissociation/association :The Bjerrum length lB Cargse, 2002 - *In water, lB = 7.14 For a multivalent salt zi : zj

|zizj | lB

if d < lB thenion pairing occurs

d(Oosawa, 1971; Bloomfield, 2000)Similar notionfor the coupleIon - charged monomerSite bound :ion immobilizedand dehydratedpKa and weak electrolytesEx. : acrylate group - COO- and Mg2+ Cargse, 2002 - *(Sabbagh & Delsanti, 2000)strong electrolytesEx. : sulfonate group SO3-

Cargse, 2002 - *18 23 water molecules per nucleotide(from IRSpectroscopy)(Bloomfield, 2000)NMR relaxation :

Na+ associated with DNAare not dehydrated orimmobilized(Bleam et al., 1980)

Lost of Mn2+ hydrationupon binding to DNA(Granot & Kearns, 1982)1_b) DNA

Cargse, 2002 - *For the phosphate group,For the bases,

Neutral at pH 7

Cargse, 2002 - *Solution of strong electrolytes (McQuarrie, 1976) :Positive and negative charges in a continuum medium of dielectric constant ee0rijri

rj

The Coulombic potential acting upon the ith ion located at rior at the arbritary point r :with r (r) the total charge density at r2_ Simple salts : the Debye-Hckel model (1923)How does the charges distribution r(r) change with respect tothe distance r from a central ion ?

Cargse, 2002 - *r

r(r) is related to the electrostatic potential y(r) by the Poissons law and by the Boltzmanns distribution :- Ionic size : s- cations and anions charges : z+e and z- e- concentration of the bulk solution : C+0, C-0 z+ e C+0 + z- e C-0 = 0 (Electroneutrality)r (r ) = z+e C+0 e-z+ey(r)/kT + z-e C-0 e-z-ey(r)/kT with Fi = ziey(r) the energy of the charge zie located at the distance r from the central ion or equivalently the electrostatic work required to bring the charge from the reference state to the distance r.Dy(r) = -r(r )/ee02_a) spatial distribution

Solving the Poisson-Boltzmann equation :ziey(r)

Cargse, 2002 - *for a 1:1 salt (NaCl,)s = 5 1/k s = 6 1/k s

Csalt (mM)1/k ()11001030100101033

Cargse, 2002 - *Knowing the ion spatial distribution and the electrostatic potential and charging the ions, one can calculate the electrostatic free energies of the system :

electrostatic energy

Felec / V kT= - (k3/12p) t(k s)

Felec < 0 due to the ionic atmospherePredict then the thermodynamic coefficients2_b) thermodynamic consequences

Cargse, 2002 - *3_ Polyelectrolyte and the counterions distributionrSimple saltrA polyelectrolyteinto an ionic solutionApplysimilarcalculation ?Yes, we can start from the Poisson-Boltzmann equation No, the approximation ziey(r)

3_a) the Poisson-Boltzmann equationCargse, 2002 - *The central ion becomes the polyelectrolyte :

y(r) its electrostatic potential at a distance r r (r) the charge density of the surrounded ionsa non-linear differential equation

Neglected : * the finite size and spatial correlations of the small ions* the non-uniformity of the dielectric constant (water) Cargse, 2002 - * a exact solution if there is no added salt :(Fuoss et al., 1951); (Zimm & Le Bret, 1983); (Deserno et al., 2000)The chain :an infinitely long cylinder of radius r02r0The monomeric unit : Size b and valence zmFor B DNA,r0 = 10 a dinucleotideb = 3.4 and zm = -2-2e /3.4Or equivalently :-1e /1.7with b = 1.7 and zm = -1The linear charge density : zme/b

Cargse, 2002 - *Number of charges per Bjerrum length lB :xM = lB / b the Manning parameterFor B DNA, b = 1.7 :xM = 4.22r0bWithout added salt :Only the counterions are presentzc e Cctotal + zm e Cmtotal = 0Counterions

zc = + 1monovalent (Na+, K+,)Monomers

zm = - 1a nucleotide(Phosphate-)

Cargse, 2002 - *The Cell model : each chain enclosed in a coaxial cylindric cell2r02R- ionic distribution :

the whole solution = a single cylindric cell- concentration effect :

DNA dilution= R increase

Cmtotal = Cctotal = 1/pbR2

Cargse, 2002 - *Returning to the Poisson-Boltzmann equation,

Dy(r) = - r(r) /ee0 with r(r) the counterions charge density at a radial distance r from the chainr(r) = zce C(R) e-zce y(r) /kT 0r0RrAssumption : y(r = R)=0 f(r)= k2 ef(r) with f(r)= - zcey(r) / kTThe screening length 1/k becomes :k2 = 4plB zc2C(R)[ for the simple salt, k2 = 4p lB (z-2C-0 + z+2C+0 ) ]

Cargse, 2002 - *

Boundary conditions :

a whole cell is neutral,for r = R f(r)/r = 0

the chain is charged with xM = lB / b = 4.2,for r = r0 f(r)/r = - 2 xM /r0

f(r) = -2 ln( [kr /g2] cos (g ln[r/RM])) !!!with g and RM two numerical constantsdependent on xM , r0 , Rr0Rrf(r)r0 = 10 R = 50 (40 g/l DNA)1/ k 25 xM =4.21.50.8

Cargse, 2002 - *Fraction of counterions within a cylinder of radius r0r0Rrr/r0R = 50 xM =4.21.50.8Counterions accumulationMonotonic increase for xM < 1

Cargse, 2002 - *R = 250 (1.7 g/l DNA)Counterions accumulationsimilar variationSimilar to simple salt;Debye Hckel or linearised Poisson-Boltzmann equationzey(r)

Cargse, 2002 - *Using the linearised equation,f(r)= k2f(r)

f(r)= - [2 xM / kr0 K1(kr0) ] K0(kr)

with f(kr 1, a strong counterion condensation occurs near the chain; only the counterions far from the chain are submitted to a low electrostatic potential and may be treated like simple salts.

Cargse, 2002 - *C (r) (M)

local concentrationof countreions

0r0Rr regular spacing ofthe phosphate chargeon a cylinder

charges set along thedouble-helixMonte-Carlo simulation (Le Bret & Zimm, 1984) :

Cargse, 2002 - *Experimentally :X-Ray scattering (Chang et al., 1990)

Heavy counterions (Thallium+)Neutron scattering (van der Maarel et al., 1992)

Contrast matching

3_b) the Manning model and the limiting lawsAn infinite line charge with a charge density xM Simple salt is present in excess over polyelectrolyteTwo-states model for xM > 1: Cargse, 2002 - *free ions treated in theDebye-Hckel approximationcondensed ions reducing the monomer chargeto an effective charge qeff. (Manning, 1969; 1978)

Cargse, 2002 - *Calculation of the effective charge qeff. :The condensed ions decrease the repulsions between the chain charges :gel. = (q eff. 2 /4pee0) e-krij/ rij

By summing over all pairs of effective charges of the chain, the electrostatic contribution to the free energy per monomer becomes for kb

Cargse, 2002 - *- The mixing entropy decreases the number of condensed ions :Volume of the region where ions are said to be condensed : VpThe ionic local concentration : Clocal = 103 q /VpConcentration of the free salt : CsAs q ions are confined in Vp,the lost of mixing energy is

gmix../ kT = q ln (Clocal/Cs)

Cargse, 2002 - *Minimization of the free energy g el. + g mix. with respect to q :

1 - ln(Vp Cs/103) - zczm [1+ (zc/zm)q ] xM ln (kb)2 =0

Focus on the salt concentration Cs dependence with k2 ~ Cs

- ln Cs ~ zczm [1+ (zc/zm)q ] xM ln Cs

The only solution to cancel the Cs dependence (in particular when Cs 0) :-1 = zczm [1+ (zc/zm)q ] xM For zm = zc = 1q = 1- 1/ xM

Cargse, 2002 - *radial extension of thecondensation region76 % of the phosphateshave a condensed counterion or the effective charge is reduced by 1/4 For a long DNA chain (of size L) in a monovalent salt,its structural charge is proportional to L/band its effective charge to (L/b) (1-q) = (L/b)/xM = L/lB !

The DNA chain behaves like a chain where the distance between charged monomers is equal to lB

b () xM saltzc : 1zcqClocal (M)Dx ()B-DNA1.74.210.761.177.420.880.3911.240.940.1216.9Single-stranded DNA41.810.440.2112.540.860.02132.2

Cargse, 2002 - *4_ Thermodynamic coefficients and the free counterionsfree in the Manning sense :

For xM < 1, all the counterions are freeFor xM > 1, only 1/ xM counterions are free

But they are submitted to the electrostatic potential created by the DNA chains;Similar to the ionic atmosphere of the Debye-Hckel interations for simple salts

thermodynamically free : where ions are only submitted to the thermal agitation

C(R)y(r = R) = 0 Cargse, 2002 - *- For the Poisson-Boltzmann cell model without added salt,Only the counterions far from the chainor located at r = Rare thermodynamically free- For DNA in excess of salt, only the ions far from the chain are thermodynamically free For xM > 1 and the limiting law,

Only 1/ 2xM counterions are thermodynamically free

Cargse, 2002 - *Na-DNA +Salt CsSalt Cs0Cs0 = C Na+0 = C Cl-0Cs = C Cl-C Na+ = Cs + Cc Na+

Counterionscoming from Na-DNASemi-permeable membraneAt the thermodynamical equilibrium,the ions chemical potentialsfrom both sides are equal :

m(Cc Na+ , Cs ) = m( Cs0 )

The Donnan effect :(NaCl)reservoir

4_a) the radius effectCargse, 2002 - *In an excess of salt (NaCl) with diluted Na-DNA,Cs0 - Cs ( 1/ 2xM ) Cc Na+ /2 and2G 1/ 2xM The salt exclusion factor : G = lim (Cs0 - Cs )/ CDNA for Cc Na+ or CDNA 0

Information on the fraction of thermodynamically free counterions

Cargse, 2002 - *Theoretically,by solving the P-B equation and averaging C Na+ (r) and C Cl-(r)

2G = (1/ 2xM ) f( kr0 ) withf(kr0 0) = 1 the limiting law2Gkr01/ 2xMCs010 mM0.1 M1 M(Strauss et al., 1966-67; Shack et al., 1952)Simple salt limit 2G =1Experimentally,the salt concentration in the DNA compartment Cs is determined by selective electrodes

Cargse, 2002 - *with fosm the osmotic coefficientThe osmotic pressure p :Na-DNA +Salt CsSalt Cs0hp = r g hUsed by T. Odijk for the free energy of DNA in bacteriophage (1998)In excess of Na-DNA compared to NaCl,Cs 0 the salt is excludedp / kT = fosm Cc Na+Returning to the P-B equation in salt-free conditions,fosm = C(R) / Cc Na+ ;fosm = ( 1+g(kr0 ) ) / 2xMwith g(kr0 ) 0

Cargse, 2002 - *2Gfosm1/ 2xMkr0100 g/l1 g/lCDNA(Auer & Alexandrowicz, 1969); (Raspaud et al., 2000) Strong deviation from the limiting law due to the finite radius r0

Theories and Experiments arequalitatively in good agreement.Quantitatively..

Cargse, 2002 - *(Stigter, 1978)(Nicolai & Mandel, 1989)(Liu et al., 1998) q/qeff= 5.9(Griffey, unpublished) 4.2(Padmanabhan et al., 1988) 2.1(Braulin et al., 1987) 2.5 To be compared with xM = 4.2 Permeability of the double-helix Specific interactions different sensibilities ofthe different technicsq/qeff

Cargse, 2002 - *End effectsL1/kCounterions evaporation4_b) the length effect

1/k1/kCargse, 2002 - *fraction of condensed counterions(Gonzalez-Mozuelos & Olvera de la Cruz, 1995)dilutionqeff ~ - ln C

Cargse, 2002 - *Returning to DNA, Importance of end effects

for short oligonucleotide helices : Simulations of Olmsted et al. (1991)

Cs = 1.8 mM1/k = 73 2G = (1/ 2xM ) f( kr0 )1/kCs (mM)k Lend effects

Cargse, 2002 - *(Olmsted et al., 1989)The localconcentrationat the DNA surfaceN = 12N = 24N 72Ends effectbecomes negligeable on averagebut evaporation stilloccurs at the ends.1/k

Cargse, 2002 - *5_ Conformation and Interaction5_a) Thermal denaturationCalculation of the free energy using the charge renormalisation (Manning, 1972)Non-electrostatic term+L/lB or np/xM charged phosphateIonic atmosphere (ns salt + np/xM free ions)(salt in excess but at low concentration)Interaction energyof the chain with its ionic atmosphere+Entropic termof the salt, of the free ions- (np/xM ) kT ln k+ (np/xM ) kT ln CsDestabilizesStabilizesthe double-helixAdditionof salt(np/xM ) denaturated > (np/xM )native DNA +

Cargse, 2002 - *+ (np/2xM ) kT ln Cs- (np/xM ) kT ln k+ (np/xM ) kT ln Csk ~ Cs 1/2The dominant term is entropicFinally, the electrostatic free energy Gele= H TS > 0and dominated by the entropy S

Entropic cost to condense or confine the counterions

Gain in electrostatic free energy to denaturate DNAD Gele = Gelesingle-stranded Geledouble-stranded < 0

Cargse, 2002 - *At the melting temperature Tm, D Gnon-ele + D Gele = 0 After manipulation :

d Tm /d log Cs A [ (1/2xM )denaturated-(1/2xM )native]

with A = k Tm2/DHmT4 DNA (Record, 1975)

From experimental Tm and calculated D GeleGnon-elepernucleotide* Tm = k log Cs + 0.41 XGC + 81.5(Korolev et al., 1998; see also Rouzina & Bloomfield, 1999)(Schildkraut & Lifson, 1965)* d Tm /d log Cs ~ D (1/2xM ) ~ D (2G )(Olmsted et al., 1991)Cargse, 2002 - *

Cargse, 2002 - *5_b) Interaction and chain conformationIntermolecular interactions :

between short DNA L = Lp = 50 nm (150 bp)For neutral rods,excluded volume v0 L2 dd = 2r0

For DNA rods,v L2deff

with an effective diameterdeff = 2r0 + (1/k) f( xM , k r0 )At low salt , deff >> d and v >> v0 In the high salt limit, deff = d and v = v0 e-kr(Brenner & Parsegian, 1972)

Cargse, 2002 - *vv0CNaCl (M)(Nicolai & Mandel, 1989)50 v0v L2deffisotropicanisotropicC i - afi-a ~ deff / LCi a ~ 1 / v CDNA increases with Cs

Cargse, 2002 - *Intramolecular interactions :The persistence length Lp : the bending flexibility

Bending makes the phosphate charges lie closerfrom one another : electrostatic contribution to Lp

Lp = l0 + lele Using the Debye-Hckel approximation and the limiting law (k Lp >>1),lele = lOSF = lB / 4 (kb)2withblB

(Odijk, 1977; Skolnick & Fixman, 1977) Using the P-B equation, lele = lOSF f(k r0 ) (Fixman, 1982; Le Bret, 1982)

Cargse, 2002 - *Lp (nm)k r01 M0.10.01CNaCllinear Col E1 plasmid(Borochov et al., 1981) Lp = l0 + lOSF

Lp = l0 + lOSF f(k r0 )l0 = 50 nm

Rg : radius of gyrationRg (nm) of T7 DNA (40 103 bp)Cargse, 2002 - *CNaClExperimental data(Sobel & Harpst, 1991)Including : excluded volumewith deff

persistence lengthwith lOSF f(k r0 ), l0 = 40 nm

(Stigter, 1985)Random coil

6_ Multivalent Counterions6_a) collapse and phase diagramCargse, 2002 - *In the Manning approach,The charge fraction zcq of condensed ions increases like zcq = 1- 1/ zcxM the effective charge is reduced by 94 %Experimentally, electrophoretic mobility m (cm2/V.s. 10-4) :mzcLinear plasmid in 50 mM monovalent salt+ 0.2 mM of zc-valent (Ma & Bloomfield, 1994)

b () xM saltzc : 1zcqClocal (M)Dx ()B-DNA1.74.210.761.177.420.880.3911.240.940.1216.9Single-stranded DNA41.810.440.2112.540.860.02132.2

1.4711.3120.923

Cargse, 2002 - *Low monovalent salt (1 mM) + Cobalt Hexamine zc = 3 on l DNACobalt Hexamine ConcentrationLight scattering (Widom & Baldwin, 1980-83)Toroidal globuleScatteredIntensity

Diffusion coefficientRH 40 nmwith spermidine (zc = 3),RH 50 nm(Wilson & Bloomfield, 1979)

Cargse, 2002 - *For higher DNA concentrations :aggregationMy first clichs !Cryo-electron microscopy

Cargse, 2002 - *Phase diagramSpermineConcentration (M)DNA concentration (M)DNA redissolutionIn excess ofspermineDNA precipitation(Raspaud et al., 1998-99)

Cargse, 2002 - *Attractive interaction : in violation of the basic P-B theory

a simple charge (quasi)-neutralization with a non-Coulombic attractive force ?

doesnt explain the redissolution in excess of multivalent salt

Cargse, 2002 - * a salt bridge model : ion-bridging model ?

Intermolecular interactions in dilute solutions with an excluded volume v :electrostatic repulsionbetween like-charged chainsshort-range attraction----localchargeinversionThe reduced effective charge : - qeff xM11 / zcCmultivalent saltCspermine (mM)CDNA (mM)(Olvera de la Cruz et al., 1995)

Cargse, 2002 - *Fraction of counterionswithin a cylinder of radius r(from the P-B equation)Few AngstromsStrong charge accumulationnear the surface ( one layer)6_b) ionic correlation

Cargse, 2002 - *Coulomb energy between neighboring ions azWc = (zce)2/4pee0 azCoupling parameter :

W = Wc / kT For multivalent ions, zc > 1 and W > 1 strong coupling Important longitudinal correlations between ionsstrongly correlated liquid (Rouzina & Bloomfield, 1996; Grosberg et al., 2002 and references therein)

Cargse, 2002 - *Two-dimensional lattice ofthe multivalent counterionsaround the surfaceCorrelation Energy ecper ion < 0 (attractive) ec Wc If n : local concentration ofthe correlated liquidec(2n) < ec(n ) Attraction between DNA

Cargse, 2002 - * In the dissolved state,Free energy per DNA molecule with qeff

- In the condensed state, using ec(Nguyen et al., 2000 ; Nguyen & Shklovskii, 2001)

Cargse, 2002 - *About the charge inversion,Spermidine concentration (zc = + 3)---+Spermine concentration (zc = + 4)NucleosomeCore Particleszc = + 2+ 3+ 4-+(Yamasaki et al., 2001)(Raspaud et al., 1999; De Frutos et al., 2001)Short DNAfragmentsT4 DNA

Cargse, 2002 - * specific effects : polyamine is a polycation

at long length scale, behaves like a zc-valent ionssimilar shape of the phase diagramat short length scale (az), each of the zc charged groupsmay be considered as a monovalent ionattenuated correlation effect(Lyubartsev & Nordenskild, 1997)

- the electrophoretic mobility and its associated zeta potentialSlipping surface located at rzduring the electrophoretic migration

Comparison betweenrz and the ionic size srz rz = s rz = 2s+-r / s(Deserno et al., 2001)

Cargse, 2002 - * mixture of mono and zc-valent cations condensed onto the DNAattenuated correlation effect

- non-ideal behavior of the free multivalent saltDebye-Hckel ionic atmosphere and the ionic size effectreduced and may suppressed the DNA overcharging(Solis & Olvera de la Cruz, 2001)

Cargse, 2002 - *DNA IS A POLYELECTROLYTEAs a conclusion,which behaves like others charged systems (for instance, synthetic materials)and which behavior may be understood in a first approach by electrostatics notion.

Salt may regulate - DNA-DNA interactions and their conformational changeswithout specific effects

- DNA - charged proteins interactions

Course limited to Simple Naked DNA study with ions,more complex behavior for Chromatin,

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For multivalent, dont forget that it also depends on the ionic size via the Born energy (ze/a)