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DNA AS A POLYELECTROLYTE

Cargèse, 2002 - 1. DNA AS A POLYELECTROLYTE. Eric Raspaud Laboratoire de Physique des Solides Bât. 510, Université Paris –Sud 91405 Orsay, France Email : raspaud@lps.u-psud.fr. Cargèse, 2002 - 2.

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DNA AS A POLYELECTROLYTE

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  1. Cargèse, 2002 - 1 DNA AS A POLYELECTROLYTE Eric Raspaud Laboratoire de Physique des Solides Bât. 510, Université Paris –Sud 91405 Orsay, France Email : raspaud@lps.u-psud.fr

  2. Cargèse, 2002 - 2 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 Å ~ 6 e / nm equivalently, 1 e / 1.7 Å One of the most highly charged systems

  3. Cargèse, 2002 - 3 For proteins, 5 / 20 basic units may be ionized : basic : lysine, arginine, histidine (-NH2+) acidic : aspartic, glutamic (-COO-) Amino-acid size ~ 4 Å Typical Linear Charge density ~ 1 /( 4  4 Å) ~ 0.6 e / nm For the Cell membrane, Typical Surface Charge Density~ 0.1- 1 e / nm2

  4. 105 Å 60 Å Proteins + (RNA or DNA) complexes : Cargèse, 2002 - 4 Nucleosome Core Particle Tobacco Mosaic Virus Linear Charge density TMV ~ 2 e / nm fd-virus ~ 5 e / nm Surface Charge Density~ 6 e / nm2

  5. Cargèse, 2002 - 5 (Bloomfield, 1999)

  6. Cargèse, 2002 - 6  3

  7. Menu Cargèse, 2002 - 7 1_ Brief Introduction on the Ionic dissociation and hydration 1_a) simple salt 1_b) DNA 2_ Simple salts : the Debye-Hückel model 2_a) spatial distribution 2_b) thermodynamic consequences 3_ Polyelectrolyte and the counterions distribution 3_a) the Poisson-Boltzmann equation 3_b) the Manning model 4_ Thermodynamic coefficients and the « free » counterions 4_a) the radius effect 4_b) the length effect 5_ Conformation and Interaction 5_a) Thermal denaturation 5_b) Interaction and chain conformation 6_ MultivalentCounterions 6_a) collapse and phase diagram 6_b) ionic correlation

  8. 1_ Brief Introduction on the Ionic dissociation and hydration1_a) simple salt Cargèse, 2002 - 8 Thermal Energy, kT 10-21Joules (0.6 kcal/mol) Coulomb’s law(1780) Attraction force between two elemantary charges e separated by a distance d Roughly and its associated energy : The salt dissociates in water In water, For a typical ionic crystal,

  9. Cargèse, 2002 - 9 In general, one defines a characteristic length of the ions dissociation/association : The Bjerrum lengthlB In water, lB = 7.14 Å if d < lB then ion pairing occurs For a multivalent salt zi : zj |zizj | lB 

  10. Cargèse, 2002 - 10 Similar notion for the couple Ion - charged monomer d Site – bound : ion immobilized and dehydrated (Oosawa, 1971; Bloomfield, 2000) strong electrolytes Ex. : sulfonate group – SO3- pKa and weak electrolytes Ex. : acrylate group - COO- and Mg2+ (Sabbagh & Delsanti, 2000)

  11. 1_b) DNA Cargèse, 2002 - 11 NMR relaxation : Na+ associated with DNA are not dehydrated or immobilized (Bleam et al., 1980) Lost of Mn2+ hydration upon binding to DNA (Granot & Kearns, 1982) 18 – 23 water molecules per nucleotide (from IR Spectroscopy) (Bloomfield, 2000)

  12. Cargèse, 2002 - 12 For the phosphate group, For the bases, Neutral at pH 7

  13. 2_ Simple salts : the Debye-Hückel model (1923) Cargèse, 2002 - 13 Solution of strong electrolytes (McQuarrie, 1976) : Positive and negative charges in a continuum medium of dielectric constant ee0 rj The Coulombic potential acting upon the ith ion located at ri rij ri or at the arbritary point r : with r (r) the total charge density at r How does the charges distribution r(r) change with respect to the distance r from a central ion ?

  14. 2_a) spatial distribution Cargèse, 2002 - 14 - 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) is related to the electrostatic potential y(r) by the Poisson’s law and by the Boltzmann’s distribution : r Dy(r) = -r(r )/ee0 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.

  15. k2 = 4p lB (z-2C-0+ z+2C+0) 1/k : the Debye screening length • (r) = A e-kr/ r a screened coulombic potential with A = (z0 lB eks)/(1+ks) Cargèse, 2002 - 15 Solving the Poisson-Boltzmann equation : ziey(r) << kT (kT/e  25mV) Expanding the exponential, one gets the linearized Poisson-Boltzmann or the Debye-Hückel equation : Df (r) = k2f(r) withf(r)= ey(r) / kT Boundary conditions : r→ , f → 0 r = s, E = - df /dr = - z0 lB /s2 (from the Gauss theorem with z0thevalence of the central ion)

  16. Fraction of charge within a sphere of radius r and concentric to the central ion r f(r) monotonic increase Charge distribution r(r) opposite sign r/s r same sign z0= -1 « Ionic atmosphere » around the central ion Cargèse, 2002 - 16 for a 1:1 salt (NaCl,…) 1/k s s= 5 Å 1/k s = 6

  17. 2_b) thermodynamic consequences Cargèse, 2002 - 17 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 < 0due to the ionic atmosphere Predict then the thermodynamic coefficients

  18. 3_ Polyelectrolyte and the counterions distribution Cargèse, 2002 - 18 A polyelectrolyte into an ionic solution Simple salt r r Apply similar calculation ? • Yes, we can start from the Poisson-Boltzmann equation • No, the approximation ziey(r) << kT is no more valid

  19. 3_a) the Poisson-Boltzmann equation Dy(r) ~r(r) In cylindrical geometry, Dy(r) = (1/r)  (r  y/  r) /  r Dy(r)~ e-zey(r) /kT r (r) ~ e-zey(r)/kT Cargèse, 2002 - 19 The central ion becomes the polyelectrolyte : y(r) its electrostatic potential at a distance r r (r)the charge density of the surrounded ions a non-linear differential equation

  20. Cargèse, 2002 - 20 • a exact solution if there is no added salt : (Fuoss et al., 1951); (Zimm & Le Bret, 1983); (Deserno et al., 2000) Neglected : * the finite size and spatial correlations of the small ions * the non-uniformity of the dielectric constant (water) The monomeric unit : Size b and valence zm The chain : an infinitely long cylinder of radius r0 The linear charge density : zme/b For B – DNA, a dinucleotide b = 3.4 Å and zm = -2 -2e /3.4 Or equivalently : -1e /1.7 with b = 1.7 Å and zm = -1 r0 = 10 Å 2r0

  21. Counterions distribution ? Cargèse, 2002 - 21 Number of charges per Bjerrum length lB : xM = lB /b the Manning parameter For B – DNA, b = 1.7 Å: xM = 4.2 Without added salt : Only the counterions are present b zc e Cctotal + zm e Cmtotal = 0 Counterions zc = + 1 monovalent (Na+, K+,…) Monomers zm = - 1 a nucleotide (Phosphate-) 2r0

  22. The Cell model : each chain enclosed in a coaxial cylindric cell Cargèse, 2002 - 22 - ionic distribution : the whole solution = a single cylindric cell - concentration effect : DNA dilution = R increase Cmtotal = Cctotal = 1/pbR2 2r0 2R

  23. Cargèse, 2002 - 23 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 chain r(r) = zce C(R) e-zce y(r) /kT Assumption :y(r = R)=0 f(r)= k2 ef(r) with f(r)= - zcey(r) / kT The screening length 1/k becomes : k2 = 4plBzc2C(R) [ for the simple salt,k2 = 4p lB (z-2C-0+ z+2C+0) ] 0 r r0 R

  24. Cargèse, 2002 - 24 Boundary conditions : a whole cell is neutral, for r = Rf(r)/r = 0 the chain is charged withxM = lB /b = 4.2, for r = r0f(r)/r = - 2 xM /r0 r0 = 10 Å R = 50 Å (40 g/l DNA) 1/ k 25 Å xM = 4.2 f(r) 1.5 f(r) = -2 ln( [kr /g2] cos (g ln[r/RM])) !!! 0.8 with g and RM two numerical constants dependent on xM , r0 , R r r0 R

  25. Cargèse, 2002 - 25 Fraction of counterions within a cylinder of radius r R = 50 Å xM = 4.2 1.5 Counterions accumulation 0.8 r/r0 0 Monotonic increase for xM < 1 r0 R r

  26. Cargèse, 2002 - 26 R = 250 Å (1.7 g/l DNA) similar variation xM = 4.2 Similar to simple salt; Debye – Hückel or linearised Poisson-Boltzmann equation zey(r) << kT orf(r) <<1 1.5 Counterions accumulation 0.8 r/r0

  27. Cargèse, 2002 - 27 Using the linearised equation, f(r)= k2f(r) f(r)= - [2 xM/ kr0K1(kr0) ]K0(kr) withf(kr <0.5) ln(kr) 0.8 xM= 0.8 For xM < 1, the electrostatic potential y(r) is lower than kT and no counterions accumulation or condensation occurs. Counterions may be treated like simple salts. For DNA xM= 4.2 > 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.

  28. Cargèse, 2002 - 28 Monte-Carlo simulation (Le Bret & Zimm, 1984) : C (r) (M) local concentration of countreions • regular spacing of • the phosphate charge • on a cylinder • charges set along the • double-helix 0 r r0 R

  29. Experimentally : Cargèse, 2002 - 29 Neutron scattering(van der Maarel et al., 1992) Contrast matching X-Ray scattering (Chang et al., 1990) Heavy counterions (Thallium+)

  30. 3_b) the Manning model and the limiting laws Cargèse, 2002 - 30 • An infinite line charge with a charge density xM • Simple salt is present in excess over polyelectrolyte (Manning, 1969; 1978) Two-states model for xM> 1: « free » ions treated in the Debye-Hückel approximation « condensed » ions reducing the monomer charge to an effective charge qeff.

  31. Cargèse, 2002 - 31 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 << 1: • gel. = - (q eff.2 /4pee0b) ln kb Setqthe number of condensed ions per monomer, the effective charge per monomer becomes: q eff. = zme + zce q = zme [1+ (zc/zm)q] with zczm<0 gel. / kT = - zm2 [1+ (zc/zm)q]2 xMln kb

  32. Cargèse, 2002 - 32 -The mixing entropy decreases the number of condensed ions : Concentration of the free salt : Cs As q ions are confined in Vp, the lost of mixing energy is gmix../ kT = q ln (Clocal/Cs) Volume of the region where ions are said to be condensed : Vp The ionic local concentration : Clocal = 103 q /Vp

  33. Cargèse, 2002 - 33 Minimization of the free energy g el. + g mix. with respect to q : 1 - ln(Vp Cs/103) - zczm [1+ (zc/zm)q ]xMln (kb)2=0 Focus on the salt concentration Cs dependence with k2~ Cs - ln Cs ~ zczm [1+ (zc/zm)q ]xMln Cs The only solution to cancel the Csdependence (in particular when Cs0) : -1 = zczm [1+ (zc/zm)q ]xM q = 1- 1/xM For  zm =  zc = 1

  34. radial extension of the condensation region Cargèse, 2002 - 34 76 % of the phosphates have 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/b and 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

  35. 4_ Thermodynamic coefficients and the « free » counterions Cargèse, 2002 - 35 « free » in the Manning sense : For xM < 1, all the counterions are « free » For 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-Hückel interations for simple salts  thermodynamically « free » : where ions are only submitted to the thermal agitation

  36. Cargèse, 2002 - 36 - For the Poisson-Boltzmann cell model without added salt, Only the counterions far from the chain or located at r = R are thermodynamically « free » C(R) y(r = R) = 0 - 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 »

  37. Cargèse, 2002 - 37 The Donnan effect : Semi-permeable membrane reservoir At the thermodynamical equilibrium, the ions chemical potentials from both sides are equal : m(Cc Na+ , Cs ) = m( Cs0 ) Na-DNA + Salt Cs Salt Cs0 (NaCl) Cs= C Cl- C Na+= Cs + Cc Na+ Cs0 = C Na+0 = C Cl-0 Counterions coming from Na-DNA

  38. 4_a) the radius effect Cargèse, 2002 - 38 In an excess of salt (NaCl) with diluted Na-DNA, Cs0 - Cs’ ( 1/ 2xM)Cc Na+ /2 The salt exclusion factor : G = lim (Cs0 - Cs )/ CDNAfor Cc Na+ or CDNA 0 and 2G 1/ 2xM Information on the fraction of thermodynamically « free » counterions

  39. Cargèse, 2002 - 39 Experimentally, the salt concentration in the DNA compartment Cs isdetermined by selective electrodes Simple salt limit 2G =1 2G • Theoretically, • by solving the P-B equation and averaging C Na+(r)and C Cl-(r) • 2G = (1/ 2xM) f( kr0 ) • with • f(kr0 0) = 1 the limiting law 1/ 2xM kr0 Cs0 10 mM 0.1 M 1 M (Strauss et al., 1966-67; Shack et al., 1952)

  40. Cargèse, 2002 - 40 The osmotic pressure p : p = r g h h Na-DNA + Salt Cs Salt Cs0 In excess of Na-DNA compared to NaCl, Cs’0 the salt is excluded with fosm the osmotic coefficient p / kT = fosmCc Na+ Returning to the P-B equation in salt-free conditions, fosm = C(R) / Cc Na+;fosm = ( 1+g(kr0 ) ) / 2xM with g(kr0 ) 0 Used by T. Odijk for the free energy of DNA in bacteriophage (1998)

  41. 1/k r0 Cargèse, 2002 - 41 fosm 2G • Strong deviation from the limiting law due to the finite radius r0 1/ 2xM kr0 CDNA 1 g/l 100 g/l • Theories and Experiments are • qualitatively in good agreement. • Quantitatively….. (Auer & Alexandrowicz, 1969); (Raspaud et al., 2000)

  42. 2 Cargèse, 2002 - 42 q/qeff (Nicolai & Mandel, 1989) (Stigter, 1978) (Liu et al., 1998) q/qeff= 5.9 (Griffey, unpublished) 4.2 (Padmanabhan et al., 1988) 2.1 (Braulin et al., 1987) 2.5 • Permeability of the double-helix • Specific interactions… • different sensibilities of • the different technics To be compared withxM = 4.2

  43. 4_b) the length effect (Alexander et al.,1984) For charged spheres, ion chemical potential : far from the spheres, mainly entropic kT ln C on its surface, mainly enthalpic (electrostatic) -qeffe2/4pee0a qeff ~ - ln C The effective charge increases with dilution Counterions evaporation Cargèse, 2002 - 43 Counterions evaporation End effects 1/k L

  44. Cargèse, 2002 - 44 fraction of condensed counterions qeff ~ - ln C 1/k 1/k dilution (Gonzalez-Mozuelos & Olvera de la Cruz, 1995)

  45. 2G 1 Nucleotides behave like simple salt denaturated DNA native DNA ( N = number of nucleotides per chain)  N = 25 N = 10 5 base-pairs Returning to DNA, Importance of end effects for short oligonucleotide helices : Simulations of Olmsted et al. (1991) Cs= 1.8 mM 1/k = 73 Å Cargèse, 2002 - 45 kLend effects 2G = (1/ 2xM) f( kr0 ) 1/k Cs (mM)

  46. Cargèse, 2002 - 46 N  72 The local concentration at the DNA surface N = 24 Ends effect becomes negligeable on average but evaporation still occurs at the ends. N = 12 1/k (Olmsted et al., 1989)

  47. 5_ Conformation and Interaction5_a) Thermal denaturation Cargèse, 2002 - 47 Calculation of the free energy using the charge renormalisation (Manning, 1972) Interaction energy of the chain with its ionic atmosphere Entropic term of the salt, of the « free » ions Non-electrostatic term + +… + - (np/xM )kT ln k +(np/xM )kT ln Cs Addition of salt Destabilizes Stabilizes Ionic atmosphere (ns salt + np/xM« free » ions) (salt in excess but at low concentration) the double-helix L/lB or np/xM charged phosphate (np/xM ) denaturated > (np/xM )native DNA

  48. Cargèse, 2002 - 48 - (np/xM )kT ln k +(np/xM )kT ln Cs k ~ Cs 1/2 The dominant term is entropic +(np/2xM )kT ln Cs • Finally, the electrostatic free energy Gele= H –TS > 0 • and dominated by the entropy S • Entropic cost to condense or confine the counterions • Gain in electrostatic free energy to denaturate DNA D Gele=Gelesingle-stranded– Geledouble-stranded< 0

  49. Tm (°C) CNa+(M) Cargèse, 2002 - 49 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 = kTm2/DHm T4 DNA (Record, 1975)

  50. kT D G non-ele > 0 G non-eledenaturated > G non-elenative Cargèse, 2002 - 50 From experimentalTm and calculated D Gele • Gnon-ele per nucleotide *Tm= k log Cs + 0.41 XGC + 81.5 (Schildkraut & Lifson, 1965) *d Tm /d log Cs ~D (1/2xM ) ~ D (2G ) (Olmsted et al., 1991) (Korolev et al., 1998; see also Rouzina & Bloomfield, 1999)

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