1 / 72

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.

Download Presentation

DNA AS A POLYELECTROLYTE

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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)

More Related