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DNA coby poly elektrolyt : Man n ingova teorie kondenzace protiiontů

DNA coby poly elektrolyt : Man n ingova teorie kondenzace protiiontů.

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DNA coby poly elektrolyt : Man n ingova teorie kondenzace protiiontů

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  1. DNA coby polyelektrolyt: Manningova teorie kondenzace protiiontů Jiří KozelkaUniversité Paris DescartesLaboratoire de Chimie et Biochimie Pharmacologiques et ToxicologiquesUMR CNRS 860145 rue des Saints-Pères75270 Paris FRANCE******Masaryk University of BrnoInstitute of Condensed Matter PhysicsFaculty of ScienceKotlářská 261137 Brno CZECH REPUBLICTel. +420 549 49 4048Port. +420 720 379 379 Manning, G. S. (1978). "The molecular theory of polyelectrolyte solutions with aplications to the electrostatic properties of polynulceotides." Quart. Rev. Biophys.11: 179-246.

  2. 1 segment: charge -1e b = 1.7 Å Double-stranded DNA can be considered as a linear long rod composed of segments, each carrying one unit charge. The segments repel each other. The repulsion is diminished if positively charged counter-ions associate with the DNA surface. DNA double helix distance between segments b defines the charge densityx of the polymer x  b-1

  3. Univalent salt in water (e.g., NaCl) VP + + + + + + + + + + + + At each segment, a fraction of positive counterions „condenses“ within a volume VP DNA double helix The charge of 1 segment is reduced to (-1+q)e Gain in enthalpy DH, since repulsion between segments diminished Loss in entropy DS, since order increases Nature will „choose“ such a q fraction, that DG = DH – TDS minimal 1 equation 2 unknowns  dDG/d = 0 dDG(, VP ) /d = 0

  4. Calculation of the electrostatic repulsion term DGel 2b efekt soli efekt vody Debye-Hückel screening factor Coulomb |i-j| = 1: number of pairs = np-1 ≈ np |i-j| = 2: number of pairs = np-2 All pairs contribute: ≈ np ≈ np Phosphate N°: 1 2 3……………..s………………………np qnet qnet qnet qnet qnet qnet qnet qnet Charge: b qnet = qe(1- ) Each pair contributes: if np>> s rD = „Debye screening length“ proportional to c-1/2 c: concentration of univalent salt . . |i-j| = s: number of pairs = np-s DGij≈ 0 |i-j| > s: interaction negligible

  5. f(x) = Simplifying the sum over pairwise electrostatic repulsion terms Looks like a Taylor series…. Cvičení 1: odvoďte Taylorův rozvoj pro funkci f(x)=ln(1+x) (vyžadováno je řešení, nikoliv jen výsledek) Taylorův rozvoj :

  6. f(x) = with a=0: series converges for x<0 with

  7. List of Taylor series of some common functions http://en.wikipedia.org/wiki/Taylor_series

  8. Simplifying the sum over pairwise electrostatic repulsion terms Cvičení 2: vypočtěte sumu = Cvičení 3: zjednodušte (rD>>b)

  9. x is dimensionless charge density Simplifying the sum over pairwise electrostatic repulsion terms Sum has become a simple logarithmic term!

  10. Cvičení 4: Vypočtěte x pro dvojšroubovicovou DNA (b = 1.7 Å) při 25 °C x = 4.2 (single-stranded DNA: x = 2.1) k = 1.38x10-23 J.K-1T = 298 K ke = 9.0x109 Nm2C-2 e =78qe = 1.60x10-19 C

  11. cbulk VP + + + + + + + + + + + + clocal= nP(nPVP)-1= VP-1 A fraction of positive counterions „condenses“ within a volume VP DNA double helix Loss in entropy DS, since order increases

  12. Mixing entropy term dependent on q* number of moles of couterions that is confined from cbulk to clocal cbulk = c = concentration [mol/L] of univalent saltKNOWN VP [L/mol] = volume to which the q counterions/phosphate are confined Vp UNKNOWN → HOPELESS CALCULATION??? Let‘s minimize the total free energy with respect to q *i.e., loss in entropy when the fraction q of counterions associates with each segment

  13. Cvičení 5: Vypočtěte první derivaci následují funkce podle x :

  14. Minimizing total free energy Dependence of DGmix on c Dependence of DGel on c Cvičení 6: vypočtěte derivaci. Naznačte postup výpočtu. From Debye-Hückel Theory of electrolytes

  15. ln c ↓ -∞ ln c ↓ -∞ k Boltzmann‘s constant x dimensionless charge density of polymer ke electrostatic constant c concentration of univalent electrolyte qe electron charge e dielectric coefficient of electrolyte e0 vacuum permittivity T absolute temperature LAV Avogadro‘s number qfraction of counterions condensed VP volume to which q counerions condense knowns unknowns We want to know q at the low concentration limit, c→0 →Singular ln c terms must vanish! Now we have got a second equation!

  16. ln c ↓ -∞ ln c ↓ -∞ We want to know q at the low concentration limit, c→0 →Singular ln c terms must vanish! Now we have got a second equation! 76% of the negative charge of the phosphates is neutralized by condensed counterions! x < 1  q < 0 no association of counterions x > 1  q > 0 association of counterions

  17. Now we can determine VP, the volume to which the θ counterions/nucleotide condense in double-stranded DNA base of natural logarithms

  18. = 6.46x1026LAV-1Å3/nucleotide = 1073 Å3/nucleotide k = 1.38x10-23 J.K-1e0 = 8.85x10-12 C2N-1m-2 DNA in water at 25°C ke = 9.0x109 Nm2C-2 LAV = 6.02x1023 mole-1 x = 4.2 qe = 1.60x10-19 C e = 2.718e =78 T = 298 K q= 0.76

  19. 7 Å 20 Å 7 Å The volume to which the condensed univalent counter-cations are confined, VP Per mole: 646 cm3 Per nucleotide: 1073 Å3 This corresponds to a layer of ~7 Å around the DNA double-helix (which corresponds ~to a cylinder with a diameter of 20 Å) Counterion Condensation Theory: Manning, G. S. (1978) Quart. Rev. Biophys.11: 179-246

  20. Experimental and theoretical results supporting the Counterion Condensation Theory 1. Titration of Carboxymethylcellulose (CMC)with different hydroxides monitored by ultrasonic absorption CMC Titration with XOH; X = Na, K, Li, N(CH3)4……. Upon titration, the COOH groups are successively deprotonated, become COO-, and the negative charge density increases. This modifies the structure of the water layer at the surface of the polymer, which can be monitored by measuring the absorption of ultrasound. If there is no association of cations with the polyanion, the structure of the water layer should be independent of the cation.

  21.  > 1: great differences between the various hydroxides  association between the counterions and the polymer, as predicted by the Manning theory  < 1:no difference between the various hydroxides  no association between the counterions and the polymer R. Zana et al., J. Chim. Phys, 68, 1258-1266 (1971)

  22. 2. Molecular dynamics simulations of a solvated DNA duplex 7 Å 20 Å 7 Å % phosphate charge [Å] Radial distribution of Na+ counterions around a DNA duplex 12 base-pairs long, from a molecular dynamics simulation (Yang et al., J. Am. Chem. Soc. 1997, 119, 59-69). The calculation predicts 82% of the phosphate charge to be neutralized, with cations moving within a radius of 15 Å from the helix axis. Manning theory predicts 76% of cations within 17 Å from the helix axis.

  23. = fraction of monovalent cations predicted by Counterion Condensation Theory to condense on each unit charge of the poly-anion • If x > 1 then a fraction of 1-ξ-1 counterions condense at each unit charge • If x <1 , there is no counterion condensation. duplex DNA in H2O at 298 K: x = 4.2 single-stranded DNA: x = 2.1 Remember: In a diluted solution of counterions, q is concentration-independent!

  24. (ax)*lna http://matematika-online-a.kvalitne.cz/derivace-funkce.htm

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