DNA coby polyelektrolyt: Manningova teorie kondenzace protiiontů Manning, G. S. (1978). "The molecular theory of polyelectrolyte solutions with aplications to the electrostatic properties of polynulceotides." Quart. Rev. Biophys. 11: 179-246. Prof. Jiří Kozelka, Biofyzikální Laboratoř, Ústav fyziky kondenzovaných látek, PřF MU, Kotlářská 2, kozelka.jiri@gmail.com F5351 Základy molekulární biofyziky Masarykova Univerzita Podzimní semestr 2024 29.11.2024 DNA double helix Double-stranded DNA can be considered as a long linear 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 each segment on the DNA surface. 1 segment: charge -1e b = 1.7 Å The distance b between segments harboring a unit charge (1 qe=1.60x10-19 C) defines the charge density of the polymer. For single-stranded DNA, b = 3.4 Å For double-stranded DNA, b =1.7 Å Before we continue, let us learn about two definitions: Bjerrum* length lB: the distance between two elementary charges (qe) at which their repulsion energy is equal to the thermal energy kBT Manning** structural parameter x: the ratio between lB and b, that is, the product of the Bjerrum length (a constant [Å]) and the linear charge density ([Å-1]). It is thus a dimensionless parameter proportional to the charge density. *Niels Jannicksen Bjerrum (1879-1958), dánský chemik ** Gerald S. Manning, professor emeritus, Rutgers University, New Jersey, USA + + + + + + + + + + + + At each segment, a fraction q of positive counterions „condenses“ within a volume VP ß dDG/dq = 0 VP 1 equation 2 unknowns Univalent salt in water (e.g., NaCl) DNA double helix dDG(q, VP ) /dq = 0 The charge of 1 segment is reduced to (-1+q)e Gain in enthalpy DH, since repulsion between segments is diminished Loss in entropy DS, since order increases Nature will „choose“ such a q fraction, that DG = DH – TDS is minimal qnet qnet qnet qnet qnet qnet qnet qnet Calculation of the electrostatic repulsion term DGel Phosphate N°: 1 2 3……………..s………………………np Each pair contributes: b |i-j| = 1: number of pairs = np-1 |i-j| = 2: number of pairs = np-2 2b . . |i-j| = s: number of pairs = np-s |i-j| > s: interaction negligible if np>> s ≈ np ≈ np ≈ np DGij≈ 0 Charge: rD = „Debye screening length“ proportional to c-1/2 c: concentration of univalent salt qnet = qe(1- q) All pairs contribute: Coulomb Debye-Hückel screening factor efekt vody efekt soli Simplifying the sum over pairwise electrostatic repulsion terms Looks like a Taylor series…. f(x) = Cvičení 1: odvoďte Taylorův rozvoj pro funkci f(x)=ln(1+x) (vyžadováno je řešení, nikoliv jen výsledek) Taylorův rozvoj : f(x) = with a=0: with series converges for x<0 > List of Taylor series of some common functions \sqrt{x+1} = \sum_{n=0}^\infty \frac{(-1)^n(2n)!}{(1-2n)n!^24^n}x^n \quad\mbox{ for } |x|<1 \mathrm{e}^{x} = \sum^{\infin}_{n=0} \frac{x^n}{n!}\quad\mbox{ for all } x \ln(1+x) = \sum^{\infin}_{n=0} \frac{(-1)^n}{n+1} x^{n+1}\quad\mbox{ for } \left| x \right| < 1 \frac{x^m}{1-x} = \sum^{\infin}_{n=m} x^n\quad\mbox{ for } \left| x \right| < 1 http://en.wikipedia.org/wiki/Taylor_series Simplifying the sum over pairwise electrostatic repulsion terms \ln(1+x) = \sum^{\infin}_{n=0} \frac{(-1)^n}{n+1} x^{n+1}\quad\mbox{ for } \left| x \right| < 1 \mathrm{e}^{x} = \sum^{\infin}_{n=0} \frac{x^n}{n!}\quad\mbox{ for all } x Cvičení 2: vypočtěte sumu = Cvičení 3: zjednodušte (rD>>b) Simplifying the sum over pairwise electrostatic repulsion terms Sum has become a simple logarithmic term! x is the dimensionless «Manning structural factor» proportional to charge density Cvičení 4: Vypočtěte x pro dvojšroubovicovou DNA (b = 1.7 Å) při 25 °C k = 1.38x10-23 J.K-1 T = 298 K ke = 9.0x109 Nm2C-2 e = 78 qe = 1.60x10-19 C x = 4.2 (single-stranded DNA: x = 2.1) > + + + + + + + + + + + + A fraction q of positive counterions „condenses“ within a volume VP VP Loss in entropy DS, since order increases DNA double helix cbulk clocal= nPq(nPVP)-1 = qVP-1 cbulk = c = concentration [mol/L] of univalent salt KNOWN Let‘s minimize the total free energy with respect to q Mixing entropy term dependent on q* Vp UNKNOWN → HOPELESS CALCULATION??? VP [L/mol] = volume to which the q counterions/phosphate are confined * i.e., loss in entropy when the fraction q of counterions associates with each segment number of moles of couterions that is confined from cbulk to clocal Cvičení 5: Vypočtěte první derivaci následují funkce podle x : > Minimizing total free energy From Debye-Hückel Theory of electrolytes Cvičení 6: vypočtěte derivaci. Naznačte postup výpočtu. Dependence of DGel on c Dependence of DGmix on 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 q fraction 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 terms containing ln c must vanish! Now we have got a second equation! x(1- q) ln c ↓ -∞ ln c ↓ -∞ We want to know q at the low concentration limit, c→0 Now we have got a second equation! x < 1 ® q < 0 no association of counterions x > 1 ® q > 0 association of counterions 76% of the negative charge of the phosphates is neutralized by condensed counterions! x(1- q) ln c ↓ -∞ ln c ↓ -∞ →Singular terms containing ln c must vanish! Now we can determine VP, the volume to which the θ counterions/nucleotide condense in double-stranded DNA base of natural logarithms =1 =0 k = 1.38x10-23 J.K-1 e0 = 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 = 2.718 e =78 T = 298 K q = 0.76 = 6.46x1026LAV-1 Å3/nucleotide = 1073 Å3/nucleotide Counterion Condensation Theory: Manning, G. S. (1978) Quart. Rev. Biophys. 11: 179-246 20 Å 7 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) 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. manning x < 1: no difference between the various hydroxides Þ no association between the counterions and the polymer x > 1: great differences between the various hydroxides Þ association between the counterions and the polymer, as predicted by the Manning theory R. Zana et al., J. Chim. Phys, 68, 1258-1266 (1971) 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. 2. Molecular dynamics simulations of a solvated DNA duplex [Å] % phosphate charge 20 Å 7 Å 7 Å q = 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 In a diluted solution of counterions, q is concentration-independent! Remember: http://matematika-online-a.kvalitne.cz/derivace-funkce.htm (ax) * lna