Bchem 675 Lecture 9 Electrostatics-Lecture 2 Debye-Hückel: Continued Counter ion condensation

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1 Bchem 675 Lecture 9 Electrostatics-Lecture 2 Debye-Hückel: Continued Counter ion condensation

2 Ion:ion interactions What is the free energy of ion:ion interactions ΔG i-i? Consider an ion in a solution with many other ions: ΔG i i = n i µ i i n i is the number of ions that the reference ion interacts with and µ i-i is the chemical potential for each ion that interacts with the reference. The chemical potential of a single ionic species will be related to the work of charging up a mole of ions of interest while in the proximity of other ions. w ch arg ing = (z i e o )2 8πε o εr i In terms of chemical potential: Δµ i i = N A w ch arg ing Δµ i i = N A z i e o 2 ψ ψ is the electrostatic potential of the ion

3 In order to determine the chemical potential change for the interaction between the total ionic assembly and the ion of interest we must know the electrostatic field at each individual ion that is a result of the other ions in solution. This requires information about the spatial distribution of the ions in the solution relative to the reference ion.

Erich Armand Arthur Joseph Hückel (in Engl. often spelled Huckel or Hueckel) b. August 9, 1896, Berlin, Germany d. 1980, Marburg, Germany A German physicist and physical chemist.

5 Erich Armand Arthur Joseph Hückel (in Engl. often spelled Huckel or Hueckel) b. August 9, 1896, Berlin, Germany d. 1980, Marburg, Germany A German physicist and physical chemist. He is known for two major contributions: (a) The Debye-Hückel theory of electrolytic solutions, (b) The Hückel method of approximate molecular orbital (MO) calculations on p-electron systems.

$through his investigations on dipole moments and on the diffraction of X-rays and electrons in gases".$

6 Petrus (Peter) Josephus Wilhelmus Debye was born March 24, 1884, at Maastricht, the Netherlands. Debye won Nobel Prize in Chemistry, 1936, "for his contributions to our knowledge of molecular structure through his investigations on dipole moments and on the diffraction of X-rays and electrons in gases". Debye died on November 2, 1966, and was buried on Pleasant Grove Cemetery, Cayuga Heights, Tompkins County, New York, U.S.A.

7 Debye-Hückel Model 1923 Treatment: A reference ion residing in a dielectric continuum. Dielectric constant is 80 Reference ion is charged-therefore, in its immediate vicinity there will be neutralization by the continuum of charge that surrounds it. At equilibrium the charge on the central ion will be exactly counterbalanced by the counter charge atmosphere: Principle of Electroneutrality: z i e o X i = 0

9 Assumptions of Debye Huckel 1. A central reference ion of a specific charge can be represented as a point charge. 2. This central charge is surrounded by a cloud of smeared-out charge contributed by the participation of all other ions in solution. 3. The electrostatic potential field in the solution can be described by an equation that combines and linearizes the Poisson and Boltzmann equations. 4. No ion-ion interactions except the electrostatic interaction given by the 1/r 2 dependence are to be considered. 5. The solvent simply provides the dielectric medium, and the ion-solvent interactions can be ignored, so the bulk permitivity of the solvent can be used.

10 Figure The excess charge in a given volume, dv, of the cloud can be related to the electrostatic potential existing between the central ion and the small volume element under consideration.

11 Poisson Equation 1 d r dr r2 ( dψ dr ) = ρ r ε o ε Relates the charge distribution, ρ, to the electrostatic potential, Ψ, in a spherically symmetrical system. The total charge in a given volume element is found by adding together all of the charges that reside in the volume: ρ r = n i z i e o The total charge available in the system will be described by the sum of all charged elements in solution.

12 Ionic Strength provides a means of quantifying the charge in an electrolyte solution I = 1 2 n i z i 2 Return to charge excess, ρ, in the volume element Each of the particles n i can be characterized using the n Boltzmann distribution: i = n o U / kt i e U>0:distribution of the ions is less than the bulk U<0: distribution of ions is increased relative to the bulk. Since all interactions are electrostatic: U=z i e o ψ r, the Boltzmann n relationship becomes i = n o i e z i e oψ r / kt and ρ r = ρ r = n i o z i e o e z i e oψ r / kt n i z i e o

13 ρ r = n i o z i e o e z i e oψ r / kt Assume that z i e o Ψ r /kt<<1 can be written as a Taylor series in which only the first 2 terms are considered. e z i e o ψ r /kt ρ r = ρ r = n o i z i e o (1 z i e oψ r ) kt n o i z i e o n i o z i 2 e o 2 ψ r kt The first term is zero-electroneutrality-and ρ r = n o i z 2 i e 2 o ψ r kt This is the linearized Boltzmann equation

14 Combine the linearized Boltzmann equation with the Poisson equation-each relating charge density in the volume element, dv, to the distance from the reference ion,r. In this equation all of the terms on the right hand of the previous equation are collected into the single variable, κ 2. Solve the equation by integrating from r=0 to r=. Next page 1 r 2 r r2 ( ψ dr ) = 1 ε o εkt 1 r 2 r r2 ( ψ dr ) = κ 2 ψ r n i o z i 2 e o 2 ψ r kt

15 ψ r = z i e o 4πε o εr The Debye Length or κ -1 : The effective radius of the charge atmosphere surrounding the central ion. What parameters contribute to this effective radius: κ 2 = κ 2 = 1 ε o εkt e o 2 i o n i z 2 2 i e o i o n i z i 2 ε o εkt κ 2 = e o 2 N A o 2 n i z i ε o εkt i IonicStrength I = 1 2 n i z i 2 κ 2 = 2e o 2 N A I ε o εkt

16 ω-ionic strength Charge distribution as a function of distance from the central ion.

17 Implications for biomolecular interactions

18 Double stranded nucleic acids and counterion Condensation (vh, J&H-Ch. 3, Section 3.8) Nucleic acids are polyanions-therefore salts will interact with them in a more specialized manner than with garden variety proteins. These interactions are important both for the stability of the nucleic acid and its interactions with other biomolecules.

19 Treat double stranded DNA as a cylinder with regularly Spaced charges. What is the spacing of the charges? Helical rise (B-DNA)=0.34 nm 2 phosphate groups/base pair Average charge spacing=0.34nm/2=0.17nm Cations will, of course, associate with the phosphates of the Backbone. To what extent do cations associate with the Backbone? What is the effective charge on Double Stranded DNA?

20 Start with Coulomb s Law Consider all of the charges on the backbone: There are N charges and each charge j interacts with all other charges k along the backbone. N N e 2 r jk is the distance between the j=1 k j εr jk charges. Debye Hückel can be used to introduce the effect of cation Screening-use κ 8πe 2 κ = ( 100εk B T )1/2 I 1/2 This describes a diffuse conterion atmosphere around the helix.

21 Counterion Condensation Direct condensation of small ions on a polyelectrolyte: Gerald Manning Depends on the valence of the ion-z and b-the average spacing of the charges along the helical axis. ξ = e 2 (εbk B T) Where ξ is a dimensionless parameter. If ξ>1 condensation occurs.

22 Polyelectrolyte in water at 25 o C, ξ=0.71/b B-DNA b=0.17nm ξ=4.2 Cations condense onto DNA What is the extent of condensation? Or What fraction of the charge on DNA remains uncompensated? 1/Zξ=0.24 for DNA in aqueous Na + environment. 76% of the phosphate charges are compensated. Theory vs. experiment: Good agreement (NMR measurements of counterions)

23 Repercussions for nucleic acid stability and its interactions with other molecules.

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