Lecture 5: Electrostatic Interactions & Screening

Transcription

1 Lecture 5: Electrostatic Interactions & Screening Lecturer: Prof. Brigita Urbanc PHYS 461 & 561, Fall

2 A charged particle (q=+1) in water, at the interface between water ( =80) and protein ( =3) Find the electric field produced by the charge q at an arbitrary point 2: 1 q=+1 2 water //////////////////////////// protein // How do we solve for the electric field and why does the presence of water protein interface matter? in the absence of the interface, electric field E = q/4 0 r 12 PHYS 461 & 561, Fall

3 Method of Image Charges Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN X. problem solution Coulomb's law: normal distance from the y=0 plane: = (x 2 +z 2 ) 1/2 PHYS 461 & 561, Fall

4 A similar problem of an interface between vaccum ( =0) and a metal ( =ꝏ): electric field locally perpendicular to the interface NO field in a metal (or else current) 2 r 12 r 02 SOLUTION (above the interface): +q 1 vaccum ( =1) metal ( =ꝏ) q 0 E = q/4 0 r 12 + ( q)/4 0 r 02 reflection effect: +q induces the shift of electrons on the high metal side PHYS 461 & 561, Fall

5 Water protein interface: mirror charge approach inverse problem the original charge in water (high permittivity 1 =80 side of the interface) force lines are repelled from the low =3 (protein) side a charged atom on the water side gets surrounded by electronegative parts of polar water molecules positive mirror charge q' = q ( 1 2 )/( ) ~ q (if 1» 2 ) E = q/4 0 r 12 + q'/4 0 r 02 PHYS 461 & 561, Fall

6 We can use the generalized form of the field around the charge q: E = q/4 eff 0 r 12 = q/4 1 0 r 12 (1 + r 12 /r 02 ) With effective permittivity eff depending on the position r: eff = 80 for the point 2 close to the charge 1 eff = 40 everywhere else in water eff = ( 1 + )/2 ~ 40 for any point 2 below the surface eff = ( 1 + )/2 ~ 40 for point 1 inside the protein except if r 12 «a, then eff = water ///////////////////////////// protein PHYS 461 & 561, Fall

7 Effective permittivity across the protein: water protein 2 water eff = 200 due to water electric dipole and induced polarization PHYS 461 & 561, Fall

8 Values of effective permittivity eff around and inside a protein: water protein protein Except inside the protein very close to the charge, the electric field is strongly reduced due to screening by polar water molecules PHYS 461 & 561, Fall

9 The medium of high permittivity (water) attracts the charge: a charge on the protein surface is repelled from the protein a charge inside the protein is attracted to water Why does water have a high permittivity? permittivity determined by atomic structure of water permittivity proportional to the polarization induced in the medium by an external electric field polarity of H 2 O Induced polarization produces effective internal field of the opposite sign, thereby diminishing the total field (relative to vaccum) PHYS 461 & 561, Fall

10 Electrostatic Interaction Between Two Oppositely Charged Atoms consider distances 3 4 A no water molecules inbetween possible! What is the value of eff at such small distances? example: Na + Cl in water dissolves (Van de Waals distance between Na and Cl ~ 3 A ) EI potential energy: 1.5 kcal/mol ( eff =80) 3.0 kcal/mol ( eff =40) 6.0 kcal/mol ( eff =20) > E HB PHYS 461 & 561, Fall

11 Free energy change associated with dissociation: oxalic acid (diprotic acid 2 H atoms per molecule): Step 1: H 2 C 2 O 4 HC 2 O 4 + H + 2 Step 2: HC 2 O 4 C 2 O 4 + H + Step 1 occurs at ph ~ 2 & Step 2 at ph ~ 4.5 ph ~ 2.5 ph value associated with H + concentration, [H + ]: [H + ] = 10 ph = exp(2.3 ph) ph [H + ] change in the Gibbs free energy G G = RT {ln([h + ] b ) ln([h + ] a )} = RT (2.3 ph) For oxalic acid ph = 2.5 G~ 3.5 kcal/mol ( eff ~40) similar result for dissociation of the carbonic acid PHYS 461 & 561, Fall

12 + Physical Interpretation: Neighboring water molecules pulling charges from sides PHYS 461 & 561, Fall

13 Experimental Determination of the Effective Permittivity (A. Fersht protein engineering) enzymes exhibit optimal activity at a ph optimum mutation of the active site amino acid to a charged amino acid shifts the ph optimum (mutation at the protein surface) ph optimum happens because the active site needs a fixed concentration of protons, [H + ] = 10 ph (by definition) [OH ] = pH (by definition) active site (AS) accepts a proton H +, AS + H + = ASH + : ([ASH + ] = [AS][H + ] probability for H + binding) [ASH + ]/[AS] = exp( F ASH+ /RT) [H + ] PHYS 461 & 561, Fall

14 F ASH+ = Free Energy of H + binding [ASH + ]/[AS] = exp( F ASH+ /RT) 10 ph = exp( F ASH+ /RT) exp ( 2.3 ph) = exp{ ( F ASH+ /RT ph)} Mutation induced charge induces the potential eu (e=+1, the charge of H + ): eu = F ASH+ (M) F ASH+ (0) M with mutation 0 without mutation (a) F ASH+ (M)/RT ph M = F ASH+ (0)/RT ph 0 (b) eu = F ASH+ (M) F ASH+ (0) = 2.3 RT (ph M ph 0 ) (c) eu = eq/4 eff 0 r, q mutation introduced charge PHYS 461 & 561, Fall

15 Equation for eff : eq/4 eff 0 r = 2.3 RT ph Results of Fersht's experiments: eff ~40 to ~120 Protein Engineering (Fersht, the father ): changing a codon on the protein gene induces the mutation at an exact site of the protein globule structural changes monitored by X ray & NMR protein as microscopic electrometer NEGLECTED: INTs between dipoles & quadrupoles (smaller than INTs between charges, decrease faster with r) PHYS 461 & 561, Fall

16 Electrostatic Interaction Between Two Free Charges in Water U(r) = q 1 q 2 / 4 eff 0 r exp( r/d) (exponential decay) eff and D depend on the properties of the medium (water) D Debye Hückel radius: D = 3/I 1/2 A also on ionic strength I I = ½ c i z i 2 c i concentration of the ion type i z i charge of the ion type I I = [mol/l] (physiological conditions) For water at physiological conditions D ~ 8 A PHYS 461 & 561, Fall

17 The origin of the screened electrostatic interaction temperature dependence of electrostatic effects: eff (T=273) = 88 & eff (T=373) = 55 88/55 ~ 1.6 & 373/273 ~ 1.4 The electrostatic interactions in water decreases with absolute temperature almost linearly entropic effect! Electrostatic INTs in water are caused by the ordering of water molecules around the charges & variation of this ordering with distance (similar to the hydrophobic effect). PHYS 461 & 561, Fall

18 Disulfide (S S) bonds formed between the side chains of two cysteines (Cys) two Cys side chains ( C H 2 SH ) release two H atoms: C H 2 SH + C H 2 SH C H 2 S S C H 2 + H 2 during formation of a disulfide bond formation and breakdown of S S bond in cells is catalyzed by an enzyme called disulfide isomerase (only to accelerate the processes) & reversible absence of disulfide isomerase freezes the formed S S bonds, S S bonds typical of secreted proteins PHYS 461 & 561, Fall

19 Coordinate bonds formed by N, O, S atoms of the protein & O atom of water to di and trivalent ions of Fe, Zn, Co, Ca, Mg (metals) metal ions characterized by vacant orbits of low energy, capable of bonding an electron pair N, O, S atoms are electron donors (radius ~1.5 A ) : their electrons occupy the vacant orbits of the metal ion (radius ~0.7 A ), forming a coordinate bond (only 1 bonded atom), several kcal/mol (similar to hydrogen bonding) if the donor atoms in the protein conformation are in a proper position for coordinate bonding, the ion gets released from water, bonds to protein (S of water increases!) chelate complex PHYS 461 & 561, Fall