Lecture 5:
Electrostatic Interactions & Screening
Lecturer:
Prof. Brigita Urbanc (brigita@drexel.edu)
10/06/2009
PHYS 461 & 561, Fall 2009-2010
1
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 0r12
➔
10/06/2009
PHYS 461 & 561, Fall 2009-2010
2
Method of Image Charges
Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0­13­805326­X.
problem
solution
Coulomb's law:
 … normal distance from the y=0 plane:  = (x2+z2)1/2 10/06/2009
PHYS 461 & 561, Fall 2009-2010
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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⨯
r12
r02
SOLUTION (above the interface):
E = q/4   0r12 + (­q)/4   0r02
➔
➔
+q
1
vaccum (=1)
­q
0
metal (=ꝏ)
reflection effect: +q induces the shift of electrons on the high  metal side
10/06/2009
PHYS 461 & 561, Fall 2009-2010
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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)/( 1 +  2)
➔
~ q (if  1 »  2 )
10/06/2009
E = q/4   0r12 + q'/4   0r02
PHYS 461 & 561, Fall 2009-2010
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We can use the generalized form of the field around the
charge q:
E = q/4 eff 0r12 = q/4 1 0r12 (1 + r12/r02)
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 ➔ 
= ( 1 +   )/2 ~ 40 for point 1
eff
inside the protein except if r12 « a, then  eff =  
10/06/2009
­
­
­ +1 ­­
­­ ­
+ + + + + water
///////////////////////////// protein
PHYS 461 & 561, Fall 2009-2010
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Effective permittivity across the protein:
1
+ + + + +
water
protein
­ ­ ­ ­ ­
water
2
 eff = 200
due to water electric dipole and induced polarization
10/06/2009
PHYS 461 & 561, Fall 2009-2010
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Values of effective permittivity  eff around and inside a protein:
+1
40
80
40
80
80
protein
40
water
80
80
60
80
+1
3
protein
80
200
Except inside the protein very close to the charge, the electric field
is strongly reduced due to screening by polar water molecules 10/06/2009
PHYS 461 & 561, Fall 2009-2010
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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 H2O ➔
Induced polarization produces effective internal field of the
opposite sign, thereby diminishing the total field (relative to
vaccum) 
➔
10/06/2009
PHYS 461 & 561, Fall 2009-2010
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Electrostatic Interaction Between Two Oppositely Charged Atoms
consider distances 3—4 Å → no water molecules in­ between possible! What is the value of  effat such small distances?
➔
example: Na+Cl­ in water dissolves (Van de Waals distance
between Na and Cl ~ 3 Å)
➔
EI potential energy:
­1.5 kcal/mol ( eff=80)
­3.0 kcal/mol ( eff=40)
­6.0 kcal/mol ( eff=20) > EHB
10/06/2009
PHYS 461 & 561, Fall 2009-2010
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Free energy change associated with dissociation:
oxalic acid (diprotic acid – 2 H­atoms per molecule): Step 1: H2C2O4 ⇄ HC2O4− + H+ ➔
−
2−
+
Step 2: HC
O
⇄
C
O
+ H
2 4 2 4
➔
Step 1 occurs at pH ~ 2 & Step 2 at pH ~ 4.5 → pH ~ 2.5
pH value associated with H+ concentration, [H+]:
[H+] = 10pH = 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
➔
10/06/2009
PHYS 461 & 561, Fall 2009-2010
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Physical Interpretation:
Neighboring water molecules “pulling charges” from sides
-
+1
-
+
-
-
+
10/06/2009
+
-
+
+
+
-
+
-
+
­1
+
+
-
PHYS 461 & 561, Fall 2009-2010
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Experimental Determination of the Effective Permittivity
(A. Fersht – protein engineering)
enzymes exhibit optimal activity at a pH­optimum
➔
mutation of the active site amino acid to a charged amino acid shifts the pH­optimum (mutation at the protein surface)
➔
pH­optimum happens because the active site needs a fixed
concentration of protons, [H+] = 10­pH (by definition)
[OH­] = 10­14+pH (by definition)
➔
active site (AS) accepts a proton H+, AS + H+ = ASH+:
([ASH+] = [AS][H+] ⨯ probability for H+ binding)
[ASH+]/[AS] = exp(­FASH+/RT) ⨯ [H+]
➔
10/06/2009
PHYS 461 & 561, Fall 2009-2010
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FASH+ = Free Energy of H+ binding [ASH+]/[AS] = exp(­FASH+/RT) ⨯ 10­pH = exp(­FASH+/RT) ⨯ exp (­2.3 ⨯ pH) = exp{­(FASH+/RT + 2.3 ⨯ pH)}
Mutation—induced charge induces the potential eU (e=+1, the charge of H+): eU = FASH+ (M) – FASH+(0)
M ­­­ with mutation
0 ­­­ without mutation
↓
(a) FASH+(M)/RT + 2.3 ⨯ pHM = FASH+(0)/RT + 2.3 ⨯ pH0
(b) eU = FASH+(M) – FASH+(0) = ­ 2.3 ⨯ RT (pHM ­ pH0)
(c) eU = eq/4 eff 0r, q ­­­ mutation introduced charge 10/06/2009
PHYS 461 & 561, Fall 2009-2010
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Equation for  eff : eq/4 eff 0r = ­ 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 X­ray & NMR
­ protein as microscopic electrometer
NEGLECTED: INTs between dipoles & quadrupoles
(smaller than INTs between charges, decrease faster with r)
10/06/2009
PHYS 461 & 561, Fall 2009-2010
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Electrostatic Interaction Between Two Free Charges in Water U(r) = q1 q2 / 4 eff 0r ⨯ exp(­r/D)
(exponential decay)
 eff and D depend on the properties of the medium (water)
D ­­­ Debye­Hückel radius: D = 3/I1/2Å also on ionic strength I
I = ½  cizi2
ci ­­­ concentration of the ion type i
zi ­­­ charge of the ion type I
I = 0.1—0.15 [mol/l] (physiological conditions)
↓
For water at physiological conditions D ~ 8 Å
10/06/2009
PHYS 461 & 561, Fall 2009-2010
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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).
10/06/2009
PHYS 461 & 561, Fall 2009-2010
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Disulfide (S—S) bonds
formed between the side chains of two cysteines (Cys)
➔
two Cys side chains ( –C H2–SH ) release two H­atoms:
➔
–C H2–SH + –C H2–SH → –C H2–S–S–C H2– + H2
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
➔
10/06/2009
PHYS 461 & 561, Fall 2009-2010
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Coordinate bonds
➔
formed by N, O, S­atoms of the protein & O­atom 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, S­atoms are electron donors (radius ~1.5 Å) : their electrons occupy the vacant orbits of the metal ion (radius ~0.7 Å), 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
➔
10/06/2009
PHYS 461 & 561, Fall 2009-2010
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