lecture 5: electrostatic interactions & screeningbrigita/courses/biophys...10/06/2009 phys 461 &...
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PHYS 461 & 561, Fall 2009-2010 110/06/2009
Lecture 5:Electrostatic Interactions & Screening
Lecturer:Prof. Brigita Urbanc ([email protected])
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PHYS 461 & 561, Fall 2009-2010 210/06/2009
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/40r
12
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PHYS 461 & 561, Fall 2009-2010 310/06/2009
Method of Image ChargesGriffiths, 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
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PHYS 461 & 561, Fall 2009-2010 410/06/2009
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)
➔ SOLUTION (above the interface): E = q/4
0r
12 + (q)/4
0r
02
➔ reflection effect: +q induces the shift of electrons on the high
metal side
1
0
2⨯ r12
r02
+q
q metal (=ꝏ) vaccum (=1)
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PHYS 461 & 561, Fall 2009-2010 510/06/2009
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 )
E = q/40r12 + q'/40r02
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PHYS 461 & 561, Fall 2009-2010 610/06/2009
We can use the generalized form of the field around thecharge q:
E = q/4eff
0r
12 = q/4
1
0r
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 =
+1
water
protein
+ + + + +/////////////////////////////
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PHYS 461 & 561, Fall 2009-2010 710/06/2009
Effective permittivity across the protein:
1
2
water
water
protein
eff
= 200due to water electric dipole and induced polarization
+ + + + +
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PHYS 461 & 561, Fall 2009-2010 810/06/2009
Values of effective permittivity eff
around and inside a protein:
+1
+1water
protein protein
8040 40
80
200
80
40
80 3
60
8080
80
Except inside the protein very close to the charge, the electric fieldis strongly reduced due to screening by polar water molecules
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PHYS 461 & 561, Fall 2009-2010 910/06/2009
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
2O
➔ Induced polarization produces effective internal field of the opposite sign, thereby diminishing the total field (relative to vaccum)
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PHYS 461 & 561, Fall 2009-2010 1010/06/2009
Electrostatic Interaction Between Two Oppositely Charged Atoms
➔ consider distances 3—4 Å → no water molecules in between 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 Å)
EI potential energy: 1.5 kcal/mol (
eff=80)
3.0 kcal/mol (eff
=40)
6.0 kcal/mol (eff
=20) > EHB
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PHYS 461 & 561, Fall 2009-2010 1110/06/2009
Free energy change associated with dissociation:
➔ oxalic acid (diprotic acid – 2 Hatoms per molecule): Step 1: H
2C
2O
4 ⇄ HC
2O
4− + H+
Step 2: HC
2O
4−
⇄ C
2O
42− + H+
➔ 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 GG = 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
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PHYS 461 & 561, Fall 2009-2010 1210/06/2009
Physical Interpretation:Neighboring water molecules “pulling charges” from sides
+1 1+
-
+
-
+ -
+
-+
-
+ -
+-
+
-
+
-+
-
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PHYS 461 & 561, Fall 2009-2010 1310/06/2009
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+]
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PHYS 461 & 561, Fall 2009-2010 1410/06/2009
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
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PHYS 461 & 561, Fall 2009-2010 1510/06/2009
Equation for eff
: eq/4eff
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)
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PHYS 461 & 561, Fall 2009-2010 1610/06/2009
Electrostatic Interaction Between Two Free Charges in Water
U(r) = q1 q
2 / 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 II = ½ c
iz
i2
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
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PHYS 461 & 561, Fall 2009-2010 1710/06/2009
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).
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PHYS 461 & 561, Fall 2009-2010 1810/06/2009
Disulfide (S—S) bonds➔ formed between the side chains of two cysteines (Cys)
➔ two Cys side chains ( –CH
2–SH ) release two Hatoms:
–CH
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
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PHYS 461 & 561, Fall 2009-2010 1910/06/2009
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
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