Download (Electrostatics in Biology)

„Practice-oriented, student-friendly modernization of the biomedical
education for strengthening the international competitiveness of the rural
Hungarian universities”
TÁMOP-4.1.1.C-13/1/KONV-2014-0001
(Electrostatics in Biology)
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Alajos Bérczi
Institute of Biophysics, BRC HAS, Szeged, Hungary
14th September, 2016
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..... Most biophysics courses and textbooks (at least in Hungary) deal with electric
potential differences (e.g. diffusion potential, Donnan potential, action potential,
Goldman-Hodgkin-Katz potential, etc.) but neglect the electrostatics of point
charges. This lecture is intended to make up for this lack by giving a selected
introduction into the parts of electrostatics of point charges that will then be
employed in explaining some experimental observations. ....
Introduction
- Why electrostatics?
- Science-historical background (Maxwell equations)
Electrical charges in aqueous media
- The Poisson-Boltzmann equation
- The Debye-Hückel approximation
- Electric dipoles in aqueous media
- Counterion condensation
Surface charges and surface potentials at interface
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- The Gouy-Chapman theory
- Surface charge layers
Employing electrostatics in biology
- Determination of surface charge density and surface potential of membrane vesicles
(electrophoresis, fluorescence method)
- Monovalent salt dependence of enzyme reactions
- Like-charge attraction in polyelectrolytes
- Variety of redox potential of the very same redox center
- Zn2+-transporting ATPase
Outlook
Introduction
Why “electrostatics”?
Martin Rees (1999): Just six numbers -/- Csak hat szám
 – =0.007; 2p+2n→He -- but m=-0.7%
 – actual/critical – (?) ~0.04 (1)
l – the “cosmological constant” – (?)
very small
Q – ratio of two energies
(gravitational/”rest-mass”) – ~105
D – spatial dimension – 3
N – the ratio of Fel to Fgr – ~1036
http://helikon7.files.wordpress.com/2008/06/rees_martin_csak_hat_szam.pdf
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Introduction
Gravitational field:
=
=
= 6.672 ∙ 10
Electrostatic field:
N
m
kg
always attractive
=
1
4
= 8.988 ∙ 10 N
m
C
attractive or repulsive
For two protons or two electrons at l = 1 nm (=10 Å) distance:
2 protons
= 1.865 ∙ 10
N
= 2.307 ∙ 10
N
= 1.237 ∙ 10
2 electrons
= 5.537 ∙ 10
N
= 2.307 ∙ 10
N
= 4.166 ∙ 10
Introduction
Fundamental
interactions in nature
Relative
strength
Gravitational
Electromagnetic
Weak
Strong
???”dark”???
1
1036
1025
1038
???
http://en.wikipedia.org/wiki/Fundamental_interaction
http://hyperphysics.phy-astr.gsu.edu/hbase/forces/funfor.html#c3
Why electrostatics in biology?
Biological objects:
 mass (m),
 electrical charge(s) (q,
Q),
 typical distances and sizes are in
0.5 nm to 1 μm range (R, r),  colloid system,
 dielectric media (m,
Example:
p, a),
Na+ and Cl, r =2 nm (20 Å), in water (r=80),
gravitation (Newton’s law):
m1m2
r2
Fg = 3.7·1044 N
1 q1q2
4 0 r r 2
Fe = 7.2·1013 N
Fg  f g
Na+
electrostatic (Coulomb’s law):
r
Fe 
Cl-
Fe >>> Fg
(Fe = 1.9·1031Fg)
Why electrostatics?
motion by diffusion: 0.1 - 0.01 m·s1
propagation of the electromagnetic effect: 3·108 m·s1
Why charge-charge interactions firstly?
Let T=298 K (25 °C), r=80, q =1.60·1019 C, r =1 nm (5 nm), l =0.3 nm (0.5 nm),  =90°,
+
-
thermal energy (for each degree of freedom):
potential energy in charge-charge interaction:
+
ET 
E qq
-
-
1
kT
2
ET=2.06·1021 J
+
1 q1 q 2

4 0  r r
Eq-q=2.88·1021 J
(Eq-q=5.75·1022 J)
potential energy in dipole-charge interaction:
Ed  q 
1
p1q 2
where p  ql
4 0 r r 2
Ed-q=8.631022 J
(Ed-q=5.751023 J)
potential energy in dipole-dipole interaction:
Ed  d 
1
p1 p2
1  3 cos 2 
3
4 0 r r


Ed-d=2.59·1022 J
(Ed-d=5.75·10–24 J)
+
+

+
•
•
-
Charge  induced dipole and dipole  induced dipole
interactions are weaker (depend on polarizability a of
molecules) and their distance dependence is r4 and r6,
respectively.

-
Electromagnetism: Maxwell equations
Formulation in terms of total charge and current;
The Maxwell’s „microscopic” equations in vacuum
Name
Differential form
Integral form
James Clark Maxwell
1831-1879
Gauss' law:
Gauss' law for magnetism:
Maxwell-Faraday equation
(Faraday's law of induction):
Ampère's circuital law
(with Maxwell's correction):
The four Maxwell’s „microscopic” equations describe the electric and magnetic fields arising from
varying distributions of electric charges and currents, and how those fields change in the vacuum in
time.
In the presence of matter, however, the following equations will modify the Maxwell’s „microscopic”
equations:
The Poisson-Boltzmann equation
If (1) there are no time-dependent changes of the field parameters, (2)
there is no magnetic field (B = 0) and (3) no electric current (J = 0),
then the following 3 equations will describe the electrostatic field:
(a) No curl
(b) No divergence
The combination of the above 4 equations will result the most general form of the Poisson equation
that can be used as strating point in the discussions of almost all biophysical problems:
If, furthermore, (4) there is no polarization (P = 0), (5) the aqueous medium is homogeneous and
isotropic ( = 0r = const.), (6) the distribution of ions are according to the Boltzmann statistics, and
(7) under the influence of electrostatic interactions only,
then the Poisson equation gets its most frequently used form  the Piosson-Boltzmann equation:
It should be noted that general solution of this equation
does not exists.
3D-Coordinate systems
The Debye-Hückel approximation
The Debye-Hückel (DH) limiting law for electrolytes. For small potentials, i.e.
where ey<<kT, the exponential factor in the Boltzmann equation can be linearized
(i.e. application of Taylor series expansion for all exponential terms in the
Boltzmann equation),
ex ~1+x+…
and the Poisson-Boltzmann equation can be solved. If
(i) only Coulomb interaction exists between ions (the other types of interaction are
neglected),
(ii) spherical symmetry can be applied for all ions,
(iii) which are considered as point charges and ions around a point charge is
approximated by ionic clouds with continuous charge distribution, and
(iv) the medium is considered as a homogenous and isotropic continuum,
we obtain the following equation for the potential distribution around a charge
„Ze”
where
The Debye length is a useful
measure of the spatial extent of
the diffuse charge layer. The
Debye lengths are in the nanoscale
regime and significantly reduced
by increasing ionic strength.
and  is the reciprocal Debye length, (1/ D).
A Debye sphere is a volume whose radius is the
Debye length (D), outside of which charges are
electrically screened.
When the size of the central ion „Ze” is also taken into account, the
potential distribution at r > R is:
Ion association – temporary ion pair formation
Ion association (temporary ion pair formation): it occurs when
- the electrostatic energy of attraction between unlike charges is larger than the thermic energy;
or with other words,
- cations (z(+)) and anions (z(-)) in an aqueous medium are mutually present in each other’s
electrostatic field and the centre of an anion is closer to the centre of a cation than a critical
distance (lcrit):
lB
+
where
—
is the Bjerrum length.
2lB
2—
4lB
2+
2+
Since lB  7 Å in an aqueous phase and at room
temperature, ion pairs play important role in the
properties of polyelectrolytes.
4lB
Dipoles and set of point charges in a well-defined space
The electrostatic field of two point charges with
opposite sign is the superposition of the two
electrostatic fields of the point charges. Far from
the dipole (i.e. when L<<r) the potential distribution
will be
ri
Since
R
–qi
di
If
+qi
then
The last equation would describe the superposition of the
electrostatic fields of an electric point charge Q and of a
permanent electric dipole p. In such case the p is called
the dipole moment of a space charge.
Counterion condensation theory
Polyelectrolytes (or polyions) are macromolecules in which a substantial portion
of the constitutional units have ionizable and/or ionic groups (proteins, DNA,
RNA). Polyelectrolytes which bear both cationic and anionic groups are called
polyampholytes (proteins).
Ion association theory predicts that such molecules exist always together with
associated counterions.
According to the original counterion condensation theory (G.S. Manning, 1969), if
(i) the contour length of a linear polyelectrolyte is L, and
(ii) this polymer chain bears N charged groups of valence zN, (the linear charge
density of a polyelectrolyte molecule is zNeN/L), and
(iii) the valence of counterions is zci,
then counterions should condense around the polyelectrolyte molecules as long as
the reduced charge spacing, l fulfils the following inequality:
Remarks:
(1) While the condensation of counterions is an assumption in the Manning’s
theory it automatically shows up in the solution of the PoissonBoltzmann equation in cylindrical coordinates employed for rod-like
polyelectrolytes.
(2) While the original counterion condensation theory deals only with the
polyelectrolytes (without any additional solutes in the medium), the
second counterion condensation theory (Manning et al. 1994-2000,
Manning 2011) study the problem in the presence of a bias electrolyte
solution.
Charge renormalization:
where  is the fraction of condensed zci-valent counterions per
polymer charged site. Thus, for instance, on B-DNA (L/N)1.7 Å
and since lB7 Å, zci  0.8 in case of zci=1 (Na+), and zci  0.9 in
case of zci=2 (Mg2+).
Counterion concentration profile vs. radial distance
around an infinitely long rod with radius of 5 Å and
linear charge density equal to the B-DNA in 0.01 M
electrolyte of 1:1 type. The Poisson-Boltzmann
model (solid line) and the counterion condensation
model (broken line) is compared.
The Gouy-Chapman theory
The Gouy-Chapman theory for charged surfaces preceded the Debye-Hückel theory for
electrolytes by about 10 years. The basic assumptions are identical in the two theories.
Let us assume that
(i) there is an infinite surface with uniform surface charge density of  at the
YZ plane and
(ii) a homogeneous and isotropic aquous medium in the X>0 half space with
(iii) point charges and the distribution of charges follows the Boltzmann
statistics.
In such case the Poisson-Boltzmann equation
can be integrated once and we obtain the following one-dimensional form:
Under the following boundary conditions
the relationship between the surface charge density (), surface potential (0)
and the ion concentrations (ci) in the medium will be given as:
Red curve refers to ionic medium of 20 mM
1:1 electrolyte. Blue curve refers to a
medium of 20 mM (M+), 30 mM (A–) and 5
mM (D2+).  is about 10 times that of the
thylakoid membranes.
The Gouy-Chapman theory
If there is only one biner z:z electrolyte present in the aqueous medium (e.g. z = 1 in
case of NaCl and z = 2 in case of MgSO4), and the bulk concentration of the electrolyte
is c(∞), the one dimensional potential distribution, ψ(x), is given by the following
equations:
where
2 mM MgATP-2 - anion
r0=5 Å
T=300 K
0.15 M NaCl
r>R (rion)
(1/)=0.256 nm
0.1 As m-2
Red arrows and curve 1 – Gouy-Chapman theory
Blue arrows and curve 2 – Debye-Hückel theory
Distribution of the potential in the electric double layer
at 4 different surface charge densities in a 100 mM 1:1
electrolyte solution (left panel) and in 4 different 1:1
electrolyte solutions at fixed surface charge density
(right panel)
Surface charge layers
The Gouy-Chapman theory has
been improved by taking into
account

the spatial distribution
of surface charges at
the interface and

the penetration of
electrolytes into the
membrane phase.
The interface region between aqueous
and membrane phases is a region of
continuous changes from point of view of
both size and composition. Charged
groups (e.g. phosphate), dipoles (e.g.
=C=O and water), and induced dipoles
determine both the relative permittivity
(r) and the Debye length (–1) in this
region. Analytical form for the potential
function thus can hardly be derived,
however, numerical approximations do
exist for some special cases.
Surface charge layers
z:z electrolite
With the following boundary conditions of
and (–) = 0
the problem can be solved completely.
After introducing the dimensionless potential (y) and the
reciprocal Debye length ( and ’) in both compartments
the potential distribution will be given as
Potential distribution at membrane surface with space charge layer
of thickness d = 10, 5, 0 Å (curves 1, 2, and 3, respectively). z = 1,
c() = 0.1 M, edd =  = –0.023 Cm–2 and m = a (solid lines) or m
= 0.5a (broken lines).
Surface charge density and surface potential
determination (1)
---- -- ------ ---Charge distribution around an electrophoretic
particle (liposome) in electrolyte.
According
to
the Helmholtz-Smoluchowski equation, a particle’s
electrophoretic velocity (vel) is proportianal to the applied electric field
strength (E), the zeta potential (the potential at the hydrodynamic plane of
shear – SE on the picture - z), and inversely proportianal to the dynamic
viscosity of the medium ()
The equation has been derived with the following assumptions:
 the radius of the vesicle is much higher than the Debye length (generally
higher than 100 nm; e.g. colloid particles),
 the charges are at the particle/solution interface,
 the surface is smooth, and
 there is no specific adsorption influencing the hydrodynamic plane of shear
(this is hardly fulfilled).
A more general formulation is the Henry’s equation which takes into account
the ionic strength of the medium (via the Debye length 1/) and can be used
for charged particles with radius R
Uncertainity of surface potential with identical
zeta potential; the real surface charge density
cannot be inferred from zeta potential.
where fH(R) is the Henry’s function (which approaches 1 for small R and 3/2
for large R).
Problem unsolved here: the relation between  and 0 ?
Surface charge density and surface potential
determination (2)
The simplest spectroscopic method for determining the surface charge density of
membrane vesicles uses the relative fluorescence intensity values of a water soluble
dye which are obtained when the quenched fluorescence in the electric double layer
is released by replacing the dye molecules with monovalent or divalent cations.
We have seen that
With
 a proper choice of fluorescent dye (e.g. 9-aminoacridine: 9AA),
 and ionic solutions [e.g. KCl and (HM)Br2 or (DM)Br2],
 and assuming that fluorescence quenching of fluorescent molecules in the electric
double layer is independent of the valence of cations used for the titration,
both s and y 0 can be determined as follows .......
PM vesicles (bar 1 nm)
Flourescence titration
Surface charge density and surface potential
determination (3)
Let us introduce
=
2000
and
=
−

then the equation for surface charge density given before will be as follows:
=
−1
where ci and zi is the concentration and the valency of the i-th ion in the solution.
Titration of the surface potential with 1:1 and 2:1 electrolytes in the presence of some
buffer solution we may write for the surface charge density and surface potential
=
+
=
+
+
for the 1:1 electrolyte, and
−2
−2 +
+2
for the 2:1 electrolyte.
−3
The surface potential (o) and the surface charge density () can be calculated as
=−
where
2
arcsinh 2
−3
=
and = −
+
and cm and cd are the monovalent and divalent cation concentrations, respectively.
−2
Bérczi A and Møller IM (1993) Eur. Biophys. J. 22: 177-183.
Monovalent salt dependence of enzyme reactions
According to the Michaelis-Menten kinetics
We have seen that
The surface potential depends both on the surface charge density and on the
concentration of all ions in the medium. Thus any change in the concentration of any
ion will affect the apparent value of the Michaelis-Menten constants.
where
and zS is the valence of the substrate.
(1) Shift of the pH optimum by monovalent salts.
Let 0, 010, 1:1 electrolyte, zS0, and let change the monovalent salt
concentration cm2=10cm1 . As consequence, 0201 but 020, and thus the surface pH
value increases (the surface concentration of H+ decreases) and the optimum pH
value will shift to a lower pH value.
(2) Activation, inhibition by monovalent salts.
Let 0, 010, 1:1 electrolyte, zS0, and let change the monovalent salt
concentration cm2=10cm1 . As consequence, 0201 but 020, and thus [S]0 increases
and KM,app decreases.
Like charge attraction of polyelectrolytes (1)
M13 virus
At the beginning it was known that multivalent ions generate attractions between like-charged polyelectrolytes in
a wide range of systems, while monovalent ions do not. It was also known that different ion valences are required
to condense different polyelectrolytes. How multivalent does an ion have to be before it can condense a given
polyelectrolyte? Using the charge-tunable M13 filamentous virus system and a family of artificial homologous
divalent ions of different effective sizes, a multivalent ion-polyelectrolyte phase diagram was constructed, and
an experimentally motivated general criterion for like-charged attraction based on the ion size and the Debye
length was established.
 M13 virus with ~2700 major coat proteins
 Dimethonium cations (n=1,2,3,4)
 5 mM Tris-HCl, 1 mM NaN3
 Small Angle X-ray Scattering (SAXS)
measurements
When  >d (the Gouy-Chapman length around the
polyelectrolyte charges is larger than the “size” of
divalent cations), local charge inversion is possible. This
may result in condensation (“association”) of
polyelectrolytes.
Like charge attraction of polyelectrolytes (2)
The basic elements of the second counterion condensation theory (Ray and Manning,
1994-2000) are as follows. For two parallel polyions (separated by distance r and in z:z’
electrolyte), there are three components of the free energy:
- the ionic interaction within each polyion, Wp(r); summation over all Debye-Hückel
pairwise interactions among the renormalized charges within each polyion,
- the ionic interaction between the two polyions, Wpp(r) K0(r); summation over all
Debye-Hückel pairwise interactions among the renormalized charges, one member of
the pair on one polyion, the other on the other polyion,
- the work of transfer of a subpopulation of counterions from the bulk solution to
the condensed layer on the two polyions, Wtransfer(r);
According to the second counterion condensation theory, W(r) comes in three
separated pieces:
- the near distances: a < r < (e)–1 where a is the cutoff distance of the closest
approach,  –1is the Debye length,
- the intermediate distance: (e)–1 < r <  –1 where  –1 is the Debye length,
- the far region: r >  –1 where  –1 is the Debye length.
In the three different regions different approximations are taken for the expansion
of the zeroth-order modified Bessel function of the second kind, K0(r).
Born model for solvation ... and the redox potential of
proteins
The simples model for charcterizing a charged particle in a medium is based on the
Born model. According to this model the electrostatic self-energy of an ion with
charge Q and effective radius R (Born radius of ion) in free space (vacuum) is given
as
and in a medium as
Thus the Born free energy of hydration or solvation (i.e. the energy needed for moving the charge from the free
space into the medium) can be defined as the difference of Wm and Wv
Similarly to this, the energy needed for moving the charge Q from the aqueous medium into a hydrophobic medium
can be defined as the difference of Wh and Wa
Problems:
- infinite media,
- the dipole character
of the “medium” is
neglected,
Zn2+-transporting ATPase (ZntA); a putative mechanism
(Nature 2014, doi:10.1038/nature13618) Putative zinc transport mechanism of
ZntA from Shigella sonnei.
Phosphorylation events in the intracellular domains drive large conformational
changes that permit alternating access to transport sites in the membrane
about 50 Å from the ATP-targeted catalytic aspartate. According to the model,
(1) a high-affinity state (E1), which is open to the intracellular space, binds to
Zn2+ and enters an occluded state.
(2) This state then undergoes phosphorylation.
(3) Completion of this event (E1P) triggers the release of the Zn2+, establishing
an outward-facing, low-affinity state (E2P).
(4) Release of the inorganic phosphate (Pi) yields the fully dephosphorylated
conformation (E2), which is followed by ...
(5) ... restoration of the inward-facing conformation (E1), which initiates a new
reaction cycle.
Zn2+-transporting ATPase (ZntA); a putative mechanism
(Nature 2014, doi:10.1038/nature13618) Putative zinc transport
mechanism of ZntA from Shigella sonnei.
A transport cycle based on schematic models of the E1 and E1P
states and the E2P and E2-Pi structures.
(1) In the presence of intracellular zinc, Zn2+ enters the ATPase
through the electronegative funnel (red).
(2) Upon Zn2+ binding to the intramembranous ion-binding site
(grey circle), ...
(3) ... F210 and M187 occlude the ion entry funnel, preventing
backflow of Zn2+.
(4) Substantial domain rearrangements in transition to the E2P
state open the extracellular pathway, lowering the affinity
for Zn2+ and ...
(5) ... mediating Zn2+ release, ...
(6) ... possibly stimulated by K693.
(7) Dephosphorylation triggers closure of the transmembrane
domain, in which K693 (as a built-in counter ion) forms a salt
bridge with D714. Upon dephosphorylation, the side chains
move to their initial positions before an E2 to E1 transition is
stimulated by the presence of intracellular Zn2+.
Outlook
Deficiencies in and/or problems with the standard Poisson-Boltzmann approaches (the linearized and the non-linearized
forms) are well known and originate from (or go back to) the two basic assumptions of the theory. The PoissonBoltzmann theory, for instance, underestimates the actual potential at highly charged surfaces by 15–25% for
monovalent and divalent electrolytes. Neglecting of (a) ion correlation, (b) dielectric saturation and (c) finite size of ions
and solvent molecules largely contribute to the inaccuracy of the theory.
The way to overcome the problem is possible only after
1) abandoning the assumption (constraint) for the dielectric permittivity of the medium; as soon as the length scale
of the order of a few Ångstroms becomes important, approximation of aqueous solution as a uniform dielectric medium
is no longer sufficient and the Poisson-Boltzmann equation has the following – more general, nonhomogeneous – form:
·
· −
+
=
r
= −
r
−
1000
∞ exp −
where f(r) is the charge density function of fixed charges (on molecules), r(r) refers to the electrostatic
inhomogeneity of the medium (space-dependent relative dielectric constant), otherwise as in earlier,
2) taking into account the finite size of ions and the charge distribution inside the macromolecules (see the term
f
 (r) in the above equation), and
3) introducing non-electrostatic component in the model.
All of these three modifications have been introduced by nowadays and modeling of any biological processes on the basis
of electrostatic interactions between the participants has been the most promising and fruitful in theoretical biophysics
and provides the background for molecular modelling. The past 100+ years have proven that electrostatic interaction is
the only interaction among the 4 fundamental interactions in nature that governs the biological processes and provide
perfect starting point for quantitative discussion of any biological problems.
We must, however, also be aware of that as soon as we want to go as
deep as atomic level, the only proper theoretical background is
quantum physics (quantum electro-dynamics).
Thank you for your attention!
This work is supported by the European Union, co-financed by
the European Social Fund, within the framework of " Practiceoriented, student-friendly modernization of the
biomedical education for strengthening the international
competitiveness of the rural Hungarian universities "
TÁMOP-4.1.1.C-13/1/KONV-2014-0001 project.