2/7/2016
Electrophoresis Techniques
Lecture 5/6
Methods in Molecular Biophysics D5
Intermolecular and Surface Forces 14
adahlin@chalmers.se
http://www.adahlin.com/
2016-02-08
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Outline
Recall the mantra of this course: Biology is complicated, but we can understand it, at
least to some extent. One important component is the role of interfaces.
A biological sample will contain many different molecules. How can one separate them
from one another? (Necessary if we are to study them!)
Electrophoresis is one such separation technique, which is based on charge.
We will also look closer at gel electrophoresis techniques, which is the standard for
separation by size.
By using dielectrophoresis, one can even control the position of molecules and direct
them to certain regions. (Accumulate molecules and counteract diffusion!)
Finally, you will learn what electroosmotic flow is. This relates to the lectures on
microfluidics.
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Interfaces are Everywhere
Inside the cell you are never more than ~100 nm away from an interface of some kind!
transmission electron microscopy
image of a cell
Rachel Edmonds Animal Cell
http://www.thinglink.com/
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The Electric Double Layer
The standard theory for the charged interface is a diffuse Gouy-Chapman layer and a
Helmholtz-Stern layer with physically adsorbed ions.
Usually you have a bit of both, at least at high potentials.
–
–
+
–
+
–
+
+
–
–
Helmholtz-Stern model,
adsorbed ions.
2015-09-10
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+
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–
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+
–
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+
+
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–
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Gouy-Chapman model,
diffuse layer.
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The Diffuse Layer
We want to know the potential ψ and the ion
concentration C as a function of distance from the
planar surface z.
The potential energy change when moving an ion a
distance z from the location where the diffusive
layer starts (z = 0) is:
ψ=0
z
+
+
E z   Q z   0 
Here ψ0 is the potential at z = 0 and Q is the charge
of the ion, which is determined by the valency ν
(…, -2, -1, 1, 2, …) by Q = νe.
+
–
–
–
–
–
+
–
–
+
+
–
+
ψ0
–
+
+
–
–
+
+
(The elementary charge is e = 1.602×10-19 C.)
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Poisson-Boltzmann Equation
To get ψ(z) at equilibrium, we use Poisson’s equation from electrostatics:
eC z    0
 2
z 2
Here ε0 = 8.854×10-12 Fm-1 is the permittivity of free space and ε is the relative
permittivity of the medium (for a static field).
We use Boltzmann statistics for ion concentration:
 e z  

C z   C0 exp 
k BT 

Note that C0 is the concentration in the bulk (not at the surface). We can now combine
these into the Poisson-Boltzmann equation with boundary conditions:
 e z  
eC0
 2


exp 
z 2
 0
k BT 

 z  0   0
 z     0

z
0
z 
Yikes…
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Approximate Solution
For low potentials (|ψ| < 80 mV) the equation has a very simple approximate solution:
 z    0 exp z 
Clearly, a very important parameter for the solution is κ which is given by:

e2
 
  0 k BT

2
 i C0i 
i
1/ 2
bulk solution, bulk
properties

One refers to κ-1 as the Debye length. It shows
how far into a solution a “surface effect” extends!
κ-1
changes in ion concentration,
potential and all kinds of
weird things…
For a solution containing only a monovalent salt:
 2C e 2 
charged interface
1/ 2
  0 
  0 k BT 
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Diffuse Layer Exercise
Assume we have a water solution with 150 mM NaCl (physiological) at room
temperature. Calculate the concentration of Cl- 0.5 nm from a surface with a potential of
+200 mV using the Gouy-Chapman model (no adsorbed ions). Comment on the result!
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Diffuse Layer Exercise
First calculate the Debye length, for monovalent salt:
C0 = 150 mmolL-1 = 150 molm-3 = 150×6.022×1023 m-3
e = 1.602×10-19 C, kB = 1.381×10-23 JK-1, ε0 = 8.854×10-12 Fm-1
Water means ε = 80, room temperature is T = 300 K.
 2C0 e 2 

  0 k BT 
1/ 2
 
 1.257... 109 m 1
The potential at z = 0.5 nm is then:
 z  0.5 nm   0.2  exp   0.5 10 9   0.106... V
The sought ion concentration is thus:
 e z  0.5 nm  
  9.28... M
C  0.15  exp 
k BT


So we get C = 9.3 molL-1, but the maximum solubility of NaCl in water is 6.2 M-1 at
room temperature, so the model is not realistic for this surface potential.
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Grahame Equation
How can we relate surface potential to charge density σ (C/m2)? The charges inducing
the diffusive layer must compensate the net charge of the ions inside it. This gives the
Grahame equation:


 0 2  2 0 k BT  Ci z  0   C0 i 

i
i

Remember that we know C if we know ψ! For very
low potentials (<25 mV) an approximation is:
+
 0   0 0
Very important: We are still only considering the
diffuse layer! The charge density you get will
generally not be that at the actual surface.
Soft Matter Physics
+
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–
+
–
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–
–
+
σ0
–
+
+
–
+
+
–
–
σs ???
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σ=0
–
+
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Adsorbed Ions
The Helmholtz-Stern layer can be thought of as a plate capacitor. The field between two
charged plates is E = σ/[εε0] = V/d and thus:
 s  0 
dΓ ion e
 0
Here Γion is the surface coverage of adsorbed ions (inverse area).
Simple, but the values are very hard to know. The distance d can be approximated with
the radius of the adsorbed ion. However, the permittivity will be very different from that
of the bulk liquid because the water molecules are highly oriented.
d
Again very important: Only a part
of the surface charges are
compensated by ions in the
adsorbed layer!
ψ0
ψs
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–
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+
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DLVO Theory
Colloid stability (double layer repulsion) depends on ionic strength!
Includes van der Waals attraction force which scales with separation d as ~1/d2.
Simplified interaction energy U for two spheres:
U d   2πR 0 0 exp d  
2
RZ
12d
just proof of principle,
only accurate when d << R
15
kinetic
barrier
10
van der Waals
Example for:
R
d
ψ0 = 20 mV
T = 300 K
ε = 80
R = 50 nm
Z = 10-19 J (Hamaker constant)
2015-09-11
R
U/[kBT]
5
0
κ-1 = 100 nm
κ-1 = 50 nm
κ-1 = 20 nm
κ-1 = 1 nm
-5
-10
-15
0
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100
200
300
d (nm)
400
500
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Zeta Potential
The zeta potential ζ is defined as the potential at the “no slip” position or the “shear
plane” within the electric double layer. This is the distance at which ions and water
molecules no longer are “stuck”. When the particle moves, water molecules and ions
closer than the point of the zeta potential will move with the particle.
+
The zeta potential is sometimes assumed
to be equal to the potential at which the
diffusive layer starts (ζ = ψ0).
ζ
ψ0
ψs
–
+
–
+
–
–
Can be measured for surfaces and for
particles!
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Soft Matter Physics
flow
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no flow
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Electrochemistry Experiments
Simplest way to control a surface potential: Set a potential against a reference electrode
by running a circuit via a counter electrode.
← e-
Pt
Ag/AgCl
A
V
–
–
–
electrolyte
+
+
potentiostat
+
sample
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Electrochemical Detection
If the target can induce a current, it can be detected!
For glucose, an enzyme (usually glucose oxidase) converts glucose into gluconic acid
and hydrogen peroxide. A mediator oxidizes the enzyme. The mediator is then oxidized
at the electrode.
A steady current is generated as long as there is glucose. The magnitude gives the
glucose concentration in the sample.
e-
glucose
oxidase
electrochemical
mediator
glucose
gluconic acid, H2O2…
e-
e-
Fe(CN)63- + e- ↔ Fe(CN)64electrode
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ferrocyanide as mediator
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Assumptions in Electrophoresis
• Homogenous electric field (E).
• Particle that carries a charge.
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+
External field (N/C or V/m) is
given by:
–
V
E
d
+
+
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+
d
–
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+
+
+
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+
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+
• Ions in the surrounding solution.
Counterions will on average be
closer to the particle and in higher
concentration than ions that carry
the same charge.
–
–
+
–
V
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Electrophoretic Mobility
Assume the particle moves at constant velocity. It is reasonable that the steady-state
velocity is acquired fast on the nanoscale due to low inertia.
We can define the electrophoretic mobility μ as the velocity per field strength:
v

E
Obviously we want to know roughly how large μ is when we design an electrophoresis
experiment! This will depend on the forces acting on the charged object. Two forces are
obvious: The force from the field and the friction from the liquid.
Ffield  QE
The total charge of the object is Q. The friction force is given by:
Fdrag  fv
For low Reynold’s numbers and spherical objects we have Stokes drag:
f  6πR
Here η is the dynamic viscosity and R is the particle radius.
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Smoluchowski Approximation
If the Debye length is much shorter than the particle size (κ-1 << R), the Smoluchowski
equation for the mobility can be used:

 0

The model assumes a simple force balance at constant velocity: Ffield = Fdrag
The electric double layer theory for a planar surface is used! This is possible if the
curvature is low compared with the double layer thickness.
Note that zeta potential appears because this is the potential that the external field
“senses”, so it will determine the effective charge of the particle.
Also, the radius of the particle no longer appears in the equation. The Smoluchowski
approximation actually works for particles of arbitrary shape!
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More Forces in Electrophoresis
All literature agrees that there is (at least) one additional retardation force from the
accumulation of counterions around the particle. These ions want to move in the
opposite direction! They will attempt to drag the particle with them, resulting in an
additional friction-type force:
Ftotal  Ffield  Fdrag  Fretardation
κ-1
Taking retardation into account is generally
very difficult…
However, if R >> κ-1 it seems reasonable to use
the Stokes friction coefficient. In this manner,
the Smoluchowski approximation
“automatically” takes retardation into account.
R
R does not have to be so large for the
approximation to be valid at physiological
conditions!
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Electrophoresis Exercise
Nanoparticles covered with 0.1 -NH3+ groups per nm2 undergo electrophoresis with a
voltage of 20 V applied over a distance of 10 cm. Make a rough estimate how long it will
take for the particles to move this distance in 100 mM NaCl (water at room
temperature). Can the rate be comparable with Brownian motion?
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Electrophoresis Exercise
The field is E = 200 Vm-1. The high ionic strength means a short Debye length, so the
Smoluchovski model should work.
vE
 0

We have σ = 0.1×1.602×10-19×1018 = 0.01602 Cm-2. The inverse Debye length is:
 2C0 e 2 

  0 k BT 
 
1/ 2
 1.026... 109 m 1
If there are no ions we can approximate zeta potential as the surface potential:
  

 0.022... V
 0
This is quite low, which is a good sign. Smoluchowski gives v = 3.12…×10-5 ms-1 (η =
10-3 Pas). Moving 10 cm takes ~9 h. Even a very small nanoparticle would only diffuse
~1 mm during this time so the electrophoretic mobility dominates.
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Gel Electrophoresis
Now we can describe separation based on charge (more specifically ζ potential) by
electrophoresis. However, one usually also lets the electrophoresis occur in a gel.
The gel is a network of linked polymers. When objects move through the gel they are
less likely to be able to pass through if they are larger. Now size will influence mobility
directly, not only the total charge.
The most common gels are:
• Agarose (inhomogenous but large
pore size, good for larger molecules).
• Polyacrylamide (homogenous but
small pore size, good for smaller
molecules).
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smaller
molecule
larger
molecule
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Denaturation
DNA and RNA strands are quite “homogenous” molecules. They have a total charge
which is simply proportional to their molecular weight. Mobility is thus simply
determined by molecular weight.
Proteins are more complicated due to their chemical diversity and complicated structure.
It is often preferable to chemically denature proteins that undergo electrophoresis. This
means that only their amino acid sequence matters.
By using charged surfactants for denaturation, the charge can be controlled. (Negatively
charged sodium dodecyl sulphate.) Influence from amino acid type is then minor and
electrophoretic separation occurs on the basis of molecular weight. Also, everything
moves in the same direction!
+
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–
+
+
–
–
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–
–
–
–
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Visualization and Calibration
In an electrophoresis experiment, the molecules that undergo separation need to be
stained somehow to visualize them in the gel.
Despite our fancy models for
electrophoretic mobility, it is
hard to predict the velocity
inside a gel.
Usually a size standard is
included for calibration. This
gives bands that corresponds to
the movement of molecules with
known molecular weight (and
charge).
DNA stained with ethidium bromide,
UV light
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Wikipedia: Gel Electrophoresis
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Two Dimensional Electrophoresis
A pH gradient can be maintained along the field. In a second run after the initial ordinary
separation a protein will stop moving when it comes to a pH region where it has neutral
charge.
2D electrophoresis makes separation much more efficient since two proteins are unlikely
to have both similar mass and isoelectric point.
Invented in 1975 and still heavily used today in proteomics!
high pH
protein
spots
first electrophoresis
gel
second
electrophoresis
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low pH
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Background: Isoelectric Point
Proteins have amino acids that can be basic or acidic. The isoelectric point of a protein is
the pH at which it carries no net charge. The electrophoretic mobility is then zero!
+
+
+
+
+ +
+
low pH
basic side chains protonated
positive charge
–
–
+
+
–
pH = isoelectric point
no net charge
– –
–
–
–
–
high pH
acidic side chains deprotonated
negative charge
Wikipedia: Amino Acid
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Dielectrophoresis
In an inhomogenous field, an additional electrophoretic force FDEP appears.
The force acts along the gradient of the field (zero gradient means homogenous field).
Field gradients appear at any type of pointy electrode geometry!
∂E/∂z = 0
+
+
+
–
–
–
+
–
–
–
+
∂E/∂z > 0
+
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The Dielectrophoretic Force
An object does NOT have to be charged to feel the force.
The force depends on the polarizability of the object.
net force
Since the field is not uniform, one pole will experience a
greater electric field and thus a higher force!
For a spherical object, the time averaged force is (for
induced dipole):
FDEP
     2
 E
 2πR 3 m 0 Re
   2 



p

p

m

m
–
–
–
–
+
+
+
+
Here ε* represents complex relative permittivity:
  
i
 0
Here σ is the conductivity of the surrounding medium
and ω is the angular frequency of the field.
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DEP Applications
It is possible to use DC fields and still get a DEP force, so why use AC fields? First,
periodically reversing the field allows elimination of ordinary electrophoretic motion due
to inherent charge. One can also determine electrical properties of particles by varying ω.
The main applications of DEP is to separate cells and particles like lipid vesicles.
Macromolecules like DNA or proteins can also be manipulated.
electrode↓ field→
DC
AC
Planar
Ordinary electrophoretic mobility
only.
No net movement from electrophoresis,
no dielectrophoretic force either.
Structured
Both electrophoretic and
dielectrophoretic mobility.
No net movement due to
electrophoresis, but movement due to
dielectrophoresis.
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Electroosmotic Flow
Remember the counterions that leads to an additional force hindering the movement of
an object in electrophoresis. These ions are dragging liquid with them! This principle can
be used to generate flow in narrow channels that have charged walls.
There must be an excess of
counterions inside the channel. The
flow will follow the field lines
(positive to negative) if the walls are
negative and vice versa.
–
–
–
–
–
+
–
–
+
–
–
–
–
+
–
+
–
+
If κ-1 << channel width, the velocity
can be approximated as:
 E
v 0
4π
+
+
–
–
+
–
–
+
–
+
–
+
–
–
–
+
–
Here ζ is the zeta potential of the
channel surface!
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Capillary Electrophoresis
The electroosmotic flow gives a constant flow profile (plug flow)! This is in contrast to
the parabolic velocity profile from pressure driven flow.
Naturally, a charged object placed inside a capillary channel will experience the
electroosmotic velocity together with the ordinary electrophoretic force. The
electroosmotic flow is always strongest. (Not obvious why…)
In capillary electrophoresis one takes advantage of the flow. All objects go through the
channel in the same direction with the flow, but those that carry charge will either go
slower or faster depending on the charge. (A type of chromatography!)
–
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+
0
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Nanopipettes
Very small openings at the end of a tip which can be moved with high precision.
A nice mixture of everything:
• Electrophoretic mobility.
• Dielectrophoretic trapping.
• Electroosmotic flow.
Can be used for delivery or trapping at the tip!
L.M. Ying
Biochemical Society Transactions 2009
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Reflections & Questions
?
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