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2/7/2016

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1Biotechnical Physics

Electrophoresis Techniques

2016-02-08

Lecture 5/6Methods in Molecular Biophysics D5Intermolecular and Surface Forces 14

[email protected]

http://www.adahlin.com/

2016-02-08 Biotechnical Physics 2

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!

Rachel Edmonds Animal Cell

http://www.thinglink.com/

transmission electron microscopy

image of a cell

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.

2015-09-10 Soft Matter Physics 4

The Electric Double Layer

+

–

+ + + + +

– – – –

–

–

–

–

+

+

+

+

–

+ + + + +

–

–

––

–

–

––

+

+

+

Helmholtz-Stern model,

adsorbed ions.

Gouy-Chapman model,

diffuse layer.

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The Diffuse Layer

+

–

+ + + + +

–

–

–

–

–

–

–

–

+

+

+–

++

–ψ0

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:

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.

(The elementary charge is e = 1.602×10-19 C.)

zψ = 0

0 zQzE


Poisson-Boltzmann Equation

To get ψ(z) at equilibrium, we use Poisson’s equation from electrostatics:

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:

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:

Yikes…

2

2

0z

zeC

Tk

zeCzC

B

0 exp

Tk

zeeC

z B0

0

2

2

exp

0

zz

00 z 0z

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Approximate Solution

zz exp0

For low potentials (|ψ| < 80 mV) the equation has a very simple approximate solution:

Clearly, a very important parameter for the solution is κ which is given by:

2/1

B0

2

02

Tk

eC

2/1

0

2

B0

2

i

ii CTk

e

κ-1

bulk solution, bulk

properties

changes in ion concentration,

potential and all kinds of

weird things…

charged interface

One refers to κ-1 as the Debye length. It shows

how far into a solution a “surface effect” extends!

For a solution containing only a monovalent salt:

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!


Diffuse Layer Exercise

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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.

The potential at z = 0.5 nm is then:

The sought ion concentration is thus:

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.


Diffuse Layer Exercise

19

2/1

B0

2

0 m 10...257.12

Tk

eC

V ...106.0105.0exp2.0nm 5.0 9 z

M ...28.9

nm 5.0exp15.0

B

Tk

zeC


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:

i

i

i

i CzCTk 0B0

2

0 02

+

–

+ + + + +

–

–

–

–

–

–

–

–

+

+

+–

++

–

σs ???

Remember that we know C if we know ψ! For very

low potentials (<25 mV) an approximation is:

Very important: We are still only considering the

diffuse layer! The charge density you get will

generally not be that at the actual surface.σ0

σ = 0

000

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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:

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

0

ion0s

eΓd

ψ0

ψs

Again very important: Only a part

of the surface charges are

compensated by ions in the

adsorbed layer!


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:

Example for:

ψ0 = 20 mV

T = 300 K

ε = 80

R = 50 nm

Z = 10-19 J (Hamaker constant)0 100 200 300 400 500

-15

-10

-5

0

5

10

15

κ-1 = 100 nm

κ-1 = 50 nm

κ-1 = 20 nm

κ-1 = 1 nm

d (nm)

U/[

k BT

]

kinetic

barrier

dR R

just proof of principle,

only accurate when d << R

d

RZdRdU

12expπ2

2

00

van der Waals

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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.


Zeta Potential

+

–

+ + + + +

–

–

–

–

–

–

–

–

+

+

+–

++

–

The zeta potential is sometimes assumed

to be equal to the potential at which the

diffusive layer starts (ζ = ψ0).

Can be measured for surfaces and for

particles!ψ0

ψs

ζno flow

flow


Electrochemistry Experiments

Simplest way to control a surface potential: Set a potential against a reference electrode

by running a circuit via a counter electrode.

A

VAg/AgCl

Pt

potentiostat

sample

electrolyte

–

+

–

–

++

← e-

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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.


Electrochemical Detection

glucose

oxidase

Fe(CN)63- + e- ↔ Fe(CN)6

4-

electrochemical

mediator

e-

glucose

electrode

gluconic acid, H2O2…

ferrocyanide as mediator

e-

e-

• Homogenous electric field (E).

• Particle that carries a charge.

• 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.

External field (N/C or V/m) is

given by:


Assumptions in Electrophoresis

++

+

+ +

++

–

+

d

VE

Vd

–

–

––

–

–

–

–

––

–

–

–

––

–

+

+

+

+

+

++

–

– –

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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:

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.

The total charge of the object is Q. The friction force is given by:

For low Reynold’s numbers and spherical objects we have Stokes drag:

Here η is the dynamic viscosity and R is the particle radius.


Electrophoretic Mobility

E

v

fvF drag

QEF field

Rf π6

If the Debye length is much shorter than the particle size (κ-1 << R), the Smoluchowski

equation for the mobility can be used:

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!


Smoluchowski Approximation

0

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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:


More Forces in Electrophoresis

nretardatiodragfieldtotal FFFF

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 does not have to be so large for the

approximation to be valid at physiological

conditions!

κ-1

R

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?


Electrophoresis Exercise

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The field is E = 200 Vm-1. The high ionic strength means a short Debye length, so the

Smoluchovski model should work.

We have σ = 0.1×1.602×10-19×1018 = 0.01602 Cm-2. The inverse Debye length is:

If there are no ions we can approximate zeta potential as the surface potential:

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.


Electrophoresis Exercise

0Ev

19

2/1

B0

2

0 m 10...026.12

Tk

eC

V ...022.00

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.


Gel Electrophoresis

×

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).larger

molecule

smaller

molecule

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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!


Denaturation

+

––

+

–+

–

––

– –

–– –

–

In an electrophoresis experiment, the molecules that undergo separation need to be

stained somehow to visualize them in the gel.


Visualization and Calibration

DNA stained with ethidium bromide,

UV lightWikipedia: Gel Electrophoresis

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).

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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!


Two Dimensional Electrophoresis

first electrophoresis

low pH

high pH

second

electrophoresis

gel

protein

spots

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!


Background: Isoelectric Point

+

++

++

+ +

––

+

–+

–

–

–– –

–

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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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!


Dielectrophoresis

+

–

–

–

––

–

+

+

+

+

+

∂E/∂z = 0

∂E/∂z > 0

An object does NOT have to be charged to feel the force.

The force depends on the polarizability of the object.

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):

Here ε* represents complex relative permittivity:

Here σ is the conductivity of the surrounding medium

and ω is the angular frequency of the field.


The Dielectrophoretic Force

2

mp

mp

0m

3

DEP2

Reπ2 ERF

+++

–

–

–

–

+

net force

0

i

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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.


DEP Applications

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.

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.


Electroosmotic Flow

– – – – – – – –

– – – – – – – –+

+ + ++

+ +

+

+

+

+

+

–

–

–

–

–

–

–

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:

Here ζ is the zeta potential of the

channel surface!

π4

0Ev

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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!)


Capillary Electrophoresis

+– 0

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!


Nanopipettes

L.M. Ying

Biochemical Society Transactions 2009

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Reflections & Questions

?

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