I . Separation Techniques (chapter 5 of V-H and notes from elsewhere)
A. General Principles
1. –fv + F = 0 or F = fv
(terminal velocity, -Fb = -bv, a = 0, F = eE or “centrifugal +
buoyant”).
2. For sphere, Stoke’s law
f0 = 6R, where  = viscosity (g/cm.s = poise). Water = 1 cp.
f
f
For others, f =
f0, where
is obtained from table 5.1 The
f0
f0
f
smallest frictional drag is from a sphere and
gives the drag
f0
for another type of particle with an equivalent volume (4/3 Re3).
e.g. Prolate ellipsoid.
1


1

f
P 3 P2 1 2
, f0 = 6Re, Re = (ab2)1/3, P = a/b.

1
f0


ln  P  P 2  1 2 


b
a
B. Sedimentation
1. Forces
Fc = "centrifugal" force, really acceleration, mv2/r.
v = r  Fc = mr2
Fb = buoyant force = “weight” of displaced fluid = mor2
Fd = frictional force = -fv
Ignore gravity
Fb
Fd
a
-Fb-Fd = - mv2/r or Fc-Fb-Fd = 0.
 mr2 - mor2 –fv = 0.
 mr2 (1- v ) -fv = 0, with v = specific volume ~
volume/mass of substance in solution (V/m).
 = mo/V, V = volume.
Mulitply by Avogadro’s number, R = NA.
mNA = M (molecular weight) [m = g/molecule, M = g/mole, NA
= # molecules/mole].
 M(1- v ) r2 – NAfv = 0
M1  vρ 
v

 s ***

NAf
2 r
1- v  is the buoyancy factor ~ 1- o/ (if o >  then it floats).
s = sedimentation coefficient, [s] = Svedberg, 1 x 10-13 sec = 1
Svedberg.
2. Determining s
a. Analytical centrifuge – absorption
b. vb = drb/dt = 2srb, where the subscript b signifies the
boundary (so rb is the boundary between solvent and
solution).
drb
r t 
  2st  t  ***
  2sdt  ln  b


0


r
t
b 0 
rb

So plot ln(rb) vs t and get slope which is equal to 2s.
c. Diffusion blurs boundary (discussed in detail below)
Simply: D = RT/NAf, ([D] = cm2/s as RT has units of(g
cm2)/(s2 . mole) and f has units of g/s)
x ~ Dt
d. Variance of f w/ T, due to variance of (T). Define w/r to
standard conditions: 20oC in water:
1  vρ20, w ηT, b
s 20 ,w  s T ,b
1  vρT, b η20, w
e. s depends on M and f (shape).
sphere: f0 = 6R0 and 4/3.R03 = V0 = M v /NA
1
 s* =
s 020 ,w v 3
1  vρ
2

M 3
2
 4π  3
6πηN A3 3
1
,
here, 0 is s 020 means 0 concentration, that is low concentrations
so you avoid concentration effects
See Figure 5.14, variance due to non-sphericity.(only globular
proteins fit curve).
f. If know M, can use s to get shape
factor
f = 6Rs, where s is the stokes radius
g. Density gradient – use buoyant
force, can separate different
densities (example sickle red cells).
C. Electrophoresis
Now have F = ZeE, fv = ZeE.
Mobility, U = v/E = Ze/f
Sphere: U = Ze/(6R)
D. Movement on a gel. (see math supplement)
1.
Cx , t 
 2 Cx , t 

D
  Fx Cx , t 
2
t
x
x
D = diffusion constant
 = mobility of molecule
F = force of molecule at position x
2. C(x, t=0) = N.(x-x0), where .(x-x0) is the dirac delta function.
F(x) = F, constant (ZeE) for electrophoresis


Define Ck , t    dxe  ikx Cx , t 

 dk
ikx 
C(x,t) = 
e
  2π
ikx 0
Ck , t 

C (k,0) = N e
Put in differential equation
 dk


 

0 
e ikx  Ck , t   Dk 2 Ck , t   μFik Ck , t 
2π
 t


Since FT of [] is zero then []  0


 
0  Ck , t   Dk 2 Ck , t   μFik Ck , t 
t

dC
     Dk 2  μFik dt
C


 Ck , t   Ck ,0 exp - t Dk 2  μFik




2
 Ne  ikx 0 e  t Dk  μFik
Transform back
Cx, t  

N
2π


N   ik x  x 0   tDk 2  tμFik
e
e
 dke
2π  
 x  x 0  μFt 

 
2
tD


e
u  tD k  i
2
x  x 0  μFt 


  tDk 

i2
tD


dke


x  x 0  μFt
2 tD

du  tD dk ,  e- u du  π
2

N
 Cx , t  
2 π
 x  x 0  Ft 

2 tD 
1  
e
tD
2
2
This is a gaussian centered around x = xo + Ft***
With rms of tD , velocity = F,
 = 1/(6r) with r = radius and  = viscosity
D = kBT
Do simple examples of 1D diffusion
See Maple animations (gelg and explain, newgelg)
E. Running DNA on a gel
1. Closed small plasmids run as rigid particles
- See discrete bands (more below)
2. Long DNA (10-50kb)
Persistence length = 500-600 angstroms, goes as 1/mass (tunnels)
Say #cations = DNA charge
Separation causes force – more work to separate
- If lots of cations get screnning
1 Xκr 
μ 
1  κr
6ππη
Xκr   some function
1
 8πN o e 2  2 1
 I 2
κ 
 1000ε 0 k B T 


1
I   C i Zi 2  ionic strength
2
Ci = concentration of ion, Zi = charge of ion
3. 2-D Electrophoresis  Pulsed Field electrophoresis
DNA snakes its way through
Migrates as 1/mass, i.e. 1/length
 Genome project (was $1 per base pair)
Using Saenger method
P
P
P
OCH 2O
Base
H
HH
H
H
3'-GAATCTAGCTC –
Primer
5'-CTTAG
Mix DNA polymerase I, nucleotides and labeled didexoy base
analogs (dATP, dTTP, dCTP, dGTP, each labeled with different
fluorescent dye). Used to mix with one didexoy at a time.
Run gel.
C
T
A
G
4. Running plectonemic helices
a. Review from chapter1
L, T, W, L = T + W
Supercoiling important in vivo
b. For DNA of several Kbp main effect on mobility is radius,
as radius decreases it becomes more mobile.
Since twist is approximately constant, mobility goes as
writhe and hence L which is an integer (for same length).
c. If nick DNA they all run together
d. For a large number of base pairs, if add a few more, doesn’t
make a difference
e. If have DNA plectonemic helices of a defined length, and
run on gel, get a gaussian distribution of bands due to
topoisomers, difference in writhe due to difference in
linking number.
F. Determination of Helical Repeat of DNA (Wang, 1979)
1. Definitions
# base pairs = n + x
l0 = distance between bands
h = helical repeat
2. Hypothesis
a. If x is an integral number multiple of h then no change in
pattern
L = T + W  L+x = T + x + W, W stays same
b. If x is not an integral multiple of h then pattern shifted
by(x/h) l0
T = integrated twist angle/2if increase T by 1 you have
added h base pairs. T  T + 22= T + 1, if increase
T
90
90
90
Original
W
L
10
100
11
101
12
102
by P/2 T  T + ½  Writhe must go down by ½ , shift
= (x/h) l0 = l0/2
3. Example, Say we have
Increase x by integer
Increase x by non-integer
Position
T
W
L
Position
T
W
L
Position
-9
91
9
100
-9
90.2
9.8
100
-8.8
-10
91
10
101
-10
90.2
10.8
101
-9.8
-11
91
11
102
-11
90.2
11.8
102
-10.8
Wang found that h = 10.4 ± 0.1
Slightly different than W-C (crystal)
Is twist constant?
II.
Light Scattering (Chapter 7 (Van Holde) and more)
A. Single Particle – Rough Treatment
 
Isotropic small particle P  E ; excite molecule and it accelerates and
reradiates.
 
 

1 d 2P
Incident: E  E 0 ei k 0  r  t  , Scattered : E s is propotiona l to
and
c 2 dt 2
falls off as 1/r.
The scattering angle, , is
defined w/r transmitted light
k0
in the scattering plane. The

scattering plane is defined by
Sample
Light Source
the vectors k and k0.
dipole
For Horizontally polarized light:
4 2 E 0 cos  ik  r  it
16 4  2 cos 2 
Es 
e
, IH = Is/I0 =
r2
4 r 2
For Vertically polarized light:

IV = Is/I0 =
16 4  2
4 r 2
For unpolarized light:
Is/I0 = ½ (IV + IH) =
8 4  2
4 2
1  cos 2 
 r
B. Dilute “gas” of small particles.
Classius-Mosotti: n2 – 1 = 4N, where n is the index of refraction, N is
the number of particles per unit volume. For a gas n ~ 1.
Taylor Series expansion: n  1 + (dn/dc)c + …
dn/dc ~ n/c – how n changes w/c, c being the mass concentration.
 n2 = 1 + 2(dn/dc)c + (dn/dc)2c2 + ...  1 + 2(dn/dc)c
c dn
M dn

 2c(dn/dc) = 4N,  =
, with M = molecular
2N dc 2N A dc
weight and NA = avogadro’s number. [c/N = M/NA:
(mass/volume)/(moles x NA/volume) = (mass/mole)/NA]


Is 2 2 M 2 1  cos 2   dn 


  (unpolarized)
I0
 dc 
N 2A 4 r 2
2
Intensity  molecular weight 2
scattered light/volume  is


is
Is N A c Is 2 2 1  cos 2 
 dn 
N 

cM  , so goes as M.
4
2
I0
I0
M I0
 dc 
NA r
2
C. Macromolecules in solution.
1. Without interaction
n2 – 1  n2 – no2, where no is the index of refraction of the
medium.
Take n  no  n2 – no2 = (n + no)(n – no)
 (n+no)dn  2 no(dn/dc)c
Note: If n = no  no scattering
See demo of disappearing beaker.


i  2 2 n o 2 1  cos 2 
 dn 

cM 
4
2
I0
 dc 
NA r
2
i
2 2 n o 2
r2
 dn 

cM 
R = Rayleigh ratio = 
I 0 1  cos 2 
 dc 
N A 4

R = KcM

2 2 n o 2  dn 
K=
 
N A 4  dc 
2
2
2. Add small interaction
Kc/ R = 1/M  Kc/ R = 1/M + 2Bc
B = 2nd virial coefficient – fudge factor.
D. Large Macromlecules s > 25 nm (polymers)
Can get size and molecular weight
Phase difference
k
particle
ko
At  = 0, get no phase difference – have scattering from different parts of
the particle but the difference in spatial phase within the particle is made
up by temporal phase at  = 0. At large  have big phase difference.
4   

sin  
Remember, phase difference is q. r and q 

2
So we get lots of forward scattering
Define P() = ratio of scattering of extended particle to an isolated small
particle (dipole) where interference is ignored.
1 N N sin qR ij
P() =
with N = number of scattering points, Rij =

N 2 i 1j1 qR ij


distance between them and q 

sin qR ij
qR ij
4   
sin   .

2
  1  qR ij 2  qR ij 4  ...
6
120
For small particles, or small , qRij  0 and P() = 1 so we have Raleigh
scattering (small particle means we have max Rij << )
q2 N 2
First two terms: P() = 1 
 R ij
6 N 2 ij
Define radius of gyration Rg2 =
1 N 2
 R ij
2 N 2 ij
Shape
Sphere
Rg
3 5 Rs
Ellipsoid
a 2a 2  b 2 / 5
L
12
Rod
q 2R g 2
16 2 R g 2
 
1
sin 2 
2
2
3
3
thus the angular dependence gives Rg.
 P() = 1 
Now have
Kc
1 1


  2Bc 
R  P  M

Kc
1
1



  2Bc 
2
2
R
M

16 R g
1
sin 2 
2
32
Use 1/(1-x)  1 + x 
2
2

Kc  16 R g

2   1
 1
sin
  2Bc 
2
2


R
M

3


 
 
Get Rg and M from Zimm Plot – Remember, for a given concentration, c,
we measure ias a function of angle and calculate R. Then calculate
Kc/R.
Curve A shows what you would get at a constant concentration, cA. One
would need to extrapolate to  = 0. Do this for several concentration 
Kc
1

 2Bc (straight line)
R  0 M
Intercept = 1/M, slope  2B
Curve B, , keep fixed, extrapolate to C=0 
2
2

Kc
1  16 R g

1
sin 2  
2
R  c0 M 
32


slope gives Rg, intercept gives 1/M
 
E. Polarized Light Scattering (and a more rigorous formulation of scattering).
1. The First Born Approximation and the coupled dipole
approximation
In general,


 
  k 2 e i k .r 
i k .r   
ˆ
ˆ
E( r )  E 0  r  
(1  kk )  dVe  V .E r 
r

 V is the polarizability per unit volume
ˆkˆk involves the outer product.
 


Tij  A i B j for Tij  A  B
A B
A x B y A x Bz 

  x x
T   A y B x A y B y A y Bz 


 A z B x A z B y A z Bz 

1  ˆkˆk is the transversality condition.

Example. If ˆk  ˆi , show ( 1  kˆ kˆ )E is perpendicular to k̂ .
Work out in class
Born Approximation

  
 s  k 2eik .r
i k  k o .r    0
E (r) 
 V r' .E
 dV' e
r
V'
, integral is over
particle – no interaction (trasversality condition is implicit).
Coupled-Dipole Approximation.


 
k 2eik .r N ik .r j  
Es (r ) 
 e  j .E j
r
j1
where the sum is taken over each point polarizable group (dipole)
and the field at the ith dipole is
N

 0 ik .r
 
 
E i  E i e 0 i   a ij  j .E j  bij (  jE j .ˆn ij )ˆn ij
j i
and
a ij 
e

ik .rij
rij

ik .r
1 ik
e ij
3 3ik  0
( k 2  2  ) and b ij 
(k 2  2 
) , Ei
rij
rij
rij rij
rij
is the incident electric field, n ij is a unit vector pointing from the
ith to the jth dipoles and rij is the distance between these dipoles.
When interaction between the subunits can be ignored (when the
particle is weakly polarizable) then only the first term need be
included and we get the first Born approximation.
2. The Stokes Vector (see polarization Bkgd)
the intensity and polarization state of light is fully described by
the four elements of the Stokes vector,
 I   E 2h  E 2v 
Q   2
2
E

E

h
v
 
 U  E 2  E 2 

   2
2
V   E r  E l 
where v, h, +, -, r, and l refer, respectively, to the vertical linear,
horizontal linear, positive diagonal linear, negative diagonal
linear, and right and left circular polarization components for
light of arbitrary polarization, and I, Q, U, and V are, respectively,
the total intensity, the difference between vertical and horizontal
polarized intensities, the difference between diagonal intensities,
and the difference between the circularly polarized intensities.
The Stokes’ vector is usually normalized and can be defined
operationally (eg. I is the intensity with no filters, Q is the
difference in intensity when a vertical vs horizontal polarizer is
used, etc.)
 1  1   1 
 0  1   0 
Examples   ,   ,   : what do these represent? How would
0 0  1
     
0  0   0 
we represent right circularly polarized light?
Work out in class.
3. Mueller Matrices.
a. Definition
The effect of the optical properties of a substance or optical
element on the Stokes vector can be written in terms of the
Mueller matrix,
 M 11
M
M   21
 M 31

 M 41
M 12
M 22
M 13
M 23
M 32
M 42
M 33
M 43
M 14 
M 24 
M 34 

M 44 
I
I
Q 
 
   M Q 
U
U
 
 
V  f
V  i
b. Mueller matrices for optical elements.
Linear Polarizer:
cos 2
sin 2
0
 1
cos 2
2
cos 2
cos 2 sin 2 0
1
P
sin 2 2
0
2  sin 2 cos 2 sin 2


0
0
0
 0
For vertical  = 90, horizontal  = 0, and diagonal  = 45.
Circular polarizer:
 1 0 0  1
0 0 0 0

 , right is upper, left is lower
0 0 0 0


 1 0 0 1 
Retarder at ±45o:
0
0
0
1

0 Cos(  ) 0  Sin (  )

 , where  is the strain.
0

0
1
0


0  Sin (  ) 0 Cos(  ) 
Examples: Show that a vertical, horizontal, circular polarizer do
what they are supposed to for incident un- and linearly polarized
lights. Show that a vertical polarizer + 90 degree retarder at 45o
can act as a circular polarizer, with the polarizer first.
Work out in class. (See Maple).
Significance of Muller Matrices
i)
Each element describes a change in polarization
Eg. Ir = ½ (M11 + M14) Io
IL = ½ (M11 - M14) Io
 M14 = (Ir – IL)/Io
Where Ir is the intensity of the light you get
1 
0 
when the incident light is   Io
0 
 
1 
A  ln 10 ( Ah  Av ) / 2,
LB  2  ( nh  nv )l /  ,
and IL is the intensity of the light you get when
1
0
the incident light is   Io. Thus M14 is CD in
0
 
 1
absorption and CIDS in transmission
(absorption).
In general, in absorption, each element has a direct
interpretation:
 LD  LD' CD 
 1
  LD
1
CB
LB' 
A 
Me
 LD'  CB
1
 LB 


1 
 CD  LB' LB
where
LB'  2  ( n  n )l /  , CB  2  ( nl  nr )l /  ,
A
LD  ln 10 ( Ah  Av ) / 2, LD'  ln 10 ( A  A ) / 2, CD  ln 10 ( Al  Ar ) / 2,
denotes an absorbance, LB refers to linear birefringence, LD refers to linear dichroism,
CB denotes circular birefringence (or ORD) and , n, and l are the wavelength of light,
index of refraction, and path length of the sample. The subscripts h, v, l, r, + and - refer
to horizontal, vertical, left circular, right circular, +45o, and –45o polarizations.
In scattering,
The Mueller scattering matrix elements, for a randomly
oriented sample, can be written (Tian and McClain, 1989)
A B
B C

 h  j
 f
g

h
j
D
k
f
g
.
k
E 
The symmetry of this scattering matrix, which disappears
for samples containing preferred orientation, was first
demonstrated by Perrin (1942). The uppercase elements
have been called dipole elements because they do not
disappear in the dipole limit (Harris and McClain, 1985).
That is, these elements are non-zero even for small
particles of low polarizability where the electric field
within the scattering particle is essentially equal to the
incident electric field. The lower case elements are called
retardation elements or non-dipole elements because of
their greater sensitivity to polarizability. The elements h,
j, and k are identically zero in the orientation average
unless the induced electric field, due to interactions
within the particle, is accounted for (Harris and McClain,
1985). The elements f and g are not necessarily zero in
the dipole limit, but they are very small and hence have
large contributions resulting from interactions within the
particle (McClain and Ghoul, 1986). Perrin showed that
the elements h, f, j, and g are zero in the orientation
average unless the particles have some inherent chirality.
Thus, these off-diagonal block elements are called
helicity elements (Tian and McClain, 1989).
The element M11 describes the total intensity of
scattered light.
This element is most commonly
measured to provide general size information. The other
dipole elements can also be used for this purpose. In
addition, the deviation of M22 from unity and M33 from
M44 is indicative of the non-spherical symmetry of the
particle (Quinby-Hunt et al., 1989). The element M14 is
also known as circular intensity differential scattering
(CIDS) and has been used to probe the chiral nature of
particles. The other helical elements can also be used for
probing chiral structures. In addition, these elements are
useful to ascertain whether the scattering medium has any
preferred orientation, as they are zero unless the particles
in the medium are chiral or have some preferred
orientation. M34 can give especially sensitive information
regarding particle size and polarizability, since it is the
only retardation element that is not also a helical element.
The first and most difficult part of calculating the Mueller
matrix for a particle is to calculate the scattered electric
field for a given incident electric field. Once the parallel
and perpendicular components of the scattered field are
calculated for incident parallel and perpendicular electric
fields, the scattering amplitude matrix is calculated which
relates these components of the incident and scattered
fields as follows (Shapiro et al., 1994b):

E sh'  1
 J11
  s   exp(ikr)
E v'  kr
J 21

J12  E ih 
 
J 22   E iv 
where i and s are now superscripts referring to scattered or
incident. The prime qualifiers for h and v are necessary to
distinguish the vertical and horizontal directions for the
scattered light from those of the incident light. If the
particle is not spherically symmetric, the orientation
average of the scattering amplitude matrix elements is
calculated by integrating them in terms of Euler angles
over all possible orientations. It is straightforward to
calculate the Mueller matrix elements from the scattering
amplitude elements, Jij (Bohren and Huffman, 1983).
4. Measurement of the Mueller Matrix
Many elements are very small, including interesting ones like
CIDS. One method of detecting small signal in large background
is modulating the signal. The method described here works for
CD and CIDS.
Light
source
90o
Polarizer
PEM
at 45o
Sample
Filter?
detector
PEM
Quartz
(strain)
Apply ac voltage –
piezoelectric crystal
Polarization State:
The Mueller matrix of PEM is the same as that of a retarder at
45o with  =  Sin(t) where  is the frequency of
modulation (usually 50 kHz), and A is the amplitude of the
modulation. The amplitude of modulation is wavelengthdependent,  = 2ds/, with d being the width of the crystal, s
is a strain factor that depends on the material and the applied
voltage, and is the wavelength.
We get
1
1
 1 
 
 


 a  Polarizer  1  PEM  cos   
 b     0   0 
 
 


c
0
  sin  
 
 


 M11  M12 cos    M14 sin   


sample  M 21  M 22 cos    M 24 sin  
 

M  M 32 cos    M 34 sin   
 31

 M  M cos    M sin  
42
44
 41

We can expand the cosine and sine terms as Bessel Functions:
Cos(Asin(wt)) = J0(A) + J2(A)cos(2wt) + J4(A)cos(4wt) + …
Sin(Asin(wt))= J1(A)sin(wt) + J3(A)cos(3wt) + …
A lock-in amplifier can select signals that are oscillating either
at the same or twice the frequency of a reference signal,
thereby selecting either the sine or cosine terms. A feedback
servo circuit is used to keep the DC current constant,
effectively normalizing by M11.
So if no final filter is used,
The DC current is M11
The 1f signal is M14/M11
The 2f signal is M12/M11
Ex – How would you get M22 and M13?
Work out in class.
F. Octopus Sperm Chromosome.
Used PLS to show DNA is perpendicular to the fiber axis.
See SEM and TEM of fiber
a. TEM shows thick helical fiber with a 200 nm radius, 100 nm thick.
b. Model
r' = rh cos(h) i+ rh sin(h) j + (Ph/(2 k

    pp pp
    tt 
   nn nn
tt
The polarizability tensor is defined by its magnitude along the
principle axes, tangent (t), normal (n), and binormal (p) to the
polymer. Thus, where the strengths along each principle axis have a
real and imaginary component.
Normally, t = derivative, normal = 2nd derivative and binormal is
ˆt  ˆn . Show these are perpendicular for r'.
Used  to move between these.
C. Results – Figures 7 and 8 – DNA is perpendicular to fiber.
D. Consistent with cholesteric crystal model.
30 nm
fiber
G. Higher order DNA structure and function
1. Eukaryotic structure (Prokaryotes probably have some proteins
too).
a. DNA is highly condensed (mitosis: 5000 times shorter).
b. As degree of condensation increases, knowledge of
structure decreases.
2. Nucleosome
a. Evidence
 Maurice Wilkins’ X-Ray structure gave
100 angstrom helical repeat
 Histone proteins found in chromatin
 Cut by DNAse every 10 nucleotides
wrapping
b. Structure
 146 bp wrapped around core particle (110
x 55 angstroms)
 with H1 have 166 bp involved in double
wrap
 Recently the amount of DNA around core
particle has been found to vary between
100 and 170 bp
c. 30 nm fiber (see pictures)
 Solenoid model
- 6-7 nucleosomes/turn
- nucleosomes parallel to helix axis
- space in center
 Crossed linker model
- Alternating sequence of nucleosomes
- Variable diameter – depends on
linker length
- 26 nm pitch
- filled center
d. Protein scaffold
30 nm fiber attached to protein matrix
e. Function
 efficient compaction
 Txn control
- Domains: genetically distinct regions
- In dipterian polytene chromosomes
see band structure: compacted vs
condensed regions, number of bands
corresponds to number of genetic
functions
- Lampbrush
chromosomes
–
unlooped active regions
-
IN other eukaryotes have differential
DNAse sensitivity
- Eg globin gene
Early embryo – adult and embryo
region sensitive
Later inactive embryo region is
insensitive
 Replication and Topoisomerases
Knots and Catanes distinguished by
electrophoresis
Necessary enzymes: Topo I SS DNA and
Topo II for ds DNA.