Crystallographic Analysis of Protein Structure

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Crystallographic Analysis of Protein Structure
BCBP 4870/6870 Spring 2003
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Materials
Jan Drenth Principles of Protein X-ray
Crystallography (in the Bookstore)
graph paper straight edge protractor compass calcu
lator w/trig functions
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Supplementary reading
Matrix algebra
An Introduction to Matrices, Sets and Groups for
Science Students by G. Stephenson (7.95)
Wave physics
Physics for Scientists and Engineers by Paul A.
Tipler
Protein structure
Introduction to Protein Structure-- by
Carl-Ivar Branden and John Tooze Introduction to
Protein Architecture The Structural Biology of
Proteins -- by Arthur M. Lesk
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The method
X-rays
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Electromagnetic spectrum
Wavelength of X-rays used in crystallography 1Å
- 3Å (Å 10-10m) most commonly 1.54Å (Cu
) Frequency c/l (3x108m/s) /(1.54x10-10m)
2x1018 s-1
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One way to make X-rays
sealed tube copper anode X-ray source
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Cu-anode emission spectrum
Cu-Ka is nearly monochromatic
1.5418 Å
Ni filter absorbs everything up to here
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The general equation for wave
Photons are oscillating electric fields.
E
Amplitude
t
a
(phase)
E(t) A cos(wt a)
wavelength
w2pc/l
The instantaneous electric field at time t
oscillation rate in cycles/second
also an oscillating magnetic field of the same
frequency, 90 degrees out of phase.
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An electric field accelerates charged particles

e-
--------------------------------------
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e- oscillates in an electric field...

• e- oscillation is the same frequency as the
  X-rays
• e- oscillation is much faster that orbiting
  motion.
• The amplitude of the e- oscillation is large
  because the mass of an e- is small. Atomic nuclei
  dont oscillate much.

e-
e-
e-
e-
e-
e-
e-
e-
e-
t
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Electron are slow compared to Xrays
Most e- orbit at a speed around 1/100th c
(2x106m/s), So in one X-ray cycle, e- would
travel 2x106m s-1 / 2x1018s-1 10-12m 0.01Å
(not much compared to the size of an atom.)
1Å atom
amount an electron moves in one xray cycle
In other word, X-rays see e- as if they were
standing still.
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oscillating e- create photons
in all directions to the oscillation.
oscillation
e-
emission
Thats scattering.
In practice, this means all directions, since
X-rays are not polarized.
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Constructive interference
e-
e-
e-
e-
Two electrons oscillating in the same place
scatter twice as much.
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Destructive interference
e-
e-
e-
e-
2 electrons separated by l/2 cancel.
but only in the horizontal direction.
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Wave addition
Sum the electric fields at each point in time.

Rule 1The sum of two waves with wavelength l
always produces a wave of wavelength l.

Constructive interference amplitude increases.
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Wave addition
e-

e-

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The phase of a wave
E(t) A cos(wt a)
5.0
-60
A 5.0, a -60 or -1/3p radians
phase where we are at t0, relative to a
cosine wave
The reference wave (cosine) has phase 0 The
sine wave has phase 90
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Decomposing a wave into two parts
a wave cosine part sine part
5.0
-60
2.5
4.33
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Decomposing a wave into two parts
5.0 cos(wt - 1/3p) 2.5 cos wt 4.33 sin wt
5.0
-60
2.5
4.33
5.0 cos(wt - 1/3p) 5.0 cos(-1/3p) cos wt - 5.0
sin(-1/3p) sin wt 2.5 cos wt 4.33 sin wt
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The sum of angles rule
cos(a b) cos a cos b sin a sin b
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Using the sum of angles rule on the wave equation
decomposes a wave into sine and cosine parts
E(t) A cos(wt a)
Using the sum of angles rule
A cos(wt a) A cosa coswt - A sina sinwt
amplitude of cosine part
amplitude of sine part
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Adding two waves by parts
cosine
sine
4.33

2.5

-3.46
-2.0


Add amplitudes of cosine and sine parts, then
recombine them.
0.87
0.5

1.0
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Adding waves by parts
Split waves into cosine and sine parts, add
them5 cos(wt - 60) 4 cos(wt 120)
5cos(-60) cos(wt ) - sin(-60) sin(wt )
4cos(120) cos(wt ) - sin(120) sin(wt )
5(0.5) 4(-0.5)cos(wt) - 5(-0.866)4(0.866)
sin(wt) 0.5 cos(wt ) - (-0.866) sin(wt )
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Adding waves by parts, part2
0.5 cos(wt ) - (-0.866) sin(wt ) Bcosb cos(wt )
- Bsinb sin(wt ) B cosb 0.5 B sinb
-0.866 phase b arctan(Bsinb/Bcosb)
arctan(-0.866/0.5) -60 amplitude B
0.5/cosb 0.5/0.5 1.0
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note on arctan function
Most calculators return a number between -90 and
90for tan-1(x/y) regardless of the sign of
y. (example tan-1(0/-1) 180, not 0) For y lt
0, add 180 to the calculators answer.
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In class exercise add two waves
A12.0 a1 90
A24.0 a2 -60
t
Split waves into cosine and sine parts, add
them Bcos(wtb ) 2 cos(wt 90) 4 cos(wt -
60) ... phaseb?? amplitudeB??
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In class exercise add two waves
A12.0 a1 90
A24.0 a2 -60
t
(1)Separate each wave into sine and cosine parts
A cos(wt a) A cosa coswt - A sina sinwt
(2) Add cosine parts, sine parts
A1cosa1 A2cosa2cos(wt) - A1sina1 A2sina2
sin(wt)
(3)Solve for new phase
(4)Solve for new amplitude
Bcoefficient of coswt term/cosb
...or...Bcoefficient of sinwt term/sinb
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answer
A12.0 a1 90
A24.0 a2 -60
t
A1 cosa1 2.0(0.0) 0.0 A2 cosa2 4.0(0.5)
2.0
A1 sina1 2.0(1.0) 2.0 A2 sina2
4.0(-0.866) -3.46
Bcos(wt b) A1cosa1 A2cosa2cos(wt)
A1sina1 A2sina2 sin(wt)
b arctan((2.0-3.46)/(2.00.0)) -36.1
B2.0/cosb 2.0/0.808 2.47
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Using the sum of angles rule on the wave equation
decomposes a wave into sine and cosine parts
E(t) A cos(wt a)
Using the sum of angles rule
A cos(wt a) A cosa coswt - A sina sinwt
amplitude of cosine part
amplitude of sine part
sine part
A
a
cosine part
Reference wave sinwt
Reference wave coswt
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Adding waves using vector addition
A12.0 a1 90
A24.0 a2 -60
t
-60
90
B2.5
b-36
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Waves represented as complex exponentials
Proof
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A wave as a complex exponential
E(t) A cos(wt a)
1
-i
A cosa coswt - A sina sinwt
A eia
-i
A
a
r
Redefinitions
Reference cosine wave real part Reference sine
wave imaginary part
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Adding complex exponentials wave addition
A1eia1 A2eia2
Multiplying complex exponentials phase shift
-60
A1eia1 eia2 A1ei(a1a2)
90
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Add these waves using a protractor and ruler
3.0, -30 2.0, 180 0.5, 90 2.0, 45 1.0,
135 2.0, 120 0.5, -120
0.5
1.0
2.0
3.0
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X-ray diffractometer
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Schematic diffractometer
beamstop
crystal
X-ray source
x-ray beam
diffracted x-rays
goniostat
X-ray detector
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The path from the source to the detector has a
length
X-raydetector
X-raysource
e-
length/l the number of oscillations completed
when hitting the detector The phase is the
non-integer part
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Distance traveled and phase
Phase D/ l nearest integer(D/l)
Same for scattered path
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Scattering by 2 e-
X-raydetector
X-raysource
Difference in pathlength rs - rs0Relative
phase a 2p(rs - rs0)/l
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Dot product
a
q
b
ab abcos(q)
ab axbxaybyazbz
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Scattering by 2 e-
rs0
s
e-
s0
r
piece of crystal, greatly magnified
e-
rs
Difference in pathlength rs - rs0Relative
phase a 2p(rs - rs0)/l
If e-(1) scatters with amplitude A1, and e-(2)
scatters with amplitude A2, then the sum of their
scattered waves is
i
A1 A2eia
a
A2
R
A1
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Use path difference to get phase
measure this...
s
e-
s0
r
e-
and this...
Get the difference, divide by the wavelength.
Multiply by 2p.Thats the phase difference.
Add the two waves using vectors.
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Exercise Sum the scattered waves
scattered xrays 90
incoming xrays, 0
Length of box l
e-
Add 4 waves using the vector method. Hint
amplitudes are all identical.