Transcript a Dx

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Diffraction patterns
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Electron densities (one unit cell)
Match diffraction patterns to electron densities
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Diffraction patterns
Y
Y
Electron densities (one unit cell)
Match diffraction patterns to electron densities
10
8
6
2
0
-2
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-6
-4
-2
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8
6
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-2
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-4
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Diffraction patterns
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Electron densities (one unit cell)
Match diffraction patterns to electron densities
Response to yesterday’s feedback:
One way to picture the light paths
for Photo 51
Path of
incident
x-rays
How will light deflect off a
(2d) sphere?
tangent line
at surface
incident
light
electron
density
of atom
q
q
The angle between the
incident light and the
surface tangent line
is the same as
the angle between the
reflected light and the
tangent line.
How will light deflect off a
(2d) sphere?
tangent line
at surface
incident
light
q
electron
density
of atom
q
The angle between the
incident light and the
surface tangent line
is the same as
the angle between the
reflected light and the
tangent line.
A recurring question from last time:
what are the incident/diffracted light
angles from Photo 51?
Path of
incident
x-rays
Consider diffraction
between two atoms
~ 3.4 Å apart on the axis
This is just one angle at which
light will be reflected off of these
spheres of electron density
A recurring question from last time:
what are the incident/diffracted light
angles from Photo 51?
Path of
incident
x-rays
Consider diffraction
between two atoms
~ 3.4 Å apart on the axis
Here is another angle.
Which angles will result in
constructive interference?
A recurring question from last time:
what are the incident/diffracted light
angles from Photo 51?
Dx: equals ml for constructive interference
Photon 1
a
Photon 2
What is the extra length of the path taken by photon 1?
A recurring question from last time:
what are the incident/diffracted light
angles from Photo 51?
Photon 1
q
q
a
Photon 2
2q
Dx = a sin 2q
= 2a cos q sin q
≈ 2a sin q (for small q)
We have constructive interference
when
2a sin q = ml
The smaller the distance a, the larger
sin q must be to satisfy the equality!
Reminder from last time: distances on
the diffraction pattern are inversely
proportional to distances in real space
c = 20 Å
a
b = 34 Å
b
c
a = 3.4 Å
The crystal’s electron density is a
periodic function
We can represent it as a sum of sinusoidal terms (a Fourier series):
Phase
Amplitude
of nth term
of nth term
Added note: the version for 3D electron density appears below
In theory, both formulas have (countably) infinite A, f. In practice,
the upper bounds of these sums are set by the limits of diffraction
spot detection.
Diffraction spots do not contain all info
needed to reconstruct the electron density
The intensity of a given
diffraction spot is its
square amplitude.
To reconstruct the electron density,
we need to go in reverse, i.e.
perform an inverse Fourier
transform on the diffraction data.
Unfortunately, we have not
measured the phases, so we cannot
perform the inverse transform.
This is the phase problem of
x-ray crystallography.
How bad is it to not have
the phase information?
Top row: Jerome Karle and
Herbert Hauptman, who
developed direct methods for
inferring the phases (in small
molecules)
Bottom: their images Fouriertransformed, the phases
switched, and then inverse
Fourier-transformed
How do we get the phases?
• Molecular replacement
– Uses existing structure information (of a homolog
or a subdomain)
• Multiple isomorphous replacement
– Add heavy atoms that bind specific sidechains
(e.g. lead binds Cys, platinum chloride binds His)
• Multi-wavelength anomalous diffraction
– Based on absorption and re-emission of x-rays at
certain wavelengths (causing a phase shift)
Electron density can be calculated from
amplitude and phase data by IFT
Threading a peptide through the
electron density
Evaluating the quality
of a structure model
• From the maximum usable angle of the
diffraction data, can calculate resolution in
Angstroms using Bragg’s law.
• We can calculate the diffraction spot
amplitudes we expect if our model is accurate,
|Acalc|, and compare to |Aobs|:
Note: in this lecture this
formula appeared with “F”
instead of “A”; the notation
has been changed to match
the earlier formula.
Evaluating the quality
of a structure model
Note: in this lecture this
formula appeared with “F”
instead of “A”; the notation
has been changed to match
the earlier formula.
• Notice that data that was used to create the
model is being used to evaluate the model
with Rcrystal!
• To avoid overfitting, some diffraction spot
data is withheld while the model is being
developed. This data is used to calculate Rfree
(same formula).
What we hope you learned about
x-ray crystallography
• How to apply what you know about pH,
mixing to crystallize a protein (or rock candy)
• How to use Bragg’s law to interpret Photo 51
• How data withheld from model training can
be used to evaluate a model’s quality
Once we get a structure, what
can we learn from it?
Lecture 54:
Enzyme catalytic mechanisms
Review: Enzymes catalyze reactions by
decreasing the activation energy, DG‡
According to the
Arrhenius rate law,
decreasing activation
energy increases
reaction rate:
Enzymes increase reaction rates by
several major mechanisms
• Positioning substrates to react with each other
Enzymes increase reaction rates by
several major mechanisms
• Positioning substrates to react with each other
• Donating or accepting a proton (acid-base
catalysis)
• Positioning a metal ion to react with a
substrate
• Reacting covalently with a substrate, then
releasing it
Proteases are enzymes that break
peptide bonds in other proteins
They catalyze the following hydrolysis reaction:
DG for this reaction is negative, but DG‡ is large (why?)
At neutral pH, the half-life of a peptide bond is ~ 100 years!
Why would we want an enzyme to
catalyze peptide bond breakage?
• Digestion
– Convert proteins into small peptides/amino acids
– Biggest names: trypsin and chymotrypsin
• Fast activation of a precursor protein
– Blood clotting by fibrin
Why would we want an enzyme to
catalyze peptide bond breakage?
• Digestion
– Convert proteins into small peptides/amino acids
– Biggest names: trypsin and chymotrypsin
• Fast activation of a precursor protein
– Blood clotting by fibrin
• Post-translational modification of a protein
Why would we want an enzyme to
catalyze peptide bond breakage?
• Digestion
– Convert proteins into small peptides/amino acids
– Biggest names: trypsin and chymotrypsin
• Fast activation of a precursor protein
– Blood clotting by fibrin
• Modification of a protein’s shape
• Localized activation of a precursor protein
– e.g., activation only after secretion
– Good for activating the proteases themselves!
Breaking peptide bonds
To break the peptide bond, we need a new atom to form a
bond with the carbonyl carbon.
This carbon does not have any free electrons, so the new
atom needs to supply both.
Ideally this new atom would “want” to form a new bond
(e.g. an atom might alleviate a net - charge by bonding).
This type of atom is called a nucleophile.
Breaking peptide bonds
Which of these molecules do you think would be a
stronger nucleophile?
1) H3O+, H2O, or HO- ?
1) HO- or H2N- ?
In serine proteases, a serine side chain
serves as the nucleophile
That’s odd: the serine side chain alcohol’s pKa is 13!
How can the serine side chain alcohol
lose its proton?
How can the serine side chain alcohol
lose its proton?
Typically, the histidine side chain pKa is ~6.
In this context, histidine is more likely to be charged
because of the stabilizing interaction with aspartate.
Interactions b/t these side chains also help to orient them.
Step one: the serine alcohol
nucleophile binds the carbonyl carbon
O
O…
C
…
N
OSer 195
…
C
…
N
O
H
H
Ser 195
We now need to break the peptide bond.
Just one problem: HN- is a better nucleophile than O-!
We have to make the amide into a worse nucleophile.
Step two: Give the
amide another proton
His 57 supplies this proton
O-
O-
H+
C
…
O
Ser 195
..
N
H
…
C
…
N+
O
H
H
Ser 195
Now, by breaking its bond with carbon, the amide can get a
free electron pair back and neutralize its positive charge.
…
Step three: let the amide leave
O-
O
H
C
…
N+
O
Ser 195
H
…
C
…
N
O
H
H
Ser 195
We can’t just leave one half of the substrate attached to serine!
Fortunately, a water molecule will break the ester bond and
donate a proton back to His 57/Ser 195.
…
What we hope you learned
• Post-translational modifications by proteases
can provide fast and spatially-limited changes
in protein shape/function
• Careful positioning of side chains far apart on
a protein’s peptide backbone, but close
together in the folded structure, can allow
them to do unusual chemistry
• Nucleophilicity can be used to predict which
atoms will stay bound to a carbon or leave