1. Match diffraction patterns to electron densities Electron densities (one unit cell) Diffraction patterns

2. Match diffraction patterns to electron densities Electron densities (one unit cell) Diffraction patterns

3. Match diffraction patterns to electron densities Electron densities (one unit cell) Diffraction patterns

4. Response to yesterday’s feedback: One way to picture the light paths for Photo 51 Path of incident x-rays

5. How will light deflect off a (2d) sphere? electron density of atom tangent line at surface   incident light 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.

6. How will light deflect off a (2d) sphere? electron density of atom tangent line at surface   incident light 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.

7. A recurring question from last time: what are the incident/diffracted light angles from Photo 51? Path of incident x-rays This is just one angle at which light will be reflected off of these spheres of electron density Consider diffraction between two atoms ~ 3.4 Å apart on the axis

8. A recurring question from last time: what are the incident/diffracted light angles from Photo 51? Consider diffraction between two atoms ~ 3.4 Å apart on the axis Path of incident x-rays Here is another angle. Which angles will result in constructive interference?

9. A recurring question from last time: what are the incident/diffracted light angles from Photo 51? What is the extra length of the path taken by photon 1? Photon 1 Photon 2 a  x:equals m for constructive interference

10. A recurring question from last time: what are the incident/diffracted light angles from Photo 51? Photon 1 Photon 2 a     x= a sin 2  = 2a cos  sin  ≈ 2a sin  (for small  ) We have constructive interference when 2a sin  = m The smaller the distance a, the larger sin  must be to satisfy the equality!

11. Reminder from last time: distances on the diffraction pattern are inversely proportional to distances in real space a b c a = 3.4 Å c = 20 Å b = 34 Å

12. The crystal’s electron density is a periodic function We can represent it as a sum of sinusoidal terms (a Fourier series): Amplitude of nth term Phase of nth term Added note: the version for 3D electron density appears below In theory, both formulas have (countably) infinite A, . In practice, the upper bounds of these sums are set by the limits of diffraction spot detection.

13. 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.

14. 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 Fourier- transformed, the phases switched, and then inverse Fourier-transformed

15. 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)

16. Electron density can be calculated from amplitude and phase data by IFT

17. Threading a peptide through the electron density

18. 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, |A calc |, and compare to |A obs |: Note: in this lecture this formula appeared with “F” instead of “A”; the notation has been changed to match the earlier formula.

19. Evaluating the quality of a structure model Notice that data that was used to create the model is being used to evaluate the model with R crystal ! To avoid overfitting, some diffraction spot data is withheld while the model is being developed. This data is used to calculate R free (same formula). Note: in this lecture this formula appeared with “F” instead of “A”; the notation has been changed to match the earlier formula.

20. 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

21. Lecture 54: Enzyme catalytic mechanisms Once we get a structure, what can we learn from it?

22. Review: Enzymes catalyze reactions by decreasing the activation energy,  G ‡ According to the Arrhenius rate law, decreasing activation energy increases reaction rate:

23. Enzymes increase reaction rates by several major mechanisms Positioning substrates to react with each other

24. 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

25. Proteases are enzymes that break peptide bonds in other proteins They catalyze the following hydrolysis reaction:  G for this reaction is negative, but  G ‡ is large (why?) At neutral pH, the half-life of a peptide bond is ~ 100 years!

26. 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

27. 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

28. 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!

29. 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.

30. Breaking peptide bonds Which of these molecules do you think would be a stronger nucleophile? 1)H 3 O +, H 2 O, or HO - ? 1)HO - or H 2 N - ?

31. In serine proteases, a serine side chain serves as the nucleophile That’s odd: the serine side chain alcohol’s pK a is 13!

32. How can the serine side chain alcohol lose its proton?

33. Typically, the histidine side chain pK a 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.

34. Step one: the serine alcohol nucleophile binds the carbonyl carbon N C O H O-O- N C O-O- H O … … … … 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. Ser 195

35. Step two: Give the amide another proton N C O-O- H O … … Now, by breaking its bond with carbon, the amide can get a free electron pair back and neutralize its positive charge. Ser 195.. H+H+ N+N+ C O-O- H O … … Ser 195 H His 57 supplies this proton

36. Step three: let the amide leave 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. N+N+ C O-O- H O … … Ser 195 H N C O H O … … H

38. 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