Protein Crystallography Part II

Transcription

1 Molecular Biology Course 2007 Protein Crystallography Part II Tim Grüne University of Göttingen Dept. of Structural Chemistry November

2 Overview Overview 1/48

3 Reminder: Reflections vs. Map A crystallographic experiment provides rather uninterpretable data, namely a list of reflections with their intensities. After some extra effort, the (initial, approximate) phases φ(hkl) of the structure factors F (hkl). By a Fourier Transformation, this result in something more imaginable, the electron density map ρ(x, y, z): ρ(x, y, z) = 1 V unit cell h,k,l F (hkl) e iφ(hkl) e 2πi(hx+ky+lz) From Map to Model 2/48

4 From Map to Model An initial electron density (and also the final one) looks quite messy and is difficult to interpret. The final coordinate model contains more useful information, e.g. where is the position of what type of atom. The molecule model is the final target of crystallography. From Map to Model 3/48

5 Storing Structural Data the PDB file Protein models are stored e.g. in the Protein Data Bank, PDB, They do not represent the mere experimental data. From the experiment we get diffraction intensities and after some work the electron density ρ within the unit cell. The model is the best match (from the author s point of view) that explains the experimental data. A typical PDB-file contains a header with supplemental information (authors, compound, publication, etc.), the crystallographic space group and unit cell dimensions. The main part of the file are ATOM entries, one per line. An atom entry contains atom type, atom name, residue type it belongs to, and coordinates, occupancy, and B-factor. HEADER LIGASE 28-APR-99 1CLI TITLE X-RAY CRYSTAL STRUCTURE OF AMINOIMIDAZOLE RIBONUCLEOTIDE AUTHOR C.LI,T.J.KAPPOCK,J.STUBBE,T.M.WEAVER,S.E.EALICK REMARK 2 RESOLUTION ANGSTROMS.... CRYST P ATOM 1 N THR A N ATOM 2 CA THR A C ATOM 3 C THR A C... 4/48

6 Occupancy and B factor of an Atom Occupancy A typical crystal consist of a large number of unit cells (>> ), and the resulting model is therefore only an average of all these cells. Some atoms, especially those of large side chains (Arginine, Phenylalanine,... ) can be partially disordered, others can have several but fixed orientations. An occupancy lower than 1 indicates that an atom occupies this position in only a fraction of all unit cells. This affects mostly residues at the surface, pointing into the surrounding solvent region. Most atoms, however, have an occupancy of 1. B factor Even though data are usually collected at 100 K, atoms are not immobile but vibrate thermal motion. The temperature or B factor describes the vibration as a sphere within which the atom oscillates. For high resolution, when enough data are available, the vibrations in each of the three directions can be described separately. The B-factor splits up into a (symmetric) 3x3 matrix that describes anisotropic thermal motion in three dimensions. 5/48

7 Illustration of the B factor Isotropic B factors Anisotropic B factors Spherical movement of atoms Ellipsoidal movement of atoms more exact 6/48

8 Occupancy: An Example of Multiple conformation Initially the model contained only one position for the Tyrosine. But the electron density map suggests that in about half the molecules in the crystal, the side chain of the Tyrosine points in a different direction this can be modelled by setting the occupancies for both orientations of the side chain to 0.5 7/48

9 Visualising a Model ball and stick CPK (space filling) C α trace(smooth) C α trace (coloured by B-factor) ball-and-stick (coloured by B-factor) ribbons 8/48

10 Data Reliability: The Data to Parameter Ratio Data to Parameter Ratio 9/48

11 Reliability of Data: The Data to Parameter Ratio No measurement can be exact and is only an approximation to the true value. It is therefore important to have enough data to support the deduced model. In protein crystallography we want to determine at least the coordinates for every atom of the structure, i.e., we require 3 data points for every position. If more data are available, we add the isotropic B-value, and at best we can even determine an anisotropic B-value. Our data are the unique reflections the number of which is determined by the resolution, the space group, and the unit cell dimensions. Res.[Å] parameters data/parameters 3.0 x,y,z 0.9:1 2.3 x,y,z; B 1.5:1 1.8 x,y,z; B 3.1:1 1.5 x,y,z; B 5.4:1 1.5 x,y,z; U 11 U 12 U 13 U 23 U 22 U :1 1.1 x,y,z; U 11 U 12 U 13 U 23 U 22 U :1 0.8 x,y,z; U 11 U 12 U 13 U 23 U 22 U 33 16:1 G. Sheldrick These ratios, up to about 1.8Å, would be much too low to allow building of a proper model. The effective number of data points is increased by the incorporation of additional (bio ) chemical etc. information. Data to Parameter Ratio 10/48

12 An Example: Data to Parameter Ratio Scenario Experiment 1: Experiment 2: High resolution, 21 data points with errors Low Resolution, 3 data points with errors data points 3 data points f(x)=x Data to Parameter Ratio 11/48

13 An Example: Data to Parameter Ratio Two Models Model 1: g(x) = g 2 x 2 + g 1 x + g 0 Model 2: h(x) = h 3 x 3 + h 1 x + h 0 Both Models contain three parameters, i.e., at least three data points are required for their unambiguous determination. Data to Parameter Ratio 12/48

14 An Example: Data to Parameter Ratio Fitting High Resolution Data data 1.19x x x x x x 0.51 χ 2 = x x χ 2 = Remarks: χ 2 is a common error estimator in statistics. χ 2 should be close to 1 for a good model. χ 2 makes a clear distinction between the two models. The reliability of χ 2 depends on a good estimate of the errors of the data points. Data to Parameter Ratio 13/48

15 An Example: Data to Parameter Ratio Fitting Low Resolution Data data 0.72x x x x x x x x Remarks: Both Models fit the data perfectly. No error estimates because #data = #parameters. Additional knowledge is required to decide about the correct model. Data to Parameter Ratio 14/48

16 An Example: Data to Parameter Ratio Fitting Low Resolution Data Constraints Assuming Constraint: data passes through (0, 0) Model 1: g(x) = g 2 x 2 + g 1 x +g 0 Model 2: h(x) = h 3 x 3 + h 1 x +h data Model 1: g 2 x 2 +g 1 x Model 2: h 3 x 3 +h 1 x 0.94x x χ 2 = x x 2 χ 2 = χ 2 favours model 1 One constraint makes the difference between Model 1 and Model 2 a lot more striking than in the previous, non-restrained example. Data to Parameter Ratio 15/48

17 16/48

18 : Getting Started The first steps in building the model consist of finding larger groups of residues with special features. In proteins this is the (C α ) main chain, in nucleic acids the position of the bases. α helices are particularly easy to locate, even at medium to low resolution (2.5 4Å). 17/48

19 Directionality of α Helices From the main chain (C α chain) one cannot determine the direction, nor which part of the sequence it covers. One gets help from the so-called Christmas tree: the side chains of an α helix point towards the N terminal end of the protein chain. Selenomethionine substituted proteins have become very popular for MAD experiments. The heavy selenium atoms are easy to find in the electron density map and help docking the sequence to the map. Disulphide bridges or metals bound to an active centre can also be helpful. 18/48

20 β Strands The other secondary structure element of proteins, β strands are also striking but more difficult to build. Especially the direction of the peptide chain can be difficult to find. 19/48

21 Automated At resolution better than, say, 2.5Å building is extremely facilitated by programs like Arp/Warp (A. Perrakis, V. Lamzin), Buccaneer (K. Cowtan), or Resolve (T. Terwilliger), which automatically build large parts of the structure. These programs can even overcome local minima. Refinement programs (i.e. the programs that calculate the electron density map) cannot cross this barrier they would get stuck in the local minimum and could not move the Phenylalanine into the right position. 20/48

22 Manual Computer programs do not know about biology, certainly not of a specific molecule/structure. Human interaction is therefore required to pay attention to: presence and identification of ligands and/or metal ions (from crystallisation or protein preparation) special interaction for complexes exceptions from standard values used in refinement correct placement of solvent (water) molecules Biochemical knowledge about the features adds valuable information to the model building process. This becomes especially important at medium or low resolution (2.5Å and worse). Additional information acts like additional data points and therefore improves the reliability of our model 21/48

23 Hydrogen Atoms? X-rays interact with the electron shell of atoms. The strength of interaction is proportional to the total number of electrons. Hydrogen atoms only have one electron. They cannot be detected by X-ray diffraction (unless with very high resolution data < 1Å). During refinement, hydrogens are treated as riding atoms, that is, in a fixed position relative to the groups they belong to (like the carbons of a phenylalanine ring). Instead of completely ignoring hydrogens, this method improves the quality of the model and also aids to keep the correct distances to neighbouring groups. Because of the fixed position, riding atoms do not increase the number of parameters. 22/48

24 Empty Space? The Solvent Region Arrangement of molecules in the unit cell Electron density map The holes in both pictures are not vacuum. They are filled with solvent, i.e., mostly water molecules. They are disordered, therefore one does not see explicit density in these parts of the crystal. Yet, they still contribute (a little) to the diffraction pattern at low resolution. How to treat the solvent region? 23/48

25 The Solvent Model Protein crystals are not very tightly packed. The space between the molecules is filled with solvent, 50 70% of the total volume on average. Because it is disordered, it contributes mostly to reflections below 6Å resolution (d>6å). Possible ways to treat the solvent are: 1. ignore the solvent results in high R-value: Not liked by crystallographers and journal editors. 2. ignore data with d>6å, i.e. only use high-resolution data better R-value but worse maps: difficult to interpret. 3. consider the solvent region as a flat lake of electron density, i.e. with a low but constant average contribution to the scattering. 24/48

26 Refinement 25/48

27 The X ray experiment can also be inverted : From an atomic model, a space group, and a unit cell, one can calculate the data that are produced from the corresponding crystal. exploits this fact to modify the structure so that it better matches the data. It does so by small changes of the coordinates and modifcation of the temperature factors to minimize the difference between calculated and measured amplitudes. Refinement 26/48

28 Excursion: Crystallographic Theory Given the structure factors F (hkl) F (hkl) exp iφ(hkl), the electron density at position (x, y, z) is given by the Fourier transformation ρ(x, y, z) = 1 V unit cell h,k,l F (hkl) e iφ(hkl) e 2πi(hx+ky+lz) Once a model is known, the structure factors can be calculated from the spherical atomic scattering factors f j by F (h, k, l) = j f j e 2πi(hx j+ky j +lz j ) (1) Finally, the spherical atomic scattering factors f j can be calculated from per atom properties, but they are also tabulated (e.g. in the International Tables for Crystallography, Volume C). They include the effect of the temperature factor. We can compare F (hkl) 2, which are measured from the X-ray experiment with F (hkl) 2 calculated from a model. Refinement Excursus 27/48

29 Initial Map Generation Amplitudes F (hkl) initial Map initial Model Phases φ(hkl) For the first map, phases were determined with MAD, or SIR, or Molecular Replacement, etc. These phases are generally of low quality, i.e., they have large errors compared to the real values. Refinement 28/48

30 The Vicious Circle of Refinement model refinement by program (checks chemical correctness) φ calculate map new model better w.r.t. map! F build model data The new model was made using the map. The map was made using the previous model. Therefore, the new model is biased against the old model: errors may persist. Refinement 29/48

31 and Refinement Creating a model from X-ray data is an iterative process consisting of model building and refinement. Refinement means global improvement of the model with respect to the experimental data. Coordinates of all atoms together with their temperature factors (and sometimes, at very high resolution, even the occupancy), are moved in order to minimise the difference between the measured intensities and the ones calculated from the model. means local improvement of the model with respect to the experimental data. Atoms are added, removed, or moved in order to ensure that 1. the model makes sense bio chemically (proximity of atoms, H-bonding, position of solvent molecules, etc.) 2. the model fits the calculated electron density (e.g. check for multiple conformations) Refinement 30/48

32 Local Minima and Traps Refinement can only find the next minimum of its target function. refinement model building model quality bad model best model good model different models Depending on the starting point (red crosses), this might result in a good or a bad model. One of the reasons one has to validate a model is to tell whether a model is good or bad (usually it is never the best ). Refinement 31/48

33 Quality Figures: the R value One measure to distinguish a good model from a bad one is the R value. It describes the agreement between measured amplitudes ( F obs (hkl) ) and those calculated from the model ( F calc (hkl), see equation 1) R = hkl ( F obs k F calc ) hkl ( F obs ) F obs are represented by the reflection data (observations), F calc are calculated from (x,y,z) and B-values of the atoms of the model. k is an overall scale factor because the values of F calc are not absolute. For small molecules, R values between 2% and 5% are normal, for macromolecules, the range is approximately 15% 25%. As a rule of thumb one can expect an R value about 1/10 of the resolution: a 2.5Å structure should have an R value of 25%. Refinement 32/48

34 Refinement and Overfitting For macromolecular molecules, the data to parameter ratio is not very high at a normal resolution range. Therefore, the R value can be nearly arbitrarily reduced by adding more and more atoms that were not really present in the crystal structure or allowing positions that chemically do not make much sense (stereochemical clashes). This is called overfitting the data. One measure to reduce overfitting is the R free value. About 5% 10% of the reflections are excluded from minimisation of the R value. They remain unconsidered and are like an independent judge : after refinement, the R free value is calculated like the R value, but with the excluded reflections. The two values must not differ too much. The R free value is common in statistics, but was only introduced to crystallography in the mid 90 s by A. Brünger. More importantly, refinement has to take restraints and constraints into account. Refinement 33/48

35 Restraints and Constraints The reflection data alone would not be sufficient to create a trustworthy model at worse than, say, 1.5Å. There are too many parameters. Therefore it is necessary to incorporate additional information. This is done by using restraints and constraints. Small molecules at high resolution can be refined unrestrained. Macromolecules are almost always refined by restrained refinement, i.e. additional information like ideal bond lengths and angles are taken into account. Constraints reduce the number of parameters. They are expression like Property X must have value Y e.g.: temperature factor is isotropic instead of anisotropic : 4 parameters per atom instead of 9 parameters per atom Restraints increase the number of data. Should be or should be approximately expressions, e.g. distance (N, C α ) = 1.458ű0.019Å. Refinement 34/48

36 Structure 35/48

37 Why? Scientific (experimental) results are always afflicted with prejudice and bias be it deliberately or by accident and ignorance. Even though articles are proof read by referees, the experiment itself will hardly ever be repeated by an independent person before publications. Protein crystallography is no exception. However, crystallographic results are most often presented by colourful pictures that can easily make the reader over interpret their meaning. Since such models are used by non crystallographers, it is important for them to be able to check their quality. 36/48

38 Caveat: The Two Faces of Photoactive Yellow Protein This model was published in 1989 (PDB entry 1phy) The correct version: published six years later (PDB entry 2phy) Kleywegt, Acta D(2000), D56 NB: The first structure was published before usage of the R free reflections and other means of validation. It is nowadays very unlikely that such coarse misinterpretations happen. 37/48

39 Caveat: Modelling Models The structure of TBP, the TATA-box binding protein (TBP or TFIIDτ) was published in 1992 (Nikolov et al., Nature 360, pp.40 46). The shape of the molecule suggested that the TATA box sits straight in the groove of the protein. The structure of the complex, published a year later by Kim et al. (Nature 365, pp ) revealed that the DNA was actually heavily bent. 38/48

40 Caveat: What You See Is What You Get? Another issue with PDB files is that they contain more information than a graphical viewer might be able to display. Many crystallographers include atoms/residues into their structures without experimental support and set their occupancy to zero. This could be justified because they know the residues were present in the molecule (at least for recombinant proteins). Personally, though, I believe that this procedure is too error prone for later users of the structure and ought to be avoided. 39/48

41 Means for Structure means estimation of the model in comparison with the data. However, since the model was created by refinement against the data, the model is biased. Therefore, there is need for an independent judge. All information can be used 1. that did not participate in the creation of the model/ minimisation of the model data difference 2. of which ideal values are known. This means that these information must be the same or similar for all proteins. 40/48

42 : Model vs. Data Data collected from the crystal are of course the first source one would think of when it comes to validation. Unfortunately, in calculating the electron density, amplitudes from the data were mixed with phases from the model. This means that our model is already heavily biased against the data. This is why the 5 10% of all reflections never used for refinement in order to be able to calculate the R free value: hkl F R free = obs F calc hkl ( F obs ) 95% of all reflections (h,k,l) are used in order to calculate the R value, which is used for model refinement and optimisation. The remaining 5% of reflections are NOT used for refinement/optimisation, and the R free is calculated from them with the same formula above. Therefore, these 5% of reflections are independent from the model. 41/48

43 : The Real Space R factor R and R free are calculated from reflection data. They are calculated in reciprocal space. These two numbers are global figures of merit: one number tries to describe the quality of the total structure. A rather local figure of merit is the real space R factor or real space correlation coefficient. It expresses the fit between the electron density calculated from the data (reflections) and model (phases) and model only (reflections and phases). The electron density around a residue does not depend much much on residues. The resulting figures are local quality indicators. 42/48

44 The Real Space R Factor: an Example 43/48

45 : Dihedral Angles and the Ramachandran Plot A quantity that was not used in refinement, and therefore is mostly unbiased, are angles. The most famous ones are the dihedral angles ψ and ϕ, defined by the peptide main chain. Φ is the angle between the two planes defined by C i 1 N i C α and N i C α i C i, whereas Ψ is the angle between the two planes of N i C α i C i and C α i C i N i+1. Because of energetic (stereochemical) reasons, these two angles are not independent. Their dependency is drawn in the Ramachandran plot. 44/48

46 : The Ramachandran Plot The Ramachandran plot shows the φ vs. ψ angles for a structure and the most probable regions derived from the 500 best determined protein structures. β strand Interactive Ramachandran window of the model building program Coot left handed α helix α helix 45/48

47 : The Kleywegt Plot Even more information can be read from the Ramachandran plot, if more than one copy of a molecule live in the asymmetric unit: the two (or more) copies should be rather similar to each other. If one plots the Ramachandran plot for all molecules into the same diagram and connects corresponding residues, one should NOT obtain a picture like this. Kleywegt, Acta D(2000), D56 46/48

48 Tools Various programs are available to check the quality of a PDB-file, e.g. WhatIF SFcheck ProCheck MolProbity The MolProbity program is available online One can upload a PDF-file or enter a PDB ID-code and various plots. It even checks the flip states of Asn, Gln, His-residue based on possible hydrogen bondings. 47/48

49 : Summary Most of the pretty pictures about proteins represent structures determined by X ray diffraction. But do not be deceived by colours and artistic compositions. Everyone who make use of PDB files / structural data should be aware of possible pitfalls. 1. Read the header information. 2. Consider the resolution and data quality 3. Does the quality and resolution match allow for the details you want to extract? 4. Make use of programs that examine structure and (if available/possible) data Interpretation of data is important for science, but one must not exaggerate and stay close to the facts. 48/48