UNIT III Protein Engineering.pdf

UNIT – III Prediction of Protein Structure and Method  Homology Modeling Computational method for predicting the 3D structure of a target protein using its amino acid sequence. Based on the concept of sequence homology, similar sequences have similar struc...

UNIT – III Prediction of Protein Structure and Method  Homology Modeling Computational method for predicting the 3D structure of a target protein using its amino acid sequence. Based on the concept of sequence homology, similar sequences have similar structures, even if they're distantly related. One of the most accurate methods for predicting protein structure. Homology models are used in many applications, including virtual screening, designing mutagenesis experiments, and rationalizing the effects of sequence variations. Also known as comparative modeling & sequence-sequence alignment method  Definitions Pairwise alignment The process of lining up two or more sequences to achieve maximal levels of identity (and conservation, in the case of amino acid sequences) for the purpose of assessing the degree of similarity and the possibility of homology. Homology - Similarity attributed to descent from a common ancestor. Identity - The extent to which two (nucleotide or amino acid) sequences are invariant. RBP: 26 RVKENFDKARFSGTWYAMAKKDPEGLFLQDNIVAEFSVDETGQMSATAKGRVRLLNNWD- 84 + K ++ + + + GTW++ MA + L + A V T + +L+ W+ glycodelin: 23 QTKQDLELPKLAGTWHSMAMA-TNNISLMATLKAPLRVHITSLLPTPEDNLEI V LHRWEN 81  Pairwise alignment of retinol-binding protein and -lactoglobulin 1 MKWVWALLLLAAWAAAERDCRVSSFRVKENFDKARFSGTWYAMAKKDPEG 50 RBP. ||| |. |... | :.||||.:| : 1...MKCLLLALALTCGAQALIVT..QTMKGLDIQKVAGTWYSLAMAASD. 44 lactoglobulin 51 LFLQDNIVAEFSVDETGQMSATAKGRVR.LLNNWD..VCADMVGTFTDTE 97 RBP : | | | | :: |.|. || |: || |. 45 ISLLDAQSAPLRV.YVEELKPTPEGDLEILLQKWENGECAQKKIIAEKTK 93 lactoglobulin 98 DPAKFKMKYWGVASFLQKGNDDHWIVDTDYDTYAV...........QYSC 136 RBP || ||. | :.|||| |..| 94 IPAVFKIDALNENKVL........VLDTDYKKYLLFCMENSAEPEQSLAC 135 lactoglobulin Identity 137 RLLNLDGTCADSYSFVFSRDPNGLPPEAQKIVRQRQ.EELCLARQYRLIV 185 RBP. | | | : ||. (bar) | || | 136 QCLVRTPEVDDEALEKFDKALKALPMHIRLSFNPTQLEEQCHI....... 178 lactoglobulin  Pairwise alignment of retinol-binding protein and -lactoglobulin 1 MKWVWALLLLAAWAAAERDCRVSSFRVKENFDKARFSGTWYAMAKKDPEG 50 RBP. ||| |. |... | :.||||.:| : 1...MKCLLLALALTCGAQALIVT..QTMKGLDIQKVAGTWYSLAMAASD. 44 lactoglobulin 51 LFLQDNIVAEFSVDETGQMSATAKGRVR.LLNNWD..VCADMVGTFTDTE 97 RBP : | | | | :: |.|. || |: || |. 45 ISLLDAQSAPLRV.YVEELKPTPEGDLEILLQKWENGECAQKKIIAEKTK 93 lactoglobulin 98 DPAKFKMKYWGVASFLQKGNDDHWIVDTDYDTYAV...........QYSC 136 RBP || ||. | :.|||| |..| 94 IPAVFKIDALNENKVL........VLDTDYKKYLLFCMENSAEPEQSLAC 135 lactoglobulin Somewhat Very 137 RLLNLDGTCADSYSFVFSRDPNGLPPEAQKIVRQRQ.EELCLARQYRLIV 185 RBP. | | similar | : ||. | || | similar (one dot) (two dots) 136 QCLVRTPEVDDEALEKFDKALKALPMHIRLSFNPTQLEEQCHI....... 178 lactoglobulin  Types of homology Orthologs Homologous sequences in different species that arose from a common ancestral gene during speciation; may or may not be responsible for a similar function. Paralogs Homologous sequences within a single species that arose by gene duplication.  common carp Orthologs: members of a zebrafish gene (protein) family in various organisms. This tree shows rainbow trout RBP orthologs. teleost African clawed frog chicken human mouse horse rat pig cow rabbit 10 changes  apolipoprotein D Paralogs: members of a retinol-binding protein 4 gene (protein) family within a Complement species component 8 Alpha-1 Microglobulin /bikunin prostaglandin D2 synthase progestagen- associated neutrophil endometrial gelatinase- protein associated lipocalin Odorant-binding protein 2A Lipocalin 1 10 changes  Homology Modeling Process ❑ Template recognition ❑ Alignment ❑ Determining structurally conserved regions ❑ Backbone generation ❑ Building loops or variable regions ❑ Conformational search for side chains ❑ Refinement of structure ❑ Validating structures  Template Recognition First, we search the related proteins sequence(templates) to the target sequence in any structural database of proteins The accuracy of model depends on the selection of proper template FASTA and BLAST from EMBL-EBI and NCBI can be used This gives a probable set of templates, but the final one is not yet decided After initial alignments and finding structurally conserved regions among templates, we choose the final template  Structurally Conserved Regions ❑ SCRs are region in all proteins of There are generally two main a particular family that are nearly approaches identical in structures. ❑ Constructing c-alpha distance ❑ Tend to be at inner cores of the matrix proteins ❑ Usually contains alpha-helices ❑ Aligning vectors of secondary and beta sheets structure units ❑ No SCR can span more than one secondary structure  Challenges ❑ To model proteins with lower similarities( eg < 30% sequence identity) ❑ To increase accuracy of models and to make it fully automated ❑ Improvements may include simulataneous optimization techniques in side chain modeling and loop modeling ❑ Developing better optimizers and potential function, which can lead the model structure away from template towards the correct structure ❑ Although comparative modelling needs significant improvement, it is already a mature technique that can be used to address many practical problems Automated Web-Based Homology Modeling ❑ SWISS Model : http://www.expasy.org/swissmod/SWISS-MODEL.html ❑ WHAT IF : http://www.cmbi.kun.nl/swift/servers/ ❑ The CPHModels Server : http://www.cbs.dtu.dk/services/CPHmodels/ ❑ 3D Jigsaw : http://www.bmm.icnet.uk/~3djigsaw/ ❑ SDSC1 : http://cl.sdsc.edu/hm.html ❑ EsyPred3D : http://www.fundp.ac.be/urbm/bioinfo/esypred/  Protein Threading The goal: find the “correct” sequence-structure alignment between a target sequence and its native-like fold in PDB MTYKLILN …. NGVDGEWTYTE Energy function – knowledge (or statistics) based rather than physics based – Should be able to distinguish correct structural folds from incorrect structural folds – Should be able to distinguish correct sequence-fold alignment from incorrect sequence-fold alignments  Process Threading - A protein fold recognition technique that involves incrementally replacing the sequence of a known protein structure with a query sequence of unknown structure. The new “model” structure is evaluated using a simple heuristic measure of protein fold quality. The process is repeated against all known 3D structures until an optimal fit is found.  Fold recognition methods 3D-PSSM http://www.sbg.bio.ic.ac.uk/~3dpssm/ Fugue http://www-cryst.bioc.cam.ac.uk/~fugue/ HHpred http://protevo.eb.tuebingen.mpg.de/toolkit/index.php?view=hhpred  Isothermal Titration Calorimetry Isothermal titration calorimetry (ITC) is one of the physical techniques that directly measures the heat discharged or consumed all along a bimolecular reaction. It is an analytical method where the ligand comes in contact with a macromolecule under constant temperature. It works on the basic principle of thermodynamics where contact between two molecules results in either heat generation or absorption, depending on the type of binding, that is, exothermic or endothermic. The instrument consists of two cells; one is the main cell for the macromolecule of concern and the other cell is called a reference cell, which is meant for the solvent. Both cells are kept at steady temperature and pressure. The ligand is sucked through a syringe and titrated into the main cell. Macromolecular binding with the ligand results either in heat discharge or consumption, which causes the change of temperature within the main cell. However, the instrument will always maintain the constant temperature in the main cell equivalent to that of the reference cell. For maintaining the temperature, the instrument gives relevant power (higher or lower) depending on interaction. The heat change is then simply calculated by integrating the power over the time (seconds), which gives us the enthalpy of the reaction. The heat discharged or consumed all along the calorimetric reaction corresponds to the fraction of bound ligand and increased ligand concentration leads to saturation of substrate and finally less heat is discharged or consumed. Calorimetry (Latin calor - heat, Greek metry - to measure) is the termodynamic technique based on the measurement of heat that may be generated (exothermic process), consumed (endothermic process) or simply dissipated by a sample. A calorimeter is an instrument used for measuring the quantity of heat absorbed or released in process of a chemical reaction. 1 calorie - express quantity of heat necessary to raise the temperature of 1 g of water by 1°C. Heat is generated by almost all processes (physical, chemical or biological)  Isothermal Titration Calorimetry ▪ constant temperature ▪ ligand titration ▪ what happend when two (bio)molecules interact? (constant temperature) ▪ Heat is released or absorbed as a result of the redistribution and formation of non- covalent bonds when the interacting molecules go from the free to the bound state. ▪ Enzyme kinetics, biological activity or the effect of molecular structure changes on binding mechanism ▪ characterization of biomolecular interactions of small molecules, proteins, antibodies, nucleic acids, lipids and others ▪ Complete thermodynamic profile of the molecular interaction in a single experiment (stoichiometry, Ka, enthalpy ∆H and entropy ∆S values) or kinetics parameters Km and kcat  Instrumentation A constant temperature is controlled by two main heaters - one for each cell. Each heater is controlled by a power feedback sensor. In case of exothermic reaction - the sample cell gets warmer than reference cell - less power supplied to sample cell heater Feedback to the heaters which ITC monitors these heat changes by compenstate the measuring the differential power, temperature applied to the cell heaters difference Power feedback sensor which detects Reference calibration heater temperature difference between sample and reference cell and control the temperature Sample calibration heater maintenance Cell main heater Goal T 0 In the calorimetric experiment, ligand is titrated to the sample cell (receptor sample) in a number of small aliquots. In the calorimetric experiment, ligand is titrated to the receptor in the sample cell in a number of small aliquots. When substances bind, heat is either generated or absorbed.  The Raw ITC Data Raw ITC data is a measure of the power difference supplied to each cell The raw signal in the power compensation calorimeter is the power ( cal/sec) applied to the control heater that is required to keep the calorimeter cell from changing temperature as a function of time. Start of titration large peaks – lots of complex formed on each injection equal height – virtually every ligand molecule becomes bound to receptor  The Raw ITC Data Raw ITC data is a measure of the power difference supplied to each cell Start of titration large peaks – lots of complex formed on each injection equal height – virtually every ligand molecule becomes bound to receptor  The Raw ITC Data Raw ITC data is a measure of the power difference supplied to each cell Start of titration large peaks – lots of complex formed on each injection equal height – virtually every ligand molecule becomes bound to receptor  The Raw ITC Data Raw ITC data is a measure of the power difference supplied to each cell Around equivalence point heat change decreases as binding sites fill up Start of titration large peaks – lots of complex formed on each injection equal height – virtually every ligand molecule becomes bound to receptor  The Raw ITC Data Raw ITC data is a measure of the power difference supplied to each cell End of titration all binding sites occupied – no further binding only “dilution peaks” after addition of more ligand Around equivalence point heat change decreases as binding sites fill up Start of titration large peaks – lots of complex formed on each injection equal height – virtually every ligand molecule becomes bound to receptor These heat flow peaks are integrated with The pattern of the heat effects/mol of respect to time, giving the total heat titrant as a function of the molar ratio released/absorbed after each injection [ligand]/[macromolecule] can then be point. analysed to give the thermodynamic parameters of the interaction. H = Q / concentration of titrant These heat flow peaks are integrated with The pattern of the heat effects/mol of respect to time, giving the total heat titrant as a function of the molar ratio released/absorbed after each injection [ligand]/[macromolecule] can then be point. analysed to give the thermodynamic parameters of the interaction.  Evaluation of ITC Data Ka P + L PL Kd Affinity Enthalpy In one ITC experiment: ▪ Enthalpy H Stoichiometry ▪ Equilibrium binding constant Ka ▪ Stoichiometry …. calculate: ligand / macromolecule ∆G = -RT ln Ka ∆G = ∆H - T∆S Kd = 1 / Ka  Ka + Kd M X MX ∆G = -RT ln Ka ∆G = ∆H - T∆S Free energy Enthalpy Entropy  ∆G = ∆H - T∆S G ≤ 0 for spontaneous process High affinity = high Ka, low Kd, high - G  + - water molecules ∆G =M∆H - T∆S X MX - ionts - protons Enthalpy : System has a tendency to reach the minimum energetic state. H has tendency to be negative. ….bonds are formated Entropy : At the molecular level, Brown´s motion rises the entropy. Entropy rises with temperature. T S has tendency to be positive. Formation of bonds means that entropy is decreasing  ∆G = ∆H - T∆S Enthalpy Entropy Changes in heat Changes in disorder Structure of the complex Independent rotational and translational ▪ Hydrogen bonding degrees of freedom ▪ Van der Waals ▪ A complex is less disordered than two Structure of the solvent molecules ▪ i.e. water Internal conformational dynamics ▪ Flexible molecules lose entropy in the Enthalpy change - energy content of the process binding bonds broken and created. The dominant Dynamics of the solvent contribution is from hydrogen bonds. ▪ i.e. water Negative value indicates enthalpy change favoring the binding. Bonds formation means higher order of the system therefore entropy decreases Negative is favourable Positive favourable Unfavorable enthalpy positive for entropically driven reactions: Hydrophobic interactions Solvation entropy due to release of water Pictorial representation showing the progress of ITC experiment with result Pictorial representation showing the stepwise progress of ITC experiment result.  Advantages Label-free In-solution No molecular weight limitations Optical clarity unimportant Minimal or no assay development X-ray Crystallography, and X-ray Diffraction Overview of Protein Structure Determination Different methods provide complementing protein structural data. Biochemical methods determine the primary structure by directly determining the amino acid sequence from the protein or indirectly, but more rapidly, from the gene or cDNA nucleotide sequence. The quaternary structure of large proteins or aggregates such as virus particles, ribosomes, or gap junctions can be determined by electron microscopy. This method provides structural information at very low resolution without atomic details, but if ordered two-dimensional arrays of the object can be obtained, the noise in the electron microscopic image can be reduced enough to reveal the shape of individual subunits or, in rare cases, the polypeptide chain path within a protein molecule. To obtain the secondary and tertiary structure, which requires detailed information about the arrangement of atoms within a protein, the main method so far has been x-ray crystallography. X-ray crystallography  X-ray Diffraction Well-ordered crystals are crucial for x-ray crystallography, as the diffraction pattern created by x-rays interacting with a repeating array of molecules reveals the protein's structure (Figure 1). Growing crystals of globular proteins is difficult due to their irregular shapes, leading to gaps filled with disordered solvent molecules, which complicates crystallization (Figure 2). Crystallization depends on various factors like pH, temperature, protein concentration, and solvent. Finding the right conditions for Figure 1: Repeating array of many identical molecules crystal growth can be slow, sometimes taking months. Crystallization robots and kits automate the process of setting up numerous experiments, accelerating the discovery of suitable crystallization conditions. Highly pure proteins (≥97%) are essential for successful crystallization. Variations in crystal packing, caused by factors like pH, can affect diffraction quality, with tightly packed crystals typically producing better results. The most frequently used procedure for making protein crystals is the hanging-drop method Figure 2: Protein crystals contain large channels and holes filled with solvent molecules, as shown in this diagram of the molecular packing in crystals of the enzyme glycolate oxidase  The first prerequisite for solving the three-dimensional structure of a protein by x-ray crystallography is a well-ordered crystal (Figure 3) that will diffract x-rays strongly. X-rays are produced when electrons drop from a higher to a lower energy state. Monochromatic x- rays are generated by bombarding a metal anode with electrons in high-voltage tubes, while rotating anode x-ray generators are commonly used in protein crystallography (Figure 4). Synchrotron storage rings produce more powerful x-ray beams, emitting radiation across a wide range of wavelengths. Synchrotrons can produce both polychromatic beams for broad wavelength exposure and highly intense monochromatic beams, which allow crystallographers to capture diffraction patterns from very small crystals and obtain high-resolution structural details. In x-ray diffraction experiments, the crystal is rotated in the x-ray beam to capture diffraction spots from all angles. To extend the crystal’s life during exposure to x-rays, it is cryocooled to -150°C, slowing free radical diffusion. Cryoprotectants are used to prevent ice formation, and the crystal is cooled with liquid nitrogen, a technique now widely used in crystallographic labs. The first prerequisite for solving the three-dimensional structure of a protein by x-ray crystallography is a well-ordered crystal that will diffract x-rays strongly. Crystallographic method Directing a beam of x-rays onto a regular, repeating array of many identical molecules X-rays are diffracted from it in a diffraction pattern The crystal (Unit Cell) of Structure of an individual molecule can be retrieved the enzyme RuBisCo from spinach The repeating unit forming the crystal is called the unit cell, and each unit cell may contain one or more molecules. Difficult to get well-ordered crystals are difficult to grow because globular protein molecules are large, spherical, or ellipsoidal objects with irregular surfaces. Diffraction pattern of a similar crystal of the same enzyme Figure 3: Well-ordered protein crystals diffract x-rays and produce diffraction patterns that can Figure 4: (a) A narrow beam of x-rays (red) is taken out from the x-ray source through a be recorded on film. The crystal shown in (a) is of the enzyme RuBisCo from spinach and the collimating device. The crystal diffracts some of the primary beam, but most passes through. photograph in (b) is a recording (Laue photograph) of the diffraction pattern of a similar crystal These diffracted beams leave the crystal in several directions and are recorded on an x-ray of the same enzyme. film or area detector. (b) A diffraction pattern from a crystal of the enzyme RuBisCo using monochromatic radiation  Four major steps in crystallization ▪ Obtain large amounts of pure protein samples ▪ Choose a protein buffer in which the protein is both soluble and stable ▪ Bring protein solution to supersaturation where spontaneous nucleation can take place ▪ Growth of crystal  Solubility As a rule, protein solubility will usually increase as you add salt to your aqueous solution, then begin to decrease when the salt concentration gets high enough to compete with the protein for hydration (interaction with water molecules). HbCO (carboxyhemoglobin) solubility as a function of ionic strength in the presence of several different types of salts Diagram from the website of Alan Clark, Victoria University of Wellington, New Zealand http://www2.vuw.ac.nz/staff/alan_clark/teaching/index.htm  Supersaturation Supersaturation can be achieved by adding more of a substance (to a solution) than can normally be dissolved. This is a thermodynamically unstable state, achieved most often in protein crystallography by vapor diffusion or other slow evaporation techniques. Zone 1 - Metastable zone. The solution may not nucleate for a long time but this zone will sustain growth. It is frequently necessary to add a seed crystal. Zone 2 - Nucleation zone. Protein crystals nucleate and grow. Zone 3 - Precipitation zone. Proteins do not nucleate but precipitate out of solution. Diagram from the website for The University of Reading, Course FS460; Investigating Protein Structure and Function  Nucleation A phenomenon whereby a “nucleus”, such as a dust particle, a tiny seed crystal, or commonly in protein crystallography, a small protein aggregate, starts a crystallization process. Nucleation poses a large energy barrier, which is easier to overcome at a higher level of supersaturation. Common difficulties If supersaturation is too high, too many nuclei form, hence an overabundance of tiny crystals In supersaturated solutions that don’t experience spontaneous nucleation, crystal growth often only occurs in the presence of added nuclei or seeds.  Major factors that affect crystallization Purity of proteins Protein concentration Starting conditions (make-up of the protein solution) Precipitating agent (precipitant) Temperature pH Additives: Detergents, reducing agents, substrates, co-factors, etc.  Purity of proteins Sources of heterogeneity (other than unrelated proteins and nucleic acids as contaminants) Partial proteolysis products Oxidation of cysteines Deamidation of Asn and Gln to Asp and Glu Post-translational modifications Oligomerization Isoforms Misfolded population Structural flexibility  Protein concentration Consistency and reproducibility are the major issues with protein concentration - you should have a reliable assay for determining the concentration. Extinction coefficient for tryptophan Bradford Assay (BSA is used as a standard) E. Coli expression systems are crystallographers’ most commonly used method of obtaining protein. Problems can arise from low expression yields: Cytotoxic - your protein is killing your E. coli Unstable plasmid or mRNA Protein is misfolded (coexpress with GroEL?) Some common eukaryotic codons are rare in E. coli  Starting conditions Precipitating agent (make-up of the protein solution) (precipitant) Salts – Ammonium sulfate The main point is to KNOW what your – Sodium chloride starting conditions are for purposes of – Potassium phosphate reproducibility. Organic reagents – MPD – Isopropanol Polyethylene glycol – PEG 4000 – PEG 6000 – PEG 8000  Temperature Temperature affects protein stability and also the dynamics of how protein solution reaching supersaturated states. Ideally: An individual crystal screen should be kept at constant temperature Each set of conditions should be screened at several temperatures The easiest are 4 C and room temperature, also try 12 or 15 C  pH Surface charges affect “crystal packing”. Crystal packing refers to the spatial arrangement of molecules within the crystal, particularly in reference to their relationships to one another Hydrophobic interactions are less important than electrostatic interactions in crystal packing Additives Sometimes you can increase the stability of your protein, and/or the homogeneity of its conformation by having relevant additives present in the crystal screen: – Detergents – Reducing agents – Substrates – Co-factors – etc. Common Methods for Crystallization Vapor Diffusion Slow Evaporation Dialysis Hanging Drop Vapor Diffusion Most popular method among protein crystallographers Crystal screen buffer is the well solution (0.5 - 1 mL) Drop (on siliconized glass cover slip) is 1/2 protein solution, 1/2 crystal screen buffer (6-10 L). So, the concentration of precipitant in the drop is 1/2 the concentration in the well. Cover slip is inverted over the top of the well and sealed with vacuum grease (airtight). The precipitant concentration in the drop will equilibrate with the precipitant concentration in the well via vapor diffusion. Sitting Drop Vapor Diffusion Same basic principles as in hanging drop method, except the drop containing your sample sits on a bridge within the well. This allows for a larger sample size (20 - 40 L), however protein is frequently precious to the crystallographer, so there isn’t that much demand for a larger sample size. Oil Immersion Micro Batch Modern method rising rapidly in popularity Typical sample size 1-6 L Figure 1- Paraffin oil allows for little to no diffusion of water through the oil. This is a true batch experiment because all the reagents are present at a specific and relatively unchanging concentration. Figure 2- Al’s oil is a 1:1 mixture of silicon oil and paraffin oil which allows for evaporation through slow diffusion through the oil. This is an evaporation Method, and the concentration of the protein and reagents in the drop does increase over time. Microdialysis Dialysis buttons can be purchased for a wide range of sample sizes (~ 5 - 350 L). In the dialysis experiment, the sample is often introduced to high salt concentrations within the button that are allowed to equilibrate with lower salt concentrations in the buffer over time. This is known as a “salting-in” method. It exploits the fact that not only does protein solubility tend to decrease with very high ionic strengths, it also has a minimum at very low ionic strength.  Properties of protein crystals Soft, easy to crush Contain large solvent channels Relatively large organic and inorganic molecules can diffuse inside Anisotropic physical properties Birefrigence due to anisotropic refraction indices Ability to diffract X-ray due to regular spaced lattices A typical diffraction pattern from a protein crystal  The oscillation equipment Rotates the crystal about an axis () perpendicular to the x-ray beam (and normal to the goniometer). The diffraction pattern from a crystal is a 3-D pattern, and the crystal must be rotated in order to observe all the diffraction spots. Capillary tubes Cryo-loops (thin (Glass or Quartz) nylon) This nice diagram also comes from Bernhard Rupp’s Crystallography 101 website: http://www-structure.llnl.gov/Xray/101index.html Crystal Mounting What happens to electron when it is hit by x-rays? Primary beam The electron starts vibrating with the same Secondary frequency as the x-ray beam beams As a result, secondary beams will be scattered in all directions Scattering from a molecule Molecule is composed of many electrons Each electron will scatter secondary radiation uppon exposure to x- rays The scattered secondary beams will interact and cause interference The scattering from a molecule is dependent on number of and distances between electrons In other words, scattering from molecule is dependent on its structure If we would know the amplitudes and phases of scattered molecule, we could calculate the structure of molecule...  Electromagnetic waves E = A cos wt E = A cos (a+wt) w = 2pn a = 2pZ/l E- electromagnetic field strength A- amplitude a - phase w- angular velocity n- frequency l - wavelenght  Braggs law Scattered beams in phase, they add up Scattered beams not in phase, they cancel each other n = 2d sin http://www.eserc.stonybrook.edu/ProjectJava/Bragg/ Bragg’s Law When a primary beam of X-rays strikes a crystal, most rays pass through, but some interact with the crystal's electrons, causing them to oscillate and emit X-rays in various directions, a process referred to as scattering. In a crystal, the electrons are arranged in a regular 3D array, leading to interference of emitted X-rays. Most X-rays cancel each other out, but some align to produce diffracted beams, which can be recorded as a pattern (Figure 5a). Sir Lawrence Bragg demonstrated that crystal diffraction can be considered as the reflection of X-rays by parallel planes within the crystal’s unit cells, similar to reflections from a set of mirrors (Figure 5b and 5c). Diffraction occurs when the difference in distance traveled by X-rays reflected from adjacent planes equals the wavelength of the X-rays. This is dependent on the reflection angle between the X-ray beam and the crystal planes. The relationship between the reflection angle (θ), the distance between the planes (d), and the wavelength (λ) is given by Bragg’s law: 2d·sinθ = λ. The diffraction pattern’s positions on the detector correspond to specific planes in the crystal, and using Bragg’s law, these positions allow for the calculation of the unit cell's size (Figure 6). Figure 5: (a) When X-rays hit a crystal, all atoms scatter them in all directions. Most Figure 5: The reflection angle, q, for a diffracted beam can be calculated from the distance scattered rays cancel out, but in specific directions, they reinforce each other to form a (r) between the diffracted spot on a film and the position where the primary beam hits the diffracted beam. (b) Parallel planes can be arranged through the crystal, where each corner of film. From the geometry shown in the diagram the tangent of the angle 2q = r/A. A is the unit cells aligns with these planes. This is similar to seeing rows of trees when passing by a distance between crystal and film that can be measured on the experimental equipment, while plantation. (c) X-ray diffraction can be viewed as reflection from parallel planes in the r can be measured on the film. Hence q can be calculated. The angle between the primary crystal. The distance traveled by X-rays reflected from different planes must satisfy Bragg's beam and the diffracted beam is 2q as can be seen on the enlarged insert at the bottom. It law: 2d·sinθ = λ. By measuring the reflection angle and using the known wavelength, the shows that this angle is equal to the angle between the primary beam and the reflecting plane distance between planes (d) can be calculated, helping determine the size of the unit cell. plus the reflection angle, both of which are equal to q Thomson Scattering The X-ray scattering is determined by the density of electrons within the crystal. Since the energy of an X-ray is much greater than that of a valence electron, the scattering may be modeled as Thomson scattering, the interaction of an electromagnetic ray with a free electron. The intensity of Thomson scattering for one particle with mass m and charge q is: Phase determination is the major crystallographic problem According to Bragg’s law, only those x-rays that positively interfere with one another give rise to diffracted beams that can be recorded as a distinct diffraction spot above background. Each diffraction spot is the result of interference of all x-rays with the same diffraction angle emerging from all atoms. The mathematical tool that is used to handle such problems is called the Fourier transform, Invented by the French mathematician Jean Baptiste Joseph Fourier. Each diffracted beam, which is recorded as a spot on the film, is Figure 6: Two diffracted beams (purple and orange), each of which is defined by three properties: amplitude, which is a measure of the defined by three properties: the amplitude, which we can strength of the beam and which is proportional to the intensity of the recorded spot; phase, which is related to its interference, positive or measure from the intensity of the spot; the wavelength, which is negative, with other beams; and wavelength, which is set by the x-ray source for monochromatic radiation. set by the x-ray source; and the phase, which is lost in x-ray experiments (Figure 6).  For larger molecules, protein crystallographers have stayed at the laboratory bench using a method called multiple isomorphous replacement (MIR), which requires the introduction of new x-ray scatterers into the unit cell of the crystal. Introduction of Heavy Atoms: In Multiple Isomorphous Replacement (MIR), heavy atoms are introduced into the crystal to enhance the diffraction pattern, as they scatter X-rays more strongly than light atoms. Isomorphous Crystals: The crystal should remain isomorphous (same structure) despite the addition of heavy atoms, to ensure accurate phase determination. This is typically achieved by diffusing heavy-metal complexes into the crystal’s solvent channels or replacing light metals in metalloproteins with heavier ones. Intensity Changes: The introduction of heavy metals changes the diffraction pattern, with some spots increasing in intensity and others decreasing, due to positive and negative interference.  Patterson Maps: Intensity differences caused by heavy- metal substitution are used to create Patterson maps, which show the vector relationships between the heavy atoms in the unit cell (Figure 7). Determining Heavy Atom Positions: From Patterson maps, the positions of the heavy atoms are deduced, which helps calculate their contribution to the diffraction pattern. Figure 7: Patterson Maps Phase Calculation: Using known amplitudes and phases of the heavy metals, and the amplitude of the protein-heavy metal complex, one can determine the phase of the protein's contribution by analyzing the interference effects (Figure 8). Phase Estimation: The extent of positive or negative interference, along with the known phase of the heavy metals, allows estimation of the phase of the protein, crucial Figure 8: Fourier summations of intensity differences create vector maps showing for determining its structure. heavy atom relationships. For example, vectors between atoms A, B, and C are depicted in dark and light colors. The vector map includes a peak at the origin. Deducing atomic arrangements from the map is straightforward for few atoms but challenging with many due to the increased number of vectors.  The Phase Problem With detector you can measure only the intensity of reflections The information about phases is lost – there is no such thing as “phase meter” This means, you must obtain phase information in some other way For small molecules (