\chapter{Introduction} X-ray crystallography is a powerful technique for examining the structure of crystalline solids. It is currently the most successful method of finding protein structures. In this section, a short introduction to protein structure will be given. X-ray methods of structure determination will be examined. The motivation for improving existing methods will be explained. \section{Protein structure} Proteins are large biological molecules, made up of \emph{amino acids}. There are twenty different amino acids. All of them have certain features in common. They have a central carbon atom ($\text{C}_\alpha$). Attached to this are: a hydrogen atom, an amino group ($\text{NH}_2$), a carboxyl group (COOH) and a side chain. This side chain is what distinguishes one amino acid from another. The amino acids are often grouped based on similarity of side chain properties. Table~\ref{tab:amino} shows the twenty amino acids and their groups. Amino acids form bonds with each other called \emph{peptide bonds}. These are formed when the carboxyl group of one amino acis condenses with the amino group of the next acid, eliminating water and forming a bond. Many amino acids can bind together in this way, forming a \emph{chain} or \emph{backbone} from which protude various side chains. The atoms on the main chain are the C$_\alpha$, to which the side chain is attached; an NH group attached to C$_\alpha$; and a carbonyl group C$^{'}$=O, where C$^{'}$ is bound to C$_\alpha$. This unit is known as a \emph{residue} and is the basic building block of a protein. \begin{table}[htbp] \begin{center} \begin{tabular}{llll} \hline Name & Abbr. & Structural Formula & Type \\ \hline Alanine & Ala & CH$_3$-CH(NH$_2$)-COOH & Hydrophobic\\ Arginine & Arg & HN=C(NH$_2$)-NH-(CH$_2$)3-CH(NH$_2$)-COOH & Basic \\ Asparagine & Asn & H$_2$N-CO-CH$_2$-CH(NH$_2$)-COOH & Polar \\ Aspartic acid & Asp & HOOC-CH$_2$-CH(NH$_2$)-COOH & Acidic \\ Cysteine & Cys & HS-CH$_2$-CH(NH$-2$)-COOH & Polar \\ Glutamine & Gln & H$_2$N-CO-(CH$_2$)$_2$-CH(NH$_2$)-COOH & Polar\\ Glutamic acid & Glu & HOOC-(CH$_2$)$_2$-CH(NH$_2$)-COOH & Acidic \\ Glycine & Gly & NH$_2$-H-C-H-COOH & Polar \\ Histidine & His & N=C-NH-C=C-CH$_2$-CH(NH$_2$)-COOH & Basic\\ Isoleucine & Ile & CH$_3$-CH$_2$-CH(CH$_3$)-CH(NH$_2$)-COOH & Hydrophobic \\ Leucine & Leu & CH$_3$-CH(CH$_3$)-CH$_2$-CH(NH$_2$)-COOH & Hydrophobic \\ Lysine & Lys & H$_2$N-(CH$_2$)$_4$-CH(NH$_2$)-COOH & Basic \\ Methionine & Met & CH$_3$-S-(CH$_2$)$_2$-CH(NH$_2$)-COOH & Hydrophobic \\ Phenylalanine & Phe & C$_6$H$_5$-CH$_2$-CH(NH$_2$)-COOH & Hydrophobic \\ Proline & Pro & NH-(CH$_2$)$_3$-CH-COOH & Hydrophobic \\ Serine & Ser & HO-CH$_2$-CH(NH$_2$)-COOH & Polar \\ Threonine & Thr & CH$_3$-CH(OH)-CH(NH$_2$)-COOH & Polar \\ Tryptophan & Trp & C$_6$H$_5$-NH-CH-C-CH$_2$-CH(NH$_2$)-COOH & Hydrophobic \\ Tyrosine & Tyr & HO-C$_6$H$_5$-CH$_2$-CH(NH$_2$)-COOH & Polar \\ Valine & Val & CH$_3$-CH(CH$_3$)-CH(NH$_2$)-COOH & Hydrophobic \\ \hline \end{tabular} \caption{The twenty amino acids} \label{tab:amino} \end{center} \end{table} Finding the sequence of the residues is relaively straightforward. This is also known as the \emph{primary structure}. This provides very limited information about the chemical and biological properties of the protein, as these are both very dependent on the shape of the protein. Thus it is important to know the structure of the protein. X-ray crystallography has been important in this kind of research since the beginning. The first structure solved was that of myoglobin (\cite{Kendrew58},\cite{Kendrew60}). To date, a few thousand structures have been solved and placed in the Protein Data Bank (PDB), a central repository for structure information. From these structures, it has been possible to document the main features of proteins. Proteins, for the most part, form a roughly spherical shape when in solution. It has been found that the surface of proteins consist mainly of hydrophilic (polar, basic or acidic) residues, while the core of a protein is mainly made from hydrophobic residues. The main chain is strongly hydrophilic, so there is a structural problem in bringing the main chain into the core. The hydrophilic nature of the main chain is neutralised by the formation of hydrogen bonds. Each peptide unit has one donor and one acceptor site for a hydrogen bond, which are neutralised by the formation of \emph{secondary structure}. There are two main type of secondary structure: \emph{$\alpha$-helix} and \emph{$\beta$-sheet}. Both types are characterised by having the main chain NH and C$^{'}$=O hydrogen bonded to each other. Linus Pauling first predicted the \helix{} would be a stable structure in proteins in 1951 (\cite{Pauling51b}). This was supported experimentally almost immediately by Max Perutz (\cite{Perutz51}), who was working on haemoglobin crystals and fibres of keratin. A high resolution study of myoglobin by Kendrew (\cite{Kendrew61}) completely confirmed the existens of the \helix. The \helix{} has 3.6 residues per turn, with hydrogen bonds between C$^{'}$=O of residue $n$ and NH of residue $n+4$. This makes the ends of the helix polar, which means that they are almost always at the surface of the molecule. There is a large variation in the lengths of a \helix{}, ranging from four residues to over forty. The average length is about ten residues, or three turns. In proteins, the \helix{} is almost always right handed. They are a distinctive feature in electron density maps, and are therefore very useful in structure determination. The \sheet{} was again proposed by Pauling (\cite{Pauling51a}). This is built up from a combination of several sections of the protein chain. This is in contrast to the \helix{}, which is made from sequential residues. The strands within a \sheet{} are usually from five to ten residues long, and are nearly fully extended. The strands align adjacent to each other, but may be parallel or anti-parallel. Each form has a different form of hydrogen bonding. The sheet is \emph{pleated}, with the \calpha{}s successively above and below the plane of the sheet. The stable secondary structure forms the hydrophobic core of the molecule. It is connected by loop regions of irregular shape, and of various lengths. These loops form the surface of the molecule, and are the most important for determining the behaviour of the molecule. The main chain C$^{'}$=O and NH do not tend to hydrogen bond with each other, leaving them free to bond with the solvent, and form the binding or enzyme active sites. The secondary structure is usually grouped into globular domains, which form the \emph{tertiary structure} of the protein. Many proteins have only one domain, and are known as \emph{monomeric}. Other proteins are formed from a group of identical chains, making a \emph{multimeric} molecule. The arrangement of the chains within the multimer is known as the \emph{quaternary structure}. \subsection{Protein crystals} Proteins need to be crystallised before x-ray diffraction can be used. Protein crystals are different to normal inorganic crystals in a number of ways. Inorganic crystals are held together by strong covalent or ionic bonds. Protein crystals are held together by hydrogen bonds between the surface groups of the molecule. This means that protein crystals are very much more fragile than normal crystals and require very delicate handling. They also do not grow to sizes much greater than a few millimetres. X-ray diffraction requires moderately sized crystals, so this is a problem. The fragility is also a problem, as the crystals tend to be destroyed by the x-ray beam. \subsubsection{Solvent content} Another important point about protein crystals is the amount of solvent present. They are normally prepared for experiment in solution called \emph{mother liquor}, which is the solvent that they were crystallised in. Within the crystal itself, there is a lot of solvent, typically filling about half the volume of the crystal. There are normally highly ordered solvent molecules on the surface of the protein, with disordered solvent filling the gaps between molecules. Both ordered and disordered solvent molecules are important to crystal integrity, and diffraction will not occur without them. \subsubsection{Protein structures in solution and crystal form} If the structure can be solved, is there any evidence that the crystalline protein structure is the same as the structure in solution, which is where proteins undergo most of their reactions? This is worrying to most crystallographers. It has been found that most proteins have the same structure in crystal and solution. However, this is not always true so tests need to be done to ensure the similarity. The most compelling evidence for the similarity of protein structures in crystal and solution is the fact that many proteins retain their function when they are in crystal form. This can only be the case when the structures are identical, as the function of a protein is based mainly on its shape. \section{X-ray diffraction} Von Laue was the first person to propose the scattering of x-rays by crystals in 1912. He believed that a crystal lattice would act like a diffraction grating for x-rays. This is because the repeat distance of the lattice is of the order of the wavelength of an x-ray. X-ray diffraction was first observed by Friedrich and Knipping. This prompted a great deal of interest, culminating in the publication of Bragg's law (1913): \begin{equation} \label{eq:bragg} 2d\sin\theta=n\lambda \end{equation} where $d$ is the separation between the planes of the crystal lattice, $\theta$ is the angle of diffraction, $\lambda$ is the wavelengh of the x-ray and $n$ is an integer. This holds as long at there is one atom per unit cell. More atoms produces the effect of having many reflecting planes, with its own reflecting power. These will interfere with each other, creating a unique pattern. It is this unique pattern that makes x-ray diffraction a powerful tool in structure determination. The pattern consiats of an array of discrete spots of differing magnitude, the spacing of which is related to the reciprocal lattice of the crystal. It is three dimensional, and the spots are indexed by three integers: $h$, $k$ and $l$. These are known as the \emph{Miller indices}. The intensity of the spot forms part of the associated \emph{structure factor}. The other part is the \emph{phase} of the structure factor. It is this that determines how the spot has been affected by interference. This quantity is not directly measurable, but is important in structure determination. A method of determining phases is necessary for a solution, and a few methods have been developed. This will be discussed, but first some mathematical background is necessary. \subsection{Mathematical background} \subsubsection{Reciprocal lattice} A unit cell may be defined by the vectors $\mbf{a}$, $\mbf{b}$ and $\mbf{c}$. A new vector $\mbf{a}^{*}$ may be defined by \begin{equation} \label{eq:reciprocal} \mbf{a}^{*}\mbf{a}=1; \mbf{a}^{*}.\mbf{b}=0; \mbf{a}^{*}.\mbf{c}=0 \end{equation} Similar relationships exist for the vectors $\mbf{b}^{*}$ and $\mbf{c}^{*}$. These new vectors define the reciprocal lattice of the crystal. The diffraction spots occur at the grid points. The Miller indices of a structure factor therefore define its location in \emph{reciprocal space}. Thus a \emph{scattering vector}, $\mbf{s}$, may be defined \begin{equation} \label{eq:scattering} \mbf{s}=h\mbf{a}^{*}+k\mbf{b}^{*}+l\mbf{b}^{*} \end{equation} The magnitude of $\mbf{s}$ gives the diffraction angle of the particular structure factor according to \begin{equation} \label{eq:scattering2} \left|\mbf{s}\right|=\frac{2\sin\theta}{\lambda} \end{equation} The magnitude of a structure factor may be calculated from the Laue equations: \begin{equation} \label{eq:laue} F(\mbf{h})=\sum_{j=1}^{N}f_j\exp(2\pi i \mbf{x}_j.\mbf{s}) \end{equation} where $f_j$ and $\mbf{x}_j$ are the atomic scattering factor and the position vector of the $j$th atom respectively. \subsubsection{Normalised structure factors} Tthe scattering of x-rays by the spherical atomic potential introduces a falloff in intensity proportional to $\sin\theta$. It is convenient to correct for this so that the most important structure factors can be identified. One way of doing this is to calculated the structure factors as if scattered by point atoms. They may be calculated from the $F$s as follows: \begin{equation} \label{eq:efromf} \left|E(\mbf{h})\right|=\frac{\left|F(\mbf{h})\right|}{\epsilon\sum_{j=1}^{N} f_j^{1/2}} \end{equation} where $\epsilon$ is a parameter dependent on the space group of the crystal. It is necessary to include this because some areas of the diffraction pattern are more intense than others due to symmetry. These structure factors are known as normalised structure factors. Their intensities have some interesting properties, namely that $\left<\left|E(\mbf{h}\right|^2\right>=1$. This enables them to be efficiently calculated from the $F$s, using either linear regression to fit the intensities as a function of resolution or to scale the $\left|F\right|^2$ in resolution ``shells''. Normalised structure factors are heavily used in direct methods calculations. \subsubsection{Scattering from atoms} The positions of the atoms can be expressed as \emph{fractional coordinates} within the unit cell: \begin{equation} \label{eq:fraction} \mbf{x}=x\mbf{a}+y\mbf{b}+z\mbf{c} \end{equation} The structure factor equation (\ref{eq:laue}) may then be written as \begin{equation} \label{eq:fraclaue} F(\mbf{h})=\sum_{j=1}^N f_j \exp\left[2\pi i \bh.\bx \right] \end{equation} This is the most useful form of the structure factor equation. \subsubsection{Crystallographic symmetry} Protein crystals form in one of 230 different \emph{space groups}. The space groups determine how the protein molecules are arranged within the unit cell. The simplest space group is $P1$, where there is no symmetry in the cell, therefore only one molecule per cell. All other space groups have two or more symmetry related molecules. This has consequences for the structure factors of the crystal, as they will also be related by symmetry. All crystals have a centre of inversion at the origin. This means that $F(\bar{\bh})$ is related to $F(\bh)$. In fact, they have the same magnitude and opposite phases. This relation is known as \emph{Friedel's Law}, and the two related structure factors are known as a \emph{Friedel pair}. A general space group operator will consist of a rotation, $\mbf{C}_p$ and a translation, $\mbf{d}_p$. The Miller indices transform under symmetry as: \begin{equation} \bh^{'}=\bh^{T}\mbf{C}_p \end{equation} There is a phase shift dependent on the translation: \begin{equation} \label{eq:symm1} \Delta\phi=2\pi\bh^{T}\mbf{d} \end{equation} \subsubsection{Structure factors and Fourier transforms} The contents of a unit cell may be regarded as a smoothly varying distribution of electron density, rather than a collection of discrete atoms. The structure factor equation (\ref{eq:fraclaue}) then becomes an integral: \begin{equation} \label{eq:fourier} F(\mbf{h})=\int_v\rho(\mbf{x}) \exp(2\pi i \bh.\bx) \text{d}\mbf{x} \end{equation} This is a Fourier transform of the electron density. The electron density, may then be obtained by inverse Fourier transforming the structure factors: \begin{equation} \label{eq:invfourier} \rho(\mbf{x})=\sum_\mbf{h}F(\mbf{h})\exp(-2\pi i \bh.\bx) \end{equation} The intensities of the $F$s are observed experimentally, but again phase information is needed to properly calculate the electron density. There are two main approaches to this. The first is to try and extract some phase information experimentally. The second method is to recreate the phase information mathematically given just the structure factor magnitudes. These methods are known as \emph{direct methods}. Experimental methods are very successful at phasing protein structures. Typically, multiple diffraction experiments are performed with the same protein or derivatives, and the data combined to provide estimates of the phases. However, protein x-ray diffraction experiments are very difficult, and it would be good if only one diffraction per protein were needed. This is the strength of direct methods. However, they are difficult to apply to proteins because of the large size of the molecules. It is possible to use experimental phase information within direct methods to provide a starting point for \emph{phase refinement}, which is a process during which existing phase estimates are improved. Both methods produce \emph{estimates} for the phases. Some way of gauging how reliable a phase estimate is needed. This is known as a \emph{figure of merit} (FOM). They are used to weight the magnitude of a structure factor when calculating an electron density map, thus they take a value between 0 and 1. This removes unreliable phase estimates from the electron density map, reducing the level of noise. This means that good way of calculating a FOM is almost more important than producing a good estimate for a phase. The FOM can generate a probability function for the phase. \subsection{Phase probability functions} \subsubsection{The von Mises distribution} The von Mises distribution is an angular probability distribution based on the Normal distribution: \begin{equation} \label{eq:vonmises} P(\phi)=\frac{\exp(\kappa\cos(\phi-\left<\phi\right>))}{2\pi I_0(\kappa)} \end{equation} where $P(\phi)$ is the probability of the phase having a particular value $\phi$, and $I_0$ is a modified Bessel function. It is centred on the phase estimate $\left<\phi\right>$, with a width governed by $\kappa$. \begin{figure}[htbp] \begin{center} \epsfig{file=figs/vonmises01.eps,width=0.5\linewidth} \caption{The von Mises distribution} \label{fig:vonmises} \end{center} \end{figure} The distribution is symmetrical about $\left<\phi\right>$. It can be seen that the distrbution is sharper for higher values of $\kappa$. A FOM can be obtained from $\kappa$: \begin{equation} \label{eq:fomkappa} FOM=\frac{I_1(\kappa)}{I_0(\kappa)} \end{equation} \subsubsection{Bimodal probability distribution} Some experimental phasing methods produce a bimodal phase distribution. This can be represented by the Hendrickson and LAttman distribution: \begin{equation} \label{eq:hendlatt} P(\phi)=N\exp(A\cos(\phi)+B\sin(\phi)+C\cos(2\phi)+D\cos(2\phi)) \end{equation} where $N$ is a normalising constant, $\phi$ is the phase assuming that the distribution is centred about zero, and $A$,$B$,$C$ and $D$ are constants known as the \emph{Hendrickson-Lattman coefficients}. They describe the bimodal distribution in a similar way to $\kappa$ in the von Mises distribution. \section{Experimental phase determination} There are several successful methods of determining phase information experimentally. They mostly involve diffracting x-rays from derivatives of the protein under study. \subsection{Isomorphous replacement} In isomorphous replacement, a change is made to the crystal that will affect the structure factors. By the effect of these changes, some conclusions may be drawn about the values of the phases. Typically, a heavy atom is introduced into the structure. This changes the intensities of the structure factors significantly. This is because a heavy atom is a much more significant scatterer than a light atom. The difference in the structure factors between the native protein and the heavy atom derivative will be due to the scattering contribution of the heavy atom. These differences may be used to calcualate a Patterson map, which will give the positions of the heavy atoms. This allows some predictions to be made about the protein phase angles. It is assumed that the shape of the protein is unaffected by the addition of the heavy atom. This means that the structure factor of the derivative is equal to the sum of the protein structure factor and the heavy atom structure factor: \begin{equation} \label{eq:heavy1} F_{\text{PH}}(\mbf{h})=F_{\text{P}}(\mbf{h})+F_{\text{H}}(\mbf{h}) \end{equation} As the structure factors are complex, this describes a triangle in the complex plane. There are two ways to construct such a triangle, which corresponds to two values of the protein phase. This produces a bimodal distribution from (\ref{eq:hendlatt}). Further derivatives can be prepared to resolve this ambiguity, as only one phase solution will be common to both derivatives. This is known as \emph{multiple isomorphous replacement} (MIR). It is quite difficult, and often impossible to prepare the necessary derivatives, which is the main weakness of this technique. \subsection{Anomalous dispersion} Most electrons oscillate with the same phase, as the driving force of the x-ray is very different to the natural frequency of the oscillation. However, for some electrons, this is not true, and there is a shift in the amplitude and phase of the oscillation. This is true for inner shell electrons where the x-ray energy is close to the transition energy. This shift in amplituse and phase is known as anomalous scattering. This leads to a breakdown of Friedel's law, in that the amplitudes of the Friedel mates will be different. The anomalous effect depends on the wavelength being similar to the natural frequency of the atom. So diffraction experiments with multiple x-ray wavelength near the absorption energy acan be carried out. From this, phase estimates can be obtained in a similar way to MIR. This technique is known as \emph{multiple anomalous dispersion} (MAD). \section{Direct methods} Ideally, direct methods require only one set of experimental magnitudes, and can recreate the phase information using the statistical properties of the electron density. A method capable of this is known as an \emph{ab initio} direct method. These are very powerful methods when used with small molecules (up to around 200 atoms in the asymmetric unit). They are of less use with large molecules like proteins (typically with thousands of atoms in the unit cell), as the strength of the information decreases proportionally with $\sqrt{N}$. With proteins, it is more typical to use direct methods to refine the phase estimates obtained from MAD or MIR experiments. The original direct method is the \emph{Patterson function} (\cite{Patterson35}). This is a way of obtaining the interatomic vectors from the structure factor amplitudes and Miller indices alone: \begin{equation} \label{eq:patterson} P(u,v,w)=\frac{1}{V}\sum_\mbf{h}\left|F(\mbf{h})\right|^2 \cos 2\pi(hu+kv+lw) \end{equation} The peaks in the Patterson function correspond to an interatomic vector in the original map, thus there are $N(N-1)$ of them. The height of the peaks is approximately proportional to $Z_iZ_j$, where $Z$ is the atomic number of the two relevent atoms. If there are a few atoms, it is relatively straightforward to find the atomic positions given the Patterson. However, for large molecules, there are a lot of peaks present, and interpretation becomes impossible, unless there are a few heavy atoms present. These will give much higher peaks than the rest of the structure. This is the technique used to find atomic positions in MIR phasing. Initial work on direct methods was done on \emph{centrosymmetric} structures. A trial and error method was used to calculate structures, by systematically searching every possible phase value. This is very time consuming, as there are $2^n$ different maps, where $n$ is the number of structure factors involved in the calculation. Even for as few as twenty reflections, there are well over one million different maps. Most interesting structures have thousands of structure factors, so trial and error methods were quickly abandoned. There are two main assumptions common to all direct methods. The first is that the obsrved intensities of the structure factors contain structural information. Secondly, the electron density is assumed to be non-negative at all points within the structure. This second assumption lead to the development of relationships between the phases, starting with \cite{Harker48}, who found a relationship between the signs of three structure factors: \begin{equation} \label{eq:harker1} s(\mbf{h}\bar)s(\mbf{k})s(\mbf{h}-\mbf{k})=+1 \end{equation} where $\mbf{h}$ and $\mbf{k}$ are different points in reciprocal space. This set of three structure factors is known as a \emph{triplet}, and is very important in direct methods. This relationship is probably true, as shown by \cite{Sayre52}, using the Sayre equation: \begin{equation} \label{eq:sayre1} F(\mbf{h})=\frac{\theta_\mbf{h}}{v}\sum_\mbf{k}F(\mbf{k})F(\mbf{h}-\mbf{k}) \end{equation} where $\theta_\mbf{h}$ is a scale factor. This applies to any equal-atom structure, whether centrosymmetric or not. Most proteins are approximately equal atom structures, so (\ref{eq:sayre1}) is generally applicable. From this, Sayre derived a similar relationship to (\ref{eq:harker1}), only he found that it was only approximately true. It is more accurate for larger structure factors. \cite{Cochran52} showed that this equation corresponded to maximising the value of \begin{equation} \label{eq:cochran} \int_v\rho^3\text{d}v \end{equation} This is known as \emph{Cochran's condition}. It means that the correct density will be arranged in atomic peaks, rather than a flat map. \cite{Karle50} derived a set of inequalities, based on determinants, to solve the non-centrosymmetric phase problem. The first three of these can be written \begin{eqnarray} \label{eq:kh1} F(\mbf{0})&\geqslant&0 \\ \label{eq:kh2} \left|F(\mbf{h})\right|&\leqslant&F(\mbf{0}) \\ \label{eq:kh3} F(\mbf{h})-\delta(\mbf{h},\mbf{k})&\leqslant&\mbf{r} \end{eqnarray} where \begin{equation} \label{eq:delta} \delta(\mbf{h},\mbf{k})=F(\mbf{h}-\mbf{k})F(\mbf{k})/F(\mbf{0}) \end{equation} and \begin{equation} \label{eq:r} \left| \mbf{r} \right|=\frac{ \left|\begin{array}{cc}F(\mbf{0})&F^*(\bh-\bk)\\F(\bh-\bk)&F(\mbf{0})\end{array} \right| \left|\begin{array}{cc}F(\mbf{0})&F^*(\bk)\\F(\bk)&F(\mbf{0})\end{array}\right|} {F(\mbf{0})} \end{equation} The foundation for the \emph{tangent formula} has been laid. Its derivation and modification is the focus of the next chapter.