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Acta Crystallogr A. Sep 1, 2011; 67(Pt 5): 435–446.
Published online Jul 6, 2011. doi:  10.1107/S0108767311021003
PMCID: PMC3171898
Mathematical aspects of molecular replacement. I. Algebraic properties of motion spaces
Gregory S. Chirikjiana*
aDepartment of Mechanical Engineering, Johns Hopkins University, 223 Latrobe Hall, 3400 N. Charles Street, Baltimore, Maryland, MD 21218, USA
Correspondence e-mail: gregc/at/jhu.edu
Received October 26, 2010; Accepted June 9, 2011.
Molecular replacement (MR) is a well established method for phasing of X-ray diffraction patterns for crystals composed of biological macromolecules of known chemical structure but unknown conformation. In MR, the starting point is known structural domains that are presumed to be similar in shape to those in the macromolecular structure which is to be determined. A search is then performed over positions and orientations of the known domains within a model of the crystallographic asymmetric unit so as to best match a computed diffraction pattern with experimental data. Unlike continuous rigid-body motions in Euclidean space and the discrete crystallographic space groups, the set of motions over which molecular replacement searches are performed does not form a group under the operation of composition, which is shown here to lack the associative property. However, the set of rigid-body motions in the asymmetric unit forms another mathematical structure called a quasigroup, which can be identified with right-coset spaces of the full group of rigid-body motions with respect to the chiral space group of the macromolecular crystal. The algebraic properties of this space of motions are articulated here.
Keywords: rigid-body motion, coset space, quasigroup, fundamental domain, molecular replacement
Over the past half century, X-ray crystallography has been a wildly successful tool for obtaining structures of biological macromolecules. Aside from finding conditions under which crystals will grow (which largely has been reduced to automated robotic searches) the major hurdle in determining a three-dimensional structure when using X-ray crystallography is that of phasing the diffraction pattern. And while experimental methods such as multiple isomorphous replacement (MIR) and multiple-wavelength anomalous dispersion (MAD) phasing are often used, if the macromolecular system under study is known a priori to consist of components that are similar in structure to solved structures, then the phasing problem can be reduced to a purely computational one, known as a molecular replacement (MR) search. In this article, six-dimensional MR searches for single-domain structures are formulated using the language and tools of modern mathematics. A coherent mathematical description of the MR search space is presented. It is also shown that more generally the An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi1.jpg-dimensional search space that results for a multi-domain macromolecule or complex constructed from An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi2.jpg rigid parts is endowed with a binary operation. This operation is shown not to be associative, and therefore the resulting space is not a group. However, as will be proven here, the result is a mathematical object called a quasigroup.
This concept can be understood graphically at this stage without any notation or formulas. Consider a planar rigid-body transformation applied to the particular gray letter ‘Q’ in the upper-right cell in Fig. 1 [triangle]. The transformation moves that ‘Q’ from its original (gray) state to a new (black) state. The change in position resulting from the translational part of the transformation can be described by a vector originating at the center of the gray ‘Q’ and terminating at the center of the black one. In this example the translation vector points up and to the right. The transformation also results in an orientational change, which in this case is a counterclockwise rotation by about 25°. If the other gray ‘Q’s are also moved from their initial state in an analogous way so that the relative motion between each corresponding pair of gray and black ‘Q’s is the same, the result will be that shown in Fig. 1 [triangle], which represents four cells of an infinite crystal. This is the same as what would result by starting with the cell in the upper right together with both of its ‘Q’s, and treating these three objects as a single rigid unit that is then translated without rotation and copied so as to form a crystal. The resulting set of black ‘Q’s is not the same as would have resulted from the single rigid-body motion of all of the gray ‘Q’s as one infinite rigid unit.
Figure 1
Figure 1
Rigid-body motion of an object in a crystal with An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi437.jpg space group.
In the scenario in Fig. 1 [triangle] there is exactly one ‘Q’ in each unit cell before the motion and exactly one in each cell after the motion, where ‘being in the unit cell’ is taken here to mean that the center point of a ‘Q’ is inside the unit cell. It just so happens in the present example that the same ‘Q’ is inside the same cell before and after this particular motion. But this will not always be the case. Indeed, if each new ‘Q’ is moved from its current position and orientation by exactly the same relative motion as before (i.e. if the relative motion in Fig. 1 [triangle] is applied twice), the result will be the black ‘Q’ in Fig. 2 [triangle]. In this figure the lightest gray color denotes the original position and orientation, the middle-gray ‘Q’ that is sitting to the upper right of each light one is the same as the black one in Fig. 1 [triangle], and now the new black one has moved up and to the right of this middle-gray one. This is the result of two concatenated transformations applied to each ‘Q’. Note that now each black ‘Q’ has moved from its original unit cell into an adjacent one. But if we focus on an individual unit cell, we can forget about the version that has left the cell, and replace it with the one that has entered from another cell. In so doing, the set of continuous rigid-body motions within a crystal becomes a finite-volume object, unlike continuous motions in Euclidean space. This finite-volume object is what is referred to here as a motion space, which is different from the motion group consisting of all isometries of the Euclidean plane that preserve handedness.
Figure 2
Figure 2
Concatenation of the motion in Fig. 1 [triangle] with itself.
Each element of a motion space can be inverted. But this inverse is not simply the inverse of the motion in Fig. 1 [triangle]. Applying the inverse of each of the rigid-body transformations for each ‘Q’ that resulted in Fig. 1 [triangle] is equivalent to moving each light-gray ‘Q’ in Fig. 3 [triangle] to the position and orientation of the new black ones to the lower left. This does not keep the center of the resulting ‘Q’ in the same unit cell, even though the original motion did. But again, we can forget about the version of the ‘Q’ that has left the unit cell under this motion, and replace it with the one that enters from an adjacent cell.
Figure 3
Figure 3
The inverse of the motion in Fig. 1 [triangle].
If we were doing this all without rotating, the result simply would be the torus, which is a quotient of the group of Euclidean translations by primitive lattice translations. But because orientations are also involved, the result is more complicated. The space of motions within each unit cell is still a coset space (in this case, of the group of rigid-body motions by a chiral crystallographic space group, due to the lack of symmetry of ‘Q’ under reflections), and such motions can be composed. But unlike a group, this set of motions is non-associative as will be shown later in the paper in numerical examples. This non-associativity makes these spaces of motions a mathematical object called a quasigroup.
The concept of quasigroups has existed in the mathematics literature for more than half a century (see e.g. Bruck, 1958 [triangle]), and remains a topic of interest today (Pflugfelder, 1990 [triangle]; Sabinin, 1999 [triangle]; Smith, 2006 [triangle]; Vasantha Kandasamy, 2002 [triangle]; Nagy & Strambach, 2002 [triangle]). Whereas the advanced mathematical concept of a groupoid has been connected to problems in crystallography (Weinstein, 1996 [triangle]), to the the author’s knowledge connections between quasigroups and crystallography have not been made before. Herein a case is made that a special kind of quasigroup (i.e. a motion space) is the natural algebraic structure to describe rigid-body motions within the crystallographic asymmetric unit. Therefore, quasigroups and functions whose arguments are elements of a quasigroup are the proper mathematical objects for articulating molecular replacement problems. Indeed, the quasigroups shown here to be relevant in crystallography have properties above and beyond those in the standard theory. In particular, the quasigroups presented here have an identity and possess a continuum of elements similar to a Lie group.1
1.1. Literature review
The crystallographic space groups have been cataloged in great detail in the crystallography literature. For example, summaries can be found in Bradley & Cracknell (2009 [triangle]), Burns & Glazer (1990 [triangle]), Hahn (2002 [triangle]), Hammond (1997 [triangle]), Julian (2008 [triangle]), Janssen (1973 [triangle]), Ladd (1989 [triangle]), Lockwood & MacMillan (1978 [triangle]), Evarestov & Smirnov (1993 [triangle]) and Aroyo et al., (2010 [triangle]), as well as in various online resources. Treatments of space-group symmetry from the perspective of pure mathematicians can be found in Conway et al. (2001 [triangle]), Engel (1986 [triangle]), Hilton (1963 [triangle]), Iversen (1990 [triangle]), Miller (1972 [triangle]), Nespolo (2008 [triangle]) and Senechal (1980 [triangle]).
Of the 230 possible space groups, only 65 are possible for biological macromolecular crystals (i.e. the chiral/proper ones). The reason for this is that biological macromolecules such as proteins and nucleic acids are composed of constituent parts that have handedness and directionality (e.g. amino acids and nucleic acids, respectively, have C–N and 5′–3′ directionality). This is discussed in greater detail in McPherson (2003 [triangle]), Rhodes (2000 [triangle]), Lattman & Loll (2008 [triangle]) and Rupp (2010 [triangle]). Of these 65, some occur much more frequently than others and these are typically non-symmorphic space groups. For example, more than a quarter of all proteins crystallized to date have An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi3.jpg symmetry, and the three most commonly occurring symmetry groups represent approximately half of all macromolecular crystals (Rupp, 2010 [triangle]; Wukovitz & Yeates, 1995 [triangle]).
The number of proteins in a unit cell, the space group An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg and aspect ratios of the unit cell can be taken as known inputs in MR computations, since they are all provided by experimental observation. From homology modeling, it is often possible to have reliable estimates of the shape of each domain in a multi-domain protein. What remains unknown are the relative positions and orientations of the domains within each protein and the overall position and orientation of the protein molecules within the unit cell.
Once these are known, a model of the unit cell can be constructed and used as an initial phasing model that can be combined with the X-ray diffraction data. This is, in essence, the molecular replacement approach that is now more than half a century old (Rossmann & Blow, 1962 [triangle]; Hirshfeld, 1968 [triangle]; Lattman & Love, 1970 [triangle]; Rossmann, 2001 [triangle]). Many powerful software packages for MR include those described in Navaza (1994 [triangle]), Collaborative Computational Project, Number 4 (1994 [triangle]), Vagin & Teplyakov (2010 [triangle]) and Caliandro et al. (2009 [triangle]). Typically these perform rotation searches first, followed by translation searches.
Recently, full six-degrees-of-freedom rigid-body searches and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi1.jpg degree-of-freedom (DOF) multi-rigid-body searches have been investigated (Jogl et al., 2001 [triangle]; Sheriff et al., 1999 [triangle]; Jamrog et al., 2003 [triangle]; Jeong et al., 2006 [triangle]) where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi2.jpg is the number of domains in each molecule or complex. These methods have the appeal that the false peaks that result when searching the rotation and translation functions separately can be reduced. This paper analyzes the mathematical structure of these search spaces and examines what happens when rigid-body motions in crystallographic environments are concatenated. It is shown that unlike the symmetry operations of the crystal lattice, or rigid-body motions in Euclidean space, the set of motions of a domain (or collection of domains) within a crystallographic unit cell (or asymmetric unit) with faces ‘glued’ in an appropriate way does not form a group. Rather, it has a quasigroup structure lacking the associative property.
1.2. Overview
The remainder of this paper (which is the first in a planned series) makes the connection between molecular replacement and the algebraic properties of quasigroups. §2 provides a brief review of notation and properties of continuous rigid-body motions and crystallographic symmetry. §3 articulates MR problems in modern mathematical terminology. §4 explains why quasigroups are the appropriate algebraic structures to use for macromolecular MR problems, and derives some new properties of the concrete quasigroup structures that arise in MR applications. Examples illustrate the lack of associativity. §5 focuses on how the quasigroups of motions defined earlier act on asymmetric units. §6 illustrates the non-uniqueness of fundamental domains and constructs mappings between different choices, some of which can be called quasigroup isomorphisms. §7 develops the special algebraic relations associated with projections from quasigroups to the asymmetric units on which they act. §8 returns to MR applications and illustrates several ways in which the algebraic constructions developed in the paper can be used to describe allowable motions of macromolecular domains while remaining consistent with constraints imposed by the crystal structure. Future papers in this series will address the geometric and topological properties of these motion spaces, and connections with harmonic analysis.
This section establishes common notation and reviews the properties of continuous and discrete motions.
2.1. Rigid-body motions and semi-direct products
The special Euclidean group, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi7.jpg, consists of all rotation–translation pairs An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi8.jpg where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi9.jpg is an An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi10.jpg rotation matrix, the set of which forms the special orthogonal group An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi11.jpg, and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi12.jpg is a translation vector. The group operation for this group is defined for every An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi13.jpg as
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd1.jpg
From this it is easy to calculate that for any An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi14.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi15.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi16.jpg where
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd2.jpg
Here An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi17.jpg is the An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi10.jpg identity matrix and 0 is the null translation vector.
The group law for An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi7.jpg in equation (1) is that of a semi-direct product, so that
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd3.jpg
An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi20.jpg is a Lie group, i.e. it consists of a continuum of elements and satisfies other formal properties described in Chirikjian & Kyatkin (2000 [triangle]). Two Lie subgroups of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg are
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd4.jpg
These are the continuous groups of pure translations and pure rotations. The group of pure translations is isomorphic with An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi22.jpg with the operation of addition, i.e. An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi23.jpg, and the group of pure rotations is isomorphic with An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi11.jpg, i.e. An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi25.jpg, where the operation for An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi11.jpg is matrix multiplication. These subgroups are special because any element An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi14.jpg can be written as a product of pure translations and rotations as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi28.jpg.
Let An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg denote the chiral group of discrete symmetries of a macromolecular crystal. An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg, though discrete, always has an infinite number of elements and can be viewed as a proper subgroup of the group of rigid-body motions, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi20.jpg, which is written as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi32.jpg, with An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi33.jpg denoting proper subgroup.
2.2. Actions, subgroups and coset spaces
The group An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi20.jpg acts on the set An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi35.jpg as
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd5.jpg
for all position vectors An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi36.jpg. Any such position can be expressed as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi37.jpg where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi38.jpg is the natural basis for An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi22.jpg consisting of orthogonal unit vectors. Alternatively, in crystallographic applications it can be more convenient to write An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi40.jpg where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi41.jpg are the directions from one lattice point to the corresponding one in an adjacent primitive unit cell. Sweeping through values An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi42.jpg defines a primitive crystallographic unit cell. Whereas An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi43.jpg denotes any of a continuum of positions, the set of all discrete translations of the form An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi44.jpg for all An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi45.jpg forms the Bravais lattice, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi46.jpg, and for any two fixed An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi47.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi48.jpg is also in the lattice. The lattice together with addition is the group of primitive lattice translations, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi49.jpg, which is infinite but discrete. An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg is the whole group of crystallographic symmetry operations that includes both lattice translations and a chiral point group as subgroups. The space group of a Bravais lattice is a semi-direct product and can be thought of as a discrete version of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi7.jpg. However, a crystal consists of both a Bravais lattice and a motif repeated inside the unit cells. This changes the symmetry, by possibly removing some rotational symmetry operations and possibly introducing some discrete screw displacements.
In general, given any proper subgroup An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi52.jpg contained in a group An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg (which is denoted as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi54.jpg), including (but not limited to) the case when An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi52.jpg is An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi56.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi57.jpg or An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg, and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg is An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi7.jpg, left and right cosets are defined, respectively, as
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd6.jpg
It is well known that a group is divided into disjoint left (or right) cosets, and that only for a normal subgroup, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi2.jpg, is it the case that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi62.jpg for all An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi63.jpg. More generally, the left- and right-coset (or quotient) spaces that contain all left or right cosets are denoted, respectively, as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi64.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi65.jpg. Normal subgroups are special because An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi66.jpg and a natural group operation, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi67.jpg, can be defined so that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi68.jpg is also a group. For example, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi56.jpg in equation (4) is a normal subgroup of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg, meaning that for all An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi71.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi72.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi73.jpg. This condition is written as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi74.jpg, and in fact it can be shown that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi75.jpg.
2.3. Unit cells as fundamental domains of orbits
A space, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi76.jpg, on which a group, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg, acts can be divided into disjoint orbits. The set of all of these orbits is denoted as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi78.jpg, as this is a kind of quotient space.2 An immediate crystallographic consequence of these definitions is that if An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg is the full chiral symmetry group of a crystal and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi35.jpg, then An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi88.jpg can be identified with the asymmetric unit. Moreover, if An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi89.jpg is the largest discrete translation group of the crystal (and so An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi90.jpg also), then An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi91.jpg can be identified with the primitive unit cell, and so too can the coset space An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi92.jpg. Since An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi93.jpg is a normal subgroup of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi56.jpg, the unit cell is actually endowed with a group structure, namely periodic addition. For this reason, a unit cell, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi95.jpg, in An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi96.jpg-dimensional space with its opposing faces glued is equivalent to an An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi96.jpg-dimensional torus,
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd7.jpg
This can be identified with the box An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi98.jpg with the operation of addition An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi99.jpg for all An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi100.jpg. This fact is implicitly and extensively used in crystallography to expand the density in a unit cell in terms of Fourier series. Furthermore, the translational motion of the contents of a unit cell is easy to handle within the framework of classical mathematics. However, if one wishes to focus attention in MR searches on the asymmetric unit An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi88.jpg, then there is no associated group operation. An advantage of using An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi88.jpg is that it is smaller (in terms of volume) than An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi103.jpg, and therefore when discretizing this space for numerical computations the number of grid points required for a given resolution will be smaller. Furthermore, even in the case when the whole unit cell is considered, though periodic translations are handled in an effortless way within the context of classical Fourier analysis, rotations of the rigid contents within a unit cell of a crystal are somewhat problematic within the classical framework, which provides the motivation for the current work.
The set of orbits An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi104.jpg can be viewed as a region in An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi76.jpg, denoted as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi106.jpg (or An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi107.jpg for short when the connection between An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi107.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi104.jpg is clear from the context). Here An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi107.jpg stands for ‘fundamental domain’. A point in An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi106.jpg is denoted as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi112.jpg, and serves as a representative for each orbit generated by the application of all elements of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg to a particular An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi36.jpg. Each point An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi36.jpg can be thought of as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi116.jpg for a unique An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi117.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi118.jpg, where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi119.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi106.jpg can be chosen as the unit cell and the asymmetric unit, respectively.
Typically MR searches are performed by reducing the problem of first finding the orientation/rotation of a homologous component, followed by a translational/positional search. This method works extremely well for single-domain proteins because the signal-to-noise ratio (SNR) is very high. However, in crystals composed of complex multi-body proteins or complexes, the SNR can be quite low.3
3.1. Crystallographic symmetry in molecular replacement
Suppose that a single copy of a macromolecular structure of interest has an electron density An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi123.jpg. That is, there exists a function An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi124.jpg. This says nothing more than that the density is non-negative. This function may be constructed by adding densities of individual domains within the structure. And if thermal motions are taken into account, each of these component densities can be motionally blurred as described in Chirikjian (2010 [triangle]).
This means that the total electron density of the non-solvent part of the crystal will be4
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd8.jpg
The symmetry group, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg, and number of copies of the molecule in a given unit cell can both be estimated directly from the experimental data (Matthews, 1968 [triangle]). Note that such a function An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi127.jpg is ‘An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg-periodic’ in the sense that for any An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi129.jpg,
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd9.jpg
Now suppose that before constructing symmetry-related copies of the density An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi123.jpg, we first move it by an arbitrary An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi63.jpg. The result will be
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd10.jpg
There should be no confusion between the single-argument and two-argument versions of the density function; they are actually different functions which are easily distinguished by their arguments. They share the same name ‘An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi132.jpg’ to avoid a proliferation of notation.
It is easy to see that for any fixed An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi63.jpg
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd11.jpg
The An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi80.jpg in each of these expressions can be taken to be in An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg, but this is wasteful because An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg extends to infinity, and the same result appears whether An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi80.jpg or An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi138.jpg is used for any An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi129.jpg. Therefore, the rigid-body motions of interest are those that can be taken one from each coset An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi140.jpg. In contrast to an element of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg, which is denoted as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi80.jpg, an element of the fundamental region An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg corresponding to the coset space An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi144.jpg is denoted as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi145.jpg. In other words, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi145.jpg is an element of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg as well as of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi144.jpg. The notation is similar to An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi112.jpg used earlier, but unlike spaces of orbits, since it is possible to have both left-coset spaces and right-coset spaces, the subscript r is used to restrict the discussion to the ‘right’ case, as well as to distinguish An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi150.jpg from An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi151.jpg.
There is never any need to consider An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi80.jpg outside of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi153.jpg since
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd12.jpg
which follows from equation (9) and the invariance of this sum under shifts of the form An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi154.jpg. Moreover, since An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi155.jpg is An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg-periodic in An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi43.jpg, there is no need to consider any An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi43.jpg outside of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi159.jpg, since
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd13.jpg
In an X-ray diffraction experiment for a single-domain protein, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi123.jpg is not obtained directly. Rather, the magnitude of the Fourier transform of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi155.jpg is obtained with An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi80.jpg held fixed by the physics of the crystal. In general, if An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi163.jpg are the vectors describing lattice directions, so that each element of the group An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi93.jpg consists of translations of the form
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd14.jpg
then the classical Fourier series coefficients for An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi155.jpg (which for each fixed An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi63.jpg is a function on An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi92.jpg) are denoted as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi168.jpg. There is duality between the Fourier expansions for An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi93.jpg and for the unit cell An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi170.jpg, and likewise An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi171.jpg is the unitary dual of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi95.jpg.
A goal of molecular replacement is then to find the specific An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi173.jpg such that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi174.jpg best matches with the diffraction pattern, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi175.jpg, which is provided from X-ray crystallography experiments.5 In other words, a fundamental goal of molecular replacement is to minimize a cost function of the form
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd15.jpg
where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi178.jpg is some measure of distance, discrepancy or distortion between densities or intensities. For example, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi179.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi180.jpg or An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi181.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi182.jpg. Of these, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi183.jpg is by far the most popular because it lends itself to computation in either Fourier space or real space via Parseval’s equality. Less detailed versions of equation (12) use An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi184.jpg in place of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi185.jpg, in which case the translational part of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi80.jpg shows up as a phase factor that disappears when computing magnitude.
No matter what the choice of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi178.jpg, the cost functions An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi188.jpg in equation (12) inherit the symmetry of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi127.jpg in equation (8) in the sense that
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd16.jpg
when An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi190.jpg is extended to take values in An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg. This makes them functions on An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi144.jpg (or, equivalently, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg), in analogy with the way that a periodic function on the real line can be viewed as a function on the circle.
Though the discussion here treats translations and rotations together, the standard approach in molecular replacement is to break up the right-hand side of equation (12) into a part that depends only on the rotational part of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi145.jpg, and then a term that depends on a combination of the translational and rotational parts of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi145.jpg. This second term is discarded and a pure rotational search is performed. Computationally this is advantageous because the dimensions of the search space are reduced from An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi196.jpg to An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi197.jpg, but since the term that is thrown away depends on the rotational part of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi145.jpg, this introduces a larger degree of ‘noise’ into the cost function, thereby introducing spurious false peaks in the rotation function that would otherwise not need to be investigated.
3.2. Visualizing F Γ\G in the case when Γ = P1
In this section an example is used to illustrate An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg graphically. Let An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi200.jpg be shorthand for An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi201.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi202.jpg where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi203.jpg is a counterclockwise rotation around the An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi204.jpg axis by angle An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi205.jpg and the composition of two motions is defined in equation (1). When An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi206.jpg, the An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi207.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi208.jpg components of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi209.jpg span a finite range, which we can take to be a unit square in the plane. Then An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi210.jpg can be viewed as a box, with the vertical direction denoting the rotation angle An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi205.jpg. The height of the top horizontal face of the box relative to its bottom is defined by An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi212.jpg radians. All opposing faces of the box are glued directly to each other with corresponding points defined by the intersection of lines parallel to coordinate axes and the faces.
This is illustrated in Fig. 4 [triangle] in which the points on opposing faces in each box are identified. This means that in Fig. 4 [triangle](a) the following sets each describe the same point: An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi213.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi214.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi215.jpg where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi216.jpg. Similarly, in Fig. 4 [triangle](b) An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi217.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi218.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi219.jpg where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi220.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi221.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi222.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi223.jpg. As a consequence, all eight of the extreme vertices in each figure correspond to the same point.
Figure 4
Figure 4
The space of motions, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi491.jpg, for a body in the planar An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi437.jpg unit cell: (a) origin of the coordinate axes in the lower-left corner; (b) origin of the coordinate axes in the center.
If we choose An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi224.jpg as in Fig. 4 [triangle](a), it becomes clear that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi225.jpg does not mean that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi226.jpg. For example, taking An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi227.jpg, then An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi228.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi229.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi230.jpg. However, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi231.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi232.jpg.
Fig. 4 [triangle](b) is a definition of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi107.jpg that has better closure properties under inversion. For example, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi234.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi235.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi236.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi237.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi238.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi239.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi240.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi241.jpg. But this An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi107.jpg is not fully closed under inversion either. For example, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi243.jpg but An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi244.jpg.
The algebraic properties established in the following section build on these ideas and will assist in the further mathematical characterization of the MR problem.
Though An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg is a group and G is a group, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg is not a normal subgroup of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg (and neither is An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi93.jpg). Therefore, unlike the situation in which An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi249.jpg or An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi250.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi251.jpg, which are again groups, the right-coset spaces An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi252.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi144.jpg are not groups. However, as will be shown here, it is possible to define a non-associative binary operation for these spaces, which turns them into quasigroups.
4.1. The quasigroup operation
As demonstrated in the previous section, the choice of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg is not unique. Given any An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi63.jpg and a fixed choice of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi153.jpg, we can define An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi257.jpg to be such that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi258.jpg for some An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi259.jpg. Therefore, we can think of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi260.jpg as a mapping that selects one representative of each coset An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi261.jpg that has the following properties,
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd17.jpg
With these three properties, it is possible to define a binary operation between any two elements An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi262.jpg. Namely,
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd18.jpg
This application of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi263.jpg to the product An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi264.jpg in equation (14) is important to ensure that the result is back inside An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg.
A right (group) action of G on An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi144.jpg can be defined as
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd19.jpg
Then, when this expression is evaluated with An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi267.jpg in place of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi268.jpg,
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd20.jpg
The relationships between An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi269.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi270.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi271.jpg are described by the commutative diagram below, where id is the identity map, and id, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi272.jpg applied to An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi273.jpg means that id is applied to An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi274.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi272.jpg is applied to An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi276.jpg.
An external file that holds a picture, illustration, etc.
Object name is a-67-00435-scheme1.jpg Object name is a-67-00435-scheme1.jpg
4.2. Lack of associativity
If An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi277.jpg, then An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi278.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi279.jpg. Furthermore, if in addition An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi280.jpg, then An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi281.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi282.jpg. However, if An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi283.jpg, then an additional An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi284.jpg operation would be required to ensure that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi285.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi286.jpg. And herein lies the reason why motions in An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg are a quasigroup rather than a group. Namely, in general
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd21.jpg
That is, the associative property fails.
Consider an example of this when An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi288.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi206.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi159.jpg is the unit square with the center at the origin and hence An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg is visualized as in Fig. 4 [triangle](b). If An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi292.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi293.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi294.jpg, then these motions all are within the fundamental region and so An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi295.jpg. However,
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd22.jpg
and
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd23.jpg
Therefore,
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd24.jpg
and
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd25.jpg
which are clearly not equal.
4.3. Left inverses are not necessarily right inverses
When it comes to computing inverses, we seek an inverse of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi257.jpg that is also in An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg. Unlike a group, there is no a priori guarantee that the left inverse exists, the right inverse exists, and that they are the same. Here we show that indeed left inverses exist, how to compute them, and that in general the left inverse is not a right inverse.
Since we would always define An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg such that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi299.jpg, it follows that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi300.jpg. Since An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi258.jpg for some An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi117.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi303.jpg and
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd26.jpg
Therefore, applying An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi284.jpg to both sides gives
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd27.jpg
But this means that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi305.jpg is the left inverse of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi306.jpg with respect to the operation An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi270.jpg. This is true regardless of whether or not An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi80.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi306.jpg are equal.
As an example, consider the case when An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi206.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi288.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg is as in Fig. 4 [triangle](a). If An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi313.jpg, then An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi314.jpg. It is easy to compute An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi315.jpg. Similarly, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi229.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi317.jpg. But, by definition, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi318.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi319.jpg. That this serves as a left inverse is demonstrated as follows:
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd28.jpg
But this left inverse is not a right inverse:
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd29.jpg
On the other hand, if An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi206.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi288.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg is as in Fig. 4 [triangle](b), and again An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi313.jpg, then An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi324.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi325.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi326.jpg. It is easy to compute An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi327.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi328.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi329.jpg. Similarly, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi229.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi331.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi332.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi318.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi334.jpg. That this serves as a left inverse is demonstrated as before:
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd30.jpg
And it still fails to be a right inverse.
In the special case when An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi335.jpg, it follows that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi336.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi337.jpg. Combining these then gives An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi338.jpg. Furthermore, in this special case, the left inverses computed above also will be right inverses. For example, if An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi339.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg is as in Fig. 4 [triangle](b), then An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi341.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi327.jpg An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi343.jpg is the same as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi344.jpg, which serves as both a left and right inverse, since in this context An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi345.jpg holds, as usual in a group. Note that if instead we used An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg as in Fig. 4 [triangle](a), then in the above example An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi347.jpg.
4.4. Solving equations
In any quasigroup the following equations can be solved for An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi306.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi349.jpg for any given An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi350.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi351.jpg that are in the quasigroup:
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd31.jpg
These solutions are denoted as
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd32.jpg
(where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi352.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi353.jpg are division on the right and left, respectively). But, since the associative law does not hold, we cannot simply apply the inverse of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi350.jpg or An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi351.jpg to obtain the answer. Instead, using the rules established in §4.1,
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd33.jpg
where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi356.jpg is the special element of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg chosen to ensure that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi257.jpg. Similarly,
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd34.jpg
Here no special choice of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi359.jpg is required, and when An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi360.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi349.jpg is simply the left inverse of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi350.jpg.
As stated in §2, the group of rigid-body motions, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg, acts on points in Euclidean space, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi76.jpg, by moving them as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi365.jpg. So too, the quasigroup An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi144.jpg acts on points in An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi88.jpg to move them to other points in the same space. However, the usual property of a group action,
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd35.jpg
does not apply for a quasigroup.
If An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi63.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi257.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi370.jpg we can define a (quasigroup) action of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi144.jpg on An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi88.jpg as
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd36.jpg
This is illustrated in the diagram below.
An external file that holds a picture, illustration, etc.
Object name is a-67-00435-scheme2.jpg Object name is a-67-00435-scheme2.jpg
Note that since
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd37.jpg
it follows that
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd38.jpg
and
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd39.jpg
Since An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg acts from the left on An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi76.jpg, and since An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi375.jpg, it follows that
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd40.jpg
Then, upon the application of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi151.jpg to both sides,
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd41.jpg
Also, combining the properties of group and quasigroup actions,
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd42.jpg
This can be written as
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd43.jpg
And though it would be too much to expect that the properties of a group action would hold for a quasigroup action, the fact that
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd44.jpg
means that
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd45.jpg
Though equations (21) and (22) are not the same in general, in the special case when An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi377.jpg they will be the same.
As depicted in Fig. 4 [triangle], the definition of a fundamental domain An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg is not unique. And since the definition of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi270.jpg depends on how An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg is defined, it too is not unique. Let An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi381.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi382.jpg denote an allowable alternative to An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi270.jpg. When examining relationships between candidate fundamental domains, it makes sense to consider allowable mappings of the form
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd46.jpg
For example, in addition to the two cases shown in Fig. 4 [triangle] when An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi206.jpg, valid fundamental domains for An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi386.jpg can be obtained by translating each horizontal slice in those figures by some continuous An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi387.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi388.jpg. Hence a continuum of different fundamental domains can exist that correspond to one coset space An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi144.jpg. Corresponding to each choice, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi306.jpg is replaced by a different An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi391.jpg.
From an algebraic perspective, it is interesting to ask when such domains are equivalent as quasigroups. In other words, we seek special bijections of the form An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi392.jpg where
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd47.jpg
Such mappings can be called quasigroup isomorphisms.
The existence of bijections is clear in the example of Fig. 4 [triangle], since it is possible to divide up the two fundamental domains into octants, and generate a mapping by permuting these octants and gluing them appropriately. However, it is not clear a priori whether or not such a bijection will preserve the quasigroup operation in the sense of equation (24).
In contrast, the conjugation of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi306.jpg by some fixed An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi71.jpg can be used to define
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd48.jpg
Then if An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi395.jpg,
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd49.jpg
and
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd50.jpg
Therefore, if we define An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi382.jpg by the equality
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd51.jpg
it is easy to see that
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd52.jpg
This is expressed in the following commutative diagram.
An external file that holds a picture, illustration, etc.
Object name is a-67-00435-scheme3.jpg Object name is a-67-00435-scheme3.jpg
In other words, for any fixed An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi71.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi398.jpg is a quasigroup since An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi399.jpg is, and the above diagram commutes. But unlike in equation (23) where the quasigroup corresponds to the same coset space, here the coset spaces are different since in general An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi400.jpg. But this discussion becomes relevant to the issue of constructing different fundamental domains for the same coset space if we restrict the choice of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi401.jpg such that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi402.jpg. This is achieved easily by restricting An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi403.jpg, the normalizer of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg in An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg. When choosing An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi406.jpg, it follows that
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd53.jpg
and, therefore, the set of all mappings An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi407.jpg forms a group under the operation of composition in equation (26), An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi408.jpg, and this group is isomorphic with An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi409.jpg where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi410.jpg is the centralizer of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg in An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg. Recall that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi413.jpg is the largest subgroup of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg in which An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg is a normal subgroup, and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi410.jpg is the subgroup of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg consisting of all elements that commute with every element of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg.
Additional algebraic properties result from the special role that translations play, both in space groups and in continuous Euclidean motions. These are explored in this section.
7.1. General relationships
When viewed as a set rather than a group, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi419.jpg. Then a natural projection operator is An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi420.jpg that simply picks off the translational part of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi421.jpg as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi422.jpg. When this projection is applied after multiplying two group elements, the result is An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi423.jpg This is of the same form as the action in equation (5). Therefore, we can write the following diagram, which is equivalent to the equation
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd54.jpg
An external file that holds a picture, illustration, etc.
Object name is a-67-00435-scheme4.jpg Object name is a-67-00435-scheme4.jpg
This algebraic property gives An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi20.jpg the geometric structure of a trivial principal fiber bundle, which will have implications for possible geometric interpretations of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg, which will be explored in the second paper in this series.
Until now, no specific choice was made to identify which representatives of the cosets An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi140.jpg are used to define An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg. Such a choice would fix the geometric structure of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg. The general discussion of this is postponed until the second paper in this series. But the case when An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi206.jpg is now addressed, and it is closely related to the properties of the projection operator discussed previously.
7.2. The case when Γ = P1
Two possible choices for An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi430.jpg when An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi288.jpg were illustrated in Fig. 4 [triangle]. More generally, the choice of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi430.jpg is partially constrained by identifying An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi430.jpg with An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi434.jpg. This does not fully define An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi430.jpg because An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi436.jpg can be defined in multiple ways (for example in Figs. 4 [triangle] a and 4 [triangle] b, this is, respectively, the unit square contained in the first quadrant and centered at the origin).
The (partial) definition
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd55.jpg
is acceptable because An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi437.jpg has no rotational or screw symmetry operators, and therefore its action from the left has no effect on the An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi11.jpg part of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi20.jpg. Then it is clear that for any An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi440.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi441.jpg and
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd56.jpg
Since a pure translation is of the form An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi442.jpg, it is possible to compute An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi443.jpg. Similarly, a translation can be identified as a position via the action An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi444.jpg where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi445.jpg is the origin in An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi35.jpg. The projection operator relates An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi447.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi448.jpg as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi449.jpg as a special case of equation (29).
In addition, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi450.jpg Note also that when viewing An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi430.jpg as in equation (28)
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd57.jpg
and
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd58.jpg
Equating the above results gives
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd59.jpg
These, together with equalities presented earlier in the paper, lead to the following commutative diagram.
An external file that holds a picture, illustration, etc.
Object name is a-67-00435-scheme5.jpg Object name is a-67-00435-scheme5.jpg
This section first reviews the multi-domain molecular replacement problem and then illustrates the applicability of the algebraic concepts developed earlier in this paper.
8.1. Multi-domain molecular replacement
Consider a multi-domain protein or complex that is known to consist of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi2.jpg rigid components, each of which has a high degree of homology to a known protein. Some of these components might also be homologous to each other, but in the absence of any evidence otherwise, the domains will be treated as having different density functions. If the An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi453.jpgth body/domain in the assemblage has density An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi454.jpg when described in its own body-fixed reference frame, then for some unknown set of rigid-body motions An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi455.jpg, the density of the whole unknown structure must be of the form
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd60.jpg
where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi456.jpg. Here An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi457.jpg are relative rigid-body motions between sequentially numbered bodies. Such a numbering does not require that the bodies form a kinematic chain, though such topological constraints naturally limit the volume of the search space.
If the assemblage/complex/multi-body protein that is formed from these individual domains/bodies is rigid, then symmetry mates in the crystal will all have the same values of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi458.jpg. Here An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi459.jpg takes the place of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi80.jpg in the earlier discussion of single-body molecular replacement, and the density becomes
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd61.jpg
Cost functions analogous to equation (12) follow naturally, but now become functions of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi461.jpg, and therefore represent a An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi1.jpg-dimensional search. Direct grid searches of very high dimensional spaces will always be inadvisable, no matter how rapidly computer technology advances. However, by taking advantage of the quasigroup structure of this search space, gradient descent methods may be appropriate. Whereas such methods are inadvisable when seeking optima in the rotation function (since there is tremendous ‘noise’ that results from discarding non-pure-rotation terms), the high-dimensional search space is far less noisy since the high-dimensional model that is matched to the diffraction pattern (or in real space, the Patterson function) has built into it a higher-fidelity model where all variables are simultaneously present, rather than sequential searches over each domain.
8.2. Applicability of quasigroup properties
The properties of quasigroups of motions and their actions on points in an asymmetric unit, as well as actions of motion groups on quasigroups, will play a role in various aspects of MR that will be explored in later papers in this series. These include modeling motional smearing such as is the case in static disorder and thermal motion in crystals, and the formulation of optimization problems such as minimizing the cost in equation (12). Such applications involve both the algebraic properties discussed here, and the geometric ones that will be described in the second paper in this series. Nevertheless, it is possible to illustrate at this stage how the concepts of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi112.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi306.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi143.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi159.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi269.jpg, An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi271.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi469.jpg interact naturally in a particular MR-related problem, as discussed below.
Consider a macromolecular structure consisting of two rigid domains. Let An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi470.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi471.jpg denote the densities of these bodies, each relative to its own body-fixed reference frame. In the case when these locally defined densities have their body-fixed frames coincident with the identity reference frame An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi472.jpg, then An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi473.jpg for An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi474.jpg. If the frame attached to body 2 has a position and orientation of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi475.jpg relative to the frame attached to body 1, then the density function for the composite structure (when the reference frame attached to body 1 is the identity) will be
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd62.jpg
Then, if body 1 is itself moved and body 2 retains its relative spatial relationship to body 1, the result will be
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd63.jpg
Using the notation for a periodic density from §3.1 and the concept of the action An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi271.jpg from §5, the resulting density of a crystal consisting of two-domain macromolecules will be
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd64.jpg
Using the algebraic rules established earlier, the second term can be written as
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd65.jpg
The extension to the multi-domain case follows in a similar way, and does not require the introduction of new concepts of action.
Unlike the step from equations (32) to (33), which is valid in the context of group actions, in general
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd66.jpg
This is because, in the case of group actions, the solution to An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi477.jpg is An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi478.jpg. But in the quasigroup case, the solution to An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi479.jpg is not An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi480.jpg But a solution can be constructed using the algebraic concepts discussed earlier. Namely, if An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi481.jpg, then
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd67.jpg
and
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd68.jpg
Hence
A mathematical equation, expression, or formula.
 Object name is a-67-00435-efd69.jpg
and similarly for An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi482.jpg. Therefore, the algebraic constructions presented earlier provide a tool for manipulating different descriptions of densities that arise in MR applications.
The algebraic structure of the molecular replacement problem in macromolecular crystallography has been articulated here. This includes enumerating the quasigroup structure of the coset space An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi483.jpg, where An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi4.jpg is the space group of the crystal and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi21.jpg is the continuous group of rigid-body motions. Equipped with these properties of the space An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi486.jpg articulated here, it becomes possible to formulate codes for searching the space of motions of macromolecules in asymmetric units in a way that is not subject to the arbitrariness of a choice of coordinates such as Euler angles, and the inescapable distortions and singularities that result from coordinate-dependent approaches. Geometric and numerical aspects of the formulation presented here will be investigated in follow-on papers. In such applications, it is important to fix a geometric interpretation of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi210.jpg. It will be shown that the algebraic concept of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi488.jpg discussed here provides insights into concrete choices for An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi210.jpg, and the mappings and quasigroup isomorphisms discussed here provide the means to convert between different choices for these domains.
Acknowledgments
This work was supported by NIH grant No. R01 GM075310. The suggestions by W. P. Thurston, S. Zucker and the anony­mous reviewer are greatly appreciated.
Footnotes
1In the mathematics literature a quasigroup with identity is called a loop (Sabinin, 1999 [triangle]; Smith, 2006 [triangle]; Vasantha Kandasamy, 2002 [triangle]), but since the word ‘loop’ is used in biological contexts to mean a physical serial polymer-like structure with constrained ends, the word ‘quasigroup’ will be used here instead of mathematical ‘loop’.
2Some books denote this as An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi79.jpg, but to be consistent with the definition of action in equation (5), in which An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi80.jpg acts on the left of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi43.jpg, it makes more sense to write An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi78.jpg in analogy with the way that An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi65.jpg preserves the order of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi84.jpg in the definition of the coset An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi85.jpg.
3It should be pointed out that the ‘noise’ here is not noise in the true sense, but rather results from false peaks in rotational correlations arising from restricting the search from a high-dimensional space (e.g. An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi1.jpg for a system composed of An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi2.jpg rigid bodies) to an initial three-dimensional orientational search.
4Though this is an infinite sum, each An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi125.jpg has compact support because each protein domain is a finite body, and so convergence is not an issue.
5Here An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi80.jpg and An external file that holds a picture, illustration, etc.
Object name is a-67-00435-efi145.jpg can be used interchangeably because of equation (10).
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