Using molecular principal axes for structural comparison: determining the tertiary changes of a FAB antibody domain induced by antigenic binding
© Silverman; licensee BioMed Central Ltd. 2007
Received: 26 June 2007
Accepted: 09 November 2007
Published: 09 November 2007
Comparison of different protein x-ray structures has previously been made in a number of different ways; for example, by visual examination, by differences in the locations of secondary structures, by explicit superposition of structural elements, e.g. α-carbon atom locations, or by procedures that utilize a common symmetry element or geometrical feature of the structures to be compared.
A new approach is applied to determine the structural changes that an antibody protein domain experiences upon its interaction with an antigenic target. These changes are determined with the use of two different, however comparable, sets of principal axes that are obtained by diagonalizing the second-order tensors that yield the moments-of-geometry as well as an ellipsoidal characterization of domain shape, prior to and after interaction. Determination of these sets of axes for structural comparison requires no internal symmetry features of the domains, depending solely upon their representation in three-dimensional space. This representation may involve atomic, Cα, or residue centroid coordinates. The present analysis utilizes residue centroids. When the structural changes are minimal, the principal axes of the domains, prior to and after interaction, are essentially comparable and consequently may be used for structural comparison. When the differences of the axes cannot be neglected, but are nevertheless slight, a smaller relatively invariant substructure of the domains may be utilized for comparison. The procedure yields two distance metrics for structural comparison. First, the displacements of the residue centroids due to antigenic binding, referenced to the ellipsoidal principal axes, are noted. Second, changes in the ellipsoidal distances with respect to the non-interacting structure provide a direct measure of the spatial displacements of the residue centroids, towards either the interior or exterior of the domain.
With use of x-ray data from the protein data bank (PDB), these two metrics are shown to highlight, in a manner different from before, the structural changes that are induced in the overall domains as well as in the H3 loops of the complementarity-determining regions (CDR) upon FAB antibody binding to a truncated and to a synthetic hemagglutinin viral antigenic target.
Comparison of different protein x-ray structures has previously been made in a number of different ways; for example, by visual examination, by differences in the locations of secondary structures, by explicit superposition of structural elements, e.g. α-carbon atom locations, or by procedures that utilize a common symmetry element or geometrical feature of the structures to be compared. This latter procedure has been utilized in connection with the identification of the structurally conserved residues within the core of the immunoglobulin variable domains . A singular advantage of such procedure, compared with the other procedures, is that it provides additional information that relates the location of the residues to attributes of the geometrical feature to which these locations have been aligned or referenced. For example, it has been pointed out that the alignment, based on the pseudo 2-fold symmetry axis of the variable domains of known immunoglobulin structures, provides information about the possible structural or functional roles of residues (italics quoted verbatim in the reference) .
The overall shape or distribution of the amino acid residues of a protein domain may also be considered a geometric invariant of a set of structures undergoing comparison when the differences in their global geometries are small and involve only a minor fraction of the residues comprising the domains. The representation of such shape may be given by the distribution of atomic, Cα, or residue centroid locations in three-dimensional space. Such representation, generating an ellipsoidal characterization of the shape of a domain, had previously provided useful information in connection with drug discovery  and with the spatial distribution of residue hydrophobicity within protein domains . This characterization of domain shape provides two spatial metrics, one of which references the location of a residue to the ellipsoidal principal axes of the domain and the other which yields information detailing the proximity of a residue to either the interior or exterior of the protein domain. The present paper describes how the changes in antibody structure that occur upon binding to an antigenic target are characterized by the consequent changes of these two metrics
One limitation of the present procedure is that the unliganded antibody structure (domain) is required as well as its antibody structure (domain) in the complex. While the Protein Data Bank (PDB)  has numerous antibodies complexed with their viral or chemical targets, there are many fewer unliganded structures listed. The number of PDB structures satisfying our requirements is further reduced since interest will be focused on antibody binding to an influenza viral hemagglutinin antigenic target. Furthermore, we require 100% sequence identity between the unliganded and complexed heavy and light chain domains of the FAB (antibody fragment). Two PDB antibody structures that satisfy these requirements are antibody HC19 complexed with a truncated hemagglutinin structure , PDB id 2VIR, and its unliganded antibody structure , PDB id 1GIG; and FAB 17/9 complexed with a peptide hemagglutinin mimetic , PDB id 1IFH, and its unliganded antibody structure , pdb id 1HIL. Interest will focus on the two distance metrics of the ellipsoidal characterization of protein domain structure and on the complementary information they present that describes the structural changes that occur upon the antigenic binding of these two antibodies. Hopefully, such information involving a different perspective from that provided previously may assist in the attempts to design synthetic vaccines on the basis of X-ray structures of anti-body-peptide complexes .
The ellipsoidal characterization of a protein domain has been previously described , however, it will be useful to indulge in a degree of redundancy to smoothly illustrate the appropriate extension required for the present application. The present calculations are based upon the residue side-chain centroids of the protein. However, as mentioned previously the distribution of points in three-dimensional space chosen to represent a protein structure may well be that of the Cα coordinates, of the atomic coordinates, or of any other set of points in space chosen to detail protein structure.
The residue centroids are calculated with inclusion of only the heavy atoms of the side-chain and without the backbone atoms. One could have included the backbone atoms as well in calculating the residue centroids which would yield minor modifications of the present results. Not including the backbone atoms places the residue centroids at a greater distance from the backbone and provides somewhat greater emphasis with regard to differences in side chain location and orientation.
n is the total number of residues.
Where is the unit dyadic.
The moments-of-geometry tensor is analogous to the moments-of-inertia tensor, however, with each point assigned a mass of one. The diagonalization of provides the moments-of-geometry, g1, g2, and g3.
The x p , y p , z p , are the coordinates in the frame of the ellipsoidal principal axes with the centroid of the structure as origin. If the magnitudes are ordered as,g1 <g2 <g3
the major semi-principal axis is of length, d/g11/2.
enables to be used as a measure of the radial fractional distance of the i th residue from the center of the protein to the protein surface. This distance, which will be called the ellipsoidal distance, is used in the calculations. It is just the value of the semi-principal major axis of the ellipsoid upon which the residue centroid is found. It provides a more accurate characterization of the amino acid proximity to the protein exterior than the radial distance from the protein center to the residue centroid, as well as providing a distance that correlates more closely with residue solvent accessibility .
To calculate the displacements of the residues in the liganded compared with the unliganded structure, the calculations are performed twice; once inclusive of all residues of the unliganded domain which we will designate by "a" and once inclusive of all residues of the domain in the complex which we will designate by "b".
The magnitude of the displacement of the i th residue centroid of the complexed domain with respect to its location in the unliganded domain, D i , is given by the distance between the coordinates of the centroids with respect to the two different sets of principal axes.D i = [(x bip - x aip )2 + (y bip - y aip )2 + (z bip - z aip )2]1/2
The subscript, with either an "a" or "b", designates whether the coordinate is referenced to the principal axes of the unliganded or of the liganded domain, respectively.
When the difference between the antibody structures of the liganded and unliganded domains is minimal this procedure will provide a relatively accurate characterization of the displacements and changes in the ellipsoidal distances that occur. However, if the liganded and unliganded structures differ sufficiently, the calculated differences may then be anomalous. For example residues far from the binding site should exhibit minimal displacements upon complexing. If this is not observed then the liganded and unliganded structures would be sufficiently different and not provide principal axes that are comparable and consequently appropriate to be used for structural comparison. If, however, only a minor region or part of the liganded and unliganded structures differs, e.g., perhaps only differing in the vicinity of the binding site, such difficulty may be circumvented by the choice of comparable substructures to reference the displacements and the changes in ellipsoidal distances. The substructures chosen, for example, may involve the elimination of residues that exhibit significant displacements between the liganded and unliganded structures. In pursuit of such strategy, after diagonalization of the tensor, all residue locations of the substructures will be provided; however, locations of the residues that have been eliminated in the choice of the substructures would then have to be calculated by translating the location of these residues to the centers-of-geometry of each of the substructures and then by rotating into the orientation of the principal axes of the substructures. This procedure will be demonstrated in the comparison between the residue locations of the 1IFH and 1HIL pdb viral structures.
Finally, it should be noted that this strategy of referencing structures undergoing comparison to the sets of principal axes of relatively invariant substructures represents a more general and inclusive strategy than referencing the structures to sets of symmetry axes, e.g., alignments based on the pseudo 2-fold symmetry axes of the variable domains of known immunoglobulin structures. In the present case the invariance of the axes is a consequence of the invariance of the substructures and need not be related to any explicit structural symmetry.
Results and discussion
Moments of the 1GIG and 2VIR structures obtained by the diagonalization of equation 2 differ by a few percent and the two sets of principal coordinates yield coordinate frames with axes alignments that differ by at most several degrees.
Certain enhanced displacements apparently identify residues that have been spatially shifted due to crystal packing. Figure 1A shows an enhanced displacement, with respect to the local background, of the residue GLN16 of the heavy chain N-terminal domain. Such displacement, clearly unrelated to antibody binding, appears to arise from crystal packing. Residues significantly displaced, while not in the vicinity of the region of binding and also observed to be considerably solvent exposed in the free state of the antibody can be so identified.
Such difference in the orientation of the two sets of principal axes can be significantly reduced by determining the axes for substructures from which significantly displaced distant residues from the center of the domain have been eliminated. While there is a degree of freedom in the choice of such elimination and one may be motivated to optimize the correspondence between the two sets of principal axes used for comparison, the substructures presently chosen will simply involve the elimination of only the two residues ASP99 and ASN100A from the H3 CDR loops, namely, the residues that exhibit the greatest displacements shown in figures 7A and 7B. With the principal axes obtained for both reduced liganded and unliganded substructures one would then rotate the original sets of residue centroids eliminated in the determination of the substructure, into the substructure principal axis orientations after translations to the substructure centers-of-geometry.
Changes in the magnitudes of the ellipsoidal distances contrast with what had been found for the displacements. Comparison of the figures 9C and 7C surprisingly shows comparable ranges of the values of these changes. This is also seen in the expanded scales of figures 9D and 7D which detail the region of interaction and consequently of the region of greatest change. So, one might conclude that the ellipsoidal distances are relatively insensitive to rotations of the principal axes. This is apparently a consequence of the proportionality of the ellipsoidal distances to the radial fractional distances from the center of the domain to the ellipsoidal surface or exterior. Such proportionalities are relatively unchanged as the principal axes are slightly rotated with respect to each other. This would be especially true for a domain approximately spherical in shape.
A new approach, enabling comparison between different, however, structurally related domains, has been applied in determining the structural changes that an antibody protein domain experiences upon its interaction with an antigenic target. The present procedure, while analogous to previous procedures that utilize common symmetry elements for comparison, utilizes, instead, the sets of principal axes of the relatively invariant global structures or substructures of the domains undergoing comparison. An ellipsoidal characterization of these structures yields two spatial metrics that provide complementary information; one, detailing the magnitude of the residue displacements and the other; their direction of their displacement with respect to either the domain exterior or interior. The information provided by the present procedure should augment related information provided by more customary procedures. Hopefully such information will contribute to the attempts to design synthetic vaccines on the basis of X-ray structures of anti-body-peptide complexes .
I thank Daniel Platt for a useful discussion that assisted in responding to the request of one of the reviewers of the present paper.
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