A Spatial Indexing Approach for Protein Structure Modeling

Abstract

This paper explores the use of spatial indices for the modeling and retrieval of protein structures. With two existing spatial indices, a preliminary framework for protein structure modeling that uses a spatial index is proposed. It provides a novel technique for modeling. In addition, it provides additional flexibility with respect to modeling granularity and structure manipulation. It is expected that this modeling approach will lead to new ways of analyzing protein structures.

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A Spatial Indexing Approach for Protein Structure Modeling
Wendy Osborn
Department of Mathematics and Computer Science
University of Lethbridge
Lethbridge, Alberta
T1K 3M4 Canada
wendy.osborn@uleth.ca
Abstract
This paper explores the use of spatial indices for the
modeling and retrieval of protein structures. With two ex-
isting spatial indices, a preliminary framework for protein
structure modeling that uses a spatial index is proposed. It
provides a novel technique for modeling. In addition, it pro-
vides additional flexibility with respect to modeling granu-
larity and structure manipulation. It is expected that this
modeling approach will lead to new ways of analyzing pro-
tein structures.
1. Introduction
Protein structure analysis is an exciting and challenging
research area in the area of bioinformatics [1]. Many repos-
itories exist today that maintain three-dimensional protein
structures and provide tools for search and retrieval. The
first such repository is the Protein Data Bank (PDB) [4]. It
remains a very popular source for information. The PDB
currently maintains over 20,000 protein structures.
current representation of the three-dimensional structure in
PDB is very inflexible and archaic [1]. Therefore, it is im-
portant to explore how three-dimensional protein structures
are modeled for future analysis.
One promising approach is in the area of spatial
databases [11]. Spatial indexing [5, 11] provides a tech-
nique for modeling and retrieval of data based on its loca-
tion in multidimensional space. It also provides a frame-
work that can be used in conjunction with existing protein
analysis strategies.
The application of spatial indexing to protein structure
modeling has not received significant attention. Srinivasa
and Kumar [12] propose a platform for modeling three-
dimensional structures in a database. They also present
strategies for search and retrieval. One limitation of their
The
work is that their use of spatial indexing is limited to a strat-
egy for point data only. Yan, Yu and Han [13] propose a
strategy for indexing a graph representation of a biomolec-
ular structure. Although their approach is promising, it has
a limitation of only being applicable to substructures.
The application of the approximation spatial index for
the modeling protein structures is investigated. Given the
properties of existing approximation strategies, a prelimi-
nary framework for protein structure modeling is proposed.
This approach works across all descriptions of a protein
structure, and also with different granularities of a model.
For example, one can model at the atomic level, at the sub-
structure level, or anywhere in between.
The remainder of this paper proceeds as follows. Sec-
tion 2 presents some required background information on
protein structure modeling and spatial indexing. Section 3
presents an application of existing spatial indexing struc-
tures for the modeling and retrieval of protein structures.
Section 4 concludes with future research directions that
need to be considered for this modeling strategy.
2. Preliminaries
This section presents some required background infor-
mation that is required for this investigation. First, the hi-
erarchical structure description, which is the common tech-
nique for describing proteins at multiple levels, is summa-
rized with an example. Second, spatial indexing is summa-
rized, with a focus on two existing strategies.
2.1. Hierarchical Structure Description
A protein structure can be described using a hierarchical
method [1]. This layered approached is composed of four
levels: primary, secondary, tertiary and quaternary. The
primary level consists of the sequence of amino acids that
make up the protein. The secondary level consists of the
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MSQPIFNDKQFQEALSRQWQRYGLNSAAEMTPRQWWLAVSEALAEMLRAQPFAKPV
ANQRHVNYISMEFLIGRLTGNNLLNLGWYQDVQDSLKAYDINLTDLLEEEIDPALGA
GGLGRLAACFLDSMATVGQSATGYGLNYQYGLFRQSFVDGKQVEAPDDWHRSNYP
WFRHNEALDVQVGIGGAVTKDGRWEPEFTITGQAWDLPVVGYRNGVAQPLRLWQAT
HAHPFDLTKFNDGDFLRAEQQGINAEKLTKVLYPNDNHTAGKKLRLMQQYFQCACS
VADILRRHHLAGRELHELADYEVIQLNDTHPTIAIPELLRVLIDEHQMSWDDAWAITS
KTFAYTNHTLMPEALERWDVKLVKGLLPRHMQIINEINTRFKTLVEKTWPGDEKVWA
....................
Figure 1. Hierarchical Description for 1AHP
substructures that the protein is composed of. Two common
protein substructures are the helix (α) and the beta sheet (β-
sheet). The tertiary level consists of the final protein struc-
ture, which is composed of all substructures from the sec-
ondary level. Finally, the quaternary contains a structure of
multiple proteins, each from a different tertiary level.
Figure 1 depicts an example of a hierarchical descrip-
tion at the primary, secondary and tertiary levels for the pro-
tein 1AHP, which is retrieved from the Protein Data Bank
[4] and viewed using RasMol [2]. The primary level con-
tains a portion of the amino acid sequence for 1AHP. The
secondary level contains the secondary structures, each of
which is a helix or a β-sheet. The tertiary level contains the
protein structure in its entirety.
2.2. Spatial Indexing
A spatial index (or spatial access method) [5, 11] pro-
vides access to data based on its location in multidimen-
sional space. Data that is indexed based on location usually
consists of points, lines and objects of arbitrary shape. In
addition, data with non-spatial information can be stored
along with spatial data. For example, a map consists of
towns (points), roads (lines) and cities (arbitrarily-shaped
objects). Each datum on a map can have non-spatial data
associated with it, such as a city name or a road number.
A spatial index supports many types of searches [5]. One
common search type is the region search. Given an object
that represents a region of space (usually a rectangle), the
goal is to find all data that overlap the region.
Many spatial indices are proposed in the literature. A
comprehensive survey is available in [5].
lar class of spatial indices that is of interest for this pro-
posed framework are approximation strategies.
eral overview of an approximation spatial index is given
next. Following this, two different approaches that adopt
the approximation strategy - the R-tree [7] and the 2DR-tree
[10, 9] are summarized.
An approximation spatial index maintains a hierarchy
of approximations for objects and the subregions contain-
ing one or more objects. Most approximation strategies
are based on the B+-tree [3], so they are height balanced
(or, every path from the root node to leaf node is the same
length).
Anapproximationforbothobjectsandsubregionsisusu-
ally represented in the form of a minimum bounding rectan-
gle (MBR). A minimum bounding rectangle defines the
extent of an object along each dimension in space.
The hierarchy is maintained using nodes. Each node can
contain a minimum of m records and a maximum of M
records, where M is the total number of records allowed
in the node, and m is a user defined value. Usually, m =
M/2. The exception is the root note, which can contain at
minimum of two records. Each record maintains:
One particu-
A gen-
(MBR(i,j),ptr(i,j))
where MBR(i,j) is an approximation and ptr(i,j) is a
pointer. In a leaf node, MBR(i,j)approximates an object
and ptr(i,j)references the actual object on secondary stor-
age. In a non-leaf node, MBR(i,j)encloses all approxima-
tions in the subtree referenced by ptr(i,j).
2.3. R-tree
The R-tree [7] is the first approximation spatial index
proposed in the literature. As with the B+-tree, the R-tree
uses linear nodes to organize approximations into a hierar-
chy. An example is described first, followed by a brief de-
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m9
m8
m5
m3
m1
m2
m4
p9
p5
m2m4
p9
m7
m5
m8
m9
m1
p5
m7
m3
Figure 2. The R-tree
scription of the region search, insertion and deletion strate-
gies.
Figure 2 depicts an example of an R-tree that is indexing
a spatial data set (the data set comes from [5]). The approx-
imations for actual objects are represented by points p5 and
p9, and rectangles m1 to m5 and m7 to m9. In addition,
there are three approximations that represent subregions -
one that contains m1, m2, m4, and p9; one that contains m3,
m5, m8 and p5; and one that contains m7 and m9. The leaf
level contains the object approximations, while the non-leaf
level (also the root level) contains the subregion approxima-
tions.
The R-tree region search begins at the root by examining
all approximations to find the ones that overlap the query
region. For all qualifying approximations, the search con-
tinues in the corresponding subtrees until it reaches zero or
more leaf nodes. The approximations for all qualifying leaf
nodes are retrieved and tested for overlap with the query
region.
The insertion algorithm first identifies the appropriate
leaf node for storing the new entry. The insertion path con-
tains minimum bounding rectangles that require the least
enlargement to include the new object. After inserting the
new approximation into the chosen leaf node, the approxi-
mations along the insertion path are updated.
If overflow of the leaf node occurs, it is handled by split-
ting the node into two new nodes. The node is split so
that the new nodes cover the smallest amount of area and
both nodes are not checked for the same query. Three split-
ting strategies are proposed: exhaustive, quadratic, and lin-
ear. The exhaustive approach finds all candidate splits and
chooses the best one. The quadratic and linear approaches
identify the two objects that are the farthest apart and cluster
theremainingobjectsintotwonodes. Thequadraticandlin-
ear approaches differ in how the two objects are identified.
If a split causes an overflow in the parent node, the split is
propagated up the tree as far as necessary. If it propagates
to the root, a new root node is created.
The R-tree deletion strategy removes the approximation
of the object to be deleted and adjusts the minimum bound-
ingrectanglesalongthedeletionpath. Underflowishandled
using a forced reinsertion strategy.
2.4. 2DR-tree
The 2DR-tree [10, 9] extends the one-dimensional struc-
tureoftheR-tree[7]totwodimensions. Itsstructureandor-
ganization is summarized first, followed by an example and
a description of the search, insertion and deletion strategies.
The nodes used in the 2DR-tree to organize approxima-
tions are two-dimensional in nature. The M locations in a
node are organized in the following manner. For each node
N, X is the number of locations along the x-axis of the
node, and Y is the number of locations along the y-axis.
An approximation is stored in an appropriate location
with respect to all other approximation in the node. Us-
ing two-dimensional nodes allows spatial relationships be-
tween objects to be preserved. The spatial relationships that
can be maintained in the 2DR-tree are north, northeast, east,
southeast, south, southwest, west and northwest. A spatial
relationship is defined between two objects using the cen-
troids of their approximations. For example, approximation
MBR 1 is northeast of MBR 2 if the centroid of MBR 1 is
northeast of the centroid of MBR 2.
Organization of approximations is dictated by the fol-
lowing validity rules [10]. For each approximation MBR:
• All approximations located directly north of MBR in
the node have a centroid that is north, northwest, or
west of the centroid for MBR in space,
• All approximations located northeast of MBR in the
node have a centroid that is northeast of the centroid
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m9
m8
m5
m3
m1
m2
m4
p9
p5
m3
m2
p9
m7
m4
m5
m8
p5
m9
m1
m7
Figure 3. The 2DR-tree
for MBR in space, and
• All approximations located directly east of MBR in
the node have a centroid that is east, southeast or south
of the centroid for MBR in space.
Figure 3 shows an example of a 2DR-tree that is index-
ing the same spatial data set from [5]. As with Figure 2,
the leaf level contains the approximations for all points and
objects, while the subregions are maintained in the non-leaf
level. The difference here is how the spatial relationships
are maintained. If we look at the subregion enclosing m4
and m3, we observe that the centroid of m3 is southeast of
the centroid for m4, so both they are placed in appropriate
locations with respect to each other in the leaf node. Sim-
ilarly, the spatial relationships are also maintained between
subregions in the non-leaf level.
Since approximations are organized, a binary search
strategy [9] can be applied in the following manner. The
search region is applied recursively to half of the objects in
the node until all overlapping approximations are found.
The insertion strategy employs a greedy search for locat-
ing an insertion path contains a minimal (if not the absolute
minimal) area increase required to insert a new approxima-
tion. At the leaf level, a location is found for the approxima-
tion that obeys all validity rules. Any overflow that occurs
is handled using one of the splitting strategies proposed in
[9, 10]. Deletion simply requires the removal of the objects
and the updating of the approximations along the insertion
path.
3. Structure Modeling Using Spatial Indices
Thissectionpresentstheproposedframeworkforprotein
structure modeling. The framework can utilize either the R-
tree or the 2DR-tree for modeling a protein structure. The
hierarchical model described earlier consists of four levels
- primary, secondary, tertiary, and quaternary. We discuss
how a spatial index can be applied to each of the secondary,
tertiary and quaternary levels. In addition, we discuss how
a spatial index can be applied in different ways within the
same level using different amounts of detail.
3.1. Secondary Level Modeling
Protein secondary structures can be modeled in differ-
ent ways. One approach is to group helices and β-sheets
that are adjacent to each other into substructures. Each sub-
structure can be represented with an approximation. This
approximation in turn can be placed in a spatial index struc-
ture. These substructures can be formed by inserting helices
and β-sheets by location into the spatial index structure.
This can potentially lead to the discovery of new substruc-
tures that serve an important purpose in the function of a
protein. An entire substructure can be discovered indepen-
dently, and can be inserted as a single unit into the spatial
index structure.
Figure 4 depicts an example of substructure modeling
using the R-tree, while Figure 5 depicts an example of sub-
structure modeling using the 2DR-tree. In both cases, each
substructure is enclosed with an approximation that can be
accessed from the index structure. The difference between
the R-tree and 2DR-tree approaches is the following. Al-
though the R-tree is a one-dimensional structure, it can be
applied to data residing in any dimension. However, in its
current form, the 2DR-tree represents a two-dimensional
topological modeling of the protein structure, since it is de-
signed to work in two-dimensional space.
A protein structure can also be indexed at the atomic
level, where atoms and bonds are grouped into leaf nodes.
If a 2DR-tree is applied at the atomic level, the bonds be-
tween atoms can be implicitly represented by maintaining
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Figure 4. R-tree Substructure ModelingFigure 5. 2DR-tree Substructure Model-
ing
the spatial relationships between the atoms, therefore elim-
inating the need to store them. A protein structure can also
be indexed at the helix/β-sheet level. In this configuration,
leaf nodes can contain helices, β-sheets, and other known
and unknown secondary structures.
In addition, multiple protein secondary structures can be
modeled within the same spatial index. This allows for pro-
tein structure homology comparisons to be carried out with
ease, since potentially related substructures are stored to-
gether. Alternatively, secondary structures can be modeled
with different spatial indices (but of the same index type)
but still compared for structural homology.
3.2. Tertiary Modeling
Similarly to secondary modeling, the tertiary represen-
tation of a protein structure can be represented at the sub-
structure level and the atomic level. Also, it is possible to
apply the same spatial index to multiple tertiary structures
for the purposes of structure homology comparison.
3.3. Quaternary Modeling
One more option is added to the existing levels of mod-
eling resolution given above. The quaternary description
of multiple tertiary structures can be successfully modeled
with one spatial index structure. This will provide support
for modeling across proteins. Also, if one is interested, it
is possible to model multiple quaternary descriptions with
one index to provide further opportunities for the analysis
of protein structures.
3.4. Searching and Analysis Strategies
One can use existing spatial searching strategies, such as
the region search, or the binary or greedy search techniques
of the 2DR-tree [9, 10], on protein structures. In addition,
one can apply other search and analysis techniques as the
basis for data retrieval. For example, strategies for struc-
ture homology comparision such as VAST [8, 6] can be
incorporated into a spatial index model. Structure homol-
ogy comparisons can be applied not only between existing
structures, but also for the prediction of a newly-discovered
protein structure based on existing structures.
3.5. Inclusion of Non-Spatial Data
Descriptive non-spatial data can be stored alongside an
approximation. This comes in handy for storing data from
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