A tool for structure alignment of molecules

Abstract

In this paper, a novel tool is proposed to align two molecules (not just proteins) based on their 3D structural data, and the user can observe the result of alignment visually via the tool. Most existing tools are designed only for alignment of proteins. Here, a new tool is developed to address shared structural features between protein structures and tRNA structures, that is, molecular mimicry, although they are two very different types of molecules. In order to align two molecules A and B, geometric hashing is applied to globally find initial matching of approximately overlapped atoms, thus parts of molecule A can be matched to parts of molecule B. Next, a fine tuning process is introduced, which is based on local optimization of overlapped parts, and the iterative closest point (ICP) is used until the number of overlapped atoms within a given distance threshold can not be increased any more. The results show that our method is useful to structurally align two molecules, not restricted to align two proteins only. Besides, our tool outperforms in terms of RMSD and number of matched atom pairs in comparison to other tools.

Full text

A Tool for Structure Alignment of Molecules
Pei-Ken Chang, Chien-Cheng Chen and Ming Ouhyoung
Department of Computer Science and Information Engineering, National Taiwan University
{zick, ccchen}@cmlab.csie.ntu.edu.tw, ming@csie.ntu.edu.tw
Abstract
In this paper, a novel tool is proposed to align two
molecules (not just proteins) based on their 3D
structural data, and the user can observe the result of
alignment visually via the tool. Most existing tools are
designed only for alignment of proteins. Here, a new
tool is developed to address shared structural features
between protein structures and tRNA structures, that is,
molecular mimicry, although they are two very
different types of molecules.
In order to align two molecules A and B, Geometric
Hashing is applied to globally find initial matching of
approximately overlapped atoms, thus parts of
molecule A can be matched to parts of molecule B.
Next, a fine tuning process is introduced, which is
based on local optimization of overlapped parts, and
the Iterative Closest Point (ICP) is used until the
number of overlapped atoms within a given distance
threshold can not be increased any more. The results
show that our method is useful to structurally align
two molecules, not restricted to align two proteins only.
Besides, our tool outperforms in terms of RMSD and
number of matched atom pairs in comparison to other
tools.
1. Introduction
Search engines for 3D models have been developed
in recent years [1] [2], however, can similar techniques
be used in molecules? If so, the benefit can be great.
The reason is that large number of protein structures
can be determined by high throughput machines,
classifying proteins into families and assigning
functions to those novel proteins become major tasks
in recent years. The Protein Data Bank (PDB)
currently contains more than 25,000 structures and it is
estimated that the number of structures in the PDB
may exceed 35,000 by 2005. Though proteins have
been grouped together on the basis of structural
similarities in the FSSP [3], CATH [4], and SCOP
databases [5], much effort still has been put into
finding the similarities among proteins. Moreover, the
rapid growth in the amount of structural data of
proteins far exceeds the ability of experimental
techniques to identify the locations and key amino
acids of active sites. Although the structural genomics
initiative (SGI) proposes to solve 10,000 protein 3D
structures in this decade, however, many biological
functions still remain unknown.
With the help of alignment tools, the structural
similarity between proteins is revealed, as well as the
functional and evolutional relationships. Holm and
Sander [6] mentioned that structural similarities among
distantly related proteins are often preserved in the
process of evolution, but very little similarity at the
sequence level.
There is an interesting problem studied, that is
molecular mimicry. The molecular mimicry problem [7]
is that a protein and a nucleic acid share a similar
substructure, and sometimes it will even extend to
similarity in interaction. Nissen et. al [8] indicated that
the structure of Elongation Factor-G is similar to that
of the complex of Elongation Factor-Tu and tRNA.
Selmer et. al [9] mentioned that Ribosomal Recycling
Factor looks like tRNA. In addition, exploitation of 3D
structural data is a key factor to enhance structure-
based drug design (SBDD), and the prediction of
protein functions and possible active sites in proteins
have become quite popular in SBDD, especially at
front-ends to molecular docking [10] [11] or
alternative active sites are sought otherwise.
This paper is organized as follows. Some related
works are discussed in section 2. The geometric
hashing algorithm and ICP algorithm we use are
detailed in section 3. The experimental results are
provided in section 4 while conclusion is given in
section 5.
2. Previous Work
In general, structure alignment based on 3D
structure has been shown to be NP complete by

Lathrop [12] and so heuristics are used to simplify the
problem. Therefore, better methods for structure
alignment are needed. Fisher et al. [13] used geometric
hashing for a C
α
-only representation of protein
structure, and a follow-up is described in Tsai et al.
[14]. Their method is based on preprocessing and
recognition algorithms of complexity O(n
3
), where n is
the number of residues of interest. Later, Pennec and
Ayache [15] [16] introduced a 3D reference frame
attached to each residue, which reduces the complexity
of recognition to O(n
2
). Shindyalov and Bourne [17]
proposed a method that involves a combinatorial
extension (CE) of an alignment path defined by
aligned fragment pairs (AFPs) rather than the more
conventional techniques which use dynamic
programming and Monte Carlo optimization.
Combinations of AFPs that represent possible
continuous alignment paths are selectively extended or
discarded thereby leading to a single optimal alignment.
Zemla [18] proposed LGA (local-global alignment)
algorithm, where longest continuous sequence is first
found, and then a second step called GDT (global
distance test) is applied. Both longest segment of
residues under selected RMSD (root mean square
distance) and largest set of equivalent residues that
deviate less than a given distance threshold are
obtained. Blankenbecler et al. [19] proposed to use
fuzzy alignment variables and iterative minimization of
a cost function. Milik et al. [20] used graph matching
and represented atoms as nodes and bond distance as
edge labels. The search method is based on
comparison of local structure features of proteins that
share a common biochemical function, and so does not
depend on overall similarity of structures and
sequences of compared proteins.
From the above survey, it is clear that all the above
papers are concerned with proteins, and complexity
reduction in alignment according to features of
proteins or segments of aligned one dimensional
sequence. Therefore, they can not solve the general
molecule alignment problem unless the tools are
modified.
3. Algorithms
In this paper, we propose a tool to align two
molecules based on their 3D structural data. The
alignment problem between two molecules A and B is
solved in two steps: Geometric Hashing and a fine
tuning process. Geometric Hashing globally finds
initial matching of approximately overlapped atoms.
Thus, parts of molecule A can be matched to parts of
molecule B. Secondly, the fine tuning process is based
on local optimization of overlapped parts, and the
Iterative Closest Point (ICP) algorithm is used until the
number of overlapped atoms within a given distance
threshold can not be increased any more.
3.1. Geometric Hashing: Step One
Geometric hashing algorithm is introduced to
structurally align two molecules. Geometric hashing
algorithm is a technique originally developed in
computer vision for object recognition and can easily
be made parallel [21] [22]. In short, the geometric
hashing algorithm is composed of two stages:
preprocessing and recognition. The basic idea is to
store in a database at preprocessing time a redundant
representation of the models by rigid transformation.
By doing so, the representation of the query object
processed at recognition time will present some
similarities with that of some database models.
Matching is possible even when the recognizable
database objects have undergone transformations or
when only partial information is present.
Often the two interesting molecules are both
proteins, so we will illustrate the solution in such a
situation first. For some cases, e.g. molecular mimicry,
two molecules belong to different type, there would be
some variance while calculating, and we will describe
later.
The three atoms N, C
α
and C in each amino acid
form a triangle which uniquely defines the position and
orientation of the amino acid in the three-dimensional
structure of a protein. Since the length of N−C
α
and
C
α
−C are fixed, and N−C
α
−C bond angle is also
changeless. As alignment considered, the
correspondence between two triplets of points in three-
dimensional space is sufficient to uniquely determine a
rigid transformation. With this mechanism, we can
choose a single residue as a basis. A basis is calculated
by the following steps and illustrated in Figure 1(a).
1. Normalize
NC
α
J
JJJJK
to
1
e
J
K
2.
1
2
1
eCC
e
eCC
α
α
×
=
×
J
K JJJJK
J
JK
J
K JJJJK
3.
321
eee
=
×
J
JKJJKJK
There are two phases, preprocessing and
recognition, in the geometric hashing algorithm. To
solve the problem of representation by different
reference coordinates, coordinate information based on
different reference frame of a model is encoded in the
preprocessing phase and stored in a large memory, in
this case, a hash table. The contents of the hash table

are independent of the scene and thus can be computed
offline to reduce the time needed for recognition.
Accessing to the memory is based on geometric
information that is invariant of the object’s pose and
computed directly from the scene. During the
recognition phase, the method accesses the previously
constructed hash table using the indices of the encoded
coordinate information of the input object and finds
their common spatial features.
In the phase of preprocessing, we calculate one
basis for each residue to generate coordinates for each
atom in a protein. In the phase of recognition, we
choose a reference frame of the protein B. For each
different reference frame of protein A in the hash table,
we accumulate the number of matched atoms by
checking whether there are two atoms close enough.
We set a threshold distance MatchThres (MatchThres
= 1 to 2Å is proper), beyond which atoms will not be
considered as a match. If no atoms can be matched
within MatchThres, we assign the score to 0. If there is
an atom within MatchThres, we assign the score to 1.
The process is repeated with each reference frame of
the protein B until all the reference frames of these two
proteins have been tested.
In the case of aligning two different kinds of
molecules, the algorithm is slightly modified while
creating the bases. For each atom whose coordinate is
P, select two atoms connected with the atom, assuming
that the coordinates for these two atoms are Q
1
and Q
2
respectively. The rule for constructing basis is
1. Normalize
1
PQ
JJJJK
to
1
e
JK
2.
12
2
12
ePQ
e
ePQ
×
=
×
J
K JJJJK
J
JK
J
K JJJJK
3.
321
eee
=
×
J
JKJJKJK
and is illustrated in Figure 1(b). The origin of the
new coordinate frame is P. If an atom is connected
with
n atoms, there would be )1( −× nn coordinate
frames made for this atom. In this way, the number of
constructed coordinate frames is too large so that the
execution is not efficient. In order to decrease the
execution time, the criteria for selecting atoms to
create bases is listed in Table 1. Then we calculate two
bases for each residue, while we calculate four bases
for each nucleotide. In proteins, the
“ CA ” atom is
on the backbone and attached with a side-chain, and
the “ CB ” atom is the attached atom. In nucleic acids,
the “ C4*” atom and the “ C3*” atom are both on the
similar position as the “ CA ” atom in proteins. And
“ O4*” atom and “ C2*” are on the similar position as
the “ CB ” atom in proteins. This is illustrated in
Figure 2.
Table 1. The rule for selecting atoms to
construct coordinate frames.
Type of the
molecule
Name of the
atom lie in P
Name of the
atom lie in Q
1
Proteins “ CA ” “ CB ”
“ C4*” “ O4*” Nucleic Acids
“ C3*” “ C2*”
(a) (b)
Figure 1. Calculation of a basis. (a) The protein structure. (b) The general molecule structure.

(a) (b)
Figure 2. A sketch of molecules to explain the rule for coordinate frame construction. (a) Amino
acid. (b)Nucleotide.
3.2. Fine Tuning Process: Step Two
Once the previous process is done by geometric
hashing for global optimization with an output of
approximate alignment, the following process is a fine
tuning process based on local optimization of
overlapped parts. This step is necessary, since the 3D
structural data in PDB always involve sampling error
in X-ray crystallography in determining atom positions.
Furthermore, geometric hashing just provides initial
alignment. Therefore the alignment needs fine tuning,
and so Iterative Closest Point (ICP) algorithm [23] [24]
is chosen. As illustrated in Figure 3, ICP algorithm is
used in this process repeatedly, until the number of
overlapped atoms within a given distance threshold
can be increased no more.
The ICP algorithm proposes a solution to a key
registration problem below: given two three-
dimensional shapes, estimate the optimal translation
and rotation that register the two shapes by minimizing
the mean square distance between them. The algorithm
guarantees that a local minimum of a mean square
objective function is found [23]. In our implementation,
we select 100 rigid transformations that lead to
maximum numbers of overlapped pairs. The results
show that ICP indeed increases the number of atoms
matched.
Figure 3. The flow chart for fine tuning
process.
4. Experimental Results
4.1. The Molecular Alignment Problem
Our tool can be used in solving the comparison of
two molecules that belong to different types. The data
and the problem of molecular mimicry (Figure 4,
Figure 5) are provided by a graduate student Mr. Han
Liang from Professor Laura Landweber’s group in
Dept. of Ecology and Evolutionary Biology, Princeton
University [25].
One data set [8] consists of EFG (Elongation
Factor-G) and EF-tu (the complex of Elongation
Factor-Tu and tRNA), and the orientations of the
original data are almost the same. The other data set [9]
consists of RRF (Ribosomal Recycling Factor) and
tRNA, but they are not in the same orientation
originally. The aligning results of these two data sets
are shown in Figure 6 and Figure 7.
After calculation by our tool, the rotation matrix
between EFG and EF-tu/tRNA is
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
−−
984929.015437.00780061.0
160734.0983483.00832201.0
0638711.00945041.0993473.0
and the translation vector is
(
)
83966.6935684.009762.2
−
−
.
For the case of RRF and tRNA, the rotation matrix
is
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
−−
626458.0293322.0722159.0
776578.0314407.0545962.0
0669089.0902835.042475.0

Figure 4. EFG vs. EF-tu/tRNA complex
(Nissen et. al 1995 shows that the binding to
ribosome is at the same place and orient-
tation.) This picture is from Professor Laura
Landweber’s group of Ecology and Evolu-
tionary Biology Dept. Princeton University,
and the orientation is manually selected.
Figure 5. RRF vs tRNA (Selmer et. al 1999
shows that the binding to ribosome is at
different place and orientation.), and again
the orientation is manually selected.
and the translation vector is
()
2718.331501.656722.36−
4.2. Comparison with Other Alignment Tools
In order to compare with other tools, we will use
the same set of proteins as in the paper of
Blankenbecler et al. [19]. Note that other protein
alignment methods usually use the knowledge of
matched 1D sequence alignment for proteins, and they
are optimized for proteins only focusing on backbone
atoms C
α
matching. Our tool does not have this
assumption, and will work for arbitrary molecules,
including tRNA. Still, for comparison purpose, we use
the same set of six proteins. Figure 8 shows that our
tool is better compared to other methods, where Figure
8(a) is reported from Blankenbecler’s [19], in which
Yale [26], Dali [27] [28], CE [17] and Lund [19]
methods are compared, while Figure 8(b) is from our
tool as compared to data in Figure 8(a).
The reasons why our method is better are
1.
Given a fixed RMSD for pairs of matched
atoms, our method has the most number of
backbone C
α
atoms;
2.
Given fixed number of matched C
α
, our method
has the lowest RMSD.
In terms of computation cost, the major cost is in
the first step, the geometric hashing. In the case of
proteins, the coordinate frames are generated from the
amino acid C
α
atoms only, and thus the computation
cost is low. For the six pairs of target proteins, all
alignment calculation is done ranging from 6 seconds
to 47 seconds. Table 2 shows the computation time on
a Pentium-4 3GHz PC.
In the case of molecules such as RNA and DNA,
the nucleic acid has a carbon ring in its base, and
therefore the number of possible coordinate frames
tends to be much more than that of proteins. Certainly,
the computation time is longer. In the case of RRF vs.
tRNA, where there are over 1000 atoms in tRNA, the
computation time is around 24 minutes, while in the
case of EFG vs. EF-tu/tRNA complex (over 4000
atoms), the computation time can be as long as 36
hours on the same 3 GHz PC. Even so, our tool can
still solve this problem, which is a very important
problem called "molecular mimicry". As far as we
know, our method is the first one to solve this kind of
problems, because our algorithm is sequence
independent, and does not use the knowledge of 1D
sequence similarity in molecule pairs.
5. Conclusion
A novel tool is developed to align two molecules
based on 3D structural data. In contrast to other
algorithms, it takes more computation time to align
two molecules by our tool. However, other tools might
be restricted to align two proteins. The experiments are
conducted based on the data from the PDB and
demonstrate that the proposed tool is useful and
versatile.
The first experiment is the molecular alignment
problem. Given two molecules, our tool will generate
the rotation matrix and translation vector so that the
above two molecules are optimally aligned. In our
experiments, the results are the same, no matter where
we randomly place the molecules in a different
location with different orientation.

Figure 6: Alignment of two molecules using our tool for EFG vs. EF-tu/tRNA complex, where the
atom number is over 4000 and the computation time is about 36 hours on a Pentium-4 3GHz PC.
Figure 7: Alignment of two molecules using our tool for RRF vs. tRNA, where the atom number
is over 1000 and the computation time is about 24 minutes on a Pentium-4 3GHz PC.

(a) (b)
Figure 8: Alignment results for a set of protein pairs in terms of RMSD of matched atom pairs
and number of aligned atoms (N). In this figure, (a) is from Blankenbecler et al. fuzzy alignment
method. The results from Yale (red squares), Dali (green triangles), CE (blue circles), and Lund
method (solid lines) are also given in their paper. (b) is from our tool as a comparison. It shows
that our results are better as compared with other methods.
Table 2: Computation time of alignment of six pairs of proteins, where MatchThres means the
threshold used in initial geometric hashing, while the other columns are in seconds.
MatchThres (Å) 8DFR-4DFRa 1MBD-1MBA 1TIE-4FGF 1CID-2RHE 7FABl2-1REIa 1FXIa-1UBQ
1.0 7 5 4 3 2 1
1.5 9 6 5 5 2 1
2.0 11 7 6 5 3 1
2.5 13 10 8 7 3 2
3.0 18 12 9 9 4 3
3.5 22 16 13 11 5 3
4.0 30 20 15 13 7 3
4.5 37 26 20 18 8 6
5.0 47 34 24 22 10 6
In the second experiment, several protein pairs are
used to compare the results with four popular
alignment tools, namely Yale [26], Dali [27] [28], CE
[17] and Lund [19] methods. Our tool performs the
best in terms of RMSD and number of matched atom
pairs.
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