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EURASIP J Bioinform Syst Biol. 2007; 2007: 60723. Published online 2007 October 31. doi: 10.1155/2007/60723. | PMCID: PMC2205990 |
Compressing Proteomes: The Relevance of Medium Range Correlations Dario Benedetto,1 Emanuele Caglioti,1 and Claudia Chica2* 1Dipartimento di Matematica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, Roma 00185, Italy 2Structural and Computational Biology Unit, EMBL Heidelberg, Meyerhofstraße 1, Heidelberg 69117, Germany Received January 14, 2007; Revised May 28, 2007; Accepted September 10, 2007. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the nonrandomness of proteome sequences by analysing the correlations that arise between amino acids at a short and medium range, more specifically, between amino acids located 10 or 100 residues apart; respectively. We show that statistical models that consider these two types of correlation are more likely to seize the information contained
in protein sequences and thus achieve good compression rates. Finally, we propose that the cause for this redundancy is related to the evolutionary origin of proteomes and protein sequences. Protein sequences have been considered for a long time as nearly random or
highly complex sequences, from the informational content point of view. The
main reason for this is the local complexity of amino acid composition, that is, the type and number of amino acids found in a sequence segment, especially
inside the globular domains [ 1]. This complexity could be related to the so
called randomness of coding sequences in DNA, already pointed out in a
pioneering work [ 2] and explained by evolutionary models [ 3]. Studies on
protein sequence compression show that proteins behave as sequences of
independent characters and have a very low compressibility, around 1%
[ 4]. The ordered set of protein sequences belonging to one organism,
the proteome, was also considered to be not compressible due to this
little Markov dependency [ 5]. Improvements are obtained by [ 6, 7].
However, later studies [ 8– 10] suggest that proteomes contain
different sources of regularities, and can be compressed to rates around
30%. For a relevant discussion on the validity of these results see Cao et al.
[ 7]. In this work, we focus on the statistical study of proteome sequences, using the concept
of entropy brought into information theory by Shannon [ 11]. The Shannon entropy is
related to the amount of information of a sequence emitted by a certain source. The
entropy  of a sequence is the limit of the average amount of information per character, when
the length of the sequence tends to infinity. In particular, for a finite sequence of
length  ,
the informational content in bits is approximately Lh and so Lh is the
minimum length in bit of any sequence that contains the same information. In this
way Lh provides a theoretical lower bound for the sequence’s compression. A compression
algorithm is intended to code a sequence into a shorter one, from which it is
possible to obtain unequivocally the former. In practise, one cannot compress at
a rate equal to the Shannon entropy for the given sequence. Nonetheless, it
is possible to approximate such a limit, using an efficient compression
algorithm. Statistical compression algorithms achieve their goal by assigning shorter code
words to the most probable characters; their efficiency depends on the accuracy
of the model used to estimate each character’s probability. Models try to take
advantage of the correlations between characters considering, for example, how
the preceding characters, that is, the character’s context, determine the
probability of the next one, as in the prediction by partial matching (PPM)
scheme [ 12]. Most successful algorithms for proteome compression are based on the
identification of duplicated sequences or repeats. The compress protein (CP)
algorithm [ 5], for example, considers that duplicated sequences in proteomes are
similar but not identical because of mutation and evolutionary divergence. CP
uses a modified PPM that includes the probability of amino acid substitutions
when estimating each residue probability. The ProtComp algorithm [ 8] optimises
the use of approximate repeats by updating the amino acid substitution matrix
as the repeated similar blocks appear along the sequence. The context-tree
weighting (CTW) [ 13] is another context-based method that has been
applied for biological sequence compression. In [ 6] the authors present a
CTW-based algorithm that predicts the probability of a character by
weighting the importance of short and long contexts considering as well the
occurrence of approximate repeats or palindromes in those contexts. The XM
[ 7] is a statistical algorithm which combines, via a Bayesian average,
the probability of an amino acid calculated on a local scale with the
probability of that same residue being part of a duplicated region of the
proteome. Nonstatistical approaches, based on the Burros-Wheeler transform (BWT) [ 9],
have also been used for identifying overlapping and distant repeats in proteomes,
and efficiently use them in compression. Even simpler models, that rely on a
block code representation of the protein sequences [ 10], have proved to be
successful in some cases. All the algorithms commented above put into evidence the existence and importance
of redundancy in proteome sequences. Here we present a purely statistical study of 8 eukaryotic
and prokaryotic proteomes. Firstly, we analyse the correlation function of the whole
sequences and find evidence of medium range correlations, between amino acids
located 100 residues apart. Then we calculate the amino acid correlations considering
the protein boundaries and identify the role of the intra/interprotein
scale in determining the medium range correlations. Furthermore, we
generate groups of amino acids using their pair correlations at distance 100, that
reveal the structural meaning of the medium range correlations. Using the results
of proteome correlations, we propose a statistical model for the distribution of amino
acids in 4 proteomes: Haemophilus influenzae (bacteria), Methanococcus jannaschii
(bacteria), Saccharomyces cerevisiae (eukarya) and Homo sapiens (eukarya), and
we estimate their compression rate to compare our results against previous
works. The sources of nonrandomness studied fall into two scales: the medium range
correlations between amino acids of the same and neighboring sequences,
at distances of order 100, and the short range Markovian correlations
between the contiguous residues up to distance 10. Previous studies [ 9] show
that proteomes present repeated subsequences at
very long distances (50–300). In
this article, we do not consider these long-range correlations of the order of the
proteome length. Protein length range correlations are in agreement with the
process of sequence duplication, as it has been previously suggested for long-range correlations [ 9]; in addition to that, we show that they also contain
information about the three-dimensional structure of the proteins. Short range
correlations might instead relate to the local constraints on amino acid
distribution due to secondary structure requirements. 2. RESULTS and DISCUSSION For our statistical analysis, we used the proteomes of 4 prokaryotic
and 4 eukaryotic organisms shown in Table 1. They were retrieved from the database of
the Integr8 web portal [ 14], with exception of the Hi, Mj, Sc, and Hs
proteomes that were obtained from the protein corpus in [ 15], for the sake
of comparison of our compression rate results with previous studies on
the same proteomes. The proteomes are not complete (in particular the
version of Hs in the protein corpus) but they represent a natural set of
proteins where the redundancy has a biological meaning. It is important to
remark that the sequence of the proteins in the proteome files of the
Integr8 database is not the natural one. Those files are not useful for our
analysis. Nevertheless, using the additional information available in the
database, it is possible to order the proteins as they are found in the
chromososmes. The proteome files of the protein corpus do not present this
problem, but the sequence of the proteins is not available. Therefore, for the
analysis shown in Table 2 and in , we have used the version of Hi,
Mj, Sc in the Integr8 database. For the same reason, the data for Hs is
missing in Table 2 since the protein order is not obtainable at the Integr8
site. | Table 2Intra- and interprotein correlation. Intraprotein
correlation is always higher than interprotein
correlation, and correlation between matching halves
(−−)
is higher than that of not corresponding halves
( ). |
2.1. Correlations As a first approximation to the general trends in residue distribution, we study the
cooccurrence of amino acids. More precisely, we calculate the pair correlations at
different distances, that is, the average number of times equal residues a appear at
distance k along the whole sequence
with
where N is the
sequence length,  is the characteristic function of finding residue a at position i, and  is the relative
frequency of amino acid a in the proteome. According to this definition, a positive correlation means that, for a
distance k,
the number of pairs of equal amino acid is more frequent than expected due to
their frequency in the proteome. The resulting correlation function for the  proteomes we studied () shows that eukaryotic sequences
have stronger correlations than prokaryotic ones. Moreover, for all the
proteomes, the correlation remains positive at a medium range, for values of k bigger
than  or  ,
depending on the proteome. We notice that the natural order of proteins in the
proteomes, given by the succession of genes in the chromosomes, is relevant:
when we randomly permute proteins, the medium range correlations are lost,
both in eukaryotes and prokaryotes. The medium range correlations imply that, in proteomes, the amino acid
distribution of neighboring proteins tends to be more similar than that of
distant ones. This fact can be related to the process of duplication, recognied as
the dominant force in the evolution of protein function [ 16]. As protein
repeats have been related to duplication at different scales (genome,
gene, or exon) [ 17], it is possible that the amino acid patterns responsible
for the observed medium range correlation have the same evolutionary
origin. Due to the correlation definition used, the medium range correlations could be
caused either by pairs of amino acids belonging to the same protein, or to different
ones. Therefore, we split the nonlocal correlation into two groups and analyse
them separately: interprotein correlations (between 2 contiguous proteins) and
intraprotein correlations (inside the same protein sequence). In Table 2, we
present the results for the intraprotein correlation between the two halves of the
same protein and the interprotein correlation between corresponding and noncorresponding halves of two contiguous proteins: first half with first half
(  ) and second half
with first half (  ). These correlations are defined as follows. Let  be the number
of proteins, let  and  be the relative frequency of
the residue a in the first and
the second half of the ith
protein, respectively, and let  be the corresponding mean value. We define for instance, We also define The intraprotein correlation is The two interprotein correlations are The correlation values in Table 2 have the same trend for all the proteomes:
intraprotein correlation is always higher than interprotein correlation.
The correlation defined by means of  are different from the traditional correlation  which is the correlation
of the symbol a at distance k,
where k is the number of residues: we have calculated the correlation function
of the frequencies of the amino acids at the distance of one protein. In
, we also analyse how the interprotein correlations between
matching and nonmatching protein halves vary with the number k of
proteins separating the two halves. We compare As an extension of the results in Table 2, we find that the correlation between
matching halves is kept higher than that of noncorresponding halves along the
proteome. Analogous results to Table 2 and hold for second-second and
first-second halves. Gene duplication can explain both the existence and order dependence of
interprotein correlation, but it is not enough to justify why intraprotein
correlations remain high, because high interprotein correlations can also appear
in a low intraprotein correlations context. Indeed, the presence of intraprotein
correlations indicates a nonrandom distribution of amino acids at a protein
length scale. This nonrandomness can be related to segmental duplication, that is, duplication of segments inside the same protein; likewise, it can reflect
the maintenance of amino acid patterns during the protein divergence that
follows gene duplication as a consequence of the structural constraints imposed
upon protein sequences. As an example, extensive searches of protein databases [ 18] reveal
the high frequency of tandemly repeated sequences of approximately  amino acids, ARM and HEAT, in eukaryotic proteins. Moreover, those repeats
present a core of strongly conserved hydrophobic residues even when the other
residues start to differ at several other positions. The evidence obtained from the correlation analysis does not allow to clarify the
nature of the structural constraints measured: do they reflect the modular
repetition of secondary structure elements, caused by duplication or, perhaps,
they depend on the conservation of higher order tertiary structure units like
domains? We try to address this question by defining amino acid groups as
explained in the next section. 2.2. Grouping of amino acids In a previous study [ 4], the complexity of large sets of nonredundant protein
sequences was measured using a reduced alphabet approximation, that is,
using groups of amino acids defined by an a priori classification. The
Shannon entropy was then estimated from the entropies of the blocks of n-characters.
The authors did not find enough evidence to support the existence of short range
correlations between the amino acids of protein sequences. Conversely, given the above evidence of medium range correlations in proteome
sequences, we build groups of correlated amino acids using the correlations between the  amino acids.
We calculate  ,
the correlation between all amino acid pairs  at distances k, in the same way
we calculate  in the previous section:
A quick look at the resulting  matrix for  (), which presumably includes both intraprotein and interprotein
correlation, puts in evidence that the signs of the matrix elements, and
thus the positive and negative correlations, are not distributed randomly
among residues but, instead, in a grouped fashion: some amino acids
present positive or negative correlations with the same subset of residues. Then, we construct groups of amino acids in such a way that they maximise
the positive medium range correlation; in practical terms it means that
amino acids which are more likely to appear at distances of order 100
would be grouped together. For a given partition of the set of amino acids in  groups,
we calculate the sum of the correlation function between any pair of residues  belonging to a same group. More precisely, groups are obtained by maximising
the following quantity:
which is function of a partition  of the amino acids in  disjoint sets  .
Due to the huge number of possible choices for the groups, we maximise
this value using a simulated annealing algorithm. This is a Monte
Carlo algorithm used for optimisation [ 19]. For a given partition G, we construct a
new partition  choosing at random a residue and changing its group. If  , the
algorithm accepts the new partition. Iterating this procedure we would reach a
local maximum which may not be the absolute maximum. In order to avoid being
trapped in a local maximum, the algorithm accepts, with a small probability P, a new partition  for which  . The value of this probability P slowly decreases to zero as the number of iterations increases in such a
way that the convergence of the algorithm to the absolute maximum of F is
guaranteed. The number and the structure of the groups chosen have the highest value of  and
represent an equilibrated partition of the 20 amino acids, that is, groups
with only one element are not accepted.
The idea behind our grouping scheme is to simplify the amino acid pattern
mining by taking advantage of their synonymous relationships. It is well
known that mutations between amino acids sharing geometrical and/or
physico-chemical properties are the basis of neutral evolution at a molecular level
[ 20]; this fact also explains why there is not a one-to-one relationship between
protein sequences and structures [ 21]. Moreover, structurally neighboring
residues have been found to distribute differentially (proximally/distally) in
the protein sequences, depending on their physico-chemical properties
[ 22]. Indeed, the groups defined from the pair correlations at a medium range
( Table 3) almost correspond with the natural classification based on their
physico-chemical properties: hydrophobic, polar, charged, small, and ambiguous.
In particular, the fact that hydrophobic amino acids group together allows us to
think that the correlation function is gathering some of the three-dimensional
information contained in the protein sequence, more precisely tertiary structure
information, as hydrophobic interactions are considered the driving forces of the
protein folding process [ 23]. | Table 3Groups of amino acids determined by maximisation of the positive
medium range correlation. Amino acids that are more likely to appear at 200 residues distance are grouped together.
|
Therefore, the reason why intraprotein correlations remain high is not
only related to the repetition of secondary structure units, but is also
the conservation of the amino acids responsible for the protein tertiary
structure. Beside this, it is important to notice that, even if the amino acid usage in
eukaryotes and prokaryotes is very similar [ 24], the amino acid correlations are
not, as they collect part of the structural information, contained in the sequences.
The number of groups is also different: 3 for H. influenzae and M. jannaschii, 4 for
S. cerevisiae and H. sapiens. This could indicate a higher interchangeability of
residues in some proteomes, but further analysis is needed to confirm this
hypothesis. 2.3. Sequence entropy estimation In order to quantify the capability that a statistical model has to identify the nonrandomness of a sequence, one can use it to construct an arithmetic coding
compressor [ 25]. We estimate the compression rate of such a compressor with the
sequence entropy
using the model to calculate the probability  of
character  at position i.
The better is the model, the lower is the estimated value of the sequence entropy. We
construct three models to estimate the probability of each character, considering the
previous ones and taking into account both short and medium range correlations.
For each model, we find parameters that minimise the sequence entropy. The  value
obtained is taken as an estimate of the compression rate of a running arithmetic
codification [ 25] of the proteomes and is used to compare our results with other
compression algorithms ( Table 4).
| Table 4Compression rate in bit per character for the studied
proteomes. One-character entropy is the entropy of the sequences
considering that their residues are independently distributed. |
Previous works on protein sequence compression like [ 5] are based on
short range Markovian models. In those models, the probability of each
amino acid is calculated as a function of the context in which it appears,
considering the frequency with which this amino acid happens to be after the  previous residues. Following this idea, we start our statistical description of proteome sequences taking
into account the information given by the neighboring residues using a variation
of the interpolated Markov models [ 26]. In order to predict the probability of the ith character, we consider the contexts up to a length  (number of contexts) that precede it, that is, the substrings  for  . For any character a, we count the number  of previous occurrences
of the substring  .
The conditional frequency of finding character  after the
context  is obtained dividing by the sum over all amino acids b at
position i: Our model 1 predicts the probability of character a at
position i with
We remark that the main difference between our short range approach and
CTW is that we give a weight to the different contexts, while in [ 6] a weight is
given to their corresponding conditional probabilities. We find that the most
informative positions were the previous 8; this length is in qualitative
agreement with the results found in [ 6]. Model 1 in Table 4 indicates
the results obtained considering only the short range correlations for  . The model depends on the parameters  that
are optimised, using standard algorithms for minimisation, in order to achieve
the best estimate of the compression rate. This “entropy minimisation” stage is
very time expensive. In a real compression procedure, those parameters should be
specified and therefore would contribute to the estimated entropy. In our case
this contribution is negligible. The short range correlations support the existence of periodic patterns in protein
sequences. They can be caused by the alternation of alpha-beta secondary
structure units, as argued in other works on latent periodicity of protein
sequences [ 27, 28]. From the point of view of protein sequence evolution, the
short range parameters can also reflect the existence of constraints on the
distribution of residues. Protein sequences are modified by mutation, but still
have to cope with folding requirements that determine a nonrandom positioning
of key residues, depending on their geometrical and physico-chemical properties.
In fact, structural alphabets derived from hidden Markov models denote that
local conformations of protein structures have different sequence specificity
[ 29]. The intra/interprotein correlations identified in previous sections suggest that
the frequencies of the single residues has nonnegligible fluctuations on the
medium range. We take into account these fluctuations in our second model
(model 2 in Table 4):
Here we added This quantity is proportional to the frequency of the amino acid a
in the subsequence
of length L,
with L a distance of medium scale, starting from the position  . The factor  guarantees
that  , so that it
increases with i in the same way as the other terms of the sum (e.g.,  ). The parameter  is optimised as  . The optimal
values for  found during the entropy minimisation stage are  for Hi,  for Mj,  for Sc,
and  for Hs. Finally, in model 3, we use the groups found in Section 2.2 (see Table 3). In
particular, a contribution to the probablity of a given residue is obtained by
computing the probability of the residue to belong to a certain group
and then the conditional probability of the residue once the group is
given is
where  is the
group of a,  is the relative
frequency of a in its group, as measured up to the position  ,
and
For this model, the optimal values of the parameter L are  for Hi,  for Mj,  for Sc,
and  for Hs. As one can see in Table 4, the capability of our statistical model to represent the
nonrandom information contained in proteomes is comparable to those models
that consider repeated amino acid patterns at both short and medium scale [ 6, 7]. The improvement in the performance of models 2 and 3 is due to the fact that
they identify the short range correlations and separate them from the
fluctuations of amino acid frequencies at a protein length range. This
demonstrates that both correlation types are informative and that the statistical
significance of repetitions at those scales is enough to model the amino acid
probabilities. The compression rate achieved when the medium range correlations are modelled
with the frequency of amino acid groups (model 3) is almost equivalent to the
compression rate of model 2. From a biological perspective it indicates that
groups of amino acids are meaningful, and that the redundant information at
medium scale has a structural component might be coming from the
three-dimensional structure constraints. According to our results, there is an important difference in the compressibility
rates of the eukaryotic and prokaryotic proteomes which is in agreement with the
correlation function in . The sequences of S. cerevisiae and H. sapiens are
more redundant, and thus more compressible, than those of H. influenzae and
M. jannaschii; correspondingly, the correlation functions of Sc and Hs remain
positive for longer distances than Hi and Mj. This additional redundancy could
be related to the presence, in eukaryotic proteomes, of paralogous proteins with
very similar distribution of synonymous amino acids, but different function.
There is evidence suggesting that paralogous genes have been recruited during
evolution of different metabolic pathways and are related to the organism
adaptability to environmental changes [ 16]. On the other hand, the lower
compressibility of the Hi and Mj proteomes is in agreement with the
reduction of prokaryotic genome size as an adaptation to fast metabolic rates
[ 30, 31]. In this article, we show that the correlation function gathers evolutionary and
structural information of proteomes. Even if proteins are highly complex
sequences, at a proteome scale, it is possible to identify correlations between
characters at short and medium ranges. It confirms that protein sequences are
not completely random, indeed they present repeated amino acid patterns at
those two scales. The alternation of secondary structure units can determine the
local redundancy. This was already known and generally modelled using Markov
models. In our opinion, sequence duplication is a reasonable explanation for the
interprotein correlation. However, it does not account for the intraprotein
correlations; this can instead be related to the maintenance of the amino acid
patterns responsible for the three-dimensional structure, as the segregation
between hydrophobic and polar amino acids indicates. More elaborately, the
sampling of the space of structures during proteome evolution is determined by
the duplication processes but it is highly constrained by the structural and
functional requirements that protein sequences have to meet inside a living
system. Prokaryotic proteomes show lower correlation values, especially for distances
under  residues, and a smaller compressibility than eukaryotic proteomes. These
characteristics point at a higher redundancy of eukaryotic proteome sequences,
and suggest that the increase of proteome size does not imply de novo
generation of protein sequences, with completely different amino acid
distribution. The authors would like to thank Toby Gibson for reading and commenting the
manuscript and the reviewers for their constructive criticism that helped to
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