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Biophys J, January 2000, p. 136-149, Vol. 78, No. 1
and
*TCE Laboratory, Istituto Superiore di Sanitá, 00161 Roma,
Italy; Department of Biochemical Sciences, University of
Roma "La Sapienza," 00185 Roma, Italy; and
Department of Molecular Biophysics and Physiology, Rush
University, Chicago, Illinois 60612 USA
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Two computational methods widely used in time series analysis were applied to protein sequences, and their ability to derive structural information not directly accessible through classical sequence comparisons methods was assessed. The primary structures of 19 rubredoxins of both mesophilic and thermophilic bacteria, coded with hydrophobicity values of amino acid residues, were considered as time series and were analyzed by 1) recurrence quantification analysis and 2) spectral analysis of the sequence major eigenfunctions. The results of the two methods agreed to a large extent and generated a classification consistent with known 3D structural characteristics of the studied proteins. This classification separated in a clearcut manner a thermophilic protein from mesophilic proteins. The classification of primary structures given by the two dynamical methods was demonstrated to be basically different from classification stemming from classical sequence homology metrics. Moreover, on a more detailed scale, the method was able to discriminate between thermophilic and mesophilic proteins from a set of chimeric sequences generated from the mixing of a mesophilic (Rubr Clopa) and a thermophilic (Rubr Pyrfu) protein. Overall, our results point to a new way of looking at protein sequence comparisons.
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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The relationship of the physiological role of
proteins with their primary structure is a crucial issue in molecular
biology. It is well known that both the 3D structure and the
physiological role of proteins is heavily dependent upon the particular
linear arrangements of amino acid residues along polypeptide chains, or
primary structures (Anfinsen, 1973
; Dayhoff et al., 1978; Sweet and
Eisenberg, 1983). Despite the recent progress in the field, fully
documented in the series of Critical Assessment of Techniques of
Proteins Structure Prediction (CASP) meetings (Murzin and
Patthy, 1999), attempts to derive general rules in predicting 3D
structure and physiological features based on protein sequences cannot
be considered fully satisfactory as yet (Micheletti et al., 1998). Thus, we decided to tackle this problem with a local approach: instead
of looking for general rules, we tried to develop a methodology able to
derive local structure/sequence relationship models, with the ultimate
goal of predicting physiological properties by means of sequence
information within properly selected sets of proteins. This approach is
similar to one used in medicinal chemistry (Hansch, 1993; Martin,
1981), where quantitative models for predicting pharmacological
properties of organic molecules from their physicochemical and
structural properties have been routinely used for the past three
decades (Hansch et al., 1962; Hansch and Leo, 1995). The success of
quantitative structure activity relationship (QSAR) models has been
demonstrated (Martin, 1981; Hansch and Leo, 1995; Hansch et al., 1996)
to be closely linked to the use of strictly congeneric molecules, i.e.,
of the same class.
From an operational point of view, a QSAR procedure is based on the
correct selection of two components in a prediction model: 1) a
meaningful set of "x" variables (regressors of the
model, descriptors spanning the physicochemical space in which the
molecules are located); and 2) an adequate statistical technique to
quantitatively tackle the problem of both physicochemical space
description and biological activity prediction (Martin, 1981; Franke,
1984). In the present case, the above elements are 1) hydrophobicity,
as the physicochemical descriptor of amino acid residues, and 2) time
series analysis as a general strategy to describe the proteins' primary structures. [We note that from a practical standpoint, spatially ordered series are equivalent to time-ordered series for the
purpose of analysis; see (Zbilut et al., 1998a)].
The relevance of hydrophobicity for protein folding is well known
(Anfinsen, 1973; Li et al., 1997; Micheletti et al., 1998; Sweet and
Eisenberg, 1983). Additionally, hydrophobicity is the only
physicochemical feature that displays a nonrandom ordering along
protein sequences (Weiss and Herzel, 1998; von Heijne, 1982), and may
thus be considered the best candidate for application of time series
analysis methods. The amino acid residues along protein chains were
coded in terms of their hydrophobicity, expressed as log P,
P being the partition coefficient between octanol and water
(Franke, 1984). Thus, our analysis focused on the hydrophobicity series
with the idea of using hydrophobicity as the "semantics" attached
to the residues' ordering along a chain. [The semantic character of a
physicochemical property should reflect both energetic interchange with
elements present in the environment (e.g., water molecules) and
entropic rearrangements induced in them.] An additional goal was to
check whether our approach was able to generate different, complementary information for protein classification, with respect to
standard sequence comparison methods (Pearson and Lipman, 1988; Wilbur
and Lipman, 1983; Thompson et al., 1997).
With respect to computational aspects, the ability of recurrence
quantification analysis (RQA) to deal with a sequence/function relationship problem has been recently demonstrated (Zbilut et al.,
1998b), while almost at the same time, Mandell and his co-workers (1997; 1998; Selz et al., 1998) reported that singular value
decomposition (SVD) in concert with spectral analysis might be able to
provide useful structural 3D information from sequence data. Thus, we combined the two approaches to obtain a global representation of
hydrophobicity patterns along a sequence. These two main methods were
supplemented with three other relatively simple descriptors, namely 1)
standard deviation (SD), 2) absolute value of the Pearson correlation
coefficient between adjacent residues (R), and 3) algorithmic
complexity of the series (Kaspar and Schuster, 1987) as estimated by
the Lempel-Ziv algorithm (LZ).
The entire set of descriptors was filtered by principal component
analysis (PCA) (Harman, 1976) in order to obtain a set of orthogonal
axes on which to project the studied sequences (dynamical space). [We
recognize the essential mathematical equivalence between SVD and PCA.
In the present context, we use the term PCA to distinguish this step in
the algorithm from the SVD plus spectral analysis step.] The chosen
"congeneric series" of proteins consists of 19 rubredoxins (Sieker
et al., 1994), and corresponds to all the bacterial rubredoxins whose
primary structures were known at the time of our analysis. The
biological feature to be predicted is the exceptional stability of the
rubredoxin from Pyrococcus furiosus, a thermophile bacterium
living at very high temperatures.
Finally, the thermostability of rubredoxins was also investigated at a
much finer level of detail by using our approach to evaluate the
results by Eidsness et al. (1997), who synthesized six chimeric
proteins originating from a thermophilic sequence and a mesophilic one.
These authors observed the splitting of the six chimeric proteins into
a thermophilic subset and a mesophilic subset, although a specific
mechanistic explanation for this behavior could not be found (Eidsness
et al., 1997). Even under such stringent sensitivity requirements, the
predictive abilities of our approach were successful.
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MATERIALS AND METHODS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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The biochemical problem
Rubredoxins are probably the simplest members of the ubiquitous
and huge family of redox metalloenzymes (Sieker et al., 1994) and
consist of a relatively short polypeptide chain (~53 AA) endowed with
a prosthetic group in the form of a ferrous/ferric ion tetrahedrically bound to four cysteine (Cys) residues. Even though their exact metabolic role in anaerobic cells has not yet been fully clarified, their structural features are fairly well known, and in Fig.
1 A the primary structures of
19 bacterial rubredoxins are reported: they are quite similar, and
>20% of the residues are strictly conserved, among which are the four
Cys residues of the active site and the five aromatic residues that
constitute the hydrophobic core of the proteins. Such a high level of
homology, however, does not find a match in the huge spectrum of
thermal stability of the bacterial strains from which they are
extracted. In particular, it has been impossible up to now to find a
rationale, on the basis of the primary as well as of the tertiary
structures, for the fact that the rubredoxin from Pyrococcus
furiosus (Rubr Pyrfu), which lives normally at 90°C,
has a half-time of thermal denaturation of 400 h at 92°C, as
compared to 6 h of Clostridium pasteurianum (Rubr
Clopa) rubredoxin, which is the most similar to it in terms of 3D
structure. In the same figure the phylogenetic tree of rubredoxins corresponding to their best alignment and the structure of the chimeric
rubredoxins obtained by Eidsness et al. (1997) starting from Rubr
Clopa and Rubr Pyrfu are also reported.
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Data analysis
Dynamical methods
Each sequence was coded in terms of the amino acid hydrophobicities (Franke, 1984). The numerical discrete series
corresponding to the protein sequence was then submitted to a 4D
embedding by the method of delays (Broomhead and King, 1986). The
embedding procedure consists in building an n-columns matrix
(in our case n = 4) out of the original linear array,
by shifting the series by a fixed lag. For example, given the series
10, 11, 21, 32, 41, 35, 40, 19... the corresponding 4D embedding
space at lag of 1 (the discrete character of amino acid sequences
dictates this choice) is:
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at the same timeon a sufficiently long
series (Broomhead and King, 1986; Webber and Zbilut, 1994). Notice that
the last n values are eliminated from the analysis as an
obvious consequence of shifting the series for the embedding. Moreover,
the four-residues window was demonstrated (Strait and Dewey, 1996)
through the application of a formalism derived from information theory,
to constitute an upper limit for the information content of protein
sequences. [We note that Kumosinski and Liebman (1994) have previously
explored the use of a somewhat related methodology.] After the common
step of building an EM, RQA and SVD diverge. While RQA, being based on
the EM rows, assumes a local viewpoint over the time series (Giuliani
et al., 1998; Zbilut et al., 1998a) SVD, being based on EM columns and dealing with average regularities, provides a global view of the series
(Broomhead and King, 1986; Mandell et al., 1997). The former and the
latter method are then particularly sensitive (Trulla et al., 1996),
and relatively insensitive, respectively, to small perturbations and
provide a complementary view of the autocorrelation structure of the
time series (Zbilut et al., 1998a).
RQA-based descriptors. RQA was first introduced in
physics by Eckmann et al. (1987) as a purely graphical technique (see
the Appendix). Subsequently, Zbilut and Webber (1992) enhanced the
technique by defining five nonlinear quantitative descriptors of
the recurrence plot that were found to be diagnostically useful in
the quantitative assessment of time series structure in fields ranging
from molecular dynamics to physiology (Giuliani and Manetti, 1996;
Webber and Zbilut, 1994; Faure and Korn, 1997).
The RQA descriptors used in this work are:
| 1. | MDIST. Average Euclidean distance between the rows of EM; |
| 2. | REC (percent recurrence). This measure quantifies the fraction of the plot filled by recurrent points. It corresponds to the fraction of recurrent pairs over all the possible pairs of epochs or, equivalently, to the fraction of pairwise distances below the chosen radius among all the computed distances; |
| 3. | DET (percent determinism). This is the percentage of sequential recurrent points that form diagonal line structures in the distance matrix. DET corresponds to the amount of patches of similar hydrophobic/hydrophilic characteristics along the sequence; |
| 4. | ENT (entropy).The entropy is defined here in terms of the Shannon-Weaver formula for information entropy computed over the distribution of length of the lines of recurrent points and measures the richness of deterministic structures of the series; |
| 5. | MAXLINE (maximal line).This index is simply the length (in terms of consecutive points) of the longest recurrent line in the plot, and is inversely related to the largest positive Ljapunov exponent. |
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) and we
refer to other papers for its implementation in the case of
hydrophobicity plots (Mandell et al., 1997, 1998; Selz et al., 1998).
Here it is sufficient to say that the technique involves the
computation of the first three eigenfunctions (principal components) of
the EM and the subsequent projection of the original series on the component space. The projection is then submitted to a complex pole
maximum entropy spectral analysis.
In order to derive quantitative descriptors of these spectra, oblique
principal components analysis (OPC, VARCLUS procedure in SAS)
(Anderberg, 1973; Harman, 1976) was applied to the digitized SVD
spectra of the 19 sequences under study (see the Appendix). OPC
subdivided the 19 spectra into four clusters. Consequently, for each
sequence, four coefficients (SP1-SP4) were computed, which correspond
to the Pearson correlation coefficients of the respective spectrum with
the four clusters that provide a global description of spectral shape.
SP1-SP4 were used to parameterize the spectral information.
Other sequence descriptors. The SD of the residue
hydrophobicities, the absolute value of the Pearson correlation
coefficient (R) of the hydrophobicity of adjacent residues, and the LZ
parameter (Kaspar and Schuster, 1987) of the sequences constituted the
last three descriptors of the sequence space (Table
1). SD measures the relative
heterogeneity of the amino acid hydrophobicity, and is a purely
statistical (shuffling-resistant, order independent) index; in
contrast, R is order-dependent and estimates the short-range regularities in the sequence. The LZ index is an easily calculable estimate of the algorithmic complexity of a symbolic sequence (Kaspar
and Schuster, 1987), and is strictly dependent upon the ordering of
amino acids in the primary structure (see the Appendix).
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Sequence homology methods
The final output of any quantitative sequence comparison method is a distance matrix whose elements contain the estimated divergence between the sequences in the corresponding row and column (Pearson and Lipman, 1988). A popular graphical representation of such matrices,
especially useful to visualize evolutionary relationships, is in the
form of dendrograms (Sneath and Sokal, 1973). The distance matrices
relative to symbolically coded primary structures used in this work
have been generated through the Clustal X (Thompson et al.,
1997) program. The standard alignment algorithm in Clustal X
can be enriched by two optional refinements: the NoGap (NG) and the
multiple substitution (MS) options. The former does not allow any gap
in the global alignment: this guarantees that "like" is always
compared to "like," although it may exclude much of the information
in the global alignment if many gaps are present. The latter (MS) is
particularly useful for largely divergent sequences. In such a case, in
fact, it becomes increasingly likely (as a result of equally
time-spaced random events) that more than one substitution will have
happened at the same site (Saitou and Nei, 1987; Higgins and Sharp,
1989). In any case, the strong correlation we observed among all the
three sequence homology metrics in our data set (HOMOL1-HOMOL3 in Table
6), points to a substantial stability and algorithm-independence of the
sequence alignments for rubredoxins.
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RESULTS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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The RQA and SVD methods generated 13 variables (Table 1) describing the protein hydrophobicity sequences from a dynamical perspective. In order to reduce the number of variables to a more manageable size, and to identify the effective (orthogonal) axes spanning the space under study, the data set was analyzed by PCA. In order to check for consistency and robustness of the dynamical descriptors, two separate PCAs were performed using 1) all 13 variables and 2) a "reduced" set lacking the SP1-SP4 variables.
The first four principal components extracted from the complete set of variables were used to define a suitable space in which the structure-function relationships of interest could be identified. The correlation between classical sequence comparison and the dynamical methods was performed by a simple Pearson coefficient between the distance matrices corresponding to 1) each of the three investigated sequence homology metrics, and 2) the 4D space of the major principal components extracted from the dynamical descriptors. The data matrix for the global dynamical characterization of the 19 sequences is reported in Table 2.
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The set of variables shown in Table 2 without the SP1-SP4 variables (thus mainly based on RQA), and submitted to PCA, produced a four-component solution explaining 92% of the total variability: such a compression is due to the high correlation among descriptors. The factor loading matrix, containing the correlation coefficients between original variables and the four components, is reported in Table 3. Sketching an interpretation of the components is made possible by considering which original variables obtain the larger loadings on each component (see the legend to the table).
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The SP1-SP4 variables, excluded from the above analysis, deal with the classification of the sequences into four well separated families of spectra (Fig. 4) and, from a computational point of view, are based on a completely different approach (see the Appendix). Moreover, since FD provides a poor representation of the spectral shapes, including SP1-SP4 in the PCA analysis implies the use of brand-new information. This is true even from a purely statistical point of view, given that SP1-SP4 are "almost" mutually orthogonal (see Methods). The "spectral shape" information has approximately the same dimensionality (four) as the PC1r, ... , PC4r space. Thus its addition to the reduced set of variables could, in principle, strongly perturb its PCA solution. If, however, the PCA solution of the complete set of variables remains substantially the same, this suggests an internal consistency of the dynamical descriptors.
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Table 4 shows that PCA of the reduced set of variables produces a practically equivalent solution as compared to the complete set. The latter solution explains 86% of total variability and shows a one-to-one correspondence (Pearson r) with the components of the reduced set (Table 5). Hence, the new set of components (PC#f) can be faithfully used as dynamical descriptors of the protein data set (for the meaning of the components see the legend to Table 4).
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The relationship between the dynamical description of proteins and
their sequence homology description has been checked by correlating two
sets of "between-sequences" distance matrices. One set includes
three distance matrices (HOMOL1, HOMOL2, HOMOL3) generated by three
different sequence homology metrics; the other set includes just one
member, i.e., the Euclidean distance matrix between the same 19 sequences in the 4D PC1f/PC4f space (DYNAM). The four distance matrices
are directly comparable by means of a Pearson coefficient (Table
6). The three sequence-matching algorithms are completely equivalent (Pearson r 
1),
while the dynamical description is proven to be practically independent from them (Pearson r 0.22).
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The internal consistency of the dynamical description of the hydrophobicity profiles provides a solid basis for the attempt to recognize known structural and/or functional features inside the "dynamical" space. The most relevant feature to look for is, in the present case, the high thermal stability of Rubr Pyrfu rubredoxin. Looking at the rubredoxins' location in the PC2f-PC3f plane, shown in Fig. 5, Rubr Pyrfu rubredoxin is located at an extreme of both the second (PC2f) and the third (PC3f) axes of the dynamical space, whereas such a peripheral location does not find a counterpart in the sequence alignment metrics (Fig. 1 B) where Rubr Pyrfu rubredoxin is located well inside the mesophilic sequence space.
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Since Rubr Clopa rubredoxin is the nearest neighbor of
Rubr Pyrfu rubredoxin in the PC2f-PC3f plane, the 3D
structural similarities between Rubr Pyrfu and Rubr
Clopa rubredoxins is recognized by the dynamical description (Fig.
5), whereas the sequence alignment metrics fail in this respect. The
modes of the hydrophobicity spectrum were interpreted (Selz et al.,
1998) as secondary and supersecondary structural features of the
protein molecule; from a purely computational point of view, the modes
correspond to peaks in the autocorrelation structure of the series. In
the RQA perspective, the presence of peaks of autocorrelation at
well-defined scales can be appreciated by plotting the amount of
recurrence (i.e., the number of recurrent pairs) on the relative
displacement along the chain of all the residue pairs. The
recurrences' displacement histograms of Rubr Pyrfu and
Rubr Clopa reported in Fig. 3 provide a much closer look at
the recurrence structure of the two sequences as compared to REC or
DET. We relied on these histograms to tackle the "high resolution"
part of our analysis; i.e., the discrimination between mesophilic and
thermophilic structures in the space spanned by Rubr Clopa,
Rubr Pyrfu and the six chimeras (Eidsness et al., 1997).
This problem is on a much more detailed scale than the previous one,
since all the sequences are located in a small fraction of the
component space (Fig. 6). Thus we needed
a finer look at the recurrence plots in order to solve it.
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A simple inspection of Fig. 2 reveals a macroscopic difference between "thermophilic" and "mesophilic" recurrence plots: while thermophilic hydrophobicity series display a regular distribution of recurrences along lines parallel to the main diagonal, reminiscent of a quasi-periodic distribution of similar hydrophobicity patterns; mesophilic series display a dense clustered distribution of recurrences. Such a difference is present in all the examined structures and, in order to quantify it, we computed the kurtosis of the recurrence displacement distribution for all the sequences. The results, reported in Fig. 7, show that the thermophilic proteins have a low kurtosis corresponding to a quasi-normal distribution of recurrences (a normal distribution has a kurtosis equal to 3) as compared to the relatively higher kurtosis of mesophilic structure corresponding to a peaked (or clustered) distribution of recurrences. Thus the kurtosis of recurrence displacement distributions is a simple index allowing for a clearcut separation of the two stability behaviors on a quantitative, completely data-driven, basis.
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From a structural point of view, this means that the thermophilic sequences have a "multiple scale" correlation structure made up of both short and long range correlations as compared to the single-scale correlation (just one peak of recurrences) in mesophilic sequences. The 3D structural counterpart of such a pattern are shown in Fig. 8 and are further discussed below.
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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The main result of the present study is that consideration of the physicochemical semantics (hydrophobicity profiles) of protein primary structures, together with the computation of order-dependent dynamical descriptors, generates completely novel information with respect to classical homology analysis, and indicates new exploration pathways for studying sequence/function relationships of proteins. This has been demonstrated by showing that 1) at difference with classical best-alignment methods, a classification of 19 bacterial rubredoxin primary structures is able to unequivocally single out the only thermophilic element of the set; and that 2) thermal sensitivity can be modeled, on a finer scale of six chimeric sequences derived from a thermophilic and a mesophilic protein, by the analysis of recurrence plots of the corresponding hydrophobicity profiles.
It is worth noting the different assumptions at the basis of sequence
alignment and dynamical metrics for the classification of protein
sequences. Sequence alignment metrics generate a "phylogenetic space" (Kimura, 1983; Saitou and Nei, 1987) in which proteins are
compared in terms of the number and location of mutations needed to go
from one structure to another and, in so doing, implicitly measure the
underlying evolutionary process. On the contrary, the dynamical
approach compares sequences in terms of their resemblance in the global
ordering of hydrophobicity values along a chain that 1) emphasizes a
physiological, synchronous view; and 2) implies that a similar kind of
ordering can be achieved even by a relatively diverse use of amino acid
residues in different proteins. These features may explain the
statistical independence of the two metrics in the considered data set.
The premise at the basis of the dynamical approach is that the primary
structure "encodes" the folding features of proteins and that the
"folding code" can be reconstructed by order-dependent descriptors
of polypeptide sequences (Mandell et al., 1997; Selz et al., 1998). In
this respect, the dynamical approach can be considered a statistical
"mean field" approximation, which, through the agency of time
series analysis methods, phenomenologically captures an
"equilibrium" folding state. This assumes that the individual amino
acid residues are part of an interacting system with long-range
correlations, which results in scaling and critical behavior. [Some
evidence for such a view are demonstrated by Manetti et al. (1999).]
In such a frame, hydrophobicity has been considered the variable of
election for energy minimization, due to the vast amount of
experimental and theoretical evidence (Weiss and Herzel, 1998; Li et
al., 1997; von Heijne, 1982; Sweet and Eisenberg, 1983) pointing to it
as the most crucial parameter in determining protein 3D structures.
Focusing on global properties of a sequence (dynamical approach), as
opposed to local features (sequence homol-ogy), changes the point
of view on protein sequence/function relationships. The holistic
character of the dynamical approach forces the analysis to the
congeneric series level; i.e., at the level of proteins having the same
kind of activity and comparatively similar structures. In this common
frame of sequence/activity relationships, the dynamical approach can be
very useful (Zbilut et al., 1998b) to model the modulation of such
relationships. The need for studying such homogeneous classes is a
direct consequence of the fact that the same values of dynamical
descriptors can be reached by completely different sequences pertaining
to unrelated proteins, which is another important resemblance between
the proposed approach and medicinal chemistry QSAR studies. Even in
this field, in fact, the global values of physicochemical descriptors
used to model and predict biological activity of organic molecules
[e.g., octanol/water partition coefficient, highest occupied molecular
orbital (HOMO), etc.] can assume the same value for very different
structures (Franke, 1984) and constrains the analysis within a
particular congeneric series of molecules with the same basic
"skeleton" and different substituents (Franke, 1984; Martin, 1981;
Hansch, 1993). The congeneric series paradigm allows, however, for a
nonequivocal definition of the biological activity to be modeled: all
the considered organic molecules are potentially active, since all of
them possess the basic determinants of that biological activity and
only differ from each other in comparatively minor elements
(substituents) modulating the activity. This allows for the use of
observed differences in the physicochemical descriptors to model such
modulation (Franke, 1984; Hansch and Leo, 1995; Martin, 1981).
In the present case we adopted the same paradigm, taking as a
congeneric series the rubredoxin family, within which we modeled the
thermophilic character. The problem of predicting the thermal stability
of proteins could be posed in many general ways, although to our
knowledge no general determinant of thermal stability, independent from
a specific context, has yet been identified (Adams and Keltzin, 1996).
As a matter of fact, the congeneric series paradigm showed itself to be
amazingly successful even in the difficult case of the chimeric protein
set, which spanned a much more limited portion of the sequence space as
compared to the rubredoxin classifications (Fig. 6), and needed a much
more detailed dynamical description. This was obtained by shifting from
an average description of recurrence plots based upon synthetic
indexes, to a more detailed description of the plots' shape
(distribution of recurrences) which only required a change in the
quantification of the same basic dynamical description (recurrence
plots), without shifting to a brand new parameterization. More
importantly, the description maintained its "holistic" character by
taking into account the entire sequence. Fig. 8 provides some help in
exploiting the bulk of structural information embedded in the
recurrence plots of Rubr Clopa and Rubr Pyrfu
(Fig. 2), and emphasizes the different distribution of recurrent
fragments, in their 3D representations, over essentially identical
backbones. In the Rubr Clopa case, the concentration of
recurrence lines in the same area of the recurrence plot (Fig. 2) is
paralleled by a marked concentration of the long-range recurrences that
mainly occur between two well-defined locations on the polypeptide
chain (Fig. 8). In the Rubr Pyrfu case, there is no
preferentially populated area in the recurrence plot, and Fig. 8 shows
that recurrent fragments are widespread over the whole backbone. The
two situations can be viewed as a different spatial distribution of the
same amount of REC (3.84 and 3.37 in Rubr Pyrfu and
Rubr Clopa, respectively) both in 1 and in 3D spaces. In
either case, however, it is impossible to identify one (or a few)
localized residues specifically responsible for the different thermal
stability of the two proteins.
If it is yet difficult to infer in general terms a well-defined set of stabilizing interactions from a recurrence distribution of hydrophobicity along primary structures, it is worth noting that the information conveyed by recurrence patterns well exceeds the simple observation of repetitive chemical motifs. In the specific case of rubredoxins, in fact, the four strictly conserved short fragments centered on cysteins in positions 6, 9, 39, and 41, which bind the prosthetic groups, are all included into the deterministic fraction of recurrences both in Rubr Clopa and in Rubr Pyrfu (see Fig. 8); only in Rubr Pyrfu, however, similar deterministic patterns not immediately perceivable from periodic chemical identity of residues can be found in completely different regions.
In any case, our conclusions are in agreement with the work of Eidsness
et al. (1997), who state:
"... Since our results do not identify a few dominant localized interactions, we suggest that the extraordinary thermostability of Rubr Pyrfu may involve a precise optimal alignment of a large number of residues, whose network of interactions are very sensitive to small structural changes dictated by the context of the sequence."
This statement corresponds to the peculiar quasi-periodic pattern in the recurrence plot of Rubr Pyrfu (Fig. 2), which can be, in principle, destroyed by a few mutations interrupting the lines of determinism, but cannot be considered as a "local" feature (i.e., specific amino acids responsible for thermostability do not exist), since it spans the entire sequence in the form of long range correlations. On completely different, and exclusively methodological, grounds, it is worth noting that time series analysis methods, given their holistic character, pose no constraints for the relative length of the sequences to be compared. Finally, besides the possibility of deriving useful sequence/function relationships for specific situations, the independence between dynamical and sequence homology metrics opens the way to the exploration of larger data bases, looking for functional similarities among proteins by a method complementary to sequence homology analyses.
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APPENDIX |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Computing dynamical descriptors of hydrophobicity sequences
RQA descriptors
RQA is based on the computation of a distance matrix, DM, between any possible pair-combination of rows (epochs) of an EM. The distance matrix is then colored, darkening the pixels located at specific (i, j) coordinates, corresponding to distance values between ith and jth rows (epochs) lower than a predefined radius (for details see Giuliani and Manetti, 1996;
Webber and Zbilut, 1994)].
The features of the distance function make the plot symmetric
(DMi, j = DMj, i) and with
a darkened main diagonal corresponding to the identity line
(DMi, j = 0 when j = i) (Fig. 2).
The darkened (recurrent) points single out recurrences within the
series and the plot can be considered as a global picture of the
autocorrelation structure of the system (Webber and Zbilut, 1994;
Giuliani et al., 1998). In other words, the recurrence plot visualizes
the distance matrix between the epochs (rows) of the embedding
matrix and consequently the autocorrelation present in the signal at any possible scale. Since, in fact, distances are computed for all the
possible pairs of epochs, the elements close to the main diagonal of
the plot correspond to short-range correlations (the diagonal marks the
identity in time), while long-range correlations correspond to points
distant from the main diagonal. Besides the global impression given by
the graphic appearance of the plot (Fig. 2), the indexes developed by
Zbilut and Webber (1992; Webber and Zbilut, 1994; Trulla et al., 1996)
allow for a quantitative description of the recurrence structure of the
plot (see Methods). The computation of such indexes implies the setting
of a recurrence threshold (radius) that in our case was set to 3 and a
line length (minimum number of subsequent recurrent points to be
considered as deterministic) that in our case was set to 3.
SVD-based descriptors
SVD spectral analysis (Mandell et al., 1997; Broomhead and King,
1986) is based on the computation of a correlation matrix (CM) among
the columns of an EM. The first three eigenvectors of the CM are then
computed by SVD and used to estimate the original series S
(primary structure) by a linear least-squares fit. It is worth noting
that the eigenvectors are extracted in order of explained variance, so
that the first vectors are more representative of the correlated,
signal-like part of the information contained in S
(Broomhead and King, 1986). The estimated series, S',
maintains all the general features of S filtered by noise,
and are subsequently submitted to a maximum entropy, complex poles,
power spectrum analysis (Selz et al., 1998). The result is
parameterized in terms of the dominant frequency (FD) and used as a
whole (with a 1000-point sampling). In order to derive quantitative
synthetic parameters of the proteins' whole spectra, the 19 1000-point-long arrays corresponding to the digitized spectra were
analyzed by means of oblique principal component analysis (OPC)
(Anderberg, 1973; Harman, 1976). OPC analysis attempts to divide a set
of variables (in our case digitized spectra) into nonoverlapping
clusters such that each cluster can be considered as essentially
unidimensional. The clusters are formed with the goal to include in
each of them variables that are both highly intercorrelated (maximal
variance explained by the cluster first component), and as much as
possible independent from those included in the other clusters (minimal correlation between clusters). The procedure consists in a stepwise increase of the clusters' number until a user-specified
criterion, involving either the percentage of variation accounted for
by the global solution or the between-clusters relative independence, is fulfilled. In our case OPC generated a four-cluster partition of the
spectra. The correlation coefficients between the original spectra
and the center of mass of the clusters (SP1-SP4) were then used
to quantitatively parameterize spectral information for the subsequent analysis.
LZ parameter
LZ transforms the representation of a numerical sequence into a binary format, substituting 1 for the higher-than-median values and 0 otherwise. This binary sequence is then analyzed trying to generate any subsequent configuration of 1's and 0's from the previous one using just two operators: copy and insert acting on the initial sequence. Starting from an initial random sequence, Sr, the procedure progressively reconstructs any predefined series: the number of instructions (copy plus insert operations) needed to produce the series, normalized by the number of instructions needed to generate the corresponding random sequence, constitutes the LZ index (Kaspar and Schuster, 1987).
Software
RQA was performed by using the original Webber and Zbilut programs that can be freely downloaded from http://homepages.luc.edu/~cwebber. SVD analysis was performed by CDA (Chaos Data Analyzer) software by J.C. Sprott of the University of Wisconsin and George Rowlands of the University of Warwick. The same program was used for the computation of the LZ index and of the r value between adjacent residues. All statistical routines were performed by SAS (SAS Institute Inc., NY, 1990), Version 6.2 (1998) for Unix systems.| |
ACKNOWLEDGMENTS |
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This work has been partly supported by Italian M.U.R.S.T. (40% and 60%) grants to A. Colosimo and C.N.R. (Grant CTB CNR 96.03746.CT114-INV557)
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FOOTNOTES |
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Received for publication 31 December 1998 and in final form 20 October 1999.
Address reprint requests to Dr. Alfredo Colosimo, Department of Biochemical Sciences, University of Roma "La Sapienza," 00185 Roma, Italy. Tel.: 39-06-499-10957; Fax: 39-06-444-0062; E-mail: a.colosimo{at}caspur.it.
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REFERENCES |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Biophys J, January 2000, p. 136-149, Vol. 78, No. 1
© 2000 by the Biophysical Society 0006-3495/00/01/136/14 $2.00
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