Universal sequence map (USM) of arbitrary discrete sequences
 Jonas S Almeida^{1, 2}Email author and
 Susana Vinga^{2}
DOI: 10.1186/1471210536
© Almeida and Vinga; licensee BioMed Central Ltd. 2002
Received: 02 November 2001
Accepted: 05 February 2002
Published: 05 February 2002
Abstract
Background
For over a decade the idea of representing biological sequences in a continuous coordinate space has maintained its appeal but not been fully realized. The basic idea is that any sequence of symbols may define trajectories in the continuous space conserving all its statistical properties. Ideally, such a representation would allow scale independent sequence analysis – without the context of fixed memory length. A simple example would consist on being able to infer the homology between two sequences solely by comparing the coordinates of any two homologous units.
Results
We have successfully identified such an iterative function for bijective mappingψ of discrete sequences into objects of continuous state space that enable scaleindependent sequence analysis. The technique, named Universal Sequence Mapping (USM), is applicable to sequences with an arbitrary length and arbitrary number of unique units and generates a representation where map distance estimates sequence similarity. The novel USM procedure is based on earlier work by these and other authors on the properties of Chaos Game Representation (CGR). The latter enables the representation of 4 unit type sequences (like DNA) as an order free Markov Chain transition table. The properties of USM are illustrated with test data and can be verified for other data by using the accompanying webbased tool:http://bioinformatics.musc.edu/~jonas/usm/.
Conclusions
USM is shown to enable a statistical mechanics approach to sequence analysis. The scale independent representation frees sequence analysis from the need to assume a memory length in the investigation of syntactic rules.
Background
For over a decade the idea of representing biological sequences in a continuous coordinate space has maintained its appeal but not been fully realized [1–3]. The basic idea is that sequences of symbols, such as nucleotides in genomes, aminoacids in proteomes, repeated sequences in MLST [Multi Locus Sequence Typing, 4], words in languages or letters in words, would define trajectories in this continuous space conserving the statistical properties of the original sequences [3, 5–9]. Accordingly, the coordinate position of each unit would uniquely encode for both its identity and its context, i.e. the identity of its neighbors [10]. Ideally, the position should be scaleindependent, such that the extraction of the encompassing sequence can be performed with any resolution, leading to an oligomer of arbitrary length. The pioneer work by Jeffrey published in 1990 [5] achieved this for genomic sequences by using the Chaos Game Representation technique (CGR), defining a unitsquare where each corner corresponds to one of the 4 possible nucleotides. Subsequent work further explored the properties of CGR of biological sequences, but two main obstacles prevented the realization of its early promise – lack of scalability with regard to the number of possible unique units and inability to represent succession schemes. Meanwhile, Markov Chain theory already offered a solid foundation for the identification of discrete spaces to represent sequences as crosstabulated conditional probabilities – Markov transition tables. This Bayesian technique is widely explored in bioinformatic applications seeking to measure homology and align sequences [11]. In a recent report [12] we have shown that, for genomic sequences, Markov tables are in fact a special case of CGR, contrary to what had been suggested previously [13]. This raised the prospect of an advantageous use of iterative maps as state spaces not only for representation of sequences but also to identify scale independent stochastic models of the succession scheme. That work [12] is hereby extended and further generalized to be applicable to sequences with arbitrary numbers of unique component units, without sacrificing the inverse correlation between distance in the map and sequence similarity independent of position. Accordingly, the technique is named Universal Sequence Map (USM).
Results
The Results are divided in two sections. The first section presents the foundations for identifying an iterative function with the desired properties. The second section describes algorithm implementation illustrated with a sample data set. Both sections are best understood by using the accompanying webbased tool (see Abstract for address) where the different steps of the procedure can be verified and reproduced with the test data or the reader's own data.
Conceptual foundations
The USM generalization proposed here is achieved by observing two stipulations: Aalternative units in the iterative map are positioned in distinct corners of unit block structures', and B – sequence processing is bidirectional.
 A.
Each unique unit is referenced in the map for positions that are at equal ndistances from each other, and possibly, but not necessarily, defining a complete block structure[3]. ndistances are defined as the maximum distance along any dimension, e.g. ndistance between [a_{1}, a_{2}, ...,a_{ n }] and [b_{1}, b_{2}, ...,b_{ n }] is max(b_{1}  a_{1}, b_{2} a_{2}, ..., b_{ n }  a_{ n }), see also Equation 3. It will be shown that this stipulation leads to the definition of spaces where distance is inversely proportional to sequence similarity, independent of position. In this respect, USM departs from previous attempts to generalize Chaos Game Representation that conserve the bidimensionality of the original CGR representation [8, 15–17].
 B.
The iterative positioning is performed in both directions. Therefore, there will be two sets of coordinates, the result of forward and backward iterative operations. It will be shown that, by adding backward and forward map distances between two positions, the number of identical units in the encompassing sequences can be extracted directly from the USM coordinates. As a consequence, two arbitrary positions can be compared, and the number of contiguous similar units is extracted by an algebraic operation that relies solely on the USM coordinates of those very two positions.
Implementation of USM algorithm
 1.
Identification of unique sequence units – e.g. these two stanzas have 19 unique characters, (table 1), i.e. uu = 19.
 2.
Replacement of each unique unit (in this case units are alphabetic characters) by a unique binary number – e.g. in table 1 each of the 19 unique units is replaced by its rank order minus one, represented as a binary number. Other arrangements are possible leading to the same final result as discussed below. The minimum number of dimensions necessary to accommodate uu unique units, n, is the upper integer of the length of its binary representation: n = ceil(log_{2}(uu)). For W. Cope's stanzas, n = ceil(log_{2}(19)) = 5. The binary reference coordinates for the unique units are defined by the numerals of the binary code – for example, a will be assigned to the position U_{'a'} = [0,0,1,0,1]. Each symbol is represented as a corner in a ndimensional cube (Table 1). The purpose of these first two steps is to guarantee that the reference positions for each unique sequence unit component are equidistant (stipulation A) in the nmetric defined above. Any other procedure resulting in equidistant unique positions will lead to the same final results independently of the actual binary numbers used or the number of dimensions used to contain them.
Binary codes for the 19 possible units occurring in the two stanzas. The first unit is a space character " ".
Unit  Bin. Code 

00000  
.  00001 
?  00010 
A  00011 
A  00101 
B  00110 
D  00111 
E  01000 
F  01001 
I  00100 
M  01010 
N  01011 
O  01100 
P  01101 
R  01110 
S  01111 
T  10000 
V  10001 
Y  10010 
 3.
The CGR procedure [5] (Eq. 1) is applied independently to each coordinate, j = 1,2, ...,n, for each unit, i, in the sequence of length k, u_{ j }^{ (i) } with i = 1,2, ...,k, and starting with a random map position taken from a uniform distribution in [0,1]^{ n }, i.e. Unif([0,1]^{ n }). The random seed is not fundamentally different from using the middle position in the map as is conventional in CGR and it has the added feature that it prevents the invalidation of the inverse logarithmic proportionality of ndistance to sequence similarity [12] for sequences that start or end with the same motif.
For a sequence with k units, the USM positions i = 1,..., k for the j = 1,..., n dimensions are determined as follows:
 4.
The previous step generated k positions in a ndimension space by processing the sequence forward (Eq. 1). This subsequent step adds an additional set of n dimensions by implementing the same procedure backward (Eq. 2), again starting at random positions for each coordinate. Consequently the first n dimensions of USM will be referred as defining a forward map and the second set of n dimensions will define a backward map. Put together, the bidirectional USM map defines a 2nunit block structure.
The n additional backward coordinates are determined as follows:

using forward coordinates alone [0.0156 0.0138 0.6314 0.0001 0.5338]

using backward coordinates alone [0.0703 0.3004 0.5169 0.2742 0.5652]
The length of the sequence that can be recovered from a position in the CGR or USM space is only as long as the resolution, in bits, of the coordinates themselves. In addition, the relevance of these iterative techniques is not associated with the property of recovering sequences as much as with the ability to recover the succession schemes, e.g. the Markov probability tables. It has been recognized for almost a decade that the density of positions in unidirectional, bidimensional, iterated CGR maps (e.g. of genomic sequences, uu = 4 >n = 2) defines a Markov table [12, 13]. The complete accommodation of Markov chains in unidirectional USM (i.e. either forward or backward, which is an equivalent to a multidimensional solution for CGR) can be quickly established by noting that the identity of a quadrant is set by its middle coordinates[13]. In order to extract the Markov format, for an arbitrary integer order ord, each of the two nunit hypercubes, the set of n forward or backward coordinates, would be divided in q = 2^{ n.(ord+1) } equal quadrants and the quadrant frequencies rearranged [12]. The use of quadrant to designate what is in fact a subunit hypercube is a consonance with the preceding work on bidimensional CGR maps [12], where it was shown that since any number of subdivisions can be considered in a continuous domain, the density distribution becomes an orderfree Markov Table that accommodates both integer and fractal memory lengths. The extraction of Markov chain transition tables from USM representations, both forward and backward, is included in the accompanying webbased application (see Abstract).
The USM algorithm determines that similar sequences, or segments of sequences, will have converging iterated trajectories: the distance will be cut in half for every consecutive similar unit. This property was notice before for CGR of genomic sequences [12], and will be further explored here for USM generalization. In that preceding work it was shown that the number of similar consecutive units can be approximated by a symmetrical logarithmic transformation of the maximum distance between two positions in either of the dimensions (ndistance), d.
d = log_{2} (MaxΔUSM_{ undirectional }) (Eq. 3)
However, the values of d necessarily overestimate the number of similar contiguous units preceding (d_{ f }, illustration for stanza comparison in Fig. 2a) or succeeding (d_{ b }, illustration for stanza comparison in Fig. 2b) the positions being compared. The value of d would be the exact number of contiguous similar units, h, if the starting positions for the similar segments where at a ndistance of 1, e.g. if they were in different corners of the unit hyperdimensional USM cube. Since the initial distance is always somewhat smaller, the homology, h, measured as the number of consecutive similar units, will be smaller than d (Eq. 5).
The contribution of φ to the similarity distance, d, can be estimated from the distribution of positions in the USM map of a random sequence. A uniformly random sequence [3, 19, 20] will occupy the USM space uniformly, and, for that matter, so will the random seed of forward and backward iterative mapping, respectively equations 1 and 2. Therefore, a uniform distribution is an appropriated starting point to estimate the effect of (p, the overdetermination of h by d (Eq.5). Accordingly, for a given x∈ [0,1], the probability, P_{ o }, that any two coordinates, x_{1} and x_{2}, are located within a radius r ∈ (0,1) is given by Equation 6.
Since P_{ o }(r) is the probability of two points chosen randomly from a uniform distribution Unif([0,1]) being at a distance less than r from each other, for any set of n coordinates in the USM, the likelihood of finding another position within a block distance of r would be described by raising equation 6 to the n exponent. Finally, recalling from equation 3 that sequence similarity can be obtained by a logarithmic transformation of r, the probability that the unidirectional coordinates of two random sequences are at a similar length d > φ is described by equation 7. The simplicity of the expansion for higher dimensions highlights the orderstatistics properties [21] of the nmetric introduced above (Eq. 3). It is noteworthy that the model for the likelihood of overdetermination is the nullmodel, e.g. the comparison of actual sequences is evaluated against the hypothesis that the similarity observed happened by chance alone.
Finally, it is also relevant to recall that the null model for d (Eq.7 for unidirectional comparisons, bidirectional null models are derived below) allows the generalization for noninteger dimensions. For example, the 19 unique unites found in the two stanzas (Table 1), define forward and backward USM maps in 5 dimensions each. However the 5^{th} dimension is not fully utilized, as that would require 2^{5} = 32 unique units. Therefore, if there is no requirement for an integer result, the effective value of n for the two stanzas can be refined as being n = log_{2}(19) = 4.25.
An estimation of bidirectional similarity will now be introduced that adds the forward and backward distance measures d_{ f } and d_{ b }. The motivation for this new estimate is the the determination of the similar length of the entire similar segment between two sequences solely by comparing any two homologous units. Accordingly, since d_{ f } is an estimate of preceding similarity and d_{ b } provides the succeeding similarity equivalent the sum of the two similar distances, D, (Eq. 8) will estimate of the bidirectional similarity, e.g. the length of the similar segment, H.
As illustrated later in the implementation, for pairwise comparisons of homologous units of similar segments, all values of D and, consequently, of φ, are exactly the same. This result could possibly have been anticipated from the preceding work [12] by noting that the value of d between two adjacent homologous units differs exactly by one unit. However, this result was in fact a surprise and one with far reaching fundamental and practical implications.
Similarly to unidirectional similarity estimation, d, the bidirectional estimate, D, being the sum of two overestimates, is also overestimated by a quantity to be defined, φ (Eq. 8). The derivation of an expression for the bidirectional overestimation will require the decomposition of P_{1} (Eq. 7) for two cases, comparison between unidirectional coordinates of similar quadrants, P_{1a}, and of opposite quadrants, P_{1b}, as described in equation 9. Recalling from equation 2, positions in the same quadrant correspond to sequence units with the same identity, and positions in opposite quadrants correspond to comparison between coordinates of units with a different identity.
The need for the distinction between same and opposite quadrant comparison, which is to say between similar and between dissimilar sequence units, is caused by the fact that same quadrant comparisons are more likely to lead to higher values of d. As illustrated above for the 16^{th} unit of the first stanza, the forward and backward coordinates must fall in the same quadrant. Consequently, the similar pattern of same and opposite quadrant comparisons for each dimension will be reflected as a bias in the bidirectional overestimation. The determination of probability, P_{2}, of overdetermination between sums of independent unidirectional similarity estimates is derived in equation 10.
The probability of bidirectional overdetermination, can now be established by using the same and opposite unidirectional comparison expressions presented in Equation 9. The resulting expression for similarity overdetermination by the distance between bidirectional USM coordinates, P_{3}, is presented in equation 11.
Discussion
Additional measures of similarity can be derived for specific practical purposes using bidirectional and unidirectional d values. For example, the use of docking algorithms to align sequences would benefit from a measure with a maximum value in the center of the similar segments. This could be provided by defining a compounded similarity measure, Hc, as suggested in equation 12. The behavior of Hc, which would be obtained by the overestimated value of dc, is illustrated for the two test stanzas in Figure 2.d.
The detection of similar segments in arbitrary sequences using D becomes very effective as the length of the similar segment increases. This was clear in the distribution of overdetermination in Fig. 3 but it is even more so when the distances between sequences with homologous segments are represented. In figure 4 the distances between the two stanzas are represented alongside the distances to be expected if no homology existed, apart from the coincidental (random null model, using Eq. 11). It can be observed for the comparison of the two stanzas (Fig. 4, gray lines) that H values above 4 units occur with higher frequency than allowed by the random distribution model, reflecting the presence of real homologous segments (similar words).
USM of biological sequences
The representation of biological information as discrete sequences is dominated by the fact that genomes are sequences of discrete units and so are the products of its transcription and translation. However, not all biological sequences are composed of units that are functionally equally distinct from each other, as is the case of proteomic data and Multilocus sequence typing [MLST, 4]. To avoid the issue of unit inequality and highlight the general applicability of the USM procedure, stanzas of a poem were used to illustrate the implementation instead. Nevertheless the original motivation of analyzing biological sequences is now recalled.
In the preceding report the authors have illustrated the properties of unidirectional nmetric estimation of similarity for the threonine operon of E. coli[12]. The same two two regions of thrA and thrB sequences of E. coli K12 MG1655 are compared in Figure 5 to highlight the advancement achieved by USM. It should be recalled that the particular dimensionality of DNA sequences, n = 2, allows a very convenient unidirectional bidimensional representation, which is in fact the Chaos Game Representation procedure (CGR) [5]. Consequently, CGR is a particular case of USM, obtained when n = 2 and only the forward coordinates are determined. This can also be verified by comparing Figure 5a with a similar representation reported before [12], obtained with the same data using CGR [Fig. 10 of that report]. The advantageous properties of full (bidirectional) USM become apparent when Fig. 5a is compared with Fig. 5b. It is clearly apparent for bidirectional USM (Fig. 5b) that all pairwise comparisons of units of identical segments now have the same D values. This coverts any individual homologous pairwise comparison into an estimation of the length of the entire similar segment. The conservation of statistical properties by the distances obtained, D, can also be confirmed by comparing observed values with the corresponding null models (Fig. 4). For the analysis of this figure it is noteworthy to recall that the statistical properties of prokaryote DNA are often undistinguishable from uniform randomness [12, 18, 19]. The genomic sequence of the first gene of the threonine operon of E. coli, thrA, is compared with that of the second, thrB. The distribution of the resulting D values is represented in figure 4 (solid black line), alongside with the null model for that dimensionality (Eq. 11, with n = log_{2}(4) = 2, gray dotted line). The genomic sequences of thrA and thrB were translated into proteomic sequences using SwissProt's on line translator, applied to the 5'3' first frame http://www.expasy.ch/tools/dna.html. Similarly, the distribution of D values for the comparison of the proteomic thrA and thrB sequences is also represented in Figure 4, alongside with the null model, Eq. 11, for its dimensionality (n = log_{2}(uu = 20 possible aminoacids) = 4.32), which is graphically nearly undistiguishable from that of the comparison between the stanzas, with n = log_{2}(uu = 19 possible letters) = 4.25 (dotted gray line for the rounded value, n = 4.3). Both the genomic and the proteomic distribution of D values is observed to be contained by the null model, unlike the comparison between the stanzas discussed above, where the existence of structure is clearly reflected by its distribution. The genomic and proteomic of thrA and thraB, used to illustrate this discussion, are provided with the webbased implementation of USM (see Methods for URL).
Conclusions
The mounting quantity and complexity of biological sequence data being produced [22] commands the investigation of new approaches to sequence analysis. In particular, the need for scale independent methodologies becomes even more necessary as the limitations of conventional Markov chains are increasingly noted [6]. These limitations are bound to become overwhelming when signals such as succession schemes of the expression of over 30,000 human genes [23] become available. This particular signal would be conveniently packaged within a 30 dimension USM unit block (n = ceil(log_{2}(3 10^{3}) = 15).
In addition, the advances in statistical mechanics for the study of complex systems, particularly in nonlinear dynamics, have not been fully utilizable for the analysis of sequences due to the missing formal link between discrete sequences and trajectories in continuous spaces. The properties of USM reported above suggest that this may indeed be such a bridge. For example, the embedding of dimensions, a technique at the foundations of many timeseries analysis techniques offers a good example of the completeness of USM representation of sequences. By embedding the forward and backward coordinates separately, at the relevant memory length, the resulting embedded USM is exactly what would be obtained by applying USM technique to the embedded dimeric sequence itself.
Methods
Computation
The algorithms described in this manuscript were coded using MATLAB™ 6.0 language (Release 12), licensed by The MathWorks Inc http://www.mathworks.com. An internet interface was also developed to make them freely accessible through userfriendly webpages http://bioinformatics.musc.edu/~jonas/usm/.
Source code
In order to facilitate the development of sequence analysis applications based on the USM state space, the software library of functions written to calculate the USM coordinates is provided with the webbased implementation (see address above). The code is provided in MATLAB format, which is general enough as to be easily ported into other environments. These functions process sequences provided as text files in FASTA format. In addition to the functions, the test datasets and a brief readme.txt documentation file are also included.
Test data
The USM mapping proposed is applicable to any discrete sequence, even if the primary goal is the analysis of biological sequences. For ease of illustration and to emphasize USM's general validity, the test dataset used to describe implementation of the algorithm consists of two stanzas of a Poem by Wendy Cope, "The Uncertainty of the Poet" [14]. In the Discussion section, USM was also applied to the DNA sequence of the threonine operon of Escherichia coli K12 MG1655, obtained from the University of Winsconsin E. coli Genome Project http://www.genetics.wisc.edu, and to its 5'3' first frame proteomic translation obtained by using SwissProt on line translator http://www.expasy.ch/tools/dna.html. The three test sequence datasets are also included in the webbased USM application.
Declarations
Acknowledgments
The authors thank Dr Santosh Mishra, Eli Lilly Co., for the insightful suggestions about the applicability of USM, and John H. Schwacke, at the Department of Biometry and Epidemiology of the Medical University of South Carolina for revising the coherence of mathematical deduction. The authors thankfully acknowledge financial support by grant SFRH/BD/3134/2000 to S. Vinga and project SAPIENS34794/99 of Fundação para a Ciência e Tecnologia of the Portuguese Ministry of Science and Technology.
Authors’ Affiliations
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