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* Graduate Group in Biophysics, and Department of Pharmaceutical Chemistry, University of California, San Francisco, California
Correspondence: Address reprint requests to Dr. Irwin D. Kuntz, Dept. of Pharmaceutical Chemistry, University of California at San Francisco, San Francisco, CA 94143-0446. Tel.: 415-476-1937; Fax: 415-502-1411; E-mail: kuntz{at}cgl.ucsf.edu.
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ABSTRACT |
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INTRODUCTION |
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–7), there is a need to seek a broader approach that would permit the evaluation and comparison of such methods. A further related concern is the additivity of information when different techniques are combined. Previously, we demonstrated (8) that the basic tenets of information theory (9) can be used to quantify the information content of distance constraints. In this and a companion article (10), we apply the same general principles to simple exact models (SEMs) to draw inferences about two important tools of computational biology, sequence analysis, and force fields.
Sequence alignment is an integral part of comparative modeling protocols. Aside from ab initio methods (11), theoretical structure prediction is generally approached in two steps:
). Because sequence space is poorly characterized, it is difficult to either evaluate or improve overall performance except in the context of specific training sets. What is needed is a unified picture of the fundamental issues.
A standard way to gain insight into complex problems is through SEMs. In the protein-folding field, these models use simplified representations of sequences and structures to mimic sequence and structure interactions in real systems. Thus, self-avoiding two-dimensional lattice walks and simplified alphabets have long been used to evaluate and understand the principles of protein folding (13,14). The ability to exhaustively enumerate all states of the system affords the opportunity to describe the system's behavior unambiguously, and it can provide a clear path relationship between assumptions and consequences. Thus, SEMs are well suited for formulating and evaluating general concepts: a task that may be much more difficult with real-world examples because of their heavy parameter dependence and need for approximations (15).
In this article, we combine the use of simplified systems with information theory to derive the costs of alignment procedures, scoring matrices, and gap penalties of idealized models. We then consider the applicability of the insights gained to the current approaches to sequence alignment. As noted above, our modeling choices are chosen to illuminate the underlying issues. We consider force fields in the companion article (10).
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METHODS |
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). The normal use of sequence-alignment procedures is considered to increase the information associated with a given probe sequence. That is, one queries a database of sequences and assigns properties (structure, function) to the probe sequence based on the statistically valid matches that are found. This information increase arises because a single sequence is placed into a cluster smaller than the full set of sequences from which it was indistinguishable before the alignment procedure. However, we can equally well treat alignment as a clustering procedure in which a number of sequences are grouped together as indistinguishable. Such clustering reduces the effective number of distinguishable objects compared to the full set of unique sequences. From this point of view, information is lost because a number of sequences that were treated as distinct from one another are now considered the same (i.e., similar sequence, function, or structure). In this context, gap penalties are a direct reflection of the price that must be paid for the information loss.
Of course, it is not feasible to write out all sequences for all proteins and nucleic acids. Nor can we advance a comprehensive model for the evolutionary and structural constraints that give rise to the sequences that form the current pan-genomic database. Rather, our strategy will be to uncover general properties by making use of model systems and simplified alphabets (14,16). However, we are also interested in exploring the implications of such models for the real world of sequences and conformations. This relationship is not a formal part of information theory, and will involve additional assumptions or hypotheses, the truth of which must be established by other methods. For example, it is straightforward to evaluate the informational consequences of the proposition that the known sequences are a random subset of all possible sequences; this proposal can be directly tested statistically, but information theory, alone, cannot determine its validity.
Shannon information of a set W
The information required to select an individual entity from a set of W objects is defined as
 | (1) |
). Given a metric set, M, that partitions the objects into subsets, the information content can be measured in bits using Shannon's formulation (9),
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As mentioned previously, clustering can lead to a change in information. We relate Eqs. 1 and 2 to yield the information gain/loss of a clustering procedure as
 | (3) |
Sequence alignment
Overview
Sequences of proteins or nucleic acids of unknown structure and function are sources of information through association with other sequences whose functions/structures are already known. The most widely used associative process is alignment. Alignment algorithms can be divided into two categories, global and local. A global alignment (17) looks for the best overall similarity among sequences, whereas a local alignment (18) searches for similar sub-sequences between two proteins. Both of these algorithms make use of a variety of scoring matrices and gap penalties (19–24). Sequence-alignment problems are underdetermined, having multiple optimal solutions depending on the parameters used. Thus far, there has not been a quantitative analysis of the parameter dependence, one reason being the absence of a standard comparison metric. With an information theoretic approach, we are able to formalize the effects of parameters such as sequence length, alphabet size, etc. We consider the sets of sequences of length N, drawn from an alphabet of A characters. Assume that the characters are used with equal frequency (effects of character correlation can be readily included at a later stage). With this simplifying assumption, each sequence has equal weight and there are AN unique sequences. The information content of the set is simply Nlog2A. Alignment procedures require the definition of a template of length T < N. The template may contain gaps—that is, the string for the template may contain one or more positions that match any character. Alternatively, the template may be considered continuous and the sequences with which it is being compared can contain gaps. We ask how many sequences of an exhaustive list match a specific template. Most generally, because there is nothing of special interest for any given template, we are interested in the information content averaged over all templates of a certain type.
We begin with the case of gapless pairwise alignments and then move to multigapped alignments. We will use both exhaustive and stochastic data sets, along with simple alignment models, to provide insight into the informational issues associated with sequence comparisons.
Statistics from alignments are gathered under two scenarios:
Gapless alignments
For an A-letter alphabet, the total number of possible N-letter sequences is AN,
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For single-occurrence, ungapped alignments, we have an alternative approach using 1), the contiguous string; and 2), the standard formula for the probability of failure to match, pF. Given the probability of occurrence of the template in a single sequence, pT = (1/A)T, and the number of independent attempts (K+1), pF = (1 – pT)K+1. The probability of a hit for a sequence, pH, is then (1 – pF) and the total number of hits for the set is AN x pH,
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However, Eq. 6 provides useful values for I for the full range of K for single occurrence of templates (see Results and Discussion, below).
Gapped alignments
For more general gap distributions, in which all templates of length T = N – K are aligned against a probe of length N, we need to consider the combinatorial arrangement of gaps of varying length. For gaps of minimum-length one, there will be C (N, K) = N!/K!(N–K)! ways to arrange the gaps in an N-long sequence. However, if we require the minimum-gap size, Gmin, to be greater than one, then the effective length of the sequence is reduced to Neffective = N – K x Gmin + K. There are AK sequences for each arrangement:
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We have also found a formulation leading to an exact solution for the number of gapped matches for single occurrences of the template. The number of hits to match a given template of length T, where K = N – T, against an exhaustive set of sequences becomes
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Gap penalties
The formulas above quantify the amount of information associated with successful alignments when an exhaustive basis set of all possible sequences is available. They also can be used to set bounds on gap penalty values (see Results and Discussion).
Gap penalties can be derived by examining the length distributions of gaps in systems where structural alignment is possible. This approach is based on a general affine model of gap penalties (25) and uses a geometric distribution to assess the probability distribution of gaps, yielding, in the Qian and Goldstein treatment (26), the formula for gap initiation (
I) and gap extension (E) penalties of
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is the half-decay length of the gap length distribution. The values for Pg and are determined in a similar way to Qian and Goldstein (26). For a given sequence-length and template, we plot the distribution of gap lengths versus the probability for the observed hits (as an example, see Fig. 1). We then fit the data to an exponential of the form
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)/[1–exp(–1/)].
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) to calculate gap initiation (I) and extension(E) penalties.
Search algorithm methods
First-occurrence alignments
For every sequence, S, in the set, given a template T, we look for the first occurrence of the symbol in the first position of the template. Looking forward, we search for the first instance of the symbol in the second position of the template and so forth, until the last position in T. If a symbol is not observed in S in order of appearance in T, the search is terminated. Indices of each hit in the sequence are tabulated to determine the length of the gap among instances of each symbol present in the template.
Multiple-occurrence alignments
Fig. 2 shows how the occurrences of a template T are sought in a sequence, S, consisting of an alphabet of size A. The final list contains all the occurrences of T in S by specifying the indices of the symbols in S. The positional indices for each occurrence are used to determine the distribution of gap lengths.
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). Higher-order terms dealing with joint probability of multiple characters can also be considered as corrections to the simple assumption of equal frequencies (28). In the same way, scoring matrices with partial weights for alternative characters reduce the effective alphabet below the limit set by equal utilization of all characters. A correction term can be generated for any scoring matrix of interest. A harder question is the relationship of the observed sequences to the exhaustive set. Many hypotheses can be put forward about the mechanisms of evolution and the types of structural constraints imposed upon both nucleic acids and proteins. We do not propose in this article to select among them. Instead, we provide simple examples to illustrate how the mapping from exhaustive sequences to sequence subsets changes the information content of alignment operations and hence changes the values of gap penalties. These simple hypotheses are:
To examine the information content of a random subset of the exhaustive sequences, we generated sets of 10,000–100,000 random sequences of lengths N = 10, 20, 30 for A = 2. These were scanned with templates of various lengths and gap lengths. The number of hits was recorded with each of the sequences as a starting point and the probabilities of clustering were calculated. The information content for each alignment procedure was tabulated.
To generate a simple evolutionary model, we used the constraint that L of the N positions would not vary. This assumption produces a subset of sequences that are in exact correspondence to sequences from the exhaustive list for N' where N' = N – L. Equations 5– 9 can then be applied directly to this subset.
Correlation of sequence alignment and conformational resolution
Of course, one random model and one simple evolutionary model just begin to explore the sequence constraints operating on the natural sequences. Presumably, one of the critical limits is that most of the experimental sequences arise from sequence subsets that provide stable three-dimensional (tertiary) structures for some range of physical variables. We have not attempted to construct such a model in this article, but others have approached the problem (29,30).
In a seminal article, Chothia and Lesk (31) provided the first quantitative relationship between sequence identity and structural similarity. In recent work, this relationship has been revisited in great detail (32). For our purposes, we use the methods above and those of Sullivan and Kuntz (33) to compare the information content of sequence and structural alignments, as follows. As a model of real proteins, we choose the backbone conformations for 100-mer compact polyalanine chains, a 20-letter alphabet, and a multiple-occurrence gapped alignment model. For a given homology level, we then compare the information of sequence and structural alignments. For our example, we select a similarity level of 90%. We use Eqs. 2, 3, and 7 to determine the information from sequence alignments. From Chothia and Lesk, a 90% identity yields a root-mean-square deviation (RMSD) of 
0.5 Å. Therefore, we ask: What is the conformational probability of 100-mer compact polyalanine models falling within 0.5 Å RMSD, as derived by Sullivan and Kuntz, to determine the information required for a corresponding structural alignment?
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RESULTS AND DISCUSSION |
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) (see Fig. 3).
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). We gathered these data either from short exhaustive binary sequences or from stochastic samples of longer sequences with larger alphabets. There are two ways that we can count gap frequencies and gap lengths (see Methods, above). First, for the equations given above, we have used a first-occurrence model in which the gap-length data are taken from the initial successful match of a template to a sequence. Alternatively, we can identify all matches of a template with a specific sequence, and for each match, tabulate the gap-length information (multiple-occurrence) model. This model appears to be closer to the empirical data reported by Qian and Goldstein (26). Our basic findings are:
), we return to the relationship between the set of observed sequences and the exhaustive reference state based on the two scenarios described earlier. Random models are easily constructed and tested. The stochastically derived probabilities for the alignments of random sequences show no surprises and are equal to those from the exhaustive set of sequences within the expected statistical variation (Table 7). Evolutionary models for the observed sequences can also be constructed. Two explicit models would be: 1), use of a full alphabet for a subset of the sequence positions with the other positions fixed; and 2), restricted alphabets at all sequence positions. In the former case, we would expect Eqs. 5–9 to apply directly, but with a reduced chain length (see Methods); in the latter, Eq. 2 can be used to compensate for the unequal probabilities of each character. To test the first evolutionary model numerically, we took the exhaustive set of fully variable 15-mers embedded in 20-mers with the first five L-positions invariant. The results (Fig. 4) for first-occurrence gapped hits closely correspond to the exhaustive 15-mer data, suggesting that Eqs. 8 and 9 are good approximations for sequences generated by evolutionary relationships. However, when we compute the multiple-occurrence gap-length distributions for the same data set, the situation is more complicated. The distribution functions are intermediate between the 15-mer and 20-mer data, with the distributions closer to the 20-mer results (Fig. 5). Others have also studied the evolutionary relationships among protein and model sequences using simple models. For example, Irback and Sandelin (34) show that real sequences deviate from random sequences in the distribution of hydrophobic residues in the chains, whereas Cui et al. (35) show that crossovers and nonhomologous combinations are favored in the evolution of low energy states. However, neither study explores information content for their systems.
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The empirical gap penalties currently in use are obtained from training sets on homologous proteins. Our approach in this article has been to explore simple, exhaustive treatments of sequence alignment with a much broader reference state. We show that these efforts lead directly to a priori gap penalties that depend on models of the protein and nucleic-acid sequence universe. Our formulas suggest alphabet-size and sequence-length dependencies that are not included in current methods. They lead directly to two practical suggestions:
Most of our results are at the low end of reported range of gap penalties. One reasonable explanation is that the set of known sequences is significantly nonrandom because of some combination of evolutionary and structural constraints; for example, reduced gap probabilities inside secondary structure elements, which would lead to higher-than-random gap penalties for gaps in loops. Although we cannot say that gap penalties based on SEMs will lead to better alignments than current methods, we hope that exploring the underlying relationships in simple systems will lead to improved understanding of the basic principles.
At a higher level, we can use the machinery set up above to ask how the information content of sequence alignment compares to the information content of structural alignments. To do this we draw on the basic formulas derived above. We also use the work of Chothia and Lesk (31), which establishes the relationship between sequence identity and structural variation, and the article of Sullivan and Kuntz (36), which gives log odds probabilities of structural variation. For this purpose, we treat a specific example of a compact 100 polyalanine backbone conformations because the Chothia-Lesk relation applies to native proteins. We ask what is the information content of a 90% identity match using a 20-letter, equal frequency, alphabet. From Eq. 7 we get 87 bits of required information (IM) for a 90% identity alignment. The full protein requires 432 bits (IS) (Eqs. 1 and 2), so the information gain from a 90% alignment is 345 bits (Eq. 3). A correction for the known frequency of use of the amino acids is –17 bits, so that a 90% homology match using realistic frequencies contains something approaching 3.5 bits/residue of information. From Chothia and Lesk, as well as later work (32), we see that 90% homology implies a structural variance of 0.5 Å. Using the data from the cumulative distribution function of Sullivan and Kuntz (36) and Eq. 2 of this article, we can estimate how much information is required to achieve an RMSD of 0.5 Å for a stochastic population of compact polyalanine chains. We get 2.4–2.5 bits/residue required to select this RMSD distribution from a stochastic set of compact chains. This calculation indicates that, roughly speaking, high-end sequence alignment combined with homology modeling could approach the quality of direct structural measurements for determining backbone geometries. It also implies that the designability hypothesis (i.e., many sequences per structure) derived from lattice models (37) can also be supported from information content assessments of off-lattice conformational estimates.
In the companion article following, we extend these calculations to include comparison with force fields as well, showing how the use of information theory allows direct comparison of quite diverse techniques.
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ACKNOWLEDGEMENTS |
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This work was supported by a grant from the National Science Foundation (No. CHE-0118481), R. Kip Guy, Principal Investigator; a National Institutes of Health grant (No. RR019864), C. Pancerella, Principal Investigator; and a University of California President's Dissertation Year Fellowship (to T.A.).
Submitted on October 12, 2004; accepted for publication August 15, 2005.
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–) exhaustive sets. Some of the distributions from the evolutionary set are almost identical to the distributions from the 15-mer and 20-mer exhaustive sets.