Information theory

Information theory

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This is an introduction to information theory for the bioinformatics lab.

 

Contents

$H=-\sum _{i=0}^{n}p_{i}\log _{2}p_{i}$

$I=H_{ref}-H_{obs}$

• Information theory
• Entropy in thermodynamics and information theory
• Mutual information

   

Further reading and resources

Shannon's "Mathematical Theory of Communication" (at Bell labs)

WebLogo server

Hopf et al. (2012) Three-dimensional structures of membrane proteins from genomic sequencing. Cell 149:1607-21. (pmid: 22579045)

[ PubMed ] [ DOI ] We show that amino acid covariation in proteins, extracted from the evolutionary sequence record, can be used to fold transmembrane proteins. We use this technique to predict previously unknown 3D structures for 11 transmembrane proteins (with up to 14 helices) from their sequences alone. The prediction method (EVfold_membrane) applies a maximum entropy approach to infer evolutionary covariation in pairs of sequence positions within a protein family and then generates all-atom models with the derived pairwise distance constraints. We benchmark the approach with blinded de novo computation of known transmembrane protein structures from 23 families, demonstrating unprecedented accuracy of the method for large transmembrane proteins. We show how the method can predict oligomerization, functional sites, and conformational changes in transmembrane proteins. With the rapid rise in large-scale sequencing, more accurate and more comprehensive information on evolutionary constraints can be decoded from genetic variation, greatly expanding the repertoire of transmembrane proteins amenable to modeling by this method.

Marks et al. (2011) Protein 3D structure computed from evolutionary sequence variation. PLoS ONE 6:e28766. (pmid: 22163331)

[ PubMed ] [ DOI ] The evolutionary trajectory of a protein through sequence space is constrained by its function. Collections of sequence homologs record the outcomes of millions of evolutionary experiments in which the protein evolves according to these constraints. Deciphering the evolutionary record held in these sequences and exploiting it for predictive and engineering purposes presents a formidable challenge. The potential benefit of solving this challenge is amplified by the advent of inexpensive high-throughput genomic sequencing.In this paper we ask whether we can infer evolutionary constraints from a set of sequence homologs of a protein. The challenge is to distinguish true co-evolution couplings from the noisy set of observed correlations. We address this challenge using a maximum entropy model of the protein sequence, constrained by the statistics of the multiple sequence alignment, to infer residue pair couplings. Surprisingly, we find that the strength of these inferred couplings is an excellent predictor of residue-residue proximity in folded structures. Indeed, the top-scoring residue couplings are sufficiently accurate and well-distributed to define the 3D protein fold with remarkable accuracy.We quantify this observation by computing, from sequence alone, all-atom 3D structures of fifteen test proteins from different fold classes, ranging in size from 50 to 260 residues, including a G-protein coupled receptor. These blinded inferences are de novo, i.e., they do not use homology modeling or sequence-similar fragments from known structures. The co-evolution signals provide sufficient information to determine accurate 3D protein structure to 2.7-4.8 Å C(α)-RMSD error relative to the observed structure, over at least two-thirds of the protein (method called EVfold, details at http://EVfold.org). This discovery provides insight into essential interactions constraining protein evolution and will facilitate a comprehensive survey of the universe of protein structures, new strategies in protein and drug design, and the identification of functional genetic variants in normal and disease genomes.

Thomsen & Nielsen (2012) Seq2Logo: a method for construction and visualization of amino acid binding motifs and sequence profiles including sequence weighting, pseudo counts and two-sided representation of amino acid enrichment and depletion. Nucleic Acids Res 40:W281-7. (pmid: 22638583)

[ PubMed ] [ DOI ] Seq2Logo is a web-based sequence logo generator. Sequence logos are a graphical representation of the information content stored in a multiple sequence alignment (MSA) and provide a compact and highly intuitive representation of the position-specific amino acid composition of binding motifs, active sites, etc. in biological sequences. Accurate generation of sequence logos is often compromised by sequence redundancy and low number of observations. Moreover, most methods available for sequence logo generation focus on displaying the position-specific enrichment of amino acids, discarding the equally valuable information related to amino acid depletion. Seq2logo aims at resolving these issues allowing the user to include sequence weighting to correct for data redundancy, pseudo counts to correct for low number of observations and different logotype representations each capturing different aspects related to amino acid enrichment and depletion. Besides allowing input in the format of peptides and MSA, Seq2Logo accepts input as Blast sequence profiles, providing easy access for non-expert end-users to characterize and identify functionally conserved/variable amino acids in any given protein of interest. The output from the server is a sequence logo and a PSSM. Seq2Logo is available at http://www.cbs.dtu.dk/biotools/Seq2Logo (14 May 2012, date last accessed).

Johansson & Toh (2010) A comparative study of conservation and variation scores. BMC Bioinformatics 11:388. (pmid: 20663120)

[ PubMed ] [ DOI ] BACKGROUND: Conservation and variation scores are used when evaluating sites in a multiple sequence alignment, in order to identify residues critical for structure or function. A variety of scores are available today but it is not clear how different scores relate to each other. RESULTS: We applied 25 conservation and variation scores to alignments from the Catalytic Site Atlas (CSA). We calculated distances among scores based on correlation coefficients, and constructed a dendrogram of the scores by average linking cluster analysis. The cluster analysis showed that most scores fall into one of two groups--substitution matrix based group and frequency based group respectively. We also evaluated the scores' performance in predicting catalytic sites and found that frequency based scores generally perform best. CONCLUSIONS: Conservation and variation scores can be classified into mainly two large groups. When using a score to predict catalytic sites, frequency based scores that also consider a background distribution are most successful.

Dou et al. (2010) Several appropriate background distributions for entropy-based protein sequence conservation measures. J Theor Biol 262:317-22. (pmid: 19808039)

[ PubMed ] [ DOI ] Amino acid background distribution is an important factor for entropy-based methods which extract sequence conservation information from protein multiple sequence alignments (MSAs). However, MSAs are usually not large enough to allow a reliable observed background distribution. In this paper, we propose two new estimations of background distribution. One is an integration of the observed background distribution and the position-specific residue distribution, and the other is a normalized square root of observed background frequency. To validate these new background distributions, they are applied to the relative entropy model to find catalytic sites and ligand binding sites from protein MSAs. Experimental results show that they are superior to the observed background distribution in predicting functionally important residues.

Capra & Singh (2007) Predicting functionally important residues from sequence conservation. Bioinformatics 23:1875-82. (pmid: 17519246)

[ PubMed ] [ DOI ] MOTIVATION: All residues in a protein are not equally important. Some are essential for the proper structure and function of the protein, whereas others can be readily replaced. Conservation analysis is one of the most widely used methods for predicting these functionally important residues in protein sequences. RESULTS: We introduce an information-theoretic approach for estimating sequence conservation based on Jensen-Shannon divergence. We also develop a general heuristic that considers the estimated conservation of sequentially neighboring sites. In large-scale testing, we demonstrate that our combined approach outperforms previous conservation-based measures in identifying functionally important residues; in particular, it is significantly better than the commonly used Shannon entropy measure. We find that considering conservation at sequential neighbors improves the performance of all methods tested. Our analysis also reveals that many existing methods that attempt to incorporate the relationships between amino acids do not lead to better identification of functionally important sites. Finally, we find that while conservation is highly predictive in identifying catalytic sites and residues near bound ligands, it is much less effective in identifying residues in protein-protein interfaces. AVAILABILITY: Data sets and code for all conservation measures evaluated are available at http://compbio.cs.princeton.edu/conservation/

Wang & Samudrala (2006) Incorporating background frequency improves entropy-based residue conservation measures. BMC Bioinformatics 7:385. (pmid: 16916457)

[ PubMed ] [ DOI ] BACKGROUND: Several entropy-based methods have been developed for scoring sequence conservation in protein multiple sequence alignments. High scoring amino acid positions may correlate with structurally or functionally important residues. However, amino acid background frequencies are usually not taken into account in these entropy-based scoring schemes. RESULTS: We demonstrate that using a relative entropy measure that incorporates amino acid background frequency results in improved performance in identifying functional sites from protein multiple sequence alignments. CONCLUSION: Our results suggest that the application of appropriate background frequency information may lead to more biologically relevant results in many areas of bioinformatics.

Vingron & Sibbald (1993) Weighting in sequence space: a comparison of methods in terms of generalized sequences. Proc Natl Acad Sci U.S.A 90:8777-81. (pmid: 8415606)

[ PubMed ] [ DOI ] Four methods for weighting aligned biological sequences have recently appeared that differ mathematically, philosophically, and in their results. Thus, while there is consensus about the need to weight sequences, the method to use is contentious. A geometric analysis based on a continuous sequence space is presented that provides a common framework in which to compare the methods. It is concluded that there are two "best" methods. When the sequences are known to be phylogenetically related and a tree can be generated without introducing excessive stress into the data, the method of Altschul et al. [Altschul, S. F., Carroll, R. J. & Lipman, D. J. (1989) J. Mol. Biol. 207, 647-653] is appropriate. When the sequences are not known to be phylogenetically related or a tree cannot be produced without unduly distorting the distances between the sequences, a modification of the method of Sibbald and Argos [Sibbald, P. R. & Argos, P. (1990) J. Mol. Biol. 216, 813-818] is preferable.

Henikoff & Henikoff (1994) Position-based sequence weights. J Mol Biol 243:574-8. (pmid: 7966282)

[ PubMed ] [ DOI ] Sequence weighting methods have been used to reduce redundancy and emphasize diversity in multiple sequence alignment and searching applications. Each of these methods is based on a notion of distance between a sequence and an ancestral or generalized sequence. We describe a different approach, which bases weights on the diversity observed at each position in the alignment, rather than on a sequence distance measure. These position-based weights make minimal assumptions, are simple to compute, and perform well in comprehensive evaluations.

Schneider & Stephens (1990) Sequence logos: a new way to display consensus sequences. Nucleic Acids Res 18:6097-100. (pmid: 2172928)

[ PubMed ] [ DOI ] A graphical method is presented for displaying the patterns in a set of aligned sequences. The characters representing the sequence are stacked on top of each other for each position in the aligned sequences. The height of each letter is made proportional to its frequency, and the letters are sorted so the most common one is on top. The height of the entire stack is then adjusted to signify the information content of the sequences at that position. From these 'sequence logos', one can determine not only the consensus sequence but also the relative frequency of bases and the information content (measured in bits) at every position in a site or sequence. The logo displays both significant residues and subtle sequence patterns.