01.4.pssm theory
- 1. Jacques.van.Helden@ulb.ac.be Université Libre de Bruxelles, Belgique Laboratoire de Bioinformatique des Génomes et des Réseaux (BiGRe) http://www.bigre.ulb.ac.be/ 1 Position-specific scoring matrices (PSSM) Regulatory sequence analysis
- 2. 2 . Binding sites for the yeast Pho4p transcription factor (Source : Oshima et al. Gene 179, 1996; 171-177) Alignment of transcription factor binding sites Gene Site Name Sequence Affinity PHO5 UASp2 ---aCtCaCACACGTGGGACTAGC- high PHO84 Site D ---TTTCCAGCACGTGGGGCGGA-- high PHO81 UAS ----TTATGGCACGTGCGAATAA-- high PHO8 Proximal GTGATCGCTGCACGTGGCCCGA--- high group 1 consensus ---------gCACGTGgg------- high PHO5 UASp1 --TAAATTAGCACGTTTTCGC---- medium PHO84 Site E ----AATACGCACGTTTTTAATCTA medium group 2 consensus --------cgCACGTTtt------- medium Degenerate consensus ---------GCACGTKKk------- high-med Non-binding sites PHO5 UASp3 --TAATTTGGCATGTGCGATCTC-- No binding PHO84 Site C -----ACGTCCACGTGGAACTAT-- No binding PHO84 Site A -----TTTATCACGTGACACTTTTT No binding PHO84 Site B -----TTACGCACGTTGGTGCTG-- No binding PHO8 Distal ---TTACCCGCACGCTTAATAT--- No binding IUPAC ambiguous nucleotide code A A Adenine C C Cy tosine G G Guanine T T Thy mine R A or G puRine Y C or T pYrimidine W A or T Weak hy drogen bonding S G or C Strong hy drogen bonding M A or C aMino group at common position K G or T Keto group at common position H A, C or T not G B G, C or T not A V G, A, C not T D G, A or T not C N G, A, C or T aNy
- 3. Jacques.van.Helden@ulb.ac.be Université Libre de Bruxelles, Belgique Laboratoire de Bioinformatique des Génomes et des Réseaux (BiGRe) http://www.bigre.ulb.ac.be/ 3 From alignments to weights Regulatory sequence analysis
- 4. 4 Sequence logo Tom Schneider’s sequence logo (generated with Web Logo http://weblogo.berkeley.edu/logo.cgi) Count matrix (TRANSFAC matrix F$PHO4_01) Residueposition 1 2 3 4 5 6 7 8 9 10 11 12 A 1 3 2 0 8 0 0 0 0 0 1 2 C 2 2 3 8 0 8 0 0 0 2 0 2 G 1 2 3 0 0 0 8 0 5 4 5 2 T 4 1 0 0 0 0 0 8 3 2 2 2 Sum 8 8 8 8 8 8 8 8 8 8 8 8
- 5. 5 Frequency matrix Pos 1 2 3 4 5 6 7 8 9 10 11 12 A 0.13 0.38 0.25 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.13 0.25 C 0.25 0.25 0.38 1.00 0.00 1.00 0.00 0.00 0.00 0.25 0.00 0.25 G 0.13 0.25 0.38 0.00 0.00 0.00 1.00 0.00 0.63 0.50 0.63 0.25 T 0.50 0.13 0.00 0.00 0.00 0.00 0.00 1.00 0.38 0.25 0.25 0.25 Sum 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 A alphabet size (=4) ni,j, occurrences of residue i at position j pi prior residue probability for residue i fi,j relative frequency of residue i at position j Reference: Hertz (1999). Bioinformatics 15:563-577. ! fi, j = ni, j ni, j i=1 A "
- 6. 6 Corrected frequency matrix P r Pos 1 2 3 4 5 6 7 8 9 10 11 12 A 0.15 0.37 0.26 0.04 0.93 0.04 0.04 0.04 0.04 0.04 0.15 0.26 C 0.24 0.24 0.35 0.91 0.02 0.91 0.02 0.02 0.02 0.24 0.02 0.24 G 0.13 0.24 0.35 0.02 0.02 0.02 0.91 0.02 0.58 0.46 0.58 0.24 T 0.48 0.15 0.04 0.04 0.04 0.04 0.04 0.93 0.37 0.26 0.26 0.26 Sum 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 A alphabet size (=4) ni,j, occurrences of residue i at position j pi prior residue probability for residue i fi,j relative frequency of residue i at position j k pseudo weight (arbitrary, 1 in this case) f'i,j corrected frequency of residue i at position j Reference: Hertz (1999). Bioinformatics 15:563-577. ! fi, j ' = ni, j + k /A ni, j i=1 A " + k ! fi, j ' = ni, j + pik ni, j i=1 A " + k 1st option: identically distributed pseudo-weight 2nd option: pseudo-weight distributed according to residue priors
- 7. 7 Weight matrix (Bernoulli model) Prior Pos 1 2 3 4 5 6 7 8 9 10 11 12 0.325 A -0.79 0.13 -0.23 -2.20 1.05 -2.20 -2.20 -2.20 -2.20 -2.20 -0.79 -0.23 0.175 C 0.32 0.32 0.70 1.65 -2.20 1.65 -2.20 -2.20 -2.20 0.32 -2.20 0.32 0.175 G -0.29 0.32 0.70 -2.20 -2.20 -2.20 1.65 -2.20 1.19 0.97 1.19 0.32 0.325 T 0.39 -0.79 -2.20 -2.20 -2.20 -2.20 -2.20 1.05 0.13 -0.23 -0.23 -0.23 1.000 Sum -0.37 -0.02 -1.02 -4.94 -5.55 -4.94 -4.94 -5.55 -3.08 -1.13 -2.03 0.19 A alphabet size (=4) ni,j, occurrences of residue i at position j pi prior residue probability for residue i fi,j relative frequency of residue i at position j k pseudo weight (arbitrary, 1 in this case) f'i,j corrected frequency of residue i at position j Wi,j weight of residue i at position j ! Wi, j = ln fi, j ' pi " # $ % & ' ! fi, j ' = ni, j + pik nr, j r=1 A " + k Reference: Hertz (1999). Bioinformatics 15:563-577. The use of a weight matrix relies on Bernoulli assumption If we assume, for the background model, an independent succession of nucleotides (Bernoulli model), the weight WS of a sequence segment S is simply the sum of weights of the nucleotides at successive positions of the matrix (Wi,j). In this case, it is convenient to convert the PSSM into a weight matrix, which can then be used to assign a score to each position of a given sequence.
- 8. 8 Properties of the weight function ! Wi, j = ln fi, j ' pi " # $ % & ' ! fi, j ' = ni, j + pik ni, j i=1 A " + k fi, j ' i=1 A " =1 The weight is positive when f’i,j > pi (favourable positions for the binding of the transcription factor) negative when f’i,j < pi (unfavourable positions)
- 9. Jacques.van.Helden@ulb.ac.be Université Libre de Bruxelles, Belgique Laboratoire de Bioinformatique des Génomes et des Réseaux (BiGRe) http://www.bigre.ulb.ac.be/ 9 Information content Regulatory sequence analysis
- 10. 10 Shannon uncertainty Shannon uncertainty Hs(j): uncertainty of a column of a PSSM Hg: uncertainty of the background (e.g. a genome) Special cases of uncertainty (for a 4 letter alphabet) min(H)=0 • No uncertainty at all: the nucleotide is completely specified (e.g. p={1,0,0,0}) H=1 • Uncertainty between two letters (e.g. p={0.5,0,0,0.5}) max(H) = 2 (Complete uncertainty) • One bit of information is required to specify the choice between each alternative (e.g. p={0.25,0.25,0.25,0.25}). • Two bits are required to specify a letter in a 4- letter alphabet. Rseq Schneider (1986) defines an information content based on Shannon’s uncertainty. R* seq For skewed genomes (i.e. unequal residue probabilities), Schneider recommends an alternative formula for the information content . This is the formula that is nowadays used. Adapted from Schneider (1986) ! Hs j( )= " fi, j log2( fi, j ) i=1 A # Hg = " pi log2(pi) i=1 A # Rseq j( ) = Hg " Hs j( ) Rseq = Rseq j( ) j=1 w # Rseq * j( ) = fi, j log2 fi, j pi $ % & ' ( ) i=1 A # Rseq * = Rseq * j( ) j=1 w #
- 11. 11 Schneider logos Schneider (1990) proposes a graphical representation based on his previous entropy (H) for representing the importance of each residue at each position of an alignment. He provides a new formula for Rseq Hs(j) uncertainty of column j Rseq(j) “information content” of column j (beware, this definition differs from Hertz’ information content) e(n) correction for small samples (pseudo-weight) Remarks This information content does not include any correction for the prior residue probabilities (pi) This information content is expressed in bits. Boundaries min(Rseq)=0 equiprobable residues max(Rseq)=2 perfect conservation of 1 residue with a pseudo-weight of 0, Sequence logos can be generated from aligned sequences on the Weblogo server http://weblogo.berkeley.edu/ ! Hs j( )= " fij log2( fij ) i=1 A # Rseq j( ) = 2 " Hs j( )+ e n( ) hij = fijRseq ( j) Pho4p binding motif
- 13. 13 Information content Prior Pos 1 2 3 4 5 6 7 8 9 10 11 12 0.325 A -0.12 0.05 -0.06 -0.08 0.97 -0.08 -0.08 -0.08 -0.08 -0.08 -0.12 -0.06 0.175 C 0.08 0.08 0.25 1.50 -0.04 1.50 -0.04 -0.04 -0.04 0.08 -0.04 0.08 0.175 G -0.04 0.08 0.25 -0.04 -0.04 -0.04 1.50 -0.04 0.68 0.45 0.68 0.08 0.325 T 0.19 -0.12 -0.08 -0.08 -0.08 -0.08 -0.08 0.97 0.05 -0.06 -0.06 -0.06 1.000 Sum 0.11 0.09 0.36 1.29 0.80 1.29 1.29 0.80 0.61 0.39 0.47 0.04 ! Imatrix = Ii, j i=1 A " j=1 w " A alphabet size (=4) ni,j, occurrences of residue i at position j w matrix width (=12) pi prior residue probability for residue i fi,j relative frequency of residue i at position j k pseudo weight (arbitrary, 1 in this case) f'i,j corrected frequency of residue i at position j Wi,j weight of residue i at position j Ii,j information of residue i at position j ! fi, j ' = ni, j + pik ni, j i=1 A " + k ! Ii, j = fi, j ' ln fi, j ' pi " # $ % & ' Reference: Hertz (1999). Bioinformatics 15:563-577. ! Ij = Ii, j i=1 A "
- 14. 14 Information content Iij of a cell of the matrix For a given cell of the matrix Iij is positive when f’ij > pi (i.e. when residue i is more frequent at position j than expected by chance) Iij is negative when f’ij < pi Iij tends towards 0 when f’ij -> 0 (because limitx->0 x*ln(x) = 0)
- 15. 15 Information content of a column of the matrix For a given column i of the matrix The information of the column (Ij) is the sum of information of its cells. Ij is always positive Ij is always positive Ij is 0 when the frequency of all residues equal their prior probability (fij=pi) Ij is maximal when • the residue im with the lowest prior probability has a frequency of 1 (all other residues have a frequency of 0) • and the pseudo-weight is 0 ! Ij = Ii, j i=1 A " = fi, j ' ln fi, j ' pi # $ % & ' ( i=1 A " ! im = argmini (pi ) k = 0 max(Ij )=1*ln( 1 pi ) = "ln(pi )
- 16. 16 ! Imatrix = Ii, j i=1 A " j=1 w " ! P site( ) " e#Imatrix Information content of the matrix The total information content represents the capability of the matrix to make the distinction between a binding site (represented by the matrix) and the background model. The information content also allows to estimate an upper limit for the expected frequency of the binding sites in random sequences. The pattern discovery program consensus (developed by Jerry Hertz) optimises the information content in order to detect over-represented motifs. Note that this is not the case of all pattern discovery programs: the gibbs sampler algorithm optimizes a log-likelihood. Reference: Hertz (1999). Bioinformatics 15:563-577.
- 17. 17 Information content: effect of prior probabilities The upper bound of Ij increases when pi decreases Ij -> Inf when pi -> 0 The information content, as defined by Gerald Hertz, has thus no upper bound.
- 18. 18 References - PSSM information content Papers by Tom Schneider Schneider, T.D., G.D. Stormo, L. Gold, and A. Ehrenfeucht. 1986. Information content of binding sites on nucleotide sequences. J Mol Biol 188: 415-431. Schneider, T.D. and R.M. Stephens. 1990. Sequence logos: a new way to display consensus sequences. Nucleic Acids Res 18: 6097-6100. Tom Schneider’s publications online • http://www.lecb.ncifcrf.gov/~toms/paper/index.html Papers by Gerald Hertz Hertz, G.Z. and G.D. Stormo. 1999. Identifying DNA and protein patterns with statistically significant alignments of multiple sequences. Bioinformatics 15: 563-577.