6.047 / Computational Biology: Genomes, Networks, Evolution Fall 2008

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1 MIT OpenCourseWare / Computational Biology: Genomes, Networks, Evolution Fall 2008 For information about citing these materials or our Terms of Use, visit:

2 Computational Biology: Genomes, Networks, Evolution Computational Gene Prediction and Generalized Hidden Markov Models Lecture 8 September 30, 2008

3 Today Gene Prediction Overview HMMs for Gene Prediction GHMMs for Gene Prediction Genscan

4 Genome Annotation Genome Sequence RNA Transcription Translation Protein

5 Eukaryotic Gene Structure ATG complete mrna coding segment TGA exon intron exon intron exon ATG... GT AG... GT AG... TGA start codon donor site acceptor site donor site acceptor site stop codon Courtesy of William Majoros. Used with permission.

6 Translation Chain of amino acids transfer RNA one amino acid Anti-codon (3 bases) Codon (3 bases) messenger RNA (mrna) Courtesy of William Majoros.Used with permission. Ribosome (performs translation)

7 Genetic Code Acid A C GCA GCC GCG GCT G H GGA GGC GGG GGT M N ATG S T AGC AGT TCA TCC TCG TCT ACA D E F Codons Acid Codons Acid Codons Acid Codons TGC TGT CAC CAT AAC AAT ACT GAC GAT I K ATA ATC ATT P Q CCA CCC CCG CCT V W GTA GTC GTG GTT TGG GAA GAG TTC TTT L AAA AAG CTA CTC CTG CTT TTA R CAA CAG AGA AGG CGA CGC CGG CGT Y ACC ACG TAC TAT Each amino acid is encoded by one or more codons. Each codon encodes a single amino acid. The third position of the codon is the most likely to vary, for a given amino acid. Figure by MIT OpenCourseWare.

8 Gene Prediction as Parsing Given a genome sequence, we wish to label each nucleotide according to the parts of genes Exon, intron, intergenic, etc The sequence of labels must follow the syntax of genes e.g. exons must be followed by introns or intergenic not by other exons We wish to find the optimal parsing of a sequence by some measure

9 Features A feature is any DNA subsequence of biological significance. For practical reasons, we recognize two broad classes of features: signals short, fixed-length features content regions variable-length features Courtesy of William Majoros. Used with permission.

10 Signals Associated with short fixed(-ish) length sequences Start Codon - ATG Stop Codons TAA, TAG,TGA 5 3 E I E I E 5 Splice Site (Acceptor) 3 Splice Site (Donor)

11 Content Regions Content regions often have characteristic base composition Example Recall: often multiple codons for each amino acid All codons are not used equally Characteristic higher order nucleotide statistics in coding sequences (hexanucleotides) P exon (X i X i-1, X i-2, X i-3, X i-4, X i-5 ) 5 3 E I E I E P(X i X i-1, X i-2, X i-3, X i-4, X i-5 ) = P(X i )

12 Extrinsic Evidence intron exon Gene Gene Prediction Algorithms BLAST Hits HMMer Domains EST Alignments Neurospora crassa (a fungus)

13 HMMs for Gene Prediction States correspond to gene and genomic regions (exons, introns, intergenic, etc) State transitions ensure legal parses Emission matrices describe nucleotide statistics for each state

14 A (Very) Simple HMM Donor T Intron Acceptor A Donor G Acceptor G the Markov model: Start Codon G Start Codon T Exon Stop Codon G Stop Codon T Start Codon A Intergenic Stop Codon A q 0 Courtesy of William Majoros. Used with permission.

15 A Generative Model We can use this HMM to generate a sequence and state labeling The initial state is q 0 Donor T Intron Acceptor A Choose a subsequent state, conditioned on the current state, according to a jk =P(q k q j ) Choose a nucleotide to emit from the state emissions matrix e k (X i ) Donor G Start Codon G Start Codon T Exon Acceptor G Stop Codon G Stop Codon T Repeat until number of nucleotides equals desired length of sequence Start Codon A Intergenic Stop Codon A q 0

16 But We Usually Have the Sequence Donor T Intron Acceptor A Donor G Acceptor G the Markov model: Start Codon G Start Codon T Exon Stop Codon G Stop Codon T Start Codon A Intergenic Stop Codon A q 0 the input sequence: AGCTAGCAGTATGTCATGGCATGTTCGGAGGTAGTACGTAGAGGTAGCTAGTATAGGTCGATAGTACGCGA What is the best state labeling? Courtesy of William Majoros. Used with permission.

17 Finding The Most Likely Path A sensible choice is to choose π that maximizes P[π x] This is equivalent to finding path π* that maximizes total joint probability P[ x, π ]: P(x,π) = a 0π1 * Π i e πi (x i ) a πi π i+1 start emission transition How do we select π efficiently?

18 A (Very) Simple HMM Donor T Intron Acceptor A Donor G Acceptor G the Markov model: Start Codon G Start Codon T Exon Stop Codon G Stop Codon T Start Codon A Intergenic Stop Codon A q 0 the input sequence: the most probable path: the gene prediction: AGCTAGCAGTATGTCATGGCATGTTCGGAGGTAGTACGTAGAGGTAGCTAGTATAGGTCGATAGTACGCGA Courtesy of William Majoros. Used with permission. exon 1 exon 2 exon 3

19 A Real HMM Gene Predictor Title page of journal article removed due to copyright restrictions. The article is the following: Krogh, Anders, I. Saira Mian, and David Haussler. "A Hidden Markov Model That Finds Genes in E.coli DNA." Nucleic Acids Research 22, no. 22 (1994):

20 HMM Limitations The HMM framework imposes constraints on state paths Donor T Intron Acceptor A Donor G Acceptor G Start Codon G Exon Stop Codon G Start Codon T Stop Codon T Start Codon A Intergenic Stop Codon A q 0

21 Human Exon Lengths Courtesy of Christopher Burge. Used with permission. Burge, MIT PhD Thesis

22 Fungal Intron Lengths 50% 40% N. crassa 50% 40% S. cerevisiae 30% 30% 20% 20% 10% 10% 0% 50% 40% C. neoformans 0% 50% 40% S. pombe 30% 30% 20% 20% 10% 10% 0% 0% Nucleotides Nucleotides Figure by MIT OpenCourseWare. Kupfer et al., (2004) Eukaryotic Cell

23 HMM Emissions Donor T Intron Acceptor A HMMs typically emit a single nucleotide per state Donor G Acceptor G Exon P(T)=1 P(A),P(G),P(C)=0 Intergenic q 0

24 Generalized HMMs (GHMMs) GHMMs emit more than one symbol per state Emissions probabilities modeled by any arbitrary probabilistic model Feature lengths are explicitly modeled W.H. Majaros ( Courtesy of William Majoros. Used with permission. Human Introns Courtesy of Elsevier, Inc. Used with permission. Burge, Karlin (1997)

25 GHMM Elements States Observations Initial state probabilities Transition Probabilities Q V π i = P(q 0 =i) a jk = P(q i =k q i-1 =j) Like HMMs Duration Probabilities f k (d)=p(state k of length d) Emission Probabilities e k (X α,α+d )=P k (X α X α+d q k,d) Now emit a subsequence

26 Model Abstraction in GHMMs W.H. Majaros ( Models must return the probability of a subsequence given a state and duration Courtesy of William Majoros. Used with permission.

27 GHMM Submodel Examples 1. WMM (Weight Matrix) L 1 i=0 P i (x i ) 2. Nth-order Markov Chain (MC) n 1 i=0 L 1 P(x i x 0...x i 1 ) P(x i x i n...x i 1 ) i=n 3. Three-Periodic Markov Chain (3PMC) 5. Codon Bias n 1 i=0 P(x α+3i x α+3i+1 x α+3i+2 ) L 1 i=0 P ( f + i)(mod 3) (x i ) 6. MDD Ref: Burge C (1997) Identification of complete gene structures in human genomic DNA. PhD thesis. Stanford University. 7. Interpolated Markov Model Ref: Salzberg SL, Delcher AL, Kasif S, White O (1998) Microbial gene identification using interpolated Markov models. Nucleic Acids Research 26: P IMM e (s g 0...g k 1 ) = λ G k P e (s g 0...g k 1 )+ (1 λ G k )P IMM e (s g 1...g k 1 ) if k > 0 P e (s) if k = 0 Courtesy of William Majoros. Used with permission.

28 GHMMs as Generative Models Like HMM, we can use a GHMM to generate a sequence and state labeling Choose initial state, q 1, from π i Choose a duration d 1, from length distribution f q1 (d) Generate a subsequence from state model e q1 (X 1,d ) Choose a subsequent state, q 2, conditioned on the current state, according to a jk =P(q k q j ) Repeat until number of nucleotides equals desired length of sequence Given a sequence, how do we pick a state labeling (segmentation)?

29 The Viterbi Algorithm - HMMs State 1 2 V k (i) K x 1 x 2 x 3..x N Input: x = x1 xn Initialization: V 0 (0)=1, V k (0) = 0, for all k > 0 Iteration: V k (i) = e K (x i ) max j a jk V j (i-1) Termination: P(x, π*) = max k V k (N) Traceback: Follow max pointers back In practice: Use log scores for computation Running time and space: Time: O(K 2 N) Space: O(KN)

30 HMM Viterbi Recursion hidden states V j (i-1) j a jk k e k V k (i) observations x i-1 x i Assume we know V j for the previous time step (i-1) Calculate V k (i) = e k (x i ) * max j ( V j (i-1) a jk ) current max this emission max ending in state j at step i-1 Transition from state j all possible previous states j

31 GHMM Recursion hidden states V j (i-1) j a jk k e k V k (i) observations x i-1 x i We could have come from state j at the last position

32 GHMM Recursion But we could also have come from state j 4 positions ago (State k could have duration 4 so far)

33 GHMM Recursion In general, we could have come from state j d positions ago (State k could have duration d so far)

34 GHMM Recursion This leads to the following recursion equation: () max max ( ) ( ) ( ) V i = P x x k V i d P d k a k i i d j jk j d Max over all prev states and state durations Prob of subsequence given state k Max ending in state j at step i-d Prob that state k has duration d Transition from j->k

35 Comparing GHMMs and HMMs HMM O(K 2 L) V k (i) = e k (x i ) * max j ( V j (i-1) a jk ) current max this emission max ending in state j at step i-1 Transition from state j GHMM O(K 2 L 3 ) all possible previous states j Similar modifications needed for forward and backward algorithms () max max ( ) ( ) ( ) V i = P x x k V i d P d k a k i i d j jk j d Max over all prev states and state durations Prob of subsequence given state k Max ending in state j at step i-d Prob that state k has duration d Transition from j->k

36 Courtesy of Elsevier, Inc. Used with permission.

37 Genscan - Burge and Karlin, (1997) Explicit State Duration GHMM 5 th order markov models for coding and non-coding sequences Each CDS frame has own model WAM models for start/stop codons and acceptor sites MDD model for donor sites Separate parameters for regions of different GC content Model +/- strand concurrently Courtesy of Elsevier, Inc. Used with permission.

38 Types of Exons Three types of exons are defined, for convenience: initial exons extend from a start codon to the first donor site; internal exons extend from one acceptor site to the next donor site; final exons extend from the last acceptor site to the stop codon; single exons (which occur only in intronless genes) extend from the start codon to the stop codon: Courtesy of William Majoros. Used with permission.

39 Intron and Exon Phase Codon Phase 0 Intron Exon Intron Exon Phase 0 Exon Phase 1 Intron Phase 1 Exon Exon Intron Exon Phase 2 Intron Phase 2 Exon Exon Intron Exon

40 Two State Types in Genscan D-type represented by diamonds C-type represented by circles D states are always followed by C and vice versa C states always preceded by same D- state Courtesy of Elsevier, Inc. Used with permission.

41 Two State Types in Genscan D-Type geometric length distribution Intergenic regions UTRs Introns P Xa, c = P ( Xa, b) P ( Xb+ 1, c) ( ) k k k f ( d) = P(state duration d state k)=p f ( d 1) k Sequence models are factorable : k k Increasing duration by one changes probability by constant factor

42 Two State Types in Genscan C-Type general length distributions and sequence generating modes Exons (initial, internal, terminal) Promotors Poly-A Signals

43 Genscan Inference Genscan uses same basic forward, backward, and viterbi algorithms as generic GHMMs But assumptions about C, and D states reduce algorithmic complexity

44 Genscan Inference C-state List D states State 1 2 V k (i) C states K C2 C1 x 1 x 2 x 3..x N L k (i) C1, duration 2, previous D-state=1 C2, duration 3, previous D-state=2

45 Genscan Viterbi Induction Max from previous step in this state Extend D-type state by one step (factorable state) Probability of emiting one more nucleotide from state k Was in state K in previous step ( ) ( ) V i+ 1 = max [ V i p e ( X ), k k k k i+ 1 c c-type states ( ) c max { V i d 1 P(y c)(1-p ) P(d c)p(x..x c) P(k c) p e ( X )}] Max probability of ending D type state y c at position i-d-1 y c y i-d i k k i+ 1 c Probability of transition from y c to c Terminate D- type state length Probability that C state was duration d Probability of subsequence of length d from state c Probability of transition from c to k Probability of nucleotide from state k Just transitioned from c-type state c of duration d which previously transitioned from D-type state y c

46 The gene finder will later be deployed for use in predicting the rest of the organism s genes. The way in which the model parameters are inferred during training can significantly affect the accuracy of the deployed program. Courtesy of William Majoros. Used with permission. Training A Gene Predictor During training of a gene finder, only a subset K of an organism s gene set will be available for training:

47 Training A Gene Predictor Courtesy of William Majoros. Used with permission.

48 Gene Prediction Accuracy Gene predictions can be evaluated in terms of true positives (predicted features that are real), true negatives (non-predicted features that are not real), false positives (predicted features that are not real), and false negatives (real features that were not predicted: These definitions can be applied at the whole-gene, whole-exon, or individual nucleotide level to arrive at three sets of statistics. Courtesy of William Majoros. Used with permission.

49 Sn = Accuracy Metrics SMC = TP TP + FN Sp = TP TP + FP TP +TN TP +TN + FP + FN F = 2 Sn Sp Sn + Sp CC = (TP TN ) (FN FP) (TP + FN ) (TN + FP) (TP + FP) (TN + FN ). ACP = 1 n TP TP + FN + TP TP + FP + TN TN + FP + TN TN + FN, AC = 2(ACP 0.5). Courtesy of William Majoros. Used with permission.

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