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Approaches to Sequence Analysis

Approaches to Sequence Analysis. Data {GTCAT,GTTGGT,GTCA,CTCA}. Parsimony, similarity, optimisation. . TKF91 - The combined substitution/indel process. Acceleration of Basic Algorithm Many Sequence Algorithm MCMC Approaches Statistical Alignment and Footprinting. GT-CAT GTTGGT GT-CA-

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Approaches to Sequence Analysis

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  1. Approaches to Sequence Analysis Data {GTCAT,GTTGGT,GTCA,CTCA} Parsimony, similarity, optimisation. TKF91 - The combined substitution/indel process. Acceleration of Basic Algorithm Many Sequence Algorithm MCMC Approaches Statistical Alignment and Footprinting GT-CAT GTTGGT GT-CA- CT-CA- Ideal Practice: 1 phase analysis. Actual Practice: 2 phase analysis. statistics s1 s2 s3 s4

  2. T= 0 # - - - ## # # # T = t # # # # s1 r s2 s1 s2 s1 s2 Thorne-Kishino-Felsenstein (1991) Process A # C G * • (birth rate) < m(death rate) 1. P(s) = (1-l/m)(l/m)l pA#A* .. *pT #T l =length(s) 2. Time reversible:

  3. # - - - - - # # # # k * - - - - * # # # # k l & m into Alignment Blocks A. Amino Acids Ignored: # - - - ## # # k e-mt[1-lb](lb)k-1 [1-lb-mb](lb)k [1-lb](lb)k p’k(t) pk(t) p’’k(t) b=[1-e(l-m)t]/[m-le(l-m)t] p’0(t)= mb(t) B. Amino Acids Considered: T - - - RQ S W Pt(T-->R)*pQ*..*pW*p4(t) 4 • T - - - - • R Q S WpR *pQ*..*pW*p’4(t) • 4

  4. One block derivation # - - ... - # #*# ... # 1 k-1 # - - ... - # # # ... # 1 k+1   # - - ... - # # # ... # 1 k pk   # - - ... - # #*# ... # 1 k-1 # - - ... - # # # ... # 1 k+1 Dpk = Dt*[l*(k-1) pk-1 + m*k*pk+1 - (l+m)*k*pk]

  5. Dpk = Dt*[l*(k-1) pk-1 + m*k*pk+1 - (l+m)*k*pk] Dp’k=Dt*[l*(k-1) p’k-1+m*(k+1)*p’k+1-(l+m)*k*p’k+m*pk+1] Dp’’k=Dt*[l*k*p’’k-1+m*(k+1)*p’’k+1- [(k+1)l+km]*p’’k] Differential Equations for p-functions # - - ... - # # # ... # # - - - ... - - # # # ... # * - - - ... - * # # # ... # Initial Conditions: pk(0)= pk’’(0)= p’k (0)= 0 k>1 p1(0)= p0’’(0)= 1. p’0 (0)= 0

  6. Basic Pairwise Recursion (O(length3)) i j Survives: Dies: i-1 i i-1 i j-1 j j i-1 i i j-2 j i-1 j j-1 …………………… …………………… …………………… e-mt[1-lb](lb)k-1, where …………………… …………………… b=[1-e(l-m)t]/[m-le(l-m)t] 0… j (j+1) cases 1… j (j) cases

  7. Basic Pairwise Recursion (O(length3)) survive death j (i-1,j) j-1 (i-1,j-1) Initial condition: p’’=s2[1:j] ………….. (i-1,j-k) ………….. ………….. i-1 i (i,j)

  8. Fundamental Pairwise Recursion. P(s1i->s2j) = p’0P(s1i-1->s2j) + Initial Condition P(s10 ->s2j) = pj’’ps2[1:j] Simplification: Ri,j=(p1f(s1[i],s2[j])+p’1ps2j[j])P(s1i-1->s2j-1) + lb ps2[j]Ri,j-1 P(s1i->s2j) = Ri,j + p’0 P(s1i->s2j-1) P(s1i->s2j) = p’0P(s1i-1->s2j)+  lbP(s1i->s2j-1) + (p1f(s1[i],s2[j]+p’1ps2j[j]- lb ps2j[j] ))P(s1i-1->s2j-1) Probability of observationP(s1,s2) = P(s1) P(s1 ->s2)

  9. Ancestral Sequence Generator # E * l/m 1- l/m #l/m 1- l/m * # # # # p’’ function generator - # E * - - - - * # # # # lb 1- lb lb 1- lb * * - # p’/p function generator # - - - - # # # # # - # E lb 1- lb 1-mb mb # # # - # - - - - - # # # # lb 1- lb - # Markov Chains Generating the p-functions

  10. Statistical Alignment Steel and Hein,2001 + Holmes and Bruno,2001 T An HMM Generating Alignments - # # E # # - E * * lb l/m (1- lb)e-m l/m (1- lb)(1- e-m) (1- l/m) (1- lb) - # lb l/m (1- lb)e-m l/m (1- lb)(1- e-m) (1- l/m) (1- lb) _ #lb l/m (1- lb)e-m l/m (1- lb)(1- e-m) (1- l/m) (1- lb) # - lb C C A C Emit functions: e(##)= p(N1)f(N1,N2) e(#-)= p(N1),e(-#)= p(N2) p(N1) - equilibrium prob. of N f(N1,N2) - prob. that N1 evolves into N2

  11. Corner Cutting ~100-1000 Better Numerical Search ~10-100 Ex.: good start guess, 28 evaluations, 3 iterations Accelleration of Pairwise Algorithm (From Hein,Wiuf,Knudsen,Moeller & Wiebling 2000) Simpler Recursion ~3-10 Faster Computers ~250 1991-->2000 ~106

  12. a-globin (141) and b-globin (146) (From Hein,Wiuf,Knudsen,Moeller & Wiebling 2000) 430.108 : -log(a-globin) 327.320 : -log(a-globin -->b-globin) 747.428 : -log(a-globin, b-globin) = -log(l(sumalign)) l*t: 0.0371805 +/- 0.0135899 m*t: 0.0374396 +/- 0.0136846 s*t: 0.91701 +/- 0.119556 E(Length) E(Insertions,Deletions) E(Substitutions) 143.499 5.37255 131.59 Maximum contributing alignment: V-LSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTTKTYFPHF-DLS--H---GSAQVKGHGKKVADALT VHLTPEEKSAVTALWGKV--NVDEVGGEALGRLLVVYPWTQRFFESFGDLSTPDAVMGNPKVKAHGKKVLGAFS NAVAHVDDMPNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPAVHASLDKFLASVSTVLTSKYR DGLAHLDNLKGTFATLSELHCDKLHVDPENFRLLGNVLVCVLAHHFGKEFTPPVQAAYQKVVAGVANALAHKYH Ratio l(maxalign)/l(sumalign) = 0.00565064

  13. VLSPADNAL.....DLHAHKR 141 AA long *########### …. ### 141 AA long 2 108 years 2 107 years 2 109 years *########### …. ### *########### …. ### ???????????????????? k AA long 109 years The invasion of the immortal link

  14. Long Insertion-Deletions can model overlapping indels more involved dynamic programming:

  15. Homology test. (From Hein,Wiuf,Knudsen,Moeller & Wiebling 2000) D(s1,s2) is evaluated in D(s1,s2*) a-, myoglobin homology tests Random s1 = ATWYFC-AKAC s2* = LTAYKADCWLE * Real s1 = ATWYFCAK-AC s2 = ETWYKCALLAD *** ** * Wi,j= -ln(pi*P2.5i,j/(pi*pj)) 1. Test the competing hypothesis that 2 sequences are 2.5 events apart versus infinitely far apart. 2. It only handles substitutions “correctly”. The rationale for indel costs are more arbitrary.

  16. Sample random alignments from real sequences Sample random alignments from random sequences cgtgttacatatatatagccgatagccg cgtgttacatatatatagccgatagccg cgtgttacatatatatagccgatagccg cgtgttacatatatatagccgatagccg Compare real and random distribution using Chi-square statistic. Goodness-of-fit of TKF91

  17. TKF92 - Unbreakable fragments • Fragments evolve into fragments. • All possible tilings of the sequences with geometric length fragments are considered.

  18. Algorithm for alignment on star tree (O(length6))(Steel & Hein, 2001) *ACGC *TT GT s2 s1 a s3 *ACG GT *###### * (l/m)

  19. Binary Tree Problem a1a2 * * # # # - - # # # - # TGA ACCT s1 s3 a1 a2 s2 s4 GTT ACG • The ancestral sequences & their alignment was known. ii. The alignment of ancestral alignment columns to leaf sequences was known The problem would be simpler if: How to sum over all possible ancestral sequences and their alignments?: A Markov chain generating ancestral alignments can solve the problem!!

  20. - # # E # # - E * * lb l/m (1- lb)e-m l/m (1- lb)(1- e-m) (1- l/m) (1- lb) # # lb l/m (1- lb)e-m l/m (1- lb)(1- e-m) (1- l/m) (1- lb) _ #lb l/m (1- lb)e-m l/m (1- lb)(1- e-m) (1- l/m) (1- lb) # - lb Generating Ancestral Alignments a1 * a2 * # # l/m (1- lb)e-m E E (1- l/m) (1- lb) - # lb

  21. The Basic Recursion ”Remove 1st step” - recursion: S E ”Remove last step” - recursion: Last/First step removal are inequivalent, but have the same complexities. First step algorithm is the simplest.

  22. # # # # - # = Where P’(kS i,H) = F(kSi,H) Sequence Recursion: First Step Removal Pa(Sk): Epifixes (S[k+1:l]) starting in given MC starts in a.

  23. Contrasting Probability versus Distance Recursions Probability: Distance (Sankoff, 1973): A # # # # - # C = = + - A 15 cases

  24. Maximum likelihood phylogeny and alignment Gerton Lunter Istvan Miklos Alexei Drummond Yun Song Human alpha hemoglobin;Human beta hemoglobin; Human myoglobin Bean leghemoglobin Probability of data e-1560.138 Probability of data and alignment e-1593.223 Probability of alignment given data 4.279 * 10-15 = e-33.085 Ratio of insertion-deletions to substitutions: 0.0334

  25. Gibbs Samplers for Statistical Alignment Holmes & Bruno (2001): Sampling Ancestors to pairs. Jensen & Hein (in press): Sampling nodes adjacent to triples Slower basic operation, faster mixing

  26. The phylogeny moves: As in Drummond et al. 2002 Metropolis-Hastings Statistical Alignment. Lunter, Drummond, Miklos, Jensen & Hein, 2005 The alignment moves: QST--QCC-S S------CCS ---QST--QC ---QST--QC TNQHVSCTGN GN-HVSCTGK TNQH-SCTLN TNQHVSCTLN ALITL---GG ALLTLTTLGG ---TLTSLGA ALLGLTSLGA We choose a random window in the current alignment Then delete all gaps so we get back subsequences QSTQCCS SCCS QSTQC QSTQC TNQHVSCTGN GN-HVSCTGK TNQH-SCTLN TNQHVSCTLN ALITL---GG ALLTLTTLGG ---TLTSLGA ALLGLTSLGA QSTQCCS -S--CCS QSTQC-- QSTQC-- TNQHVSCTGN GN-HVSCTGK TNQH-SCTLN TNQHVSCTLN ALITL---GG ALLTLTTLGG ---TLTSLGA ALLGLTSLGA Stochastically realign this part

  27. Metropolis-Hastings Statistical Alignment Lunter, Drummond, Miklos, Jensen & Hein, 2005

  28. The Basics of Footprinting I • Many aligned sequences related by a known phylogeny: positions HMM: 1 n 1 sequences k slow - rs HMM: fast - rf • Two un-aligned sequences: A C G T ATG A-C A

  29. Many un-aligned sequences related by a known phylogeny: • Conceptually simple, computationally hard • Dependent on a single alignment/no measure of uncertainty • Statistical Alignment • Explicit stochastic model of substitution and indel evolution A C sometimes - # # # # - HMM: A T G • Advantages: Summing over uncertainty + confidence on inference The Basics of Footprinting II

  30. Statistical Alignment andFootprinting. acgtttgaaccgag---- 1 acgtttgaaccgag---- sequences sequences 1 k k Comment:The A-HMM * S-HMM is an approximate approach as S-HMM does not include an evolutionary model acgtttgaaccgag---- 1 sequences Alignment HMM k Ex.: nnnnnnnnnnn Alignment HMM Signal HMM nnnnnnnnnnn

  31. Structure HMM S F F F S S 0.1 0.1 0.1 0.1 0.9 0.9 F S SF FS SS FF (A,S) F F S S F Alignment HMM ? Structure HMM “Structure” does not stem from an evolutionary model • The equilibrium annotation • does not follow a Markov Chain: • Each alignment in from theAlignment HMM • is annotated by the Structure HMM: • No ideal way of simulating: using the HMM at the alignment will give other distributions on the leaves using the HMM at the root will give other distributions on the leaves

  32. Structure Description • Simple Promotor/Enchancer Structure: only Fast/Slow Start M2 = M2 M3 M1 Stop Alignment HMM Structure HMM • Advanced Promotor/Enchancer Structure: General HMM (J. Liu) • De novo cis-regulatory module elicitation for eukaryotic genomes. Proc Nat’l Acad Sci USA, 102, 7079-84 • For instance, different nature of indel process • The substitution process • Other possibilities: • Gene Structure/RNA Structure

  33. An example Previously-identified binding sites indicated by colored boxes • Predicted functional elements in RED BOLD TEXT • In overall region, program correctly identified 8 out of 11 binding sites with 4 false positives • Overlapping binding sites may indicate repressor relationships • False positives show lesser degree of conservation, could be undetected binding sites

  34. An example further • 8 out of 11 binding sites correctly identified, total of 4 false identifications, one of which lay adjacent to the true binding site. • Issues with the highly analyzed regions as gold standards - probably only find very strong regulatory regions.

  35. References Statistical Alignment • Fleissner R, Metzler D, von Haeseler A.Simultaneous statistical multiple alignment and phylogeny reconstruction.Syst Biol. 2005 Aug;54(4):548-61. • Hein,J., C.Wiuf, B.Knudsen, Møller, M., and G.Wibling (2000): Statistical Alignment: Computational Properties, Homology Testing and Goodness-of-Fit. (J. Molecular Biology 302.265-279) • Hein,J.J. (2001): A generalisation of the Thorne-Kishino-Felsenstein model of Statistical Alignment to k sequences related by a binary tree. (Pac.Symp.Biocompu. 2001 p179-190 (eds RB Altman et al.) • Steel, M. & J.J.Hein (2001): A generalisation of the Thorne-Kishino-Felsenstein model of Statistical Alignment to k sequences related by a star tree. ( Letters in Applied Mathematics) • Hein JJ, J.L.Jensen, C.Pedersen (2002) Algorithms for Multiple Statistical Alignment. (PNAS) 2003 Dec 9;100(25):14960-5. • • Holmes, I. (2003) Using Guide Trees to Construct Multiple-Sequence Evolutionary HMMs.Bioinformatics, special issue for ISMB2003, 19:147i–157i. • • Jensen, J.L. & Hein, J. (2004) A Gibbs sampler for statistical multiple alignment. Statistica Sinica, in press. • • Miklós, I., Lunter, G.A. & Holmes, I. (2004) A 'long indel' model for evolutionary sequence alignment. Mol. Biol. Evol. 21(3):529–540. • • Lunter, G.A., Miklós, I., Drummond, A.J., Jensen, J.L. & Hein, J. (2005) Bayesian Coestimation of Phylogeny and Sequence Alignment. BMC Bioinformatics, 6:83 • • Lunter, G.A., Miklós, I., Drummond, A., Jensen, J.L. & Hein, J. (2003) Bayesian phylogenetic inference under a statistical indel model. pspdfLecture Notes in Bioinformatics, Proceedings of WABI'03, 2812:228–244. • • Lunter, G.A., Miklós, I., Song, Y.S. & Hein, J (2003) An efficient algorithm for statistical multiple alignment on arbitrary phylogenetic trees.J. Comp. Biol., 10(6):869–88Miklos, Lunter & Holmes (2002) (submitted ISMB) • Miklos, I & Toroczkai Z. (2001) An improved model for statistical alignment, in WABI2001, Lecture Notes in Computer Science, (O. Gascuel & BME Moret, eds) 2149:1-10. Springer, Berlin • Metzler D. “Statistical alignment based on fragment insertion and deletion models.” Bioinformatics. 2003 Mar 1;19(4):490-9. • Miklos, I (2002) An improved algorithm for statistical alignment of sequences related by a star tree. Bul. Math. Biol. 64:771-779. • Miklos, I: Algorithm for statistical alignment of sequences derived from a Poisson sequence length distribution Disc. Appl. Math. accepted. • Thorne JL, Kishino H, Felsenstein J.Inching toward reality: an improved likelihood model of sequence evolution.J Mol Evol. 1992 Jan;34(1):3-16. • Thorne JL, Kishino H, Felsenstein J.An evolutionary model for maximum likelihood alignment of DNA sequences.J Mol Evol. 1991 Aug;33(2):114-24. Erratum in: J Mol Evol 1992 Jan;34(1):91. • Thorne JL, Churchill GA.Estimation and reliability of molecular sequence alignments.Biometrics. 1995 Mar;51(1):100-13. TKF92, Long Indel, Explain HMM, Multiple Recursion, Hidden State Space, 1-state recursion and other reductions, competing algorithms,

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