RNA Secondary Structure aagacuucggaucuggcgacaccc uacacuucggaugacaccaaagug aggucuucggcacgggcaccauuc ccaacuucggauuuugcuaccaua aagccuucggagcgggcguaacuc RNA and Translation CS262 Lecture 19 Win07 Batzoglou RNA and Splicing CS262 Lecture 19 Win07 Batzoglou Hairpin Loops Interior loops Stems Multi branched loop Bulge loop CS262 Lecture 19 Win07 Batzoglou Tertiary Structure Secondary Structure CS262 Lecture 19 Win07 Batzoglou CS262 Lecture 19 Win07 Batzoglou CS262 Lecture 19 Win07 Batzoglou Modeling RNA Secondary Structure Context Free Grammars CS262 Lecture 19 Win07 Batzoglou A Context Free Grammar S AB A aAc a B bBd b Nonterminals Terminals Production Rules S A B a b c d 5 rules Derivation Start from the S nonterminal Use any production rule replacing a nonterminal with a terminal until no more nonterminals are present S AB aAcB aaaacccB aaaacccbBd aaaacccbbbbbdddd Produces all strings ai 1cibj 1dj for i j 0 CS262 Lecture 19 Win07 Batzoglou Example modeling a stem loop S a W1 u W1 c W2 g W2 g W3 c W3 g L c L agugc What if the stem loop can have other letters in place of the ones shown CS262 Lecture 19 Win07 Batzoglou ACGG UGCC AG U CG Example modeling a stem loop S a W1 u W1 c W 2 g W2 g W3 c W3 g L c L agucg g W1 u g W3 u a L u agccg ACGG UGCC gW1u gW2u gW3u gLu gW1c gW2c gW3c gLc cW1g cW2g cW3g cLg uW1g uW2g uW3g uLg L aL1 cL1 gL1 uL1 L1 aL2 cL2 gL2 uL2 L2 a c g u aa uu aaa uuu CS262 Lecture 19 Win07 Batzoglou U CG cugugc More general Any 4 long stem 3 5 long loop S aW1u W1 aW2u W2 aW3u W3 aLu AG uW1a uW2a uW3a uLa GCGA UGCU GCGA UGUU AG C CG CUG U CG A parse tree alignment of CFG to sequence S a W1 u W1 c W2 g W2 g W3 c W3 g L c L agucg ACGG UGCC S W1 W2 W3 L A C G G A G U G C C C G U CS262 Lecture 19 Win07 Batzoglou AG U CG Alignment scores for parses We can define each rule X s where s is a string to have a score Example W g W c W a W u W g W u W x W z 3 forms 3 hydrogen bonds 2 forms 2 hydrogen bonds 1 forms 1 hydrogen bond 1 when x z is not an a u g c g u pair Questions How do we best align a CFG to a sequence DP How do we set the parameters Stochastic CFGs CS262 Lecture 19 Win07 Batzoglou The Nussinov Algorithm Let s forget CFGs for a moment Problem Find the RNA structure with the maximum weighted number of nested pairings A C C A G C C G G C A U A U U A A AG G U A A C U C G C A G C G A G C G A G G C G A C G A U G C A U C A A U U G A U A C A CC A G U G U G UUC U G G ACCACGCUUAAGACACCUAGCUUGUGUCCUGGAGGUCUAUAAGUCAGACCGCGAGAGGGAAGACUCGUAUAAGCG CS262 Lecture 19 Win07 Batzoglou The Nussinov Algorithm Given sequence X x1 xN Define DP matrix i j F i j maximum number of weighted bonds if xi xj folds optimally Two cases if i j xi is paired with xj F i j s xi xj F i 1 j 1 i j xi is not paired with xj F i j max k i k j F i k F k 1 j CS262 Lecture 19 Win07 Batzoglou i k j The Nussinov Algorithm Initialization F i i 1 0 F i i 0 for i 2 to N for i 1 to N Iteration For l 2 to N For i 1 to N l j i l 1 F i 1 j 1 s xi xj F i j max max i k j F i k F k 1 j Termination Best structure is given by F 1 N Need to trace back refer to the Durbin book CS262 Lecture 19 Win07 Batzoglou The Nussinov Algorithm and CFGs Define the following grammar with scores S gSc 3 aSu 2 gSu 1 cSg 3 uSa 2 uSg 1 SS 0 aS 0 cS 0 gS 0 uS 0 0 Note is the string Then the Nussinov algorithm finds the optimal parse of a string with this grammar CS262 Lecture 19 Win07 Batzoglou The Nussinov Algorithm Initialization F i i 1 0 F i i 0 for i 2 to N for i 1 to N S a c g u Iteration For l 2 to N For i 1 to N l j i l 1 F i 1 j 1 s xi xj S aSu F i j max max i k j Termination Best structure is given by F 1 N CS262 Lecture 19 Win07 Batzoglou F i k F k 1 j S SS Stochastic Context Free Grammars In an analogy to HMMs we can assign probabilities to transitions Given grammar X1 s11 sin Xm sm1 smn Can assign probability to each rule s t P Xi si1 P Xi sin 1 CS262 Lecture 19 Win07 Batzoglou Example S aSb a b Probability distribution over all strings x x anbn 1 then P x 2 n 2 n 2 x an 1bn same Otherwise P x 0 CS262 Lecture 19 Win07 Batzoglou Computational Problems Calculate an optimal alignment of a sequence and a SCFG DECODING Calculate Prob sequence grammar EVALUATION Given a set of sequences estimate parameters of a SCFG LEARNING CS262 Lecture 19 Win07 Batzoglou Normal Forms for CFGs Chomsky Normal Form X YZ X a All productions are either to 2 nonterminals or to 1 terminal Theorem technical Every CFG has an equivalent one in Chomsky Normal Form The grammar in normal form produces exactly the same set of strings CS262 Lecture 19 Win07 Batzoglou Example of converting a CFG to C N F S S ABC A Aa a B Bb b C CAc c A a Converting CS262 Lecture 19 Win07 Batzoglou a B B b S S AS S BC A AA a B BB b C DC c C c D CA B A S A A A a a B B B C B B b b b D C A c a C c C b b C A c a c Another example S ABC A C aA B bB b C cCd c Converting S AS S BC A C C c A A A a B B B b B b C C C c C c C CD D d CS262 Lecture 19 Win07 Batzoglou Decoding the CYK algorithm Given x x1 xN and a SCFG G Find the most likely parse of x the most likely alignment of G to x Dynamic programming variable i j V likelihood of the most likely parse of xi xj rooted at nonterminal V Then 1 N S CS262 Lecture 19 Win07 Batzoglou likelihood of the …
View Full Document
Unlocking...