McCaskill: Efficient Base Pair Probabilities Idea: Compute p kl := Pr[( k , l ) | S ] recursively (DP!), recurse from long base pairs (outside) to small ones (inside) 1) simple case (external base pair) Definition (Probability of external base pair) p E kl := Pr[ P E kl ] , where P E kl := { P | P ∈ P , ( k , l ) is external base pair in P } , where ( k , l ) is external base pair in P iff ( k , l ) ∈ P and � ∃ ( i , j ) ∈ P : i < k < l < j . kl = Q 1 k − 1 Q b Z P E kl Q l +1 n S.Will, 18.417, Fall 2011 Z P E = Q 1 k − 1 Q b kl Q l +1 n p E kl = kl Z Q 1 n
McCaskill: Efficient Base Pair Probabilities 2) general case a) ( k , l ) is external base pair b) ( k , l ) limits stacking, bulge, or interior loop closed by ( i , j ) c) ( k , l ) inner base pair of multiloop closed by ( i , j ) S.Will, 18.417, Fall 2011
McCaskill: Efficient Base Pair Probabilities 2) general case a) ( k , l ) is external base pair � b) ( k , l ) limits stacking, bulge, or interior loop closed by ( i , j ) p SBI kl ( i , j ) := p ij Pr[loop i , j , k , l | ( i , j )] Z { P ∈P ij | P has loop i , j , k , l } = p ij Z P b ij exp( − β eSBI( i , j , k , l )) Q b kl = p ij . Q b ij S.Will, 18.417, Fall 2011 c) ( k , l ) inner base pair of multiloop closed by ( i , j )
McC: Base Pair Probabilities — Multiloop Case 2) general case c) ( k , l ) inner base pair of multiloop closed by ( i , j ) p M kl ( i , j ) := p ij Pr[multiloop with inner base pair ( k , l ) closed by ( i , j ) | ( i , j )] Three cases: position of ( k , l ) in the multiloop (i) (k,l) leftmost base pair (ii) (k,l) middle base pair (iii) (k,l) rightmost base pair S.Will, 18.417, Fall 2011
McC: Base Pair Probabilities — Multiloop Case 2) general case c) ( k , l ) inner base pair of multiloop closed by ( i , j ) p M kl ( i , j ) := p ij Pr[multiloop with inner base pair ( k , l ) closed by ( i , j ) | ( i , j )] Three cases: position of ( k , l ) in the multiloop (i) (k,l) leftmost base pair Q b kl Q m l +1 j − 1 exp( − β ( a + b + ( k − i − 1) c )) (ii) (k,l) middle base pair Q m i +1 k − 1 Q b kl Q m l +1 j − 1 exp( − β ( a + b )) (iii) (k,l) rightmost base pair S.Will, 18.417, Fall 2011 Q m i +1 k − 1 Q b kl exp( − β ( a + b + ( j − l − 1) c ))
McC — Multiloop Case (Ctd.) Recall p M kl ( i , j ) := p ij Pr[multiloop with inner base pair ( k , l ) closed by ( i , j ) | ( i , j )] putting the three cases of ( k , l ) position together p M kl ( i , j ) = p ij [ Q b kl Q m l +1 j − 1 exp( − β ( a + b + ( k − i − 1) c )) + Q m i +1 k − 1 Q b kl Q m l +1 j − 1 exp( − β ( a + b )) kl exp( − β ( a + b + ( j − l − 1) c ))] Q − 1 + Q m i +1 k − 1 Q b ij S.Will, 18.417, Fall 2011
McCaskill — Base Pair Probabilities — Summary p kl = p E � p SBI � p M kl + kl ( i , j ) + kl ( i , j ) i < k , l < j i < k , l < j Remarks • Recursive formula for p kl furnishes DP • Efficient calculation of all p kl in O ( n 4 ) time/ O ( n 2 ) space • Time reduction to O ( n 3 ) possible (not shown, but you learned the “trick”) • The algorithm by the p kl recursion is an outside algorithm; in contrast the algo for computing Z and the Q ij is inside. S.Will, 18.417, Fall 2011 For getting the probabilities, we combined inside and outside.
Summary Part I Algorithms • Nussinov • Zuker • McCaskill Common • O ( n 3 ) time, O ( n 2 ) space • non-crossing structure (= “no pseudoknots”) Differences • realism: base pairs ↔ free energy (loop-based) S.Will, 18.417, Fall 2011 • mfe ↔ ensemble
Next? Comparing RNAs S.Will, 18.417, Fall 2011
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