1. RNA secondary structure RNA is (usually) single-stranded The nucleotides ‘want’ to pair with their Watson-Crick complements (AU, GC) They may ‘settle’ for a wobble pair (GU) The set of such pairs is the secondary structure of the sequence

2. The problem Input: An RNA sequence R Output: A predicted secondary structure S = {(r i1, r j1 ), (r i2, r j2 ).. (r in, r jn )}

3. Faustian bargain for RNA secondary structure prediction Assume an RNA strand folds to lowest energy Three types of base pair interactions (G-C, A-U, G-U) All structures are nested Energies are strictly additive

4. Lowest energy assumption There are exponentially many configurations for a nucleic acid (or protein) sequence A sequence assumes its stable configuration quickly (seconds) How does it find global minimum quickly? (Levenstein paradox) Alternative: it finds local minimum

5. Base pair interactions Atomic interactions too complex to calculate Base pair interactions are ‘reasonable’ compromise

6. Nested structures More than 95% of structures are nested, but.. Those that aren’t may well be significant Non-nested structures include pseudoknots and kissing loops

7. Pseudoknot

8. Kissing loops

9. Loops Nested structures are loops The size of a loop is the number of unpaired nucleotides it contains The arity of a loop is the number of interior pairs it contains Every loop has exactly one closing pair

10. Hairpin loop Arity 0 (always), size 4 (variable)

11. Hairpin loop -- size unpaired nucleotides

12. Hairpin loop - closing pair Closing pair

13. Bulge Arity 1 (always), size 1 (variable

14. Interior loop Arity 1 (always), size 2

15. Multiloop Arity 2 (variable), size 3 (variable)

16. Composite RNA secondary structure

17. Nearest neighbor formula Energy is a function of closing base pair and its nearest neighbors Non-symmetric ( E(XYZW)  E(WZYX) ) Energies of nested loops are additive E(R) = {E(r i1, r i+11, r j-11 ), (r i+12, r j-12 ).. (r i+1n, r j-1n )}

18. Loop recurrence relation E(S i,j ) = E(S i+1,j ) E(S i,j+1 ) min{E(S i,k ) + E(S i,k ), i < k < j} E(L i,j ) where...

19. L i,j recurrence relation E(L i,j ) = L i,j is a hairpin L i,j is a stacked pair L i,j is an i-bulge L i,j is a j-bulge L i,j is an interior loop min

20. Hairpin loop Nearest neighbors