RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Counting RNA Secondary Structures of Arbitrary Pseudoknot Type Review of Jin, Qin, & Reidys (2008) Bull. Math. Biol. 70

Berton A. Earnshaw Department of Mathematics, University of Utah Salt Lake City, Utah 84112

February 26, 2008

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

A single strand of RNA

http://ilab.cs.ucsb.edu/projects/helly/rna.jpg

Primary structure: sequence of bases (A,G,U,C) Secondary structure: pairing of bases Watson-Crick pairs: A-U, G-C (less often U-G)

Tertiary structure: resulting 3D molecule Different tertiary structures ⇒ different enzymatic properties

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

A single strand of RNA: An example

Primary structure: AACCAUGUGGUACUUGAUGGCGAC

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

A single strand of RNA: An example

Primary structure: AACCAUGUGGUACUUGAUGGCGAC Secondary structure:

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

A single strand of RNA: An example

Primary structure: AACCAUGUGGUACUUGAUGGCGAC Secondary structure:

Tertiary structure: extremely difficult to predict (probably NP-hard)

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

RNA secondary structure as k-noncrossing arch diagram k-noncrossing arch diagram of order n graph on vertex set {1, . . . , n} all vertices have degree ≤ 1 there do not exist k arches {i1 , j1 }, . . . , {ik , jk } such that i1 < · · · < ik < j1 < · · · < jk

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

RNA secondary structure as k-noncrossing arch diagram k-noncrossing arch diagram of order n graph on vertex set {1, . . . , n} all vertices have degree ≤ 1 there do not exist k arches {i1 , j1 }, . . . , {ik , jk } such that i1 < · · · < ik < j1 < · · · < jk

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

RNA secondary structure as k-noncrossing arch diagram k-noncrossing arch diagram of order n graph on vertex set {1, . . . , n} all vertices have degree ≤ 1 there do not exist k arches {i1 , j1 }, . . . , {ik , jk } such that i1 < · · · < ik < j1 < · · · < jk

RNA secondary structure of n bases, pseudoknot type k − 2 k-noncrossing (but not k − 1) arch diagram of order n no 1-arches {i, i + 1} “abstract” secondary structure (no primary structure)

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Counting RNA secondary structures

Establish bijection between k-noncrossing arch diagrams and certain walks in Zk−1 Count walks via reflection principle (Weyl groups) Enumerate restricted walks (RNA secondary structures)

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

k-noncrossing arch diagrams and walks in Weyl chamber

Walk in Zm of length n sequence of vectors x0 , x1 , . . . , xn ∈ Zm s.t. |xi +1 − xi | = 0 or 1

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

k-noncrossing arch diagrams and walks in Weyl chamber

Walk in Zm of length n sequence of vectors x0 , x1 , . . . , xn ∈ Zm s.t. |xi +1 − xi | = 0 or 1

Weyl chamber subset of vectors x = (x1 , . . . , xm ) ∈ Zm s.t. x1 > · · · > xm > 0

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

k-noncrossing arch diagrams and walks in Weyl chamber

Walk in Zm of length n sequence of vectors x0 , x1 , . . . , xn ∈ Zm s.t. |xi +1 − xi | = 0 or 1

Weyl chamber subset of vectors x = (x1 , . . . , xm ) ∈ Zm s.t. x1 > · · · > xm > 0

Theorem (Chen et al. (2007) Trans. Am. Math. Soc. 359) There exists a bijection between k-noncrossing arch diagrams of order n and walks of length n in Zk−1 which start and end at a = (k − 1, k − 2, . . . , 1) and remain in the Weyl chamber.

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Idea of proof: Oscillating Young diagrams, RSK algorithm Young diagram collection of squares µ arranged in left-justified rows number of squares in each row weakly decreasing

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Idea of proof: Oscillating Young diagrams, RSK algorithm Young diagram collection of squares µ arranged in left-justified rows number of squares in each row weakly decreasing

Oscillating Young diagrams sequence of Young diagrams ∅ = µ0 , µ1 , . . . , µn = ∅ µi and µi +1 differ by at most one square

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Idea of proof: Oscillating Young diagrams, RSK algorithm Young diagram collection of squares µ arranged in left-justified rows number of squares in each row weakly decreasing

Oscillating Young diagrams sequence of Young diagrams ∅ = µ0 , µ1 , . . . , µn = ∅ µi and µi +1 differ by at most one square

Young tableau filling of Young diagram with positive integers numbers weakly increasing in each row numbers strictly decreasing in each column

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Idea of proof: Oscillating Young diagrams, RSK algorithm Young diagram collection of squares µ arranged in left-justified rows number of squares in each row weakly decreasing

Oscillating Young diagrams sequence of Young diagrams ∅ = µ0 , µ1 , . . . , µn = ∅ µi and µi +1 differ by at most one square

Young tableau filling of Young diagram with positive integers numbers weakly increasing in each row numbers strictly decreasing in each column

RSK algorithm method for creating sequences of Young tableaux

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Idea of proof: The bijection

(4,3,2,1),(5,3,2,1),(6,3,2,1),(6,4,2,1),(6,4,2,1),(6,4,2,1), (6,4,3,1),(6,4,3,1),(5,4,3,1),(5,4,3,2),(6,4,3,2),(6,4,3,1), . (6,4,3,1),(6,4,2,1),(6,4,2,1),(6,3,2,1),(5,3,2,1),(4,3,2,1)

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Counting walks in Weyl chamber: Weyl group Set ∆m = {em } ∪ {ej−1 − ej | j = 2, .., m} Each α ∈ ∆m called a (simple) root Hyperplane Pα normal to α ∈ ∆m called a wall Weyl chamber ⊆ region of Rm bounded by walls

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Counting walks in Weyl chamber: Weyl group Set ∆m = {em } ∪ {ej−1 − ej | j = 2, .., m} Each α ∈ ∆m called a (simple) root Hyperplane Pα normal to α ∈ ∆m called a wall Weyl chamber ⊆ region of Rm bounded by walls

∆1 = {1}, P1 = {0} ∆2 = {(0, 1), (1, −1)}, P(0,1) = h(1, 0)i, P(1,−1) = h(1, 1)i

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Counting walks in Weyl chamber: Weyl group Set ∆m = {em } ∪ {ej−1 − ej | j = 2, .., m} Each α ∈ ∆m called a (simple) root Hyperplane Pα normal to α ∈ ∆m called a wall Weyl chamber ⊆ region of Rm bounded by walls

∆1 = {1}, P1 = {0} ∆2 = {(0, 1), (1, −1)}, P(0,1) = h(1, 0)i, P(1,−1) = h(1, 1)i Weyl group Bm : generated by reflections through walls D E α·x Bm = x 7→ x − 2 α | α ∈ ∆m α·α B1 ∼ = Z2 , B2 ∼ = D4

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Counting walks in Weyl chamber: Reflection principle wn (x, y) = # walks x → y of length n wn+ (x, y) = # walks x → y of length n remaining in Weyl chamber

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Counting walks in Weyl chamber: Reflection principle wn (x, y) = # walks x → y of length n wn+ (x, y) = # walks x → y of length n remaining in Weyl chamber Theorem (Gessel & Zeilberger (1992) Proc. Am. Math. Soc. 115) If x, y ∈ Zk−1 are in the Weyl chamber, then X sgn(β)wn (β(x), y). wn+ (x, y) = β∈Bk−1

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Counting walks in Weyl chamber: Reflection principle wn (x, y) = # walks x → y of length n wn+ (x, y) = # walks x → y of length n remaining in Weyl chamber Theorem (Gessel & Zeilberger (1992) Proc. Am. Math. Soc. 115) If x, y ∈ Zk−1 are in the Weyl chamber, then X sgn(β)wn (β(x), y). wn+ (x, y) = β∈Bk−1

Theorem (Grabiner & Magyar (1993) J. Algebr. Comb. 2) If x = (x1 , . . . , xk−1 ), y = (y1 , . . . , yk−1 ) are in the Weyl chamber, ∞ X n=0

wn+ (x, y)

k−1 xn = ex det[Ixi −yj (2x) − Ixi +yj (2x)] i ,j=1 n!

P 2r +j /(j!(r + j)!) is hyperbolic Bessel where Ir (2x) = ∞ j=0 x function of 1st kind of order r .

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Counting k-noncrossing arch diagrams Set fk (n, l ) = # k-nc arch diagrams of order n with l isolated nodes

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Counting k-noncrossing arch diagrams Set fk (n, l ) = # k-nc arch diagrams of order n with l isolated nodes With a = (k − 1, k − 2, . . . , 1), we have shown that n X fk (n, l ) wn+ (a, a) = l=0

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Counting k-noncrossing arch diagrams Set fk (n, l ) = # k-nc arch diagrams of order n with l isolated nodes With a = (k − 1, k − 2, . . . , 1), we have shown that n X fk (n, l ) wn+ (a, a) = l=0

∞ X n X n=1 l=0

fk (n, l )

xn n!

= ex det[Ii −j (2x) − Ii +j (2x)]|ik−1 ,j=1

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Counting k-noncrossing arch diagrams Set fk (n, l ) = # k-nc arch diagrams of order n with l isolated nodes With a = (k − 1, k − 2, . . . , 1), we have shown that n X fk (n, l ) wn+ (a, a) = l=0

∞ X n X n=1 l=0

fk (n, l )

xn n!

= ex det[Ii −j (2x) − Ii +j (2x)]|ik−1 ,j=1

     n n 2 f2 (n, l ) = C n−l , f3 (n, l ) = C n−l C n−l − C n−l +1 2 2 2 l l 2   2m 1 Cm = , mth Catalan number m+1 m

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Counting k-noncrossing RNA secondary structures Set Sk (n, l ) = # k-nc RNA structures of n bases with l isolated nodes n X Sk (n, l ) Sk (n) = # k-nc RNA structures of n bases = l=0

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Counting k-noncrossing RNA secondary structures Set Sk (n, l ) = # k-nc RNA structures of n bases with l isolated nodes n X Sk (n, l ) Sk (n) = # k-nc RNA structures of n bases = l=0

Theorem (Jin, Qin & Reidys, 2008) (n−l)/2

Sk (n, l ) =

X

b

(−1)

b=0

⌊n/2⌋

Sk (n) =

X b=0

b

(−1)





 n−b fk (n − 2b, l ) b

n−b b

 n−2b X l=0

fk (n − 2b, l )

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Idea of proof Set Gk (n, l , j) = # k-nc arch diagrams of order n with l isolated nodes, j 1-arches (n−l)/2

Fk (x) =

X j=0

Gk (n, l , j)x j

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Idea of proof Set Gk (n, l , j) = # k-nc arch diagrams of order n with l isolated nodes, j 1-arches (n−l)/2

Fk (x) =

X

Gk (n, l , j)x j

j=0

Note

  (n−l)/2   (b) X Fk (1) j n−b = Gk (n, l , j) = fk (n − 2b, l ) b! b b j=b

Both count (with multiplicity) all k-nc arch diagrams with l isolated nodes constructed by:
specifying b 1-arches (can be done in n−b ways) b filling n − 2b remaining nodes with k-nc arch diagram having l isolated nodes (can be done in fk (n − 2b, l) ways)

Each of Gk (n, l , j) arch diagrams counted

j
b

times

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Idea of proof Taylor expanding Fk about x = 1 gives (n−l)/2

Fk (x) =

X F (b) (1) (x − 1)b b! b=0

=

(n−l)/2 

X b=0

 n−b fk (n − 2b, l )(x − 1)b b

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Idea of proof Taylor expanding Fk about x = 1 gives (n−l)/2

Fk (x) =

X F (b) (1) (x − 1)b b! b=0

=

(n−l)/2 

X b=0

 n−b fk (n − 2b, l )(x − 1)b b

Therefore Sk (n, l ) = Gk (n, l , 0) = Fk (0)   (n−l)/2 X b n−b = (−1) fk (n − 2b, l ) b b=0

RNA Structures

Counting k-nc arch diagrams

Counting RNA 2nd structures

Idea of proof Taylor expanding Fk about x = 1 gives (n−l)/2

X F (b) (1) (x − 1)b b!

Fk (x) =

b=0

=

(n−l)/2 

X b=0

 n−b fk (n − 2b, l )(x − 1)b b

Therefore Sk (n, l ) = Gk (n, l , 0) = Fk (0)   (n−l)/2 X b n−b = (−1) fk (n − 2b, l ) b b=0

Table 1 The rst 15 numbers of 3-noncrossing RNA structures n

1 2 3 4 5

6

7

8

9

10

11

12

13

14

15

S3 (n) 1 1 2 5 13 36 105 321 1018 3334 11216 38635 135835 486337 1769500

RNA Structures

Counting k-nc arch diagrams

The end

Thank you!

Counting RNA 2nd structures