Combinatorics of RNA Secondary Structures
Christine E. Heitsch School of Mathematics Georgia Institute of Technology
The 23rd Clemson Mini-Conference on Discrete Mathematics and Algorithms – October 3, 2008
Discrete Mathematical Biology
Combinatorics. . . “as motivated by”
and
“with applications to” . . . molecular biology.
C. E. Heitsch, GaTech
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Combinatorics on (Biological) Words
DNA −→
←− RNA
DNA and RNA are (oriented, biochemical) sequences over (nucleotide) alphabets with the complementary Watson-Crick base pairing. C. E. Heitsch, GaTech
2
RNA: More Than Just the Messenger The structure of Pariacoto virus reveals a dodecahedral cage of duplex RNA, by Tang et. al. in Nat Struct Biol.
Breakthrough of the year: Small RNAs Make Big Splash Published in Science Issue of 20 Dec 2002.
C. E. Heitsch, GaTech
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RNA Structural Hierarchy
Three Dimensional RNA Molecular Structure Tertiary: all other intra−molecular interactions
tRNA
Secondary: set of base pairs induced by self−bonding Primary: linear sequence of nucleotide bases
C. E. Heitsch, GaTech
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Folded RNA Molecules Form Structures Selective base pair hybridization ⇐⇒ structure and function GCGGAUUUAG CUCAGUUGG GAGAGCGCCA GCCUGAAGA UCUGGAGGUC CUGGUUCGA UCCACAGAAU UCGCACCA
Primary sequence −→ secondary structure −→ 3D molecule C. E. Heitsch, GaTech
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Important Biomathematical Questions Prediction? GCGGAUUUAG CUCAGUUGG GAGAGCGCCA GCCUGAAGA UCUGGAGGUC CUGGUUCGA UCCACAGAAU UCGCACCA
Analysis?
Design? How do RNA sequences encode secondary structures?
C. E. Heitsch, GaTech
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Representing RNA Configurations by Trees Abstract folded sequence to its graphical “skeleton”: 1−loop
4−loop
leaf vertex
vertex of degree 3 5 5 4
6 stacked base pairs external loop
edge of weight 6
6
root vertex
stacked base pairs −→ edges, single-stranded regions −→ vertices. C. E. Heitsch, GaTech
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Designing RNA Secondary Structures
Given a plane tree T with edge weights W , produce (by a deterministic algorithm) an RNA sequence R such that S(R) is represented by T . R=
T 5 5 4 6
C. E. Heitsch, GaTech
S(R)
gcggau uua
Design
gcuc aguuggga gagc g
Predict
=⇒
ccaga cugaaga ucugg
=⇒
agguc cugug uucgauc cacag a auucgc acca
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Sequence to Structure: A One-to-Many Mapping Output of sir_graph by D. Stewart and M. Zuker
Output of sir_graph by D. Stewart and M. Zuker
A
A 40
A A A G G A C G 30 C G C A A A A G 20 C G C C
Output of sir_graph by D. Stewart and M. Zuker
C C C C
A
G
G C
A
A A
C G
3’
5’
G A A A C C 60
G G G
C
G
C C
dG 30= -39.02 [initially -43.4]
05Dec16-11-06-00
A A A G 50
C G C G A
A A A
A
A
G C
C
G C
C C
G
A G C
A
G C G C
A A
G C
05Dec16-11-25-58
G C A A
G C
A
40
A
A
A
A
5’
A
G C
C
G
A
A 50
G C
40
C C
G G
A
A A
A
A A
A
A
A A 5’
05Dec16-11-06-00
dG = -42.3 [initially -42.3]
A
60
A
A 3’
05Dec16-11-06-00
30
A
A 10
A
dG = -41.7 [initially -41.7]
C G
A
G C
G
dG = -42.4 [initially -42.4]
C G C G
G C
A
A
50
A
A A A A 10
G A A
C
20
40
A
C G C G
C G
A A A C G
A
C G C G
C G
A
A A
A
A A C G 60 C G
C G
A
A
A A
C G
A
A
3’
G G G G G C C G C C C C A
A
C G
A
A
A
A
3’
A
20
A A A
20
60
A
5’
40
A
A A A A
G C
A
A
G C
A A
Output of sir_graph by D. Stewart and M. Zuker
A
A
G C
A
30
20
C A A
A
G C
G C
G C
G
G C
G C
C
G
G C
G C
C
G
G C
30
C
G A
A
G G
C
G
A A
G
G C
G C
5’
G
A
G C
10
G C
A
10
A
A
A
G
C
G
A A
A 50 A
A A A A 10 G C
G C
C
C
A A
A
A A
Output of sir_graph by D. Stewart and M. Zuker
A
A
3’
G G A A A C G C 60 G C 50 G C G C A C A A A
dG = -42.4 [initially -42.4]
05Dec16-11-06-00
Secondary structures for the combinatorial RNA sequence R = aaaa gggggg aaaa cccccc aaaa gggggg aaaa cccccc aaaa gggggg aaaa cccccc aaaa. C. E. Heitsch, GaTech
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Plane / Ordered / Linear Trees From Stanley’s Enumerative Combinatorics, Vol. 2 : Definition. A plane tree T is a rooted tree whose subtrees at any vertex are linearly ordered. A vertex with k children has degree k.
,
,
,
,
T3 =
Tn = {T | n edges (and n + 1 vertices)} and |Tn| = C. E. Heitsch, GaTech
1 n+1
·
2n n
= Cn 10
Local Moves (“as motivated by”) Intuitively, switch the “pairing” between the “half-edges” of any two adjacent edges.
C. E. Heitsch, GaTech
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Local Moves Give Global Structure
G3:
Theorem (H). The graph Gn is connected with diameter n − 1 n and n-partite with disjoint sets Tn,k where |Tn,k | = n1 nk k−1 . C. E. Heitsch, GaTech
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Catalan Numbers and Noncrossing Partitions
1
1 2
4
2
4
3
3
P = {1}, {2, 3}, {4}
P 0 = {1, 3, 4}, {2}
Theorem (Fomin, H). The lattice (Tn, ) is isomorphic to NC(n) — the lattice of noncrossing partitions of [n] ordered by refinement.
C. E. Heitsch, GaTech
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Open Enumeration Problem Solved Complementation operation: 4’
1
n
1’
4
2 3’
3
2’
[Kreweras, 1972]
C. E. Heitsch, GaTech
Counting orbits in the Catalan lattice
2 3 4 5 6 7 8 9
Orbits 2 4 (1) 0 1 0 1 (1) 1 0 1 1 1 0 1 1 1 0
of Length 6 n 2n 0 1 0 0 1 0 0 1 1 0 2 3 0 3 9 0 5 28 0 8 85 3 14 262
Total
Cn
1 2 3 6 14 34 95 280
2 5 14 42 132 429 1430 4862
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Discrete Mathematical Biology
Combinatorics. . . “as motivated by”
and
“with applications to” . . . molecular biology.
C. E. Heitsch, GaTech
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Important Biomathematical Questions Prediction? GCGGAUUUAG CUCAGUUGG GAGAGCGCCA GCCUGAAGA UCUGGAGGUC CUGGUUCGA UCCACAGAAU UCGCACCA
Analysis?
Design? How do RNA sequences encode secondary structures?
C. E. Heitsch, GaTech
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Predicting Nested RNA Base Pairings Let R = b1b2 . . . bn ∈ {a, u, c, g}+ be a 50 to 30 RNA sequence. Definition. Let S(R) be a set of base pairs S(R) = {bi − bj | 1 ≤ i < j ≤ n} where all bi − bj and bi0 − bj 0 are distinct, i = i0 ⇐⇒ j = j 0, and either i < i0 < j 0 < j or i < j < i0 < j 0. Then S(R) is a nested secondary structure of R. C. E. Heitsch, GaTech
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Thermodynamics of RNA Folding RNA secondary structures are balanced between energetically favorable helices (stacked base pairs) and destabilizing loops (single-stranded regions). ‘‘Helices’’
Loops Helices
‘‘Loops’’
C. E. Heitsch, GaTech
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Nearest Neighbor Energy Model
S. cerevisiae Phe-tRNA at 37◦
∆G = -22.7 kcal/mole
C. E. Heitsch, GaTech
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RNA Viral Genomes (“with applications to”) Selective base pair hybridization ⇐⇒ structure and function The armyworm Pariacoto virus (or PaV) consists of 2 ssRNA sequences: RNA1 with 3,011 nucleotides and RNA2 which has 1,311 nucleotides.
? 100’s of different 2nd−ary structures with free energy near minimal dG = −1669.4 kcal/mole.
Primary sequence −→ secondary structure ?!? −→ 3D molecule C. E. Heitsch, GaTech
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Identifying Significant Base Pairings Two mfold predicted secondary structures: a PaV RNA2 configuration (right) and the base pairing of a “scrambled” RNA2 sequence (below) which is much more extended and less extensively branched.
C. E. Heitsch, GaTech
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Analyzing Vertex Energies For a tree T , let EL(T ) be the sum of the vertex energies. Vertex degree Related loop Minimal energy
0 1-loop 4.10
1 2-loop 2.30
k − 1, k ≥ 3 k-loop, k ≥ 3 3.40 - 1.50 k
root, degree j ≥ 1 external -1.90 j
Minimal Energy
EL(T) = 6.6
EL(T) = 6.7
EL(T) = 6.7
EL(T) = 5.2
EL(T) = 6.8
Plane trees T with 3 edges and their total loop energies EL(T ). C. E. Heitsch, GaTech
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Minimal Loop Energy Configurations
Minimal Energy
EL(T) = 6.6
EL(T) = 6.7
EL(T) = 6.7
EL(T) = 5.2
EL(T) = 6.8
Plane trees T with 3 edges and their total loop energies EL(T ). Theorem (H). For plane trees T with n edges, the total loop energy EL(T ) is minimal when T has the maximal number of vertices with degree 2. (When n is odd, the root has degree 1.) C. E. Heitsch, GaTech
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Designing, Analyzing, and Predicting. . . Mathematical Result: Associated loop energies are minimized by maximizing vertices with three edges.
Biological Hypothesis: Branching degree in viral RNA loops correlates with functional significance. C. E. Heitsch, GaTech
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In Conclusion
There is much to explore at the interface of discrete mathematics and molecular biology.
C. E. Heitsch, GaTech
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Acknowledgments
• Our Clemson hosts and conference organizers! • NIH R01, Joint DMS/NIGMS Initiative to Support Research in the Area of Mathematical Biology. • Burroughs Wellcome Fund Career Award at the Scientific Interface. • Computation and Informatics in Biology and Medicine Training Program. • Anne Condon (CS) and Holger Hoos (CS), University of British Columbia. • Sergey Fomin (Math), University of Michiagn, Ann Arbor. • Steve Harvey (Bio) and David Bader (CS), Georgia Institute of Technology. • Michael Zuker’s mfold algorithm.
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