Math 8803/4803, Spring 2008: Discrete Mathematical Biologypeople.math.gatech.edu/~heitsch/Teaching/Sp08/Lectures/lect9.pdfMath 8803/4803, Spring 2008: Discrete Mathematical Biology

Math 8803/4803, Spring 2008:Discrete Mathematical Biology

Prof. Christine Heitsch

School of Mathematics

Georgia Institute of Technology

Lecture 9 – January 27, 2008

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, GA Tech 1

The “Central Dogma of Molecular Biology”

“Information flows as:

where RNA mediates the production of proteins from DNA.”


RNA: more than just the messenger

Breakthrough of the year:Small RNAs Make Big SplashPublished in Science Issue of 20 Dec 2002.

The structure of Pariacoto virus revealsa dodecahedral cage of duplex RNA,by Tang et. al. in Nat Struct Biol.


Biological function follows form

“Over the past two decades it has become clear that a variety of

RNA molecules have important or essential biological functions in

cells, beyond the well-established roles of ribosomal, transfer and

messenger RNAs in protein biosynthesis. . . . Each class of RNA is

likely to have a unique fold that confers biochemical function.”

From “Structural Genomics of RNA” by Jennifer A. Doudna,

published in Nature Structural Biology, Nov. 2000.


Levels of RNA structure I

Primary: linear sequence of nucleotide bases

Tertiary: all otherintra−molecular

interactions

Secondary: set of base pairsinduced by self−bonding

Three DimensionalRNA MolecularStructure

tRNA


Levels of RNA structure II

Selective base pair hybridization ⇐⇒ structure and function

GCGGAUUUAG

UCGCACCA

GCCUGAAGAUCUGGAGGUCCUGGUUCGAUCCACAGAAU

CUCAGUUGGGAGAGCGCCA

Primary sequence −→ secondary structure −→ 3D molecule


Important biomathematical questions

GCGGAUUUAG

UCGCACCA

GCCUGAAGAUCUGGAGGUCCUGGUUCGAUCCACAGAAU

CUCAGUUGGGAGAGCGCCA

Prediction?

Analysis?

Design?

How do RNA sequences encode secondary structures?


Sequence to structure: a one-to-many mapping

R = gcgga uuuagcuc aguuggga gagc g ccaga cugaa

gaucugg agguc cugug uucgauc cacag a auucgc acca

S1(R) =

S2(R) =

Hypothesis: RNA sequences fold with minimal free energy.


Thermodynamics of RNA folding

‘‘Helices’’

‘‘Loops’’

RNA secondary structures are balanced between

energetically favorable helices (stacked base pairs)

and destabilizing loops (single-stranded regions).

Loops

Helices


Predicting nested RNA base pairings

Let R = b1b2 . . . bn ∈ {a,u,c,g}+ be a 5′ to 3′ 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 bi′ − bj′ are

distinct, i = i′ ⇐⇒ j = j′, and

either i < i′ < j′ < j or i < j < i′ < j′.

Then S(R) is a nested secondary structure of R.





S(R) = {bi − bj | 1 ≤ i < j ≤ n}

where for all distinct bi − bj and

bi′ − bj′, i = i′ ⇐⇒ j = j′,







S(R) = {bi − bj | 1 ≤ i < j ≤ n}

where for all distinct bi − bj and

bi′ − bj′, i = i′ ⇐⇒ j = j′,




Components of RNA secondary structures I

Definition. Let i.j denote a base pair bi − bj in a a nested RNA secondarystructure S(R).

• A base bi′ or base pair i′.j′ ∈ S(R) is accessible from i.j if i < i′ (< j′) < jand if there is no other base pair i′′.j′′ ∈ S(R) such thati < i′′ < i′ (< j′) < j′′ < j.

• A stacked pair is formed if exactly the base pair (i + 1).(j − 1) is accessiblefrom i.j. Successive stacked pairs form a stem.

• Otherwise, i.j closes a k-loop with k − 1 base pairs for k ≥ 1 and l ≥ 0unpaired bases accessible from i.j.

• The external loop, denoted Le(k− 1), is the set of l > 0 unpaired bases andk − 1 base pairs without a closing base pair.


Components of RNA secondary structures II

Assumption 1. The free energy of S(R) is the sum of loop free energies.Assumption 2. The free energy of a loop is independent of all other loops.

http://www.cs.washington.edu/education/courses/527/00wi/


Nearest neighbor energy model

∆G = -22.7 kcal/mole

S. cerevisiaePhe-tRNAat 37◦


Free energy parameters

Version 3.0 for RNA Folding at 37◦

Available through http://www.bioinfo.rpi.edu/∼zukerm/


Algorithmic question

Problem. How to find a RNA folding with minimal free energy?

Solution. Dynamic programming.

Reason. Recursive prediction of RNA secondary structures.

Let R = b1b2 . . . bi . . . bj . . . bn and W (0) = 0.

W (j) = min(W (j − 1), min1≤i<j

(V (i, j) + W (i− 1))) for j > 0

W (j) is the minimal free energy of an optimal structure for the first j residues. V (i, j) is as

W (j), but assuming i.j forms a base pair. A recursive calculation of V (i, j) depends on

V (i + 1, j − 1) and four other functions.


Recursive RNA secondary structure prediction

Let R = b1b2 . . . bi . . . bj . . . bn and W (0) = 0.

W (j) = min(W (j − 1), min1≤i<j

(V (i, j) + W (i− 1))) for j > 0

V (i, j) =∞ for i ≥ j

min(eH(i, j), eS(i, j) + V (i + 1, j − 1), V BI(i, j), V M(i, j)) for i < j

V BI(i, j) = mini′, j′

i < i′ < j′ < j

(eL(i, j, i′, j′) + V (i

′, j′))

V M(i, j) =

mink, i1, j1, i2, j2, . . . , ik, jk

i < i1 < j1 < i2 < j2 < . . . < ik < jk < j

k ≥ 2

(eM(i, j, i1, j1, i2, j2, . . . , ik, jk) +kX

h=1

V (ih, jh))


Secondary structure prediction software

• The DINAMelt web server by Nick Markham & Michael Zuker –

http://www.bioinfo.rpi.edu/applications/hybrid/

• Michael Zuker’s mfold Server –

http://www.bioinfo.rpi.edu/applications/mfold/rna/form1.cgi

• Vienna RNA Package – http://www.tbi.univie.ac.at/∼ivo/RNA/

• RNAsoft – http://www.rnasoft.ca/

• GTfold – coming soon!


Open problem: suboptimal foldings

The ∆E = 9.5 energy dot plot forthe cdk2 gene of Xenopus leavis.

Dots represent base pairs, colorcoded by energy (in kcal/mole).

Black: in optimal folding

Red: within 3.1

Blue: from 3.2-6.2

Yellow: from 6.3-9.5


Open problem: motif recognition

∆G = −2976.7 kcal / mole

3126 base pairs: 868 a - u, 1890 g - c, 368 u - g

Palmenberg & Sgro, unpublished.

Hepatitis C China Virus Complete Genome

9400 bases: 1920 a, 2796 c, 2650 g, 2034 u

Mfold secondary structure prediction


Open problem: pseudoknots

i

ji’

j’Pseudoknotted structure:

i < i′ < j < j′. 3D pseudoknot model


Acknowledgments

• Access Excellence Graphics Gallery –http://www.accessexcellence.com/RC/VL/GG/index.html

• Predicted RNA foldings courtesy of Michael Zuker’s mfold algorithm.

• CSE 527, Winter 2000 – Computational Biology – Martin Tompahttp://www.cs.washington.edu/education/courses/527/00wi/

• Bio-5495 – RNA Secondary Structure – Prof. Michael Zuker, Department ofMathematical Sciences, Rensselaer Polytechnic Institute.http://www.bioinfo.rpi.edu/ zukerm/Bio-5495/RNAfold-html/rnafold.html

• Prof. Ann Palmenberg (Dept of Biochemistry & Institute for MolecularVirology) and Dr. Jean-Yves Sgro (Institute for Molecular Virology),University of Wisconsin – Madison.


References

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Biochemistry, 31(47):11665–11676, 1992.

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Comput Biol, 7(3):409 – 427, 2000.

[3] D. Mathews, J. Sabina, M. Zuker, and D. Turner. Expanded sequence dependence of

thermodynamic parameters improves prediction of RNA secondary structure. J. Mol. Biol.,

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[4] S. B. Needleman and C. D. Wunsch. A general method applicable to the search for

similarities in the amino acid sequence of two proteins. J Mol Biol, 48(3):443–53, March

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[5] A. C. Palmenberg and J.-Y. Sgro. Topological organization of picornaviral genomes:

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[8] P. Schuster, W. Fontana, P. Stadler, and I. Hofacker. From sequences to shapes and back:


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[11] S. Wuchty, W. Fontana, I. L. Hofacker, and P. S. chuster. Complete suboptimal folding of

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secondary structure predi ction: A practical guide. In J. Barciszewski and B. Clark, editors,

RNA Biochemistry and Biotechnology, NATO ASI Series, pages 11–43. Kluwer Academic

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Documents

Math 8803/4803, Spring 2008: Discrete Mathematical Biologypeople.math.gatech.edu/~heitsch/Teaching/Sp08/Lectures/lect9.pdfMath 8803/4803, Spring 2008: Discrete Mathematical Biology