RNA Structure more

7/25/2019 RNA Structure more

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RNA Matrices and RNARNA Matrices and RNA

Secondary StructuresSecondary Structures

Institute for Mathematics and Its

Applications: RNA in Biology,Bioengineering and Nanotechnology,

University of Minnesota

Octoer !" # Novemer !, !$$%

Asamoah N&'anta, Morgan State University

N&'anta()e'el*morgan*edu


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RNA SecondaryRNA Secondary StructureStructure+rediction+rediction

Given a primary sequence, we want to findthe biological function of the relatedsecondary structure. To achieve this goal

we predict itssecondary structureusing alattice walk orpath approach.

This walk approach involves enumerative

combinatorics and is connected to infinitelower triangular matrices called RNmatrices.


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RNA Secondary StructureRNA Secondary Structure

!rimary "tructure# The linear sequence ofbases in an RN molecule

"econdary "tructure# The foldingorcoiling of the sequence due to bondednucleotide pairs$ %&, G%'

Tertiary "tructure # The three dimensionalconfiguration of an RN molecule. Thethree dimensional shape is important forbiological function, and it is harder topredict.


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RNA MoleculeRNA Molecule

Ribonucleic acid (RNA) molecule: Three main

categories

mRNA (messenger) carries genetic information fromgenes to other cells

tRNA (transfer) carries amino acids to a ribosome(cells for making proteins)

rRNA (ribosomal) part of the structure of a ribosome


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RNA Molcule cont*-RNA Molcule cont*-

(ther types )RN* molecules$

snRN )small nuclear RN* # carries geneticinformation from genes to other cells

miRN )micro RN* # carries amino acids to

a ribosome )cells for making proteins* iRN )immune RN* # part of the structure

of a ribosome )+mportant for +- studies*


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+rimary RNA Se.uence+rimary RNA Se.uence

'G'&''&''G'GGGG&'G'&

Nucleotide ength, /0 bases


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/eometric Representation/eometric Representation

"econdary structure is a graph

defined on a set of n labeled points

)1.". 2aterman, 3405*

6iological

'ombinatorial7Graph Theoretic

Random 2alk

(ther Representations


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RNA StructureRNA Structure

012 structure of

Haloarcula marismortui

3S riosomal RNA in

large riosomal suunit


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RNA NUMB4RSRNA NUMB4RS

3,3,3,/,8,5,30,90,5/,35:,8/9,405,;

These numbers count RNsecondary structures of length n.


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RNA 5ominatoricsRNA 5ominatorics

Recurrence Relation$

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

=+=+

===6

6

!,66

6!6$

n

j

njnsjsnsns

sss

M. Waterman!ntroduction to "omputational #iolog$:Maps se%uences and genomes &''.

M. Watermanecondar$ structure of single*stranded

nucleic acids Ad+. Math. (suppl.) &',-.


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5ounting Se.uence 2ataase5ounting Se.uence 2ataase

The (n%line


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RNA 5ominatorics cont*-RNA 5ominatorics cont*-

The number of RN secondary structures

for the sequence A3,nB is counted by the

coefficients of ")C*$

'oefficients of the power series$

)3,3,3,/,8,5,30,90,5/,35:,8/9,405,;*

( ) ! 0 7 3 8 %6 ! 7 9 06% %s z z z z z z z z= + + + + + + + +L


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( ) ! 0 7 3 8 %6 ! 7 9 06% %s z z z z z z z z= + + + + + + + +L


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6ased on the coefficients of the generating

function there are appro>imately 3.9 billion

possible RN structures of length n D /0.

( ) ! 0 !%6,0"!,!6 ! 36,$6!s z z z z z= + + + + + +L L


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&sing the recurrence relation we can find the

closed form generating function associated

with the RN numbers.

( ) ( )! ! 0 7

!

6 6 ! !

!

z z z z z z

s z

z

+ +

=

( ) ! 0 7 3 8 !%6%6 6,0"!, !36, $ !! 7 9 6s z z z z z z z z= + + + + + + + + +L L


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act Eormula, and symptotic


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")n,k* is the number of structures of

length n with e>actly k base pairs$

Eor n,k F ,


( ) 66

, 6 6

n k n k

s n k k kk

= +


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( ) ( )8,6 6$ and 8,! 8s s= =


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RN hairpin combinatorics.

( )

( )

( )

( )

!

6

!

6$

7

!$

! 6, numer of hairpins of length n

; ! 6, numer of hairpins 'ith m or more ases in

the loop, n m


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Random =al&Random =al&

random walk is a lattice pathfrom

one point to another such that stepsare allowed in a discrete number of

directions and are of a certain length


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RNA =al& # >ype IRNA =al& # >ype I

N"%a>is and there

are no consecutive N" steps


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>ype I, RNA Array n ? &->ype I, RNA Array n ? &-

6863!70$!96%

$636$60609

$$67887

$$$600!

$$$$6!6

$$$$$66

$$$$$$6


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>ype I, RNA Array n ? &->ype I, RNA Array n ? &-

$ $ $ $ $

6 $ $ $ $

! 6 $ $

0 6 $ $

8 7 $ $

60 3 6 $

!9 0$ !7 63 8 6

$

$

$ $

0

6

6

6

!

7

9

6%

$

8 6

60 6$


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>ype I, @ormation Rule Recurrence->ype I, @ormation Rule Recurrence-

( )

( ) ( )

( ) ( ) ( )

( ) 6,$,6

,6,,6

,$,6

6$,$

$

$

+>=+

++=+

=+

=

nkknm

jkjnmknmknm

jjnmnm

m

j

j

Note. () can be deri+ed using this recurrence.


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@irst Moments=eighted Ro' Sums@irst Moments=eighted Ro' Sums

6 $ $ $ $ $ 66 6 $ $ $ $ !

6 ! 6 $ $ $ 0

! 0 0 6 $ $ 77 8 8 7 6 $ 3

9 60 60 6$

60

9

!633

6773 6 8

=

"omputing the a+erage height of the /alks abo+e the

0*a0is is gi+en b$ the alternate 1ibonacci numbers


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RNA =al& # >ype IIRNA =al& # >ype II

N"%a>is and

there are no consecutive "N steps


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>ype II , RNA Array n ? &->ype II , RNA Array n ? &-

$ $ $ $ $ $

6 $ $ $ $ $

! 6 $ $ $ $

7 0 6 $ $ $

" % 7 6 $ $

6% 66 3 6 $

73 76 !" 68 8 6

!$

6

6

!

7

9

6%

0%


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4?amples4?amples

Type +$


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RNA =al& Bi)ectionRNA =al& Bi)ection

Theorem$ There is a bi=ection between theset of N"


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Main >heoremMain >heorem

Theorem$ There is a bi=ection between the

set of RN secondary structures of length

n and the set of N"


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Main >heorem cont*-Main >heorem cont*-

!roof )sketch*$ 'onsider an RN

sequence of length n and convert it

to the non%intersecting chord form.'onsider the following rules$

( ) ( )SNji

Ek

,,

A li i ;I 6 + di iA li ti ;I 6 + di ti


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Given primary RN sequences and using RN

combinatorics, the goal of this pro=ect is to model

components of an +-%3 RN secondary structure

)namely "/ and "9 domains*. The ma=orconcentration of this pro=ect is on reducing the

minimum free energy to form an optimum +-%3 RN

secondary structure.

"ource$ +-%3 sequence prediction, /0, in progress

Application: ;I16 +redictionApplication: ;I16 +rediction


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2!3*! 4 RNA tructural 5lements. Illustration of a 'or&ing model of the ;I1I 3C

U>R sho'ing the various stem1loop structures important for virus replication* >heseare the >AR element, the polyA- hairpin, the U31+BS comple?, the stem1loops 617

containing the 2IS, the ma)or splice donor, the ma)or pac&aging signal, and the gag

start codon, respectively* Nucleotides and numering correspond to the ;I1I ;DB!

se.uence* Adapted from 5lever et al* 6""3- and Ber&hout and van =amel !$$$--

A li i ;I 6 + di i -A li ti ;I 6 + di ti t -


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Application: ;I16 +rediction cont*-Application: ;I16 +rediction cont*-

The following sequence was obtained from the N'6+ website.The first 9K9 nucleotides were e>tracted from the entire +-%3RN genomic sequence$

GG&'&'&'&GG&&G''G&'&GG''&GGGG'&'&'&GG'&'&GGG''''&G'&&G''&'&G'&&G''&&GG&G'&&'G&G&G&G&G'''G&'&G&&G&G&G'&'&GG&'&GG&'''&'G'''&&&&G&'G&G&GG&'&'&G'G&GG'G'''G'GGG''&GG'GGGG''GGGG'&'&'&'G'G'GG'&'GG'&&G'&GG'G'G''GG'GGG'GGGGG'GG'G

'&GG&GG&'G''&&&&G'&G'GGGG'&GGGGGG&GGG&G'GGG'G&'G&&&G'G

'olor key$"/ # yellow

"9 % red


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@uture Research: 5enters @or@uture Research: 5enters @or

6iological and 'hemical "ensors Research

icology and 6iosensorsResearch

The mission is to advance the fundamentalscientific and technological knowledge needed to

enable the development of new biological andchemical sensors.


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Math1Bio 5ollaoratorsMath1Bio 5ollaorators

Jwayne ill, 6iology Jept., 1"&

lvin Lennedy and Richard 2illiams,'hemistry Jept., 1"&

2ilfred Ndifon,

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