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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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RNA 5ominatorics cont*-RNA 5ominatorics cont*-
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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RNA 5ominatorics cont*-RNA 5ominatorics cont*-
&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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RNA 5ominatorics cont*-RNA 5ominatorics cont*-
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 ,
RNA 5ominatorics cont*-RNA 5ominatorics cont*-
( ) 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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RNA 5ominatorics cont*-RNA 5ominatorics cont*-
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,