partition function for circular rna and applicationronny/chribska2006.pdf · partition function for...

Partition function for linear RNAsExtension for circular RNAs

Application

Partition function for circular RNA and Application

Ronny Lorenzronny@bioinf.uni-leipzig.de

Department of BioinformaticsUniversity of Leipzig

October 30, 2006

Ronny Lorenz Partition function for circular RNA and Application

Application

1 Partition function for linear RNAsStructure decomposition

Application

2 Extension for circular RNAsStructure decomposition

Application

2 Extension for circular RNAsStructure decomposition

3 ApplicationOccurence of circular RNAsMetastable Structures

ApplicationStructure decomposition

let S be the set of all possible secondary structures for a givenRNA sequence

each s ∈ S is associated with a free energy Gs

The partition function Q for a given sequence

Q = ∑s∈S

e−GsRT (1)

i and j do not pair

i and j form a hairpin loop, with closing pair [i,j],

there exist two positions k,l with i < k < l < j so that thesubsequence i,j forms an interior loop with closing pairs i,j andk,l

i and j are the closing pair of a multiloop

dynamic programming approach, that calculates the partitionfunction for all subsequences x [i, j] and ends with the partitionfunction for the whole sequence...

Decomposition of paired/unpaired i,j

��

i i+1 j|

i jk+1k

��

i i+1 j|

i jk+1k

Qi,j = 1+ ∑1<k<j

Qbi,k ·Qk+1,j (2)

��

i i+1 j|

i jk+1k

Qi,j = 1+ ∑1<k<j

Qbi,k ·Qk+1,j (2)

��

i i+1 j|

i jk+1k

Qi,j = 1+ ∑1<k<j

Qbi,k ·Qk+1,j (2)

Decomposition with constraint that i pair with j

��

| |=i j i j i ij jd e u u+1

��

Qbi,j = e

−GHi ,j

∑i<d<e<j

Ii ,d ,e,jRT ·Qb

d ,e +

∑i<u<j

Qmi+1,u ·Q

m1u+1,j−1 ·e

−aRT (3)

��

Qbi,j = e

−GHi ,j

∑i<d<e<j

Ii ,d ,e,jRT ·Qb

d ,e +

∑i<u<j

Qmi+1,u ·Q

m1u+1,j−1 ·e

−aRT (3)

��

Qbi,j = e

−GHi ,j

∑i<d<e<j

Ii ,d ,e,jRT ·Qb

d ,e +

∑i<u<j

Qmi+1,u ·Q

m1u+1,j−1 ·e

−aRT (3)

��

Qbi,j = e

−GHi ,j

∑i<d<e<j

Ii ,d ,e,jRT ·Qb

d ,e +

∑i<u<j

Qmi+1,u ·Q

m1u+1,j−1 ·e

−aRT (3)

��

Qbi,j = e

−GHi ,j

∑i<d<e<j

Ii ,d ,e,jRT ·Qb

d ,e +

∑i<u<j

Qmi+1,u ·Q

m1u+1,j−1 ·e

−aRT (3)

��

Qbi,j = e

−GHi ,j

∑i<d<e<j

Ii ,d ,e,jRT ·Qb

d ,e +

∑i<u<j

Qmi+1,u ·Q

m1u+1,j−1 ·e

−aRT (3)

��

Qbi,j = e

−GHi ,j

∑i<d<e<j

Ii ,d ,e,jRT ·Qb

d ,e +

∑i<u<j

Qmi+1,u ·Q

m1u+1,j−1 ·e

−aRT (3)

Decomposition of multiloop

��

= | |i j ji i ij jj−1 uu+1 u+1u

��

Qmi,j = Qm

i,j−1 ·e−cRT +

∑i<u<j

e−(u−i−1)·c−b

RT ·Qbu+1,j +

∑i<u<j

Qmi,u ·Q

bu+1,j ·e

−bRT (4)

��

Qmi,j = Qm

i,j−1 ·e−cRT +

∑i<u<j

e−(u−i−1)·c−b

RT ·Qbu+1,j +

∑i<u<j

Qmi,u ·Q

bu+1,j ·e

−bRT (4)

��

Qmi,j = Qm

i,j−1 ·e−cRT +

∑i<u<j

e−(u−i−1)·c−b

RT ·Qbu+1,j +

∑i<u<j

Qmi,u ·Q

bu+1,j ·e

−bRT (4)

��

Qmi,j = Qm

i,j−1 ·e−cRT +

∑i<u<j

e−(u−i−1)·c−b

RT ·Qbu+1,j +

∑i<u<j

Qmi,u ·Q

bu+1,j ·e

−bRT (4)

��

Qmi,j = Qm

i,j−1 ·e−cRT +

∑i<u<j

e−(u−i−1)·c−b

RT ·Qbu+1,j +

∑i<u<j

Qmi,u ·Q

bu+1,j ·e

−bRT (4)

��

Qmi,j = Qm

i,j−1 ·e−cRT +

∑i<u<j

e−(u−i−1)·c−b

RT ·Qbu+1,j +

∑i<u<j

Qmi,u ·Q

bu+1,j ·e

−bRT (4)

��

Qmi,j = Qm

i,j−1 ·e−cRT +

∑i<u<j

e−(u−i−1)·c−b

RT ·Qbu+1,j +

∑i<u<j

Qmi,u ·Q

bu+1,j ·e

−bRT (4)

��

Qmi,j = Qm

i,j−1 ·e−cRT +

∑i<u<j

e−(u−i−1)·c−b

RT ·Qbu+1,j +

∑i<u<j

Qmi,u ·Q

bu+1,j ·e

−bRT (4)

Decomposition of multiloop part with exactly one component

��

j jii j−1

��

j jii j−1

Qm1i,j = Qm1

i,j−1 ·e−cRT +

Qbi,j ·e

−bRT (5)

��

j jii j−1

Qm1i,j = Qm1

i,j−1 ·e−cRT +

Qbi,j ·e

−bRT (5)

��

j jii j−1

Qm1i,j = Qm1

i,j−1 ·e−cRT +

Qbi,j ·e

−bRT (5)

��

j jii j−1

Qm1i,j = Qm1

i,j−1 ·e−cRT +

Qbi,j ·e

−bRT (5)

��

j jii j−1

Qm1i,j = Qm1

i,j−1 ·e−cRT +

Qbi,j ·e

−bRT (5)

Similarities to linear case:

Q is also computed by previously computing Qi,j for allsubsequences

structure decomposition for Qi,j∀1 ≤ i < j ≤ n

Difference:

computing sum of weighted energies for all loops that containbases xn and x1

Difference:

computing sum of weighted energies for all loops that containbases xn and x1

leads to the computation of so called ”exterior loops”

Computation of M2 Array

��

k nu u+1

Computation of M2 Array

��

k nu u+1

QM2k ,n = ∑

Qm1k ,u ·Q

m1u+1,n (6)

Computation of exterior hairpin loop

QoH = ∑

p,q ·e−GHq,p

RT (7)

QoH = ∑

p,q ·e−GHq,p

RT (7)

QoH = ∑

p,q ·e−GHq,p

RT (7)

Computation of exterior interior loop

��

QoI = ∑

k<l<p<q

Qbk ,l ·Q

bp,q ·e

−GIk ,l ,p,qRT (8)

��

QoI = ∑

k<l<p<q

Qbk ,l ·Q

bp,q ·e

��

QoI = ∑

k<l<p<q

Qbk ,l ·Q

bp,q ·e

��

QoI = ∑

k<l<p<q

Qbk ,l ·Q

bp,q ·e

Computation of exterior multiloop

1n��

��

1n��

��

QoM = ∑

Qm1,k ·Q

M2k+1,n ·e

−aRT (9)

1n��

��

QoM = ∑

Qm1,k ·Q

M2k+1,n ·e

−aRT (9)

1n��

��

QoM = ∑

Qm1,k ·Q

M2k+1,n ·e

−aRT (9)

1n��

��

QoM = ∑

Qm1,k ·Q

M2k+1,n ·e

−aRT (9)

Partition function for circular case:

Qo = QoH +Qo

I +QoM (10)

This is the step where i am actually in my diploma thesis

This is the step where i am actually in my diploma thesisI will skip

equations for base pairing probability matrix

way of backtracking to obtain secondary structures

Application

Occurence of circular RNAsMetastable Structures

Most known RNAs are linear and circular RNAs are not found veryoftenBut they are found in:

Application

Viroids and satellite RNAs (sizes arround 300-400 nt)

Application

Genome of Hepatitis δ virus

Application

spliced Group I Introns

Application

Box C/D RNAs in Pyrococcus furiosus

Application

In almost all cases, circular RNAs code not for any mRNA but fortheir own structure

Application

What can be the aim for application of CPF?

Application

finding metastable structure states

Application

better prediction of circular RNA structures

Application

better prediction of circular RNA structures

other things I did not have thought about yet

Application

Thank you for your attention

partition function for circular rna and applicationronny/chribska2006.pdf · partition function for...

Documents

transcriptome-wide discovery of circular rnas in...

circular rnas are a large class of animal rnas with

disease-associated circular rnas: from biology to...

short hairpin rnas dave lyons may 2009. short hairpin rnas...

long non-coding rnas and enhancer rnas regulate the...

exon-intron circular rnas regulate transcription in the...

(long) non-coding rnas

research paper circular rna signature in hepatocellular...

energy release rate and mode partition for …energy release...

refinement of some partition inequalities partitions, the...

merkel cell polyomavirus encodes circular rnas (circrnas...

problem 59: most circular partition of a square

th on small rnas

non-coding rnas or rnas come more than in three flavours

non-coding rnas ( ncrnas )

man cue rnas

circinteractome: a web tool for exploring circular rnas ......

circular rnas are a large class of animal rnas with...

long noncoding rnas in c....

circular rnas in gastric cancer: potential biomarkers and...