Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for...
Transcript of Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for...
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Partition function for linear RNAsExtension for circular RNAs
Application
Partition function for circular RNA and Application
Ronny [email protected]
Department of BioinformaticsUniversity of Leipzig
October 30, 2006
Ronny Lorenz Partition function for circular RNA and Application
![Page 2: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/2.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
1 Partition function for linear RNAsStructure decomposition
Ronny Lorenz Partition function for circular RNA and Application
![Page 3: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/3.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
1 Partition function for linear RNAsStructure decomposition
2 Extension for circular RNAsStructure decomposition
Ronny Lorenz Partition function for circular RNA and Application
![Page 4: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/4.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
1 Partition function for linear RNAsStructure decomposition
2 Extension for circular RNAsStructure decomposition
3 ApplicationOccurence of circular RNAsMetastable Structures
Ronny Lorenz Partition function for circular RNA and Application
![Page 5: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/5.jpg)
Partition function for linear RNAsExtension for circular RNAs
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)
Ronny Lorenz Partition function for circular RNA and Application
![Page 6: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/6.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
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...
Ronny Lorenz Partition function for circular RNA and Application
![Page 7: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/7.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of paired/unpaired i,j
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i j=
i i+1 j|
i jk+1k
Q
Ronny Lorenz Partition function for circular RNA and Application
![Page 8: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/8.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of paired/unpaired i,j
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i j=
i i+1 j|
i jk+1k
Q
Qi,j = 1+ ∑1<k<j
Qbi,k ·Qk+1,j (2)
Ronny Lorenz Partition function for circular RNA and Application
![Page 9: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/9.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of paired/unpaired i,j
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������������������������������
i j=
i i+1 j|
i jk+1k
Q
Qi,j = 1+ ∑1<k<j
Qbi,k ·Qk+1,j (2)
Ronny Lorenz Partition function for circular RNA and Application
![Page 10: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/10.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of paired/unpaired i,j
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i j=
i i+1 j|
i jk+1k
Q
Qi,j = 1+ ∑1<k<j
Qbi,k ·Qk+1,j (2)
Ronny Lorenz Partition function for circular RNA and Application
![Page 11: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/11.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition with constraint that i pair with j
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| |=i j i j i ij jd e u u+1
Qb
Ronny Lorenz Partition function for circular RNA and Application
![Page 12: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/12.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition with constraint that i pair with j
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| |=i j i j i ij jd e u u+1
Qb
Qbi,j = e
−GHi ,j
RT +
∑i<d<e<j
e−G
Ii ,d ,e,jRT ·Qb
d ,e +
∑i<u<j
Qmi+1,u ·Q
m1u+1,j−1 ·e
−aRT (3)
Ronny Lorenz Partition function for circular RNA and Application
![Page 13: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/13.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition with constraint that i pair with j
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| |=i j i j i ij jd e u u+1
Qb
Qbi,j = e
−GHi ,j
RT +
∑i<d<e<j
e−G
Ii ,d ,e,jRT ·Qb
d ,e +
∑i<u<j
Qmi+1,u ·Q
m1u+1,j−1 ·e
−aRT (3)
Ronny Lorenz Partition function for circular RNA and Application
![Page 14: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/14.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition with constraint that i pair with j
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| |=i j i j i ij jd e u u+1
Qb
Qbi,j = e
−GHi ,j
RT +
∑i<d<e<j
e−G
Ii ,d ,e,jRT ·Qb
d ,e +
∑i<u<j
Qmi+1,u ·Q
m1u+1,j−1 ·e
−aRT (3)
Ronny Lorenz Partition function for circular RNA and Application
![Page 15: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/15.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition with constraint that i pair with j
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| |=i j i j i ij jd e u u+1
Qb
Qbi,j = e
−GHi ,j
RT +
∑i<d<e<j
e−G
Ii ,d ,e,jRT ·Qb
d ,e +
∑i<u<j
Qmi+1,u ·Q
m1u+1,j−1 ·e
−aRT (3)
Ronny Lorenz Partition function for circular RNA and Application
![Page 16: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/16.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition with constraint that i pair with j
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| |=i j i j i ij jd e u u+1
Qb
Qbi,j = e
−GHi ,j
RT +
∑i<d<e<j
e−G
Ii ,d ,e,jRT ·Qb
d ,e +
∑i<u<j
Qmi+1,u ·Q
m1u+1,j−1 ·e
−aRT (3)
Ronny Lorenz Partition function for circular RNA and Application
![Page 17: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/17.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition with constraint that i pair with j
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| |=i j i j i ij jd e u u+1
Qb
Qbi,j = e
−GHi ,j
RT +
∑i<d<e<j
e−G
Ii ,d ,e,jRT ·Qb
d ,e +
∑i<u<j
Qmi+1,u ·Q
m1u+1,j−1 ·e
−aRT (3)
Ronny Lorenz Partition function for circular RNA and Application
![Page 18: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/18.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition with constraint that i pair with j
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| |=i j i j i ij jd e u u+1
Qb
Qbi,j = e
−GHi ,j
RT +
∑i<d<e<j
e−G
Ii ,d ,e,jRT ·Qb
d ,e +
∑i<u<j
Qmi+1,u ·Q
m1u+1,j−1 ·e
−aRT (3)
Ronny Lorenz Partition function for circular RNA and Application
![Page 19: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/19.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop
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= | |i j ji i ij jj−1 uu+1 u+1u
mQ
Ronny Lorenz Partition function for circular RNA and Application
![Page 20: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/20.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop
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= | |i j ji i ij jj−1 uu+1 u+1u
mQ
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)
Ronny Lorenz Partition function for circular RNA and Application
![Page 21: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/21.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop
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= | |i j ji i ij jj−1 uu+1 u+1u
mQ
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)
Ronny Lorenz Partition function for circular RNA and Application
![Page 22: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/22.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop
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= | |i j ji i ij jj−1 uu+1 u+1u
mQ
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)
Ronny Lorenz Partition function for circular RNA and Application
![Page 23: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/23.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop
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= | |i j ji i ij jj−1 uu+1 u+1u
mQ
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)
Ronny Lorenz Partition function for circular RNA and Application
![Page 24: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/24.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop
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= | |i j ji i ij jj−1 uu+1 u+1u
mQ
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)
Ronny Lorenz Partition function for circular RNA and Application
![Page 25: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/25.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop
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= | |i j ji i ij jj−1 uu+1 u+1u
mQ
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)
Ronny Lorenz Partition function for circular RNA and Application
![Page 26: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/26.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop
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= | |i j ji i ij jj−1 uu+1 u+1u
mQ
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)
Ronny Lorenz Partition function for circular RNA and Application
![Page 27: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/27.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop
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= | |i j ji i ij jj−1 uu+1 u+1u
mQ
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)
Ronny Lorenz Partition function for circular RNA and Application
![Page 28: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/28.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop part with exactly one component
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i j|=
j jii j−1
Qm1
Ronny Lorenz Partition function for circular RNA and Application
![Page 29: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/29.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop part with exactly one component
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����������������������������������������
i j|=
j jii j−1
Qm1
Qm1i,j = Qm1
i,j−1 ·e−cRT +
Qbi,j ·e
−bRT (5)
Ronny Lorenz Partition function for circular RNA and Application
![Page 30: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/30.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop part with exactly one component
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����������������������������������������
i j|=
j jii j−1
Qm1
Qm1i,j = Qm1
i,j−1 ·e−cRT +
Qbi,j ·e
−bRT (5)
Ronny Lorenz Partition function for circular RNA and Application
![Page 31: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/31.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop part with exactly one component
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i j|=
j jii j−1
Qm1
Qm1i,j = Qm1
i,j−1 ·e−cRT +
Qbi,j ·e
−bRT (5)
Ronny Lorenz Partition function for circular RNA and Application
![Page 32: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/32.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop part with exactly one component
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����������������������������������������
i j|=
j jii j−1
Qm1
Qm1i,j = Qm1
i,j−1 ·e−cRT +
Qbi,j ·e
−bRT (5)
Ronny Lorenz Partition function for circular RNA and Application
![Page 33: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/33.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Decomposition of multiloop part with exactly one component
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����������������������������������������
i j|=
j jii j−1
Qm1
Qm1i,j = Qm1
i,j−1 ·e−cRT +
Qbi,j ·e
−bRT (5)
Ronny Lorenz Partition function for circular RNA and Application
![Page 34: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/34.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Similarities to linear case:
Ronny Lorenz Partition function for circular RNA and Application
![Page 35: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/35.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Similarities to linear case:
Q is also computed by previously computing Qi,j for allsubsequences
Ronny Lorenz Partition function for circular RNA and Application
![Page 36: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/36.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Similarities to linear case:
Q is also computed by previously computing Qi,j for allsubsequences
structure decomposition for Qi,j∀1 ≤ i < j ≤ n
Ronny Lorenz Partition function for circular RNA and Application
![Page 37: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/37.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
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:
Ronny Lorenz Partition function for circular RNA and Application
![Page 38: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/38.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
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
Ronny Lorenz Partition function for circular RNA and Application
![Page 39: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/39.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
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
leads to the computation of so called ”exterior loops”
Ronny Lorenz Partition function for circular RNA and Application
![Page 40: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/40.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of M2 Array
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����������������
����������������
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����������������
k n=
k nu u+1
Ronny Lorenz Partition function for circular RNA and Application
![Page 41: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/41.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of M2 Array
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k n=
k nu u+1
QM2k ,n = ∑
k<u<n
Qm1k ,u ·Q
m1u+1,n (6)
Ronny Lorenz Partition function for circular RNA and Application
![Page 42: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/42.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of exterior hairpin loop
1n
1n
=
p
q
Ronny Lorenz Partition function for circular RNA and Application
![Page 43: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/43.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of exterior hairpin loop
1n
1n
=
p
q
QoH = ∑
p<qQb
p,q ·e−GHq,p
RT (7)
Ronny Lorenz Partition function for circular RNA and Application
![Page 44: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/44.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of exterior hairpin loop
1n
1n
=
p
q
QoH = ∑
p<qQb
p,q ·e−GHq,p
RT (7)
Ronny Lorenz Partition function for circular RNA and Application
![Page 45: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/45.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of exterior hairpin loop
1n
1n
=
p
q
QoH = ∑
p<qQb
p,q ·e−GHq,p
RT (7)
Ronny Lorenz Partition function for circular RNA and Application
![Page 46: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/46.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of exterior interior loop
1n
1n
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k
l
p
q
=
Ronny Lorenz Partition function for circular RNA and Application
![Page 47: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/47.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of exterior interior loop
1n
1n
������������������������������
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���������������
���������������
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���������������
k
l
p
q
=
QoI = ∑
k<l<p<q
Qbk ,l ·Q
bp,q ·e
−GIk ,l ,p,qRT (8)
Ronny Lorenz Partition function for circular RNA and Application
![Page 48: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/48.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of exterior interior loop
1n
1n
������������������������������
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���������������
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k
l
p
q
=
QoI = ∑
k<l<p<q
Qbk ,l ·Q
bp,q ·e
−GIk ,l ,p,qRT (8)
Ronny Lorenz Partition function for circular RNA and Application
![Page 49: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/49.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of exterior interior loop
1n
1n
������������������������������
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���������������
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k
l
p
q
=
QoI = ∑
k<l<p<q
Qbk ,l ·Q
bp,q ·e
−GIk ,l ,p,qRT (8)
Ronny Lorenz Partition function for circular RNA and Application
![Page 50: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/50.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of exterior interior loop
1n
1n
������������������������������
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���������������
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k
l
p
q
=
QoI = ∑
k<l<p<q
Qbk ,l ·Q
bp,q ·e
−GIk ,l ,p,qRT (8)
Ronny Lorenz Partition function for circular RNA and Application
![Page 51: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/51.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of exterior multiloop
1n
1n������������������������������������������������������������
������������������������������������������������������������
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������������������������������������������������������������
=
kk+1
Ronny Lorenz Partition function for circular RNA and Application
![Page 52: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/52.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of exterior multiloop
1n
1n������������������������������������������������������������
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������������������������������������������������������������
=
kk+1
QoM = ∑
1<k<n
Qm1,k ·Q
M2k+1,n ·e
−aRT (9)
Ronny Lorenz Partition function for circular RNA and Application
![Page 53: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/53.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of exterior multiloop
1n
1n������������������������������������������������������������
������������������������������������������������������������
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=
kk+1
QoM = ∑
1<k<n
Qm1,k ·Q
M2k+1,n ·e
−aRT (9)
Ronny Lorenz Partition function for circular RNA and Application
![Page 54: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/54.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of exterior multiloop
1n
1n������������������������������������������������������������
������������������������������������������������������������
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=
kk+1
QoM = ∑
1<k<n
Qm1,k ·Q
M2k+1,n ·e
−aRT (9)
Ronny Lorenz Partition function for circular RNA and Application
![Page 55: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/55.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Computation of exterior multiloop
1n
1n������������������������������������������������������������
������������������������������������������������������������
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������������������������������������������������������������
=
kk+1
QoM = ∑
1<k<n
Qm1,k ·Q
M2k+1,n ·e
−aRT (9)
Ronny Lorenz Partition function for circular RNA and Application
![Page 56: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/56.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
Partition function for circular case:
Qo = QoH +Qo
I +QoM (10)
Ronny Lorenz Partition function for circular RNA and Application
![Page 57: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/57.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
This is the step where i am actually in my diploma thesis
Ronny Lorenz Partition function for circular RNA and Application
![Page 58: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/58.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
This is the step where i am actually in my diploma thesisI will skip
Ronny Lorenz Partition function for circular RNA and Application
![Page 59: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/59.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
This is the step where i am actually in my diploma thesisI will skip
equations for base pairing probability matrix
Ronny Lorenz Partition function for circular RNA and Application
![Page 60: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/60.jpg)
Partition function for linear RNAsExtension for circular RNAs
ApplicationStructure decomposition
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
Ronny Lorenz Partition function for circular RNA and Application
![Page 61: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/61.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
Occurence of circular RNAsMetastable Structures
Most known RNAs are linear and circular RNAs are not found veryoftenBut they are found in:
Ronny Lorenz Partition function for circular RNA and Application
![Page 62: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/62.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
Occurence of circular RNAsMetastable Structures
Most known RNAs are linear and circular RNAs are not found veryoftenBut they are found in:
Viroids and satellite RNAs (sizes arround 300-400 nt)
Ronny Lorenz Partition function for circular RNA and Application
![Page 63: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/63.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
Occurence of circular RNAsMetastable Structures
Most known RNAs are linear and circular RNAs are not found veryoftenBut they are found in:
Viroids and satellite RNAs (sizes arround 300-400 nt)
Genome of Hepatitis δ virus
Ronny Lorenz Partition function for circular RNA and Application
![Page 64: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/64.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
Occurence of circular RNAsMetastable Structures
Most known RNAs are linear and circular RNAs are not found veryoftenBut they are found in:
Viroids and satellite RNAs (sizes arround 300-400 nt)
Genome of Hepatitis δ virus
spliced Group I Introns
Ronny Lorenz Partition function for circular RNA and Application
![Page 65: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/65.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
Occurence of circular RNAsMetastable Structures
Most known RNAs are linear and circular RNAs are not found veryoftenBut they are found in:
Viroids and satellite RNAs (sizes arround 300-400 nt)
Genome of Hepatitis δ virus
spliced Group I Introns
Box C/D RNAs in Pyrococcus furiosus
Ronny Lorenz Partition function for circular RNA and Application
![Page 66: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/66.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
Occurence of circular RNAsMetastable Structures
Most known RNAs are linear and circular RNAs are not found veryoftenBut they are found in:
Viroids and satellite RNAs (sizes arround 300-400 nt)
Genome of Hepatitis δ virus
spliced Group I Introns
Box C/D RNAs in Pyrococcus furiosus
other
Ronny Lorenz Partition function for circular RNA and Application
![Page 67: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/67.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
Occurence of circular RNAsMetastable Structures
Most known RNAs are linear and circular RNAs are not found veryoftenBut they are found in:
Viroids and satellite RNAs (sizes arround 300-400 nt)
Genome of Hepatitis δ virus
spliced Group I Introns
Box C/D RNAs in Pyrococcus furiosus
other
In almost all cases, circular RNAs code not for any mRNA but fortheir own structure
Ronny Lorenz Partition function for circular RNA and Application
![Page 68: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/68.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
Occurence of circular RNAsMetastable Structures
What can be the aim for application of CPF?
Ronny Lorenz Partition function for circular RNA and Application
![Page 69: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/69.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
Occurence of circular RNAsMetastable Structures
What can be the aim for application of CPF?
finding metastable structure states
Ronny Lorenz Partition function for circular RNA and Application
![Page 70: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/70.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
Occurence of circular RNAsMetastable Structures
What can be the aim for application of CPF?
finding metastable structure states
better prediction of circular RNA structures
Ronny Lorenz Partition function for circular RNA and Application
![Page 71: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/71.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
Occurence of circular RNAsMetastable Structures
What can be the aim for application of CPF?
finding metastable structure states
better prediction of circular RNA structures
other things I did not have thought about yet
Ronny Lorenz Partition function for circular RNA and Application
![Page 72: Partition function for circular RNA and Applicationronny/chribska2006.pdf · Partition function for linear RNAs Extension for circular RNAs Application Partition function for circular](https://reader031.fdocuments.net/reader031/viewer/2022022519/5b16e8497f8b9a6d6d8e94bc/html5/thumbnails/72.jpg)
Partition function for linear RNAsExtension for circular RNAs
Application
Occurence of circular RNAsMetastable Structures
Thank you for your attention
Ronny Lorenz Partition function for circular RNA and Application