Gallager, R. G. (2012). Discrete stochastic processes,
Chapter 4, volume 321. Springer Science & Business
Media.
Geman, S. and Geman, D. (1984). Stochastic relaxation,
gibbs distributions, and the bayesian restoration of
images. Pattern Analysis and Machine Intelligence,
IEEE Transactions on, (6):721–741.
Hastings, W. K. (1970). Monte carlo sampling methods us-
ing markov chains and their applications. Biometrika,
57(1):97–109.
Huang, F. W., Qin, J., Reidys, C. M., and Stadler, P. F.
(2009). Partition function and base pairing probabili-
ties for rna–rna interaction prediction. Bioinformatics,
25(20):2646–2654.
Jaccard, P. (1901). Etude comparative de la distribution
ﬂorale dans une portion des Alpes et du Jura. Impr.
Corbaz.
Kolb, F. A., Engdahl, H. M., Slagter-J
¨
ager, J. G., Ehres-
mann, B., Ehresmann, C., Westhof, E., Wagner, E.
G. H., and Romby, P. (2000a). Progression of a loop–
loop complex to a four-way junction is crucial for the
activity of a regulatory antisense rna. The EMBO jour-
nal, 19(21):5905–5915.
Kolb, F. A., Malmgren, C., Westhof, E., Ehresmann, C.,
Ehresmann, B., Wagner, E., and Romby, P. (2000b).
An unusual structure formed by antisense-target rna
binding involves an extended kissing complex with
a four-way junction and a side-by-side helical align-
ment. Rna, 6(3):311–324.
Li, A. X., Marz, M., Qin, J., and Reidys, C. M. (2011). Rna–
rna interaction prediction based on multiple sequence
alignments. Bioinformatics, 27(4):456–463.
Liu, J. S. (1994). The collapsed gibbs sampler in bayesian
computations with applications to a gene regulation
problem. Journal of the American Statistical Associa-
tion, 89(427):958–966.
McCaskill, J. S. (1990). The equilibrium partition function
and base pair binding probabilities for rna secondary
structure. Biopolymers, 29(6-7):1105–1119.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N.,
Teller, A. H., and Teller, E. (1953). Equation of state
calculations by fast computing machines. The journal
of chemical physics, 21(6):1087–1092.
Metzler, D. and Nebel, M. E. (2008). Predicting rna sec-
ondary structures with pseudoknots by mcmc sam-
pling. Journal of mathematical biology, 56(1-2):161–
181.
Meyer, I. M. (2008). Predicting novel rna–rna interactions.
Current opinion in structural biology, 18(3):387–393.
Mneimneh, S. (2009). On the approximation of optimal
structures for rna-rna interaction. IEEE/ACM Trans-
actions on Computational Biology and Bioinformatics
(TCBB), 6(4):682–688.
Mneimneh, S. and Ahmed, S. A. (2015). Multiple rna inter-
action: Beyond two. To appear in IEEE Transactions
on NanoBioscience.
Mneimneh, S., Ahmed, S. A., and Greenbaum, N. L.
(2013). Multiple RNA interaction - formulations, ap-
proximations, and heuristics. In BIOINFORMATICS
2013 - Proceedings of the International Conference
on Bioinformatics Models, Methods and Algorithms,
Barcelona, Spain, 11 - 14 February, 2013., pages 242–
249.
M
¨
uckstein, U., Tafer, H., Hackerm
¨
uller, J., Bernhart, S. H.,
Stadler, P. F., and Hofacker, I. L. (2006). Ther-
modynamics of rna–rna binding. Bioinformatics,
22(10):1177–1182.
Newby, M. I. and Greenbaum, N. L. (2001). A conserved
pseudouridine modiﬁcation in eukaryotic u2 snrna in-
duces a change in branch-site architecture. RNA,
7(06):833–845.
Pervouchine, D. D. (2004). Iris: intermolecular rna interac-
tion search. Genome Informatics Series, 15(2):92.
Powers, D. M. (2011). Evaluation: from precision, recall
and f-measure to roc, informedness, markedness and
correlation.
Rousseeuw, P. J. (1987). Silhouettes: a graphical aid to
the interpretation and validation of cluster analysis.
Journal of computational and applied mathematics,
20:53–65.
Salari, R., Backofen, R., and Sahinalp, S. C. (2010). Fast
prediction of rna-rna interaction. Algorithms for
molecular Biology, 5(5).
Sashital, D. G., Cornilescu, G., and Butcher, S. E. (2004).
U2–u6 rna folding reveals a group ii intron-like do-
main and a four-helix junction. Nature structural &
molecular biology, 11(12):1237–1242.
Sun, J.-S. and Manley, J. L. (1995). A novel u2-u6 snrna
structure is necessary for mammalian mrna splicing.
Genes & Development, 9(7):843–854.
Tong, W., Goebel, R., Liu, T., and Lin, G. (2013). Ap-
proximation algorithms for the maximum multiple rna
interaction problem. In Combinatorial Optimization
and Applications, pages 49–59. Springer.
Tong, W., Goebel, R., Liu, T., and Lin, G. (2014). Approx-
imating the maximum multiple rna interaction prob-
lem. Theoretical Computer Science.
Wei, D., Alpert, L. V., and Lawrence, C. E. (2011). Rnag:
a new gibbs sampler for predicting rna secondary
structure for unaligned sequences. Bioinformatics,
27(18):2486–2493.
Zhao, C., Bachu, R., Popovi
´
c, M., Devany, M., Brenowitz,
M., Schlatterer, J. C., and Greenbaum, N. L. (2013).
Conformational heterogeneity of the protein-free hu-
man spliceosomal u2-u6 snrna complex. RNA,
19(4):561–573.
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