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# Publisher Correction: Process calculi may reveal the equivalence lying at the heart of RNA and proteins

The Original Article was published on 24 January 2019

Correction to: Scientific Reports https://doi.org/10.1038/s41598-018-36965-1, published online 24 January 2019

This Article contains errors introduced during its publication, where the wrong fonts were used to distinguish between two models.

In the HTML version, in the Results section under subheading ‘Folding step’,

“In this way the whole folding processes, $${{\mathscr{{F}}}}_{rna}$$ and $${ {\mathcal F} }_{p}$$ respectively, can be defined as following”:

“In this way the whole folding processes, $${ {\mathcal F} }_{rna}$$ and $${ {\mathcal F} }_{p}$$ respectively, can be defined as following”:

In the same section of the HTML version, ‘Theorem 1’,

“If $${{\mathscr{{F}}}}_{rna}={\mathscr{{H}}}({{\mathscr{{F}}}}_{rna})$$ and $${{\mathscr{{F}}}}_{p}={\mathscr{{H}}}({{\mathscr{{F}}}}_{p})$$ then $${{\mathscr{{F}}}}_{rna}$$ and $${{\mathscr{{F}}}}_{p}$$ are strongly bisimilar ($${{\mathscr{{F}}}}_{rna}\, \sim \,{{\mathscr{{F}}}}_{p}$$)”.

“If $${{\mathscr{{F}}}}_{rna}= {\mathcal H} ({ {\mathcal F} }_{rna})$$ and $${{\mathscr{{F}}}}_{p}= {\mathcal H} ({ {\mathcal F} }_{p})$$ then $${{\mathscr{{F}}}}_{rna}$$ and $${{\mathscr{{F}}}}_{p}$$ are strongly bisimilar ($${{\mathscr{{F}}}}_{rna}\, \sim \,{{\mathscr{{F}}}}_{p}$$)”.

“The application of $${\mathcal H}$$ to the models described in the previous section results in a new representation of the folding processes of RNAs and proteins, indicated by the symbols $${ {\mathcal F} }_{rna}$$ and $${ {\mathcal F} }_{p}$$ respectively (see Section 2 of the Supplementary Information for a complete description)”.

“The application of $${\mathcal H}$$ to the models described in the previous section results in a new representation of the folding processes of RNAs and proteins, indicated by the symbols $${{\mathscr{{F}}}}_{rna}$$ and $${{\mathscr{{F}}}}_{p}$$ respectively (see Section 2 of the Supplementary Information for a complete description)”.

“If $${ {\mathcal F} }_{rna}= {\mathcal H} ({ {\mathcal F} }_{rna})$$ and $${ {\mathcal F} }_{p}= {\mathcal H} ({ {\mathcal F} }_{p})$$ then $${ {\mathcal F} }_{rna}$$ and $${ {\mathcal F} }_{p}$$ are strongly bisimilar ($${ {\mathcal F} }_{rna}\, \sim \,{ {\mathcal F} }_{p}$$)”.

“If $${{\mathscr{{F}}}}_{rna}= {\mathcal H} ({ {\mathcal F} }_{rna})$$ and $${{\mathscr{{F}}}}_{p}= {\mathcal H} ({ {\mathcal F} }_{p})$$ then $${{\mathscr{{F}}}}_{rna}$$ and $${{\mathscr{{F}}}}_{p}$$ are strongly bisimilar ($${{\mathscr{{F}}}}_{rna}\, \sim \,{{\mathscr{{F}}}}_{p}$$)”.

In the proof for Theorem 1,

“A winning strategy of the defender starts from the pair of states ($${ {\mathcal F} }_{rna}^{s},{ {\mathcal F} }_{p}^{s}$$) of the relative LTSs, transliterated (RNAFS, PFS) as in Fig. 3”.

“A winning strategy of the defender starts from the pair of states ($${{\mathscr{{F}}}}_{rna}^{s},{{\mathscr{{F}}}}_{p}^{s}$$) of the relative LTSs, transliterated (RNAFS, PFS) as in Fig. 3”.

“The $${ {\mathcal F} }_{rna}$$ and $${ {\mathcal F} }_{p}$$ folding processes, likewise $${ {\mathcal F} }_{rna}$$ and $${ {\mathcal F} }_{p}$$, are defined as

$${\mathcal F}_{rna}\mathop{=}\limits^{\rm{def}}({\mathcal F}_{rna}^{s}|\Delta {G}_{{\mathcal{F}}^{s}})\backslash \{ndg,\,pdg,\,zdg\};$$

$${\mathcal F}_{p}\mathop{=}\limits^{\rm{def}}({\mathcal F}_{p}^{s}|\Delta {G}_{{\mathcal{F}}^{s}})\backslash \{ndg,\,pdg,\,zdg\};$$

where

$$\Delta {G}_{{{\mathcal{{F}}}}^{s}}\mathop{=}\limits^{{\rm{def}}}\overline{pdg}.\,\Delta {G}_{{{\mathcal{{F}}}}^{s}}+\overline{ndg}.\,\Delta {G}_{{{\mathcal{{F}}}}^{s}}+\overline{zdg}.\,\Delta {G}_{{{\mathcal{{F}}}}^{s}}$$

“The $${{\mathscr{{F}}}}_{rna}$$ and $${{\mathscr{{F}}}}_{p}$$ folding processes, likewise $${ {\mathcal F} }_{rna}$$ and $${ {\mathcal F} }_{p}$$, are defined as

$${{\mathscr{{F}}}}_{rna}\mathop{=}\limits^{{\rm{def}}}({{\mathscr{{F}}}}_{rna}^{s}|\Delta {G}_{{{\mathscr{{F}}}}^{s}})\backslash \{ndg,\,pdg,\,zdg\};$$

$${{\mathscr{{F}}}}_{p}\mathop{=}\limits^{{\rm{def}}}({{\mathscr{{F}}}}_{p}^{s}|\Delta {G}_{{{\mathscr{{F}}}}^{s}})\backslash \{ndg,\,pdg,\,zdg\};$$

where

$$\Delta {G}_{{{\mathscr{{F}}}}^{s}}\mathop{=}\limits^{{\rm{def}}}\overline{pdg}.\,\Delta {G}_{{{\mathscr{{F}}}}^{s}}+\overline{ndg}.\,\Delta {G}_{{{\mathscr{{F}}}}^{s}}+\overline{zdg}.\,\Delta {G}_{{{\mathscr{{F}}}}^{s}}$$

And,

“Such proof can also be obtained with the aid of an automated tool; in Fig. 4 we show the results of the bisimulation game performed with CAAL on the processes $${ {\mathcal F} }_{rna}$$ and $${ {\mathcal F} }_{p}$$, transliterated RNAFOLDING and PFOLDING respectively”.

“Such proof can also be obtained with the aid of an automated tool; in Fig. 4 we show the results of the bisimulation game performed with CAAL on the processes $${{\mathscr{{F}}}}_{rna}$$ and $${{\mathscr{{F}}}}_{p}$$, transliterated RNAFOLDING and PFOLDING respectively”.

“Winning strategy of the defender in the strong bisimulation game that compares the pair of processes ($${ {\mathcal F} }_{rna}^{s},{ {\mathcal F} }_{p}^{s}$$), transliterated (RNAFS, PFS). The results of this play proves that RNAFS ~ PFS, i.e. that the two processes are strongly bisimilar”.

“Winning strategy of the defender in the strong bisimulation game that compares the pair of processes ($${{\mathscr{{F}}}}_{rna}^{s},{{\mathscr{{F}}}}_{p}^{s}$$), transliterated (RNAFS, PFS). The results of this play proves that RNAFS ~ PFS, i.e. that the two processes are strongly bisimilar”.

“Labelled Transition Systems of (a) the redefined $${ {\mathcal F} }_{rna}^{s}$$ process, transliterated RNAFS, and of (b) the redefined $${ {\mathcal F} }_{p}^{s}$$, process, transliterated PFS, generated with the CAAL web-based tool (Concurrency Workbench, Alborg Edition)”.

“Labelled Transition Systems of (a) the redefined $${{\mathscr{{F}}}}_{rna}^{s}$$ process, transliterated RNAFS, and of (b) the redefined $${{\mathscr{{F}}}}_{p}^{s}$$, process, transliterated PFS, generated with the CAAL web-based tool (Concurrency Workbench, Alborg Edition)”.

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Correspondence to Emanuela Merelli.

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Maestri, S., Merelli, E. Publisher Correction: Process calculi may reveal the equivalence lying at the heart of RNA and proteins. Sci Rep 9, 16651 (2019). https://doi.org/10.1038/s41598-019-51082-3

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• DOI: https://doi.org/10.1038/s41598-019-51082-3