Supplement S1: RNA secondary structure. structure + sequence format

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Calvin Rogers
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1 Supplemet S1: RN seodary struture RN struture is ofte expressed shematially y its ase pairig: the Watso-rik (W) ase pairs (deie) with (rail), ad G (Guaie) with (ytosie) ad also the o-watso-rik (o-w) ase pair G with. RN sequees are typially writte from left to right. The egiig of the sequee is usually o the left had side ad is alled the 5 ed, ad the opposite ed is alled the 3 ed. umer of simple strutural motifs are geerated y the way that the RN moleule forms these ase pairs. I seodary struture of RN, the typial strutural motifs are usually lassified as stem, iteral loop (or iterior loop), ad multirah loop. This stadard eglets a large group of possile strutures alled pseudokots (PK). Stem: Whe more tha oe ase pair appears i the form of a group of otiguous ase pairs, the resultig struture motif is desried as a stem (Fig. S1a). For RN, this stem motif appears as a flat ojet. However, the atual struture i three dimesios (3D) has a twist that makes a 360 o rotatio roughly every 10 ps (for -RN: the most ommoly foud struture for RN). a G G G struture + sequee format struture format Supplemet Figure S1. example of a stem motif represeted as seodary struture. (a) stem iludig oth the seodary struture ad the sequee laels. () stem that oly iludes the ase pairig iformatio. Loops: The other simple strutural motifs, ased o the types of loops, a also e foud i RN strutures. Hairpi loop (H-loop): The simplest suh motif is the H-loop (Fig. S2). The H-loop osists of two omplimetary sequees joied y some o-pairig ases i a loop. simple example would e the sequee

2 (Fig. S2a). The Figure also otais three additioal represetatios. I Figure S2, oly the ase pairig iformatio of the struture diagram is show (i a similar way as Fig. S1). Figure S2 is a raket graph developed y Hofaker et al. [1], where the raket is mathed y a equal ad opposite raket o the 3 side. Figure S2d is a highly simplified diagram similar to Fig. S2, ut lakig speifi details aout the exat size of the loop regio. a d (((((...))))) Supplemet Figure S2. example of a hairpi loop (H-loop). (a) Seodary struture with ase idex iluded. () Seodary struture with oly the ase pairig ad ase positio iluded. () simplified represetatio of this seodary struture usig raket otatio. (d) seodary struture that oly speifies the stem ase pairs without speifyig the exat size of the loop regios. Iteral loop (I-loop): other ommo struture is the iteral loop (I-loop), Fig. S3a-h. iteral loop appears etwee two stems ad has 1 upaired ases o the 5 side ad 2 upaired ases o the 3 side; where 1 = 0,1, 2 uleotides (t) ad likewise for 2. Fig. S3a shows a symmetri iteral loop ( 1 = 2) where 1 = 2 = 1 t: i.e., the umer of ases i o eah side of the loop are equal to oe. This I-loop motif also iludes ulges whih have the property that 1 > 0 ad 2 = 0 or 1 = 0 ad 2 > 0. Large strutures of RN typially have may loops, ulges, ad iteral loops. There are may examples of symmetri I-loops foud i RN struture dataases. Some asymmetri I-loops ( ) a also e foud. 1 2

3 a e f 1 2 ((((.(((((...))))).)))) g G G G ((((...(((((...))))))))) d 1 2 h Supplemet Figure S3. Examples of seodary struture for iteral loops ad ulges (I-loops). (a) Seodary struture of iteral loops with ase idex iluded. () Seodary struture with oly the ase pairig ad ase positio iluded. () simplified represetatio of this seodary struture usig raket otatio. (d) seodary struture that oly speifies the stem loatios ad is ot speifi aout the loop regios. (e-h) The same defiitios apply for the example of a ulge. Multirah loop (M-loop): third ommo seodary struture motif is kow as a multirah loop (M-loop or MBL). These are more omplex strutures that osist of several of these previously desried stem-loop type strutures. These strutures are also quite ommo like the other seodary struture motifs. example of a multirah loop is show i Fig. S4. The stem, H-loop, I-loop ad M-loop are the four fudametal motifs that make up seodary struture. The fudametal feature of RN seodary struture is that ase idies i ad j ( i< j) are allowed to ase pair with eah other if they satisfy the followig properties with all other ase pairs ( i ' ad j ', with i' < j' ) (1) if i ad (2) if ad j are otaied withi i ' ad ' j ', the i' < i< j < j' i j are ot otaied withi i ad j ', the either i ad j are less tha i ', or i ad j are greater tha j ', (3) if either ase is true, the i = i' ad j = j'.

4 a G G G.((((.((((...))))..(((((...))))).)))). Supplemet Figure S4. Seodary struture of multirah loops (M-loop or MBL). (a) Seodary struture of M-loop with ase idex iluded. () Seodary struture with oly the ase pairig ad ase positio iluded. () simplified represetatio of this seodary struture usig raket otatio. Pseudokot (PK) ad kots: Whereas may RN strutures are kow to satisfy these rules (for example, trn), this is ot always the ase. The most ommo deviatios from these ase pairig rules is a lass of strutures alled pseudokots (PKs). pseudokot permits violatio of the aove three pairig relatioship rules. (4) For pseudokots: parts of the struture still satisfy ases 1 through 3. However, i additio, i at least oe part of a regio etwee k ad l suh that k i, j, i', j' l, there exists some ase pairs that satisfy either i' < i< j' < j or i < i' < j < j'. Hee, may more possiilities a e geerated oe we start allowig pseudokots. Pseudokots differ from real kots i the sese that the strad does ot pass ompletely through the loop ut oly eomes potetially etagled with it. yoe who has tried to utagle a pair of earphoes or tried to utagle a reetly, eatly woud up ord a realize that kots aturally our o log flexile ords. Ideed, great effort seems to e required to avoid taglig a heap of ords. Figure S5a shows a ommo pseudokot kow as a H-type pseudokot. This is also kow as a BB pseudokot. The struture is otated elow with the stadard parethesis otatio for the asi seodary struture (here show i lue) ad square rakets for the pseudokot likage (here show i red). Gree idiates the regios of free strad that are ot formig ase pairs (ps).

5 The olor distitio of the stems i this example is ot importat eause oth stems are the same legth ad oth stems are less tha 10 ps. a...(((((...[[[[[.)))))...]]]]]... e d...((((((((((((((...[[[[[[[[[[[[[[.))))))))))))))...]]]]]]]]]]]]]]... g f...(((((...[[[[[.)))))...(((((.]]]]]...)))))... h...((((((((((((((...[[[[[.))))))))))))))...]]]]]... Supplemet Figure S5. Examples of pseudokots ad kots. (a) H-type pseudokot where the likage stem is deoted i red, stadard seodary struture i lue ad free strad (regios of o ase pairig) i gree. () The same struture i (a) deoted i raket otatio. () example of a kot where the stem legth of oth the seodary struture ad the liage stem are loger tha 9 ase pairs. (d) The same struture i raket otatio. (e) exteded pseudokot, sometimes referred to as a kissig-loop ad also kow as a BB pseudokot. (f) The struture show i raket otatio. (g) pseudokot. The differee etwee () ad (g) is that the likage stem is shorter tha 9 otiguous ps. (h) The same struture i raket otatio. Figure S5 shows a kot ad the orrespodig raket struture is show elow i Fig. S5d. Both stems i Fig. S5 otai 14 ase pairs (p). Sie the helial axis makes a rotatio of 360 o every 10 ps, this meas that the struture i Fig. S5 is tagled i a kot. There is o reaso why suh a struture aot form. Ideed, kots are kow to form i some rare proteis [2,3]. However, it is ot a pseudokot ad the urret approah is ot desiged to estimate its existee or the likelihood of its formatio. Oe importat feature of a pseudokot is, therefore, that the likage stem (here show i red) must e shorter tha 10 otiguous ps. Figure S5e shows two stem-hairpis-loops (lue ad purple) that are joied y a likage stem (red). The stems are all short as i Fig. S5a. This is a pseudokot, sometimes referred to as a kissig

6 loop. It is also kow as a BB pseudokot. It is also oserved i a umer of plaes, although less frequetly tha H-type pseudokots. The orrespodig raket otatio for this struture is show elow (Fig. S5f). Figure S5g shows a struture itermediate etwee Figs. S5a ad. This struture is also a H-type pseudokot. Here we see a eessary oditio for defiig a segmet as a likage stem ad the most importat distitio etwee a kot ad a pseudokot: a likage stem aot otai more tha 9 otiguous ps. Whe a iteral loop reaks the otiuity, this rule may ot eessarily apply. Therefore, loger overall struture ould form, ut oly if the stem is ot otiguous. Moreover, this would have to e osidered o a ase y ase asis. s a fial uriosity, i Fig. S6, a seodary struture (left had side) ad its orrespodig kot (right had side) are show i equilirium. The kot reders a peuliar 2D illusio perhaps remiiset of artist M.. Esher s Belvedere (1958). The kotted struture ould oeivaly exist i hemial equilirium with a stadard seodary struture, though eletrostati effets may reder it less favorale thermodyamially. It ould result from a simple misfoldig. seodary struture Belvedere kot Supplemet Figure S6. speial type of kot (right had side) that eomes etagled due to the way the struture folds up. The two dimesioal ature of the RN shematis teds to hide urious possiilities as this. The kot is see i equilirium etwee the ukotted struture (left had side) ad the Belvedere kot (right had side). Futioal RN strutures that otai kots of the form show i Figs. S5 or S6 are urretly ukow or have ot ee reported. However, pseudokots are ofte oserved i futioal RN strutures, partiularly H-type pseudokots (Fig. S5a). Sigle strad RN sequees suh as messeger RN with itros a have sequees with legths that umer i the tes of thousads of uleotides. With suh a propesity for a few simple ords to eome tagled, ad, sie the ell

7 a have may thousads of protei ad RN strads preset withi the ellular eviromet, this suggests that there is a fair amout of effort made withi the ell to prevet or get rid of kots [2]. I this work, we are oered with the preditio of pseudokots. These strutures have the property that they a e evaluated as strutures resultig from reversile foldig. Referees 1. Hofaker IL, Fotaa W, Stadler PF, Bohoeffer S, Taker M, et al. (1994) Fast foldig ad ompariso of RN seodary strutures. Moatshefte f hemie 125: Lua R, Groserg Y (2006) Statistis of kots, geometry of oformatios, ad evolutio of proteis. PLoS omp Biol 2: e Virau P, Miry L, ad Kardar M (2006) Itriate kots i proteis: futio ad evolutio. PLoS omp Biol 2: e122.

ANOTHER PROOF FOR FERMAT S LAST THEOREM 1. INTRODUCTION

ANOTHER PROOF FOR FERMAT S LAST THEOREM Mugur B. RĂUŢ Correspodig author: Mugur B. RĂUŢ, E-mail: m_b_raut@yahoo.om Abstrat I this paper we propose aother proof for Fermat s Last Theorem (FLT). We foud