Supplementary Information for: Use of Fibonacci numbers in lipidomics. Enumerating various classes of fatty acids

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1 Supplemetary Iformatio for: Use of Fiboacci umbers i lipidomics Eumeratig various classes of fatty acids Stefa Schuster *,1, Maximilia Fichter 1, Severi Sasso 1 Dept. of Bioiformatics, Friedrich Schiller Uiversity, Erst-Abbe-Platz, Jea, Germay Istitute of Geeral Botay ad Plat Physiology, Friedrich Schiller Uiversity, Dorburger Str. 159, Jea, Germay * stefa.schu@ui-jea.de Table of Cotets Page 1. Plot of the umber of fatty acids as a fuctio of chai legth S. Earlier related work S 3. Explicit formula for the Fiboacci series S4 4. Modified fatty acids with cis- ad tras-isomers combied S6 5. Modified fatty acids with cis- ad tras-isomers cosidered separately S7 6. The Golde sectio S11 7. Further biological implicatios S11 8. Supplemetary refereces S1 S1

2 1. Plot of the umber of fatty acids as a fuctio of chai legth x (umodified FAs, cis/tras isomers combied) y (modified FAs with either oxo or hydroxy groups, cis/tras isomers combied) z (modified FAs with both oxo ad hydroxy groups, cis/tras isomers combied) u (umodified FAs, cis/tras isomers cosidered separately) v (modified FAs with either oxo or hydroxy groups, cis/tras isomers cosidered separately) w (modified FAs with both oxo ad hydroxy groups, cis/tras isomers cosidered separately) q (umodified FAs, cis/tras isomers cosidered separately, at most 6 double bods) Theoretical umber Number of carbo atoms Supplemetary Figure 1. Semi-logarithmic plot of the umber of fatty acids vs. chai legth (for = 1 to = ), cf. Table 1. As all the series (perhaps except q, for which this eeds to be ivestigated) grow asymptotically expoetially, the curves i this plot are liear for large.. Earlier related work I graph theory, a matchig i a graph is a subset of edges of the graph without commo vertices (Supplemetary Fig. ). 6 That is, these edges should ot be adjacet to each other. I FAs, they ca be iterpreted as double bods. The total umber of matchigs is the Hosoya idex 3,50 (see also Ref. 16), as illustrated i Supplemetary Fig.. For the graph show, five differet matchigs exist. I FAs, however, the bod ext to the carboxy ed must be a sigle bod. S

3 Therefore, the show matchigs are relevat for FAs ivolvig five carbos. I geeral, the total umbers of matchigs i paths of icreasig legth are give by the Fiboacci series. 7,50,51 As metioed i the mai text, aother equivalet problem is to fid the umber of biary strigs (cosistig of 0 ad 1 digits) of a give legth without adjacet 1 digits. 3 That problem goes back to the study of short ( light ) ad log ( heavy ) syllables (beig twice as log as the short syllables) i aciet Saskrit prosody (about BC) ad led to the series that is ow kow as Fiboacci series. 35 If the itervals betwee two short syllables are coded as 0 ad two short syllables liked to a log oe are coded as 1, calculatig the umbers of patters of partitioig a give umber of beats leads the above-metioed problem. Also the matchigs of a ubrached graph ca be ecoded as biary strigs without adjacet 1 digits (Supplemetary Fig. ). I several recet textbooks o discrete mathematics, as a example of applicatio, the equivalet exercise is give to fid a recurrece relatio for the umber of ways to climb stairs if oe ca take oe stair or two stairs at a time. 3,5 This ca be coded as 0 for oe step ad (the biary strig) 10 for two steps. I the case of alleic FAs, that is, whe adjacet double bods are allowed, the equivalet strig problem is to fid all biary strigs of a give legth (without ay restrictio). That leads to expoetial growth with the basis of two as give i Eq. (7). As had bee metioed by Hosoya himself, the Hosoya idex correspods to the series of Fiboacci umbers for ubrached paraffis (hydrocarbos) with icreasig chai legths. 7,50 However, he did ot metio the relatioship to sigle ad double bods ad did ot cout all ubrached alkaes ad alkees of a give legth sice he oly cosidered saturated hydrocarbos. Hosoya as well as Radić 53 used the Hosoya idex as a molecular descriptor to predict physico-chemical properties such as boilig poit, heat of formatio, etropy etc. Radić ad Pompe 54 cosidered alkees i additio, usig molecular descriptors to predict molar refractio. Breusch 55 calculated how may costitutioal isomers of uiformly polysubstituted, saturated FAs exist, which does ot lead to Fiboacci umbers. I oe of the cited papers, the coutig problem of our mai text was cosidered. S3

4 Supplemetary Figure. Illustratio of usig the Hosoya idex (by coloured graphs) ad biary strigs for coutig fatty acids. Show are the five matchigs of a exemplifyig ubrached four-vertex graph, correspodig to the side chai of a FA with five carbo atoms i total. The subsets of edges without commo vertices (which ca be iterpreted as double bods) are show i red; the subset is empty i the last case. The total umber of matchigs (Hosoya idex) equals five for the graph show. For a liear (i.e., ubrached, acyclic) graph, the matchigs ca be ecoded as biary strigs without cosecutive 1 digits. 3. Explicit formula for the Fiboacci series Although the derivatio of explicit formulas for the Fiboacci series has log bee kow i umber theory 3,4,33,5, we here show it for completeess of the presetatio. Eq. () is a liear recursio formula. The usual solutio procedure is to use a expoetial fuctio x = αλ. (S1) x I what follows, we specify the basis λ by subscripts correspodig to the symbol used for the series. Substitutig Eq. (S1) ito the recursio formula () leads to the quadratic equatio λ λ 1 = 0 (S) x x with the solutios S4

5 1± 5 λ x,1/ =. (S3) The positive solutio is the Golde ratio. The explicit formula is obtaied by a liear combiatio of two expoetial fuctios with the two bases give i Eq. (S3). x = α + 1 α (S4) The coefficiets α 1 ad α are determied by usig the iitial coditios. It is coveiet to start with = 0 rather tha = 1 because ay umber to the power of zero gives uity. x 0 is obtaied as x 0 = x x 1 = 0. Thus, Eq. (S4) gives, for = 0 ad = 1: α1 + α α α 0 = α 1 + α, 1 = (S5a,b) This leads to the Biet formula 3,4,33,5 x = (S6) Although this formula ivolves irratioal umbers, the resultig umbers are itegers. This is because the digits after the period i the two terms of the differece i Eq. (S6) cacel out whe calculatig the particular x. Eq. (S6) ca be simplified due to the observatio that the mius solutio i Eq. (S3) is less tha uity. For = 1, we have S5

6 1 1 5 = (S7) while the first term i Eq. (S6) reads = (S8) Their sum is x 1 = 1. We would obtai the same result by just roudig to a iteger value. This procedure also works for ay > 1 because the modulus (absolute value) of the egative term i Eq. (S6) is gettig smaller ad smaller for icreasig, leadig to Eq. (4) i the mai text. 4. Modified fatty acids with cis- ad tras-isomers combied Now we derive formulas for FAs with oxo groups. For FAs ivolvig hydroxy groups (but o oxo groups), the calculatio is the same. Sice carbos are of valece four, oxo groups adjacet to carbo-carbo double bods (so-called ketees) would oly be possible at the methyl ed of the side chai, but will ot be cosidered here due to the istability of this arragemet. The recursio procedure outlied i the mai text ad illustrated i Fig. ca be adapted by sayig that ot oly a methyl group but, alteratively, a -CH=O group ca be appeded to the -th carbo atom via a carbo-carbo sigle bod. To iclude a double bod betwee two carbos, we agai start at the FA with 1 carbos, but the caot add a oxo group to this carbocarbo double bod. This leads to the recursio formula (8) i the mai text, which together with the iitial values give i Eq. (9), defies the Pell umbers. That series ca be foud i the Olie Ecyclopedia of Iteger Sequeces at by searchig for idex A Usig agai a expoetial fuctio asatz (Eq. (S1)) leads to the quadratic equatio λ λ 1 = 0 (S9) y y S6

7 with the solutio λ = 1. (S10) y, 1/ ± The positive solutio is the Silver ratio. Takig ito accout the iitial coditios, we obtai the explicit formula 4,31 y ( 1+ ) ( 1 ) = (S11) Agai, the first term is always ear to the correct iteger value. Thus, we ca simplify this to Eq. (10) give i the mai text. I the case of FAs that ca have both oxo ad hydroxy groups, the recursio works as follows. A methyl group, oxo group or hydroxy group ca be appeded to the -th carbo atom via a carbo-carbo sigle bod. To iclude a double bod betwee two carbos, we agai start at the FA with 1 carbos, but the ca oly add a further methyl group. This leads to the recursio formula (11), which together with Eq. (1) defies the 3-Fiboacci umbers. That series ca be foud at (idex A006190). Aalogously as above, the followig explicit equatio for 3-Fiboacci umbers ad, thus, for FAs with two types of possible fuctioal side groups (e.g. oxo ad hydroxy) is derived: z = , (S1) which ca be simplified to Eq. (13) i the mai text. The positive basis is the Broze ratio. 5. Modified fatty acids with cis- ad tras-isomers cosidered separately We first derive a recursio formula for FAs that ca cotai either oxo or hydroxy groups. Here, we exemplify this by hydroxy groups ad illustrate it i Supplemetary Fig. 3. Assume we kow v i for all i=1 to. Now we wat to determie v +1. Startig from a FA with carbos, we ca add S7

8 either a methyl group by a sigle bod or a hydroxymethyl group (-CH OH) by a sigle bod. If the carbos 1 ad are liked by a sigle bod, the there is oly oe possible cofiguratio for the additioal sigle bod. If the carbos 1 ad are liked by a double bod, the we add the additioal sigle bod i oe of the possible cofiguratios, say cis. Thus, the trasitio from to +1 gives a term v i the recursio (Supplemetary Fig. 3, left-had side). Before geeratig the additioal structures, we defie the followig quatities. For each iteger i>1, we ote that there are as may molecules ivolvig i carbos ad edig with a sigle bod ad a methyl group as there are molecules edig with a sigle bod ad a -CH OH group. We deote this umber by a i. The remaiig umber of FAs with i carbos, which ed with a =CH group, is deoted by b i. Thus, v i = a i + b i. Supplemetary Figure 3. Illustratio of the recursive eumeratio method for modified fatty acids that ca cotai hydroxy groups, with cis- ad tras-isomers cosidered separately. Possibly occurrig additioal hydroxy groups i the two structures o the upper left are ot show. Red lies, bods added durig the procedure. Larger solid dots, variable chai legth. Further explaatios, see text. S8

9 We ow put i= 1. We ca exted the a 1 molecules edig with a sigle bod ad a methyl group by (i) a double bod ad a sigle bod to a methyl group or (ii) a double bod ad a sigle bod to a -CH OH group. I both cases, we choose the tras cofiguratio. This complemets the molecules i cis cofiguratio geerated above ad adds a term a 1 to the recursio (Supplemetary Fig. 3, right-had side). Moreover, to all a 1 FAs with 1 carbos, we ca add a sigle bod ad a double bod, geeratig two ew eds like that: -CH -CH=CH ad -CHOH-CH=CH. This adds a term a 1 to the recursio oce more. To the b 1 FAs with 1 carbos edig with a double bod, we add a viyl group (-CH=CH ) i two ways: so that the double bod betwee carbos - ad -1 is i cis or i trascofiguratio. Thus, also the umber of the molecules edig with a double bod should be doubled i the recursio. I this way, we geerate all possible exteded structures (without overlap betwee the geerated structures) ad obtai the recursio formula (14). Special attetio must be paid to the iitial values. For =1, there is oly oe possibility for the side chai: oe sigle hydroge, correspodig to formic acid. For =, v = because the side chai ca cosist of a methyl group or a -CH OH group. For +1=3, we caot yet apply recursio (14) because we excluded the molecule ivolvig oe carbo oly ad a hydroxyl group (carboic acid). Thus, oly the factor 1 rather tha should be assiged to v 1 i the recursio: v 3 = v + v 1 = * + 1 = 5. From +1=4 o, we ca apply recursio (14). This gives rise to the umber series 1,, 5, 14, 38, 104, give i Table 1 (idex A05945 at The quadratic equatio leads to the basis λ v = 1+SQRT(3) =.73 for the explicit formula (15). A simple way of writig a explicit formula is by derivig a coefficiet to λ v ad roudig. This ca be doe by dividig a sufficietly large v (e.g. v 10 ) by λ v. Thus, we obtai the coefficiet α , leadig to the explicit formula (15). For = 1, for example, the term i paretheses i Eq. (15) is , which correctly gives v 1 = 1 upo roudig. I a aalogous way, we ca derive equatios for the case where both oxo ad hydroxy groups ca occur. We agai proceed i a recursive way. Startig from a FA with carbos, we ca add either a methyl group or a -CH OH group or a -CH=O group, each by a sigle bod. If the carbos 1 ad are liked by a sigle bod, the there is oly oe possible cofiguratio for S9

10 the additioal sigle bod. If the carbos 1 ad are liked by a double bod, the we arrage the additioal sigle bod so that the double bod is i oe of the possible cofiguratios, say cis. Thus, the trasitio from to +1 gives a term 3 w i the recursio. For each iteger i>1, we ote that there are as may molecules ivolvig i carbos ad edig with a sigle bod ad a methyl group as there are molecules edig with a sigle bod ad a -CH OH group ad also as there are molecules edig with a sigle bod ad a -CH=O group. We deote this umber by c i. The remaiig umber of FAs with i carbos, which ed with a double bod, is deoted by d i. Thus, w i = 3 c i + d i. We ow put i= 1. We ca exted the c 1 molecules edig with a sigle bod ad a methyl group by (i) a double bod ad a sigle bod to a methyl group, (ii) a double bod ad a sigle bod to a -CH OH group or (iii) a double bod ad a sigle bod to a -CH=O group. I both cases, we choose the tras cofiguratio. This complemets the molecules i cis cofiguratio geerated above ad adds a term 3 c 1 to the recursio. Moreover, to all 3 c 1 FAs with 1 carbos, we ca add a sigle bod ad a double bod, geeratig three ew eds like that: -CH -CH=CH ad -CHOH-CH=CH ad C=O-CH=CH. This adds a term 3 c 1 to the recursio oce more. To the d 1 FAs with 1 carbos edig with a double bod, we add a viyl group (-CH=CH ) i two ways: so that the double bod betwee carbos - ad -1 is i cis or i trascofiguratio. Thus, also the umber of the molecules edig with a double bod should be doubled i the recursio. I this way, we geerate all possible exteded structures ad obtai the recursio formula (19). Special attetio must be paid to the iitial values. For =1, there is formic acid oly. For =, w =3 because the side chai ca cosist of a methyl group, a -CH OH group or -CH=O group. For +1=3, we caot yet apply recursio (19) because we excluded the molecule ivolvig oe carbo oly ad a hydroxy group (carboic acid); a (secod) oxo group is impossible ayway. Thus, oly a value of 1 rather tha should be assiged to w 1 i the recursio: w 3 = 3 w + w 1 = 3*3 + 1 = 10. From +1=4 o, we ca apply recursio (19). This gives rise to the umber series w give i Table 1. The quadratic equatio leads to the basis λ w = 3/+SQRT(17)/ = for the explicit formula. With a appropriate coefficiet, the explicit formula (0) is obtaied. S10

11 6. The Golde sectio It is kow from mathematics that the ratio of two cosecutive Fiboacci umbers teds to the Golde sectio. This ca be show by substitutig Eq. (4) ito x +1 /x. As the differece to the rouded value is gettig smaller ad smaller, it ca be eglected i that ratio for large. This leads to the followig observatio. I the costructio procedure of the FAs show i Fig., which we used to derive the recursio formula (), we added a termial double bod by startig from the FAs with 1 carbos, while we added a termial sigle bod by startig from the FAs with carbos. As also x /x -1 teds to the Golde sectio, the ratio of the umbers of FAs with a termial sigle bod ad a termial double bod (for a give chai legth) approximately equals the Golde sectio, 1.618, ad coverges to that umber with icreasig chai legth. The iverse ratio is Note that the digits after the period are the same, which is oe of the strikig properties of the Golde sectio. The properties of the Golde sectio the imply that the fractio of FAs with a termial sigle bod (compared to all FAs of the cosidered chai legth) teds to A aalogous calculatio ca be doe i the case where cis- ad tras-isomers are couted separately. Let d deote the umber of FAs of legth with a termial double bod. Thus, u d is the umber of FAs with a termial sigle bod. The ratio d /u shows a o-trivial patter: For = 1 7, for example, it follows the series 0, 0, 1/, ¼, 3/8, 5/16, 9/3. We deote the limit value of this series by γ. I the trasitio from to +1, a double bod ca oly be added to a FA with a termial sigle bod, ad oly i oe cofiguratio. Thus, d +1 = u d. Moreover, due to the recursio for u, we have d +1 /u +1 = γ = (u d )/(u ) = (1 γ)/ (S13) This leads immediately to γ = 1/3. Thus, the fractio of FAs with a termial double or sigle bod teds to 1/3 or /3, respectively, whe cis-/tras-isomerism is cosidered. 7. Further biological implicatios A further applicatio of this work is to estimate the time ecessary to perform, i the laboratory, the chemical sythesis of all FAs of a certai legth. This could also be of iterest i sythetic S11

12 biology, which is aimed at costructig systems (e.g. metabolic pathways) that have ever bee preset withi livig orgaisms. 56 Such egieered systems could produce FAs ot foud before. Our aalysis ca also help i uderstadig priciples of evolutio icludig prebiotic evolutio. I livig orgaisms, oly relatively few buildig blocks out of a eormous theoretical umber are used. Out of more tha 100 chemical elemets, oly six are maily used: C, H, O, N, S ad P. Oly four ucleobases appear i the DNA; proteis are built from a limited set of amio acids. For example, the umber of proteiogeic aliphatic amio acids is exceeded by far by the umber of aturally occurrig o-proteiogeic versios ad eve more so by the theoretically possible structures, for which a recursio formula ca be give. 39 Biological complexity the arises by a versatile combiatio of a few buildig blocks. As for FAs, a strikigly high umber occurs i ature, but much less tha the umber of theoretically coceivable structures. The realized umber might be that high because FAs are, i a sese, buildig blocks ad polymers at the same time. Due to the sythesis by assemblig twocarbo uits, which also applies to may polyketides, may possibilities arise i isertig double bods, hydroxy groups ad other fuctioal groups. A additioal source of complexity is the combiatio of FAs ito phospholipids ad triglycerides. Besides the more widely kow RNA world sceario for prebiotic evolutio 56, some authors have put forward the idea of a lipid world where the first self-replicatig uit was a lipid vesicle or micelle. 57 Most likely, lipid diversity has affected the course of prebiotic ad biotic evolutio may times. 8. Supplemetary refereces 50. Hosoya, H. Topological idex ad Fiboacci umbers with relatio to chemistry. Fiboacci Quart. 11, (1973). 51. Došlić, T. & Litz, M.S. Matchigs ad idepedet sets i polypheylee chais. MATCH Commu. Math. Comput. Chem. 67, (01). 5. Matoušek, J. & Nešetřil: Ivitatio to Discrete Mathematics. Oxford Uiversity Press, Oxford (003). S1

13 53. Radić, M. Wieer-Hosoya idex - a ovel graph theoretical molecular descriptor. J. Chem. If. Comput. Sci. 44, (004). 54. Radić, M. & Pompe, M. O characterizatio of the CC double bod i alkees. SAR ad QSAR Eviro. Res. 10, (1999). 55. Breusch, F. L. Azahl der Isomere vo polysubstituierte Fettsäure. Fette, Seife, Astrichm. 7, 1-6 (1970). 56. Szostak, J. W., Bartel, D. P. & Luisi, P. L. Sythesizig life. Nature 409, (001). 57. Segrè, D., Be-Eli, D., Deamer, D. & Lacet, D. The lipid world. Origis Life Evol. B. 31, (001). S13

Statistics 11 Lecture 18 Sampling Distributions (Chapter 6-2, 6-3) 1. Definitions again

Statistics 11 Lecture 18 Sampling Distributions (Chapter 6-2, 6-3) 1. Definitions again Statistics Lecture 8 Samplig Distributios (Chapter 6-, 6-3). Defiitios agai Review the defiitios of POPULATION, SAMPLE, PARAMETER ad STATISTIC. STATISTICAL INFERENCE: a situatio where the populatio parameters