Abstract
In lipid biochemistry, a fundamental question is how the potential number of fatty acids increases with their chain length. Here, we show that it grows according to the famous Fibonacci numbers when cis/trans isomerism is neglected. Since the ratio of two consecutive Fibonacci numbers tends to the Golden section, 1.618, organisms can increase fatty acid variability approximately by that factor per carbon atom invested. Moreover, we show that, under consideration of cis/trans isomerism and/or of modification by hydroxy and/or oxo groups, diversity can be described by generalized Fibonacci numbers (e.g. Pell numbers). For the sake of easy comprehension, we deliberately build the proof on the recursive definitions of these number series. Our results should be of interest for mass spectrometry, combinatorial chemistry, synthetic biology, patent applications, use of fatty acids as biomarkers and the theory of evolution. The recursive definition of Fibonacci numbers paves the way to construct all structural formulas of fatty acids in an automated way.
Introduction
Fatty acids (FAs) are of crucial importance for all organisms and many viruses. They occur, for example, within triglycerides, which serve as energy and carbon stores, and within phospholipids in biomembranes^{1}. Many lipids such as diacylglycerol play an important role in cellular signalling. The importance of FAs is further underlined by differences between healthy and diseased cells, so that FAs are medically relevant biomarkers. Several FAs with conjugated double bonds exert inhibitory effects on cancer cells^{2}. Many fatty acids and lipids are involved in fungal development and pathogenicity^{3,4}.
A high diversity of FAs is beneficial for regulating various properties of membranes such as fluidity^{5}, for optimizing packaging within lipoproteins as well as for their signalling function between plants and plant pathogens^{6}. While an enormous multitude of more than 1000 different FAs occur in living organisms, only around 20 FAs are widely found. Palmitic (16:0), oleic (18:1) and linoleic acids (18:2; Fig. 1a) account for around 80% of commodity oils and fats^{7} (numbers in parentheses indicate the numbers of carbon atoms and double bonds). There are striking differences in different kingdoms and lineages: while higher plants synthesize more than 300 different FAs, higher animals synthesize a far smaller range^{8}. Despite the wealth of known FAs, it is not immediately clear whether all theoretically possible variants of a given chain length are really used in living nature because synthesizing all of them in a coordinate way would require many different enzymes.
Most natural FAs have evennumbered chain lengths up to 22 carbon atoms, while some FAs reach chain lengths of more than 30 (e. g., on plant cuticles)^{7}. Evenchain FAs are commonly synthesized by condensing and reducing several twocarbon units from acetylcoenzyme A molecules^{1}. Oddchain FAs occur in low quantities in many different species of microorganisms, plants and some animals^{9}. For example, pentadecanoic acid reaches a level of approximately 1% in cow milk fat and is made by bacteria in the rumen^{7}. Other examples are margaric acid (17:0), a common constituent of lipids, pelargonic acid (9:0), occurring as esters in pelargonium oil, and valeric acid (5:0), occurring in valerian^{7}. Linoleic acid and αlinolenic acid (18:3) are two examples of polyunsaturated FAs (PUFAs) and are essential constituents of the human diet^{1}. Less common dietary PUFAs such as eicosapentaenoic acid (20:5) are healthpromoting and can be obtained from fish or algae^{10}. Allenic FAs (two adjacent double bonds) and cumulenic FAs (three or more adjacent double bonds) are rare in nature due to their decreased stability^{11}. At least three allenic FAs have been found in Phlomis (Lamiaceae)^{12}. One of them is phlomic acid (20:2) with double bonds at positions 7 and 8.
Propionate and butyrate are shortchain FAs (SCFAs) produced by the microbiome in the human gut^{13}. While unbranched side chains are most common, there are some examples of branched FAs such as phytanic acid, a chlorophyll catabolite^{14}. Hydroxylated FAs include ricinoleic acid (Fig. 1b), cutin acids, which are the building blocks of the polymer cutin covering the aerial surfaces of plants^{7}, and several dihydroxyoctadecadienoic acids (18:2) produced by the fungus Aspergillus nidulans and having effects on sporulation^{3}.
The goal of the present study is to elucidate the combinatorial multitude of unbranched FAs when allowing for double bonds. This covers a large range of FAs. We here derive formulas for enumerating FAs in dependence on the number of carbon atoms involved. As the number of double bonds and, thus, the number of hydrogens can vary for any given chain length, the different forms are not necessarily isomers, but can be called congeners^{15}. Many enumeration techniques in mathematical chemistry are focussed on isomers^{16}, although some of these techniques can also be used for counting congeners^{17,18,19}. In recent years, the study of congeners (which have similar structures and may or may not have different sum formulas) has attracted more and more interest^{15,20,21,22}. For example, the congeners of common persistent organic pollutants with at most p different substituents instead of hydrogens were enumerated by a graph isomorphism algorithm^{21}. Gutman^{22} calls the different Kekulé structures of aromatic compounds congeners. For the present analysis, we make use of the relatively simple chemical structure of unbranched FAs. We will exactly define in each case which congener classes of fatty acids we will analyse.
We derive both recursion and explicit formulas for unmodified and two classes of modified FAs. For each of these, we distinguish two cases depending on whether or not cis/trans isomerism is considered. Our analysis not only answers a fundamental question but may also support applications such as lipidomics, a highthroughput technology used for the simultaneous detection and quantification of a large number of lipid species.
Results
Following a broad definition that includes all chain lengths^{7}, we here define FAs as straightchain (unbranched) aliphatic monocarboxylic acids that contain carboncarbon single or double bonds (Fig. 1a). Further below, we allow for modified FAs including hydroxy and oxo groups. For the sake of easy comprehension, we deliberately build the proof on the recursive definition of Fibonacci numbers and related series rather than on more sophisticated techniques of chemical combinatorics.
Unmodified fatty acids with cis and transisomers combined
In a first approach, we do not count cis and transisomers of unsaturated FAs separately. Allenic and cumulenic FAs are first neglected as well, but will be considered in one of the classes studied below. Let x_{n} denote the number of theoretically possible, unmodified FAs involving n carbons. For n = 1, we just have the carboxy group linked to one hydrogen, which makes up formic acid (Fig. 1a). For n = 2, there is only one possibility to attach a methyl group to the carboxy group, giving rise to acetic acid. For n = 3, the saturated FA is propionic acid. However, there is also the possibility to insert a double bond, giving rise to acrylic acid (Fig. 1a). Thus,
and x_{3} = 2. A general enumeration procedure can be derived by standard methods from discrete mathematics^{23}. We can code single bonds by 0 and double bonds by 1. As no two double bonds must be adjacent to each other, we look for the number of all binary strings (consisting of 0 and 1 digits) of a given length without consecutive 1 digits. The length is n−2 because the carbon atom of the carboxy group cannot engage in a carboncarbon double bond, and the remaining n−1 carbons are connected by n−2 bonds. The corresponding number series can be calculated by the recursion formula
and the initial conditions (1)^{23}.
Eqs. (1) and (2) define the famous series of Fibonacci numbers^{23,24,25}. The series reads
For the concrete case of fatty acids, the recursion is explained in Fig. 2. For example, the three FAs for n = 4 are butyric acid (saturated, having its name because of presence in milk and butter), crotonic acid and 3butenoic acid (both unsaturated) (Fig. 1a). Table 1 shows the Fibonacci numbers for n = 1−22. The number series is illustrated in Supplementary Fig. 1.
Besides the binary string problem mentioned above, there are a number of equivalent problems in mathematics. In graph theory, a matching in a graph is a set of edges without common vertices^{26}. That edge set corresponds to the double bonds in fatty acids because the latter must not be adjacent (see also Supplementary Fig. 2). The total number of matchings is the Hosoya index, which is particularly easy to compute for linear graphs and then leads to the Fibonacci series (cf. Supplementary Information)^{27}.
It is known from number theory^{24} that an explicit formula can be derived from the recursion formula (see also Supplementary Information):
This allows one to calculate the number x_{n} directly from n without the necessity to compute previous numbers first. Note that the explicit formula involves irrational numbers although all Fibonacci numbers are integers. In fact, the ratio is the legendary Golden section^{24,25}. Eq. (4) implies that the numbers of FAs show an asymptotically exponential growth with the basis of 1.618. It is understandable that the basis lies between 1 and 2 because there are two possibilities: single bond and double bond, yet the double bonds cannot be adjacent to each other.
An alternative way of proving that the diversity of unmodified fatty acids follows the Fibonacci series is by using a formula derived by Lucas:^{24}
where m equals the largest integer that is less or equal to (n−1)/2. In this sum, each term can be interpreted such that it describes the number of possibilities of inserting k double bonds into a chain length of n, in such a way that no double bonds are adjacent to each other nor next to the carboxy end. Due to those constraints, the choice of positions is made out of n−k−1 positions. That alternative way of computation is more cumbersome than Eq. (4), though, because sums of binomial coefficients need to be computed.
When setting m to an arbitrary value less than the value mentioned above, Eq. (5) can be used to compute the number of FAs with at most m double bonds. In Table 1, we give the numbers, q_{n}, of FAs with at most six double bonds, by way of example. Two examples of unsaturated FAs with six double bonds are docosahexaenoic acid (22:6), which has cardiovascularprotective properties^{8} and is a structural component of several human organs, and nisinic acid (allcis−6,9,12,15,18,21tetracosahexaenoic acid, 24:6) in fish. Up to n = 14, the series q_{n} coincides with the Fibonacci numbers x_{n} because a FA with 14 carbons can harbour up to six carboncarbon double bonds. For higher n, the series q_{n} grows more slowly, as can also be seen in Supplementary Fig. 1. Any other upper bound m on the number of double bonds can be considered in the calculation as well using Eq. (5).
Now, we admit adjacent double bonds, that is, allenic and cumulenic FAs, and denote the number of FAs in this case by u_{n}. Axial stereoisomers are not counted separately here. Both for n = 1 and n = 2, u_{n} equals 1 because no double bond can be adjacent to the carboxy group. From n = 3 on, the number doubles for each additional carbon atom because there are two possibilities to extend the chain: u_{n+1} = 2u_{n}. The series reads
or
where the first term u_{1} needs to be defined separately as 1. In Table 1 and Supplementary Fig. 1, the series is compared to the one defined by Eq. (4).
Unmodified fatty acids with cis and transisomers considered separately
When the FAs involve nonterminal double bonds, cis and transisomers can be distinguished. For example, the cisisomer of crotonic acid is isocrotonic acid (Fig. 1a). This distinction is particularly useful when cis and transisomers exert different biological functions or different effects on the structure of lipid membranes due to their different molecular shape^{15}. Here, we exclude allenic and cumulenic FAs. Special attention needs to be paid to FAs with conjugated double bonds such as in the various isomers of conjugated linoleic acid (18:2) or sorbic acid (6:2). As the corresponding double bonds and the single bonds in between form a πsystem, the formal single bonds cannot rotate freely. This gives rise to socalled scis and strans isomers. However, the π interaction in these bonds is weaker than in the formal double bonds so that the isomers equilibrate quickly^{28}. Therefore, we will only consider cis and transisomers with respect to double bonds and neglect scis/strans isomerism here.
For n = 1 and n = 2, no double bond is possible. So, the first two numbers in the series are equal to 1. From n = 2 on, there are two cases if we add the (n+1)th carbon:
 a
There is a single bond at position n. Then we have two possibilities: Adding a carbon by a single bond or a double bond.
 b
There is a double bond at position n. Then we have again two possibilities: Adding a carbon by a single bond in cis configuration or in trans configuration.
As in both cases, the number doubles for each additional carbon atom, we obtain the same series as given in Eqs. (6) and (7).
Modified fatty acids with cis and transisomers combined
Let us now consider modified FAs, again excluding allenic FAs. Neglecting cistransisomerism first, we start by allowing oxo groups so that one or several carbons can be linked with oxygen atoms by double bonds. An example is acetoacetic acid (n = 4) (Fig. 1b). Biosynthetically, oxidized FAs can be oxylipins (oxidation products of unsaturated FAs) or polyketides^{29,30}.
Denoting the number of modified FAs by y_{n}, we obtain the recursion formula (for derivation, see Supplementary Information):
In contrast to the Fibonacci numbers, we have
because an oxo group occurs already in glyoxylic acid (Fig. 1b).
Together with these initial conditions, Eq. (8) leads to the series given in the column for y_{n} in Table 1 and plotted in Supplementary Fig. 1. In mathematics, they are known as the Pell numbers or 2Fibonacci numbers^{24,25,31} and obey the explicit formula
(cf. Supplementary Information), where the basis is the Silver ratio. They represent one instance of generalized Fibonacci numbers given by a linear combination (other than the sum) of the two preceding numbers^{32}.
The same series and formula applies to the case where hydroxy groups are allowed instead of oxo groups. In that case, hydroxy groups as part of an enol moiety are not counted because a rapid equilibrium generally favours the corresponding form with an oxo group (ketoenol tautomerism^{28}). Similarly, geminal diols are easily converted to the corresponding ketones or aldehydes by loss of one water molecule^{28}. Thus, they can be considered equivalent to those molecules. Moreover, we exclude the case where n = 1 and a hydroxy group is linked to the only carbon. The corresponding compound, carbonic acid, is an inorganic compound and not considered a FA. When hydroxy groups are included, also different stereoisomers (with R and S stereocenters) can occur, which are not, however, counted separately here.
Now we further extend this analysis to modified FAs that can contain both oxo and hydroxy groups. We obtain the recursion formula
(see Supplementary Information) and the initial conditions
This leads to the series z_{n} in Table 1 given by the explicit formula
(derivation in Supplementary Information, plot in Supplementary Fig. 1). In mathematics, these numbers are called 3Fibonacci numbers^{24,25}.
Modified fatty acids with cis and transisomers considered separately
Now, we extend the enumeration procedure for modified FAs by considering cis and transisomers separately. However, we neglect R/S stereoisomerism and allenic FAs. We start by allowing oxo groups only or hydroxy groups only and denote the number of modified FAs by v_{n}.
In the Supplementary Information, we derive both a recursion and an explicit formula, which resemble those for the Fibonacci numbers. The new formulas read
with
and
The recursion formula (14) corresponds to another instance of generalized Fibonacci numbers^{32}. The initial values can be derived from the first chemical structures,
Although Eq. (14) has recursion depth two, three initial values are needed here (see Supplementary Information). From n + 1 = 4 on, Eq. (14) generates the number series v_{n} given in Table 1. This is also obtained by Eq. (15) for all positive integers n.
Now we consider the case where oxo groups and/or hydroxy groups occur and use the symbol w_{n}. In the Supplementary Information, both a recursion formula and an explicit formula are derived:
with
and
With the relevant initial conditions
we obtain the number series given in Table 1, column for w_{n}. Eq. (19) holds from n + 1 = 4 on.
In the special case of vicinal oxo and hydroxy groups, these groups can swap places due to ketoenol tautomerism involving the corresponding enediol intermediate. This reduces the number of really different FAs. We leave it to future studies to consider this effect in enumerating FAs.
Discussion
As a starting point for the enumeration of lipid congeners, we deliberately restricted our analysis to certain classes of FAs. Our goal was to derive enumeration formulas for the biochemically most relevant classes of FAs. For example, unmodified FAs with cis and trans isomers considered separately but neglecting allenic and cumulenic FAs are the basic group of FAs that are usually considered in biochemistry textbooks. We have first derived the number of unbranched, unmodified FAs with different numbers of double bonds to be given by the Fibonacci series when cis/trans isomerism is neglected. By building the proof on the recursive definition, it is very short and easy to comprehend. Also in enumerating Kekulé structures in chemistry, recursive relations are often used^{33,34}. In the special case of polyphenanthrenes^{33} and related nanotubes^{34}, they lead to Fibonacci numbers as well.
As described above, part of our results can also be derived from the concept of Hosoya index^{27}. Nevertheless, to the best of our knowledge, the use of (usual and generalized) Fibonacci numbers for enumerating FAs has not been published before. Extending the present analysis, the Hosoya index can be used also for enumerating branched fatty acids. As shown above, the congeners of fatty acids modified by functional groups can be enumerated by generalized Fibonacci numbers.
Fibonacci numbers are named after Leonardo Pisano, called Fibonacci, although they had been found more than one millennium earlier by scholars in ancient India when studying Sanskrit poems (see Supplementary Information)^{35}. In his book “Liber abaci” from 1202, Fibonacci derived this series by studying the population dynamics of rabbits. For a sketch of his life in medieval Italy, see ref. 16. The Fibonacci series frequently occurs in biology such as in phyllotaxis and secondary structures of proteins^{25}. Interestingly, the Fibonacci sequence is also employed to model Xray diffraction patterns of films of mixed FA salts^{36}.
Although known for a long time, the Fibonacci numbers have lost nothing of their fascination, and more and more fields of application are found. Here, we have shown that fatty acids, which are important molecules in our own body, obey that appealing arithmetics. This finding also applies to analogous classes of terminally monosubstituted hydrocarbons such as aldehydes, alcohols or aliphatic amino acids^{37}.
In the case of unmodified FAs, considering cis/trans isomerism leads to a simple exponential series with the basis of 2. As far as we know, that observation has not been published before either. This number series occurs in the wellknown legend of wheat grains on the chessboard^{38}. The legend says that a Brahmin called Sessa, the inventor of the game of chess ancestor, chaturanga, was entitled to request a prize from the king. The man asked that on the first square of the chessboard, he would receive one grain of wheat (in some tellings, rice), two on the second one, and so on, doubling the amount each time.
It is somewhat surprising that considering isomerism makes formula (4) for unmodified FAs even easier. For other classes of molecules, it is the other way round. For example, considering stereoisomerism in aliphatic amino acids makes the enumeration more complicated^{39} and so does considering cis/trans isomerism in the enumeration of modified FAs.
The ratio of two consecutive Fibonacci numbers tends to the Golden section (Supplementary Information). Therefore, starting from an (unmodified) FA of a given length and investing one more carbon atom, an organism can increase the variability of the FA approximately by the Golden section factor, 1.618, or by 2 when cis and transisomers are counted separately. Furthermore, the fraction of FAs with a terminal single bond is approximately the inverse Golden ratio, 0.618, or 2/3 if cis and transisomers are counted separately (see Supplementary Information).
An interesting question is why most FAs used in living organisms have chain lengths of 16–22. A biological constraint arises from the thickness of biomembranes, while a physicochemical constraint is that melting temperatures increase with increasing chain length^{40} so that very long FAs might be too rigid to be used in organisms.
Although interesting, it is beyond the scope of this paper to study in detail which of the theoretically possible FAs are used in living organisms. An impressive number but certainly not all FAs are used in reality. For example, the Fibonacci number x_{18} for unmodified FAs is 2,584 and u_{18} = 2^{16} = 65,536 (Table 1). Several FAs with n = 18 play a role in biology: stearic acid (18:0), oleic acid and its transisomer elaidic acid (one double bond at position 9), linoleic acid (two double bonds at positions 9 and 12) (Fig. 1a), vaccenic acid (one double bond at position 11) and the various isomers of conjugated linoleic acid (two double bonds, e.g. at positions 9 and 11, or 10 and 12). A few of the latter as well as vaccenic acid occur in cow milk^{41}.
The following general observations on naturally occurring FAs are worth noting:
 I
Practically all chain lengths up to about 35 occur in biological systems^{7}.
 II
Comparing the numbers of naturally occurring FAs with the potential numbers, there appear to be peaks at chain lengths of 16 and 18^{1}. However, all shorter lengths occur as well, e.g. capric acid (10:0) occurring in coconut oil and goat milk and inhibiting the yeasttohyphae transition of the fungus Candida albicans^{4}, or cis2decenoic acid (10:1) made by Pseudomonas aeruginosa^{42} and the various mediumlength FAs mentioned above.
 III
FAs rarely involve more than six conjugated double bonds. Therefore, coloured FAs can only rarely be observed in living organisms. The yellowish colour of some adipose tissues comes from carotenoids^{43}. Longer conjugated systems show lower stability and, thus, are sensitive to light. It is believed that methyl branches and terminal cyclohexenyl groups, as observed in carotenoids, contribute to increased polyene stability^{44}. Two examples of rare modified FAs with extensive conjugated systems are laetiporic acid and its derivative 2dehydro3deoxylaetiporic acid produced by the basidiomycete Laetiporus sulphureus, with the former FA being its major orange pigment^{45}. They include 10 conjugated carboncarbon double bonds and one methyl branch each, and the latter even has a further, nonconjugated carboncarbon double bond.
 IV
Double bonds often occur at a distance of three carbons and then are called homoconjugated. Examples are provided by nisinic acid (24:6, see above) and adrenic acid (allcis7,10,13,16docosatetraenoic acid, 22:4).
Beside the academic interest^{39}, a promising field of application of this analysis is lipidomics, in which the entirety of a cell’s lipids is studied under different conditions^{46}. Lipids and their constituents are most commonly identified by mass spectrometry (MS), and quantification is typically based on comparison of massspectrometric ion intensities between individual lipids and suitable standards^{47}. Similar to the more advanced proteomics field, the generation of lipidomics data by MS relies on accurate metabolite databases. In analysing the spectra, it is very helpful to know the maximum number of compounds that can potentially appear. Lipid databases are required for the identification in highthroughput and can also guide fragmentation experiments^{48}. The formal description presented here may help refine lipid databases and thereby facilitate automated lipid identification as well as in the screening of fungistatic compounds. Moreover, in patent applications related to chemical compounds, it is crucial to know the number of (existing or potential) compounds for which a patent is filed. As outlined in the Supplementary Information, other applications of the presented results concern several aspects of synthetic biology and the understanding of how chemical complexity arose during evolution.
In future studies, it is of interest to derive formulas for larger classes of FAs, using more sophisticated and complex methods. For example, stereoisomers of hydroxylated FAs and the (rarely occurring) FAs involving triple bonds can be studied (for enumeration of amino acids involving triple bonds, see ref. 37). The number of different FAs allowing branching and only carboncarbon single bonds can be computed in the same way as that of primary alcohols or of aliphatic amino acids only involving single bonds^{39}. One suitable method for doing so is based on Pólya’s enumeration theorem^{49}. Furthermore, the probability of having double bonds at specific positions (such as ω3, 6, 9) can be studied.
The recursive definition of Fibonacci and generalized Fibonacci numbers paves the way to list all structures successively in silico. With the abovementioned coding of FAs as binary strings such as 0001010 and appropriate software, this can be translated into the chemical structures.
Additional Information
How to cite this article: Schuster, S. et al. Use of Fibonacci numbers in lipidomics – Enumerating various classes of fatty acids. Sci. Rep. 7, 39821; doi: 10.1038/srep39821 (2017).
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Acknowledgements
The authors would like to thank Gunnar Brinkmann, Ute Holtzegel, Robert Israel, Gerhard Jahreis, Daniel Merkle, Markus Nebel, Georg Pohnert, Heiko Stark, Stephan Wagner, and Silvio Waschina for helpful discussions. Ina Weiss helped with the literature search. Financial support by the German Ministry for Education and Research (BMBF) within the Virtual Liver Network and the DFG via the Jena School of Microbial Communication and the Transregio 124 (FungiNet, project B1) is gratefully acknowledged.
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S. Schuster devised the combinatorial problem, made the mathematical discovery and elaborated the mathematical solutions, with advice on chemical and biochemical aspects and interpretation from S. Sasso. M. Fichtner did the numerical calculations and helped in the interpretation. All authors wrote the manuscript.
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Schuster, S., Fichtner, M. & Sasso, S. Use of Fibonacci numbers in lipidomics – Enumerating various classes of fatty acids. Sci Rep 7, 39821 (2017). https://doi.org/10.1038/srep39821
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