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The number of cells in colonies of the cyanobacterium Microcystis aeruginosa satisfies Benford's law
Aquatic Botany 89 (2008) 341–343
Contents lists available at ScienceDirect
Aquatic Botany journal homepage: www.elsevier.com/locate/aquabot
Short communication
The number of cells in colonies of the cyanobacterium Microcystis aeruginosa satisfies Benford’s law Eduardo Costas a, Victoria Lo´pez-Rodas a, F. Javier Toro b, Antonio Flores-Moya b,* a b
Gene´tica, Facultad de Veterinaria, Universidad Complutense, E-28040 Madrid, Spain Biologı´a Vegetal (Bota´nica), Facultad de Ciencias, Universidad de Ma´laga, Campus de Teatinos s/n, E-29071 Ma´laga, Spain
A R T I C L E I N F O
A B S T R A C T
Article history: Received 11 October 2007 Received in revised form 13 March 2008 Accepted 17 March 2008 Available online 21 March 2008
Nowadays, numerous power- or scaling-laws are encountered in many fields of biology: an example is Benford’s law. According to this law, the first significant digit of any given series of numbers is figure 1 more often than figure 2, which in turn appears more often than 3, and so on. Here we show that number of cells per colony in the cyanobacterium Microcystis aeruginosa (F.T. Ku¨tzing) F.T. Ku¨tzing, randomly isolated from different reservoirs and lakes from Andalusia (S Spain), was very variable (with figures differing up to five orders of magnitude). However, the distribution of the number of cells per colony satisfies Benford’s law. This situation could be much more general in colonial cyanobacteria. Crown Copyright ß 2008 Published by Elsevier B.V. All rights reserved.
Keywords: Benford’s law Colony Microcystis aeruginosa
1. Introduction Several aspects of the biology of the cyanobacterium Microcystis aeruginosa (F.T. Ku¨tzing) F.T. Ku¨tzing have been extensively studied because this species is the most important cause of hepatotoxic blooms in freshwater ecosystems world-wide (Rinehart et al., 1994; Sivonen and Jones, 1999; Visser et al., 2005). Microcystis aeruginosa forms colonies ranging from microscopic to macroscopic free-floating groups up to ca. 8 mm in extent. Large colonies may contain a million cells, but small ones can be clusters of only a few dozen cells, and solitary cells may also occur (Bourrely, 1970; Koma´rek and Anagnostidis, 1998; Whitton, 2002; Cronberg and Annadotter, 2006; Joosten, 2006). However, the study of possible principles or constraints linked to the morphology of cells and colonies has been scarcely addressed (Rico et al., 2006) and, as far as we know, no studies have been carried out to understand whether the number of cells per colony follows any regular pattern. Nowadays, numerous power- or scaling- laws are encountered in many areas of the physical, biological, and social sciences (GellMann, 1994). An example of a general principle is Benford’s law, a phenomenological law also called the first digit law, first digit phenomenon, or leading digit phenomenon (Hill, 1995a). It might be expected that the first significant digit (i.e., the leftmost non-
* Corresponding author. Tel.: +34 952 131 951; fax: +34 952 131 944. E-mail address: fl[email protected] (A. Flores-Moya).
zero digit) of any given series of numbers, or of a set of numbers measuring any given phenomenon, is randomly distributed and, consequently, the frequency of the leading digit (from 1 to 9) tends to be represented in a similar proportion of 11.1% (i.e., one digit out of 9). However, Benford (1938) analyzed a large amount of naturally occurring data (e.g. tables of molecular weights, population sizes, river basin drainages areas, numbers appearing on newspaper front pages, and others) and demonstrated that the leadings digits tend not to be uniformly distributed. More accurately, the average number of appearances of the first digits follows a logarithmic law: 1 F d ¼ Log10 1 þ ; d 2 f1; 2; . . . ; 9g (1) d where Fd is the frequency of appearance of the first significant digit d. Consequently, frequencies of leading digits range from 30.1% for d = 1 to 4.6% for d = 9. In practice, it is difficult to apply this law universally. A satisfactory explanation for the reason that Benford’s law unquestionably applies to many situations in the real world has been given through the seminal work of Hill (1995b). Most of the previous known examples of Benford behavior in data sets are mathematical functions and operations and financial data (see recompilation in Hu¨rlimann, 2006). However, recently Nigrini and Miller (2007) provide an excellent work on Benford’s law applied to hydrological data, which includes several interesting questions. In this sense, the aim of this work was to study the pattern of number of cells per colony in M. aeruginosa. Here we show that the number of cells per colony in M. aeruginosa,
0304-3770/$ – see front matter . Crown Copyright ß 2008 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.aquabot.2008.03.011
E. Costas et al. / Aquatic Botany 89 (2008) 341–343
342
Table 1 Sampling locations of colonies from Microcystis aeruginosa in Andalusia (S Spain), and distribution of no. of cells per colony Location
Date
No. of colonies isolated
No. of cells per colony <102
La Minilla (reservoir) El Gergal (reservoir) Taraje (lake) La Rocina (pond)
XII-1999 V-2000 VI-2003 VI-2003
Total
102–103
103–104
104–105
>105
69 55 176 109
4 8 6 6
8 16 17 23
23 17 69 54
31 14 77 26
3 0 7 0
409
24
64
163
148
10
randomly isolated from different reservoirs and lakes, was very variable (with figures differing up to five orders of magnitude). However, the distribution of the number of cells per colony satisfies Benford’s law. 2. Materials and methods Four hundred and nine M. aeruginosa colonies, collected from different locations in Andalusia (S Spain; Table 1) were analyzed. Samples were fixed in formalin 4%. Colonies were randomly isolated using a micromanipulator under an inverted microscope (see details in Carrillo et al., 2003; Rico et al., 2006). The number of cells of each isolated colony was counted by using two different procedures, depending on the size of the colony. For smaller colonies, each one was isolated on a microscope slide and the cells were counted directly. In the case of the larger colonies, each one of them was isolated in a microcentrifuge tube, the cells were liberated by sonication, and they were counted in a hemacytometer. The goodness of fit of the frequency of the first significant digit in the number of cells in colonies of M. aeruginosa, to the values predicted by the Eq. (1), was checked by a x2-test (Zar, 1999), with eight degrees of freedom. 3. Results The number of cells per colony was extremely variable, from dozens to more than 105 cells, but those with orders of magnitude ranging from 103 to 105 were the most frequent at all of the sampling sites (Table 1). However, a robust pattern was found in the first significant digit in the number of cells in the colonies: it significantly fit to the Benford’s law (x2 = 0.983–0.998 in the different sampling locations, d.f. = 8; see Fig. 1).
4. Discussion Gell-Mann (1994) hypothesized that numerous robust patterns, scaling-laws and empirical theories could explain many facts encountered in everyday biology. Often, such patterns remain beyond description because they are not explored. In this sense, identification of robust patterns has been scarcely used to elucidate the biology of cyanobacteria. However, the distribution of cell numbers in colonies of M. aeruginosa is an example of a robust pattern: it obeys Benford’s law. We suggest an explanation for this pattern. Many sets of numbers, especially those that grow exponentially, fit Benford’s law (Hill, 1995a,b; Pietronero et al., 2001; Gottwald and Nicol, 2002; Engel and Leuenberger, 2003). Because the cells of M. aeruginosa divide synchronously, the number of cells in colonies grows approximately by the exponential law 2n, where n is the number of divisions. It must be highlighted that the set f2ng1 n¼1 follows Benford’s law in base 10 because Log10 2 is irrational and, if a is irrational, then na is equidistributed on [0, 1]. However, 2n is not Benford in every base (for example 2n is not Benford base 2k for any integer k, as Log2 k 2 is irrational for any integer k). Alternatively, it could seem that the number of cells per colony should fit a normal distribution, which is usually assumed by phycologists. However, Benford’s law is not applicable for variables following a normal distribution, although if one mixes numbers from different normal distributions Benford’s law sometimes reappears (Boyle, 1994; Hill, 1995a,b; Pietronero et al., 2001; Gottwald and Nicol, 2002; Engel and Leuenberger, 2003, for an extensive explanation). In practice, the distribution of the number of cells in M. aeruginosa colonies, obtained from very different localities, shows a marked asymmetry in favor of small first digits. The first three
Fig. 1. Observed frequency of the first significant digit of the no. of cells per colony in Microcystis aeruginosa and predicted frequency according to Benford’s law. The values of x2-test for the goodness of fit to the Benford’s law (d.f. = 8) are included.
E. Costas et al. / Aquatic Botany 89 (2008) 341–343
digits (from 1 to 3) have globally a frequency of 60% while the other six digits (from 4 to 9) appear only in 40% of the colonies. This situation could be much more general in colonial cyanobacteria, and very likely in all exponentially growing colonies where all the cells are retained, but it remains to be investigated. Benford’s law could be used as a tool to check several properties of cyanobacteria biology, which cannot be verified under natural conditions or laboratory studies. As an example, the conformity to Benford’s law in the number of cells in colonies of M. aeruginosa suggest that (i) all the cells of a colony have the same probability to divide following the exponential law 2n, (ii) cells divides synchronically and (iii) the colony fragmentation follow an exponential pattern. Moreover, if the number of cells in colonies of cyanobacteria satisfies Benford’s law, then some colonies could be more susceptible to survive than others due to its size. Conformity to Benford’s law suggest that M. aeruginosa colonies between 10,000 and 20,000 cells and between 100,000 and 200,000 cells are favored. More work is necessary to generalize utility of Benford’s law in other colonial and filamentous cyanobacteria. Acknowledgements This work was financially supported by CGL2004-02701/HID, CGL2005-01938/BOS, and P05-RNM-00935 grants. Dr. Eric C. Henry (Herbarium, Department of Botany and Plant Pathology, Oregon State University, USA) kindly revised the English style and usage. The suggestions from two anonymous referees helped us to improve the paper. References Benford, F., 1938. The law of anomalous numbers. Proc. Am. Phil. Soc. 78, 551–572. Bourrely, P., 1970. Les Algues d’Eau Douce. III. Les Algues Bleus et Rouges. E´ditions N. Boube´e & Cie, Paris. Boyle, J., 1994. An application of Fourier series to the most significant digit problem. Am. Math. Mon. 101, 879–886.
343
Carrillo, E., Ferrero, L.M., Alonso-Andicoberry, C., Basanta, A., Martı´n, A., Lo´pezRodas, V., Costas, E., 2003. Interstrain variability in toxin production in populations of the cyanobacterium Microcystis aeruginosa from water-supply reser˜ ana National Park (southern Spain). voirs of Andalusia and lagoons of Don Phycologia 42, 269–274. Cronberg, G., Annadotter, H., 2006. Manual on Aquatic Cyanobacteria. A Photo Guide and a Synopsis of Their Toxicology. Intergovernmental Oceanographic Commission of UNESCO, and International Society for the Study of Harmful Algae, Copenhagen. Engel, H.A., Leuenberger, C., 2003. Benford’s law for exponential random variables. Stat. Probab. Lett. 63, 361–365. Gell-Mann, M., 1994. The Quark and the Jaguar: Adventures in the Simple and the Complex. W.H. Freeman and Co., New York, NY. Gottwald, G.A., Nicol, M., 2002. On nature of Benford’s law. Physica A 303, 387–396. Hill, T.P., 1995a. Base invariance implies Benford´s Law. Proc. Am. Math. Soc. 123, 887–895. Hill, T.P., 1995b. The significant-digit phenomenon. Am. Math. Mon. 102, 322–327. Hu¨rlimann, T., 2006. Benford’s law from 1881 to 2006: a bibliography (http:// arxiv.org/ftp/math/papers/0607/0607168.pdf). Joosten, A.M.T., 2006. Flora of the Blue–Green Algae of the Netherlands. I. The Nonfilamentous Species of Inland Waters. KNNV Publishing, Utrecht. Koma´rek, J., Anagnostidis, K., 1998. Cyanoprokaryota. 1. Teil: Chroococcales. In: Ettl, H., Ga¨rtner, G., Heynig, H., Mollenhauer, H.D. (Eds.), Su¨sswasserflora von Mitteleuropa 19/1. Gustav Fischer, Jena-Stuttgart-Lu¨beck-Ulm, pp. 1–548. Nigrini, M.J., Miller, S.J., 2007. Benford’s law applied to hydrology data—results and relevance to other geophysical data. Math. Geol. 39, 469–490. Pietronero, L., Tossati, E., Tossati, A., Vespignani, A., 2001. Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf. Physica A 293, 297–304. Rico, M., Altamirano, M., Lo´pez-Rodas, V., Costas, E., 2006. Analysis of polygenic traits of Microcystis aeruginosa (Cyanobacteria) strains by Restricted Maximum Likelihood (REML) procedures. 1. Size and shape of colonies and cells. Phycologia 45, 237–242. Rinehart, K.L., Namikoshi, M., Choi, B.W., 1994. Structure and biosynthesis of toxins from blue–green algae (cyanobacteria). J. Appl. Phycol. 6, 159–176. Sivonen, K., Jones, G., 1999. Cyanobacterial toxins. In: Chorus, I., Bartram, J. (Eds.), Toxic Cyanobacteria in Water. A Guide to Their Public Health Consequences, Monitoring and Management. E&FN Spon, London, pp. 41–112. Visser, P.M., Ibelings, B.W., Mur, L.R., Walsby, A.E., 2005. The ecophysiology of the harmful cyanobacterium Microcystis. Features explaining its success and measures for its control. In: Huisman, J., Matthijs, H.C.P., Visser, P.M. (Eds.), Harmful Cyanobacteria. Aquatic Ecology Series 3. Springer, Dordrecht, pp. 109–142. Whitton, B.A., 2002. Phyllum Cyanophyta (blue–green algae/Cyanobacteria). In: John, D.M., Whitton, B.A., Brook, A.J. (Eds.), The Freshwater Algal Flora of the British Isles. An Identification Guide to Freshwater and Terrestrial Algae. Cambridge University Press, Cambridge, pp. 25–122. Zar, J.H., 1999. Biostatistical Analysis, 4th ed. Prentice Hall, Upper Saddle River, NJ.
Contents lists available at ScienceDirect
Aquatic Botany journal homepage: www.elsevier.com/locate/aquabot
Short communication
The number of cells in colonies of the cyanobacterium Microcystis aeruginosa satisfies Benford’s law Eduardo Costas a, Victoria Lo´pez-Rodas a, F. Javier Toro b, Antonio Flores-Moya b,* a b
Gene´tica, Facultad de Veterinaria, Universidad Complutense, E-28040 Madrid, Spain Biologı´a Vegetal (Bota´nica), Facultad de Ciencias, Universidad de Ma´laga, Campus de Teatinos s/n, E-29071 Ma´laga, Spain
A R T I C L E I N F O
A B S T R A C T
Article history: Received 11 October 2007 Received in revised form 13 March 2008 Accepted 17 March 2008 Available online 21 March 2008
Nowadays, numerous power- or scaling-laws are encountered in many fields of biology: an example is Benford’s law. According to this law, the first significant digit of any given series of numbers is figure 1 more often than figure 2, which in turn appears more often than 3, and so on. Here we show that number of cells per colony in the cyanobacterium Microcystis aeruginosa (F.T. Ku¨tzing) F.T. Ku¨tzing, randomly isolated from different reservoirs and lakes from Andalusia (S Spain), was very variable (with figures differing up to five orders of magnitude). However, the distribution of the number of cells per colony satisfies Benford’s law. This situation could be much more general in colonial cyanobacteria. Crown Copyright ß 2008 Published by Elsevier B.V. All rights reserved.
Keywords: Benford’s law Colony Microcystis aeruginosa
1. Introduction Several aspects of the biology of the cyanobacterium Microcystis aeruginosa (F.T. Ku¨tzing) F.T. Ku¨tzing have been extensively studied because this species is the most important cause of hepatotoxic blooms in freshwater ecosystems world-wide (Rinehart et al., 1994; Sivonen and Jones, 1999; Visser et al., 2005). Microcystis aeruginosa forms colonies ranging from microscopic to macroscopic free-floating groups up to ca. 8 mm in extent. Large colonies may contain a million cells, but small ones can be clusters of only a few dozen cells, and solitary cells may also occur (Bourrely, 1970; Koma´rek and Anagnostidis, 1998; Whitton, 2002; Cronberg and Annadotter, 2006; Joosten, 2006). However, the study of possible principles or constraints linked to the morphology of cells and colonies has been scarcely addressed (Rico et al., 2006) and, as far as we know, no studies have been carried out to understand whether the number of cells per colony follows any regular pattern. Nowadays, numerous power- or scaling- laws are encountered in many areas of the physical, biological, and social sciences (GellMann, 1994). An example of a general principle is Benford’s law, a phenomenological law also called the first digit law, first digit phenomenon, or leading digit phenomenon (Hill, 1995a). It might be expected that the first significant digit (i.e., the leftmost non-
* Corresponding author. Tel.: +34 952 131 951; fax: +34 952 131 944. E-mail address: fl[email protected] (A. Flores-Moya).
zero digit) of any given series of numbers, or of a set of numbers measuring any given phenomenon, is randomly distributed and, consequently, the frequency of the leading digit (from 1 to 9) tends to be represented in a similar proportion of 11.1% (i.e., one digit out of 9). However, Benford (1938) analyzed a large amount of naturally occurring data (e.g. tables of molecular weights, population sizes, river basin drainages areas, numbers appearing on newspaper front pages, and others) and demonstrated that the leadings digits tend not to be uniformly distributed. More accurately, the average number of appearances of the first digits follows a logarithmic law: 1 F d ¼ Log10 1 þ ; d 2 f1; 2; . . . ; 9g (1) d where Fd is the frequency of appearance of the first significant digit d. Consequently, frequencies of leading digits range from 30.1% for d = 1 to 4.6% for d = 9. In practice, it is difficult to apply this law universally. A satisfactory explanation for the reason that Benford’s law unquestionably applies to many situations in the real world has been given through the seminal work of Hill (1995b). Most of the previous known examples of Benford behavior in data sets are mathematical functions and operations and financial data (see recompilation in Hu¨rlimann, 2006). However, recently Nigrini and Miller (2007) provide an excellent work on Benford’s law applied to hydrological data, which includes several interesting questions. In this sense, the aim of this work was to study the pattern of number of cells per colony in M. aeruginosa. Here we show that the number of cells per colony in M. aeruginosa,
0304-3770/$ – see front matter . Crown Copyright ß 2008 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.aquabot.2008.03.011
E. Costas et al. / Aquatic Botany 89 (2008) 341–343
342
Table 1 Sampling locations of colonies from Microcystis aeruginosa in Andalusia (S Spain), and distribution of no. of cells per colony Location
Date
No. of colonies isolated
No. of cells per colony <102
La Minilla (reservoir) El Gergal (reservoir) Taraje (lake) La Rocina (pond)
XII-1999 V-2000 VI-2003 VI-2003
Total
102–103
103–104
104–105
>105
69 55 176 109
4 8 6 6
8 16 17 23
23 17 69 54
31 14 77 26
3 0 7 0
409
24
64
163
148
10
randomly isolated from different reservoirs and lakes, was very variable (with figures differing up to five orders of magnitude). However, the distribution of the number of cells per colony satisfies Benford’s law. 2. Materials and methods Four hundred and nine M. aeruginosa colonies, collected from different locations in Andalusia (S Spain; Table 1) were analyzed. Samples were fixed in formalin 4%. Colonies were randomly isolated using a micromanipulator under an inverted microscope (see details in Carrillo et al., 2003; Rico et al., 2006). The number of cells of each isolated colony was counted by using two different procedures, depending on the size of the colony. For smaller colonies, each one was isolated on a microscope slide and the cells were counted directly. In the case of the larger colonies, each one of them was isolated in a microcentrifuge tube, the cells were liberated by sonication, and they were counted in a hemacytometer. The goodness of fit of the frequency of the first significant digit in the number of cells in colonies of M. aeruginosa, to the values predicted by the Eq. (1), was checked by a x2-test (Zar, 1999), with eight degrees of freedom. 3. Results The number of cells per colony was extremely variable, from dozens to more than 105 cells, but those with orders of magnitude ranging from 103 to 105 were the most frequent at all of the sampling sites (Table 1). However, a robust pattern was found in the first significant digit in the number of cells in the colonies: it significantly fit to the Benford’s law (x2 = 0.983–0.998 in the different sampling locations, d.f. = 8; see Fig. 1).
4. Discussion Gell-Mann (1994) hypothesized that numerous robust patterns, scaling-laws and empirical theories could explain many facts encountered in everyday biology. Often, such patterns remain beyond description because they are not explored. In this sense, identification of robust patterns has been scarcely used to elucidate the biology of cyanobacteria. However, the distribution of cell numbers in colonies of M. aeruginosa is an example of a robust pattern: it obeys Benford’s law. We suggest an explanation for this pattern. Many sets of numbers, especially those that grow exponentially, fit Benford’s law (Hill, 1995a,b; Pietronero et al., 2001; Gottwald and Nicol, 2002; Engel and Leuenberger, 2003). Because the cells of M. aeruginosa divide synchronously, the number of cells in colonies grows approximately by the exponential law 2n, where n is the number of divisions. It must be highlighted that the set f2ng1 n¼1 follows Benford’s law in base 10 because Log10 2 is irrational and, if a is irrational, then na is equidistributed on [0, 1]. However, 2n is not Benford in every base (for example 2n is not Benford base 2k for any integer k, as Log2 k 2 is irrational for any integer k). Alternatively, it could seem that the number of cells per colony should fit a normal distribution, which is usually assumed by phycologists. However, Benford’s law is not applicable for variables following a normal distribution, although if one mixes numbers from different normal distributions Benford’s law sometimes reappears (Boyle, 1994; Hill, 1995a,b; Pietronero et al., 2001; Gottwald and Nicol, 2002; Engel and Leuenberger, 2003, for an extensive explanation). In practice, the distribution of the number of cells in M. aeruginosa colonies, obtained from very different localities, shows a marked asymmetry in favor of small first digits. The first three
Fig. 1. Observed frequency of the first significant digit of the no. of cells per colony in Microcystis aeruginosa and predicted frequency according to Benford’s law. The values of x2-test for the goodness of fit to the Benford’s law (d.f. = 8) are included.
E. Costas et al. / Aquatic Botany 89 (2008) 341–343
digits (from 1 to 3) have globally a frequency of 60% while the other six digits (from 4 to 9) appear only in 40% of the colonies. This situation could be much more general in colonial cyanobacteria, and very likely in all exponentially growing colonies where all the cells are retained, but it remains to be investigated. Benford’s law could be used as a tool to check several properties of cyanobacteria biology, which cannot be verified under natural conditions or laboratory studies. As an example, the conformity to Benford’s law in the number of cells in colonies of M. aeruginosa suggest that (i) all the cells of a colony have the same probability to divide following the exponential law 2n, (ii) cells divides synchronically and (iii) the colony fragmentation follow an exponential pattern. Moreover, if the number of cells in colonies of cyanobacteria satisfies Benford’s law, then some colonies could be more susceptible to survive than others due to its size. Conformity to Benford’s law suggest that M. aeruginosa colonies between 10,000 and 20,000 cells and between 100,000 and 200,000 cells are favored. More work is necessary to generalize utility of Benford’s law in other colonial and filamentous cyanobacteria. Acknowledgements This work was financially supported by CGL2004-02701/HID, CGL2005-01938/BOS, and P05-RNM-00935 grants. Dr. Eric C. Henry (Herbarium, Department of Botany and Plant Pathology, Oregon State University, USA) kindly revised the English style and usage. The suggestions from two anonymous referees helped us to improve the paper. References Benford, F., 1938. The law of anomalous numbers. Proc. Am. Phil. Soc. 78, 551–572. Bourrely, P., 1970. Les Algues d’Eau Douce. III. Les Algues Bleus et Rouges. E´ditions N. Boube´e & Cie, Paris. Boyle, J., 1994. An application of Fourier series to the most significant digit problem. Am. Math. Mon. 101, 879–886.
343
Carrillo, E., Ferrero, L.M., Alonso-Andicoberry, C., Basanta, A., Martı´n, A., Lo´pezRodas, V., Costas, E., 2003. Interstrain variability in toxin production in populations of the cyanobacterium Microcystis aeruginosa from water-supply reser˜ ana National Park (southern Spain). voirs of Andalusia and lagoons of Don Phycologia 42, 269–274. Cronberg, G., Annadotter, H., 2006. Manual on Aquatic Cyanobacteria. A Photo Guide and a Synopsis of Their Toxicology. Intergovernmental Oceanographic Commission of UNESCO, and International Society for the Study of Harmful Algae, Copenhagen. Engel, H.A., Leuenberger, C., 2003. Benford’s law for exponential random variables. Stat. Probab. Lett. 63, 361–365. Gell-Mann, M., 1994. The Quark and the Jaguar: Adventures in the Simple and the Complex. W.H. Freeman and Co., New York, NY. Gottwald, G.A., Nicol, M., 2002. On nature of Benford’s law. Physica A 303, 387–396. Hill, T.P., 1995a. Base invariance implies Benford´s Law. Proc. Am. Math. Soc. 123, 887–895. Hill, T.P., 1995b. The significant-digit phenomenon. Am. Math. Mon. 102, 322–327. Hu¨rlimann, T., 2006. Benford’s law from 1881 to 2006: a bibliography (http:// arxiv.org/ftp/math/papers/0607/0607168.pdf). Joosten, A.M.T., 2006. Flora of the Blue–Green Algae of the Netherlands. I. The Nonfilamentous Species of Inland Waters. KNNV Publishing, Utrecht. Koma´rek, J., Anagnostidis, K., 1998. Cyanoprokaryota. 1. Teil: Chroococcales. In: Ettl, H., Ga¨rtner, G., Heynig, H., Mollenhauer, H.D. (Eds.), Su¨sswasserflora von Mitteleuropa 19/1. Gustav Fischer, Jena-Stuttgart-Lu¨beck-Ulm, pp. 1–548. Nigrini, M.J., Miller, S.J., 2007. Benford’s law applied to hydrology data—results and relevance to other geophysical data. Math. Geol. 39, 469–490. Pietronero, L., Tossati, E., Tossati, A., Vespignani, A., 2001. Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf. Physica A 293, 297–304. Rico, M., Altamirano, M., Lo´pez-Rodas, V., Costas, E., 2006. Analysis of polygenic traits of Microcystis aeruginosa (Cyanobacteria) strains by Restricted Maximum Likelihood (REML) procedures. 1. Size and shape of colonies and cells. Phycologia 45, 237–242. Rinehart, K.L., Namikoshi, M., Choi, B.W., 1994. Structure and biosynthesis of toxins from blue–green algae (cyanobacteria). J. Appl. Phycol. 6, 159–176. Sivonen, K., Jones, G., 1999. Cyanobacterial toxins. In: Chorus, I., Bartram, J. (Eds.), Toxic Cyanobacteria in Water. A Guide to Their Public Health Consequences, Monitoring and Management. E&FN Spon, London, pp. 41–112. Visser, P.M., Ibelings, B.W., Mur, L.R., Walsby, A.E., 2005. The ecophysiology of the harmful cyanobacterium Microcystis. Features explaining its success and measures for its control. In: Huisman, J., Matthijs, H.C.P., Visser, P.M. (Eds.), Harmful Cyanobacteria. Aquatic Ecology Series 3. Springer, Dordrecht, pp. 109–142. Whitton, B.A., 2002. Phyllum Cyanophyta (blue–green algae/Cyanobacteria). In: John, D.M., Whitton, B.A., Brook, A.J. (Eds.), The Freshwater Algal Flora of the British Isles. An Identification Guide to Freshwater and Terrestrial Algae. Cambridge University Press, Cambridge, pp. 25–122. Zar, J.H., 1999. Biostatistical Analysis, 4th ed. Prentice Hall, Upper Saddle River, NJ.