Autapomorphies and the retention index: Solving a limit via l’Hôpital-Bernoulli’s rule

ABSTRACT

The retention index (r or ri) is used to compute the amount of similarity in a character that can be interpreted as a synapomorphy on a given cladogram. In the case of autapomorphies, the retention index assumes the form 0 / 0, which is a mathematically indeterminate form. Originally, this issue was resolved by assigning a value of one to the retention index in such cases. However, some authors did not follow this original interpretation and assumed that, in this case, the retention index must assume a value of zero. In this note, we present two arguments supporting the original definition. The first argument is an application of the concept of limit from calculus to determine the assumed value of the retention index function in this specific situation. The second argument is a biological one that invokes the original definitions of autapomorphy and synapomorphy.

KEY WORDS:
Calculus; homology; homoplasy; indeterminate form; mathematically undefined; synapomorphy

In phylogenetic systematics, several indices are used to measure the “fit”, i.e., the degree of homoplasy and/or the degree of synapomorphies, of a given character or phylogenetic tree, also interpreted as goodness of fit measures (Sober 2002, Turjak and Trontelj 2012). For a good summary and review, see Kitching et al. (1998). Among these indices, the retention index (r or ri) is widely applied to compute “the fraction of apparent synapomorphy in the character that is retained as synapomorphy on the tree”, as originally defined (Farris 1989). See the original definition of synapomorphy (Hennnig 1966):

“We will call the characters or character conditions from which transformation started (a, b) in a monophyletic group plesiomorphous, and the derived conditions (a’, a”, b’, b”) apomorphous. […] We will call the presence of plesiomorphous characters in different species symplesiomorphy, the presence of apomorphous characters synapomorphy, always with the assumption that the compared characters belong to one and the same transformation series. […] It is evident that the presence of corresponding characters in two or more species is a basis for assuming that these species form a monophyletic group only if the characters are apomorphous, if their correspondence rests on synapomorphy.” ( Hennnig 1966 : 89, 90).

See also a more recent and logical definition of synapomorphy:

“Synapomorphy-the quality of sister taxa sharing a derived character state-is the central tenet of phylogenetic systematics.[…] A character state represents a synapomorphy for a group if the character state’s distribution pattern does not conflict with placement of the group in the taxonomic structure ( Patterson, 1982 ). This means that, in an ideal case, a synapomorphic character state will be shared by all of the clade’s members (terminal taxa) and at the same time will not occur outside that clade.” ( Turjak and Trontelj 2012 : 627, 628).

The retention index ranges from zero (which corresponds to the maximum apparent homoplasy in the given character) to one (which corresponds to the maximum apparent synapomorphy in the given character), and it can be easily computed via the following function:

$ri=\left(g-s\right)/\left(g-m\right)$, see (Farris 1989: 417).

The variables are m, which “represents the minimum amount of change that the character may show on any tree”; s, which “denotes the amount of change in the character (for an integral character, number of steps) required parsimoniously by the considered tree”; and g, which “denotes the greatest amount of change that the character may require on any tree, that is, the greatest possible value of s” (Farris 1989). The value of g can be calculated by the number of terminals with the less prevalent character states.

The ensemble or overall retention index (R or RI) is the general summation of the retention index among all characters in a given tree topology (Farris 1989). These indices are normally computed for characters optimized on a parsimonious tree. However, it can be computed for any given tree, including suboptimal trees or trees generated by other methods for comparative purposes, such as maximum likelihood or Bayesian inference (although direct optimization is based on the maximum parsimony criterium). The same principle is also applied in the case of the consistency index (c or ci), the rescaled consistency index (rc), which is the product of ci and ri, the ensemble or overall consistency index (C or CI), and the ensemble or overall rescaled consistency index (RC) (Kluge and Farris 1969, 1989).

The problem

A binary autapomorphic character is one with a given state that is present in only one terminal (i.e., an operational taxonomic unit, OTU). In this situation, g = m = s = 1. A similar phenomenon occurs in the case of multistate autapomorphic characters, which have more than two states, all but one of which are only present in a single terminal. In this situation, g = m = s, and they have a value higher than one.

Therefore, in both of these situations (binary and multistate autapomorphic, non-homoplastic characters), since g = m = s, the retention index assumes the form 0 / 0, which is, formally, a mathematically undefined situation (specifically, an indeterminate form). There is a simple way one can understand 0 / 0 as an indeterminate form; just note that any real (ℝ) number x satisfies the following equation: 0 . x = 0 (any real number multiplied by zero equals zero).

There is also the case when two or more distantly related terminals independently present the same character state; we have a case of non-apparent autapomorphies, but in this case, it is also a homoplasy. Note that this can only be inferred after phylogenetic reconstruction. In this situation, g = s ≠ m, the retention index naturally assumes a value of zero; we do not have an indeterminate form.

One should keep in mind that the most common application of cladistics is to perform phylogenetic analyses with discrete characters, but this reasoning can be applied to continuous characters as well (Kluge and Farris 1969, Ackerly and Donoghue 1988, Klingenberg and Gidaszewski 2010). In this sense, g, m, and s are values in the domain of positive real numbers (ℝ+).

Note that in the case of multistate autapomorphic non-homoplastic characters, this description is suitable considering unordered (nonadditive) characters, i.e., the Fitch (1971) optimization. The abovementioned situation can be easily explored for other parsimony variations (Camin and Sokal 1965, Kluge and Farris 1969, Farris 1977) and character weighting schemes (Farris 1969).

In Farris (1989), this indeterminate form was clearly resolved, assuming that in the case where g = m = s, the retention index assumes the value of one: “If g = m, then s = g, and r is taken to be unity, so that d (the distortion coefficient of Farris 1973) is taken to be 0. I shall call r the retention index” (Farris 1989). This original solution to the indeterminate form of 0 / 0 was followed by some authors, such as (Kitching et al. 1998). It is also the same way the TNT 1.5 software reports (Goloboff and Catalano 2016).

However, some authors did not follow this original interpretation for the case of the retention index applied to autapomorphies. To cite just a few examples: in the textbooks of Amorim (2002), Forey (2006), Wiley and Lieberman (2011), Morrrone (2013), and Brower and Schuh (2021), it is clearly assumed that, in the case of autapomorphies, the retention index must assume the value of zero without a proper justification. In the textbook of Wiley et al. (1981) and in the PAUP* 4.0 software (Swofford and Bell 2017), a conservative approach is found, the retention index is reported as “0 / 0” and simply termed undefined, avoiding further complications. Finally, the WinClada v.1.00.08 software reports it as “uninformative” (Nixon 2002).

The confusion regarding the value of the retention index applied to autapomorphies has persisted in phylogenetic studies. Some recent studies have reported a value of zero for the retention index in the case of autapomorphies, e.g., de Medeiros and Vanin (2020) and Zeballos et al. (2023), while others stated that the retention index cannot be calculated for autapomorphies, e.g., Sayad and Yassin (2019). Therefore, we can ask the following question: What is the most suitable value for the retention index in the case of autapomorphies? One? Zero? 0 / 0?

In this rapid communication, we present two arguments supporting the original definition (Farris 1989) to use the value of one for the retention index in the case of autapomorphies. A similar rationale can be extended to the distortion coefficient, d (Farris 1973, 1989), a complement of the retention index.

Argumentation and solution

Indeterminate forms are not mathematical dead ends. They can be solved for specific cases, given a specific function. With that in mind, the first argument is straightforward, applying the calculus’ limit concept, e.g., the epsilon-delta (ε, δ) definition of limit (see Felscher 2000), to determine the assumed value of the function of the retention index in any desired case. We suspect that it is the position of Farris (1989), but it was not originally (or otherwise) addressed by the author. In this view, one can promptly compute the limit of a function of the retention index via l’Hôpital-Bernoulli’s rule, which uses derivatives in the numerator and denominator to find the limit of a particular indeterminate form (not just 0 / 0; it can also be used for all other indeterminate forms: ∞ / ∞; 0 . ∞; ∞ - ∞; 0 ⁰ ; 1 ^∞ ; and ∞ ⁰ ); this method is indeed a robust way to address this problem (see Struik 1963). Therefore, step by step:

$\underset{\left(g\to m=s\right)}{\mathit{lim}}\frac{\left(g-s\right)}{\left(g-m\right)},$,

Then,

$\underset{\left(g\to m\right)}{\mathit{lim}}\frac{\left(g-m\right)}{\left(g-m\right)}$,

The l’Hôpital-Bernoulli’s rule general definition is:

$\underset{\left(x\to c\right)}{\mathit{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\underset{\left(x\to c\right)}{\mathit{lim}}\frac{\frac{d}{dx}f\left(x\right)}{\frac{d}{dx}g\left(x\right)}$,

All four necessary conditions for l’Hôpital-Bernoulli’s rule are met:

1. $\underset{\left(x\to c\right)}{\mathit{lim}}f\left(x\right)=\underset{\left(x\to c\right)}{\mathit{lim}}g\left(x\right)=0\mathrm{o}\mathrm{r}\pm \infty$,

2. ƒ(x) and g(x) are differentiable,

3. $\frac{d}{dx}g\left(x\right)\ne 0$,

4. $\underset{\left(x\to c\right)}{\mathit{lim}}\frac{\frac{d}{dx}f\left(x\right)}{\frac{d}{dx}g\left(x\right)}$ is an existing limit.

Applying the l’Hôpital-Bernoulli’s rule, we have:

$\underset{\left(g\to m\right)}{\mathit{lim}}\frac{\frac{d}{dg}\left(g-m\right)}{\frac{d}{dg}\left(g-m\right)}$,

Applying the subtraction rule of differentiation, we have:

$\underset{\left(g\to m\right)}{\mathit{lim}}\frac{\frac{d}{dg}\left(g\right)-\frac{d}{dg}\left(m\right)}{\frac{d}{dg}\left(g\right)-\frac{d}{dg}\left(m\right)}$,

Solving the derivatives:

$\underset{\left(g\to m\right)}{\mathit{lim}}\frac{\left(1-0\right)}{\left(1-0\right)}$,

Simplify,

$\underset{\left(g\to m\right)}{\mathit{lim}}\left(\frac{1}{1}\right)$, $\underset{\left(g\to m\right)}{\mathit{lim}}\left(1\right)$,

Given that the limit of a constant function is equal to the constant itself,

$\underset{\left(x\to a\right)}{\mathit{lim}}\left(c\right)=c$,

Then,

$\underset{\left(g\to m\right)}{\mathit{lim}}\left(1\right)=1$,

Therefore,

$\underset{\left(g\to m=s\right)}{\mathit{lim}}\frac{\left(g-s\right)}{\left(g-m\right)}=1$

The second argument is biological or phylogenetic in nature. Let us assume the original formulation (Farris 1989), where in the retention index, the value of zero is equivalent to the “maximum of apparent homoplasy in the given character” and the value of one is equivalent to the “maximum of apparent synapomorphy in the given character”. Considering that an autapomorphic character is an apomorphic character that is as exclusive as possible (i.e., present in only one specific or supraspecific terminal), it is most appropriate to assume the value of one. This interpretation is consistent with the original definition of autapomorphy and synapomorphy (Hennig 1966), see also Yeates (1992):

“The apomorphous features characteristic for a particular monophyletic group (present only in it) can be ignored in discussing its relations to other groups; we will call such characters the autapomorphous characters of a monophyletic group. Naturally these characters are ‘autapomorphous’ only if the group in question is compared with other groups; so long as we are trying to show that the group itself is monophyletic, these same characters would be ‘synapomorphous characters’ of the species making up the group” ( Hennnig 1966 : 90).

This point of view is also apparently supported by Farris (1989), in which the author also traced back the homology concept, which can be logically interpreted as synonymous with synapomorphy (see de Pinna 1991, Brower and de Pinna 2012, 2013, but see also Nixon and Carpenter 2012a, 2012b, 2012c, 2013):

“The interpretation of the retention index can be seen from a simple argument. On a tree for which s = m, r = 1, there is no homoplasy, and all similarities between terminals in the character are homologous” ( Farris 1989 : 417).

Additionally, this interpretation is congruent with a proper formal logical definition of synapomorphy (see definition “1” in Turjak and Trontelj 2012).

We are fully aware of the dependence of this argumentation strategy on the applied concept/definition of synapomorphy (Hennig 1966). However, exploring the variations and validity of other understandings beyond a Hennigian argument goes far beyond the scope of this communication. However, at least to us, creating an equivalence between autapomorphy and homoplasy (i.e., the case if we assume ri = 0 / 0 = 0) seems more problematic.

Final remarks

It is well known that, when the ensemble retention index (R or RI) is calculated for a given tree, this summation is not influenced by autapomorphies (Farris 1989, Naylor and Kraus 1995). Despite that, it is not adequate just to compute an ensemble retention index for a given tree. A more complete overview is necessary by measuring and exploring several indices and properties in a phylogenetic tree and individual characters, including metrics influenced by autapomorphies. It should be noted that autapomorphies are included in more recent morphological and paleontological studies, e.g., Costa et al. (2014), Sayad and Yassin (2019), de Medeiros and Vanin (2020), Černý and Simonoff (2023), Zeballos et al. (2023).

In some older cladistic studies, autapomorphies were often neglected and altogether excluded, as they did not influence at all the topology of a tree generated via maximum parsimony (Kitching et al. 1998). We briefly present some arguments in favor of always including autapomorphic characters; see Yeates (1992) for a more complete argument: 1) autapomorphies have the potential to be diagnostic characters to terminals (higher taxa, and species), therefore, they have taxonomic importance (e.g., Ferreira et al. 2023); 2) it is well-known that the consistency index (c or ci) is “inflated” by the inclusion of autapomorphies (Brooks et al. 1986, Carpenter 1988), but there is no need to exclude them since most popular phylogenetic software, e.g., TNT 1.5 (Goloboff and Catalano 2016) and PAUP* 4.0 (Swofford and Bell 2017), report the consistency index both with and without autapomorphies (so the inclusion of autapomorphies is not detrimental in any point and also does not significantly increase tree search time); 3) in maximum likelihood (Felsenstein 1973, 1981) and Bayesian (Rannala and Yang 1996, Yang and Rannala 1997) approaches, autapomorphies inform branch lengths and equilibrium state frequencies, moreover, in the context of these methods, autapomorphies are phylogenetically informative (Felsenstein 1978, Lewis 2001).

Therefore, to close our argumentation, we emphasize that we need robust proofs and well-defined properties for a metric applied in phylogenetic systematics. In this approach, some conceptual issues were reviewed, and we properly clarified a more appropriate way to interpret the retention index applied to the case of autapomorphies and concerning the inclusion of these characters in phylogenetic analyses.

ACKNOWLEDGEMENTS

We are very grateful to Elliot Santovich Scaramal (Instituto de Filosofia e Ciências Sociais, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil) and Lucas dos Anjos (Department of Renewable Resources, University of Alberta, Edmonton, Alberta, Canada) for revising a preliminary version of the text. This work was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Education Ministry of Brazil (CAPES), Finance Code 001. We are also thankful to the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for processes 312786/2022-0 to J.R.M.M. and 310567/2018-1 to C.A.M.R. We are also thankful to Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) processes E-26/010.001887/2019, SEI-260003/001170/2020, SEI-260003/012995/2021 to C.A.M.R. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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• Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Education Ministry of Brazil (CAPES)-Finance Code 001 Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for processes 312786/2022-0 to J.R.M.M. and 310567/2018-1 Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) processes E-26/010.001887/2019, SEI-260003/001170/2020, SEI-260003/012995/2021 This work was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Education Ministry of Brazil (CAPES), Finance Code 001. We are also thankful to the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for processes 312786/2022-0 to J.R.M.M. and 310567/2018-1 to C.A.M.R. We are also thankful to Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) processes E-26/010.001887/2019, SEI-260003/001170/2020, SEI-260003/012995/2021 to C.A.M.R. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript

ADDITIONAL NOTES

• ZooBank register

  https://zoobank.org/17FDF18C-905C-4868-887B-686059B4E3C1

• How to cite this article

  Barbosa FF, Mermudes JRM, Russo CAM (2024) Autapomorphies and the retention index: Solving a limit via l’Hôpital-Bernoulli’s rule. Zoologia 41: e24014. https://doi.org/10.1590/S1984-4689.v41.e24014

• Published by

  Sociedade Brasileira de Zoologia at Scientific Electronic Library Online (https://www.scielo.br/zool)

Edited by

Editorial responsibility

Walter A.P. Boeger

Publication Dates

• Publication in this collection
  23 Sept 2024
• Date of issue
  2024

History

• Received
  20 Feb 2024
• Accepted
  21 June 2024