Hattoria 12: 9–25. 2021

Evolutionary leverage of dissilient genera of Pleuroweisieae (Pottiaceae) evaluated with Shannon-Turing analysis

Richard H. ZANDER

Missouri Botanical Garden, 4344 Shaw Blvd., St. Louis, Missouri 63116, U.S.A. Author for correspondence: Richard H. ZANDER, [email protected]

Abstract Evolutionary leverage is the average of total transformational trait changes among species in a dissilient (radiative) genus or larger, integral taxonomic group. It is proposed as a measure of evolutionary success in modern environments. Comparing evolutionary leverage aids in measuring potential evolutionary coherency and ecological success among taxonomic groups, here exemplified with Pleuroweisieae and Barbuloideae of Pottiaceae and genera of Streptotrichaceae (all Bryophyta). Shannon-Turing analysis—calculation using summed informational bits and sequential Bayes— translates to relative support for order and coherence of species in dissilient (radiative) genera. Hymenostylium gracillimum (Pottiaceae) is transferred to Ardeuma, while H. xerophilum and H. hildebrandtii (Merceyopsis hymenostylioides as synonym) are transferred to Eobryum. These three possess the primitive trait of a stem central strand in Pleuroweisieae. The genus Ozobryum is resurrected from synonymy with Molendoa.

Introduction “Evolutionary leverage” is a term used for suites of adaptations that presumably allow better competition against other species, e.g. angiosperm success over competing groups (Brodribb & Feild 2010). Evolutionary leverage is more exactly defined here as a comparison of the average number of trait transformations among newly generated species in a dissilient genus as a closed causal group (Zander 2018: 36). A dissilient genus is one characterized by a progenitor species with apparent descendant species radiating from it without reversals in key traits (reversed from those in an outgroup or the inferred progenitor of a progenitor) or at least very few reversals. When such a radiation is evident, a phylogenetically monophyletic group may be divided into smaller, empirically distinguishable genera. A closed causal group is one for which the species are indisputably a member of that group, and probabilistic evaluations of taxon order of speciation may safely ignore the possibility of membership in other groups. That is, evolutionary leverage is measured by the average number of new traits in a speciation event summed over all species in a genus in one well-characterized radiation—the traits being either adaptive or nearly neutral (e.g. from drift).

9 It is expected that many large genera require taxonomic dissection into smaller, more coherent genera that exhibit few if any reversals in major traits within a genus during speciation events. Trait changes when they occur in spite of stabilizing and purifying selection are scattered apparently randomly among many possible traits (in Pottiaceae more than 70, Zander 1993). Reversals in apparently neutral traits that have been proved to be taxonomically important for one species are revealed as taxonomically unimportant at the genus level by other conservative traits. The fact that this methodology provides evolutionary relationships that are parsimonious, matches fairly well with molecular systematics, and fits evolutionary theory is an existence proof of this variant of Dollo parsimony, and argues for a general retention of taxonomically important evolutionarily conservative traits over millions of years rather than among only a few living fossils. Gould (1970) has additional arguments in support of minimizing trait reversals in evolutionary analysis. Such dissilient (radiating) genera are mostly without changes in shared conservative, ancestral traits. The average number of trait changes in species in a genus, that is from progenitor to descendant, is an integration of two important values, the number of species in a genus, and the number of new traits associated with a speciation event in each of those species. Such new traits are commonly those used in a standard key distinguishing species. The total trait transformations and the total number of species are measures of evolutionary success in modern environments, as (1) multiple adaptations around a single set of fundamental traits, (2) a single combination of basal traits sufficiently robust as to tolerate a number of somewhat encumbering anagenetic mutations gradually accumulated among descendant species, or (3) some combination of these. High evolutionary leverage signals demonstrated evolvability, that is, the ability to respond to selective pressure. In the context of an ecosystem, large genera with well adapted traits are an empirically defined taxonomic group above that of species. Expressed as an average number of trait changes per species in a genus, evolutionary leverage reflects a degree of demonstrated persistence among competing species and changing environments. This measure works well with genera conceived as centers of radiation or dissilience, i.e., a putative progenitor species and its descendants. The rationale for naming ancestors is that if a modern species has the traits inferred for an ancestor, then that ancestor can be in practice named the same. Most of the presently recognized large genera need translation into smaller dissilient genera, and some genera may have poorly distinguished species (few trait transformations) while others have many. Past analyses of genera using principles of macroevolutionary systematics (see Zander 2019c) have pointed out that when species of a genus fit together well, particularly if generated with no reversals of important traits, then the whole genus has a better chance of modeling an integral ecological unit in evolution. The average number of trait transformations per species in a dissilient genus measures the degree of morphological heterogeneity between species. A “good” genus is one with an expected number of traits per species, such as that within two standard deviations of the number of traits per species evidenced among related dissilient genera. Past studies have shown that there are about four trait transformations on average from progenitor to descendant in taxa of the moss family Pottiaceae. Although many transformations per speciation event are possible (some punctuational, some perhaps anagenetic), retention of the immediate ancestral morphotype confirms membership in the

10 genus as a closed causal group. Easy determination of first, second, third and fourth standard deviations (sigmas) is available to taxonomists via the Running Sigmas Calculator (Zander 2020), an Microsoft Excel spreadsheet available online.

Materials and Methods Background Because macroevolutionary systematics is a new technique for generating evolution- based classifications, a summary of principles and protocols is given here distilled from more expansive presentations by Zander (2013, 2018, 2019a, 2019b, 2019c). The object is to generate from expressed traits apparently subject to adaptive or nearly neutral fixation a model parsimoniously arranging morphological trait transformations associated with speciation between sequential pairs of species. The idea is simple. Starting with a group of previously selected related species, select three closely related species and attach them serially as a (1) putative outgroup or prior ancestor, (2) a progenitor, and (3) a descendant such that there are no (or minimum) reversals of the descendant back to the outgroup or prior ancestor. Continue inserting additional species at the beginning, middle or end of this series, allowing branching when series end. This is like the construction of a “Besseyan cactus” but is composed of species. A set of diagrams illustrating this process is given by Zander (2018: 81). The requirement of no (or minimal) reversals makes this a second order Markov chain where the last element partially depends on the antepenultimate of any serial set of three species. The antepenultimate is always an outgroup for the next two species in a lineage. Arranging an evolutionary tree such that there are no reversals also models the maximum preservation of the number of conservative traits through the speciation process. The diagram of this model is a speciation tree, or caulogram. This focuses on transformations between species, dealing with each species pair, one by one. A trait involved in speciation is one commonly used in a standard key to species. Optimization is by arrangement of species such that there are no or few trait reversals in a lineage. Because standard preliminary taxonomic analysis reduces the number of species involved to very few, massive computerized analysis is generally unnecessary. Phylogenetics, on the other hand, develops dichotomous non-ultrametric dendrograms, or cladograms, based on optimized dichotomous transformations between sequentially dichotomously split sets of traits. Macroevolutionary systematics, conversely, models the inferred order of speciation between the individual elements of a lineage, that is, between two species in the context of an outgroup or inferred ancestor. Phylogenetics searches for a pattern with maximum likelihood or maximum posterior probability (Felsenstein 2004: 256, 291) of the shared ancestry of sets of whole set of sampled taxa at once. The optimum pattern stands out from a distribution curve as an outlier. Support for caulograms is by posterior probabilities of a putative ancestral species generating a descendant species given a particular outgroup or accepted shared ancestor. There may be one or more descendants, and any descendant may be an ancestor in its own right. Assuming a minimum of two morphological (or other expressed) traits are needed to define a species (one simple trait may be simply a rogue gene in a population), odds analyses

11 summarized in Table 3 allow Bayesian posterior probabilities (BPP) to be calculated. Because traits when expressed as Shannon information bits (one bit per trait) are exponential, they may be added. This is a technique known as sequential Bayesian analysis, pioneered by code- breaker Alan Turing (Gleick 2011; Zander 2014a: 12). For example, two unreversed traits in the inferred descendant species merit 0.80 BPP in support of the order of speciation, three 0.89 BPP, four 0.94 BPP, five 0.97 BPP, and six 0.98 BPP. Shannon-Turing analysis is used here to integrate and standardize character value as a measure of taxon coherence and evolutionary order. A Shannon bit is that measure of information necessary to make a decision at a minimum level of confidence. Translating traits to Shannon informational bits, which are exponents, is simply one bit per simple trait, and they may be added in the context of Turing sequential Bayesian analysis. Informational bits may be interpreted as exact values of Bayesian posterior probabilities. In practice, the average number of trait transformations per speciation event in past studies is 4.04 for Streptotrichaceae (Zander 2018) and 3.6 for 14 species of Didymodon segregates (Zander 2019b). Species with unusually large number of trait changes are either highly specialized or progenitors in their own right with their own small cloud of descendants. All species dealt with in macroevolutionary systematics are either extant or inferred “missing links”. The rationale is that if an extant species has the same traits as an inferred ancestral species from the past, then both species may be identified by the same taxonomic name. In molecular phylogenetics, patterns of transformations of sets of traits are sampled to find the optimal pattern of maximum likelihood or maximum posterior probability, and Bayesian support measures are based on the differences between the probability of maximum likelihood and the second best probability, or the maximum posterior probability and the probability of the prior. Thus, Bayesian support values from macroevolutionary systematics are calculated on data from expressed traits through sequential Bayes from only a few possible scenarios. Standard phylogenetic studies, however, are based on traits expected to track evolution of expressed traits, and one pattern is selected from a large sample of a vast number of possible patterns, all of miniscule probability, any one of which is a possible explanation for the data. Only one is commonly a clear outlier of the distribution. Molecular analysis can be extremely helpful when morphological study is not clear because of few available traits, or when convergence is suspected. The accuracy of molecular studies that track evolution has been checked by reference to known morphologically based relationships, and there is indeed a close relationship in morphological and molecular patterns of evolution. Molecular analysis can demonstrate ancestor-descendant relationships in two major ways: Molecular paraphyly, with two samples of the same species distant on a cladogram, can imply a progenitor-descendant relationship between the paraphyletic species and the apophyletic species between the paraphyletic samples. Also, a very large distance between two samples of the same taxon can imply genuine polyphyly due to perhaps convergence or lack of telling characters.

12 Table 1. Species and genera of Pleuroweisieae with caulogram, transformative traits, informational bits, and Bayesian posterior probabilities (BPP) in support of order of speciation. Bits may be summed as support for the integrity of the genus. Name Caulogram, genera boxed Traits Bits BPP UNKNOWN ● Peristomate progenitors with Pleuroweisieae gametophytes; N.A. stem central strand and hyalodermis present; leaves lanceolate, margins plane; costa with two stereid bands and epidermal cells; laminal cells with large papillae, crowded, multifid; peristome present, twisted to straight. Eobryum ● Leaves long-lanceolate, base weakly broadened; costal 5.00 0.97 anoectangioides epidermal cells present; stem central strand present; laminal papillae simple to bifid, crowded. E. hildebrandtii ● Leaves elliptical, apex rounded-acute; costa thickened, 3.00 0.89 ending 2–3 cells before apex. E. xerophilum ● Leaves elliptical, apex broadly acute; costa ending at 2.00 0.80 apex or short-excurrent. Anoectangium ● Pleurocarpous; leaves elliptic to lanceolate, acute, apiculate 5.00 0.97 aestivum to short-mucronate; laminal papillae massive, usually fused, ca. 3–4 centered over each lumen. A. euchloron ● Rosulate, leaves long-elliptic to ligulate, apex broadly 5.00 0.97 acute to rounded; distal laminal cells low spiculose- papillose, walls nearly circular in section. A. clarum ● Leaves long-lanceolate to long-triangular, distal laminal 4.00 0.94 cells very thick-walled, lumens usually oval; laminal papillae small, simple, ca. 4–5 evenly scattered across each lumen. A. incrassatum ● Leaves short-elliptic, apiculus short or absent, medial 4.00 0.93 cell lumens rounded-rectangular, distal marginal cells elongate. A. stracheyanum ● Leaves long-lanceolate to long-ligulate, length 7–10 times 3.00 0.89 greatest width above leaf base, often constricted just above the weakly sheathing leaf base. A. sikkimense ● Leaves long-triangular, base abruptly widened as a 3.00 0.89 short-sheathing skirt about 1.3 as broad as at midleaf, leaf base often serrulate-crenulate on margins. Ardeuma ● Stem central strand present, no hyalodermis, oblong- 4.00 0,.94 gracillimum lanceolate leaves; costa with poorly differentiated adaxial and also often abaxial epidermal cells. A. recurvirostrum ● Stem central strand absent, rhizoids dark. 2.00 0.80 A. annotinum ● Leaves reflexed at top of much broadened base, leaf 3.00 0.89 cells elongate. A. aurantiacum ● Leaves linear to elliptic-lanceolate; leaf cells mostly 3.00 0.89 trigonous or porose; base weakly differentiated.

13 Table 1. Continued. Name Caulogram, genera boxed Traits Bits BPP

Gymnostomum ● Leaves ovate-lanceolate, medial laminal cells quadrate, 5.00 0.97 aeruginosum papillae bifid and scattered, costa ending before the apex, adaxial cells quadrate to short-rectangular. G. calcareum ● Capsule mammillose; leaves short-elliptic; laminal cells 3.00 0.89 small. G. mosis ● Leaves very short, ovate; leaf margins bistratose. 2.00 0.80 G. viridulum ● Leaves very short; gemmae produced in axils. 2.00 0.80 Hymenostyliella ● Hyalodermis absent, distal laminal margins distantly 5.00 0.97 llanosii toothed and strongly involute, adaxially bulging medial laminal cells, and perichaetia on short lateral branches. H. alata ● Leaf apex cucullate, costa with two adaxial lamellae. 2.00 0.80 Hymenostylium ● Stem hyalodermis absent, leaves broadly lanceolate to 4.00 0.94 xanthocarpum long-elliptic, medial laminal cells bulging with one or two large simple papillae, and laminal cell walls bulging more adaxially than abaxially. H. townsendii ● Leaves 4.5–5 mm, long-acuminate, widest just above 4.00 0.94 expanded base; costa excurrent as a mucro; marginal leaf cells bistratose. Molendoa ● Lateral perichaetia with strongly sheathing leaves, stem 5.00 0.97 sendtneriana lacks hyalodermis, distal laminal cells occasionally bistratose, with low, scab-like papillae, superficial walls thicker than internal. M. hornschuchiana ● Plants large; leaves subulate, with dentate shoulders of 5.00 0.97 sheathing base; costa semilunar in section; distal laminal cells often elongate. M. peruviana ● Leaves short-ovate, distal margins bistratose; costa 3.00 0.89 lacking adaxial epidermis. Ozobryum ● Pleurocarpous; leaves ovate, margins bi- to tri-stratose; 4.00 0.94 ogalalense costa with one stereid band; laminal cells with 1(–2) large papillae. Reimersia ● Stem section triangular; leaves trifarious, squarrose, distal 4.00 0.94 inconspicua portion of the leaf base with paired concavities Tuerckheimia ● Apparently peristomate; long-triangular leaves, medial 4.00 0.94 guatemalensis laminal cells without trigones or porosity, bulging on both sides with massive multifid papillae. T. svihlae ● Eperistomate; costa excurrent as a mucro. 2.00 0.80 T. valeriana ● Eperistomate; distal leaf margins broadly sinuous-dentate. 2.00 0.80

Patriculars A spreadsheet (as the basis for Table 1) was developed to incorporate both data on Pleuroweisieae species and a caulogram (a speciation tree) of their macroevolutionary relationships using the Excel drawing character set. The species in the genera investigated

14 were limited to those with which the present author was familiar, and which allowed theoretical and parsimonious construction of dissilient (a progenitor species radiating descendants with no reversals) genera. Much information was taken from previous studies (Zander 1976, 1993, 2017a, 2017b, 2019c; Zander & Eckel 2017) of this widespread, hygrophilic tribe of mosses, and from recent studies such as that of Cano & Jiménez (2013). The results were compared with similar data on numbers of traits per species and per genus obtained from previous studies of Barbuloideae (Pottiaceae) (Zander 2014a, 2014b) and Streptotrichaceae (Zander 2018). The range of values of evolutionary leverage were calculated and compared. A study of the genus Eobryum R.H.Zander & Sollman (Zander & Sollman 2020) distinguished the monotypic taxon as progenitor or progenitor equivalent to a number of genera in the Pleuroweisieae (Pottiaceae), included in the present study for comparison of evolutionary leverage. Genera treated here are Anoectangium Schwägr., Ardeuma R.H.Zander & Hedd., Eobryum R.H.Zander & P.Sollman, Gymnostomum Hedw., Hymenostyliella E.B.Bartram, Hymenostylium Brid., Molendoa Lindb., Reimersia P.C.Chen, and Tuerckheimia Broth. The genus Ardeuma, a segregate of Hymenostylium not including the type but including the common species Ardeuma recurvirostrum (Hedw.) R.H.Zander & Hedd., has been characterized as having no central strand in the stem (Zander & Hedderson 2016), at least for the many species studied in the past (Zander 1993). For the Eobryum treatment, the types of two species recently described in Hymenostylium by Köckinger & Kučera (2011) were examined.

Taxonomic treatments ARDEUMA The generitype of Ardeuma was designated (Zander & Hedderson 2016) to be A. recurvirostrum (Hedw.) R.H.Zander & Hedd., and several combinations have been made in Ardeuma for certain of the species of Hymenostylium (see also Zander & Brinda 2016). Much work needs to be done in the taxonomy of this genus.

Ardeuma gracillimum (Nees & Hornsch.) R.H.Zander, comb. nov. Gymnostomum gracillimum Nees & Hornsch., Bryol. Germ. 1: 149. 1823, basionym; Hymenostylium gracillimum (Nees & Hornsch.) Köckinger & Jan Kučera, J. Bryol. 33: 203. 2011. Type: AUSTRIA. Pongau, Alpib. Salisburg. (Hornsch.), Arnott- collection, Hornschuch s.n. (lectotype, E-00165014!).

The lectotype by Köckingter & Kučera (2011) of Hymenostylium gracillimum clearly exhibited a stem central strand but was otherwise largely indistinguishable from H. recurvirostrum, particularly in lacking both costal epidermal cells and stem hyalodermis. Worldwide, traits distinguishing H. gracillimum from H. recurvirostrum given by Köckinger & Kučera (2011)—stem sclerodermis weak or absent, leaf cells not porose, and sporophytes with microstomous urns and smaller spores—are somewhat variable in H. recurvirostrum, a typically polymorphic hygrophilic taxon, and are of dubious value. In as much as the presence

15 of a stem central strand is of considerable importance in distinguishing taxa in the Pleuroweisieae, this taxon is recognized here at the species level. The rhizoidal tubers with lateral bulging cells illustrated for Hymenostylium gracillimum by Köckinger & Kučera (2011) were absent in the lectotype and, because they are identical with those of Gyroweisia tenuis (Hedw.) Schimp., which has similar morphology and occurs in the same habitats, their significance needs to be restudied.

EOBRYUM Eobryum xerophilum (Köckinger & Jan Kučera) R.H.Zander, comb. nov. Hymenostylium xerophilum Köckinger & Jan Kučera, J. Bryol. 33: 199. 2011, basionym. Type: AUSTRIA. Styria, Eisenerzer Alpen, Reiting mountain, Kaisertal NW of Seiz, ca. 980 m elev., Köckinger 05-954 (holotype, E).

Hymenostylium xerophilum is well described and illustrated by Köckinger & Kučera (2011) and Ignatova et al. (2019). On examination of the type it proved to be a strikingly distinctive species, immediately distinguished by the stem central strand, short-lanceolate leaved, sharply reflexed just above the leaf base, and costal epidermis differentiated both adaxially and abaxially. The ventral costal cells are short-rectangular distally, elongate proximally. Absent are the elongate, often porose medial laminal cells characteristic of A. recurvirostrum and its close relatives. The other genera of the Pleuroweisieae, at least for the well-known species considered here, each form a closed causal group that excludes H. xerophilum, but it fits well with the recently described genus Eobryum.

Eobryum hildebrandtii (Müll.Hal.) R.H.Zander, comb. nov. Fig. 1. Weissia hildebrandtii Müll.Hal., Linnaea 40: 289. 1876, basionym; Hymenostylium hildebrandtii (Müll.Hal.) R.H.Zander, Bull. Buffalo Soc. Nat. Sci. 32: 127. 1993; Ardeuma hildebrandtii (Müll.Hal.) R.H.Zander, J. Bryol. 38: 299. 2016. Type: SOMALIA. Serrut mountain, 1,800 m elev., Hildebrandt s.n. (isotype, NY-01408144!). Merceyopsis hymenostylioides Broth. & Dixon, J. Bot. 48: 302, pl. 508: f. 6. 1910, basionym; Gymnostomum hymenostylioides (Broth. & Dixon) R.H.Zander, Bull. Buffalo Soc. Nat. Sci. 32: 153. 1993; syn. nov. Type: INDIA. Simla, Long 7206 (holotype, H!).

A specimen from Yemen (Kürschner 1998) cited below matched the types of both Hymenostylium hildebrandtii (Kürschner 1998), and Merceyopsis hymenostylioides. Dixon’s (1910) concept of the genus Merceyopsis Broth. & Dixon included a statement that there was no stem central strand. Examination of the type of M. hymenostylioides proved that a distinct, brownish central strand was indeed present, as was the case with material of H. hildebrandtii. A combination in Eobryum is made here. This species is similar to E. xerophilum except that the leaves are very fragile, the much-thickened costa ends 2–3 cells before the rounded-acute apex, and the basal cells are somewhat inflated. The more acutely pointed perichaetial leaves are about twice the length of the cauline leaves although not prominent without dissection (Fig. 1A–B). The genus Merceyopsis was lectotypified with M. pellucida Broth. [=

16 Figure 1. Eobryum hildebrandtii (Müll.Hal.) R.H.Zander. A. Dry habit. B. Moist habit. C–E. Leaves. F. Leaf apex. G. Leaf base. H. Section at midleaf. I. Section near leaf base. J. Stem section. Scales: a = 1.5 mm, A–B; b = 0.5 mm, C–E; c = 100 μm, F–J. All drawn from Yemen, Raab-Straube 9605 (MO).

17 Scopelophila cataractaae (Mitt.) Broth.] by Zander (1993: 153), and is not an earlier name at the genus level. The distribution of the rare species E. hildebrandtii is now known to be Somalia, Saudi Arabia, Oman, Yemen and Morocco (Cano et al. 2002; Kürschner & Frey 2020).

Specimen cited: YEMEN. road between Schibam and Kaukaban, 100 m N of Kaukaban, 15°30′00″N, 43°54′00″E, sandstone, 2,700 m elev., Raab-Straube 9605 (MO).

HYMENOSTYLIUM The genus Ardeuma was proposed by Zander & Hedderson (2016) to accommodate all species of the genus Hymenostylium excepting the type, H. xanthocarpum (Hook.) Brid., a species that differed significantly in the presence of a stem central strand, leaves broadest about midleaf and constricted just above the base, distal laminal cells ventrally bulging and massively unipapillose, and basal cells differentiated only in the lower 1/5–1/7 of the leaf. A second quite similar species, H. townsendii R.H.Zander, was later described (Zander 2017a). This differed by longer leaves, acuminate leaf apex, costa excurrent as a sharp mucro, and distal laminal marginal cells bistratose, distinctive similarities that implied that both species comprised a good natural genus with a clear evolutionary trajectory signaled by the latter species’ uncommon and thus advanced (with respect to most trichostomoid outgroups) traits.

MOLENDOA According to Cano & Jiménez (2013), Molendoa handelii (Schiffn.) Brinda & R.H.Zander is morphologically very close to M. peruviana (Sull.) M.J.Cano & J.A.Jiménez. The latter is the earlier name, and it is used here because the transformational traits from the putative progenitor (Table 1) are much the same.

OZOBRYUM Ozobryum ogalalense G.L.Merr. has been recognized in Molendoa but is retained here in Ozobryum G.L.Merr. because it is anomalous in Molendoa, having interior laminal cell walls the same thickness as the superficial walls. Not being a member of the here considered closed causal groups, i.e., not fitting as part of other radiations, it is considered a monotypic genus in the Pleuroweisieae, differing from the putative progenitor Eobryum anoectangioides by four significant traits. More study is necessary, particularly of the somewhat divergent Mexican population (Zander 2005).

Results Using the Running Sigmas Calculator (Zander 2020), which provides ranges for first, second, third and fourth standard deviations, the numbers of informational bits (traits) per species in the Pleuroweisieae as conceived here were evaluated. The 29 species above had a bit range of 2–5 new trait transformations per species, as given in Table 1. In Table 1, the Bayesian posterior probability (BPP) of the correct order of speciation is given for each species based on summed informational bits per species (see chart of equivalencies, Zander 2018: 32). First

18 Table 2. Three taxonomic groups of mosses (partial in Barbuloideae) with average transformative traits per species (per speciation event), summed species per genus limited here to those in a conceptualized dissilient genus, summed number of transformative traits each species contributes to a genus, and evolutionary leverage or average transformations in each species per genus. Average Evol. leverage: Total spp. Total traits Family Infrafamily Genus traits per average traits per genus per genus species per genus Pottiaceae Pleuroweisieae 10.30 Anoectangium 4.0 6 24 Ardeuma 3.0 4 12 Gymnostomum 3.0 4 14 Eobryum 3.0 3 9 Hymenostyliella 3.5 2 7 Hymenostylium 4.0 2 8 Molendoa 4.3 3 13 Ozobryum 4.0 1 4 Reimersia 4.0 1 4 Tuerckheimia 2.7 3 8

Pottiaceae Barbuloideae 13.8 Didymodon 2.6 5 13 Exobryum 4.0 1 4 Fuscobryum 3.3 4 13 Geheebia 4.1 7 29 Trichostomopsis 2.0 3 6 Vinealobryum 3.6 5 18

Streptotrichaceae Streptotrichaceae 9.60 Austroleptodontium 8.0 1 8 Crassileptodontium 4.0 4 16 Leptodontiella 3.0 1 3 Leptodontium 3.7 3 11 Microleptodontium 3.5 4 14 Rubroleptodontium 5.0 1 5 Stephanoleptodontium 3.1 8 25 Streptotrichum 4.0 1 4 Trachyodontium 4.0 1 4 Williamsiella 5.8 4 6

standard deviation for the number of new traits per speciation event (as radiation away from a progenitor) was 4.7 traits, second 5.9 traits. Thus, all species were within 95 percent of expected variation in numbers of trait transformations per speciation event. A requirement of science is that analyses of similar subjects should have generally similar results, such that coherent theories can be advanced to account for such similar subjects. The

19 Table 3. Shannon informational bits calculated at one per new trait are translatable into Bayesian support. Each bit is given its value as a power of two, the odds ratio as a proportion of value to 1, the odds ratio given as a fraction, and the percent probability or Bayesian posterior probability (BPP) of that fraction. Bits 0 1 2 3 4 5 6 7 8 Value 1.00 2.00 4.00 8.00 16.00 32.00 64.00 128.00 256.00 Odds ratio 1 : 1 2 : 1 4 : 1 8 : 1 16 : 1 32 : 1 64 : 1 128 : 1 256 : 1 Fraction 1/2 2/3 4/5 8/9 16/17 32/33 64/65 128/129 256/257 BPP 0.500 0.667 0.800 0.889 0.941 0.970 0.985 0.992 0.996

average number of trait transformations per species summed within genus of the tribe Pleuroweisieae is given in Table 2, with comparisons of numbers of summed new traits per species in a genus in macroevolutionary studies of other genera in Pottiaceae and the recently segregated family Streptotrichaceae. The range of average number of transformation traits among species in each of the 10 genera of Pleuroweisieae is 2.7–4.3, with an average of 3.6. First S.D. is 4.1, second is 4.7, thus there are no outliers. The average number of new traits per species per genus measures evolutionary leverage. The range for 10 Pleuroweisieae genera was 4–24 transformational traits among species per genus, with an average of 10.30. First S.D. was of 13.9, second of 19.8; thus, only Anoectangium was outside the expected range of values. As an outlier, however, the species of Anoectangium fit together into a coherent radiative genus with expected number of transformations per speciation event (Table 1). This was compared to the results (Zander 2014a but see Zander 2019b for Exobryum R.H.Zander emendation) for six segregate genera of Didymodon Hedw. (Barbuloideae, Pottiaceae), with a range of average number of new traits in each species summed per genus in the range of 4–29, with an average of 13.83 new traits per genus. A third macroevolutionary study, that of 10 genera of the Streptotrichaceae, a family segregated from Pottiaceae by Zander (Zander 2018: 72, 201) found a range of 3–25 of transformations in each species totaled per genus, with an average of 9.60 new traits per genus. First S.D. was of the average traits per genus was 13.0, second S.D. was 20.0. The average traits per species in a genus were within two S.D.s, excepting the genus Stephanoleptodontium R.H.Zander. This genus has an average of 25 new traits per genus, well within the third standard deviation and thus a mild outlier, particularly given that the genus forms a closed causal group with species neatly nested and no reversals (Zander 2018). It is clear that the subfamily Barbuloideae of the Pottiaceae has the greatest evolutionary leverage with summed 13.83 trait transformations on average for each of the six genera, compared to 10.30 average per genus for the Pleuroweisieae, and 9.6 per genus for Streptotrichaceae. The difference, in part, may be that, in comparison with the other two groups, the Barbuloideae is more speciose per genus. Curiously, the Barbuloideae retains the ancestral twisted peristome. It may be hypothesized that Pleuroweisieae and Streptotrichaceae, given the reduced nature of the peristome, are remnants of ancient groups specialized for

20 particular environments, hygric for the former group, and arboreal for the latter group later spreading to páramo and mountain rock habitats. On the other hand, the average number of advanced traits per species per genus are rather similar, which reassures one that a similar evolutionary process is involved. Each genus evaluated here is dissilient, meaning it can be conceived parsimoniously as a radiation from a progenitor species, it nearly always has no reversals in conservative traits, and it is an integral unit of evolution, one step higher than species, which is the (more) basic unit. The genera of the Barbuloideae may be taken as evolutionarily robust such that in the event of partial extinction, the genus is more likely to be preserved as a functional unit in survival, given the success of its fundamental set of conservative traits, in the ecosystem.

Discussion Biodiversity analysis requires classifications that accurately reflect well-modeled evolutionary relationships. Following strict phylogenetic monophyly, species, genera, and even families are commonly either sunk to synonymy or invented without reasonable evolutionary evaluation of the molecular races that are expected to track speciation. Making predictions for the expected vast changes in floral and faunal composition is now difficult because of the sea of rapidly changing nomenclature that lumps disparate taxa or splits otherwise coherent taxa into unmanageable units. Systematics is a science, based on sampling nature, naming basic units and larger sets of supraspecific entities, and presenting evolution-based classifications. But it has always been vulnerable because mathematics and statistics are not reputed to be a strength of taxonomists. In phylogenetic analysis, sampling is poor, with, say, 5 or 10 molecular exemplars of a species being considered a lot; Morrison (2013) suggested an optimum sample of about 30, while Thompson (2012) offered formulae for estimating samples necessary for accurate estimation that often require hundreds of samples depending on precision and accuracy expected. Evolutionary scenarios from different data sets (morphology and DNA) are compared under Bayes Formula (by merging the data), instead of Bayes Factors, even though support values are above 50% for each of contrary topologies. Accuracy is rejected in favor of precision. Molecular studies substitute clades of molecular strains of species for actual species, when (or perhaps because) the last might require much additional sampling, assuming many information-bearing strains are not extinct. “Shared ancestors” are considered by some cladists to be interspersed as nodes in a cladogram without actually modeling any ancestral species in the constructivist sense (Kleene & Vesley 1965) (e.g., can they be fully described or not? Where are their autapomorphies? Do those reverse and disappear? How does this affect parsimony or other analytic method? Are these shared ancestors models of unknown species we should look for?). Problems in phylogenetics were presented by Brummit (2006), Hörandl & Stuessy (2010), and Sivarajan (1991), among many others, and are summarized by Zander (2013, 2018). Unfortunately, there is a bias against morphological evaluations of evolutionary relationships. Nonparametric bootstrapping (Felsenstein 1985, 2004: 340) as commonly

21 implemented is not appropriate for evaluation of support for morphological cladograms because the data (extracted from each species description) is already optimized by the prior efforts of taxonomists to distinguish taxa, create a key to them, and give a classification, i.e., a method parallel to that of cladistic analysis. An analogy to morphological bootstrapping is comparing keys generated by subsamples of the characters in a standard key to species—the result is a number of keys with too few distinguishing characters. The traits of a species in a key are simply not enough to subsample; one might complain to the Deity or Mother Nature for such a limited distinction. The proper morphological data set for bootstrapping is that of the traits of all the specimens taxonomically examined to generate the descriptions. Subsampling such a data set and implementing standard taxonomic procedures on each subsample can be expected to allow species descriptions, keys and classifications that would prove to be much the same as those of each other subsample, leading to a judgement of high support because of sets of considerable agreeable homogeneity in the available data. This is quite the contrary of the low bootstrap values derived from traits extracted from the description of species rather than specimens. On the other hand, Shannon-Turing sequential Bayes in macroevolutionary systematics provides a proper and justifiable measure of probabilistic support for a speciation tree based on traits from pre-optimized species descriptions. The face-off between morphological and molecular taxonomy is an example of the rule of the excluded middle. One or the other must be chosen as correct. Otherwise, if Bayesian posterior probabilities of both alternatives are higher than 0.50, they refute each other. A return to 3-valued logic has been advocated by Gisin (2020a, 2020b), a student of quantum entanglement, who suggested a probabilistic alternative of “both” being true—subject to further observation. Another way to approach the quandary is to turn to dialectics. One may search for an encompassing explanation as a third alternative that addresses adequately both well-supported morphological and molecular analysis of the same taxa. One such explanation is that branching molecular races in one species—such that each may generate a different descendant species— are the source of molecular paraphyly and short-range polyphyly. This includes the hypothesis of a progenitor-descendant multiplex nesting order of species generation. Morphologic and molecular evolutionary trees are then mostly congruent given acknowledgement of the measured (Zander 2019a, 2019b) average inaccuracy of molecular cladistic order of speciation of 4.5 nodes due to radiation of molecular strains and races. Much remains in classification that is incongruent between morphological and molecular classifications because much is yet unevaluated with macroevolutionary covering models. Gisin’s probabilistic conception applies to these separately well-supported but presently incongruent classifications. A model explaining supraspecific taxon coherency is that there is an empirical taxonomic unit higher than species in which such coherency is generated. That is the idea that there may be one or more descendants of some generalized progenitor. Each descendant is identical (except for minor traits commonly scattered and often reversing in related groups) to the progenitor plus at least two additional traits that make it a different species (one trait difference may just be a mutation). A coherent radiative genus consisting of well-nested descendants excludes other genera,

22 including, say, a one-species genus. The monotypic genus may be a remnant of an ancient lineage. Removing from consideration any morphological traits that are considered highly adaptive in other related genera, i.e., are found in specialized descendants, leaves us with conservative traits that are at least hints of a long-lost lineage. Such an intellectual adventure is suppressed by cladistic tree-thinking; that is, modeling sets of species each evolving into two smaller sets of species, which is not a model of evolution. Rather than wondering “where the gaps are” in difficult groups, one might search for progenitors (modern species with the same traits as a putative ancestor) and see if other species can be nested around them as adaptive or nearly neutral descendants to create two or more dissilient genera. Radiation does not need to be adaptive, which is now pretty much accepted. Modern genera, I think, have many radiative genera in them and no one has looked for them as empirical taxa higher than species. At least I have demonstrated these in my own publications on, e.g., Didymodon and Streptotrichaceae. Molecular analysis, such as that by Köckinger & Kučera (2011) is not expected to provide well-distinguished genera that may represent evolutionary radiation for these taxa, as argued by Zander (2019a, 2019b). A continued evaluation of the Zander (2019b) metadata examination of various other researchers’ molecular systematics of several species of Pottiaceae found that molecular paraphyly was significant when measured in terms of classical statistics. In that study, (1) large numbers of species were found to have molecular races, (2) those species with sufficient data to determine the numbers of molecular races averaged 4.52 molecular races, (3) a large portion (0.41) of all demonstrably multiracial species are also paraphyletic, and (4) apophyletic (descendant) species averaged 3.59 per paraphyly, implying that phylogenetic’s only two descendants from a shared ancestor is too few to be optimal, at least for some groups. Further study of the same metadata (Zander 2019b) found that there averaged 4.93 nodes between maximally distant paraphyletic exemplars of the same species. Standard deviations were calculated with an Excel spreadsheet using the 27 measures of nodal distance between most-distant paraphyletic exemplars of the same species as samples. The range was 2 to 15 nodes. One sigma (first standard deviation) was 2.92 nodes, 2 sigmas was 7.89, 3 sigmas was 12.8. Assuming a normal distribution, 1 sigma is 0.68 of variation, 2 sigmas is 0.95, and 3 sigmas is 0.997. Most of the probability distribution of apophyletic species from one paraphyletic species occurs for this data within 13 nodes of any single occurrence of any molecularly paraphyletic species. That is, 13 linked shared phylogenetic nodes (hypothetical shared ancestors) is the critical distance for 3 sigma dispersion of molecular races, that is, implying an expected large probabilistic multifurcation of descendant species of any one ancestral species. Differences between two molecular studies may be due to two different samples of the same species consisting of different races (molecular moieties) separately generating different descendant species. Agreement may be due to extinction of wayward molecular races. Thus, phylogenetic non-monophyly is not certain within any 13 continuous nodes on a molecular cladogram, being much the same as a mobile multifurcation.

23 Acknowledgements Patricia M. Eckel graciously provided the illustration. J. A. Jiménez and another reviewer provided helpful comments. The Missouri Botanical Garden is thanked for its continued support of research on bryophyte taxonomy, particularly in the application of evolutionary theory to systematics.

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24 Cambridge University Press, Cambridge. Thompson S. K. 2012. Sampling. Third Edition. John Wiley and Sons, Inc., Hoboken, New Jersey. Zander R. H. 1976. The tribe Pleuroweisieae (Pottiaceae, Musci) in Middle America. The Bryologist 80: 232–269. Zander R. H. 1993. Genera of the Pottiaceae: Mosses of harsh environments. Bulletin of the Buffalo Society of Natural Sciences 32: 1–378. Zander R. H. 2005. Molendoa ogalalensis (Bryopsida, Pottiaceae) new to Mexico. The Bryologist 108: 129–130. Zander R. H. 2013. Framework for Post-Phylogenetic Systematics. Zetetic Publications, St. Louis. Zander R. H. 2014a. Classical determination of monophyly, exemplified with Didymodon s. lat. (Bryophyta). Part 2 of 3, concepts. Phytoneuron 2014-79: 1–23. Zander R. H. 2014b. Classical determination of monophyly, exemplified with Didymodon s. lat. (Bryophyta). Part 3 of 3, Analysis. Phytoneuron 2014-80: 1–19. Zander R. H. 2017a. Hymenostylium townsendii R.H.Zander (Pottiaceae, Bryophyta), a new species from Nepal. Journal of Bryology 39: 1–4. Zander R. H. 2017b. Anoectangium sikkimense (Pottiaceae, Bryophyta) new to the New World from Alaska, and its macroevolutionary relationships. The Bryologist 120: 435–440. Zander R. H. 2018. Macroevolutionary Systematics of Streptotrichaceae of the Bryophyta and Application to Ecosystem Thermodynamic Stability, ed. 2. Zetetic Publications, St. Louis. Zander R. H. 2019a. Infraspecific molecular trees are associated with serial macroevolution in Pottiaceae (Bryophyta). Ukrainian Botanical Journal 76: 379–394. Zander R. H. 2019b. Macroevolutionary versus molecular analysis: Systematics of the Didymodon segregates Aithobryum, Exobryum and Fuscobryum (Pottiaceae). Hattoria 10: 1–38. Zander R. H. 2019c. Macroevolutionary evaluation methods extended, consolidated, and exemplified with Anoectangium (Pottiaceae) in North America and the Himalayas. Annals of the Missouri Botanical Garden 104: 324–338. Zander R. H. 2020. Running Sigmas Calculator. Res Botanica Technical Report 2020-08-2. A spreadsheet published on Res Botanica, a Missouri Botanical Garden Web Site: http://www.mobot. org/plantscience/resbot/Meth/RunningSigmasCalculator.xlsx Accessed Aug. 2, 2020. Zander R. H. & Brinda J. C. 2016. A new combination in Ardeuma (Pottiaceae, Bryophyta). Lindbergia 39: 39–40. Zander R. H. & Eckel P. M. 2017. Notes on Pleuroweisieae (Pottiaceae, Bryophyta) in North America. Evansia 34: 61–64. Zander R. H. & Hedderson T. A. 2016. A re-evaluation of Hymenostylium xanthocarpum (Hook.) Brid., and Ardeuma R.H.Zander & Hedd., a new name for all other species of Hymenostylium (Pottiaceae, Bryophyta). Journal of Bryology 38: 295–301. Zander R. H. & Sollman P. 2020. A new genus of Pottiaceae (Bryophyta) of ancient lineage from Asia. The Bryologist 123: 179–187. manuscript received March 22, 2021; accepted May 11, 2021

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