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Let “genus” refer to some well-defined collection of individual organisms. For the time being, we assume this to be reasonable. For any given genus, we wish to partition the elements into possibly smaller classes called “species.” Then the relation, R, defined by xRy if and only if x and y are in the same species is an equivalence relation. By mathematical necessity, R is reflexive, symmetric, and transitive.
The hope is that we may use inter-fertility to establish a definition of species. A natural first attempt may be to define R as follows. xRy if and only if x=y or x and y are inter-fertile. Dan puts forth several different reasons (see proviously posted link) why this relation fails to generate well-defined classes (species). By introducing some modifications, he arrives at the concept “para-coherent group” which is based upon a shared gene pool. I believe that Paul’s suggestion to use graph theory (see previous comment) quickly leads to Dan’s idea.
Following Paul’s idea, let the nodes of a graph represent all organisms within a given genus at some specific point in time. (A necessary consequence of evolution is that the graph, ie. the relation, is dynamic over time.) Put an edge joining two nodes if the organisms they represent are inter-fertile. Then the resulting graph has components (maximal connected subgraphs) that amount to para-coherent groups. The relation xSy if and only if x and y are in the same component is reflexive, symmetric, and transitive. S is our species relation.
Within a given component (species) define the relation E by xEy if and only if there is an edge joining x and y. Then E need not be transitive (nor is it reflexive… but that is easily fixed). Then what Dan calls a “ring species” amounts to either a non-transitive string or a non-transitive cycle. Dan’s crab example is a non-transitive 4-cycle. Then if non-transitive strings or cycles are “large”, organisms at opposite ends of a string or opposing sides of a cycle could end up being morphologically quite different. What happens if organisms that had been placed into different genera end up S related? Given the existence of ring species, this does not seem far-fetched. So xSy but not xEy still ends up as a potential problem. As I understand the situation, DNA classification has run into a similar problem (in addition to a sort of dual problem… morphologically virtually identical organisms put into different species). Organisms which are originally put into the same genus get shuffled into different genera. These points would appear to me to necessitate a rethinking of “genus.” One may now suppose that the entire problem of classifying organisms into each of the acceptable divisions
kingdom, phyllum etc. may be reduced to Graph Theory and/or Set Theory. I think it would be best to begin with the primitive elements (organisms) and work up through the chain of classification types (as opposed, for instance, to having some a-priori notion of “genus”). Maybe step 2 could be to make each “species” into a node, and then connect according to desirable “genus” criteria. I have virtually no background in Biology (except for some stuff I learned via my interest in fungi), but I would imagine this proposal might constitute a massive undertaking; a complete restructuring of life-form classification…?
Paul’s maximal connected subgraph (component) and Dan’s para coherent group are equivalent concepts. There does appear to me to be a chicken/egg type problem when one begins with “populations.” Thus, I think the nodes in the graph should be individual organisms.
I wrote up a Word doc with my analysis of the component/species realtionship. But I don’t know how to post a word doc to the comment. I’m such a hopeless Neandertech! :-)
I notice that neither mating types nor “ring species” present a problem for my maximal-connected-component graph theory definition. :) (They do present some trickiness for actually determining species membership empirically. Basically, mating experiments can discover that two particular nodes share or do not share an edge; with knowledge of enough edges, the maximal connected components can be eventually identified. Another tricky bit is that in practice the granularity has to be populations, though in principle it’s individuals; and in some cases, identifying if two individuals belong to the same population or not can be tricky. With mushrooms, nearby identical-looking fruiting bodies may not always even be of the same species.)
I thought I’d jump in here, a little late perhaps.
Here’s a link to something I wrote on species cohesion through mating compatibility. The strict mating compatibility of Mayr’s biological species concept only works for kinds of things with a single mating type. When applied to species with multiple mating types, the mating compatibility criterion generates paradoxes.
The article is only a couple of pages long.
It’s not the exact definition of “interfertile”, but necessary to define a biological species. The term “species” is evolving too :-)
I would only consider two organisms interfertile if they were capable of producing fertile offspring, rather than only evolutionary dead-ends like the mule.
If we go back some billion years, we all share the same ancestor..
(of course, interfertility does not automatically make two species the same, the result also has to be fertile and able to mate with its relatives, to be meaningful)
The species concept is rather clear for animals, and hybridization is rare.
Hybridization is not uncommon between different species in the same genus of plants.
Fungi do rarely create hybrids, even between very closely related species.
But doesn’t it all depend on how we have defined the species?
If we return to the Hydnum complex – to sort it out, I think quite a lot of work remains to be done, comparing samples from all continents.
“You have for example two intersterile forms of Fomitopsis pinicola in NA, but both of them are interfertile with the european pinicola. It may be the same with Ischnoderma, but in the opposite direction (one species in NA, but two intersterile forms in Europe, benzoinum on conifers and resinosum on hardwoods).”
“Viewed mathematically, we hope (if not assume) that members of a given genus may be partitioned into species. Then the species concept should be an equivalence relation. Fominotopsis pinicola appears as an example where the necessary mathematical property “transitivity” fails when applied to the “interbreeding” criterion. Thus, “species” cannot be determined according to this criterion.”
Perhaps the solution to this is a dash of graph theory. Take a graph where nodes are forms and edges join pairs that are interfertile. Now, interfertile is not transitive, but “belongs to the same maximal connected subgraph” is transitive, and would make the three Ischnoderma forms a single species, and ditto the three Fomitopsis forms.
The big question mark is whether there might be some monster chain of interbreeding links lurking somewhere in there that would make Fomitopsis cajanderi the same species as Amanita phalloides, or worse maybe even Homo sapiens.
(Okay, that last seems doubtful given that every pair consisting of a human and any nonhuman seems to be intersterile, but I trust you get my point.)
DNA tests of Fomitopsis pinicola that includes other than european samples.
I beleive that the biological species concept is the only approach that works in the long run. Phylogenetics is an important part of the work, but it can only give a hint to delimitation of species.
we hope (if not assume) that members of a given genus may be partitioned into species. Then the species concept should be an equivalence relation. Fominotopsis pinicola appears as an example where the necessary mathematical property “transitivity” fails when applied to the “interbreeding” criterion. Thus, “species” cannot be determined according to this criterion.
I suppose this sort of thing shows why some mycologists have been very keen on the DNA principle. It offers hope that the species concept (at least when applied to fungi) may yet achieve mathematical precision.
I wonder if DNA analysis has been applied to the F. pinicola problem?
Isn’t it supposed to be based on the possibility of hybrids or not? Forms that are intersterile are usually considered to be different species – but that isn’t as simple as it sounds.
You have for example two intersterile forms of Fomitopsis pinicola in NA, but both of them are interfertile with the european pinicola. It may be the same with Ischnoderma, but in the opposite direction (one species in NA, but two intersterile forms in Europe, benzoinum on conifers and resinosum on hardwoods).
I’m not familiar with how taxonomists are dealing with it. I guess there are many different views on this evolutionary thing.. But I don’t think it’s possible to draw lines between species that are based only on a certain percentage of differences between their DNA. One question is how much efforts (time and money) that can be raised to sort out every single species.
umbilicatum here in the northeast US are mostly very small, cap diamters less than 1.5 inch. The cream colored repandum type often grows to over 7 inches in diameter with thickness of stipe well over one inch. So from my point of view there seems to be two distinct types. But I agree, names should be acceptable across different regions. So lumping does seem sensible. Maybe it is better to have discussions about the differences in our various local varieties of species X rather than about a lot of different species. However the scientific concept of species is rooted in technical criteria that make it possible to understand an organism as a well defined entity. And if the technicalities say “split”, then that is what must be done. What seems interesting to me is that DNA sometimes says “lump” the different looking ones but sometimes says “split” the similar ones.
lumping is more comfortable than creating new species of every form or "sub"clade, that easily results in a never ending story of changing names on them (and makes them impossible to ID without DNA-tests).
In my area, northern Scandinavia, we have our typical pale repandum in old spruce woods, in richer parts of these woods I also find a thinfleshed form of rufescens that resembles your umbilicatum.
When I go further south, I have found a form that looks to me like an intermediate between rufescens and repandum, among hardwoods (birch/aspen). What they call rufescens in southern Sweden is generally found in beech woods. What has been called umbilicatum in our area (doubtful name, not described quite like the american form as it has been interpreted here at MO…) is very small (cap size about an inch), more brownish than orange, and supposed to be found with hardwoods in the southernmost parts.
analysis sometimes suggests “lumping” of “species.” For instance, the Morchella elata types. For what it’s worth, most of what I ID as H. repandum is found in hardwood forests, often beech. Most of what I ID as H. umbilicatum is much smaller in size, and occurs with conifers, primarily hemlock.
If anyone is interested in examining samples, I can probably get some of each type by next October.
in this article from 2006 (also including american collections of umbilicatum): http://www.plantslo.org/Kongres/PDF/1%20Filogenija-FINAL.pdf
Here’s an interesting part of the discussion:
“The distance between H. umbilicatum collections and nearest H. rufescens
clade is similar to or shorter than distances between other clades of H.
rufescens. Regarding the phylogenetic analysis the position of H. umbilicatum
as a stable species is doubtful although it represents a good example of how
little molecular variation can be observed in samples from geographically
distant locations. H. rufescens shows highest polymorphism of ITS regions
with one clade corresponding to H. ellipsosporum (Ostrow and Beenken,
2004). Based on the high molecular diversity of presumably homogeneous
rDNA region in H. rufescens this species seems to be a species in a process of
intensive speciation, not correlated to the geographical distances between
different clades obtained. An important role in this process could play the
ectomycorrhizal partner or some other, non-measured characteristics of the
origins of the collections included in the study.
There are several possible explanations for a high molecular diversity in H.
rufescens. Harrington and Rizzo (1999) reported a high importance of niche in
determining the development and maintenance of fungal species which is not
necessarily correlated to geographical distances. H. rufescens is a common
species in Europe growing next to different ectomycorrhizal partners which
could lead to a possible diversification at the molecular level.”
I think the result supports the idea of only one Hydnum species (Hydnum repandum) and keeping the different forms as varieties – in cases when it’s possible to identify them.
and spore size is similar between repandum and rufescens, but H. repandum is a paler species with decurrent teeth, rufescens with darker orange cap and teeth which are not decurrent, but leaving a bare zone around the upper part of the stem. When repandum and rufescens were described by Elias Fries, both species were said to occur in America. The one that is called umbilicatum in the US, looks a lot like rufescens..
More info (and questions..) here:
I’ve also observed the lighter variety here in the U.S.:
Don’t know if there’s ever going to be DNA analysis on this
one, both are edible so not much need to differentiate
them. We’ll see if anyone adds H. rufescens to field guides here.
I generally use U.S. names for genus and/or species on my observations,
like Verpa instead of Mitrophora.
So, is that “elephant” more like a snake, or a wall? ;)
as either a species or a subspecies variety in any of the American manuals. Phillips lists rufescens (aka. repandum var. rufescens) as a European species. I wonder if there has been any DNA analysis to support any of this?.. or maybe some other micro trait like spore size?
I tend to find two types of hedgehogs… the large cream colored one which perfectly fits the usual description of H. repandum, and a small thin orangish capped one with somewhat fragile flesh that fits the description of H. umbilicatum. But apparently there are robust orangish ones that fall into the repandum category, at least here in the US. Kuo describes the cap color of repandum as “dull orange tan or paler.”
then you do consider rufescens, as well as umbilicatum, subspecies of repandum (which of course is a european species too).
I’ve seen two references to H. rufescens that mention it being found in
Europe. If it’s shown to be identical to what I’m finding here though
comparison with a type specimen there, I would consider changing the name.
Do you consider it just a subspecies of repandum?
Created: 2009-10-25 12:40:02 EDT (-0400)
Last modified: 2009-10-25 12:40:02 EDT (-0400)
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