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# Example of hybridization as a Petri net

I am working in phylogenetic analysis, concentrating on a particular kind of hybridization called allopolyploidization. I have put what I think is the Petri net for the process here.

Example of hybridization as a Petri net

For me the interesting questions are ones like: if you start with one diploid, and end up with two diploids and three tetraploids, what are the most likely sequences of evolutionary events (speciations, exinctions and hybridizations) to have produce this? So it seems I am trying to put probabilities on Feynman diagrams.

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1.
edited September 2011

Cool! It would be great if stochastic Petri nets could help you.

Are the 'things' of type 'haploid' or 'diploid' in this Petri net organisms, or species of organisms? Without this being clarified, things will get confusing.

For example, here:

the transition 'birth' turns one rabbit into two rabbits, not one rabbit into two species of rabbit.

(Perhaps confusingly, in Part 8 I've taken to calling each type of thing a 'species'. But this is just the typical mathematicians' way of using words to mean whatever they want: don't take it literally. So, there could be a species called 'hydrogen', or 'rabbit', or 'species of rabbit', and we could have a transition where one thing of this type comes in and two go out.)

I'm guessing that by a 'diploid' you mean a diploid species, not an individual diploid organism. After all, the transition 'extinction' makes sense for species, not individual organisms.

Hmm, maybe this interpretation makes sense throughout your diagram. If so, there's no problem. At first I thought that some transitions made sense only for individual organisms.

Comment Source:Cool! It would be great if stochastic Petri nets could help you. Are the 'things' of type 'haploid' or 'diploid' in this Petri net organisms, or species of organisms? Without this being clarified, things will get confusing. For example, here: <img width = "450" src = "http://math.ucr.edu/home/baez/networks/wolf-rabbit.png" alt = ""/> the transition 'birth' turns one rabbit into two rabbits, not one rabbit into two _species_ of rabbit. (Perhaps confusingly, in Part 8 I've taken to calling each type of thing a 'species'. But this is just the typical mathematicians' way of using words to mean whatever they want: don't take it literally. So, there could be a species called 'hydrogen', or 'rabbit', or 'species of rabbit', and we could have a transition where one thing of this type comes in and two go out.) I'm guessing that by a 'diploid' you mean a diploid _species_, not an individual diploid organism. After all, the transition 'extinction' makes sense for species, not individual organisms. Hmm, maybe this interpretation makes sense throughout your diagram. If so, there's no problem. At first I thought that some transitions made sense only for individual organisms.
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2.
edited September 2011

Are the 'things' of type 'haploid' or 'diploid' in this Petri net organisms, or species of organisms? Without this being clarified, things will get confusing.

Extinction applies to entire species: one species goes in, no species come out. In contrast, if some individuals of species A have hybrid offspring with individuals of species B, then we'll have three species: the tetraploid hybrids of species C, and the diploid descendants of those organisms in A and B which did not hybridise. As a reaction equation,

$$A + B \rightarrow A + B + C,$$ leading to a "stochastic Hamiltonian" term proportional to

$$a^\dagger b^\dagger c^\dagger a b - a^\dagger a b^\dagger b.$$ The "hybridisation" box has two arrows going in from "diploid", and three arrows going out, one to "tetraploid" and the others back to "diploid". I think that makes sense.

If we're just keeping track of the number of species which are diploid and tetraploid, we could use a pair of operators for each type:

$$[a_D, a_D^\dagger] = [a_T, a_T^\dagger] = 1,$$ and then write a stochastic Hamiltonian including terms like

$$a_D^\dagger a_D^\dagger a_T^\dagger a_D a_D - a_D^\dagger a_D a_D^\dagger a_D.$$

Comment Source:John Baez asked, > Are the 'things' of type 'haploid' or 'diploid' in this Petri net organisms, or species of organisms? Without this being clarified, things will get confusing. Extinction applies to entire species: one species goes in, no species come out. In contrast, if some individuals of species _A_ have hybrid offspring with individuals of species _B,_ then we'll have _three_ species: the tetraploid hybrids of species _C,_ and the diploid descendants of those organisms in _A_ and _B_ which did not hybridise. As a reaction equation, $$A + B \rightarrow A + B + C,$$ leading to a "stochastic Hamiltonian" term proportional to $$a^\dagger b^\dagger c^\dagger a b - a^\dagger a b^\dagger b.$$ The "hybridisation" box has two arrows going in from "diploid", and three arrows going out, one to "tetraploid" and the others back to "diploid". I think that makes sense. If we're just keeping track of the number of species which are diploid and tetraploid, we could use a pair of operators for each type: $$[a_D, a_D^\dagger] = [a_T, a_T^\dagger] = 1,$$ and then write a stochastic Hamiltonian including terms like $$a_D^\dagger a_D^\dagger a_T^\dagger a_D a_D - a_D^\dagger a_D a_D^\dagger a_D.$$
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3.

Yes, I mean diploid species when I say diploid. I understood the way you were using "species" in the blog. I will try to clarify the example without talking about "species of species".

Comment Source:Yes, I mean diploid species when I say diploid. I understood the way you were using "species" in the blog. I will try to clarify the example without talking about "species of species".
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4.
edited September 2011

This is slightly off the topic, but have you (Graham) read John's post on phylogenetic trees and operads? I think it'd be cool to try to figure out enough details to apply this to some existing data to see if any interesting biologically relevant meaning can be extracted. I know a bit about, but am certainly no expert in, phylogenetic analysis.

Comment Source:This is slightly off the topic, but have you (Graham) read John's post on [phylogenetic trees and operads](http://johncarlosbaez.wordpress.com/2011/07/06/operads-and-the-tree-of-life/)? I think it'd be cool to try to figure out enough details to apply this to some existing data to see if any interesting biologically relevant meaning can be extracted. I know a bit about, but am certainly no expert in, phylogenetic analysis.
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5.

It seems that a Petri net in which every transition has one input defines a branching process. I think this is roughly true, but I don't know how to make it precise.

Comment Source:Yes, I read John's post about operads, but wasn't able to relate it to anything I know much about. It seems that a Petri net in which every transition has one input defines a [branching process](http://en.wikipedia.org/wiki/Branching_process). I think this is roughly true, but I don't know how to make it precise.
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6.
edited September 2011

In this comment I'll use 'species' in the biological sense rather than as I defined in Part 8. I'll call the yellow circles in a stochastic Petri net 'states':

Graham's example may work fine using only species, not individuals of those species, as Blake outlined. But I might want to do biology in a way that involves individuals, as in population biology. Then we might want a framework a bit like stochastic Petri nets, but more general, having not a fixed set of states. but a set of states that can change with time, together with a certain number of things in each state, that can also change with time. The states here are species, the things in some state are individuals of that species. Individuals can be born and die, but also species can come into existence and go extinct.

Perhaps a simpler way to implement this idea is to take a fixed but enormous set of 'potential species' as our states. An 'actual' species would be a state with at least one thing in that state. At any time there would be lots of states that have no things in that state; these correspond to species with no individuals of that species - species that aren't actually there yet. This stays within the stochastic Petri net framework.

Comment Source:In this comment I'll use 'species' in the biological sense rather than as I defined in Part 8. I'll call the yellow circles in a stochastic Petri net 'states': <img width = "450" src = "http://math.ucr.edu/home/baez/networks/wolf-rabbit.png" alt = ""/> Graham's example may work fine using only species, not individuals of those species, as Blake outlined. But I might want to do biology in a way that involves individuals, as in population biology. Then we might want a framework a bit like stochastic Petri nets, but more general, having not a _fixed_ set of states. but a set of states that can change with time, together with a certain number of things in each state, that can also change with time. The states here are species, the things in some state are individuals of that species. Individuals can be born and die, but also species can come into existence and go extinct. Perhaps a simpler way to implement this idea is to take a fixed but enormous set of 'potential species' as our states. An 'actual' species would be a state with at least one thing in that state. At any time there would be lots of states that have no things in that state; these correspond to species with no individuals of that species - species that aren't actually there yet. This stays within the stochastic Petri net framework.
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7.

Anyway, despite my last comment, I don't actually want to make things more complicated than necessary! If Graham thinks some model is interesting, I'll be glad to think about it.

Comment Source:Anyway, despite my last comment, I don't actually want to make things more complicated than necessary! If Graham thinks some model is interesting, I'll be glad to think about it.
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8.

I have added some more context to Example of hybridization as a Petri net explaining why I am interested in this model.

In reply to John about population biology. Yes, you could make things more complicated by modelling individuals. However most of the available data is genetic. It is easier to sequence the entire genome of a frog than to count the number of frogs of that species. So below the level of species, people are mostly interested in modelling the evolution of genes. If you want things to be nice and complicated, you can imagine a gene regulatory network evolving along the branches of a species network, gaining and losing nodes and edges as it does so, while the species network is growing itself as prescribed by a Petri net. Of course, this is a ridiculous oversimplification of what really happens.

Comment Source:I have added some more context to [Example of hybridization as a Petri net](http://www.azimuthproject.org/azimuth/show/Example+of+hybridization+as+a+Petri+net) explaining why I am interested in this model. In reply to John about population biology. Yes, you could make things more complicated by modelling individuals. However most of the available data is genetic. It is easier to sequence the entire genome of a frog than to count the number of frogs of that species. So below the level of species, people are mostly interested in modelling the evolution of genes. If you want things to be nice and complicated, you can imagine a [gene regulatory network](http://en.wikipedia.org/wiki/Gene_regulatory_network) evolving along the branches of a species network, gaining and losing nodes and edges as it does so, while the species network is growing itself as prescribed by a Petri net. Of course, this is a ridiculous oversimplification of what really happens.
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9.

Comment Source:So this page doesn't get lost, I'll add a link to it from * [[Petri net]]