Comments on: The Mathematics of Biodiversity (Part 3)

By: Petri Net Programming « Azimuth

Petri Net Programming « Azimuth — Mon, 01 Oct 2012 18:12:58 +0000

Here is an example discussed earlier on Azimuth. It describes the virus that causes AIDS. The species are healthy cell, infected cell, and virion [...]

By: Epistasis between mutations is host-dependent for an RNA virus « Tracing Knowledge … Στα ίχνη της Γνώσης

Sat, 28 Jul 2012 07:25:00 +0000

[...] The Mathematics of Biodiversity (Part 3) (johncarlosbaez.wordpress.com) [...]

By: Linas Vepstas

Linas Vepstas — Thu, 12 Jul 2012 22:57:50 +0000

In the mid 1980′s, I skimmed a preprint from some russians, applying the renormalization group to genetic problems. I don’t remember any details, but It seemed very compelling at the time. A google search today yields lots of hits for “genetic dynamics” … perhaps a topic for future azimuth posts?

By: Blake Stacey

Blake Stacey — Tue, 03 Jul 2012 15:44:09 +0000

If I could upgrade to a less defective brain, that would be a great help.

Failing that, I’m trying to bring a couple colleagues of mine in on writing the “Invasion fitness” series. They might be able to fill some of the gaps currently in it.

By: John Baez

John Baez — Mon, 02 Jul 2012 07:42:49 +0000

I keep hoping we’ll publish that essay on the blog here! Anything I can do to catalyze that reaction? If you tell me it’s done, I’ll put it on the blog. If you want comments on it first, let me know.

By: Blake Stacey

Blake Stacey — Mon, 02 Jul 2012 04:19:50 +0000

what we’re secretly doing here is studying a symmetric monoidal category generated by some ‘species’ (objects) and ‘reactions’ (morphisms).

Which reminds me: I should one day get back to the line of thought I started at the end of this essay.

By: John Baez

John Baez — Sat, 30 Jun 2012 14:40:33 +0000

Oh—thanks, Romain! It’s funny how the beautiful symmetry of your original diagram distracted me from thinking about what it means and noticing that it was wrong. There’s probably a moral for me there.

I don’t know how large the fraction of infected cells (of some relevant type) becomes in the late stage of HIV. I think the extra feature of your more complicated Petri will only matter much if that becomes large. Someone told me it never becomes large, but I don’t know if that’s true.

By: romain

romain — Sat, 30 Jun 2012 12:59:06 +0000

Then the new Petri has one less arrow… (I should have seen there was no 4th-order reaction):

Too bad the net has lost some of its symmetry.

So, as mentioned by Mike and Arrow, we may also include the possibility for virions to infect already infected cells:

and then:

with equal to the former .
At late stages of the infection, this feature may save a small fraction of healthy cells by distracting virions by already infected cells…

By: John Baez

John Baez — Sat, 30 Jun 2012 08:48:21 +0000

Okay, I get it. I think Romain may also be using ‘reaction equation’ to mean something like

To me that’s not an ‘equation’, especially when we’re treating it as irreversible. It’s a ‘reaction’—or in my heart of hearts, a ‘morphism’, since what we’re secretly doing here is studying a symmetric monoidal category generated by some ‘species’ (objects) and ‘reactions’ (morphisms).

By: John Baez

John Baez — Sat, 30 Jun 2012 08:45:03 +0000

Romain wrote:

Oops, I guess something’s wrong: the set of reactions and the set of equations do not match.

I changed the differential equations so that the reaction

with rate constant is properly taken into account:

I believe everything is okay now… maybe you can check.