The Mathematics of Biodiversity (Part 3)

We tend to think of biodiversity as a good thing, but sometimes it’s deadly. Yesterday Andrei Korobeinikov gave a talk on ‘Viral evolution within a host’, which was mainly about AIDS.

The virus that causes this disease, HIV, can reproduce very fast. In an untreated patient near death there are between 1010 and 1012 new virions per day! Remember, a virion is an individual virus particle. The virus also has a high mutation rate: about 3 × 10-5 mutations per generation for each base—that is, each molecule of A,T,C, or G in the RNA of the virus. That may not seem like a lot, but if you multiply it by 1012 you’ll see that a huge number of new variations of each base arise within the body of a single patient.

So, evolution is at work within you as you die.

And in fact, many scientists believe that the diversity of the virus eventually overwhelms your immune system! Although it’s apparently not quite certain, it seems that while the body generates B cells and T cells to attack different variants of HIV as they arise, they eventually can’t keep up with the sheer number of variants.

Of course, the fact that the HIV virus attacks the immune system makes the disearse even worse. Here in blue you see the number of T cells per cubic millimeter of blood, and in red you see the number of virions per cubic centimeter of blood for a typical untreated patient:

Mathematicians and physicists have looked at some very simple models to get a qualitative understanding of these issues. One famous paper that started this off is:

• Lev S. Tsimring, Herbert Levine and David A. Kessler, RNA virus evolution via a fitness-space model, Phys. Rev. Lett. 76 (1996), 4440–4443.

The idea here is to say that at any time t the viruses have a probability density p(r,t) of having fitness r. In fact the different genotypes of the virus form a cloud in a higher-dimensional space, but these authors are treating that space is 1-dimensional, with fitness as its one coordinate, just to keep things simple. They then write down an equation for how the population density changes with time:

\displaystyle{\frac{\partial }{\partial t}p(r,t) = (r - \langle r \rangle)\, p(r,t) + D \frac{\partial^2 }{\partial r}p(r,t) - \frac{\partial}{\partial r}(v_{\mathrm{drift}}\, p(r,t)) }

This is a replication-mutation-drift equation. If we just had

\displaystyle{\frac{\partial }{\partial t}p(r,t) = (r - \langle r \rangle)\, p(r,t) }

this would be a version of the replicator equation, which I explained recently in Information Geometry (Part 9). Here

\displaystyle{ \langle r \rangle = \int_0^\infty r p(r,t) dr }

is the mean fitness, and the replicator equations says that the fraction of organisms of a given type grows at a rate proportional to how much their fitness exceeds the mean fitness: that’s where the (r - \langle r \rangle) comes from.

If we just had

\displaystyle{\frac{\partial }{\partial t}p(r,t) =  D \frac{\partial^2 }{\partial r^2}p(r,t) }

this would be the heat equation, which describes diffusion occurring at a rate D. This models the mutation of the virus, though not in a very realistic way.

If we just had

\displaystyle{\frac{\partial}{\partial t} p(r,t) =  - \frac{\partial}{\partial r}(v_{\mathrm{drift}} \, p(r,t)) }

the fitness of the virus would increase at rate equal to the drift velocity v_{\mathrm{drift}}.

If we include both the diffusion and drift terms:

\displaystyle{\frac{\partial }{\partial t} p(r,t) =  D \frac{\partial^2 }{\partial r^2}p(r,t) - \frac{\partial}{\partial r}(v_{\mathrm{drift}} \, p(r,t)) }

we get the Fokker–Planck equation. This is a famous model of something that’s spreading while also drifting along at a constant velocity: for example, a drop of ink in moving water. Its solutions look like this:

Here we start with stuff concentrated at one point, and it spreads out into a Gaussian while drifting along.

By the way, watch out: what biologists call ‘genetic drift’ is actually a form of diffusion, not what physicists call ‘drift’.

More recently, people have looked at another very simple model. You can read about it here:

• Martin A. Nowak, and R. M. May, Virus Dynamics, Oxford University Press, Oxford, 2000.

In this model the variables are:

• the number of healthy human cells of some type, \mathrm{H}(t)

• the number of infected human cells of that type, \mathrm{I}(t)

• the number of virions, \mathrm{V}(t)

These are my names for variables, not theirs. It’s just a sick joke that these letters spell out ‘HIV’.

Chemists like to describe how molecules react and turn into other molecules using ‘chemical reaction networks’. You’ve seen these if you’ve taken chemistry, but I’ve been explaining more about the math of these starting in Network Theory (Part 17). We can also use them here! Though May and Nowak probably didn’t put it this way, we can consider a chemical reaction network with the following 6 reactions:

• the production of a healthy cell:

\longrightarrow \mathrm{H}

• the infection of a healthy cell by a virion:

\mathrm{H} + \mathrm{V} \longrightarrow \mathrm{I}

• the production of a virion by an infected cell:

\mathrm{I} \longrightarrow \mathrm{I} + \mathrm{V}

• the death of a healthy cell:

\mathrm{H} \longrightarrow

• the death of a infected cell:

\mathrm{I} \longrightarrow

• the death of a virion:

\mathrm{V} \longrightarrow

Using a standard recipe which I explained, we can get from this chemical reaction network to some ‘rate equations’ saying how the number of healthy cells, infected cells and virions changes with time:

\displaystyle{ \frac{d\mathrm{H}}{dt}  =   \alpha - \beta \mathrm{H}\mathrm{V} - \gamma \mathrm{H} }

\displaystyle{ \frac{d\mathrm{I}}{dt}  = \beta \mathrm{H}\mathrm{V} - \delta \mathrm{I} }

\displaystyle{ \frac{d\mathrm{V}}{dt}  = - \beta \mathrm{H}\mathrm{V} + \epsilon \mathrm{I} - \zeta \mathrm{V} }

The Greek letters are constants called ‘rate constants’, and there’s one for each of the 6 reactions. The equations we get this way are exactly those described by Nowak and May!

What Andrei Korobeinikov is to unify the ideas behind the two models I’ve described here. Alas, I don’t have the energy to explain how. Indeed, I don’t even have the energy to explain what the models I’ve described actually predict. Sad, but true.

I don’t see anything online about Korobeinikov’s new work, but you can read some of his earlier work here:

• Andrei Korobeinikov, Global properties of basic virus dynamics models.

• Suzanne M. O’Regan, Thomas C. Kelly, Andrei Korobeinikov, Michael J. A. O’Callaghan and Alexei V. Pokrovskii, Lyapunov functions for SIR and SIRS epidemic models, Appl. Math. Lett. 23 (2010), 446-448.

The SIR and SIRS models are models of disease that also arise from chemical reaction networks. I explained them back in Network Theory (Part 3). That was before I introduced the terminology of chemical reaction networks… back then I was talking about ‘stochastic Petri nets’, which are an entirely equivalent formalism. Here’s the stochastic Petri net for the SIRS model:

Puzzle: Draw the stochastic Petri net for the HIV model discussed above. It should have 3 yellow circles and 6 aqua squares.

20 Responses to The Mathematics of Biodiversity (Part 3)

  1. Blake Stacey says:

    Are “squares” and “circles” interchanged in the last sentence? The HIV model has 3 time-varying population sizes and 6 reaction equations.

    • romain says:

      I have a solution to the puzzle to offer:

      BTW, I have a few question regarding the model:
      – When an infected cell dies, could the virion survive and infect another healthy cell?
      – Also, are virions sufficiently “clever” not to infect an already infected cell? (I think this issue is discussed in Marc Mangel’s book.)

      • John Baez says:

        Great picture, Romain! I like its almost symmetrical form. Only I can post pictures in comments here, so I added your picture to your comment. If you don’t mind, I may use it in my talk here in Barcelona.

        I don’t know enough about viruses to answer your questions. My guesses are ‘rarely’ and ‘no’. But I’ll ask someone here in Barcelona… or maybe someone reading the blog will know the answers.

        • romain says:

          Oops, I guess something’s wrong: the set of reactions and the set of equations do not match. According to the reaction \mathrm{H} + \mathrm{V} \longrightarrow \mathrm{I}
          then
          \displaystyle{ \frac{d\mathrm{V}}{dt}  = \epsilon \mathrm{I} - \zeta \mathrm{V} }
          should be
          \displaystyle{ \frac{d\mathrm{V}}{dt}  = \epsilon \mathrm{I} - \zeta \mathrm{V} } - \beta \mathrm{H}\mathrm{V}.

          I based my picture on the set of equations so it agrees with it but not with the set of reactions…

        • Mike Stay says:

          When a virion infects a cell, it injects its genetic material into the cell, leaving an empty case behind. The cell produces many more virions internally until it dies, releasing them. So that particular virion doesn’t survive.

          It’s certainly possible for a cell to get infected by multiple virions at the same time. If a cell gets infected by two different viruses at the same time, their genomes can mix in a process called “genetic reassortment”. This is apparently how chimpanzee SIV started: it’s a combination of genetic material from the red-capped mangabey SIV and the spot-nosed guenon SIV.

          Some time in the past million years, a chimp ate two of these smaller monkeys. Usually these viruses die off, since they’re evolved to spread in monkeys, not chimps. But in this chimp, one of the possible reassortments worked and spread among chimps.

          In around 1908, a Bantu hunter probably slaughtered an infected chimp and cut himself, acquiring HIV. Scientists know of at least twelve different spillover events where simian viruses have crossed in this way to humans; only one is responsible for AIDS.

          (I got most of this information from this Radiolab episode: http://www.radiolab.org/2011/nov/14/)

        • John Baez says:

          Mike wrote:

          When a virion infects a cell, it injects its genetic material into the cell, leaving an empty case behind. The cell produces many more virions internally until it dies, releasing them. So that particular virion doesn’t survive.

          From what little I know, that sounds like a good description of how many viruses work… but HIV is a retrovirus, so there are some extra nuances as summarized here:

          The virus uses reverse transcriptase to translate its RNA into DNA, and integrase to copy that into the cell’s genome. The cell then produces copies of the virion, which can leave the cell without bursting. I think some retroviruses can stay in the cell for quite a while without killing it.

          When integrated into the host’s germ line, retroviruses can also get passed on to the next generation of the host organism! These so-called ‘endogenous retroviruses’ make up 5-8% of the human genome!

        • John Baez says:

          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

          \mathrm{H} + \mathrm{V} \longrightarrow \mathrm{I}

          with rate constant \beta is properly taken into account:

          \displaystyle{ \frac{d\mathrm{H}}{dt}  =  \alpha - \beta \mathrm{H}\mathrm{V} - \gamma \mathrm{H} }

          \displaystyle{ \frac{d\mathrm{I}}{dt}  = \beta \mathrm{H}\mathrm{V} - \delta \mathrm{I} }

          \displaystyle{ \frac{d\mathrm{V}}{dt}  = - \beta \mathrm{H}\mathrm{V} + \epsilon \mathrm{I} - \zeta \mathrm{V} }

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

      • Arrow says:

        In general completed virions of many viruses can and do survive cell death, often this is their only way of escaping infected cells.

        HIV however leaves infected cell by budding from its surface and in the process acquires a portion of its lipid membrane which is required for infecting other cells. During cell death cell membrane bursts into fragments and the whole cell falls apart so while some viruses that were already almost fully assembled and anchored to the membrane might still manage to properly bud out others which were at earlier stages won’t make it.

        As for the second question in general infection does not prevent another virus particle from entering the same cell.

        • romain says:

          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:

          \mathrm{I} + \mathrm{V} \longrightarrow \mathrm{I}

          and then:

          \displaystyle{ \frac{d\mathrm{V}}{dt} = - \beta \mathrm{H}\mathrm{V} + \epsilon \mathrm{I} - \zeta \mathrm{V} - \theta \mathrm{V} \mathrm{I} }

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

        • John Baez says:

          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.

    • John Baez says:

      Blake wrote:

      Are “squares” and “circles” interchanged in the last sentence?

      Yes, I’ll fix that. My network theory posts use the convention that in Petri nets, aqua squares are ‘states’ (also called ‘species’ in chemistry) while yellow circles are ‘transitions’ (also called ‘reactions’ in chemistry). This color convention goes back to David Tweed, who drew the first pictures of this kind for Azimuth!

      The HIV model has 3 time-varying population sizes and 6 reaction equations.

      You probably meant 6 reactions; there are just 3 equations in this model. (Ever notice how small corrections tend to contain their own, smaller, mistakes?)

      • Blake Stacey says:

        I guess when I see “reaction equation” I tend to think “chemistry-looking thing with an arrow in it” whereas a formula for a d-something-by-dt is a “rate equation” or a “dynamical rule”. (Or, in a discrete-time model, an “update equation”.) I don’t know where I get my phrasing preferences from—probably a mishmash of all sorts of sources.

        Ever notice how small corrections tend to contain their own, smaller, mistakes?

        As long as the series converges…

        • John Baez says:

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

          \mathrm{H} + \mathrm{V} \longrightarrow \mathrm{I}

          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).

        • Blake Stacey says:

          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.

        • John Baez says:

          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.

        • Blake Stacey says:

          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.

  2. 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?

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

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