biology | Azimuth

Maximum Entropy and Ecology

21 February, 2013

I already talked about John Harte’s book on how to stop global warming. Since I’m trying to apply information theory and thermodynamics to ecology, I was also interested in this book of his:

• John Harte, Maximum Entropy and Ecology, Oxford U. Press, Oxford, 2011.

There’s a lot in this book, and I haven’t absorbed it all, but let me try to briefly summarize his maximum entropy theory of ecology. This aims to be “a comprehensive, parsimonious, and testable theory of the distribution, abundance, and energetics of species across spatial scales”. One great thing is that he makes quantitative predictions using this theory and compares them to a lot of real-world data. But let me just tell you about the theory.

It’s heavily based on the principle of maximum entropy (MaxEnt for short), and there are two parts:

Two MaxEnt calculations are at the core of the theory: the first yields all the metrics that describe abundance and energy distributions, and the second describes the spatial scaling properties of species’ distributions.

Abundance and energy distributions

The first part of Harte’s theory is all about a conditional probability distribution

$R(n,\epsilon | S_0, N_0, E_0)$

which he calls the ecosystem structure function. Here:

• $S_0$ : the total number of species under consideration in some area.

• $N_0$ : the total number of individuals under consideration in that area.

• $E_0$ : the total rate of metabolic energy consumption of all these individuals.

Given this,

$R(n,\epsilon | S_0, N_0, E_0) \, d \epsilon$

is the probability that given $S_0, N_0, E_0,$ if a species is picked from the collection of species, then it has $n$ individuals, and if an individual is picked at random from that species, then its rate of metabolic energy consumption is in the interval $(\epsilon, \epsilon + d \epsilon).$

Here of course $d \epsilon$ is ‘infinitesimal’, meaning that we take a limit where it goes to zero to make this idea precise (if we’re doing analytical work) or take it to be very small (if we’re estimating $R$ from data).

I believe that when we ‘pick a species’ we’re treating them all as equally probable, not weighting them according to their number of individuals.

Clearly $R$ obeys some constraints. First, since it’s a probability distribution, it obeys the normalization condition:

$\displaystyle{ \sum_n \int d \epsilon \; R(n,\epsilon | S_0, N_0, E_0) = 1 }$

Second, since the average number of individuals per species is $N_0/S_0,$ we have:

$\displaystyle{ \sum_n \int d \epsilon \; n R(n,\epsilon | S_0, N_0, E_0) = N_0 / S_0 }$

Third, since the average over species of the total rate of metabolic energy consumption of individuals within the species is $E_0/ S_0,$ we have:

$\displaystyle{ \sum_n \int d \epsilon \; n \epsilon R(n,\epsilon | S_0, N_0, E_0) = E_0 / S_0 }$

Harte’s theory is that $R$ maximizes entropy subject to these three constraints. Here entropy is defined by

$\displaystyle{ - \sum_n \int d \epsilon \; R(n,\epsilon | S_0, N_0, E_0) \ln(R(n,\epsilon | S_0, N_0, E_0)) }$

Harte uses this theory to calculate $R,$ and tests the results against data from about 20 ecosystems. For example, he predicts the abundance of species as a function of their rank, with rank 1 being the most abundant, rank 2 being the second most abundant, and so on. And he gets results like this:

The data here are from:

• Green, Harte, and Ostling’s work on a serpentine grassland,

• Luquillo’s work on a 10.24-hectare tropical forest, and

• Cocoli’s work on a 2-hectare wet tropical forest.

The fit looks good to me… but I should emphasize that I haven’t had time to study these matters in detail. For more, you can read this paper, at least if your institution subscribes to this journal:

• J. Harte, T. Zillio, E. Conlisk and A. Smith, Maximum entropy and the state-variable approach to macroecology, Ecology 89 (2008), 2700–2711.

Spatial abundance distribution

The second part of Harte’s theory is all about a conditional probability distribution

$\Pi(n | A, n_0, A_0)$

This is the probability that $n$ individuals of a species are found in a region of area $A$ given that it has $n_0$ individuals in a larger region of area $A_0.$

$\Pi$ obeys two constraints. First, since it’s a probability distribution, it obeys the normalization condition:

$\displaystyle{ \sum_n \Pi(n | A, n_0, A_0) = 1 }$

Second, since the mean value of $n$ across regions of area $A$ equals $n_0 A/A_0,$ we have

$\displaystyle{ \sum_n n \Pi(n | A, n_0, A_0) = n_0 A/A_0 }$

Harte’s theory is that $\Pi$ maximizes entropy subject to these two constraints. Here entropy is defined by

$\displaystyle{- \sum_n \Pi(n | A, n_0, A_0)\ln(\Pi(n | A, n_0, A_0)) }$

Harte explains two approaches to use this idea to derive ‘scaling laws’ for how $n$ varies with $n$ . And again, he compares his predictions to real-world data, and get results that look good to my (amateur, hasty) eye!

I hope sometime I can dig deeper into this subject. Do you have any ideas, or knowledge about this stuff?

26 Comments | biodiversity, biology, information and entropy | Permalink
Posted by John Baez

Prospects for a Green Mathematics

15 February, 2013

contribution to the Mathematics of Planet Earth 2013 blog by John Baez and David Tanzer

It is increasingly clear that we are initiating a sequence of dramatic events across our planet. They include habitat loss, an increased rate of extinction, global warming, the melting of ice caps and permafrost, an increase in extreme weather events, gradually rising sea levels, ocean acidification, the spread of oceanic “dead zones”, a depletion of natural resources, and ensuing social strife.

These events are all connected. They come from a way of life that views the Earth as essentially infinite, human civilization as a negligible perturbation, and exponential economic growth as a permanent condition. Deep changes will occur as these idealizations bring us crashing into the brick wall of reality. If we do not muster the will to act before things get significantly worse, we will need to do so later. While we may plead that it is “too difficult” or “too late”, this doesn’t matter: a transformation is inevitable. All we can do is start where we find ourselves, and begin adapting to life on a finite-sized planet.

Where does math fit into all this? While the problems we face have deep roots, major transformations in society have always caused and been helped along by revolutions in mathematics. Starting near the end of the last ice age, the Agricultural Revolution eventually led to the birth of written numerals and geometry. Centuries later, the Enlightenment and Industrial Revolution brought us calculus and eventually a flowering of mathematics unlike any before. Now, as the 21st century unfolds, mathematics will become increasingly driven by our need to understand the biosphere and our role within it.

We refer to mathematics suitable for understanding the biosphere as green mathematics. Although it is just being born, we can already see some of its outlines.

Since the biosphere is a massive network of interconnected elements, we expect network theory will play an important role in green mathematics. Network theory is a sprawling field, just beginning to become organized, which combines ideas from graph theory, probability theory, biology, ecology, sociology and more. Computation plays an important role here, both because it has a network structure—think of networks of logic gates—and because it provides the means for simulating networks.

One application of network theory is to tipping points, where a system abruptly passes from one regime to another. Scientists need to identify nearby tipping points in the biosphere to help policy makers to head off catastrophic changes. Mathematicians, in turn, are challenged to develop techniques for detecting incipient tipping points. Another application of network theory is the study of shocks and resilience. When can a network recover from a major blow to one of its subsystems?

We claim that network theory is not just another name for biology, ecology, or any other existing science, because in it we can see new mathematical terrains. Here are two examples.

First, consider a leaf. In The Formation of a Tree Leaf by Qinglan Xia, we see a possible key to Nature’s algorithm for the growth of leaf veins. The vein system, which is a transport network for nutrients and other substances, is modeled by Xia as a directed graph with nodes for cells and edges for the “pipes” that connect the cells. Each cell gives a revenue of energy, and incurs a cost for transporting substances to and from it.

The total transport cost depends on the network structure. There are costs for each of the pipes, and costs for turning the fluid around the bends. For each pipe, the cost is proportional to the product of its length, its cross-sectional area raised to a power α, and the number of leaf cells that it feeds. The exponent α captures the savings from using a thicker pipe to transport materials together. Another parameter β expresses the turning cost.

Development proceeds through cycles of growth and network optimization. During growth, a layer of cells gets added, containing each potential cell with a revenue that would exceed its cost. During optimization, the graph is adjusted to find a local cost minimum. Remarkably, by varying α and β, simulations yield leaves resembling those of specific plants, such as maple or mulberry.

A growing network

Unlike approaches that merely create pretty images resembling leaves, Xia presents an algorithmic model, simplified yet illuminating, of how leaves actually develop. It is a network-theoretic approach to a biological subject, and it is mathematics—replete with lemmas, theorems and algorithms—from start to finish.

A second example comes from stochastic Petri nets, which are a model for networks of reactions. In a stochastic Petri net, entities are designated by “tokens” and entity types by “places” which hold the tokens. “Reactions” remove tokens from their input places and deposit tokens at their output places. The reactions fire probabilistically, in a Markov chain where each reaction rate depends on the number of its input tokens.

A stochastic Petri net

Perhaps surprisingly, many techniques from quantum field theory are transferable to stochastic Petri nets. The key is to represent stochastic states by power series. Monomials represent pure states, which have a definite number of tokens at each place. Each variable in the monomial stands for a place, and its exponent indicates the token count. In a linear combination of monomials, each coefficient represents the probability of being in the associated state.

In quantum field theory, states are representable by power series with complex coefficients. The annihilation and creation of particles are cast as operators on power series. These same operators, when applied to the stochastic states of a Petri net, describe the annihilation and creation of tokens. Remarkably, the commutation relations between annihilation and creation operators, which are often viewed as a hallmark of quantum theory, make perfect sense in this classical probabilistic context.

Each stochastic Petri net has a “Hamiltonian” which gives its probabilistic law of motion. It is built from the annihilation and creation operators. Using this, one can prove many theorems about reaction networks, already known to chemists, in a compact and elegant way. See the Azimuth network theory series for details.

Conclusion: The life of a network, and the networks of life, are brimming with mathematical content.

We are pursuing these subjects in the Azimuth Project, an open collaboration between mathematicians, scientists, engineers and programmers trying to help save the planet. On the Azimuth Wiki and Azimuth Blog we are trying to explain the main environmental and energy problems the world faces today. We are also studying plans of action, network theory, climate cycles, the programming of climate models, and more.

If you would like to help, we need you and your special expertise. You can write articles, contribute information, pose questions, fill in details, write software, help with research, help with writing, and more. Just drop us a line.

This post appeared on the blog for Mathematics of Planet Earth 2013, an international project involving over 100 scientific societies, universities, research institutes, and organizations. They’re trying to have a new blog article every day, and you can submit articles as described here.

Here are a few of their other articles:

• The mathematics of extreme climatic events—with links to videos.

• From the Joint Mathematics Meetings: Conceptual climate models short course—with links to online course materials.

• There will always be a Gulf Stream—and exercise in singular perturbation technique.

13 Comments | azimuth, biology, mathematics, strategies | Permalink
Posted by John Baez

Graduate Program in Biostatistics

7 November, 2012

Are you an undergrad who likes math and biology and wants a good grad program? This one sounds really interesting. The ad I bumped into is focused on minority applicants, maybe because U.C. Riverside is packed with students whose skin ain’t pale. But I’d say biostatistics is a good career even if you have the misfortune of needing high-SPF sunscreen:

The Department of Biostatistics, which administers PhD training at the Harvard School of Public Health, seeks outstanding minority applicants for its graduate programs in Biostatistics.

Biostatistics is an excellent career choice for students interested in mathematics applied to real world problems. The current data explosion is contributing to the rising stature of, and demand for biostatisticians, as noted in the New York Times:

I keep saying that the sexy job in the next 10 years will be statisticians … and I’m not kidding.

To date, Biostatistics has not been successful in attracting qualified minority students, particularly African Americans. Students best suited for careers in Biostatistics are those with strong mathematical abilities, combined with interests in health and biology. Unfortunately, statistics is not widely taught at the undergraduate level, and many potentially excellent candidates simply do not learn about the possibility of a valuable and fulfilling career in Biostatistics. Many minority students who could thrive in a Biostatistics program choose instead to enter medical school. Public health in general, and Biostatistics in particular, are not even considered as options. We would like your help in identifying qualified students before they make their choices regarding graduate school or other career paths.

All doctoral students accepted in our department are guaranteed full tuition and stipend support throughout their program, as long as they are making satisfactory progress towards the PhD degree. Every effort is made to meet the individual needs of each student, and to insure the successful completion of graduate work.

The web site for prospective students is here.

Please note the deadline for submitting applications to the MA and PhD programs for entry in the fall of 2013 is December 15, 2012.

We look forward to answering any questions you may have. Questions about our graduate programs can be directed to Jelena Follweiller, at jtillots@hsph.harvard.edu.

Leave a Comment » | biology, jobs | Permalink
Posted by John Baez

Azimuth News (Part 2)

28 September, 2012

Last week I finished a draft of a book and left Singapore, returning to my home in Riverside, California. It’s strange and interesting, leaving the humid tropics for the dry chaparral landscape I know so well.

Now I’m back to my former life as a math professor at the University of California. I’ll be going back to the Centre for Quantum Technology next summer, and summers after that, too. But life feels different now: a 2-year period of no teaching allowed me to change my research direction, but now it’s time to teach people what I’ve learned!

It also happens to be a time when the Azimuth Project is about to do a lot of interesting things. So, let me tell you some news!

Programming with Petri nets

The Azimuth Project has a bunch of new members, who are bringing with them new expertise and lots of energy. One of them is David Tanzer, who was an undergraduate math major at U. Penn, and got a Ph.D. in computer science at NYU. Now he’s a software developer, and he lives in Brooklyn, New York.

He writes:

My areas of interest include:

• Queryable encyclopedias

• Machine representation of scientific theories

• Machine representation of conflicts between contending theories

• Social and technical structures to support group problem-solving activities

• Balkan music, Afro-Latin rhythms, and jazz guitar

To me, the most meaningful applications of science are to the myriad of problems that beset the human race. So the Aziumuth Project is a good focal point for me.

And on Azimuth, he’s starting to write some articles on ‘programming with Petri nets’. We’ve talked about them a lot in the network theory series:

They’re a very general modelling tool in chemistry, biology and computer science, precisely the sort of tool we need for a deep understanding of the complex systems that keep our living planet going—though, let’s be perfectly clear about this, just one of many such tools, and one of the simplest. But as mathematical physicists, Jacob Biamonte and I have studied Petri nets in a highly theoretical way, somewhat neglecting the all-important problem of how you write programs that simulate Petri nets!

Such programs are commercially available, but it’s good to see how to write them yourself, and that’s what David Tanzer will tell us. He’ll use the language Python to write these programs in a nice modern object-oriented way. So, if you like coding, this is where the rubber meets the road.

I’m no expert on programming, but it seems the modularity of Python code nicely matches the modularity of Petri nets. This is something I’d like to get into more deeply someday, in my own effete theoretical way. I think the category-theoretic foundations of computer languages like Python are worth understanding, perhaps more interesting in fact than purely functional languages like Haskell, which are better understood. And I think they’ll turn out to be nicely related to the category-theoretic foundations of Petri nets and other networks I’m going to tell you about!

And I believe this will be important if we want to develop ‘ecotechnology’, where our machines and even our programming methodologies borrow ingenuity and wisdom from biological processes… and learn to blend with nature instead of fighting it.

Petri nets, systems biology, and beyond

Another new member of the Azimuth Project is Ken Webb. He has a BA in Cognitive Science from Carleton University in Ottawa, and an MSc in Evolutionary and Adaptive Systems from The University of Sussex in Brighton. Since then he’s worked for many years as a software developer and consultant, using many different languages and approaches.

He writes:

Things that I’m interested in include:

• networks of all types, hierarchical organization of network nodes, and practical applications

• climate change, and “saving the planet”

• programming code that anyone can run in their browser, and that anyone can edit and extend in their browser

• approaches to software development that allow independently-developed apps to work together

• the relationship between computer-science object-oriented (OO) concepts and math concepts

• how everything is connected

I’ve been paying attention to the Azimuth Project because it parallels my own interests, but with a more math focus (math is not one of my strong points). As learning exercises, I’ve reimplemented a few of the applications mentioned on Azimuth pages. Some of my online workbooks (blog-like entries that are my way of taking active notes) were based on content at the Azimuth Project.

He’s started building a Petri net modeling and simulation tool called Xholon. It’s written in Java and can be run online using Java Web Start (JNLP). Using this tool you can completely specify Petri net models using XML. You can see more details, and examples, on his Azimuth page. If I were smarter, or had more spare time, I would have already figured out how to include examples that actually run in an interactive way in blog articles here! But more on that later.

Soon I hope Ken will finish a blog entry in which he discusses how Petri nets fit into a bigger setup that can also describe ‘containers’, where molecules are held in ‘membranes’ and these membranes can allow chosen molecules through, and also split or merge—more like biology than inorganic chemistry. His outline is very ambitious:

This tutorial works through one simple example to demonstrate the commonality/continuity between a large number of different ways that people use to understand the structure and behavior of the world around us. These include chemical reaction networks, Petri nets, differential equations, agent-based modeling, mind maps, membrane computing, Unified Modeling Language, Systems Biology Markup Language, and Systems Biology Graphical Notation. The intended audience includes scientists, engineers, programmers, and other technically literate nonexperts. No math knowledge is required.

The Azimuth Server

With help from Glyn Adgie and Allan Erskine, Jim Stuttard has been setting up a server for Azimuth. All these folks are programmers, and Jim Stuttard, in particular, was a systems consultant and software applications programmer in C, C++ and Java until 2001. But he’s really interested in formal methods, and now he programs in Haskell.

I won’t say anything about the Azimuth server, since I’ll get it wrong, it’s not quite ready yet, and Jim wisely prefers to get it working a bit more before he talks about it. But you can get a feeling for what’s coming by going here.

How to find out more

You can follow what we’re doing by visiting the Azimuth Forum. Most of our conversations there are open to the world, but some can only be seen if you become a member. This is easy to do, except for one little thing.

Nobody, nobody , seems capable of reading the directions where I say, in boldface for easy visibility:

Use your whole real name as username. Spaces and capital letters are good. So, for example, a username like ‘Tim van Beek’ is good, ‘timvanbeek’ not so good, and ‘Tim’ or ‘tvb’ won’t be allowed.

The main point is that we want people involved with the Azimuth Project to have clear identities. The second, more minor point is that our software is not braindead, so you can choose a username that’s your actual name, like

Tim van Beek

instead of having to choose something silly like

timvanbeek

tim_van_beek

But never mind me: I’m just a crotchety old curmudgeon. Come join the fun and help us save the planet by developing software that explains climate science, biology, and ecology—and, just maybe, speeds up the development of green mathematics and ecotechnology!

5 Comments | azimuth, biology, chemistry, networks | Permalink
Posted by John Baez

Carbon Cycle Box Models

24 July, 2012

guest post by Staffan Liljegren

What?

I think the carbon cycle must be the greatest natural invention, all things considered. It’s been the basis for all organic life on Earth through eons of time. Through evolution, it gradually creates more and more biodiversity. It is important to do more research on the carbon cycle for the earth sciences, biology and in particular global warming—or more generally, climate science and environmental science, which are among the foci of the Azimuth project.

It is a beautiful and complex nonlinear geochemical cycle, I decided to give a rough outline of its beauty and complexity. Plants eat water and carbon dioxide with help from the sun (photosynthesis) and while doing so they produce air and sugar for others to metabolize. These plants in turn may be eaten by vegan animals (herbivores), while animals may also be eaten by other animals like us humans, being meat eaters or animals that eat both animals and plants (carnivores or omnivores).

Here is an overview of the cycle, where yellow arrows show release of carbon dioxide and purple arrows show uptake:

Say a plant gets eaten by an animal on land. Then the animal can use its carbon while breathing in air and breathing out water and carbon dioxide. Ruminant animals like cows and sheep also produce methane, which is a greenhouse gas like carbon dioxide. When a plant or animal dies it gets eaten by others, and any remains go down into the soil and sediments. A lot of the carbon in the sediments actually transforms into carbonate rock. This happens over millions of years. Some of this carbon makes it back into the air later through volcanoes.

Where?

Carbon is not a very abundant element on this planet: it’s only 0.08% of the total mass of the Earth. Nonetheless, we all know that many products of this atom are found throughout nature: for example in diamonds, marble, oil… and living organisms. If you remember your high school chemistry you might recall that the lab experiments with organic chemistry were the fun part of chemistry! The reason is that carbon has the ability to easily form compounds with other elements. So there is a tremendous global market that depends on the carbon cycle.

We humans are one fifth carbon. Other examples are trees, which we humans use for many things in our economic growth. But there are also fascinating flows inside the trees. I’ve read about these in Colin Tudge’s book The Secret Life of Trees – How They Live and why they Matter, so I will use this book for examples about forests and trees. You may already be familiar with these, but maybe not know a lot of details about their part in the carbon cycle.

When I stood in front of an tall monkey-puzzle tree in the genus Auracaria I was just flabbergasted by its age, and how it used to be widespread when the dinosaurs where around. But how does it manage to get the water to its leaves? Colin Tudge writes that during evolution trees invented stem-cell usage to grow the new outer layer, and developed microtechnology before we even existed as a species, where the leaves pull on several micron sized channels through osmosis and respiration to get the water up through the roots and trunk to the leaves at speeds typically around 6 meters per hour. But if needed, they can crank it up to 40 meters per hour to get it to the top in an hour or two!

Why?

Global warming is a fact and there are several remote sensing technologies that have confirmed this. You can see it nicely by clicking on this—you should see a NASA animation of satellite measurements superposed on top of Keeling’s famous graph of CO₂ measured at Mauna Loa measurements from 2002 to 2009. Here’s more of that graph:

Many of the greenhouse gases that contribute to increasing temperature contains carbon: carbon dioxide, methane and carbon monoxide. I will focus on carbon dioxide. Its behavior is vastly different in air or water. In air it doesn’t react with other chemicals so its stays around for a longer time in the atmosphere. In the ocean and on land the carbon dioxide reacts a lot more, so there’s an uptake of carbon in both. But not in the ocean where it stays a lot longer mainly due to ocean buffering. I will have a lot more to say about the ocean geochemistry in the upcoming blog postings.

The carbon dioxide levels in the atmosphere in 2011 are soon approaching 400 parts per million (ppm) and the growth is increasing for every year. The parts per million is in relation to the volume of the atmosphere. David Archer says that if all the carbon dioxide were to fall as frozen carbon dioxide—’dry ice’—it would just be around 10 centimeters deep. But the important thing to understand is that we have thrown the carbon cycle seriously out of balance with our human emissions, so we might be close to some climate tipping points.

Colin my fellow ‘tree-hugger’ has looked at global warming and its implication for the trees. Intuitively it might seem that warmer temperatures and higher levels of CO₂ might be beneficial for their growth. Indeed, the climate predictions of the International Panel on Climate Change assume this will happen. But there is a point where the micro-channels (stomata) start to close, due to too much photosynthesis and carbon dioxide. Taken together with higher temperature, this can make the trees’ respiration faster than its photosynthesis, so they end up supplying more carbon dioxide to atmosphere.

Trees also are very excellent at preventing floods, since one tree can divert 500 litres per day through transpiration. This easily adds up to 5000 cubic metres per square kilometre, making trees very good at reducing flood and and reducing our need for disaster preventions if they are left alone to do do their job.

How?

One way of understanding how the carbon cycle works is to use simple models like box models where we treat the carbon as contained in various ‘boxes’ and look at how it moves between boxes as time passes. A box can represent the Earth, the ocean, the atmosphere, or depending on what I want to study, any other part of the carbon cycle.

I’ll just mention a few examples of flows in the carbon cycle, to give you a feeling for them: breathing, photosynthesis, erosion, emission and decay. Breathing is easy to grasp—try to stop doing it yourself for a short moment! But how is photosynthesis a flow? This wonderful process was invented by the cyanobacteria 3.5 billion years ago and it has been used by plants ever since. It takes carbon out of the atmosphere and moves it into plant tissues.

In a box model, the average time something stays in a box is called its residence time, e-folding time, or response time by scientists. The rest of the flows in my list I leave up to you to think about: which are uptakes which are releases, and where do they occur?

The basic equation in a box model is called the mass balance equation:

$\dot m = \sum \textrm{sources} - \sum \textrm{sinks}$

Here $m$ is the mass of some substance in some box. The sources are what flows into that box together with any internal sources (production). The sinks are what flows out together with any internal sinks (loss and deposition).

In my initial experiments where I used the year 2008, when I looked at a 1-dimensional global box model of CO₂ in the atmosphere with only the fossil fuel as source, I get similar results to this diagram from the Global Carbon Project (petagram of carbon per year, which is the same as gigatonnes per year):

$global carbon budget 2000 - 2010$

I used the observed value from measurements at Mauna Loa. The atmosphere sink is 3.9 gigatonnes of carbon per year and the fossil fuel emission source is 8.7 GtC per year. The ocean also absorbs 2.1 GtC per year, and the land acts as a sink at 2.5 GtC per year.

I hope this will be the first of a series of posts! Next time I want to talk about a box model for the ocean’s role in the carbon cycle.

References

• Colin Tudge, The Secret Life of Trees: How They Live and Why They Matter, Penguin, London, 2005.

• David Archer, The Global Carbon Cycle, Princeton U. Press, Princeton, NJ, 2011.

19 Comments | biology, carbon emissions | Permalink
Posted by John Baez

Disease-Spreading Zombies

20 July, 2012

Are you a disease-spreading zombie?

You may have read about the fungus that can infect an ant and turn it into a zombie, making it climb up the stem of a plant and hang onto it, then die and release spores from a stalk that grows out of its head.

But this isn’t the only parasite that controls the behavior of its host.

If you ever got sick, had diarrhea, and thought hard about why, you’ll understand what I mean. You were helping spread the disease… especially if you were poor and didn’t have a toilet. This is why improved sanitation actually reduces the virulence of some diseases: it’s no longer such a good strategy for bacteria to cause diarrhea, so they evolve away from it!

There are plenty of other examples. Lots of diseases make you sneeze or cough, spreading the germs to other people. The rabies virus drives dogs crazy and makes them want to bite. There’s a parasitic flatworm that makes ants want to climb to the top of a blade of grass, lock their jaws onto it and wait there until they get eaten by a sheep! But the protozoan Toxoplasma gondii is more mysterious.

It causes a disease called toxoplasmosis. You can get it from cats, you can get it from eating infected meat, and you can even inherit it from your mother.

Lots of people have it: somewhere between 1/3 and 1/2 of everyone in the world!

A while back, the Czech scientist Jaroslav Flegr did some experiments. He found that people who tested positive for this parasite have slower reaction times. But even more interestingly, he claims that men with the parasite are more introverted, suspicious, oblivious to other people’s opinions of them, and inclined to disregard rules… while infected women, are more outgoing, trusting, image-conscious, and rule-abiding than uninfected women!

What could explain this?

The disease is carried by both cats and mice. Cats catch it by eating mice. The disease causes behavior changes in mice: they seem to become more anxious and run around more. This may increase their chance of getting eaten by a cat and passing on the disease. But we are genetically similar to mice… so we too may become more anxious when we’re infected with this disease. And men and women may act differently when they’re anxious.

It’s just a theory so far. Nonetheless, I won’t be surprised to hear there are parasites that affect our behavior in subtle ways. I don’t know if viruses or bacteria are sophisticated enough to trigger changes in behavior more subtle than diarrhea… but there are always lots of bacteria in your body, about 10 times as many as actual human cells. Many of these belong to unidentified species. And as long as they don’t cause obvious pathologies, doctors have had little reason to study them.

As for viruses, don’t forget that about 8% of your DNA is made of viruses that once copied themselves into your ancestors’ genome. They’re called endogenous retroviruses, and I find them very spooky and fascinating. Once they get embedded in our DNA, they can’t always get back out: a lot of them are defective, containing deletions or nonsense mutations. But some may still be able to get back out. And there are hints that some are implicated in certain kinds of cancer and autoimmune disease.

Even more intriguingly, a 2004 study reported that antibodies to endogenous retroviruses were more common in people with schizophrenia! And the cerebrospinal fluid of people who’d recently gotten schizophrenia contained levels of a key enzyme used by retroviruses, reverse transcriptase, four times higher than control subjects.

So it’s possible—just possible—that some viruses, either free-living or built into our DNA, may change our behavior in subtle ways that increase their chance of spreading.

For more on Jaroslav Flegr’s research, read this fascinating article:

• Kathleen MacAuliffe, How your cat is making you crazy, The Atlantic, March 2012.

Among other things you’ll read about the parasitologists
Glenn McConkey and Joanne Webster, who have shown that Toxoplasma gondii has two genes that allow it to crank up production of the neurotransmitter dopamine in the host’s brain. It seems this makes rats feel pleasure when they smell a cat!

(Do you like cats? Hmm.)

Of course, in business and politics we see many examples of ‘parasites’ that hijack organizations and change these organizations’ behavior to benefit themselves. It’s not nice. But it’s natural.

So even if you aren’t a disease-spreading zombie, it’s quite possible you’re dealing with them on a regular basis.

6 Comments | biology, risks | Permalink
Posted by John Baez

The Mathematics of Biodiversity (Part 8)

14 July, 2012

Last time I mentioned that estimating entropy from real-world data is important not just for measuring biodiversity, but also for another area of biology: neurobiology!

When you look at something, neurons in your eye start firing. But how, exactly, is their firing related to what you see? Questions like this are hard! Answering them— ‘cracking the neural code’—is a big challenge. To make progress, neuroscientists are using information theory. But as I explained last time, estimating information from experimental data is tricky.

Romain Brasselet, now a postdoc at the Max Planck Institute for Biological Cybernetics at Tübingen, is working on these topics. He sent me a nice email explaining this area.

This is a bit of a digression, but the Mathematics of Biodiversity program in Barcelona has been extraordinarily multidisciplinary, with category theorists rubbing shoulders with ecologists, immunologists and geneticists. One of the common themes is entropy and its role in biology, so I think it’s worth posting Romain’s comments here. This is what he has to say…

Information in neurobiology

I will try to explain why neurobiologists are today very interested in reliable estimates of entropy/information and what are the techniques we use to obtain them.

The activity of sensory as well as more central neurons is known to be modulated by external stimulations. In 1926, in a seminal paper, Adrian observed that neurons in the sciatic nerve of the frog fire action potentials (or spikes) when some muscle in the hindlimb is stretched. In addition, he observed that the frequency of the spikes increases with the amplitude of the stretching.

• E.D. Adrian, The impulses produced by sensory nerve endings. (1926).

For another very nice example, in 1962, Hubel and Wiesel found neurons in the cat visual cortex whose activity depends on the orientation of a visual stimulus, a simple black line over white background: some neurons fire preferentially for one orientation of the line (Hubel and Wiesel were awarded the 1981 Nobel Prize in Physiology for their work). This incidentally led to the concept of “receptive field” which is of tremendous importance in neurobiology—but though it’s fascinating, it’s a different topic.

Good, we are now able to define what makes a neuron tick. The problem is that neural activity is often very “noisy”: when the exact same stimulus is presented many times, the responses appear to be very different from trial to trial. Even careful observation cannot necessarily reveal correlations between the stimulations and the neural activity. So we would like a measure capable of capturing the statistical dependencies between the stimulation and the response of the neuron to know if we can say something about the stimulation just by observing the response of a neuron, which is essentially the task of the brain. In particular, we want a fundamental measure that does not rely on any assumption about the functioning of the brain. Information theory provides the tools to do this, that is why we like to use it: we often try to measure the mutual information between stimuli and responses.

To my knowledge, the first paper using information theory in neuroscience was by MacKay and McCulloch in 1952:

• Donald M. Mackay and Warren S. McCulloch, The limiting information capacity of a neuronal link, Bulletin of Mathematical Biophysics 14 (1952), 127–135.

But information theory was not used in neuroscience much until the early 90′s. It started again with a paper by Bialek et al. in 1991:

• W. Bialek, F. Rieke, R. R. de Ruyter van Steveninck and D. Warland, Reading a neural code, Science 252 (1991), 1854–1857.

However, when applying information-theoretic methods to biological data, we often have a limited sampling of the neural response, we are usually very happy when we have 50 trials for a given stimulus. Why is this limited sample a problem?

During the major part of the 20th century, following Adrian’s finding, the paradigm for the neural code was the frequency of the spikes or, equivalently, the number of spikes in a window of time. But in the early 90′s, it was observed that the exact timing of spikes is (in some cases) reliable across trials. So instead of considering the neural response as a single number (the number of spikes), the temporal patterns of spikes started to be taken into account. But time is continuous, so to be able to do actual computations, time was discretized and a neural response became a binary string.

Now, if you consider relevant time-scales, say, a 100 millisecond time window with a 1 millisecond bin with a firing frequency of about 50 per second, then your response space is huge and the estimates of information with only 50 trials are not reliable anymore. That’s why a lot of efforts have been carried to overcome the limited sampling bias.

Now, getting at the techniques developed in this field, John already mentioned the work by Liam Paninski, but here are other very interesting references:

• Stefano Panzeri and Alessandro Treves, Analytical estimates of limited sampling biases in different information measures, Network: Computation in Neural Systems 7 (1996), 87–107.

They computed the first-order bias of the information (related to the Miller–Madow correction) and then used a Bayesian technique to estimate the number of responses not included in the sample but that would be in an infinite sample (a goal similar to that of Good’s rule of thumb).

• S.P. Strong, R. Koberle, R.R. de Ruyter van Steveninck, and W. Bialek, Entropy and information in neural spike trains, Phys. Rev. Lett. 80 (1998), 197–200.

The entropy (or if you prefer, information) estimate can be expanded in a power series in $N$ (the sample size) around the true value. By computing the estimate for various values of $N$ and fitting it with a parabola, it is possible to estimate the value of the entropy as $N \rightarrow \infty.$

These approaches are also well-known:

• Ilya Nemenman, Fariel Shafee and William Bialek, Entropy and inference, revisited, 2002.

• Alexander Kraskov, Harald Stögbauer and Peter Grassberger, Estimating mutual information, Phys. Rev. E. 69 (2004), 066138.

Actually, Stefano Panzeri has quite a few impressive papers about this problem, and recently with colleagues he has made public a free Matlab toolbox for information theory (www.ibtb.org) implementing various correction methods.

Finally, the work by Jonathan Victor is worth mentioning, since he provided (to my knowledge again) the first estimate of mutual information using geometry. This is of particular interest with respect to the work by Christina Cobbold and Tom Leinster on measures of biodiversity that take the distance between species into account:

• J. D. Victor and K. P. Purpura, Nature and precision of temporal coding in visual cortex: a metric-space analysis, Journal of Neural Physiology 76 (1996), 1310–1326.

He introduced a distance between sequences of spikes and from this, derived a lower bound on mutual information.

• Jonathan D. Victor, Binless strategies for estimation of information from neural data, Phys. Rev. E. 66 (2002), 051903.

Taking inspiration from work by Kozachenko and Leonenko, he obtained an estimate of the information based on the distances between the closest responses.

Without getting too technical, that’s what we do in neuroscience about the limited sampling bias. The incentive is that obtaining reliable estimates is crucial to understand the ‘neural code’, the holy grail of computational neuroscientists.

9 Comments | biodiversity, biology, information and entropy | Permalink
Posted by John Baez

The Mathematics of Biodiversity (Part 6)

6 July, 2012

Here are two fun botany stories I learned today from Lou Jost.

The decline and fall of the Roman Empire

I thought Latin was a long-dead language… except in Finland, where 75,000 people regularly listen to the news in Latin. That’s cool, but surely the last time someone seriously needed to write in Latin was at least a century ago… right?

No! Until the beginning of 2012, botanists reporting new species were required to do so in Latin.

Like this:

Arbor ad 8 alta, raminculis sparse pilosis, trichomatis 2-2.5 mm longis. Folia persistentia; laminae anisophyllae, foliis majoribus ellipticus, 12-23.5 cm longis, 6-13 cm latis, minoribus orbicularis, ca 8.5 cm longis, 7.5 cm latis, apice acuminato et caudato, acuminibus 1.5-2 cm longis, basi rotundata ad obtusam, margine integra, supra sericea, trichomatis 2.5-4 mm longis, appressis, pagina inferiore sericea ad pilosam, trichomatis 2-3 mm longis; petioli 4-7 mm longi. Inflorescentia terminalis vel axillaris, cymosa, 8-10 cm latis. Flores bisexuales; calyx tubularis, ca. 6 mm longus, 10-costatus; corolla alba, tubularis, 5-lobata; stamina 5, filis 8-10 mm longis, pubescentia ad insertionem.

The International Botanical Congress finally voted last year to drop this requirement. So, the busy people who are discovering about 2000 species of plants, algae and fungi each year no longer need to file their reports in the language of the Roman Empire.

Orchid Fever

The first person who publishes a paper on a new species of plant gets to name it. Sometimes the competition is fierce, as for the magnificent orchid shown above, Phragmipedium kovachii.

Apparently one guy beat another, his archenemy, by publishing an article just a few days earlier. But the other guy took his revenge by getting the first guy arrested for illegally taking an endangered orchid out of Peru. The first guy wound up getting two years’ probation and a $1,000 fine.

But, he got his name on the orchid!

I believe the full story appears here:

• Eric Hansen, Orchid Fever: A Horticultural Tale of Love, Lust, and Lunacy, Vintage Books, New York, 2001.

You can read a summary here.

Ecominga

By the way, Lou Jost is not only a great discoverer of new orchid species and a biologist deeply devoted to understanding the mathematics of biodiversity. He also runs a foundation called Ecominga, which runs a number of nature reserves in Ecuador, devoted to preserving the amazing biodiversity of the Upper Pastaza Watershed. This area contains over 190 species of plants not found anywhere else in the world, as well as spectacled bears, mountain tapirs, and an enormous variety of birds.

The forests here are being cut down… but Ecominga has bought thousands of hectares in key locations, and is protecting them. They need money to pay the locals who patrol and run the reserves. It’s not a lot of money in the grand scheme of things—a few thousand dollars a month. So if you’re interested, go to the Ecominga website, check out the information and reports and pictures, and think about giving them some help! Or for that matter, contract me and I’ll put you in touch with him.

5 Comments | biodiversity, biology | Permalink
Posted by John Baez

The Mathematics of Biodiversity (Part 5)

3 July, 2012

I’d be happy to get your feedback on these slides of the talk I’m giving the day after tomorrow:

• John Baez, Diversity, entropy and thermodynamics, 6 July 2012, Exploratory Conference on the Mathematics of Biodiversity, Centre de Recerca Matemàtica, Barcelona.

Abstract: As is well known, some popular measures of biodiversity are formally identical to measures of entropy developed by Shannon, Rényi and others. This fact is part of a larger analogy between thermodynamics and the mathematics of biodiversity, which we explore here. Any probability distribution can be extended to a 1-parameter family of probability distributions where the parameter has the physical meaning of ‘temperature’. This allows us to introduce thermodynamic concepts such as energy, entropy, free energy and the partition function in any situation where a probability distribution is present—for example, the probability distribution describing the relative abundances of different species in an ecosystem. The Rényi entropy of this probability distribution is closely related to the change in free energy with temperature. We give one application of thermodynamic ideas to population dynamics, coming from the work of Marc Harper: as a population approaches an ‘evolutionary optimum’, the amount of Shannon information it has ‘left to learn’ is nonincreasing. This fact is closely related to the Second Law of Thermodynamics.

This talk is rather different than the one I’d envisaged giving! There was a lot of interest in my work on Rényi entropy and thermodynamics, because Rényi entropies—and their exponentials, called the Hill numbers—are an important measure of biodiversity. So, I decided to spend a lot of time talking about that.

11 Comments | biodiversity, biology, information and entropy, mathematics, physics, probability | Permalink
Posted by John Baez

The Mathematics of Biodiversity (Part 4)

2 July, 2012

Today the conference part of this program is starting:

• Research Program on the Mathematics of Biodiversity, June-July 2012, Centre de Recerca Matemàtica, Barcelona, Spain. Organized by Ben Allen, Silvia Cuadrado, Tom Leinster, Richard Reeve and John Woolliams.

Lou Jost kicked off the proceedings with an impassioned call to think harder about fundamental concepts:

• Lou Jost, Why biologists should care about the mathematics of biodiversity.

Then Tom Leinster gave an introduction to some of these concepts, and Lou explained how they show up in ecology, genetics, economics and physics.

Suppose we have $n$ different species on an island. Suppose a fraction $p_i$ of the organisms belong to the $i$ th species. So,

$\displaystyle{ \sum_{i=1}^n p_i = 1}$

and mathematically we can treat these numbers as probabilities.

People have many ways to compute the ‘biodiversity’ from these numbers. Some of these can be wildly misleading when applied incorrectly, and this has led to shocking errors. For example, in genetics, a commonly used formula for determining when plants or animals on a bunch of islands will split into separate species is completely wrong.

In fact, if we’re not careful, some measures of biodiversity can fool us into thinking we’re saving most of the biodiversity when we’re actually losing almost all of it!

One good example involves measures of similarity between tropical butterflies in the canopy (the top of the forest) and the understory (the bottom). According to Lou Just, some published studies say the similarity is about 95%. That sounds like the two communities are almost the same. However, almost no butterflies living in the canopy live in the understory, and vice versa! The problem is that mathematics is being used inappropriately.

Here are four famous measures of biodiversity:

• Species richness. This is just the number of species:

$n$

• Shannon entropy. This is the expected amount of information you gain when someone tells you which species an organism belongs to:

$\displaystyle{ - \sum_{i=1}^n p_i \ln(p_i) }$

• The inverse Simpson index. This is the reciprocal of the probability that two randomly chosen organisms belong to the same species:

$\displaystyle{ 1 \big/ \sum_{i=1}^n p_i^2 }$

The probability that two organisms belong to the same species is called the Simpson index:

$\displaystyle{ \sum_{i=1}^n p_i^2 }$

This is used in economics as a measure of the concentration of wealth, where $p_i$ is the fraction of wealth owned by the $i$ th individual. Be careful: there’s a lot of different jargon in different fields, so it’s easy to get confused at first! For example, the probability that two organisms belong to different species is often called the Gini–Simpson index:

$\displaystyle{ 1 - \sum_{i=1}^n p_i^2 }$

It was introduced by the statistician Corrado Gini a century ago, in 1912 and the ecologist Edward H. Simpson in 1949. It’s also called the heterozygosity in genetics.

• The Berger–Parker index. This is the fraction of organisms that belong to the most common species:

$\mathrm{max} \, p_i$

So, unlike the other main ones I’ve listed, this quantity tends to go down when biodiversity goes up. To fix this we could take its reciprocal, as we did with the Simpson index.

What a mess, eh? But here’s some good news: all these quantities are functions of a single quantity, the Rényi entropy:

$\displaystyle{ H_q(p) = \frac{1}{1 -q} \ln \sum_{i=1}^n p_i^q }$

for various values of the parameter $q.$

I’ve written about the Rényi entropies and their role in thermodynamics before on this blog. I’ll also talk about it later in this conference, and I’ll show you my slides. So, I won’t repeat that story here. Suffice it to say that Rényi entropies are fascinating but still a bit mysterious to me.

But one of Lou Jost’s main points is that we can make bad mistakes if we work with Rényi entropies when we should be working with their exponentials, which are called Hill numbers and denoted by a $D$ , for ‘diversity’:

$\displaystyle{ {}^qD(p) = e^{H_q(p)} = \left(\sum_{i=1}^n p_i^q \right)^{\frac{1}{1-q}} }$

These were introduced by M. O. Hill in 1973. One reason they’re good is that they are effective numbers. This means that if all the species are equally common, the Hill number equals the number of species, regardless of $q$ :

$p_i = \frac{1}{n} \; \Longrightarrow \; {}^qD(p) = n$

So, they’re a way of measuring an ‘effective’ number of species in situations where species are not all equally common.

A closely related fact is that the Hill numbers obey the replication principle. This means that if we have probability distributions on two finite sets, each with Hill number $X$ for some choice of $q,$ and we combine them with equal weights to get a probability distribution on the disjoint union of those sets, the resulting distribution has Hill number $2X.$

Another good fact is that the Hill numbers are as large as possible when all the probabilities $p_i$ are equal. They’re as small as possible, namely 1, when one of the $p_i$ equals 1 and the rest are zero.

Let’s see how all the measures of biodiversity I listed are either Hill numbers or can easily be converted to Hill numbers. We’ll also see that at $q = 0,$ the Hill number treats all species that are present in an equal way, regardless of their abundance. As $q$ increases, it counts more abundant species more heavily, since we’re raising the probabilities $p_i$ to a bigger power. And when $q = \infty$ , we only care about the most abundant species: none of the others matter at all!

Here goes:

• The species richness is the limit of the Hill numbers as $q \to 0$ from above:

$\displaystyle{ \lim_{q \to 0^+} {}^qD (p) = n }$

So, we can just call this ${}^0D(p).$

• The exponential of the Shannon entropy is the limit of the Hill numbers as $q \to 1$ :

$\displaystyle{ \lim_{q \to 1} {}^qD(p) = \exp\left(- \sum_{i=1}^n p_i \ln(p_i)\right) }$

So, we can just call this ${}^1D(p).$

• The inverse Simpson index is the Hill number at $q = 2$ :

$\displaystyle{ {}^2D(p) = 1 \big/ \sum_{i=1}^n p_i^2 }$

• The reciprocal of the Berger–Parker index is the limit of Hill numbers as $q \to +\infty$ :

$\displaystyle{ \lim_{q \to +\infty} {}^qD(p) = 1 \big/ \mathrm{max} \, p_i }$

so we can call this quantity ${}^\infty D(p).$

These facts mean that understanding Hill numbers will help us understand lots of measures of biodiversity! And the good properties of Hill numbers will help us avoid dangerous mistakes.

For mathematicians, a good challenge is to find theorems uniquely characterizing the Hill numbers…. preferably with assumptions that biologists will accept as plausible facts about ‘diversity’. Some theorems like this already exist for specific choices of $q,$ but it will be better to characterize the function ${}^q D$ for all values of $q$ in one blow. Tom Leinster is working on such a theorem now.

Another important task is to generalize Hill numbers to take into account things like:

• ‘distances’ between species, measured either genetically, phylogenetically or functionally,

• ‘values’ for species, measured either economically or
any other way.

There’s a lot of work on this, and many of the talks here conference will discuss these generalizations.

13 Comments | biodiversity, biology, information and entropy, mathematics | Permalink
Posted by John Baez

	hypergeometric on Bridging the Greenhouse-Gas Em…
	hypergeometric on Bridging the Greenhouse-Gas Em…
	hypergeometric on Bridging the Greenhouse-Gas Em…
	John Baez on Bridging the Greenhouse-Gas Em…
	hypergeometric on Bridging the Greenhouse-Gas Em…
	hypergeometric on Bridging the Greenhouse-Gas Em…
	westy31 on Graph Laplacians
	anonymous on Graph Laplacians
	John Baez on Graph Laplacians
	John Baez on Graph Laplacians
	John Baez on Network Theory (Part 12)
	Arjun Jain on Network Theory (Part 12)
	John Carlos Baez on… on Energy and the Environment…
	westy31 on Graph Laplacians
	westy31 on Graph Laplacians

Azimuth