This Week’s Finds (Week 315)

This is the second and final part of my interview with Thomas Fischbacher. We’re talking about sustainable agriculture, and he was just about to discuss the role of paying attention to flows.

JB: So, tell us about flows.

TF: For natural systems, some of the most important flows are those of energy, water, mineral nutrients, and biomass. Now, while they are physically real, and keep natural systems going, we should remind ourselves that nature by and large does not make high level decisions to orchestrate them. So, flows arise due to processes in nature, but nature ‘works’ without being consciously aware of them. (Still, there are mechanisms such as evolutionary pressure that ensure that the flow networks of natural ecosystems work—those assemblies that were non-viable in the long term did not make it.)

Hence, flows are above everything else a useful conceptual framework—a mental tool devised by us for us—that helps us to make sense of an otherwise extremely complex and confusing natural world. The nice thing about flows is that they reduce complexity by abstracting away details when we do not want to focus on them—such as which particular species are involved in the calcium ion economy, say. Still, they retain a lot of important information, quite unlike some models used by economists that actually guide—or misguide—our present decision-making. They tell us a lot about key processes and longer term behaviour—in particular, if something needs to be corrected.

Sustainability is a complex subject that links to many different aspects of human experience—and of course the non-human world around us. When confronted with such a subject, my approach is to start by asking: ‘what I am most certain about’, and use these key insights as ‘anchors’ that set the scene. Everything else must respect these insights. Occasionally, some surprising new insight forces me to reevaluate some fundamental assumptions, and repaint part of the picture. But that’s life—that’s how we learn.

Very often, I find that those aspects which are both useful to obtain deeper insights and at the same time accessible to us are related to flows.

JB: Can you give an example?

TF: Okay, here’s another puzzle. What is the largest flow of solids induced by civilization?

JB: Umm… maybe the burning of fossil fuels, passing carbon into the atmosphere?

TF: I am by now fairly sure that the answer is: the unintentional export of topsoil from the land into the sea by wind and water erosion, due to agriculture. According to Brady & Weil, around the year 2000, the U.S. annually ‘exported’ about 4×10¹² kilograms of topsoil to the sea. That’s roughly three cubic kilometers, taking a reasonable estimate for the density of humus.

JB: Okay. In 2007, the U.S. burnt 1.6 × 10¹² kilograms of carbon. So, that’s comparable.

TF: Yes. When I cross check my number combining data from the NRCS on average erosion rates and from the CIA World Factbook on cultivated land area, I get a result that is within the same ballpark, so it seems to make sense. In comparison, total U.S. exports of economic goods in 2005 were 4.89×10¹¹ kilograms: about an order of magnitude less, according to statistics from the Federal Highway Administration.

If we look at present soil degradation rates alone, it is patently clear that we see major changes ahead. In the long term, we just cannot hope to keep on feeding the population using methods that keep on rapidly destroying fertility. So, we pretty much know that something will happen there. (Sounds obvious, but alas, thinking of a number of discussions I had with some economists, I must say that, sadly, it is far from being so.)

What actually will happen mostly depends on how wisely we act. The possibilities range from nuclear war to a mostly smooth swift transition to fertility-building food production systems that also take large amounts of CO₂ out of the atmosphere and convert it to soil humus. I am, of course, much in favour of scenarios close to the latter one, but that won’t happen unless we put in some effort—first and foremost, to educate people about how it can be done.

Flow analysis can be an extremely powerful tool for diagnosis, but its utility goes far beyond this. When we design systems, paying attention to how we design the flow networks of energy, water, materials, nutrients, etc., often makes a world of a difference.

Nature is a powerful teacher here: in a forest, there is no ‘waste’, as one system’s output is another system’s input. What else is ‘waste’ but an accumulation of unused output? So, ‘waste’ is an indication of an output mismatch problem. Likewise, if a system’s input is not in the right form, we have to pre-process it, hence do work, hence use energy. Therefore, if a process or system continually requires excessive amounts of energy (as many of our present designs do), this may well be an indication of a design problem—and could be related to an input mismatch.

Also, the flow networks of natural systems usually show both extremely high recycling rates and a lot of multi-functionality, which provides resilience. Every species provides its own portfolio of services to the assembly, which may include pest population control, creating habitat for other species, food, accumulating important nutrients, ‘waste’ transformation, and so on. No element has a single objective, in contrast to how we humans by and large like to engineer our systems. Each important function is covered by more than one element. Quite unlike many of our past approaches, design along such principles can have long-term viability. Nature works. So, we clearly can learn from studying nature’s networks and adopting some principles for our own designs.

Designing for sustainability with, around, and inspired by natural systems is an interesting intellectual challenge, much like solving a jigsaw puzzle. We cannot simultaneously comprehend the totality of all interactions and relations between adjacent pieces as we build it, but we keep on discovering clues by closely studying different aspects: form, colour, pattern. If we are on the right track, and one clue tells us how something should fit, we will discover that other aspects will fit as well. If we made a mistake, we need to apply force to maintain it and hammer other pieces into place—and unless we correct that mistake, we will need ever more brutal interventions to artificially stabilize the problems which are mere consequences of the original mistake. Think using nuclear weapons to seal off spilling oil wells drilled in deep waters needed because we used up all the easily accessible high-quality fuels. One mistake begets another.

There is a reason why jigsaw puzzles ‘work’: they were created that way. There is also a reason why the dance of natural systems ‘works’: coevolution. What happens when we run out of steam to stabilize poor designs (i.e. in an energy crisis)? We, as a society, will be forced to confront our past arrogance and pay close attention to resolving the design mistakes we so far always tried to talk away. That’s something I’d call ‘true progress’.

Actually, it’s quite evident now: many of our ‘problems’ are rather just symptoms of more fundamental problems. But as we do not track these down to the actual root, we keep on expending ever more energy by stacking palliatives on top of one another. Growing corn as a biofuel in a process that both requires a lot of external energy input and keeps on degrading soil fertility is a nice example. Now, if we look closer, we find numerous further, superficially unrelated, problems that should make us ask the question: "Did we assemble this part of the puzzle correctly? Is this approach really such a good idea? What else could we do instead? What other solutions would suggest themselves if we paid attention to the hints given by nature?" But we don’t do that. It’s almost as if we were proud to be thick.

JB: How would designing with flows in mind work?

TF: First, we have to be clear about the boundaries of our domain of influence. Resources will at some point enter our domain of influence and at some point leave it again. This certainly holds for a piece of land on which we would like to implement sustainable food production where one of the most important flows is that of water. But it also holds for a household or village economy, where an important flow through the system is that of purchase power—i.e. money (but in the wider sense). As resources percolate through a system, their utility generally degrades—entropy at work. Water high up in the landscape has more potential uses than water further down. So, we can derive a guiding principle for design: capture resources as early as possible, release them as late as possible, and see that you guide them in such a way that their natural drive to go downhill makes them perform many useful duties in between. Considering water flowing over a piece of land, this would suggest setting up rainwater catchment systems high up in the landscape. This water then can serve many useful purposes: there certainly are agricultural/silvicultural and domestic uses, maybe even aquaculture, potentially small-scale hydropower (say, in the 10-100 watts range), and possibly fire control.

JB: When I was a kid, I used to break lots of things. I guess lots of kids do. But then I started paying attention to why I broke things, and I discovered there were two main reasons. First, I might be distracted: paying attention to one thing while doing another. Second, I might be trying to overcome a problem by force instead of by slowing down and thinking about it. If I was trying to untangle a complicated knot, I might get frustrated and just pull on it… and rip the string.

I think that as a culture we make both these mistakes quite often. It sounds like part of what you’re saying is: "Pay more attention to what’s going on, and when you encounter problems, slow down and think about their origin a bit—don’t just try to bully your way through them."

But the tool of measuring flows is a nice way to organize this thought process. When you first told me about ‘input mismatch problems’ and ‘output mismatch problems’, it came as a real revelation! And I’ve been thinking about them a lot, and I want to keep doing that.

One thing I noticed is that problems tend to come in pairs. When the output of one system doesn’t fit nicely into the input of the next, we see two problems. First, ‘waste’ on the output side. Second, ‘deficiency’ on the input side. Sometimes it’s obvious that these are two aspects of the same problem. But sometimes we fail to see it.

For example, a while ago some ground squirrels chewed a hole in an irrigation pipe in our yard. Of course that’s our punishment for using too much water in a naturally dry environment, but look at the two problems it created. One: big gushers of water shooting out of the hole whenever that irrigation pipe was used, which caused all sort of further problems. Two: not enough water to the plants that system was supposed to be irrigating. Waste on one side, deficiency on the other.

That’s obvious, easy to see, and easy to fix: first plug the hole, then think carefully about why we’re using so much water in the first place. We’d already replaced our lawn with plants that use less water, but maybe we can do better.

But here’s a bigger problem that’s harder to fix. Huge amounts of fertilizer are being used on the cornfields of the midwestern United States. With the agricultural techniques they’re using, there’s a constant deficiency of nitrogen and phosphorus, so it’s supplied artificially. The figures I’ve seen show that about 30% of the energy used in US agriculture goes into making fertilizers. So, it’s been said that we’re ‘eating oil’—though technically, a lot of nitrogen fertilizer is made using natural gas. Anyway: a huge deficiency problem.

On the other hand, where is all this fertilizer going? In the midwestern United States, a lot of it winds up washing down the Mississipi River. And as a result, there are enormous ‘dead zones’ in the Gulf of Mexico. The fertilizer feeds algae, the algae dies and decays, and the decay process takes oxygen out of the water, killing off any life that needs oxygen. These dead zones range from 15 and 18 thousand square kilometers, and they’re in a place that’s one of the prime fishing spots for the US. So: a huge waste problem.

But they’re the same problem!

It reminds me of the old joke about a guy who was trying to button his shirt. "There are two things wrong with this shirt! First, it has an extra button on top. Second, it has an extra buttonhole on bottom!"

TF: Bill Mollison said it in a quite humorous-yet-sarcastic way in this episode of the Global Gardener movie:

• Bill Mollison, Urban permaculture strategies – part 1, YouTube.

While the potential to grow a large amount of calories in cities may be limited, growing fruit and vegetables nevertheless does make sense for multiple reasons. One of them is that many things that previously went into the garbage bin now have a much more appropriate place to go—such as the compost heap. Many urbanites who take up gardening are quite amazed when they realize how much of their household waste actually always ‘wanted’ to end up in a garden.

JB: Indeed. After I bought a compost bin, the amount of trash I threw out dropped dramatically. And instead of feeling vaguely guilty as I threw orange peels into the trash where they’d be mummified in a plastic bag in a landfill, I could feel vaguely virtuous as I watched them gradually turn into soil. It doesn’t take as long as you might think. And it comes as a bit of a revelation at first: "Oh, so that’s how we get soil."

TF: Perhaps the biggest problem I see with a mostly non-gardening society is that people without even the slightest own experience in growing food are expected to make up their mind about very important food-related questions and contribute to the democratic decision making process. Again, I must emphasize that whoever does not consciously invest some effort into getting at least some minimal first hand experience to improve their judgment capabilities will be easy prey for rat-catchers. And by and large, society is not aware of how badly they are lied to when it comes to food.

But back to flows. Every few years or so, I stumble upon a jaw-dropping idea, or a principle, that makes me realize that it is so general and powerful that, really, the limits of what it can be used for are the limits of my imagination and creativity. I recently had such a revelation with the PSLQ integer relation algorithm. Using flows as a mental tool for analysis and design was another such case. All of a sudden, a lot made sense, and could be analyzed with ease.

There always is, of course, the ‘man with a hammer problem’—if you are very fond of a new and shiny hammer, everything will look like a nail. I’ve also heard that expressed as ‘an idea is a very dangerous thing if it is the only one you have’.

So, while keeping this in mind, now that we got an idea about flows in nature, let us ask: "how can we abuse these concepts?" Mathematicians prefer the term ‘abstraction’, but it’s fun either way. So, let’s talk about the flow of money in economies. What is money? Essentially, it is just a book-keeping device invented to keep track of favours owed by society to individuals and vice versa. What function does it have? It works as ‘grease’, facilitating trade.

So, suppose you are a mayor of a small village. One of your important objectives is of course prosperity for your villagers. Your village trades with and hence is linked to an external economy, and just as goods and services are exchanged, so is money. So, at some point, purchase power (in the form of money) enters your domain of influence, and at some point, it will leave it again. What you want it to do is to facilitate many different economic activities—so you want to ensure it circulates within the village as long as possible. You should pay some attention to situations where money accumulates—for everything that accumulates without being put to good use is a form of ‘waste’, hence pollution. So, this naturally leads us to two ideas: (a) What incentives can you find to keep money on circulating within the village? (There are many answers, limited only by creativity.) And (b) what can you do to constrain the outflow? If the outlet is made smaller, system outflow will match inflow at a higher internal pressure, hence a higher level of resource availability within the system.

This leads us to an idea no school will ever tell you about—for pretty much the same reason why no state-run school will ever teach how to plan and successfully conduct a revolution. The road to prosperity is to systematically reduce your ‘Need To Earn’—i.e. the best way to spend money is to set up systems that allow you to keep more money in your pocket. An frequent misconception that keeps on arising when I mention this is that some think this idea would be about austerity. Quite to the contrary. You can make as much money as you want—but one thing you should keep in mind is that if you have that trump card up your sleeve that you could at any time just disconnect from most of the economy and get by with almost no money at all for extended periods of time, you are in a far better position to take risks and grasp exceptional opportunities as they arise as someone would be who committed himself to having to earn a couple of thousand pounds a month.

The problem is not with earning a lot of money. The problem is with being forced to continually make a lot of money. We readily manage to identify this as a key problem of drug addicts, but fail to see the same mechanism at work in mainstream society. A key assumption in economic theory is that exchange is voluntary. But how well is that assumption satisfied in practice if such forces are in place?

Now, what would happen if people started to get serious about investing the money they earn to systematically reduce their need to earn money in the future? Some decisions such as getting a photovoltaic array may have ‘payback times’ in the range of one or two decades, but I consider this ‘payback time’ concept as a self-propagating flawed idea. If something gives me an advantage in terms of depending on less external input now, this reduction of vulnerability also has to be taken into account—’payback times’ do not do that. So—if most people did such things, i.e. made strategic decisions to set up systems so that their essential needs can be satisfied with minimal effort—especially money, this would put a lot of political power back into their hands. A number of self-proclaimed ‘leaders’ certainly don’t like the idea of people being in a position to just ignore their orders. Also note that this would have a funny effect on the GDP—ever heard of ‘imputations’?

JB: No, what are those?

TF: It’s a funny thing, perhaps best explained by an example. If you fully own your own house, then you don’t pay rent. But for the purpose of determining the GDP, you are regarded as paying as much rent to yourself (!) as you would get if you rented out the house. See:

• Imputed rent, Wikipedia.

Evidently, if people make a dedicated effort at the household level to become less dependent on the economy by being able to provide most of their essential needs themselves (housing, food, water, energy, etc.) to a much larger extent, this amounts to investing money in order to need less money in the future. If many people did this systematically, it would superficially have a devastating effect on the GDP—but it would bring about a much more resilient (because less dependent) society.

The problem is that the GDP really is not an appropriate measure for progress. But obviously, those who publish these figures know that as well, hence the need to fudge the result with imputations. So, a simple conclusion is: whenever there is an opportunity to invest money in a way that makes you less dependent on the economy in the future, that might be well worth a closer look. Especially if you get the idea that, if many people did this, the state would likely have to come up with other imputations to make the impact on the GDP disappear!

JB: That’s a nice thought. I tend to worry about how the GDP and other economic indicators warp our view of what’s right to do. But you’re saying that if people can get up the nerve to do what’s right, regardless, the economic indicators may just take care of themselves.

TF: We have to remember that sustainability is about systems that are viable in the long run. Environmental sustainability is just one important aspect. But you won’t go on for long doing what you do unless it also has economic long-term viability. Hence, we are dealing with multi-dimensional design constraints. And just as flow network analysis is useful to get an idea about the environmental context, the same holds for the economic context. It’s just that the resources are slightly different ones—money, labour, raw materials, etc. These thoughts can be carried much further, but I find it quite worthwhile to instead look at an example where someone did indeed design a successful system along such principles. In the UK, the first example that would come to my mind is Hill Holt Wood, because the founding director, Nigel Lowthrop, did do so many things right. I have high admiration for his work.

JB: When it comes to design of sustainable systems, you also seem to be a big fan of Bill Mollison and some of the ‘permaculture’ movement that he started. Could you say a bit about that? Why is it important?

TF: The primary reason why permaculture matters is that it has demonstrated some stunning successes with important issues such as land rehabilitation.

‘Permaculture’ means a lot of different things to a lot of different people. Curiously, where I grew up, the term is somewhat known, but mostly associated with an Austrian farmer, not Bill Mollison. And I’ve seen some physicists who first had come into contact with it through David Holmgren‘s book revise their opinions when they later read Mollison. Occasionally, some early adopters did not really understand the scientific aspects of it and tried to link it with some strange personal beliefs of the sort Martin Gardner discussed in Fads and Fallacies in the Name of Science. And so on. So, before we discuss permaculture, I have to point out that one might sometimes have to take a close look to evaluate it. A number of things claiming to be ‘permaculture’ actually are not.

When I started—some time ago—to make a systematic effort to get a useful overview over the structure of our massive sustainability-related problems, a key question to me always was: "what should I do?"—and a key conviction was: "someone must have had some good ideas about all this already." This led me to actually not read some well-known "environmentalist" books many people had read which are devoid of any discussion of our options and potential solutions, but to do a lot of detective work instead.

In doing so, I travelled, talked to a number of people, read a lot of books and manuscripts, did a number of my own experiments, cross-checked things against order-of-magnitude guesstimates, against the research literature, and so on. At one point—I think it was when I took a closer look into the work of the laureates of the ‘Right Livelihood award’ (sometimes called the ‘Alternative Nobel Prize’)—I came across Bill Mollison’s work. And it struck a chord.

Back in the 90s, when mad cow disease was a big topic in Europe, I spent quite some time pondering questions such as: "what’s wrong with the way farming works these days?" I immediately recognized a number of insights I independently had arrived at back then when studying Bill Mollison’s work, and yet, he went so much further—talked about a whole universe of issues I still was mostly unaware of at that time. So, an inner voice said to me: "if you take a close look at what that guy already did, that might save you a lot of time". Now, Mollison did get some things wrong, but I still think taking a close look at what he has to say is a very effective way to get a big picture overview over what we can achieve, and what needs urgent attention. I think it greatly helps (at least to me) that he comes from a scientific background. Before he decided to quit academia in 1978 and work full time on developing permaculture, he was a lecturer at the University of Hobart, Tasmania.

JB: But what actually is ‘permaculture’?

TF: That depends a lot on who you ask, but I like to think about permaculture as if it were an animal. The ‘skeleton’ is a framework with cleverly designed ‘static properties’ that holds the ‘flesh’ together in a way so that it can achieve things. The actual ‘flesh’ is provided by solutions to specific problems with long term viability being a key requirement. But it is more than just a mere semi-amorphous collage of solutions, due to its skeleton. The backbone of this animal is a very simple (deliberately so) yet functional (this is important) core ethics which one could regard as being the least common denominator of values considered as essential across pretty much all cultures. This gives it stability. Other bones that make this animal walk and talk are related to key principles. And these principles are mostly just applied common sense.

For example, it is pretty clear that as non-renewable resources keep on becoming more and more scarce, we will have to seriously ponder the question: what can we grow that can replace them? If our design constraints change, so does our engineering—should (for one reason or another) some particular resource such as steel become much more expensive than it is today, we would of course look into the question whether, say, bamboo may be a viable alternative for some applications. And that is not as exotic an idea as it may sound these days.

So, unquestionably, the true solutions to our problems will be a lot about growing things. But growing things in the way that our current-day agriculture mostly does it seems highly suspicious, as this keeps on destroying soil. So, evidently, we will have to think less along the lines of farming and more along the lines of gardening. Also, we must not fool ourselves about a key issue: most people on this planet are poor, hence for an approach to have wide impact, it must be accessible to the poor. Techniques that revolve around gardening often are.

Next, isn’t waiting for the big (hence, capital intensive) ‘technological miracle fix’ conspicuously similar to the concept of a ‘pie in the sky’? If we had any sense, shouldn’t we consider solving today’s problems with today’s solutions?

If one can distinguish between permaculture as it stands and attempts by some people who are interested in it to re-mold it so that it becomes ‘the permaculture part of permaculture plus Anthroposophy/Alchemy/Biodynamics/Dianetics/Emergy/Manifestation/New Age beliefs/whatever’, there is a lot of common sense in permaculture—the sort of ‘a practical gardener’s common sense’. In this framework, there is a place for both modern scientific methods and ancient tribal wisdom. I hence consider it a healthy antidote to both fanatical worship of ‘the almighty goddess of technological progress’—or any sort of fanatical worship for that matter—as well as to funny superstitious beliefs.

There are some things in the permaculture world, however, where I would love to see some change. For example, it would be great if people who know how to get things done paid more attention to closely keeping records of what they do to solve particular problems and to making these widely accessible. Solutions of the ‘it worked great for a friend of a friend’ sort do us a big disservice. Also, there are a number of ideas that easily get represented in overly simplistic form—such as ‘edge is good’—where one better should retain some healthy skepticism.

JB: Well, I’m going to keep on pressing you: what is permaculture… according to you? Can you list some of the key principles?

TF: That question is much easier to answer. The way I see it, permaculture is a design-oriented approach towards systematically reducing the total effort that has to be expended (in particular, in the long run) in order to keep society going and allow people to live satisfying lives. Here, ‘effort’ includes both work that is done by non-renewable resources (in particular fossil fuels), as well as human labour. So, permaculture is not about returning to pre-industrial agricultural drudgery with an extremely low degree of specialization, but rather about combining modern science with traditional wisdom to find low-effort solutions to essential problems. In that sense, it is quite generic and deals with issues ranging from food production to water supply to energy efficient housing and transport solutions.

To give one specific example: Land management practices that reduce the organic matter content of soils and hence soil fertility are bound to increase the effort needed to produce food in the long run and hence considered a step in the wrong direction. So, a permaculture approach would focus on using strategies that manage to build soil fertility while producing food. There are a number of ways to do that, but a key element is a deep understanding of nature’s soil food web and nutrient cycling processes. For example, permaculture pays great attention to ensuring a healthy soil microflora.

When the objective is to minimize the effort needed to sustain us, it is very important to closely observe those situations where we have to expend energy on a continual basis in order to fight natural processes. When this happens, there is a conflict between our views how things ought to look like and a system trying to demonstrate its own evolution. In some situations, we really want it that way and have to pay the corresponding price. But there are others—quite many of them—where we would be well advised to spend some thought on whether we could make our life easier by ‘going with the flow’. If thistles keep on being a nuisance on some piece of land, we might consider trying to fill this ecological niche by growing some closely related species, say some artichoke. If a meadow needs to be mowed regularly so that it does not turn into a shrub thicket, we would instead consider planting some useful shrubs in that place.

Naturally, permaculture design favours perennial plants in climatic regions where the most stable vegetation would be a forest. But it does not have to be this way. There are high-yielding low-effort (in particular: no-till, no-pesticide) ways to grow grains as well, mostly going back to Masanobu Fukuoka. They have gained some popularity in India, where they are known as ‘Rishi Kheti’—’agriculture of the sages’. Here’s a photo gallery containing some fairly recent pictures:

• Raju Titus’s Public Gallery, Picasa.

Wheat growing amid fruit trees: no tillage, no pesticides — Hoghangabad, India

An interesting perspective towards weeds which we usually do not take is: the reason this plant could establish itself here is that it’s filling an unfilled ecological niche.

JB: Actually I’ve heard someone say: "If you have weeds, it means you don’t have enough plants".

TF: Right. So, when I take that weed out, I’d be well advised to take note of nature’s lesson and fill that particular niche with an ecological analog that is more useful. Otherwise, it will quite likely come back and need another intervention.

I would consider this "letting systems demonstrate their own evolution while closely watching what they want to tell us and providing some guidance" the most important principle of permaculture.

Another important principle is the ‘user pays‘ principle. A funny idea that comes up disturbingly often up in discussions of sustainability issues (even if it is not articulated explicitly) is that there are only a limited amount of resources which we keep on using up, and once we are done with that, this would be the end of mankind. Actually, that’s not how the world works.

Take an apple tree, for example. It starts out as a tiny seed, and has to accumulate a massive amount of (nutrient) resources to grow into a mature tree. Yet, once it completes its life cycle, dies down and is consumed by fungi, it leaves the world in a more fertile state than before. Fertility tends to keep growing, because natural systems by and large work according to the principle that any agent that takes something from the natural world will return something of equal or even greater ecosystemic value.

Let me come back to an example I briefly mentioned earlier on. At a very coarse level of detail, grazing cows eat grass and return cow dung. Now, in the intestines of the cow, quite a lot of interesting biochemistry has happened that converted nonprotein nitrogen (say, urea) into much more valuable protein:

• W. D. Gallup, Ruminant nutrition, review of utilization of nonprotein nitrogen in the ruminant, Journal of Agricultural and Food Chemistry 4 (1956), 625-627.

A completely different example: nutrient accumulators such as comfrey act as powerful pumps that draw up mineral nutrients from the subsoil, where they would be otherwise inaccessible, and make them available for ecosystemic cycling.

Russian comfrey, Symphytum x uplandicum

It is indeed possible to not only use this concept for garden management, but as a fundamental principle to run a sustainable economy. At the small scale (businesses), its viability has been demonstrated, but unfortunately this aspect of permaculture has not received as much attention yet as it should. Here, the key questions are along the lines of: do you need a washing machine, or is your actual need better matched by the description ‘access to some laundry service’?

Concerning energy and material flows, an important principle is "be aware of the boundaries of your domain of influence, capture them as early as you can, release them as late as you can, and extract as much beneficial use out of them as possible in between". We already talked about that. In the era of cheap labour from fossil fuels, it is often a very good idea to use big earthworking machinery to slightly adjust the topography of the landscape in order to capture and make better use of rainwater. Done right, such water harvesting earthworks can last many hundreds of years, and pay back the effort needed to create them many times over in terms of enhanced biological productivity. If this were implemented on a broad scale, not just by a small percentage of farmers, this could add significantly to flood protection as well. I am fairly confident that we will be doing this a lot in the 21st century, as the climate gets more erratic and we face both more extreme rainfall events (note that saturation water vapour pressure increases by about 7% for every Kelvin of temperature increase) as well as longer droughts. It would be smart to start with this now, rather than when high quality fuels are much more expensive. It would have been even smarter to start with this 20 years ago.

A further important principle is to create stability through a high degree of network connectivity. We’ve also briefly talked about that already. In ecosystem design, this means to ensure that every important ecosystemic function is provided by more than one element (read: species), while every species provides multiple functions to the assembly. So, if something goes wrong with one element, there are other stabilizing forces in place. The mental picture which I like to use here is that of a stellar cluster: If we put a small number of stars next to one another, the system will undergo fairly complicated dynamics and eventually separate: in some three-star encounters, two stars will enter a very close orbit, while the third receives enough energy to go over escape velocity. If we lump together a large number of stars, their dynamics will thermalize and make it much more difficult for an individual star to obtain enough energy to leave the cluster—and keep it for a sufficiently long time to actually do so. Of course, individual stars do ‘boil off’, but the entire system does not fall apart as fast as just a few stars would.

There are various philosophies how to best approach weaving an ecosystemic net, ranging from ‘ecosystem mimicry‘;—i.e. taking wild nature and substituting some species with ecological analogs that are more useful to us—to ‘total synthesis of a species assembly’, i.e. combining species which in theory should grow well together due to their ecological characteristics, even though they might never have done so in nature.

JB: Cool. You’ve given me quite a lot to think about. Finally, could you also leave me with a few good books to read on permaculture?

TF: It depends on what you want to focus on. Concerning a practical hands-on introduction, this is probably the most evolved text:

• Bill Mollison, Introduction to Permaculture, Tagari Publications, Tasmania, 1997.

If you want more theory but are fine with a less refined piece of work, then this is quite useful:

• Bill Mollison, Permaculture – A Designer’s Manual, Tagari Publications, Tasmania, 1988.

Concerning temperate climates—in particular, Europe—this is a well researched piece of work that almost could be used as a college textbook:

• Patrick Whitefield, The Earth Care Manual: a Permaculture Handbook for Britain and Other Temperate Climates, Permanent Publications, East Meon, 2004.

For Europeans, this would probably be my first recommendation.

JB: Thanks! It’s been a very thought-provoking interview.

---

Ecologists never apply good ecology to their gardens. Architects never understand the transmission of heat in buildings. And physicists live in houses with demented energy systems. It’s curious that we never apply what we know to how we actually live. – Bill Mollison

Is Life Improbable?

Mine? Yes. And maybe you’ve wondered just how improbable your life is. But that’s not really the question today…

Here at the Centre for Quantum Technologies, Dagomir Kaszlikowski asked me to give a talk on this paper:

• John Baez, Is life improbable?, Foundations of Physics 19 (1989), 91-95.

This was the second paper I wrote, right after my undergraduate thesis. Nobody ever seemed to care about it, so it’s strange—but nice—to finally be giving a talk on it.

My paper does not try to settle the question its title asks. Rather, it tries to refute the argument here:

• Eugene P. Wigner, The probability of the existence of a self-reproducing unit, Symmetries and Reflections, Indiana University Press, Bloomington, 1967, pp. 200-208.

According Wigner, his argument

purports to show that, according to standard quantum mechanical theory, the probability is zero for the existence of self-reproducing states, i.e., organisms.

Given how famous Eugene Wigner is (he won a Nobel prize, after all) and how earth-shattering his result would be if true, it’s surprising how little criticism his paper has received. David Bohm mentioned it approvingly in 1969. In 1974 Hubert Yockey cited it saying

for all physics has to offer, life should never have appeared and if it ever did it would soon die out.

As you’d expect, there are some websites mentioning Wigner’s argument as evidence that some supernatural phenomenon is required to keep life going. Wigner himself believed it was impossible to formulate quantum theory in a fully consistent way without referring to consciousness. Since I don’t believe either of these claims, I think it’s good to understand the flaw in Wigner’s argument.

So, let me start by explaining his argument. Very roughly, it purports to show that if there are many more ways a chunk of matter can be ‘dead’ than ‘living’, the chance is zero that we can choose some definition of ‘living’ and a suitable ‘nutrient’ state such that every ‘living’ chunk of matter can interact with this ‘nutrient’ state to produce two ‘living’ chunks.

In making this precise, Wigner considers more than just two chunks of matter: he also allows there to be an ‘environment’. So, he considers a quantum system made of three parts, and described by a Hilbert space

Here the first corresponds to a chunk of matter. The second corresponds to another chunk of matter. The space corresponds to the ‘environment’. Suppose we wait for a certain amount of time and see what the system does; this will be described by some unitary operator

Wigner asks: if we pick this operator in a random way, what’s the probability that there’s some -dimensional subspace of ‘living organism’ states in , and some ‘nutrient plus environment’ state in , such that the time evolution sends any living organism together with the nutrient plus environment to two living organisms and some state of the environment?

A bit more precisely: suppose we pick in a random way. Then what’s the probability that there exists an -dimensional subspace

and a state

such that maps every vector in to a vector in ? Here means the 1-dimensional subspace spanned by the vector .

And his answer is: if

then this probability is zero.

You may need to reread the last few paragraphs a couple times to understand Wigner’s question, and his answer. In case you’re still confused, I should say that is what I’m calling the space of ‘living organism’ states of our chunk of matter, while is the ‘nutrient plus environment’ state.

Now, Wigner did not give a rigorous proof of his claim, nor did he say exactly what he meant by ‘probability’: he didn’t specify a probability measure on the space of unitary operators on . But if we use the obvious choice (called ‘normalized Haar measure’) his argument can most likely be turned into a proof.

So, I don’t want to argue with his math. I want to argue with his interpretation of the math. He concludes that

the chances are nil for the existence of a set of ‘living’ states for which one can find a nutrient of such nature that interaction always leads to multiplication.

The problem is that he fixed the decomposition of the Hilbert space as a tensor product

before choosing the time evolution operator . There is no good reason to do that. It only makes sense split up a physical into parts this way after we have some idea of what the dynamics is. An abstract Hilbert space doesn’t come with a favored decomposition as a tensor product into three parts!

If we let ourselves pick this decomposition after picking the operator , the story changes completely. My paper shows:

Theorem 1. Let , and be finite-dimensional Hilbert spaces with . Suppose is any unitary operator, suppose is any subspace of , and suppose is any unit vector in Then there is a unitary isomorphism

such that if we identify with using , the operator maps into .

In other words, if we allow ourselves to pick the decomposition after picking , we can always find a ‘living organism’ subspace of any dimension we like, together with a ‘nutrient plus environment’ state that allows our living organism to reproduce.

However, if you look at the proof in my paper, you’ll see it’s based on a kind of cheap trick (as I forthrightly admit). Namely, I pick the ‘nutrient plus environment’ state to lie in , so the nutrient actually consists of another organism!

This goes to show that you have to be very careful about theorems like this. To prove that life is improbable, you need to find some necessary conditions for what counts as life, and show that these are improbable (in some sense, and of course it matters a lot what that sense is). Refuting such an argument does not prove that life is probable: for that you need some sufficient conditions for what counts as life. And either way, if you prove a theorem using a ‘cheap trick’, it probably hasn’t gotten to grips with the real issues.

I also show that as the dimension of approaches infinity, the probability approaches 1 that we can get reproduction with a 1-dimensional ‘living organism’ subspace and a ‘nutrient plus environment’ state that lies in orthogonal complement of . In other words, the ‘nutrient’ is not just another organism sitting there all ready to go!

More precisely:

Theorem 2. Let , and be finite-dimensional Hilbert spaces with . Let be the set of unitary operators with the following property: there’s a unit vector , a unit vector , and a unitary isomorphism

such that if we identify with using , the operator maps into . Then the normalized Haar measure of approaches 1 as .

Here is the orthogonal complement of ; that is, the space of all vectors perpendicular to .

I won’t include the proofs of these theorems, since you can see them in my paper.

Just to be clear: I certainly don’t think these theorems prove that life is probable! You can’t have theorems without definitions, and I think that coming up with a good general definition of ‘life’, or even supposedly simpler concepts like ‘entity’ and ‘reproduction’, is extremely tough. The formalism discussed here is oversimplified for dozens of reasons, a few of which are listed at the end of my paper. So far we’re only in the first fumbling stages of addressing some very hard questions.

All my theorems do is point out that Wigner’s argument has a major flaw: he’s choosing a way to divide the world into chunks of matter and the environment before choosing his laws of physics. This doesn’t make much sense, and reversing the order dramatically changes the conclusions.

By the way: I just started looking for post-1989 discussions of Wigner’s paper. So far I haven’t found any interesting ones. Here’s a more recent paper that’s somewhat related, which doesn’t mention Wigner’s work:

• Indranil Chakrabarty and Prashant, Non existence of quantum mechanical self replicating machine, 2005.

The considerations here seem more closely related to the Wooters–Zurek no-cloning theorem.

Information Geometry (Part 8)

Now this series on information geometry will take an unexpected turn toward ‘green mathematics’. Lately I’ve been talking about relative entropy. Now I’ll say how this concept shows up in the study of evolution!

That’s an unexpected turn to me, at least. I learned of this connection just two days ago in a conversation with Marc Harper, a mathematician who is a postdoc in bioinformatics at UCLA, working with my friend Chris Lee. I was visiting Chris for a couple of days after attending the thesis defenses of some grad students of mine who just finished up at U.C. Riverside. Marc came by and told me about this paper:

• Marc Harper, Information geometry and evolutionary game theory.

and now I can’t resist telling you.

First of all: what does information theory have to do with biology? Let me start with a very general answer: biology is different from physics because biological systems are packed with information you can’t afford to ignore.

Physicists love to think about systems that take only a little information to describe. So when they get a system that takes a lot of information to describe, they use a trick called ‘statistical mechanics’, where you try to ignore most of this information and focus on a few especially important variables. For example, if you hand a physicist a box of gas, they’ll try to avoid thinking about the state of each atom, and instead focus on a few macroscopic quantities like the volume and total energy. Ironically, the mathematical concept of information arose first here—although they didn’t call it information back then; they called it ‘entropy’. The entropy of a box of gas is precisely the amount of information you’ve decided to forget when you play this trick of focusing on the macroscopic variables. Amazingly, remembering just this—the sheer amount of information you’ve forgotten—can be extremely useful… at least for the systems physicists like best.

But biological systems are different. They store lots of information (for example in DNA), transmit lots of information (for example in the form of biochemical signals), and collect a lot of information from their environment. And this information isn’t uninteresting ‘noise’, like the positions of atoms in a gas. The details really matter. Thus, we need to keep track of lots of information to have a chance of understanding any particular biological system.

So, part of doing biology is developing new ways to think about physical systems that contain lots of relevant information. This is why physicists consider biology ‘messy’. It’s also why biology and computers go hand in hand in the subject called ‘bioinformatics’. There’s no avoiding this: in fact, it will probably force us to automate the scientific method! That’s what Chris Lee and Marc Harper are really working on:

• Chris Lee, General information metrics for automated experiment planning, presentation in the UCLA Chemistry & Biochemistry Department faculty luncheon series, 2 May 2011.

But more about that some other day. Let me instead give another answer to the question of what information theory has to do with biology.

There’s an analogy between evolution and the scientific method. Simply put, life is an experiment to see what works; natural selection weeds out the bad guesses, and over time the better guesses predominate. This process transfers information from the world to the ‘experimenter’: the species that’s doing the evolving, or the scientist. Indeed, the only way the experimenter can get information is by making guesses that can be wrong.

All this is simple enough, but the nice thing is that we can make it more precise.

On the one hand, there’s a simple model of the scientific method called ‘Bayesian inference’. Assume there’s a set of mutually exclusive alternatives: possible ways the world can be. And suppose we start with a ‘prior probability distribution’: a preconceived notion of how probable each alternative is. Say we do an experiment and get a result that depends on which alternative is true. We can work out how likely this result was given our prior, and—using a marvelously simple formula called Bayes’ rule—we can use this to update our prior and obtain a new improved probability distribution, called the ‘posterior probability distribution’.

On the other hand, suppose we have a species with several different possible genotypes. A population of this species will start with some number of organisms with each genotype. So, we get a probability distribution saying how likely it is that an organism has any given genotype. These genotypes are our ‘mutually exclusive alternatives’, and this probability distribution is our ‘prior’. Suppose each generation the organisms have some expected number of offspring that depends on their genotype. Mathematically, it turns out this is just like updating our prior using Bayes’ rule! The result is a new probability distribution of genotypes: the ‘posterior’.

I learned about this from Chris Lee on the 19th of December, 2006. In my diary that day, I wrote:

The analogy is mathematically precise, and fascinating. In rough terms, it says that the process of natural selection resembles the process of Bayesian inference. A population of organisms can be thought of as having various ‘hypotheses’ about how to survive—each hypothesis corresponding to a different allele. (Roughly, an allele is one of several alternative versions of a gene.) In each successive generation, the process of natural selection modifies the proportion of organisms having each hypothesis, according to Bayes’ rule!

Now let’s be more precise:

Bayes’ rule says if we start with a ‘prior probability’ for some hypothesis to be true, divide it by the probability that some observation is made, then multiply by the ‘conditional probability’ that this observation will be made given that the hypothesis is true, we’ll get the ‘posterior probability’ that the hypothesis is true given that the observation is made.

Formally, the exact same equation shows up in population genetics! In fact, Chris showed it to me—it’s equation 9.2 on page 30 of this
book:

• R. Bürger, The Mathematical Theory of Selection, Recombination and Mutation, section I.9: Selection at a single locus, Wiley, 2000.

But, now all the terms in the equation have different meanings!

Now, instead of a ‘prior probability’ for a hypothesis to be true, we have the frequency of occurrence of some allele in some generation of a population. Instead of the probability that we make some observation, we have the expected number of offspring of an organism. Instead of the ‘conditional probability’ of making the observation, we have the expected number of offspring of an organism given that it has this allele. And, instead of the ‘posterior probability’ of our hypothesis, we have the frequency of occurrence of that allele in the next generation.

(Here we are assuming, for simplicity, an asexually reproducing ‘haploid’ population – that is, one with just a single set of chromosomes.)

This is a great idea—Chris felt sure someone must have already had it. A natural context would be research on genetic programming, a machine learning technique that uses an evolutionary algorithm to optimize a population of computer programs according to a fitness landscape determined by their ability to perform a given task. Since there has also been a lot of work on Bayesian approaches to machine learning, surely someone has noticed their mathematical relationship?

I see at least one person found these ideas as new and exciting as I did. But I still can’t believe Chris was the first to clearly formulate them, so I’d still like to know who did.

Marc Harper actually went to work with Chris after reading that diary entry of mine. By now he’s gone a lot further with this analogy by focusing on the role of information. As we keep updating our prior using Bayes’ rule, we should be gaining information about the real world. This idea has been made very precise in the theory of ‘machine learning’. Similarly, as a population evolves through natural selection, it should be gaining information about its environment.

I’ve been talking about Bayesian updating as a discrete-time process: something that happens once each generation for our population. That’s fine and dandy, definitely worth studying, but Marc’s paper focuses on a continuous-time version called the ‘replicator equation’. It goes like this. Let be the set of alternative genotypes. For each , let be the number of organisms that have the th genotype at time . Say that

where is the fitness of the th genotype. Let be the probability that at time , a randomly chosen organism will have the th genotype:

Then a little calculus gives the replicator equation:

where

is the mean fitness of the organisms. So, the fraction of organisms of the th type grows at a rate proportional to the fitness of that type minus the mean fitness. It ain’t enough to be good: you gotta be better than average.

Note that all this works not just when each fitness is a mere number, but also when it’s a function of the whole list of probabilities . That’s good, because in the real world, the fitness of one kind of bug may depend on the fraction of bugs of various kinds.

But what does all this have to do with information?

Marc’s paper has a lot to say about this! But just to give you a taste, here’s a simple fact involving relative entropy, which was first discovered by Ethan Atkin. Suppose evolution as described by the replicator equation brings the whole list of probabilities —let’s call this list —closer and closer to some stable equilibrium, say . Then if a couple of technical conditions hold, the entropy of relative to keeps decreasing, and approaches zero.

Remember what I told you about relative entropy. In Bayesian inference, the entropy relative to is how much information we gain if we start with as our prior and then do an experiment that pushes us to the posterior .

So, in simple rough terms: as it approaches a stable equilibrium, the amount of information a species has left to learn keeps dropping, and goes to zero!

I won’t fill in the precise details, because I bet you’re tired already. You can find them in Section 3.5, which is called “Kullback-Leibler Divergence is a Lyapunov function for the Replicator Dynamic”. If you know all the buzzwords here, you’ll be in buzzword heaven now. ‘Kullback-Leibler divergence’ is just another term for relative entropy. ‘Lyapunov function’ means that it keeps dropping and goes to zero. And the ‘replicator dynamic’ is the replicator equation I described above.

Perhaps next time I’ll say more about this stuff. For now, I just hope you see why it makes me so happy.

First, it uses information geometry to make precise the sense in which evolution is a process of acquiring information. That’s very cool. We’re looking at a simplified model—the replicator equation—but doubtless this is just the beginning of a very long story that keeps getting deeper as we move to less simplified models.

Second, if you read my summary of Chris Canning’s talks on evolutionary game theory, you’ll see everything I just said meshes nicely with that. He was taking the fitness to be

where the payoff matrix describes the ‘winnings’ of an organism with the th genotype when it meets an organism with the th genotype. This gives a particularly nice special case of the replicator equation.

Third, this particularly nice special case happens to be the rate equation for a certain stochastic Petri net. So, we’ve succeeded in connecting the ‘diagram theory’ discussion to the ‘information geometry’ discussion! This has all sort of implications, which will take quite a while to explore.

As the saying goes, in mathematics:

Everything sufficiently beautiful is connected to all other beautiful things.

Networks and Population Biology (Part 4)

Today was the last day of the tutorials on discrete mathematics and probability in networks and population biology. Persi Diaconis gave two talks, one on ‘exponential families’ of random graphs and one on ‘exchangeability’. Since there’s way too much to summarize, I’ll focus on explaining ideas from the first talk, leaving you to read about the second here:

• Persi Diaconis and Svante Janson, Graph limits and exchangeable random graphs.

Susan Holmes also gave two talks. The first was on metagenomics and the human microbiome—very cool stuff. Did you know that your body contains 100 trillion bacteria, and only 10 trillion human cells? And you’ve got 1000 species of bacteria in your gut? Statistical ecologists are getting very interested in this.

Her second talk was about doing statistics when you’ve got lots of data of different kinds that need to be integrated: numbers, graphs and trees, images, spatial information, and so on. This is clearly the wave of the future. You can see the slides for this talk here:

• Susan Holmes, Heterogeneous data challenge: combining complex data.

The basic idea of Persi Diaconis’ talk was simple and shocking. Suppose you choose a random graph in the most obvious way designed to heighten the chance that it contains a triangle, or some other figure. Then in fact all you’ve done is change the chance that there’s an edge between any given pair of vertices!

But to make this precise—to make it even true—we need to say what the rules are.

For starters, let me point you back to part 2 for Persi’s definitions of ‘graph’ and ‘graph homomorphism’. If we fix a finite set , there will be a big set of graphs with exactly these vertices. To define a kind of ‘random graph’, we first pick a probability measure on each set . Then, we demand that these probability measures converge in a certain sense as .

However, we can often describe random graphs in a more intuitive way! For example, the simplest random graphs are the Erdős–Rényi random graphs. These depend on a parameter . The idea here is that we take our set of vertices and for each pair we flip a coin that lands heads up with probability . If it lands heads up, we stick in an edge between those vertices; otherwise not. So, the presence or absence of each edge is an independent random variable.

Here’s a picture of an Erdős–Rényi random graph drawn by von Frisch, with a 1% chance of an edge between any two vertices. But it’s been drawn in a way so that the best-connected vertices are near the middle, so it doesn’t look as random as it is:

People have studied the Erdős–Rényi random graphs very intensively, so now people are eager to study random graphs with more interesting correlations. For example, consider the graph where we draw an edge between any two people who are friends. If you’re my friend and I’m friends with someone else, that improves the chances that you’re friends with them! In other words, friends tend to form ‘triangles’. But in an Erdős–Rényi random graph there’s no effect like that.

‘Exponential families’ of random graphs seem like a way around this problem. The idea here is to pick a specific collection of graphs and say how commonly we want these to appear in our random graph. If we only use one graph , and we take this to be two vertices connected by an edge, we’ll get an Erdős–Rényi random graph. But, if we also want our graph to contain a lot of triangles, we can pick to be a triangle.

More precisely, remember from part 2 that is the fraction of functions mapping to vertices of that are actually graph homomorphisms. This is the smart way to keep track of how often shows up inside . So, we pick some numbers and define a probability measure on as follows: the probability of any particular graph should be proportional to

If you’re a physicist you’ll call this a ‘Gibbs state’, and you’ll know this is the way to get a probability distribution that maximizes entropy while holding the expected values of constant. Statisticians like to call the whole family of Gibbs states as we vary the number an ‘exponential family’. But the cool part, for me, is that we can apply ideas from physics—namely, statistical mechanics—to graph theory.

So far we’ve got a probability measure on our set of graphs with vertices. These probability measures converge in a certain sense as . But Diaconis and Chatterjee proved a shocking theorem: for almost all choices of the graphs and numbers , these probability measures converge to an Erdős–Rényi random graph! And in the other cases, they converge to a probabilistic mixture of Erdős–Rényi random graphs.

In short, as long as the numbers are positive, exponential families don’t buy us much. We could just work with Erdős–Rényi random graphs, or probabilistic mixtures of these. The exponential families are still very interesting to study, but they don’t take us into truly new territory.

The theorem is here:

• Sourav Chatterjee and Persi Diaconis, Estimating and understanding random graph models.

To reach new territory, we can try letting some be negative. The paper talks about this too. Here many questions remain open!

Networks and Population Biology (Part 3)

I think today I’ll focus on one aspect of the talks Susan Holmes gave today: the space of phylogenetic trees. Her talks were full of interesting applications to genetics, but I’m afraid my summary will drift off into a mathematical daydream inspired by what she said! Luckily you can see her actual talk uncontaminated by my reveries here:

• Susan Holmes, Treespace: distances between trees.

It’s based on this paper:

• Louis Billera, Susan Holmes and Karen Vogtmann, Geometry of the space of phylogenetic trees, Advances in Applied Mathematics 27 (2001), 733-767.

As I mentioned last time, a phylogenetic tree is something like this:

More mathematically, what we see here is a tree (a connected graph with no circuits), with a distinguished vertex called the root, and vertices of degree 1, called leaves, that are labeled with elements from some -element set. We shall call such a thing a leaf-labelled rooted tree.

Now, the tree above is actually a binary tree, meaning that as we move up an edge, away from the root, it either branches into two new edges or ends in a leaf. (More precisely: each vertex that doesn’t have degree 1 has degree 3.) This makes sense in biology because while species often split into two as they evolve, it is less likely for a species to split into three all at once.

So, the phylogenetic trees we see in biology are usually leaf-labeled rooted binary trees. However, we often want to guess such a tree from some data. In this game, trees that aren’t binary become important too!

Why? Well, each edge of the tree can be labeled with a number saying how much evolution occurred along that edge: for example, how many DNA base pairs changed. But as this number goes to zero, we get a tree that’s not binary anymore. So, we think of non-binary trees as conceptually useful ‘intermediate cases’ between binary trees.

This idea immediately leads us to consider a topological space consisting of phylogenetic trees which are not necessarily binary. And at this point in the lecture I drifted off into a daydream about ‘operads’, which are a nice piece of mathematics that’s closely connected to this idea.

So, I will deviate slightly from Holmes and define a phylogenetic tree to be a leaf-labeled rooted tree where each edge is labeled by a number called its length. This length must be positive for every edge except the edge incident to the root; for that edge any nonnegative length is allowed.

Let’s write for the set of phylogenetic trees with leaves. This becomes a topological space in a fairly obvious way. For example, there’s a continuous path in that looks like this:

Moreover we have this fact:

Theorem. There is a topological operad called the phylogenetic operad, or whose space of n-ary operations is for and the empty set for .

If you don’t know what an operad is, don’t be scared. This mainly just means that you can glue a bunch of phylogenetic trees to the top of another one and get a new phylogenetic tree! More precisely, suppose you have a phylogenetic tree with leaves, say . And suppose you have more, say . Then you can glue the roots of to the leaves of to get a new phylogenetic tree called . Furthermore, this gluing operation obeys some rules which look incredibly intimidating when you write them out using symbols, but pathetically obvious when you draw them using pictures of trees. And these rules are the definition of an operad.

I would like to know if mathematicians have studied the operad It’s closely related to Stasheff’s associahedron operad, but importantly different. Operads have ‘algebras’, and the algebras of the associahedron operad are topological spaces with a product that’s ‘associative up to coherent homotopy’. I believe algebras of the phylogenetic operad are topological spaces with a commutative product that’s associative up to coherent homotopy. Has someone studied these?

In their paper Holmes and her coauthors discuss the associahedron in relation to their own work, but they don’t mention operads. I’ve found another paper that mentions ‘the space of phylogenetic trees’:

• David Speyer and Bernd Sturmfels, The tropical Grassmannian, Adv. Geom. 4 (2004), 389–411.

but they don’t seem to study the operad aspect either.

Perhaps one reason is that Holmes and her coauthors deliberately decide to ignore the labellings on the edges incident to the leaves. So, they get a space of phylogenetic trees with leaves whose product with is the space I’m calling . As they mention, this simplifies the geometry a bit. However, it’s not so nice if you want an operad that accurately describes how you can build a big phylogenetic tree from smaller ones.

They don’t care about operads; they do some wonderful things with the geometry of their space of phylogenetic trees. They construct a natural metric on it, and show it’s a CAT(0) space in the sense of Gromov. This means that the triangles in this space are more skinny than those in Euclidean space—more like triangles in hyperbolic space:

They study geodesics in this space—even though it’s not a manifold, but something more singular. And so on!

There’s a lot of great geometry here. But for Holmes, all this is just preparation for doing some genomics— for example, designing statistical tests to measure how reliable the phylogenetic trees guessed from data actually are. And for that aspect, try this:

• Susan Holmes, Statistical approach to tests involving phylogenies, in O. Gascuel, editor, Mathematics of Evolution and Phylogeny, Oxford U. Press, Oxford, 2007.

Networks and Population Biology (Part 2)

Yesterday I was too sick to attend this conference:

• Tutorials on discrete mathematics and probability in networks and population biology, Institute of Mathematical Sciences, National University of Singapore, 2-6 May 2011. Organized by Andrew Barbour, Malwina Luczak, Gesine Reinert and Rongfeng Sun.

But today I bounced back, and I want to tell you about the first lectures by Persi Diaconis and his wife Susan Holmes, both statisticians at Stanford University.

Since Persi Diaconis is one of those people who make it really obvious that being a mathematician is more fun than any other profession, I should say a bit about him. He left home at the age of 14 to become a professional stage magician, and only returned to school so that he could learn all the math needed to understand Feller’s famous book An Introduction to Probability Theory and Its Applications. Since then he has worked on the mathematics of shuffling cards and the physics of flipping coins. He also works on random matrix theory, the statistics of Riemann zeta zeroes, applications of Hopf algebras to probability theory, and all sorts of other things that don’t sound fun unless you know enough to realize that yes, they are.

In my last blog post on this conference Tim van Beek asked if Persi could really flip a coin and have it land with whatever side up he wanted. So, I asked him about that. He said yes, he could. He didn’t demonstrate it at the time, since we were just about to go to a talk. But he said you can see videos of other people doing it on the web.

Unfortunately, the videos I’ve found so far mainly convince me, not that there are people who have mastered the physical skill of controlling a coin flip, but that there are lots of tricksters out there! Here are two:

• Derrin Brown, The system – coin toss.

• EricSurf6, How to win a coin toss.

Can you figure out how they do it? If you don’t get the second one, you can watch this.

So, I have to wonder: can Persi really flip a coin and have it land the way he wants, or is just fooling people into thinking he can? Mind you, I wouldn’t consider the latter a bad thing, at least insofar as he’s still a practicing stage magician: a basic part of the craft is trickery of this sort. I should ask if he’s sworn the Magician’s Oath:

“As a magician I promise never to reveal the secret of any illusion to a non-magician, unless that one swears to uphold the Magician’s Oath in turn. I promise never to perform any illusion for any non-magician without first practicing the effect until I can perform it well enough to maintain the illusion of magic.”

Anyway, he is giving a series of talks on “Graph limits and exponential models”. There are lots of very large graphs in biology, so he plans to talk about ways to study large graphs by taking limits where the graphs become infinite. He began by describing a result so beautiful that I would like to state it here.

‘Graph’ means many things, but in his talk a graph is simple (no self-loops or multiple edges between vertices), undirected and labeled. In other words, something like this:

A graph homomorphism

is a map sending vertices of to vertices of , such that whenever vertices of have an edge between them, the vertices have an edge between them in .

Given this, we can define to be the fraction of the functions from vertices of to vertices of that are actually graph homomorphisms.

The famous graph theorist Lovasz said that a sequence of graphs converges if converges for all graphs .

Using this definition, we have:

Theorem (Aldous-Hoover, Lovasz et al). There is a compact metric space such that each graph gives a point in , and a sequence of graphs converges iff

for some point .

The space is called the space of graph limits, and the important thing is that one can construct it rather explicitly! Persi explained how. For a written explanation, see:

• Miklos Racz, Dense graph limits, 31 October 2010.

After a break, Susan Holmes began her series of talks on “Phylogeny, trees and the human microbiome”. I can’t easily summarize all she said, but here are the slides for the first talk:

• Susan Holmes, Molecular evolution: phylogenetic tree building.

Her basic plan today was to teach us a little genomics, a little statistics, and a little bit about Markov chains, so that we could start to understand how people attempt to reconstruct phylogenetic trees from DNA data.

A phylogenetic tree is something like this:

or something less grand where, for example, you only consider species of frogs, or even HIV viruses in a single patient. These days there are programs to generate such trees given copies of the ‘same gene’ in the different organisms you’re considering. These programs have names like PhyML, FastML, RaxML and Mr. Bayes. In case you’re wondering, ‘ML’ stands for ‘maximum likelihood’ a standard technique in statistics, while Bayes was the originator of some very important ideas in statistical reasoning.

But even though there are programs to do these things, there are still lots of fascinating problems to solve in this area! And indeed, even understanding the programs is a bit of a challenge.

Since we were talking about the genetic code here recently, I’ll wrap up by mentioning one thing learned about this from Susan’s talk.

It’s common to describe the accumulation of changes in DNA using substitution models. These are continuous time Markov chains where each base pair has a given probability of mutating to another one. Often people assume this probability for each base pair is independent of what its neighbors do. The last assumption is known to be false, but that’s another story for another day! What I wanted to say is that there are two famous models. The simplest is the Jukes-Cantor model, where each of the four bases—A, T, C, and G—has an equal probability of mutating into any other. But a somewhat better model is the Kimura model, where the transitions

A ↔ T
C ↔ G

have a different rate of occuring than all the remaining possibilities. If you look at the pictures of the A, T, C, and G molecules here you’ll instantly see why:

Since I’m busy learning about continuous-time Markov chains, it’s nice to see more good examples!

Networks and Population Biology (Part 1)

There are some tutorials starting today here:

• Tutorials on discrete mathematics and probability in networks and population biology, Institute of Mathematical Sciences, National University of Singapore, 2-6 May 2011. Organized by Andrew Barbour, Malwina Luczak, Gesine Reinert and Rongfeng Sun.

Rick Durrett is speaking on “Cancer modelling”. For his slides, see here, here and here. But here’s a quick taste:

Back in 1954, Armitage and Doll noticed that log-log plots of cancer incidence as a function of age are close to linear, except for breast cancer, which slows down in older women. They suggested an explanation: a chain of independent random events have to occur before cancer can start. A simple model based on a Markov process gives a simple formula for how many events it must take—see the first batch of slides for details. This work was the first of a series of ever more sophisticated multi-stage models of carcinogenesis.

One of the first models Durrett explained was the Moran process: a stochastic model of a finite population of constant size in which things of two types, say and are competing for dominance. I believe this model can be described by a stochastic Petri net with two states, and , and two transitions:

and

Since I like stochastic Petri nets, I’d love to add this to my collection.

Chris Cannings is talking about “Evolutionary conflict theory” and the concept of ‘evolutionary stable strategies’ for two-party games. Here’s the basic idea, in a nutshell.

Suppose a population of animals roams around randomly and whenever two meet, they engage in some sort of conflict… or more generally, any sort of ‘game’. Suppose each can choose from some set of strategies. Suppose that if one chooses strategy and the other chooses strategy , the expected ‘payoff’ to the one is , while for the other it’s .

More generally, the animals might choose their strategies probabilistically. If the first chooses the ith strategy with probability and the second chooses it with probability then the expected payoff to the first player is

where the angle brackets are the usual inner product in I’m saying this in an overly fancy way, and making it look like quantum mechanics, in the hope that some bright kid out there will get some new ideas. But it’s not rocket science; the angle bracket is just a notation for this sum:

Let me tell you what it means for a probabilistic strategy to be ‘evolutionarily stable’. Suppose we have a ‘resident’ population of animals with strategy and we add a few ‘invaders’ with some other strategy, say . Say the fraction of animals who are invaders is some small number , while the fraction of residents is .

If a resident plays the game against a randomly chosen animal, its expected payoff will be

Indeed, it’s just as if the resident was playing the game against an animal with probabilistic strategy ! On the other hand, if an invader plays the game against a randomly chosen animal, its expected payoff will be

The strategy is evolutionarily stable if the residents do better:

for all probability distributions and sufficiently small .

Canning showed us how to manipulate this condition in various ways and prove lots of nice theorems. His slides will appear online later, and then I’ll include a link to them. Naturally, I’m hoping we’ll see that a dynamical model, where animals with greater payoff get to reproduce more, has the evolutionary stable strategies as stable equilibria. And I’m hoping that some model of this sort can be described using a stochastic Petri net—though I’m not sure I see how.

On another note, I was happy to see Persi Diaconis and meet his wife Susan Holmes. Both will be speaking later in the week. Holmes is a statistician who specializes in “large, messy datasets” from biology. Lately she’s been studying ant networks! Using sophisticated image analysis to track individual ants over long periods of time, she and her coauthors have built up networks showing who meets who in ant ant colony. They’ve found, for example, that some harvester ants interact with many more of their fellows than the average ant. However, this seems to be due to their location rather than any innate proclivity. They’re the ants who hang out near the entrance of the nest!

That’s my impression from a short conversation, anyway. I should read her brand-new paper:

• Noa Pinter-Wollman, Roy Wollman, Adam Guetz, Susan Holmes and Deborah M. Gordon, The effect of individual variation on the structure and function of interaction networks in harvester ants, Journal of the Royal Society Interface, 13 April 2011.

She said this is a good book to read:

• Deborah M. Gordon, Ant Encounters: Interaction Networks and Colony Behavior, Princeton U. Press, Princeton New Jersey, 2010.

There are also lots of papers available at Gordon’s website.