Free Online Math Books?

I was poking around the web for a copy of Euclid's Elements, and came across a nice list of over 75  freely available online math books. There's a good mix of material there, ranging from centuries old classics up to modern day course topics and modern application areas - something for everybody.  Check it out!

The Mathematics of Darwin's Legacy

There's a short post over at the Origins blog at sciencemag.org that mentions a conference that had escaped my notice until now - looks like it could be pretty interesting.

Any mathematical biologists that do work with evolutionary models might have some Thanksgiving plans to reconsider (like say, celebrating in Portugal during the "Mathematics of Darwin's Legacy" conference) ?

Experiments, Mathematics and Theory in Ecology

If you check wikipedia or dust off your favorite dictionary and look up the definition of "ecology" you will find something like the following:

Ecology: The branch of biology concerned with the relations between organisms and their environment.

As a branch of biology, and thus a science, you might think that ecologists have centuries old traditions (much like physicists and chemists) - doing controlled experiments in their laboratories or gardens and using the scientific method to test hypotheses and formulate scientific theories. Right?

Well, not quite. History tells us almost the opposite has been the case up until recently... very recently, come to think of it!

I had originally set out to write a single piece, but it got a bit long so I've split it into two parts: the first basically revisits the recent conversation (from the course I TA) that prompted all this, and the second is a bit of a followup heavily seasoned with a few tangents that are likely of interest.

Earlier this week in class, a friend of mine raised a question in class that caught me a little off guard. It made me realized something I had taken overlooked or granted during the past decade or two of my science education: only very, VERY recently did we begin to develop a real understanding of how organisms interact with (and respond to) the world around them. The same could probably be said of knowledge about the natural world!

But on to our example. In lecture this week (for the course "Theoretical Ecology"), we discussed some really nice work that included research done by Dr. Jef Huismann and others, currently at the Institute for Biodiversity and Ecosystem Dynamics at the University of Amsterdam.

During the hot summer months, a lake used for recreation near Amsterdam turns into a smelly, stagnant health hazard due to blooms of toxic cyanobacteria. Understandably, local parks officials wanted to find a way to continue recreational use of this increasingly smelly and toxic body of water without the drastic measures require to stop the actual problem of nutrient pollution (e.g. fertalizers) in the lake.

His group used very controlled laboratory experiments in conjunction with mathematical models of those experimental systems to understand how mixing patterns in water (e.g. due to temperature gradients) influence competition for light among different types of algae under controlled laboratory conditions (think little green beakers). These factors are known to shape the types and numbers of algae you see in small ponds and lakes, and presumably play a role in our lake. (For the philosophically inclined reader, this is using good ol' scientific reductionism being used to lay a conceptual foundation.)

Results from those small scale experiments were then combined with more complicated computer models of the hydrodynamics of an actual lake in order to understand how mixing could be used to control the algal community residing there. With that, they were able to use the models to see how different ways of artificially changing the hydrodynamics in the lake might provide a solution to the problem.

So what was the solution? Based on all the modeling and experimental work, it turned out that a little extra mixing in the right places would cause the good algae to replace the bad. With a few properly placed pumps to bubble the lake, it was returned to its more recreation-friendly state. (More details can be found in Jef's scientific papers, and also in Chapter 7 of the book Harmful Cyanobacteria - if you're interested.)

Along with the many other scientific details uncovered along the way, this is a really cool example of using experimental findings in conjunction with mathematical and/or computer models in order to do exemplary scientific work. The models extend our reasoning and deductive abilities and combined with nice experimental results, lead to a deeper understanding of how algal communities form in these sorts of ponds and lakes.

Much as mathematical models helped Newton understand and describe the laws of motion, Huismann and many other modern day ecologists use similar mathematical models to describe and make predictions about biological systems. But if the math is so similar, why weren't Newton's biologist friends (or at least their grandchildren) doing the same sorts of thing back in the 1700s?? What's so different now that we had to wait 200 years for in order to apply these techniques to biological systems the way Newton and his colleagues applied them to planetary motion?

After the instructor finished talking about Huismann's work (and some of its more technical details), a friend of mine raised his and asked essentially this question: Why didn't someone do this 50 years ago?? It seems so... rudimentary!

Naturally, we looked to the instructor anticipating his response, which was essentially this: Physicists have been using experiments and models to understand and describe natural processes since Galileo (around 1600) - Ecologists (and their biologist and other predecessors) have only been doing it since the early 1900s or later. It just took that long for folks to embrace the idea of doing experiments and using mathematics understand and describe the natural phenomena being observed.

This, admittedly, caught me a little off-guard. I'm sure my thoughts were something like "Wait, what? But, why!?" But in truth, it is an interesting question: why has it taken so long for some of the sciences to gain prominence in recent centuries and (more generally) throughout human history? What walls were broken down recently that unleashed the flood of scientific inquiry we see today?

Well, there are of course a number of ways to answer these questions - certainly many more than I am aware of. Still, I can point to a few of them. Check back for part II of this post in the next couple of days, where I'll try to address some of them.

HIV: Modeling the experiment & the problem of evolution

I just noticed a small article in the ScienceNOW Daily News on using microbicide gels to decrease the risk of contracting HIV. Give it a read!

So why did this article (and this more detailed information from the NIH) catch my attention?

Right now, as I type this, over 9,400 women in Africa are participating in a second, even larger clinical trial - the subject of some other interesting research I'll get into below. The results of that study will in large part determine whether or not this product makes it to market. Being my usual critical self, I immediately have two questions come to mind: "Will it be effective?" and "Is it safe?"

This first question will get a strong answer via this study - after all, 'effective' is relatively straight forward thing to describe and measure. But what do we mean by "safe"?? This brings me to the other big reason this article grabbed my attention: Dr. Sally Blower.

This past fall I had the pleasure of meeting Dr. Sally Blower, a mathematical biologist at UCLA, while I was visiting Ohio State's Mathematical Biosciences Institute during a workshop. She presented some of her research taking a critical look at the second study mentioned in the ScienceNOW article. Her technical paper on the matter can be found on her website.

To briefly summarize the work she presented, she and her colleagues were interested in addressing the possible risk of drug resistant strains arising from the use of these microbicide gels. HIV has a relatively high mutation rate (leading to lots of genetic variation in a viral population) and anyone already infected with HIV who is exposed to anti-retroviral drugs (ARVs) could unknowingly be facilitating natural selection on the virus, leading to drug resistant strains of HIV. Unfortunately, this is a very real problem in the fight against HIV/AIDS, and to let a high-risk product pass clinical testing could come at a price in the long run.

So to understand how well the experiment could assess this risk, as well as the efficiency of the microbicide gels as a means of protection against HIV infection, she and her colleagues created a computer model of the experiment. They began by simulating a population of women and men in which HIV was being transmitted.

As the omniscient creators of this virtual world, they were able to include and manipulate many key factors in the transmission process, including other means of protection (e.g. condoms), the efficacy of the gels, and so on. They were able to "parameterize [the] transmission model using epidemiological, clinical, and behavioral data to predict the consequences of widescale usage of high-risk microbicides" in the population. They then collected data from a number of simulations, following the same type of protocol as the real study, which they could then compare to the actual transmission process in the simulated population.

This clever use of mechanistic models and real world data accomplished two things. First, the computer model allowed them to assess the limitations of the real world experimental protocol, which helps researchers in their interpretation of the real-world experimental results. Second, because they were free to vary the model parameters and run the simulated experiment repeatedly, they could explore the simulated transmission process under different scenarios and describe how the factors included in the model contribute to the eventual outcome.

So did we learn anything from all of this? Among their results, they found that the "planned trial designs could mask resistance risks and therefore enable high-risk microbicides to pass clinical testing" - unfortunate news. On the other hand, their findings suggest that "even if ARV-based microbicides are high-risk and only moderately efficacious, they could reduce HIV incidence."

I can't say what the future holds for these microbicide gels, although I certainly hope they prove to be another means to battle against HIV worldwide. If you'd like more information on HIV/AIDS, check out the 2008 report on the global AIDS epidemic (I'd recommend browsing the "Media kit") from the United Nations Programme on HIV/AIDS.