Mathematics Tuesday, December 16, 2003. Post by Ricky James
We’ll start at the beginning with what we know for sure. Ferns exist. Or at least, some of them do. Interestingly, as a category of plant they’ve actually evolved themselves into existence twice. During the Carboniferous Period 300 million years ago, six families of ferns evolved containing thousands of different fern species, all of which underwent a mass extinction event and do not exist today except as fossils. A second burst of evolution during the Permian, Triassic and Jurassic periods millions of years later recreated all-new families of ferns that were as different from the earlier extinct Carboniferous ferns as early extinct dinosaurs were from later mammals. Three of the Permian families of ferns (Ophioglossales, Marattiales and Filicales) containing around 12,000 species have lasted until modern times and exist today.
What characteristics are required to qualify as a fern keeps botanists arguing into the wee hours of the night. For our purposes here, ferns are plants that have roots and leaves but not seeds or flowers. Plants with seeds and flowers are in the “angiosperm” plant category that evolved after ferns did and are not a part of this discussion. If ferns don’t have seeds or flowers, how do they reproduce? With spores that grow under the leaves, as you can see in a series of photos about halfway down the webpage here.
The leaves of ferns are special in ways besides that they generate spores. The structure of many fern plants and their leaves exhibit self-similarity. You can break off a small portion of a fern plant, and what you hold in your hand looks like a smaller version of the whole plant. Self-similarity like this is a key characteristic of fractal mathematics – but I’m getting ahead of myself.
Now let’s talk about just what makes up a leaf – any leaf, but fern leaves in particular. Plant cells first evolved in the oceans. When they left the nice, bouyant, supporting water and conquered harsh, hostile land (a daunting and arduous process that was a prehistoric version of what humans face colonizing the solar system today), it was as though they had landed on a high-gravity planet like Jupiter. Them po li’l ole plant cells were squished flat as a pancake from relentless, unsupported gravity. It was a slaughter on the beaches of barren prehistoric landmasses that made the opening scene of Saving Private Ryan look like a picnic. The fate of all future life on Earth, including you and me, was poised on the edge of oblivion if aquatic plant cells hadn’t ultimately succeeded in their assault. They would have gone nowhere, least of all the beautiful Missouri Botanical Garden at St. Louis in the middle of the continent, if it hadn’t been for their secret weapon: lignin.
Lignin is the chemical that makes wood possible. After millions of years of casualties on the beaches, plant cells evolved that secreted lignin and so use wood to give them structural suppport against gravity when out of the water. Instead of a skeleton of bone like humans have on their insides, plant cells surround themselves on the outside with ultra-tiny boxes of wood called plant cell walls. Just like humans build towering skyscrapers full of tiny little rooms a hundred stories high (wow!), a growing multicellular plant is likewise a type of towering skyscraper full of tiny little cell wall chambers, each with a single plant cell inside. In this way, plant cells have defied the crush of gravity, built absolutely astounding structures literally millions of cell diameters or “stories” high (WOW!), and conquered land. Luckily for us, the continents aren’t barren anymore. Unluckily for both us and plants, skyscrapers fall, but that’s another article…
Fern leaves are made of lignin and plant cell wall boxes, too. To say we have barely a clue about how plant cell walls grow and multiply is a colossal understatement. One thing we do know, even if we don’t fully understand it, is that during growth plant cell wall boxes are flexible, not rigid. This flexibility during plant cell wall growth is a key factor in ultimately achieving the unique and familiar body shapes of plants like ferns during the growth phase. Furthermore, a group of proteins called expansins are somehow responsible for controlling plant cell wall growth. Being proteins, different expanins are of course created by different genes, and research efforts aimed specifically at ferns are underway to catalog the different DNA and genes of various ferns.
Time at last to get to the, er, point about ferns. How different living organisms translate their different sets of genes into specific body shapes is, of course, the ongoing mystery of embryogenesis and morphogenesis. Ferns offer especially insightful model systems for comparative biological studies to understand how differences in genes cause differences in body shapes. For example, it would be fairly easy to find what differences in expansin genes exist between the black spleenwort fern and the lady fern. These expansin gene differences, and their resulting differences in lignin secretion control and plant cell wall multiplication, are presumably why these two ferns have two different leaf shapes. There aren’t many organisms where we currently have a clue about how to find connections between genome and body shape. Ferns are one of the few where we do. That makes them a very special biological system for future study, and one where the insights gained could be applied later to more complex biological systems. Today understanding of fern leaf shapes, tomorrow the world, er, brain cell layout!
However, it’s important to realize that just as there is no single gene coding for five fingers in a human hand, there is presumably no single gene for leaf shape in plants, fern or otherwise. Instead, fern leaf shape almost certainly has to be caused by some interaction between multiple genes, between multiple expansin control proteins. We don’t really have a clue of how to model such gene-to-expansin-control-protein-to-body-shape interactions at all right now, even if we had maps of all the relevant fern genes and 3-D structure models of the expansin proteins they code for sitting on the Internet right now, which we don’t. Even though the necessary research steps to achieve understanding of the control-of-body-shape problem in fern leaves seem clear, there’s obviously a long, long way to go.
Or maybe there’s a shortcut! The regularity and geometric repetition of fern leaves would intuitively make mathematics a logical candidate tool to help solve the control-of-fern-leaf-shape problem. Initial work in this area occurred in the 1960s by Swedish biologist Aristid Lindenmayer who developed a funky combination of botany, mathematics and computer science called Lindenmayer or L-Systems to model plant growth. His book The Algorithmic Beauty of Plants is a classic on this subject and has spawned many easy-to-use computer applications that can produce most provocative pictures. But at its heart, L-systems is ultimately a computer modeling system to draw pictures of branches using mathematics, not a branch of pure mathematics itself.
In contrast to L-systems, fractals (and their cousins cellular automations and chaos attractors) are indisputably pure mathematics. Because of its co-invention with the rise of modern computers and resulting historical immaturity, fractal research currently resembles a bunch of science fair projects as much as an organized discipline that can be taught from beginning to end in textbook fashion.
One such science-fair project was developed in the 1980s by Michael Barnsley. He developed and patented a methodology called iterated function systems (IFS) that can take any object (like a fern: check out the bottom third of that last link) and generate a set of numbers that can be used to reconstitute an image of that object using fractal mathematics. His primary interest was to produce data compression products for visual images. This is a big bucks business in a world that wants to get more and more television channels pumped trough an existing cable television wire or satellite transponder. Today Barnsley’s company, a Georgia Tech spinoff, is deep in the art of the deal and making those image-processing bucks.
But an intriguing question remains – is the iterated function system methodology more than just a way of making a buck on oomputer image processing? Specifically, is the IFS methodology actually what fern genes and extensin proteins and cell wall boxes really, truly use as a blueprint to generate a particular fern leaf shape? If so, our recent understanding of IFS could be an extremely important shortcut to understanding the entire body-shape-control process and a ultimately a key breakthrough in embryogenesis and morphogenesis…and evidence that fractals do indeed have some Mystic Connection to nature!
Not so fast; this is an article about cognative dissonance, remember? Just because using IFS to draw ferns is now a trivial exercise that even teenagers do by hand in their math classes and something you can do yourself with free, downloadable programs, and just because minor tweaks to IFS number sets can easily generate different “species” of fern leaves, and just because this is an intuitively elegant and beautiful model that would bolster the position that Fractal Math Has A Deep and Mystical Meaning, well, NONE OF THIS means ferns are actually growing their distinctive leaf shapes in this manner.
In fact, there are some compelling reasons to believe ferns DON’T use IFS as a way to make their leaves. Consider the central stem of a fern, and something triggers the growth of a leaf at that point; call that stem location the “ground floor.” The fern genes and extensin proteins and new cell wall boxes go to work in a mysterious way we don’t understand, and they construct the leaf as yet another “skyscraper,” one “floor” at a time. A foundation of “first floor” boxes get laid down, and atop them a layer of “second floor” boxes, and on and on, with some “floors” not having as many “rooms” as others and so being “wider” or “narrower” as they define the overall leaf shape. This common-sense methodology HAS to be followed in some way even if we don’t understand it yet, with the leaf and its shape being built up upon previously built foundations.
Now look at what IFS does to form a leaf. A set of pre-defined IFS numbers is fed into a set of fractal equations and out comes a list of number after iterated number, each of which defines locations in space that are part of the fern. But here’s the key point: two fern cell locations generated mathematically one right after the other using IFS don’t touch each other in space. If you watch what happens, the points plotted for the fern jump out all over the place and aren’t built up row upon row. Using IFS methodology, it’s like the blueprint says to build room 1000 on floor 2000 and then room 3000 on floor 4000. If you have a computer with a memory that stores all of these points as they’re generated and then PLOTS THEM ALL AT ONCE, the result looks like a beautiful fern. If you are a blob of fern cells, how can you follow an IFS blueprint that says the next step is to lay down the next cell an inch or a centimeter out in space where nothing is yet?
IFS fractals generate beautiful ferns. Barnsley’s original research actually generated a computerized fern image that he went out to the library and later identified as a black spleenwort! It is SO tempting to say this has GOT to be how fern shape is controlled on a gene/protein level! It is such intuitively compelling evidence that Fractals Have A Connection To Nature, which furthers our gut belief that mathematics is the key to reality!
But how can that growing blob of fern cells possibly lay the next cell out there where it hasn’t gotten to yet, where a chunk of leaf will eventually be but isn’t yet?
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