I'm teaching my developmental biology course this afternoon, and I have a slightly peculiar approach to the teaching the subject. One of the difficulties with introducing undergraduates to an immense and complicated topic like development is that there is a continual war between making sure they're introduced to the all-important details, and stepping back and giving them the big picture of the process. I do this explicitly by dividing my week; Mondays are lecture days where I stand up and talk about Molecule X interacting with Molecule Y in Tissue Z, and we go over textbook stuff. I'm probably going too fast, but I want students to come out of the class having at least heard of Sonic Hedgehog and β-catenin and fasciclins and induction and cis regulatory elements and so forth.

On Wednesdays, I try to get simultaneously more conceptual and more pragmatic about how science is done. I've got the students reading two non-textbookish books, Brown's In the Beginning Was the Worm and Zimmer's At the Water's Edge, because they both talk about how scientists actually work, and because they do such a good job of explaining how development produces an organism or influences the evolution of a lineage. The idea is that they'll do a better job of prompting thinking (dangerous stuff, that, but I suppose it's what a university is supposed to do) than the dryer and more abstract material they're subjected to in Wolpert's Principles of Development.

Today we're going to be discussing the concept of analogies for how development works. It's prompted by the third chapter in Brown's book, titled "The Programme", where they learn all about Sidney Brenner's long infatuation with the idea that the activity of the genome can be compared to the workings of a computer program. We'll also be talking about Richard Dawkins' analogy of the genome as a recipe, from How do you wear your genes? and Richard Harter's rather better and more amusing analogy for the genome as a village of idiots. I'm also going to show them this diagram from Lewontin's The Triple Helix, to remind them that there's more to development than just gene sequences.

But I'm not going to write about any of that right now. Analogies have a troubling effect on developmental biology because the fundamental processes are so different from what we experience in day-to-day life that they are always flawed and misleading. My main message is going to be that while they may help us grasp what's going on, we have to also recognize where the analogies fail (like, everywhere!) and be mentally prepared to leap elsewhere.

That said, what I'm writing about here is my favorite analogy for development. It's flawed, as they all are, but it just happens to fit my personal interests and history, and after all, that's what analogies are—attempts to map the strange unto the familiar. And, unfortunately, my experience is a wee bit esoteric, so it's not really something I can talk about in this class unless I want to lecture for an hour or so to give all the background, violating the spirit of my Wednesday high-concept days. I can do it here, though!

I'm a long-time microscopy and image processing geek, and you know what that means: Fourier transforms (and if you don't know what it means, I'm telling you now: Fourier transforms). I'm going to be kind and spare you all mathematics of any kind and do a simplified, operational summary of what they're all about, but if bizarre transformations of images aren't your thing, you can bail out now.

A Fourier transform is an operation based on Fourier's theorem, which states that any harmonic function can be represented by a series of sine and cosine functions, which differ only in frequency, amplitude, and phase. That is, you can build any complex waveform from a series of sine and cosine waves stacked together in such a way as to cancel out and sum with one another. The variations in intensity across a complex image can be treated as a harmonic function, which can be decomposed into a series of simpler waves—a set ranging from low frequency waves that change slowly across the width of the image, to high frequency waves that oscillate many times across it. We can think of an image as a set of spatial frequencies. If there is a slight gradient of intensity, where the left edge is a little bit darker than the right edge, that may be represented by a sine wave with a very long wavelength. If the image contains very sharp edges, where we have rapid transitions from dark to light in the space of a few pixels, that has to be represented by sine waves with a very short wavelength, or we say that the image contains high spatial frequencies.

One fun thing to do (for extraordinarily geeky values of "fun") is to decompose an image into all of the spatial frequencies present in it and map those frequencies onto another image, called the power spectrum. All of the low spatial frequencies are represented as pixels near the center of the image, while the high spatial frequencies are pixels farther and farther away from the center. And the fun doesn't stop there! You can then take the power spectrum, apply a Fourier transform to it, which basically takes all the waves defined in it and sums them up, and reconstruct the original image.

I know, this all sounds very abstract and pointless. Fourier transforms are used in image processing, though, and one can do some very nifty things with them; it also turns out that one useful way to think of a microscope objective lens is as an object that carries out a Fourier transform on an image, producing a power spectrum at its back focal plane, which the second lens than transforms back into the original at the image plane.

Still lost? Check out this extremely spiffy online tutorial in Fourier imaging, then. It'll show you visually what I'm talking about.

You can select from a series of images, and here I've picked the human epithelial cell, seen on the left and labeled "specimen image":

It's not very pretty. It has a bunch of diagonal stripes imposed on it, which is something you might get if there were annoying rhythmic power line noise interfering with the video signal on your TV. Those stripes, though, represent an intensely well represented, specific spatial frequency imposed on the image, so they'll be easy to pick up in the power spectrum.

The second image is the power spectrum, the result of a Fourier transform applied to the first image. Each dot represents a wavelength present in the Fourier series for that image, with low frequencies, or slow changes in intensity, mapped to the center, and sharp-edged stuff way out on the edges. Those diagonal lines in the original are regularities that will be represented by a prominent frequency at some distance from the center; you can probably pick them out, the two bright stars in the top left and bottom right quadrant. All the speckles all over the place represent different spatial frequencies that are required to reconstitute the image.

Important consideration: the power spectrum is showing you the spatial frequency domain, not the image. A speckle in the top right corner, for instance, does not represent a single spot on the top right side of our epithelial cell; it represents a wavelength that has to be applied to the entire image. Similarly, that bright star in the lower right is saying that there is a strongly represented sine wave with a particular orientation that has to be represented over everything, just as we see in the original.

The third image is the result if we apply another transform to the power spectrum, to restore the original image.

How is this useful? In image processing, we sometimes want to filter the power spectrum, to do things like remove annoying repetitive elements, like that diagonal hash splattered all over our epithelial cell. The Fourier tutorial lets you do that, as shown below.

You can use the mouse to draw ovals over the power spectrum, and the software will then filter out all of those spatial frequencies that have been highlighted in red. I told you that those two stars were the spatial representation of the diagonal lines all over the image, so here I've gone and blotted them out of existence. Then we reconstruct the image by applying a Fourier transform to the filtered power spectrum, and voila…we get our epithelial cell back, with the superimposed noise mostly gone. It's like magic!

Try it yourself. You can wipe out speckles all over the place and see what effect they have. If you blot out the edges of the power spectrum, what you'll be doing is deleting the higher spatial frequencies, which represent the sharp edges in the image, so you'll be effectively blurring the reconstructed image (which, to all you microscopists, is what stopping down the aperture at the focal plane does, chopping out high spatial frequencies and blurring your image). Notice that blotting out some particular set of speckles may not have much of a detectable effect at all, while others may cause dramatic changes. Notice also that a filter in one place on the power spectrum won't typically have an affect on one discrete place on the reconstructed image, but will affect it virtually everywhere.

I don't know about you, but I find that playing with power spectrums is good for hours of fun. I really wish I'd had this page available years ago, when I was teaching a course in image processing!

Before everyone vanishes to play with Fourier transforms, though, let me get back to my original point—which was to make an analogy with how the genome works.

Think of the genome as analogous to the power spectrum; we'll call it the genomic spectrum. The organism is like the reconstructed image; we'll call that the phenotypic image. A mutation is like the filters applied to the power spectrum. Most discrete mutations will have small effects, and they will be expressed in every cell, while some mutations will affect prominent aspects of the phenotype and will be readily visible. Genes don't directly map to parts of the morphology, but to some abstract component that will contribute to many parts of the form to varying degrees. There is no gene for the tip of your nose, just as there is no speckle in the power spectrum responsible one of the folds in the membrane of the epithelial cell image.

And what is development? It isn't represented in any of the pictures. Development is the Fourier transform itself, or the lens of the microscope; it's the complex operation that turns an abstraction into a manifest form. What results is dependent on the pattern in the genome, but it's also dependent on the process that extracts it.

And now that I've gotten that cosmic philosophizing off of my chest, I can go teach my class without being tempted into confusing the students by explaining one novel idea with which they are unfamiliar by using an analogy to another novel idea with which I am certain they are completely unfamiliar.