"Phase"
by A.R. Ammons
To read this poem go HERE, then scroll down to page xxvi.
Chemist’s Delight: Seeing the Unseeable
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| X-ray diffraction by fibers of two forms of DNA. B-DNA is the familiar double helix. Analysis of images like these revealed its structure. Image kindly provided by Professor Kenneth Holmes. |
When it comes to the structures of molecules, the question that most people do not think to ask is this: How do we know structure? Molecules are unimaginably small. No amount of magnifying by microscopes can bring them directly to our eyes.
To illuminate small things, you need light waves that are even smaller. Visible light waves are huge compared to molecules. They pass over individual molecules with no effect whatsoever. X-rays are a form of “light” with waves smaller than the smallest molecules, so in theory, one can illuminate and see molecules with X-rays. There were many obstacles to turning theory into practice, but scientists overcame them in one of the great scientific triumphs of the twentieth century. Most of the complex structures we know were obtained from X-ray diffraction by fibers or crystals of biological substances. When applied to crystals, the method is called X-ray crystallography.
Going through a phase
Everything you learn alters the way you look at the world. My life of learning and teaching chemistry certainly changes my readings of the world around me, and it alters my readings of literature and art as well. Doing a crossword puzzle and seeing the clue "Chemical compound, 8 letters”, I quickly think, "aldehyde" or "dodecane", although I am already doubting whether most puzzle writers would expect their readers to know these words. In the New York Times, such words are fair game, but in the local newspaper, I'll probably have to come up with a less learned alternative, like kerosene.
Several years after my first encounter with “Phase”, I am just now sitting down to recall why I immediately decided to give it a prominent place in my little book (1) about X-ray crystallography, the most powerful method of figuring out the structures of molecules. Why would I want to have my readers read it just as they are entering a technical treatise in physical biochemistry? I remember simply having the feeling that it was fitting.
The title, "Phase", of course makes me wonder what crystallographic overtones I might have found within its spare lines, because measuring phases of X-ray waves are crucial to crystallography (I’ll explain). I hesitate to rule out the possibility that Ammons himself was conversant with crystallography, because he repeatedly delighted me by adeptly weaving complex scientific ideas like entropy or metabolism into his poems. I honestly can't recall my exact thoughts on that first reading, but meeting it again, it certainly looks like fertile ground for analogies with things crystallographic.
So here goes (2).
Projections and molecular models
Dropping its leaves in still, cold weather, a tree lays down a projection or silhouette of itself along a vertical axis—the axis being the trunk, if the tree is very symmetrical—a projection in the same shape as the shadow that would be cast by a light far above its crown. As the poem's narrator notes, a gentle, steady wind has skewed the projection, just as if the light casting this shadow had been slightly to one side of center, or as if the tree had leaned over, producing a projection along a tilted axis. From here, it's easy to imagine reloading the leaves, tilting the tree further, and dropping the leaves to produce an even more elongated projection; then tilting it 90 degrees, where leaf drop makes a projection perpendicular to the trunk, a profile; then turning it upside down, making a projection that's the mirror image of the first. And I am picturing how combining all the projections around their common center would reveal a three-dimensional image—a model—of the tree.
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| A snow-globe greeting card, showing how superposition of flat models can construct a three-dimensional image |
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| Pop-Up Snow Globe Greeting Card SGS020 Snowflake, © Up With Paper, www.upwithpaper.com. Used by permission. |
One very powerful way to make accurate models of molecules, or as chemists put it, “to determine their structure,” is to direct a narrow beam of X-rays through a crystal or fiber of the substance under study. Opposite the X-ray source is a detector, which was X-ray-sensitive photographic film in the old days, but today it is an array of CCDs, tiny electronic light detectors like those that serve as the film in digital cameras. The detector records the destination and intensity (dark or faint) of the X-rays that emerge from the sample (crystal or fiber). The results on the detector are many spots, called reflections, of varying intensity, but arranged in an orderly array, either rows and columns of spots from crystals, or smooth arcs of spots from fibers (see figure at top of page).
X-rays are like light beams, which means that, in theory, we could focus the rays that emerge from the sample, and get silhouettes of the molecules. With enough silhouettes from different angles, we could construct detailed images.
But alas, no material can focus X-rays the way glass lenses can focus visible light. Fortunately, there is an arcane mathematical operation called the Fourier transform, which is the theoretical equivalent of focusing an image with a lens. This mathematical operation, requiring, for large molecules, the world’s biggest and fastest computers, combines all the shadows of the enormous number of molecules in the crystal or fiber into a single image, the image of the average molecule in the sample. Individual molecules cast such weak shadows that we need the amplification of many molecules sitting in identical positions—as in a crystal—to make out the shadows.
So now, as I consider the poem, I'm thinking of how the Fourier transform of a crystal's diffraction pattern is a projection of the contents of the sample along the axis of the x-ray beam, the x-ray beam casting a shadow of the molecules as seen from a single direction. To see the image as that of a solid, three-dimensional model, crystallographers must cast these shadows from all unique orientations of the sample, and assemble these shadows around their common center into an image, a fuzzy image of the molecule.
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| Electron-density map, stereo pair. (This image is a stereo pair, like a picture for a stereopticon or View-Master. Either image will do, but if you want to seethe figure in 3D, see instructions at http://spdbv.vital-it.ch/TheMolecularLevel/0Help/StereoView.html. |
This image is called an electron-density map, because it is an image of the electron clouds that surround the molecule. It is actually these electrons that redirect the X-rays as they pass through the sample to produce the diffraction pattern. This map is still chemically rather ambiguous, like a space-filling model with no colors and boundaries to distinguish the elements. Obtaining a detailed atomic model requires scientists to fill in the map with models of the atoms that they know are there from more traditional kinds of chemical analysis, such as sequencing, which tells what building blocks are present, but not how they are arranged in space.
So there is, now that I work it out for myself, a very strong analogy between the projection that the tree produces of itself by dropping leaves, and the computed projections of molecules obtained by X-ray crystallography. This analogy was lurking, only dimly visible at best when I first read the poem. I worked it out and confirmed that it is fitting, at least to me, by a very powerful, but sometimes often arduous expedient, to wit, by writing these very words that you are reading. We often think that we learn, and then write; more often, we write to learn.
What does the term"phase" mean?
Returning to the title of the poem, I see now that one of Ammons's angles in this poem is of course that autumn, with its vivid colors and smells, and its seasonal tasks, is a phase of the year.
So now I am trying to help you, my reader, by coming up with a simple example of phases, the crystallographic kind, and thinking about properties of a calendar year as sine waves, the mathematical shape of light rays, including X-rays. The length of days during the year varies in cyclic fashion, like a sine wave, as shown in this figure, the green curve.
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| Average daily temperature (red) and length of day through one year. |
The curve’s lowest point, the shortest day, is around December 21, and days get longer until June 21 or so, then shorter again toward Christmas. Repeated year after year, a graph of day length versus time is, roughly, a sine wave.
Now for the notion of phase. The red curve on the figure tracks the average daily temperature. To draw it, I need to slide the sine wave along the year (called changing its phase in relation to the day-length curve), until the minimum value falls on January 12, the night that is, on average, the chilliest here along the Maine coast. The phase of this thermal cycle is shifted about three weeks to the right of the day-length cycle—they are about 22 days out of phase.
The green wave tracks the length of day, from the long, cozy nights of winter’s depths, to the long days of summer, with their gradual dawns and lingering dusks. The red wave, on the other hand, tracks the expected temperatures, explaining the need for many of the year’s tasks (raking leaves, shoveling snow, edging the garden), and it tracks comfort as well, which is usually greatest when the thermal curve is steepest, between the extremes of temperature, in spring and autumn. The phase difference between these two waves warns us that the earth gets colder before it recovers from the lack of light in the shortest days, and that the harshest—but for me and many other Mainers, the most beautiful—season of the year looms ahead.
In crystallography, the most difficult part of calculating those molecular silhouettes is to figure out how the phases of all those spot-producing beams are related to each other, just like the phase differences of the two curves I have described. Crystallographers call this the phase problem, a prosaic term for a difficult task that can make or break an effort to learn the structure of an important protein, like the protease with which some powerful HIV drugs interfere. Learning the structure of this molecule helped to explain how some HIV drugs worked, and guided the design of better ones. Very useful shadows, these molecular projections.
By now I am satisfied that my intuition about this poem had a solid foundation. But any sort of analysis of a poem, how it works, what it evokes, always leaves me feeling somewhat empty in comparison to the feeling that the poem itself brings. The satisfaction of fleshing out some of the connections somehow falls short of the joy that I felt when I first read the poem, and first sensed that it would sit comfortably at the beginning of a book about crystallography. It is like any brief explanation of the main point of a literary work. No matter how cogent, it can’t beat the work.
I doubt I have found all of the reasons that this poem appealed to me. Maybe, in that first encounter, I was sensing not only what I found to be apt about the poem as a lead-in to my book. Perhaps I was sensing some of the things you might find as well.
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(1) Crystallography Made Crystal Clear: A Guide for Users of Crystallographic Models, 3rd Edition, Gale Rhodes, San Diego, CA, Academic Press/Elsevier, 2006.
(2) I wrote this essay when I finally got around to fleshing out my thoughts on the poem. In writing the first draft, I was simply thinking by writing. Although I then made the figures and edited to fit them, I tried to retain as much as possible of the spontaneity of the original—so this is an episode of One Culture—Live. Those of you better versed in poetry will also see that I can be very slow on the uptake.
