The spirals and whorls seen in sunflowers, pine cones and various other plant structures have long held a special fascination for mathematicians and for biologists with a mathematical bent. After all, you can find Fibonacci numbers in those natural patterns—who could resist? But it’s not just Golden Ratio mysticism that accounts for this interest. More important, I think, is the mere fact that these patterns are simple and orderly enough that we have some hope of understanding them at a deep level. If you want to build a general theory of plant form and growth, then the process that yields the spiral patterns—called phyllotaxis—is a good place to start.
The Joint Mathematics Meetings in New Orleans had an especially good session on phyllotaxis, organized by SIAM, the Society for Industrial and Applied Mathematics. I learned a lot.
Plant stems and roots grow mainly from the tip, from a region of rapid cell division called the apical meristem. It’s not hard to see how this causes elongation of a shoot, but what about branching? Buds and florets and various other structures do not just arise at random as a plant grows; they are spaced at regular intervals, often in a helical pattern. For example, if you number the branches from top to bottom along a plant stem, then as you visit the branches in numerical order, you’ll find you are also going around the stem repeatedly. The divergence angle between successive branches is a crucial factor in determining the overall geometry of the plant. If the angle is 90 degrees, say, then the plant will have fourfold symmetry, and branch n will always lie directly above branch n+4. Supposedly, the divergence angle is often near 137.5 degrees, which is the “golden angle,” dividing a circle into two pie slices whose central angles are in Fibonacci ratio (the limit of the series 1/1, 1/2. 2/3, 3/5, 5/8…).
How do plants accomplish this trick? A basic idea formulated in the 19th century (and perhaps glimpsed even earlier) is that any emerging branch inhibits the growth of other branches nearby; thus a new branch can begin developing only after expansion or elongation has created enough space to make room for it. Alan Turing, more than 50 years ago, suggested a chemical mechanism that might account for this effect.
This approach to understanding plant growth is by now textbook material, but there were some ideas presented in the six talks of the New Orleans session that came as news to me.
For starters, Turing got it backwards (which is not at all the same as getting it wrong). Turing’s model of biological development supposed that a few isolated hotspots produce a chemical growth factor, which then diffuses throughout the tissue; the gradient of concentration controls growth, with new buds appearing only where the concentration exceeds some threshold. The evidence now suggests that plant growth factors (called auxins) are produced by all cells at roughly the same level, and the concentration gradients arise not from passive diffusion but from active transport. Cells pump the auxins “uphill,” toward regions where their concentration is already elevated. Thus there is positive feedback: Abundant auxin attracts still more of it. If this mechanism operated without opposition, all the auxin would eventually accumulate in one place; the counterbalance is the continual creation of new cells, which has the effect of diluting concentration. In New Orleans Eric Mjolsness of the University of California Irvine presented these results, which have just been published in the Proceedings of the National Academy of Sciences. In a follow-on talk Przemyslaw Prusinkiewicz of the University of Calgary presented an algorithmic model based on the experimental results; this work too has recently appeared in PNAS.
Whereas the auxin-pump mechanism fills in some intricate biomolecular details, another line of work highlights a model of phyllotaxis that is simpler and than most others. Pau Atela and Christophe Golé of Smith College illustrated the idea with a penny game. Start with some pennies at the bottom of a sheet of paper, arrayed randomly but neither overlapping nor separated by more than one diameter. Now add pennies one at a time, always at the lowest available position on the paper. Each newly placed penny will be tangent to two others already present. (Tangency to three or four neighbors is possible but vanishingly rare.) A few stages of the process are illustrated below, where each newly added penny is shown in red.
If we interpret the rectangular sheet of paper as an unrolled cylinder, then the patterns produced in this way mimic phyllotaxis. The vertical position of a penny represents the height of a branching or budding point along a cylindrical stem; horizontal position corresponds to angle around the stem. (Note that on the unrolled cylinder left and right edges are identified, so that a penny going off the right side of the sheet comes back at the same height on the left.) Atela and Golé show by analysis and by numerical simulation that periodic branching patterns generally emerge even from random starting positions. If I understand correctly, they find that the famous golden angle is not very common when they measure the angle between individual successive branch points; on the other hand, the average angle does seem to converge on a value in the neighborhood of 137 degrees.
Although the penny model of phyllotaxis was new to me, it is not really real new at all. The model originated with work by Mary and Robert Snow in the 1930s and has been studied by several others since then, including Stéphane Douady, who also spoke at the New Orleans session. For more details see the excellent phyllotaxis web site assembled by Atela and Golé and their students in conjunction with an exhibition at Smith in 2002–2003.
About the pretentiously literary title of this post: I know it’s right on the tip of your tongue…. Yes, that’s right, it’s Dylan Thomas:
The force that through the green fuse drives the flower
Drives my green age; that blasts the roots of trees
Is my destroyer.
And I am dumb to tell the crooked rose
My youth is bent by the same wintry fever.