Endless Spirals and Beautiful Patterns: Solving Math Problems Behind Cauliflowers
HHave you ever looked at a cauliflower before preparing it and got lost in its beautiful pattern? Probably not, if you’re sane, but I assure you it’s worth a try. What you will find is that what initially looks like an amorphous drop actually has a striking regularity.
If you look closely, you will see that the many flagships look alike and are made up of miniature versions of themselves. In mathematics, we call this property self-similarity, which is a characteristic of abstract geometric objects called fractals. But why do cauliflowers have this property? Our new study, published in Science, found an answer.
There are many examples of fractals found in nature, such as those in ice crystals or tree branches. In math, the number of copies of an initial pattern is infinite. Cauliflowers exhibit a high level of self-similarity, involving seven or more copies of the “same” bud.
This is especially noticeable on Romanesco cauliflower (often referred to as Romanesco broccoli), one of the first images that will appear if you search for “plant fractals” online. What is striking about Romanesco are the very well defined pyramidal buds that accumulate along endless spirals. Although less immediately obvious, a similar arrangement is also present in other cauliflowers.
Spirals are found in many plants; it is the primary organizational model of plants – a subject that has been studied for over 2,000 years. But although cauliflowers share spiraling characteristics with many other plants, their self-similarity is unique. Where does this particularity come from? And do cauliflower’s spirals come from the same mechanisms as those of other plants?
About 12 years ago, two of my mathematician colleagues in France, François Parcy and Christophe Godin, started asking exactly these questions and invited me to participate in the effort. We have spent many hours frantically disassembling the florets, counting them, measuring the angles between them, studying the literature on the molecular mechanisms underlying the growth of cauliflowers, and trying to create realistic computer models. of these mysterious cabbages.
Most of the data available concerned Arabidopsis thaliana, also known as the flowering plant “thale watercress”. Although it is a weed, it is of prime importance in modern plant biology as its genetics have been extensively studied for many years, as have its many variants. And it turns out to be related to all cabbages belonging to the Brassicaceae family. Arabidopsis actually has its own version of cauliflower, resulting from a single mutation involving a single pair of similar genes. The genetics of this mutant plant are therefore very similar to the genetics of cauliflower.
If you spend time observing the branches along the stem of, say, certain weeds in your garden (which are likely to include close relatives of Arabidopsis), you will see how they follow each other quite closely, with the same angle between each successive pair. And if there are enough organs along this spiral, you will start to see other spirals, going both clockwise and counterclockwise.
If you are successful in counting the spirals, you will usually find numbers somewhere along the Fibonacci sequence, where the next number in the sequence is found by adding the two numbers before it. This gives 0, 1, 1, 2, 3, 5, 8, 13, etc. On a typical cauliflower, expect to see five clockwise and eight counterclockwise spirals, or vice versa (see images above and below). below). But why? To understand how the geometry of plants changes over their lifetimes, we need math – and microscopes.
We now know that for each plant the main spiral is already formed on a microscopic scale. It happens very early in its development. At this stage, it includes spots, in which very specific genes are expressed (activated). The genes expressed in a spot determine whether that spot will become a branch, a leaf or a flower.
But genes actually interact with each other, in complex “gene networks”, leading to the expression of specific genes in specific areas and at specific times. It goes beyond mere intuition, so mathematical biologists rely on differential equations to write models of these gene networks to predict their behavior.
To determine how cauliflowers take on their particular shape after the first leaves formed, we built a model that had two main components. It was a reproduction of the spiral formation we see in large cauliflowers and a model of the underlying gene network we see in Arabidopsis. We then tried to match the two to determine which genetics led to the structure of the cauliflower.
We have found that four main genes are the crucial players: their initials are S, A, L and T, quite appropriately. The “A” is missing in Arabidopsis flowering plants that have mutated to look like cauliflower, and it’s also a gene that causes spots to turn into flowers.
What makes cauliflower so special is that these growing tip spots try to turn into flowers for a while (up to several hours), but keep failing for lack of “A” . Instead, they develop into stems, which turn into stems, etc. – multiplying almost endlessly without growing leaves, resulting in almost identical cauliflower buds.
The time they spend trying is fundamental – doing it correctly in our model allowed us to reproduce cauliflowers and romanescos exactly on the computer. We confirmed it was just by changing the growth in a real environment Arabidopsis mutant cauliflower plant, effectively transforming it into a miniature romanesco-like shape.
It’s amazing how complex nature is. The next time you have cauliflower for dinner, take a moment to admire it before you eat it.
Etienne Farcot is Assistant Professor of Mathematics at the University of Nottingham. This article first appeared on The Conversation.