Tag Archives: fractals

Photo: Rum Bucolic Ape/Flickr

How the cauliflower got its mesmerizing fractals

Closeup of romanesco cauliflower. Credit: Rum Bucolic Ape/Flickr

For at least two thousand years, humans have noticed that many plants grow leaves or flowers in a spiral pattern. Some of these plants, such as the cauliflower, take it a step further, growing spirals in self-repeating patterns commonly known as fractals. The resulting patterns are fascinating and intriguing, which begs the question: why do cauliflower and other fractal-like plants grow like this? French researchers at CNRS posed this exact question and identified an underlying genetic mechanism that produces fractal geometry.

Fractals were first described mathematically by Polish-born mathematician Benoit Mandelbrot in the 1970s. Their defining feature is having the same degree of non-regularity on all scales — looking up close or at the whole thing. They’re also both complicated and irregular. Clouds, mountains, coastlines, cauliflower plants, and even galaxies are all examples of natural fractals.

A small part of a cloud is strikingly similar to the whole thing. A pine tree is composed of branches that are composed of branches – which in turn are composed of branches. And a romanesco cauliflower’s pyramid buds accumulate along endless spirals, accommodating smaller buds that have the same geometry and so on. You get the idea.

If you count the spirals on a romanesco cauliflower — and Etienne Farcot, currently an assistant professor of mathematics at the University of Nottingham did just that — the values will tend to be those along the Fibonacci sequence, where the next number in the sequence is the sum of the previous two numbers before it. The typical cauliflower has five spirals growing clockwise and eight anticlockwise, for instance.

Five clockwise spirals on a cauliflower (left) and eight anticlockwise spirals on the same plant. Credit: Etienne Farcot.

Farcot teamed up with François Parcy, a geneticist at CNRS in France, and Christophe Godin, an expert in plant modeling and computer science at the French Institute for Computational Science and Mathematics, to investigate why this mesmerizing cabbage grows its fractal buds.

After years of careful mathematical and genetic analysis, as well as computer modeling, the researchers concluded that the unusual spirals are the result of the cauliflower trying to grow flowers — and failing in the process.

The cauliflower has undifferentiated cells in its branched tips that divide and want to develop into other organs. These cells produce buds that should bloom into flowers but end up producing more buds instead, which make their own buds and so on. This may be due to a self-selected mutation during the domestication of wild cauliflower.

This self-repeating process happens early in the plant’s development and is due to the action of four genes that form a complex “gene network”. In this network, the expression of the four genes is constantly being changed such that some are turned on or off at specific times.

In order to validate this theory, the researchers designed two mathematical models. One describes the formation of the spirals seen in large cauliflower plants. The other describes the gene network in Arabidopsis, a related plant of the same family with the cauliflower and one of the most widely studied plants from a genetic standpoint.

After some trial and error, the researchers managed to reproduce cauliflower and romanesco plants on their computers exactly like they look in real life. Additionally, they tweaked the growth of Arabidopsis cauliflower mutant plant, effectively turning it into a miniature romanesco.

“It is amazing how complex nature is. The next time you have cauliflower for dinner, take a moment to admire it before you eat it,” Farcot said.

The findings were described in the journal Science.

The images show the same cell in an RICM image (on the right) and a bright-field image (left). Small cell protrusions, invisible in bright-field images, can be visualised with RICM. © MPI for Intelligent Systems

Distinguishing cancer cells using fractal geometry offers faster diagnosis

Photo: Rum Bucolic Ape/Flickr

Photo: Rum Bucolic Ape/Flickr

Fractals are non-regular geometric shapes that have the same degree of non-regularity on all scales. Fractals are the kind of shapes we see in nature, basically, and even though the term was first coined only a coupled of decades ago or if this is the first time you’ve heard about fractals, chances have it that you already interact with them on a daily basis. Think of the spirals of pinecone seeds, leaves, trees and rivers (these look like trees, ever noticed that?), electricity driven patterns, galaxies or good old broccoli!

Fractal geometry, due to its ubiquity, has always fascinated scientists. Now, researchers at Max Planck Institute for Intelligent Systems in Stuttgart and the University of Heidelberg found that cancer cells can be very accurately characterized using fractal geometry.

The cell on the right displays a greater degree of fractality than that on the left, which is an indication of its stronger aggressiveness. © MPI for Intelligent Systems

The cell on the right displays a greater degree of fractality than that on the left, which is an indication of its stronger aggressiveness. © MPI for Intelligent Systems

Apparently, cancer cells exhibit a high degree of fractality – a measure of the statistical distribution of the irregularities, of the cell contours – or at least much larger than normal cells. More often than not, chaotic phenomena display an amazingly consistent order when scaled enough. Tumors generally grow chaotically, causing very irregular convexities of varying size on the cell surface.

Fractals, everywhere?

The researchers employed a mathematical method, combined with image recognition software, to see whether  progression can be reliably assessed in a cell simply by studying fractals. By studying the statistical distribution of  the occurrence of structural details on the surface of different tumour cells, the team of researchers were able to identify cells. Not only that, but the method actually proved to be faster than conventional biopsy and to top things over, it also proved capable of distinguishing between different tumours.

 The images show the same cell in an RICM image (on the right) and a bright-field image (left). Small cell protrusions, invisible in bright-field images, can be visualised with RICM. © MPI for Intelligent Systems

The images show the same cell in an RICM image (on the right) and a bright-field image (left). Small cell protrusions, invisible in bright-field images, can be visualised with RICM. © MPI for Intelligent Systems

Diagnosing cancer today is an invasive and complicated procedure. Tissue samples are collected, stained using specific antibodies and biomarkers and then studied for specific markers. This particular method is expensive and only correctly diagnoses 85% of the time.

Using fractal geometry, team of researchers were able to identify cancer cells more reliably and much faster since the cells can be studied under a microscope without requiring special preparation. Next, the scientists plan on implementing the method in clinical trials and study different malignant cell lines and primary cells.

Marker-Free Phenotyping of Tumor Cells by Fractal Analysis of Reflection Interference Contrast Microscopy Images, Nano Letters, online publication 30 September 2013; DOI: 10.1021/nl4030402

Ring patterns form in a micro-colony of engineered bacteria. Credit: Stephen Payne, Pratt School of Engineering, Duke.

Bacteria growth limited by time, not only concentration. Revises 1950’s Alan Turing theory

How do organs such as the heart or kidneys know when to stop growing? A number of theories have been proposed to answer this, the most entrenched of which dating back from 1952, when the infamous Alan Turing used math to show how biological cell patterns form and how these knew when to stop division. Turing envisioned that the cells knew when to stop growing based on their concentration in a certain location. Researchers at Duke University designed a gene circuit to coax bacteria to grow in a predictable ring pattern. Their findings suggest that the bacteria can sense their environment and that the  engineered gene circuit functions as a timing mechanism. Counter to established theories, the findings may have profound implications in biotechnology.

Ring patterns form in a micro-colony of engineered bacteria. Credit: Stephen Payne, Pratt School of Engineering, Duke.

Ring patterns form in a micro-colony of engineered bacteria. Credit: Stephen Payne, Pratt School of Engineering, Duke.

The team of researchers were led by associate professor of biomedical engineering Lingchong You.

“Everywhere you look in nature there are patterns, many of them very beautiful and even inspirational,” said You. “Our work adds another dimension to the general principles of pattern formation.”

Alan Turing was very fascinated about pattern formations in nature and fractals. The scientist was a particularly gifted mathematician with a keen eye for patterns. It’s worth noting, that Turing, among other scientific contributions of invaluable worth (Turing machine), was the man who broke the Nazi Enigma code, shortening WWII. Unfortunately, Turing was disgraced by his home country due to his sexual orientation, fact that led to his regrettable suicide.

Nevertheless, Turing’s legacy with biological patterns still lives on. In the 1950’s, he imagined that biological patterns are formed due to interactions of certain chemicals he called  “morphogens” that initiated and directed patterns by triggering on- or off-switches. Using math, Turing showed that morphogens could move in space, revealing patterns that mimic those seen in animal skins and leaf shapes. His model became the de facto leading theory regarding biological pattern formation.

A new dimension to bio-pattern growth: time

After using molecular biology lab techniques, however, You and colleagues were unable to replicate a biological pattern predicted by Turing’s model. The Duke researchers engineered a version of the favored lab pet bacteria, E. Coli, to produce two molecules. One serving as the “on” switch promoting colony growth and replication, and the other acting as an “off” switch that halted growth prompted by increase concentration of “on” molecules.

To better analyze colony growth and pattern formation, the researchers also engineered the bacteria to produce fluorescent colours. The ensuing patterns didn’t behave as the scientists initially predicted, though. Instead, the colonies were much smaller than the research team expected based on how fast the “on” signal should diffuse.

To solve the mystery, the scientists added a high concentration of the “on” signal to the growth chamber, flooding the bacteria with the signal. The bacteria formed the same distinctive ring pattern over the same time, which showed they weren’t responding to changes in the concentration of the “on” signal in space.

The only viable explanation, it seemed, was that the “on” molecules acted as a timing cue.  A mathematical model of the timing mechanism was made in order to test this idea. This model predicted how the cells would respond to changes in the size of their growth chamber. This was later confirmed by experiments.

“By serving as a timing cue, the morphogen ‘on’ signal enables the system to sense and respond to the size of the environment,” said You. “The larger the area, the longer it takes for the morphogen to accumulate to a high enough concentration to trigger pattern formation. As such, a larger area will lead to a larger ring pattern.”

Next, the researchers plan on using other gene circuits to create more intricate patterns. Using this technique and armed with new found knowledge on how bio patterns form, scientists could make finely tuned scaffolds for the production of new materials, such as thin metal films for energy applications.

The findings were presented in a paper published in the journal Molecular Systems Biology.